development, implementation and application of …iqc.udg.es/~eduard/recerca/thesis.pdf · girona,...

Institut de Química Computacional

Departament de Química

Facultat de Ciències

Universitat de Girona

DEVELOPMENT, IMPLEMENTATION AND

APPLICATION OF ELECTRONIC

STRUCTURAL DESCRIPTORS TO THE

ANALYSIS OF CHEMICAL BONDING,

AROMATICITY AND CHEMICAL REACTIVITY

Eduard Matito i Gras

Directed by:

Prof. Miquel Solà i Puig Prof. Miquel Duran i Portas

El professor Miquel Solà i Puig, catedràtic d’Universitat a l’Àrea de Química Física de la Universitat de Girona, i el professor Miquel Duran Portas, catedràtic d’Universitat a l’Àrea de Química Física de la Universitat de Girona, certifiquem que: L’Eduard Matito i Gras, llicenciat en Química per la Universitat de Girona, ha realitzat sota la nostra direcció, a l’Institut de Química Computacional i al Departament de Química de la Facultat de Ciències de la Universitat de Girona, el treball d’investigació que porta per nom: “Development, Implementation and Electronic Structural Descriptors

to the Analysis of Chemical Bonding, Aromaticity and Chemical Reactivity”

que es presenta en aquesta memòria per optar al grau de Doctor en Química. I perquè consti a efectes legals, signem aquest certificat. Girona, 5 de Maig de 2006 Dr. Miquel Solà i Puig Dr. Miquel Duran i Portas

“And, when you want something, all the universe conspires in helping you to achieve it”

The Alchemist (Paulo Coelho)

Als meus pares i a en Xevi,

a l’Anna.

Contents Preface 1.- Quantum Mechanics 1.1.- A Major Breakthrough in Physics and Chemistry....................................15 1.2.- Density Matrices........................................................................................17 2.- Methodology 2.1.- Electron Sharing Indexes..........................................................................25 2.2.- Aromaticity from ESI analyses..................................................................28 2.3.- The Electron Localization Function..........................................................31 3.- Goals of the present thesis.....................................................................................37 4.- Applications I: Electron Sharing Indexes and Aromaticity 4.1.- Comparison of AIM Delocalization Index and the Mayer and Fuzzy Atom

Bond Orders........................................................................................................43

4.2.- Exploring the Hartree-Fock Dissociation Problem in the Hydrogen

Molecule by Means of Electron Localization Measures..................................53

4.3.- The Aromatic Fluctuation Index (FLU): A New Aromaticity Index Based on

Electron Delocalization......................................................................................67

4.4.- An Analysis of the Changes in Aromaticity and Planarity Along the

Reaction Path of the Simplest Diels-Alder Reaction. Exploring the Validity of

Different Indicators of Aromaticity....................................................................79

4.5.- Aromaticity Measures from Fuzzy-Atom Bond Orders (FBO). The

Aromatic Fluctuation (FLU) and the para-Delocalization (PDI) indexes..........89

4.6.- Electron Sharing Indexes at Correlated Level. Application to Aromaticity

Calculations........................................................................................................99

4.7.- Aromaticity Analyses by Means of the Quantum Theory of Atoms in

Molecules.........................................................................................................137

5.- Applications II: The Electron Localization Function

5.1.- The Electron Localization Function at Correlated Level.......................197

5.2.- Bonding in Methylalkalimetal (CH3M)n (M = Li – K; n = 1, 4).

Agreement and Divergences between AIM and ELF Analyses......................229

5.3.- Comment on the “Nature of Bonding in the Thermal Cyclization of (Z)-

1,2,4,6-Heptatetraene and Its Heterosubstituted Analogues”.....................245

5.4.- Electron fluctuation in Pericyclic and Pseudopericyclic reactions.......251

6.- Results and discussion..........................................................................................269

7.- Conclusions............................................................................................................281

8.- Complete list of publications.................................................................................287

9.- References.............................................................................................................291

APPENDIX.....................................................................................................................297

Preface. Computational Chemistry and Theoretical Chemistry

At the outset of the 21st century, theoretical and computational chemistry

has arrived at a position of central importance in chemistry.1 The algorithms of

calculation are being improved, making calculations that were prohibitive feasible

and with accuracies sometimes competing with experimental ones.

Roughly and generally speaking, theoretical chemistry may be defined as the

use of non-experimental reasoning to explain or predict chemical phenomena.

Therefore, a theoretical chemist uses chemical, physical, mathematical and

computing skills to study chemical systems. In theoretical chemistry, chemists (or

even physicists) develop theories, algorithms and computer programs for the

prediction of atomic and molecular properties.

On its side, computational chemistry is regarded as a branch of theoretical

chemistry, and focuses on the application of the results from theoretical chemistry

to the analysis of interesting chemical problems. Although the term computational

chemistry was originally regarded as an application of quantum mechanics to the

study of chemical systems, nowadays it is understood in a wider sense,

encompassing ab initio, semi-empirical and molecular mechanics methods. Hence,

from computational chemistry, it is possible to address different problems such as

benchmark calculations for small molecules, reaction simulations, studies of large

biological systems, drug design or analyze protein folding, among others. The

development of computers is opening a wide range of possibilities for

computational and theoretical chemists to address larger and larger systems.

Nowadays, the specialization of quantum chemists, as either theoretical or

computational ones, has widely separated the research fields. Some chemists

essentially devote their research to the performance of computational experiments,

in the same fashion an experimental chemist performs a laboratory task. While

some chemists focus their research in a rather purely theoretical work, with scarce

chemical applications and development and programming of theory instead. This

thesis is neither one thing nor the other; or maybe both, as it takes a bit from each

side.

11

In the following pages the reader will find the development, implementation

and application of computational tools to the study of electronic structure of

molecules, with special emphasis in the aromaticity and the electron sharing

indexes, which help to grasp inside the nature of chemical bonding.

The main contents of this thesis are abridged in first chapters, where a brief

review of the theory used throughout is given; whereas, chapters 4 and 5 are

devoted to the applications and correspond to the works of the present thesis, as

they were accepted or submitted to publication. Some of these projects came after

and during the development and the calculation of density matrices for correlated

calculations already presented in the Master Thesis. Therefore, for the details of

these density matrices calculations, the algorithms and the formulas the reader is

driven to ref. 2; although a brief review is given in next second section of the next

chapter.

Chapter 4 collects all those works concerning electron sharing indexes and

aromaticity (vide infra). They have been ordered so that the reader will first look into

the concept of electron sharing index (ESI),3 from partitions schemes; afterwards

see its applicability to define quantitative aromaticity measures, which are finally

analyzed and tested from different perspectives. The last section of chapter 4

comprises a brief review on the most recent applications of the Quantum theory of

Atoms in Molecules (QTAIM)4 to the study of aromaticity.

On its side, chapter 5 focuses on the analysis of the Electron Localization

Function (ELF).5 In the first section is given a review on the ELF for both

monodeterminantal and many-body wave functions, whereas second section

compares the description of the chemical bonding from the QTAIM-ESI and the ELF

analyses to describe the electron structure of methyalkalimentals. The second

section is about the ELF definition and calculations at correlated level, while 4.3

and 4.4 analyze the ELF as a tool to discern between two mechanistic approaches,

the pericyclic and pseudopericyclic reactions.

12

1.- Quantum Mechanics

13

1.- Quantum Mechanics:

1.1.- A major breakthrough in physics and chemistry.

The advent of quantum mechanics supposed a Copernican turn not only for

classical physicists, but also for the whole chemistry society. It meant a general

breakdown of the understanding of rules that govern the matter behavior. Concepts

like the Max Planck’s quantization of energy, Erwin Schrödinger’s Equation or

Werner Heisenberg’s Uncertainty Principle had a huge influence in physics, quickly

moving towards chemistry.

In the framework of Quantum Mechanics concepts like chemical bonding,

atomic charge or aromaticity are simply meaningless. Quantities which are not

observable have no place in this theory. It was not until 1926, when Max Born gave

his interpretation of the square wave function as the probability of the existence of

a given state, that a statistical interpretation of the wave function was possible. It

meant the end of determinism which had been ruling the physics so far. Therefore,

although it was not possible to speak e.g. of an exact amount of electronic charge in

a given region, one could assign a probability for that amount of electronic charge of

being in a given region. This is to say, the square wave function plays the role of the

probability density of occurrence. Namely, from the square wave function one can

obtain the density and pair density, which are the one-particle and the two-particle

electron distributions, respectively. From the density and the pair density the

electron structure of given molecule can be easily depicted. Controversially, in spite

of surrounding the pillars of classical physics -the determinism-, as we will see later

on, Max Born’s interpretation of the square wave function finally served to hold part

of old conceptions of chemistry and physics.

Whereas the whole physicists’ world dramatically changed in the 20th

century, some hardly rooted conceptions in the chemistry remained. Chemists still

used (and use) bonding strength, aromaticity or atomic charges as an explanation

for a given chemical phenomena. In order to feed chemists hunger for old concepts,

several research groups3-52 have put considerable effort on reconciling quantum

mechanics with these old chemical concepts. With the probabilistic interpretation of

15

the square wave function as a starting point, the definition of atomic populations

and charges,51 bond orders of several kinds,12, 23, 25, 45, 53 atomic decompositions of

the molecular space, among others, helped bridging the classical chemistry with the

quantum mechanics. Because of their computational cost, first works were devoted

to rather approximated methods, covering only the most elementary quantum

mechanics calculations.

With the development of computers, the calculation of more accurate

structures was also brought together with the calculation of more sophisticated and

reliable analysis of the electron distribution. Despite first calculations were based

on atomic charges, currently no one bases a serious analysis of the electron

distribution on atomic charges. Several charge schemes have been put forward,

and for most calculations, it is difficult to reach an unanimous conclusion from

different charge analyses (cf. chapter 4, section 1). This is so because populations

and charges depend strongly on the model chosen to define an atom in a

molecule.54

In the literature, several electronic descriptors based on the pair or the one-

electron densities have been proposed3, 6, 7, 12, 21-25, 27, 45, 48-50, 53, 55, 56 with more or

less success in their practical applications. In order to be chemically meaningful the

descriptor must give a definition of an “atom” in a molecule, or instead be able to

identify some chemical interesting regions (such as lone pairs, bonding regions,

among others). In this line, several molecular partition schemes have been put

forward: the QTAIM,4, 8, 9 the ELF,5, 57 Voronoi cells, Hirshfeld atoms, Fuzzy-Atoms,50,

58 etc.

The goal of this thesis is to explore the density descriptors based on the

molecular partitions of AIM, ELF and Fuzzy-Atom, analyze the existing descriptors at

several levels of theory, propose new aromaticity descriptors, and study its ability to

discern between different mechanisms of reaction.

16

1.2.- Density Matrices

The wave function seems to be the angular stone of quantum chemistry. The

wave quantum mechanics was introduced by Erwin Schrödinger, who explained the

movement of the electron as a wave. At the same time, Werner Heisenberg set up a

matricial treatment to explain the atom behavior, but his work was much harder to

understand than wave mechanics used by Schrödinger and did not have much

success. However, the most important scientist to contribute to mathematical

foundations of quantum mechanics was not a physicist. The English mathematician

John Von Neumann established the basis of quantum mechanics, and it was not

done in terms of wave functions as such, but from density matrices. Let us put it

into context.

A quantum state is any possible state in which a quantum mechanical

system can be. If a quantum state is fully specified, it can be described by a state

vector, a wave function, or a complete set of quantum numbers for a given system;

such a state is said to be pure. On the other hand, a partially known quantum state,

such as an ensemble with some quantum numbers fixed, can be uniquely described

only by a density matrix; it is usually referred as a mixed quantum state, or an

ensemble state as it represents a statistical distribution of pure states.

In 1927, Von Neumann introduced the formalism of the density matrices, or

density operators, used in quantum theory to describe the statistical state of a

quantum system. Its applicability, which covers a major range that the wave

function itself, is discussed at some extent in ref 2.

A N-order density matrix, being N the number of electrons in our system, may

be constructed in the following manner:

( ) ( ) ( )NNNNN xxxxxxxx rrrrrrrr ...''......''... 11*

11 ΨΨ=ρ (1)

17

And thus, this matrix (for the sake of clarity, let us call so, even though tensor would

be a more appropriate term to describe it), depends upon 2N variables. For most of

proposes this dependence is beyond feasible computations with current computers,

and that is the reason for reducing the number of variables of which it depends by

integration. This way, one may construct the m-order reduced density matrices, by

integration of N-m of its coordinates,

( ) ( )∫ ++Δ= NmNmNNNmm

m xdxdxxxxxxxx rrrrrrrrrr ......''......''... 111111)( ργ (2)

after setting xi=x’i by means of a generalized Dirac delta:

( ) ( )∏+=

+++ −=Δ≡ΔN

miiiNmNm

Nm xxxxxx

1111 '...'...' rrrrrr δ (3)

In the case of a wave function expanded in terms of Slater determinants,

KK

Kc ψ∑=Ψ (4)

with ΨΚ being the Slater determinant in Eq. (5). Without the loss of generalization,

let us suppose that it is constructed from a set of orthonormalized spin orbitals:

)()...()(!

12211 NNK xxx

Nrrr χχχψ = (5)

In such a case, Equation (2) can be further simplified to read:

( ) ( ) ( ) ( ) ( )∑Γ=

m

m

mm

m

m

jjjiii

mjjmiijjj

iiimmm xxxxxxxx

K

K

KK

rrrrrrrr

21

21

11

21

21...'...'...''... 1

*1

*11

)( χχχχγ (6)

As a result the m-order reduced density matrix (m-RDM) is calculated as an

expansion of our basis set.

18

In our laboratory we have designed an algorithm for the calculation of these

expansion coefficients, m

m

jjjiiiK

K21

21Γ

-which we call m-order density matrices (DMm)-,

from the coefficients cK given in Eq. (4).

It is of particular usefulness to further simplify Eq. (6) by only considering the

diagonal terms of the m-RDM, which fulfill xi=x’i. When not taking into account the

off-diagonal terms, we are left with the m-order density functions:

( ) ( ) ( ) ( ) ( )∑Γ=

m

mmm

m

m

jjjiii

mjjmiijjj

iiimm xxxxxx

K

K

KK

rrrrrr

21

2111

21

21......... 1

*1

*1

)( χχχχγ (7)

The most simple of these density functions is the one-electron density, or simply the

density:

( ) ( ) ( )∑ Γ=ji

jj

ii xxx,

11*

1)1( rrr χχγ (8)

Also interesting is the definition of the pair density:

( ) ( ) ∑ Γ=

lkji

lkklijji xxxxxx

,,

212*

1*

212 )()()()(, rrrrrr χχχχγ

(9)

We can give to the pair density the significance of the probability density of

finding a certain couple of electrons (1,2), regardless of the position of the

remaining N-2 electrons. The interpretation is analogous with the one given by Born

in 1927 to the electron density in terms of the probability density of finding one

electron, irrespective of the position of the others. In this line, all m-order density

functions have a similar significance: they are related to the probability of finding m

electrons independently of the positions of the other N-m electrons.

Maybe the reader is wondering why “regardless the position of the remaining

N-2 electrons”. Let us give a simple proof. Take Eq. (1), by setting xi=x’i for all

19

variables and integrating over the whole space, we would get 1 for a normalized

wave function; which is to say, finding the N electrons over the whole space has

probability of 100%, which is an obvious statement. If we take Eq. (2), and we

consider it separately for any electron in the system, we would count separately the

probability of finding 1, 2... over x1. Summing up all terms we would have N times

Eq. (2), as the probability of finding any electron by x1. If now we integrate x1 over the

whole space, the result is 100N%. Unless we have one electron the probability is

beyond 100%, as a consequence of summing repeated probabilities; therefore each

separate one-electron distribution contains information regarding the positions of

other electrons in the system. The following Venn diagram illustrates the

phenomena for a three particle (A, B and C) system:

Further details of the way of calculating probabilities of finding an electron

(or some number of electrons) in certain position, while others are excluded from

being in that region is also given in ref. 2.

It is of common practice to multiply Eq. (7) by either the number of

unordered groups of size m taken from N elements, the binomial coefficient

⎟⎟⎠

⎞⎜⎜⎝

⎛=

mN

C Nm , or all possible groups of size m, !m

mN

P Nm ⎟⎟

⎠

⎞⎜⎜⎝

⎛= . In this way, one obtains

from the m-RDM the expected number of groups of m electrons.

Figure1. Venn diagram for a three possibilities event.

20

The density matrices (DM) are intrinsically important in the work carried out

in this thesis. For practical proposes DM are usually written in the basis of either

atomic orbitals or molecular (spin) orbitals, and its construction from those wave

functions formed as a linear combination of Slater determinants was performed

through the aforementioned algorithm, which will be submitted elsewhere.59 In

particular, sections 4.2, 4.6, 4.7 and 4.8 benefit from this algorithm to calculate the

Configuration Interaction with Simples and Doubles (CISD) second and first order

reduced density matrices.

It is worth noticing these DM are not needed when dealing with

monodeterminantal wave functions, since one can easily prove that they must be

diagonal, regardless the order; the following relationship between p-RDM and 1-

RDM holds:

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )ppp

p

ppp

xxxx

xxxxxxxx

rrL

rrMOM

rrL

rr

rrrr

''

''...''...

11

1

11

111

11

γγ

γγγ = (10)

and the analogous one for DM is as follows:

m

m

m

mm

m

m

mm

m

m

m

llkk

lk

lk

lk

lk

lk

lk

lk

lk

llkk

LL

LL

L

MOM

L

L

MOM

L1

1

1

1

1

1

1

1

1

1

1

1δ

δδ

δδ==

ΓΓ

ΓΓ=Γ (11)

where deltas refer to well known Kronecker deltas.

Finally, in this context, it is worth to mention what are known as relaxed

density matrices. The relaxed density matrices come from the non-fulfillment of the

Hellmann-Feynman theorem. This theorem guarantees that any certain property

reproduces the same value calculated from an expectation value or as an energy

derivative. It holds for exact wave functions, and some approximate theoretical

methods, such as the self-consistent field theory. However, Hellmann-Feynman

theorem does not hold for perturbative, truncated configuration interaction, or

coupled-cluster methods. Therefore, at such levels of theory one can calculate the

21

corresponding DM as either the expected value of the corresponding Dirac delta, cf.

Eq. (2), or as an energy derivative. The latter is known as response density matrix or

relaxed density, since it includes effects due to orbital relaxation through the

coupled-perturbed Hartree-Fock (CPHF) calculations; the other densities are called

unrelaxed densities.

It is widely known that one-electron properties (which need of the DM1),

such as dipole moments, produce more reliable results when computed as an

energy derivative, since they represent the wave function response with respect to a

perturbation. However, very little is said about the effect of relaxation in the electron

distribution itself. In consequence, due to practical purposes, we have preferred to

use the unrelaxed density, which is the one obtained with our algorithm. Since some

computational packages provide the relaxed DM1 and DM2, we are considering the

possibility of comparing these densities in a further study, especially their effects on

the calculation of ESI which is our primary goal.

22

2.- Methodology

23

2.- Methods

2.1.- Electron Sharing Indexes

An extensive review of the concept of sharing index was done in prevision for

the next Faraday discussion on Chemical Bonding (FD135) to be held in

Manchester, and it is given in section 6 of chapter 4. Therefore, we do not review

the concept of electron sharing index in great detail, but instead we comment on

some aspects of the formulation not included in the next chapters.

The electron sharing indexes (ESI) account to some extent for the electron

sharing of a given pair of atoms (or more generally of a pair of moieties). Hereafter

the ESI computed are calculated from the exchange correlation density (XCD), which

reads as follows:

( ) ( ) ( ) ),()()(),( 211212121 xxxxxx γγγγ −=xc (12)

and measures the divergences of the pair density —a true electron pair

distribution—, and a fictitious pair density constructed as the product of two

independent one electron distributions. Therefore, the XCD integrates to N, since

the density and the pair density integrate to P1N=N and P2N=N(N-1), respectively:

∫ ∫ = Nddxc 2121 xxxx ),(γ (13)

The XCD is an appealing function because it fulfills several interesting

properties. First of all, it is non-negative definite; the number of fictitious pairs of

electrons for a model of independent electrons is always greater than the actual

number of pairs. A simple proof of this result for monodeterminantal wave function

can be found in chapter 4.1. This proof also holds for wave functions constructed

from several determinants. (See chapter 4.6) Secondly, by definition, in the case of

25

non-interacting electrons the function is zero. And in third place, integrated over the

whole space, it gives the number of electrons in the system, N.

In the literature several authors propose ),( 21 xxxcγ as a measure of

electron sharing between points x1 and x2. The XCD ensures that a) every pair of

points in the whole space is assigned a certain electron sharing (the decomposition

is exhaustive), b) the sharing is always positive, c) for non-interacting points the

sharing is equally zero. All these properties make the XCD a suitable candidate for

the calculation of electron sharing between atoms; by integration of each

coordinate in a given finite region (which we could somehow identify with an atom),

we may get back to the classical concept of bond.

Some definitions of an atom in a molecule are explained in chapter 4.1, and

the power of the ESI described from XCD is discussed in great detail in chapter 4.6.

Usually, in the literature people use Eq. (13) as it is written, however,

providing most molecular calculations are closed-shell and monodeterminantal

(either HF or DFT within the Kohn and Sham formalism), people sometimes prefer

writing Eq. (13) further simplified and without spin dependence. Let us do this

simple transformation, which will help us to realize about an important feature of

the XCD.

For a monodeterminantal wave functions Eq. (10) indicates how to write the

pair density from the 1-RDM:

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )22

112

121

111

1

212

xxxxxxxx

xx rrrr

rrrrrr

γγγγ

γ = (14)

Splitting the latter formula on its spin cases we get:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )|()|()|()()()()()(

)()()()(),(),(),(),(11111111

11112222

212121212121

212121212121

rrrrrrr|rrrrr

rrrrrrrrrrrrββαααββα

ββααβααβββαα

γγγγγγγγ

γγγγγγγγ

−−+

++=+++ (15)

26

At this point we can realize about an interesting fact: the cross-spin terms at the

r.h.s of the last equation come entirely from the density function. An important

consequence is that the pair density calculated from Eq. (14) has its cross-spin

contribution (also known as Coulomb correlation) calculated as a fictitious pair

density constructed from two independent one electron densities.

By putting together those spin parts that depend upon the density or the pair

density we arrive at:

( ) ( ) ( ) ( ) ( ) ( ) ( ) )|()|()|()|()()(),( 1111112212121212121 rrrrrrrrrrrr ββαα γγγγγγγ −−= (16)

Notice in Eq. (15) we cannot join the out-of-diagonal terms of the 1-RDM, excepting

in a closed-shell system where we would write:

( ) ( ) ( ) ( ) ( ) ),(),(21)()(),( 11112

21212121 rrrrrrrr γγγγγ −= (17)

From Eq. (14) one may write the XCD for a monodeterminantal closed-shell wave

function:

( ) 21 ),(),( 2121 xxxx γγ =xc (18)

and by analogy with Eq. (12), we can give the spinless version of the XCD for a

monodeterminantal closed-shell wave function:

( ) 21 ),(21),( 2121 rrrr γγ =xc (19)

Therefore, it is important noticing the factor of one half, due to the fact that actually

0),( =21 rrαβγ xc within a monodeterminantal wave function. Further consequences of

27

this fact, encountered when comparing monodeterminantal and many-body wave

functions, are commented in chapter 4.6.

2.2.- Aromaticity from ESI analyses

In 1825 an English chemist, Michael Faraday, synthesized a colorless and

flammable liquid with a pleasant, sweet smell which he gave it the name bicarburet

of hydrogen. Few years after that, a German scientist, Eilhard Mitscherlich produced

the same compound via distillation of benzoic acid and lime; Mitscherlich gave the

compound the German name benzin.

Synthesized by Faraday from the decomposition of oil by heat, benzene

turned out the first of a large series of compounds with a characteristic smell.

Indeed, the term aromaticity was named after the Greek term aroma (αρωμα),

meaning “pleasant odour”. Organic textbooks are full of references to aromatic

compounds, the organic chemistry –actually the whole chemistry- is strongly ruled

by aromaticity criteria to assess the failure or the success of reactions.

Not even the advent of quantum mechanics, which consigned to oblivion a

wide range of classical chemical concepts, could make chemists forget about the

tenet of aromaticity. Widely used for both experimentalists and computational

chemists, despite lacking a clear and unambiguous definition, it has remained as

natural tool to explain several chemical phenomena. Since it is not an observable

property, aromaticity is regarded to be measured from its manifestations, which are

many. Among the most common characteristics of an aromatic compound, one can

include: bond length equalization, abnormal chemical shifts and magnetic

anisotropies, energetic stabilization and high electron delocalization. Such a wide

range of manifestations makes aromaticity quantification rather cumbersome. One

must take into account several aromaticity characteristics before stating a molecule

as aromatic. Moreover, some authors60 suggest abandoning the term aromaticity,

to use instead specific nomenclature such as energetic-aromaticity, or magnetic-

aromaticity, and so forth.

28

A statistical study of aromaticity indexes by means of a principal component

analysis served Katritzky et al.61 to show magnetic-aromaticity is orthogonal to other

aromaticity measures as energetic-aromaticity or geometrical-aromaticity. Indeed,

nowadays aromaticity is considered a multifold property, and most of researchers

involved in this field suggest several aromaticity indexes to be calculated prior to

any judgement about the actual aromaticity of a given compound. In our group we

have focused our efforts in the development of electronic aromaticity indexes,

gathered the former experience in the field of electronic structure characterization.

In this thesis we will essentially focus on the aromaticity measured from

electron delocalization. Since the study of ESI could account for the extent of

electron sharing, we thought a good idea to apply our knowledge in the field of ESI

to the study of aromaticity (cf. chapter 4.7). The first work in our group in this line

was carried out by Poater et al.62 when defining the PDI (the para delocalization

index), which measures the electron sharing between atoms in para position, in a

given ring. The idea came from the work of Bader et al,13 who investigated the

electron sharing between carbons in para and meta positions for benzene, with the

finding of a higher electron sharing in the former case, despite the larger distance. It

is worth to mention Fulton29 observed previously this feature of benzene. The main

shortcoming of PDI is the need for para-related atoms, which only exist for six-

membered rings. Therefore, aromaticity in rings of other sizes cannot be recovered

with PDI.

In this line, we tried to devise a new aromaticity index whose applicability

was extended to any given size of a ring. The aromatic fluctuation index (FLU)39 was

constructed by comparing electron sharing between bonded pairs of atoms in a

ring, with respect to a typical aromatic molecule. It is fair to say, the HOMA index63

of aromaticity, defined beforehand, was our inspiration to construct an index which

measures, not divergences in bond length with respect to aromatic molecules, but

divergences in the electron sharing with that of aromatic molecules. A detailed

definition of FLU and its motivation is extensively given in chapter 4.3.

29

Since different ESI have already been considered, a natural extension of the

work on aromaticity indexes was testing different ESI for the calculation of

aromaticity measures.44 In chapter 4.5 we discuss the applicability of the Fuzzy-

Atom ESI to the study of PDI, FLU and FLUπ measures. Similarly, chapter 4.6 collects

PDI, FLU values at CISD level of theory to analyze the effect of the inclusion of

electron correlation in these indexes, calculated from ESI defined from both

partitions, QTAIM and Fuzzy-Atom.

But it is not a matter of defining aromaticity indexes to simply broaden the

number of aromaticity measures in the literature. Although chemistry community

accepts aromaticity as a manifold property, it would be a rather awkward task to

calculate all existing indexes of aromaticity in order to evaluate such property. Some

aromaticity indexes, such as HOMA, are computationally inexpensive, while others,

as NICS, have a remarkable computational cost. On the other hand, some

aromaticity indexes may fail on giving the answer expected from most elementary

chemical grounds. Therefore, the utility of aromaticity indexes needs to be tested; a

continued revision of aromaticity and their adequacy for each chemical situation

must be also a major goal for aromaticity researchers. Since aromaticity is hardly

rooted in the chemistry community, chemists have certain preconceived ideas

about aromaticity. In chapter 4.4 we address this issue, and we examine the

simplest Diels-Alder reaction (whose transition state is said to be most aromatic

point along the reaction path) with HOMA, NICS and FLU indexes. Since SCI was not

considered when the work of chapter 4.4 was published we also re-examinate the

performance of this index for this reaction in chapter 4.6. Chapter 4.3 includes a

false-positive test on FLU index –there defined- which is shown not to be fulfilled by

all indexes of aromaticity; for instance, NICS index of aromaticity fails to reproduce

it.

30

2.3.- The Electron Localization Function

The pair density and the density can be split according to the next spin

cases:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )212

212

212

212

212 ,,,,, rrrrrrrrrr rrrrrrrrrr βααβββαα γγγγγ +++= (20)

( ) ( ) ( )111 rrr rrr βα ρρρ += (21)

The pair density contains the information about the position of electron

pairs, and thus the correlation between pairs of electrons. The pair density of

electrons of the same spin contains not only the information about the so-called

Coulomb correlation, but also the information of correlation due to the presence of

electrons with the same spin, the Fermi correlation. Whereas Coulomb correlation is

mainly introduced by many-body methods (vide supra), Fermi correlation is already

present in monodeterminantal wave functions.

We have commented in the last chapter the probabilistic interpretation of

the density functions. Playing with probabilities we can construct the conditional

probability (CP) of finding electron 2 nearby r2 when electron 1 is at r1:

( )( ) ( )( )1

212

21,,

rrrrrP r

rrrr

ργ

= (22)

Both the pair density and the CP contain all necessary information about the

correlation of electrons. The main advantage of the CP function is that it is actually

discounting irrelevant information concerning the position of the reference electron.

This is the function Becke and Edgecombe5 used to define the electron localization

function (ELF); namely the spherical average of the Fermi contribution to this

function, ( )21 , rrP rrσσ :

31

( ) ( ) ( )0

· ,sinh

, rr

rr rrrrrr=

∇ +∇∇

=+s

s srrPs

ssrrPe σσσσ , (23)

whose series of Taylor around the reference electron (s=0) reads:

( ) ( ) ( )0

22

0

22· ,61,

611, rrr

rr rrrrrrK

rrr==

∇ +∇≈+⎟⎠⎞

⎜⎝⎛ +∇+=+

sssss srrPssrrPssrrPe σσσσσσ . (24)

Becke and Edgecombe used the relative ratio of Eq. (24) with respect to the same

quantity for the homogenous electron gas (HEG) assuming a monodeterminant

situation (use Eq. (17) in Eq. (22) for the HEG64):

( ) ( ) ( ) 3535322353220 31036

53 ρρπρπ σ

σ FcD === , (25)

where ( ) 3223103 π=Fc is the Fermi constant. Hence the relative ratio reads:

( ) ( )( )[ ] 3/8

0

22

0 2

,

rc

srr

DD

F

ss

r

rrrrr

σ

σσ

σ

σ

ρ

γ=

+∇= (26)

Becke and Edgecombe chose the following scaling to define the ELF:5

( )2011

σσ DDELF

+= . (27)

Since the probability of finding an electron with spin σ when there is another

electron with the same spin nearby is lower when the former is localized, Eq. (27) is

larger for localized systems. It is straightforward noticing the ELF ranges in the

interval [0,1].

32

Chapter 5.1 contains the full development of the formulas of the ELF for

monodeterminantal and many-body methods, with special emphasis in the meaning

and the features of the ELF. Besides, chapter 5.2 examines a chemical system

where ELF and AIM lead to divergence concerning the character of bonding.

Interestingly, ELF seems to provide the correct answer. Chapter 5.3 and 4.4 are

devoted to explain how ELF can give inside into the mechanistic differences

between pericyclic and pseudopericyclic reactions.

33

3.- Goals of the present thesis

35

3.- Goals of the present thesis.

The present thesis is devoted to the study of electron sharing indexes (ESI)

and the electron localization function (ELF). The main goal of this thesis is to grasp

inside the concepts of ESI and the ELF as a tools to describe the electronic

distribution in molecules.

The potential of the ESI has been widely documented in the literature for

monodeterminantal methods, but at the post-Hartree-Fock level of theory the

studies are scarce. Since aromaticity is said to be tightly connected with the

electron delocalization in molecules, the exploration of the ESI to account for the

aromaticity of the system will be the subject of our study.

On the other hand, the ELF is a novel method for the analysis of the electron

structure. So far, most ELF analyses have been performed at HF or DFT (within the

Kohn-Sham formalism) levels of theory and thus, the analysis of the ELF for

correlated calculations is of considerable interest. Indeed, some molecular systems

need of highly correlated calculations to be properly described, and thus the

correlated ELF counterpart is needed for a good description of its electronic

structure. The definition, implementation and application to interesting chemical

systems will be the goal of our studies.

To these aims, correlated first- and second-order density matrices will be

needed, for which, the development done in the Master Thesis will be crucial.2

Nonetheless, an improvement of this algorithm will be needed to deal with large

systems.

37

4.- Applications I: Electron Sharing

Indexes and Aromaticity

39

4.1.- Comparison of AIM Delocalization Index and the

Mayer and Fuzzy Atom Bond Orders.

41

Comparison of the AIM Delocalization Index and the Mayer and Fuzzy Atom Bond Orders

Eduard Matito, † Jordi Poater,‡ Miquel Sola,* ,† Miquel Duran, † and Pedro Salvador*,†

Institut de Quı´mica Computacional and Departament de Quı´mica, UniVersitat de Girona, 17071 Girona,Catalonia, Spain, and Afdeling Theoretische Chemie, Scheikundig Laboratorium der Vrije UniVersiteit,De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands

ReceiVed: July 13, 2005

In this paper the behavior of three well-known electron-sharing indexes, namely, the AIM delocalizationindex and the Mayer and fuzzy atom bond orders are studied at the Hartree-Fock level. A large number offive-membered ring molecules, containing several types of bonding, constitute the training set chosen forsuch purpose. A detailed analysis of the results obtained shows that the three indexes studied exhibit strongcorrelations, especially for homonuclear bonds. The correlation is somewhat poorer but still significant forpolar bonds. In this case, the bond orders obtained with the Mayer and fuzzy atom approaches are normallycloser to the formally predicted bond orders than those given by the AIM delocalization indexes, which areusually smaller than those expected from chemical intuition. In some particular cases, the use of diffusefunctions in the calculation of Mayer bond orders leads to unrealistic results. In particular, noticeable trendsare found for C-C bonds, encouraging the substitution of the delocalization index by the cheaper fuzzy atomor even the Mayer bond orders in the calculation of aromaticity indexes based on the delocalization indexsuch as the para-delocalization index and the aromatic fluctuation index.

Introduction

Since the early work of Mulliken,1 when one desires tocharacterize the chemical structure of a certain molecule, thefirst attempts usually point toward density-based descriptors.Most popular indicators of electron distribution in a moleculeare the electron population and the bond order. The latter,together with the quantities more generally labeled as “electron-sharing indexes”sborrowing Fulton’s2 terminologyshave gainedfirst place in order to characterize the nature of the chemicalbond.

A number of bond orders are nowadays available for suchpurposes: delocalization index (DI),3 Mayer bond order (MBO),4

those derived from natural bond orbital (NBO) analysis,5 orfuzzy atom bond order (FBO)6 are just a few examples in anever-ending list of indexes. The question of whether theseindexes can predict the same electron-sharing between the atomsof a molecular system is still open. In this paper we choosethree of the most popular electron-sharing indexes (DI, MBO,and FBO) with the aim of assessing their similarities andanalyzing possible divergences. As shown before,3,6,7 at theHartree-Fock (HF) level of theory these three indexes have asimilar expression since all of them share the same formula butdefine the atomic regions in a different way. Thus, at least ageneral agreement in their magnitudes is expected.

These bond orders have been sometimes referred to ascovalent bond orders.8,9 In fact, it has been also shown that atthe RHF level of theory the bond order between two atoms isproportional to the expected value of the spin coupling betweentheir unpaired electrons.10

Theory. The three bond orders or electron-sharing indexesstudied in this work arise from the analysis of the so-called

exchange-correlation density:11

which is defined through the diagonal terms of the spinless first-order,γ1(r), and second-order,12 γ2(r1,r2), density matrixes. Theexchange-correlation density accounts for the difference betweenan independent electron model and a “true” electron pair oneand integrates to the number of electrons in the system:

For a single determinant wave function, hence within the HFframework, the second-order spinless density matrix can beobtained through the nondiagonal terms of the first-order one:

Thus, the HF exchange-correlation density takes the well-knownsimple form:

Note that we could have writtenγx instead ofγxc in eq 4 as atthe HF level the Coulomb correlation is not included. Substitu-tion of eq 4 into eq 2 leads to

The goal now is to exactly decompose the two-electron integralof eq 5 into one- and two-atom contributions. The correspondingdiatomic contributions give rise to the bond order between thetwo atoms involved (the one-atom contributions are called

* Address correspondence to either author. Fax:+34 972 418356.E-mail: [email protected] (P.S.), [email protected] (M.S.).

† Universitat de Girona.‡ Vrije Universiteit.

γxc(r1,r2) ) γ1(r1)γ1(r2) - γ2(r1,r2) (1)

∫∫γxc(r1,r2) dr1 dr2 ) N2 - N(N - 1) ) N (2)

γ2(r1,r2) ) γ1(r1)γ1(r2) - 1

2γ1(r1,r2)γ

1(r2,r1) (3)

γxc(r1,r2) ) 12

γ1(r1,r2)γ1(r2,r1) ) 1

2|γ1(r1,r2)|2 (4)

N ) ∫∫γxc(r1,r2) dr1 dr2 ) 12∫∫γ1(r1,r2)γ

1(r2,r1) dr1 dr2

(5)

9904 J. Phys. Chem. A2005,109,9904-9910

10.1021/jp0538464 CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 10/11/2005

43

localization indexes within the Atoms in Molecules (AIM)13,14

framework, whereas twice the diatomic contributions are oftenreferred as delocalization indexes (DIs)3,15). The strategy thatallows us to obtain such one- and two-atom contributions isbriefly discussed below.

Delocalization Index (DI). The basic idea behind a 3Dphysical space partitioning analysis is to assign every pointrof the physical space to a given atom. That is, each atom has aregion of the space assigned to it (usually called “atomicdomain” or “atomic basin”). The integral of a monoelectronicfunction can be performed, often numerically, over each atomicdomain giving the corresponding atomic contribution to it. It isevident that if the partitioning of the physical space is exhaus-tive, the total integral can be exactly recovered by summing allatomic contributions. Similarly, the integral of a two-electronfunction will be decomposed as the sum over all pairs of atomiccontributions.

The decomposition of the space into atomic domains isusually carried out within the framework of the AIM theory.13,14

In this context, the so-called DI,3,15,16δAIM(A,B), between atomsA and B is obtained by integration of the exchange-correlationdensity over the atomic domains of atoms A and B (ΩA andΩB):

By virtue of the relation (eq 1), the previous expression can bewritten as

where the quantities⟨NA⟩ and⟨NANB⟩ correspond to the expectednumber of electrons and electron pairs over the atomic domainΩA and the atomic pair domainΩA, ΩB, respectively.Equation 7 is very important as it shows that the DI can berelated to thecoVarianceof the populations in the domains ofatoms A and B:

which in turn is a measure of the correlation between bothpopulations. When the wave function is expressed as a singledeterminant, like in the present work, the DI can be easilyobtained from the numerical integration of eq 5 over the atomicdomains:

For a closed-shell system, the DI can be expressed as

whereSijA is the overlap between doubly occupied molecular

orbitals (MOs)i and j over the basin of atom A.

Using the expansion of the MOs as a linear combination ofatomic orbitals (LCAO), the first-order density matrix for asingle-determinant closed-shell wave function

can be expressed in terms of the set of basis functionsøµwhere the wave function is expanded, and the elements of theso-called density matrix,D, are defined as:

with C being the matrix containing the MO coefficients.Substituting eq 11 into eq 9, we are left with an alternativeexpression forδAIM(A,B):

where the atomic overlap matrix elements of atom A are givenby

An interesting property of the DI thus defined is that it is anonnegative quantity. Indeed, according to eq 4 the exchange-correlation density for a single-determinant wave function is anonnegative two-electron function. Even though this propertyof the exchange-correlation density does not hold at thecorrelated level, the DIs are necessarily nonnegative because,for a given pair of atoms, any increase of⟨NA⟩ necessarilyimplies a decrease of⟨NB⟩; hence the covariance17 of bothquantities is negative (negative slope).

The DI was originally defined by Bader as ameasure of theextent of the correlatiVe interactions between electrons intodifferent regions15 and was only later studied as a bond order.2,7,9

The fact that for unequally shared electron pairs the DI providesa bond order value lower than the formally expected18 supportsthe association of the DI with acoValent bond order.8,9,19 In1999, Fradera et al.3 revised the index and defined it as anelectron-sharing index, refusing the idea of using it as a bondorder (covalent or not) for several reasons.3,20 It is worthmentioning that, in the single-determinant approach, the defini-tions of the bond order given by Fulton and co-workers2,21-24

and AÄ ngyan et al.7 fully coincide with the DI defined in theframework of the AIM theory. Finally, this index has beenstudied at both correlated (CISD)3,25-27 and non-correlated levels(HF), and with the intention of computing the indexes for aDFT calculation, Kohn-Sham orbitals were also used as anapproximation to recover the pair density.26

Fuzzy Atom Bond Order (FBO). Another approach that hasbeen recently explored in the context of population analysis28,29

and later on bond order6 and energy decomposition schemes30

is the use of “fuzzy atoms”, where the atomic domains do nothave boundaries. Instead, at every pointr of the space a weightfactorwA(r) is defined for each atom (A) to measure to whichextent the given point belongs to atom A. These atomic weightfactors are chosen to be nonnegative and to satisfy the followingcondition when summing over all atoms of the system:

γ1(r,r′) ) ∑µν

Dµνøν/(r)øµ(r′) (11)

Dµν ) 2∑i

occ

CµiCνi/ (12)

δAIM(A,B) ) ∑µ,λ

(DSA)µλ(DSB)λµ (13)

SµνA ) ∫ΩA

øµ/(r)øν(r) dr (14)

∑A

wA(r) ) 1 (15)

δAIM(A,B) ) ∫ΩA∫ΩB

γxc(r1,r2) dr1 dr2 +

∫ΩB∫ΩA

γxc(r1,r2) dr1 dr2 ) 2∫ΩA∫ΩB

γxc(r1,r2) dr1 dr2 (6)

δAIM(A,B) ) 2∫ΩA∫ΩB

γ1(r1)γ1(r2) dr1 dr2 -

2∫ΩA∫ΩB

γ2(r1,r2) dr1 dr2 ) 2⟨NA⟩⟨NB⟩ - 2⟨NANB⟩ (7)

Cov[NA,NB] ) ⟨NANB⟩ - ⟨NA⟩⟨NB⟩ (8)

δAIM(A,B) ) 12∫ΩA

∫ΩBγ1(r1,r2)γ

1(r2,r1) dr1 dr2 +

12∫ΩB

∫ΩAγ1(r1,r2)γ

1(r2,r1) dr1 dr2 )

∫ΩA∫ΩB

|γ1(r1,r2)|2 dr1 dr2 (9)

δAIM(A,B) ) 4∑i,j

SijASij

B (10)

Comparison of Three Electron-Sharing Indexes J. Phys. Chem. A, Vol. 109, No. 43, 20059905

44

To decompose the exchange-correlation density into atomic paircontributions, one simply substitutes eq 11 in the right-handside of eq 5 and inserts the identity of eq 15 for the coordinatesof the two electrons in each of the integrals to obtain

where “atomic” overlap matrix elements for an atom A are now

The bond order between atoms A and B is defined analogouslyto eq 13 as

Note that eqs 13 and 18 only differ in the definition of the atomicoverlap matrix elements. Also, since the weight factors arestrictly nonnegative, the bond order within the framework ofthe fuzzy atoms is a positive quantity as well.

Finally, the DI can be understood as a special case of theFBO where the weight factors are simply defined aswA(r) ) 1∧ wB(r) ) 0 ∀ B * A if r ∈ ΩA, that is, a Bader atom.

Mayer Bond Order (MBO). Substitution of the exchange-correlation density corresponding to a closed shell monodeter-minantal wave function (eq 11) into eq 5, followed by LCAOreplacement of the MOs and substitution of the integrations overatomic basins by a Mulliken-like partitioning of the correspond-ing integrals leads to, after trivial manipulations, the well-knowndefinition of the Mayer bond order (MBO):4,31-34

where the overlap matrix elements are defined as:

It is worth noting that the expressions for the DI, FBO, andMBO given in eqs 13, 18, and 19, respectively, only differ inthe definition of the atomic overlap matrix elements.

The same expression can be derived fromthe one-atomspinless first-order density matrix,35 defined as follows:

being the total first-order density the sum of the atomiccontributions:

An effective HF exchange-correlation density between atomsA and B can be built from the corresponding pairs of one-atomdensities of atoms A and B according to

The integration of this function gives directly the sameexpression as eq 19 for the bond order between atoms A and Bin a more intuitive way:

The MBO, as most of the Hilbert-space based methods, isespecially sensitive to the basis set used in the calculation.7,36

Particularly, the inclusion of diffuse functions that lack of amarked atomic character can destroy the chemical picture ofthe molecular system. The index is also ill-defined when bondfunctions are included or for a non-LCAO-type calculation. Itis, however, very easy to calculate and computationally inex-pensive. In particular, the DI involves numerical integration overthe atomic domains, which is sometimes a CPU demandingprocedure due to the complex topology the domains can exhibit(sometimes the integration can fail). Also, if nonnuclearattractors are found in the electron density the chemical pictureof the molecule is lost, since there are regions of the physicalspace that are not assigned to any atom of the molecule. Onthe other hand, both the DI and FBO are more stable withrespect to the change in the basis set,3,6,16 at the expense of abigger computational effort.

The numerical integrations to be carried out within the fuzzyatom framework are sensibly less expensive and straightforwardthan those involving the DI. Since the expression for the DI isanalogous to that of the FBO, one is tempted to use the atomicoverlap matrix elements obtained using fuzzy atoms and henceuse the FBO for the calculation of other existing DI-basedindices such as the para-delocalization index (PDI)37,38 andaromatic fluctuation index (FLU)39 aromaticity descriptors.However, there is a certain degree of arbitrariness concerningthe definition of the weight factors that characterize the fuzzyatoms. In the actual implementation of the method, the authorsused Becke’s method40 for multicentric integration, where theweight factors are obtained through an algebraic function foreach atom that is equal to one in the vicinity of the respectivenucleus and progressively decreases to zero with the distance.This function depends on some set of atomic radii and a so-called stiffness parameter that controls the “shape” of the atom.Other weight functions have been used, including Hirshfeld’s41

original idea of promolecular densities. Nevertheless, eventhough numerical experience seems to indicate that reasonableresults can be obtained using Becke’s function, it is mandatoryto explore the effect of, for example, using a different set ofatomic radii.

Therefore, the aim of the present work is to shed light on thefollowing questions: (a) to what extent three of the most popularelectron sharing indexes can predict the same electron pairdistribution in a molecule?; (b) can the fuzzy atom-based bondorder be used instead of the more expensive AIM-basedcounterpart?; and (c) are the fuzzy atoms bond orders stableenough with respect to the change of the atomic radii used todefine the weight factors? To this end, we have computed the

N ) ∑µνλσ

DµνDλσ ∫øν/(r1)∑

A

wA(r1)øλ(r1) dr1 ×

∫øσ/(r2)∑

B

wB(r2)øµ(r2) dr2 ) ∑A

∑B

∑µνλσ

DµνDλσ ×

∫øν/(r1)wA(r1)øλ(r1) dr1 ∫øσ

/(r2)wB(r2)øµ(r2) dr2 )

∑A

∑B

∑µλ


SµνA ) ∫øµ

/(r)wA(r)øν(r) dr (17)

δFBO(A,B) )1

2∑µ,λ

(DSA)µλ(DSB)λµ +1

2∑µ,λ

(DSB)µλ(DSA)λµ )

∑µ,λ


δMBO(A,B) )1

2∑µ∈A

∑λ∈B

(DS)µλ(DS)λµ +

1

2∑µ∈B

∑λ∈A

(DS)µλ(DS)λµ ) ∑µ∈A

∑λ∈B

(DS)µλ(DS)λµ (19)

Sµν ) ∫øµ/(r)øν(r) dr (20)

γA1(r,r′) ) ∑

µ∈A,ν

Dµνøν/(r)øµ(r′) (21)

γ1(r,r′) ) ∑A

γA1(r,r′) (22)

γxcAB(r1,r2) ) 1

2γA

1(r1,r2)γB1(r2,r1) + 1

2γB

1(r1,r2)γA1(r2,r1) )

γA1(r1,r2)γB

1(r2,r1) (23)

δMBO(A,B) ) ∫ γA1(r1,r2)γB

1(r2,r1) dr1 dr2 )

∑µ∈A,ν

∑λ∈B,σ

DµνDλσSνλSσµ ) ∑µ∈A

∑λ∈B

(DS)µλ(DS)λµ (24)

9906 J. Phys. Chem. A, Vol. 109, No. 43, 2005 Matito et al.

45

DI, FBO, and MBO values for a subset of 117 five-memberedring molecules (see Scheme 1) of those 219 chosen by Cyran´skiet al.42 to test different aromaticity indexes. This set of moleculesyielded 3619 different measures of the indexes, with a richvariety of bonds (see caption of Scheme 1), including aconsiderable number of C-C bonds in different bondingsituations. With this large and diverse set of bonds we can bettergrasp reliable conclusions about Mayer, fuzzy atom, and AIMbonding description abilities.

It is worth noting that a comparison between the DI and MBOresults was already carried out by AÄ ngyan et al.7 and Bocchic-chio et al.43 for a reduced set of simple molecules. These authorsconcluded that the picture of bonding for nonpolar chemicalbonds given by DI and MBO is practically equivalent. However,for polar bonds, the difference between these two indexesbecomes significant. In particular, the DI yields bond ionicitiesmuch greater than the MBO approach. In the present work, weextend these previous investigations by performing a systematicand exhaustive study on 3619 different types of chemicalinteractions. In addition, for the first time, the analysis includesthe recently defined FBOs. An especial attention will be paidto the C-C bond orders.

Computational and Technical Details

We have performed single-point calculations of all analyzedmolecules at the HF level with the 6-311+G** basis set on thegeometries reported by Cyran´ski et al.,42 which were optimizedat the MP2 level with the same basis set. In some particularcases the 6-311G** basis has also been used (see below). Allelectronic structure calculations were performed with Gaussian98.44 The DIs were obtained using a slightly modified versionof the AIMPAC45 package. Fuzzy atom bond orders werecalculated with the FUZZY code,6,46 which implements aBecke’s multicenter integration algorithm with the Chebyshevand Lebedev radial and angular quadratures, respectively. A gridof 30 radial by 100 angular points per atom has been used inall cases. We have employed the Becke’s algebraic functionwith the recommended stiffness parameterk ) 3. We have usedthe set of atomic radii determined by Suresh and Koga,47 exceptas otherwise indicated.

From the set of 219 molecules of Cyran´ski et al.,42 we havechosen those containing two conjugated double bonds, asdetailed in Scheme 1. Additionally, the molecules were chosenon the basis of their accurate and easy numerical integrationwith program AIMPAC, so that we could get correct measuresof DI. The total number of studied molecules was finally 117.As a whole, from 122 molecules with conjugated double bondspresent in ref 42, we have ruled five out because of integrationproblems.48

Results and Discussion

Figures 1-3 show the correlation between the MBO, the bondorder derived from the partitioning of the molecular space in

fuzzy atoms (FBO) and the Bader’s AIM DI. In all of the figureswe can clearly distinguish two regions: one corresponding tothe bonded pairs of atoms, with appreciable values for the bondorder, and another containing nonbonded pairs, with valuesbelow 0.5. In the three plots, the points of nonbonded pairs ofatoms organize on a round shape fashion, showing no cleartendency; fluctuating from index to index. This lack of trendleads us to assert that, apparently, no reliable comparison ofthe indexes can be done within nonbonded pairs of atoms.Nevertheless, we are currently exploring this issue in deeperdetail.

The first plot (Figure 1) shows the correlation between theDI and the FBO indexes. The points corresponding to bondedpairs organize in a thick band centered slightly below they )x line; thus, almost all bonded pairs of atoms exhibit a highervalue of the FBO than the DI. However, the correlation is quitegood, especially for homonuclear apolar bonds. The two groupsof points enclosed with a square and a circle in Figure 1 arethe two major sets of uncorrelated bond orders. The first groupconsists of bond orders involving B, Be, or Al atoms. Thenumber of electrons shared by these atoms is drasticallydecreased within the AIM framework. This is not surprisingsince DIs of 0.272, 0.503, and 0.393 were already reported3 forthe corresponding hydrides (BeH2, BH3, AlH3) at the HF/6-311++G** level of theory (electron correlation has a minoreffect on these values3,26). At the same level of theory, the FBOsare 0.986, 0.977, and 0.949, respectively. This difference in bondorders is the result of the different space partitioning, and it isin line with the big difference in the net atomic charges givenby the two schemes. The calculated Bader atomic partial chargesof Be, B, and Al atoms in the hydrides were+1.7, +2.1, and+2.36 au while the fuzzy atom approach leads to significantlysmaller atomic charges of 0.06,-0.18, and 0.39, respectively.Indeed, in the case of BH3, when the atomic radius of B in theFBO calculation is adjusted in order to reproduce the AIMcharges, the FBO value drops down to 0.519, quite close to theDI value. This seems to indicate that, similar to DI, the FBOvalues can be very small for bonded interactions provided thatstrong ionicity is also predicted. Nevertheless, since the adjustedvalue of the B radius is as low as 0.22 au, generally one shouldnot expect FBO to provide such small values when using

SCHEME 1: Structures of the Molecules Studied withX1, X2, X3, X4 ) C, N or P, Y ) BeH-, B-, BH, BH2

-, O,S, NH, PH, CH-, CH2, CF2, N-, NH2

+, Al-, AlH,AlH 2

-, SiH+, SiH2, P-, PH2+, GaH, GaH2

-, GeH-,GeH+, GeH2, As-, AsH, AsH2

+, Se, CdCH2, CdO, CdSor CdSe

Figure 1. Plot of delocalization index (DI) against bond order fromfuzzy partitioning (FBO).


46

balanced sets of atomic radii. The fact that atomic charges andbond orders are not observables prevents distinguishing betweenany of the two possible interpretations of these results, that is,either the DI overemphasizes the ionic with respect to thecovalent character or the FBO underestimates the ionic characterof the chemical bonds. However, in this case, the FBO resultsare more consistent with the expected formal bond orders. Thedata within the square in Figure 1 correspond to P-O-typebonds. These are considered as single bonds within AIM (δAIM-(P,O)≈ 0.8), while FBO reports a bond multiplicity betweensingle and double (δFBO(P,O)≈ 1.4). Previous calculations byChesnut49,50 on P-O bonds also yield DI values lower thanexpected from the formal multiplicity of the bonds. Thus,Chesnut reports for formally single P-O bonds DI values of0.60-0.90 au and of 1.40 and 1.54 au for formal PdO doublebonds. It seems clear from the data presented that DI valuesare sometimes too small to be considered as bond order indexesfrom the classical point of view. However, even though Baderand co-workers3 refuse the idea of using the DI as a bond order,its expression at the HF level is deeply connected with otherbond order indexes. Thus, a priori one would have expected amore similar behavior between the DI and the analogous FBOindexes.

In Figure 2 the correlation of DI and MBO indexes isdepicted. The data on this plot are somewhat more spread,especially in those regions marked with circles and a square,which will be discussed in greater detail later. Again, nocorrelation is observed for the nonbonding interactions (bottomleft data) with DIs below 0.3 au. As a general trend, formalbond orders for homonuclear chemical bonds are well-reproduced by both DIs and MBOs, whereas DIs yield rathersystematically lower bond orders than MBOs for polar bonds.7,43

In this sense, most values of the bond order tend to appear belowthey ) x line. In fact, the largest calculated DI is 2.00, whereasfor MBO values up to 2.15 have been frequently observed.

Although the correlation between both indexes is rather good,there are again some particular bond types that are poorlycorrelated. The encircled points mostly contain data obtainedfor bonds involving phosphorus atom, such as P-O, P-C, orP-N. The MBO values for these bonds are, in general, below0.5, yielding negative values in some particular cases. Theexplanation for this odd behavior of the MBO in these cases is

given below. The bonding interactions involving B, Be, and Alatoms (data inside the square) are in line with the aforemen-tioned underestimation of the bond order by the DIs in polarbonds. Now this hypothesis is put forward in the light of thevalues assigned by MBO, which yields values of ca. 0.9 forsingle (B, Be, Al)-X bonds, much closer to the FBO than tothe DI. Finally, some points are clearly off of the correlation.For instance, in the top left and center of the plot, two pointswith a DI of ca. 1.8 present a corresponding MBO of 0.25 and1.1. These values correspond to CdS and CdSe bond types,respectively. The MBO values are clearly too small. We haverepeated the calculation for these two molecules with differentbasis sets in order to determine whether it can be assigned to abasis set deficiency. It has been found that increasing the basisset quality up to 6-311++G(2df,pd) barely changes the valueof the bond order. However, with the 6-311G** basis set, theMBO values increase up to 1.89 and 1.76, respectively. Thus,as already reported,7 including diffuse functions in the calcula-tion can dramatically affect the MBO values. This deficiencyis also well-known for other Hilbert-space descriptors such asMulliken charges.51 The observed discrepancies in the P-(C,N, O) bonds pointed out above can also be attributed to theeffect of the diffuse functions on the calculation of MBOs.

Finally, the correlation between MBO and FBO is shown inFigure 3. Most of the data appear below they ) x line, so thatthe FBO values are in general larger than the MBO ones. Again,the bond type that exhibits more dispersion in the calculatedvalues involves the phosphorus atoms due to poor MBO valuesinduced by the use of diffuse functions. The most dramatic case,however, corresponds to an Al-C bond with a FBO value closeto 1 that presents an unphysical negative MBO value. Thecorresponding MBO value calculated from a HF/6-311G** wavefunction yields 0.60.

In conclusion, we can say that there is a general goodagreement in the trends showed by the three indexes. This isespecially true for homonuclear interactions. In this case,unsurprisingly, the different methods of partitioning the mo-lecular space produce comparable partitioning schemes and,consequently, lead to similar bond orders. The correlationbecomes poorer for polar bonds. In particular, the DI valuesfor polar bonds were normally slightly below the expectedvalues. In addition, large deviations are found for bondsinvolving phosphorus atoms. In this latter group of molecules,the MBO method often fails to properly describe the bonding,due to the effect of diffuse functions lacking of marked atomic

Figure 2. Plot of delocalization index (DI) against Mayer bond order(MBO).

Figure 3. Plot of Mayer bond order (MBO) against bond order fromfuzzy partitioning (FBO).


47

character. For bonds involving B, Be, and Al atoms, the DIvalues differ considerably from both FBO and MBO ones,leading to unrealistically low bond orders.

From the groups of indexes studied we have extracted thoseconcerning C-C bonding interactions. Analogous correlationsto those depicted in Figures 1-3 have been constructed for thissubset, consisting of 179 values, and plotted in Figures 4-6.As expected from the fact that we restrict the comparison ofbond orders to homonuclear bonds, the agreement is betterbetween DI and FBO values, whereas some points are off ofthe correlation in Figures 5 and 6 where MBO is correlated.However, again, one has to bear in mind that these points outof the trend may be due to the use of diffuse functions. It isalso worth mentioning that the FBO values are systematicallylarger than MBO or DI ones. Nevertheless, the results areencouraging since all plots show good correlations (Pearsoncoefficient is always abover ) 0.96) between the three methods.This opens up the possibility of using either the FBO or MBOprocedures to construct reliable bond order-based indices instead

of the existing computationally demanding AIM-based ones.In particular, we plan to explore this possibility for PDI andFLU.37,39

As aforementioned, the weight factors that control the shapeand size of the fuzzy atoms are not uniquely defined. Anexhaustive analysis of the effect of different weight factors uponthe fuzzy atom derived indices probably deserves specialattention, and it is out of the scope of the present work.Nevertheless, we have tested the effect of taking a different setof atomic radii that are used to determine the weight factoraccording to Becke’s function. In particular, we have chosenthe empirical set of Slater-Bragg.52 The comparison of theFBOs computed with both sets of atomic radii is shown inFigure 7. Both sets of values are practically coincident (r )1.000), even for the “nonbonded pairs”. Indeed, contrary toelectron populations, bond orders have been proved to be verystable with respect to dramatic changes in atomic radii of atoms,such as changing the value for a particular atom and keepingthe rest invariant.53

Conclusions

Three different sharing indexes, namely, the DI, the FBO,and the MBO have been compared for a large group of five-membered rings with a rich variety of bonds. In general, thereis a good agreement with the exception of bonds involving P,

Figure 4. Correlation of delocalization index (DI) with bond orderfrom fuzzy partitioning (FBO) for the 179 different C-C bondscontained in our training set of molecules; correlation coefficientr )0.9765.

Figure 5. Correlation of delocalization index (DI) with Mayer bondorder (MBO) for the 179 different C-C bonds contained in our trainingset of molecules; correlation coefficientr ) 0.9687.

Figure 6. Correlation of Mayer bond order (MBO) with bond orderfrom fuzzy partitioning (FBO) for the 179 different C-C bondscontained in our training set of molecules; correlation coefficientr )0.9646.

Figure 7. Correlation of bond order from fuzzy partitioning (FBO)calculated using Koga vs Slater-Bragg atomic radii sets; correlationcoefficient r ) 1.0000.


48

Al, B, and Be atoms. For the last three significant differencesof the DI results and the FBO or MBO values are found. Onthe basis of previous results on the hydrides of Be, B, and Al,3

we consider that the DI values are underestimated for thesesystems because of the reduced size of the Be, B, and Al atomicbasins within the AIM partitioning of the molecular space. Asfar as P bonds are concerned, the answers differ from index toindex. For these bonds, the MBO suffers from strong basis setdependence (the use of diffuse functions dramatically decreasesthe bond order), although its values are closer to the FBO onesif more atomic(without diffuse functions) basis sets are used.With respect to the remaining types of bonds, good correlationsare achieved between the three indexes, with most pointsfollowing the expected trend despite some exceptions. Thisdrives us to conclude that, in general, the three indexes predictthe same electron pair distribution. The agreement between DI,FBO, and MBO results is particularly good for homonuclearbonds. In particular, noticeable trends are found for C-C bonds,encouraging the substitution of the DIs by the cheaper FBOsor even the MBOs in aromatic indexes based on DI such as thePDI and FLU. This issue is currently being investigated in ourlaboratory.

Note Added in Proof. Soon after the manuscript wasaccepted, a paper appeared in this journal by Torre et al. (Torre,A.; Alcoba, D. R.; Lain, L.; Bochicchio, R. C.J. Phys. Chem.A 2005, 109, 6587) that discusses the use of fuzzy atoms todetermine three- and two-center bond indices.

Acknowledgment. Financial help has been furnished by theSpanish MCyT Projects BQU2002-0412-C02-02 and BQU2002-03334. M.S. is indebted to the Departament d’Universitats,Recerca i Societat de la Informacio´ (DURSI) of the Generalitatde Catalunya for financial support through the DistinguishedUniversity Research Promotion, 2001. E.M. thanks the Secretarıáde Estado de Educacioń y Universidades of the MECD fordoctoral fellowship AP2002-0581. J.P. also acknowledges theDURSI for the postdoctoral fellowship 2004BE00028. We alsothank the Centre de Supercomputacio´ de Catalunya (CESCA)for partial funding of computer time.

References and Notes

(1) Mulliken, R. S.J. Chem. Phys.1955, 23, 1833.(2) Fulton, R. L.; Mixon, S. T.J. Phys. Chem.1993, 97, 7530.(3) Fradera, X.; Austen, M. A.; Bader, R. F. W.J. Phys. Chem. A

1999, 103, 304.(4) Mayer, I.Chem. Phys. Lett.1983, 97, 270.(5) Reed, A. E.; Curtiss, L. A.; Weinhold, F.Chem. ReV. 1988, 88,

899.(6) Mayer, I.; Salvador, P.Chem. Phys. Lett.2004, 383, 368.(7) AÄ ngyan, J. G.; Loos, M.; Mayer, I.J. Phys. Chem.1994, 98, 5244.(8) AÄ ngyan, J. G.; Rosta, E.; Surjań, P. R.Chem. Phys. Lett.1999,

299, 1.(9) Cioslowski, J.; Mixon, S. T.J. Am. Chem. Soc.1991, 113, 4142.

(10) Clark, A. E.; Davidson, E. R.J. Chem. Phys.2001, 115, 7382.(11) Ruedenberg, K.ReV. Mod. Phys.1962, 34, 326.(12) In this context, the second-order density matrix is normalized to

the number of electron pairs,N(N - 1).(13) Bader, R. F. W.Acc. Chem. Res.1985, 18, 9.(14) Bader, R. F. W.Chem. ReV. 1991, 91, 893.(15) Bader, R. F. W.; Stephens, M. E.J. Am. Chem. Soc.1975, 97,

7391.

(16) Fradera, X.; Poater, J.; Simon, S.; Duran, M.; Sola`, M. Theor. Chem.Acc.2002, 108, 214.

(17) Since it is calculated from the pair density distribution, the effecton the population of other atomic domains at the same time cannot be takeninto account.

(18) Bader, R. F. W.; Matta, C. F.Inorg. Chem.2001, 40, 5603.(19) Kar, T.; AÄ ngyan, J. G.; Sannigrahi, A. B.J. Phys. Chem. A2000,

104, 9953.(20) Poater, J.; Duran, M.; Sola`, M.; Silvi, B. Chem. ReV. 2005, 105, in

press.(21) Fulton, R. L.J. Phys. Chem.1993, 97, 7516.(22) Fulton, R. L.J. Phys. Chem. A2004, 108, 11691.(23) Fulton, R. L.; Perhacs, P.J. Phys. Chem. A1998, 102, 9001.(24) Fulton, R. L.; Perhacs, P.J. Phys. Chem. A1998, 102, 8988.(25) Bochicchio, R. C.; Lain, L.; Torre, A.Chem. Phys. Lett.2003, 374,

567.(26) Poater, J.; Sola`, M.; Duran, M.; Fradera, X.Theor. Chem. Acc.

2002, 107, 362.(27) Wang, Y. G.; Werstiuk, N. H.J. Comput. Chem.2003, 24, 379.(28) Clark, A. E.; Davidson, E. R.Int. J. Quantum Chem.2003, 93,

384.(29) Clark, A. E.; Sonnenberg, J. L.; Hay, P. J.; Martin, R. L.J. Chem.

Phys.2004, 121, 2563.(30) Salvador, P.; Mayer, I.J. Chem. Phys.2004, 120, 5046.(31) Mayer, I.Int. J. Quantum Chem.1984, 26, 151.(32) Mayer, I.Int. J. Quantum Chem.1985, 28, 419.(33) Mayer, I.Int. J. Quantum Chem.1986, 29, 477.(34) Mayer, I.Int. J. Quantum Chem.1986, 29, 73.(35) Vyboishchikov, S. F.; Salvador, P.; Duran, M.J. Chem. Phys.2005,

244110.(36) Bridgeman, A. J.; Cavigliasso, G.; Ireland, L. R.; Rothery, J.J.

Chem. Soc., Dalton Trans.2001, 2095.(37) Poater, J.; Fradera, X.; Duran, M.; Sola`, M. Chem. Eur. J.2003, 9,

400.(38) Poater, J.; Fradera, X.; Duran, M.; Sola`, M. Chem. Eur. J.2003, 9,

1113.(39) Matito, E.; Duran, M.; Sola`, M. J. Chem. Phys.2005, 122, 014109.(40) Becke, A. D.J. Chem. Phys.1988, 88, 2547.(41) Hirshfeld, F. L.Theor. Chim. Acta1977, 44, 129.(42) Cyranski, M. K.; Krygowski, T. M.; Katritzky, A. R.; Schleyer, P.

v. R. J. Org. Chem.2002, 67, 1333.(43) Bochicchio, R. C.; Ponec, R.; Lain, L.; Torre, A.J. Phys. Chem. A

2000, 104, 9130.(44) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb,

M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.;Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A.D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi,M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.;Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick,D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.;Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi,I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.;Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M.W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon,M.; Replogle, E. S.; Pople, J. A.Gaussian 98, revision A.11; Gaussian,Inc.: Pittsburgh, PA, 1998.

(45) Biegler-Konig, F. W.; Bader, R. F. W.; Tang, T.-H.J. Comput.Chem.1982, 3, 317.

(46) Mayer, I.; Salvador, P. Program ‘FUZZY’; Girona, 2003. Availablefrom http://occam.chemres.hu/programs.

(47) Suresh, C. H.; Koga, N.J. Phys. Chem. A2001, 105, 5940.(48) If the reader checks out the Supporting Information file given by

Cyranski et al.42, the computed molecules are the following: 3, 6, 7, 12,15, 17, 18, 20, 21, 23, 25, 27, 29, 31, 33, 35, 38, 42, 43, 45, 47, 49, 51,53-78, 80-83, 92-94, 96-101, 104-140, 142-146, 148-153, 198, 201,204, 207, 210, 213, and 216.

(49) Chesnut, D. B.J. Phys. Chem. A2003, 107, 4307.(50) Chesnut, D. B.Chem. Phys.2003, 291, 141.(51) Fonseca-Guerra, C.; Handgraaf, J.-W.; Baerends, E. J.; Bickelhaupt,

F. M. J. Comput. Chem.2003, 25, 189.(52) Slater, J. C.J. Chem. Phys.1964, 41, 3199.(53) Salvador, P. Unpublished results.


49

4.2.- Exploring the Hartree-Fock Dissociation Problem in

the Hydrogen Molecule by Means of Electron

Localization Measures.

51

2004-0927MS #

A Novel Exploration of the Hartree–Fock HomolyticBond Dissociation Problem in the HydrogenMolecule by Means of Electron LocalizationMeasures

Most introductory quantum chemistry textbooks discuss the well-know problem of theinappropriate description given by the restricted Hartree–Fock (HF) method of the homolyticbond dissociation. This weakness of the restricted HF method is generally addressed byanalyzing the difference between the energies at long internuclear distances obtained with theconfiguration interaction (CI) and HF methods, which requires an exhaustive understanding ofthe methodology. In this article we provide a new insight into this subject using localizationand delocalization indices defined in the framework of the atoms-in-molecules theory toanalyze the homolytic bond dissociation in the hydrogen molecule. It is shown that therestricted HF requirement of molecular orbitals to be occupied simultaneously by a couple ofelectrons with different spin is responsible for localization to hold on the same value whiledissociation is happening, thus reflecting the well-known deficiency of the HF method to dealwith bond dissociation. On the other hand, when the CI method is used, the localizability ofthe electrons in the system turns into the intuitive scheme expected for homolytic bonddissociation.

Abstract

Keywords ListComputational Chemistry; Computer-Based Learning; Graduate Education / Research;Learning Theories; Lewis Structures; Molecular Properties / Structure; Physical Chemistry;Quantum Chemistry; Textbooks / Reference Books; Theoretical Chemistry; Upper-DivisionUndergraduate

53

www.JCE.DivCHED.org • Vol. XX No. XX Month 200X • Journal of Chemical Education 1

The workhorse of the wavefunction methods is theHartree–Fock (HF) approximation. The HF procedure is im-portant not only for its own sake but as the starting pointfor other more crude or more elaborate approximations, suchas those represented by the semiempirical methods or thoseincluding to some extent electron correlation effects, respec-tively (1). For this reason, most introductory quantum chem-istry textbooks contain a full chapter dedicated to the HFmethod (2, 3). At the end of such chapter, it is usually shownthat the HF approach within the restricted formalism pro-vides an inappropriate description of the hydrogen moleculeat long bond lengths, which leads to the so-called HF ho-molytic bond dissociation problem. This weakness of the re-stricted HF method can be reasonably solved by using theunrestricted HF methodology or more properly by employ-ing configuration interaction (CI) methods. The aim of thiswork is not to dwell on how to solve the HF dissociationproblem but to provide a new view on this subject by usinglocalization and delocalization indices defined from the sec-ond-order density within the atoms-in-molecules (AIM)theory (4, 5). We have observed that many students of in-troductory courses of quantum chemistry find the conceptof localizability of electrons helpful to better grasp this well-know defect of the restricted HF method. Finally, it is worthmentioning that different authors have treated the hydrogenmolecule (although just a few studied the dissociation pro-cess) using second-order densities from other interesting per-spectives. Thus, Savin and coworkers (6) carried out adiscussion in terms of probability, while Ponec (7, 8) andother authors (9–11) present an study in terms of fluctua-tions, bond orders, or other related quantities.

Density-Based Descriptors

We begin here with a brief review of the concepts usedto quantify electron localizability in molecules. Density-baseddescriptors of electronic structure have gained the first placeas functions to characterize the molecular structure and thenature of the bonding in molecules. First attempts to elec-tronic structure characterization were done in the frameworkof Lewis bonding theory (12), that afterwards was completedby the valence shell electron pair repulsion (VSEPR) (13–15)method of Gillespie. Later on, molecular orbitals became thepreferred tool to discuss molecular electron distribution andto analyze chemical bonding. Although still widely used, inmany works this latter method has been replaced or comple-mented by the use of density-based descriptors of the elec-tronic structure owing to their simplicity and easy handtreatment. A recent review in this Journal studying molecu-lar electron density distributions states this fact (16).

The two methodologies more commonly used in recentyears to derive density-based descriptors of molecular bond-ing are the AIM theory (4, 5, 17, 18), and the electron lo-calization function (ELF) (19–21). In both approaches,subsystems (basins) are defined in terms of the vector fieldof the gradient of the function involved, namely the elec-tron density or the ELF, respectively. These basins are usedto divide the molecular space. Other partitions, like theMulliken-like partition in the Hilbert space expanded by thebasis functions are also possible. In the AIM theory, the par-tition of space is done through atomic basins that are definedas the regions in real space bound by zero-flux surfaces inthe electron density or by infinity (4, 5, 16–18). Usually, eachbasin can be assigned to one of the atoms in the molecule.

Localizability To Characterize Electronic Structure

Based on AIM (4, 5, 16–18) partition of space but eas-ily extended to any other possible subdivision of space, Baderand Stephens (22) introduced a quantity that accounts forelectron localizability; the quantity known as exchange-cor-relation density, γxc, that can be calculated from density func-tions, ρ, as follows

xc 1 2,( ) = 2( )γ γr r rr r r r1 2 1 2,( ) − ( ) ( )ρ ρ (1)

where r1 and r2 stand for the coordinates of electrons 1 and2, respectively, and the second term on the right-hand sidecontains the well-know electron density first-order densityfunction—and the first term is known as second-order den-sity function or simply electron-pair density—and resultsfrom integration of the squared wavefunction over the wholespace on n − 2 of its coordinates:

r r r r r r21 2 1

23,,( ) = ( )( )γ d dK K KΨ n n

RR 33(2)

The second [first] order density function can be written interms of the second [first] order density matrix (DM2[DM1]), which is its basis of representation, see for instanceref 23

ii,j,k,l

j2

1 2 1 2∑ ( ) ( ) = *( ) *( )γ χ χr r r r, ( ) ( )χ χk lr r1 2ij kl2( )( |Γ ))

( ) = * ( )( ) ( )∑ ρ χ χk lk,l

k lr r r1 1 ( )11|Γ

(3)

where χ(r) represents the molecular orbital i and γ(2)(ij |kl )and γ(1)(kl ) are the DM2 and DM1 written in terms of mo-

A Novel Exploration of the Hartree–Fock Homolytic Bond WDissociation Problem in the Hydrogen Molecule by Meansof Electron Localization MeasuresEduard Matito, Miquel Duran, and Miquel Solà*Institut de Química Computacional and Departament de Química, Universitat de Girona, 17071 Girona, Catalonia,Spain; *[email protected]

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http://www.jce.divched.org/Journal/Issues/2005/

http://www.jce.divched.org/Journal/

2 Journal of Chemical Education • Vol. XX No. XX Month 200X • www.JCE.DivCHED.org

lecular orbitals. The quantity in eq 1 is the difference be-tween a common electron pair distribution, that is, the sec-ond-order density function and a noninteracting electron pairdistribution constructed as two independent electron distri-butions. Thus, the exchange-correlation density measures theinfluence of the correlation between pairs of electrons, sincewe are comparing a pair interacting system with a noninter-acting one. The pair regions (r1, r2) of the space where itsvalue is greater are due to a higher influence of electron cor-relation on that sharing zone, where one can expect electronpresence to be quite important; the function value helps toquantify this importance.

Five years ago, Fradera, Austen, and Bader (24) gavename to these quantities when integrated on two certain re-gions of space, that is, the atomic basins A and B. Exchange-correlation density when integrated over these pair regionsturns into the well-known delocalization index (DI), δ(A, B ):

d dxcδ A B r rBA

, ,( ) = − ( ) −γ 1 2 r r,1 21 2r r

d dxc

BA

( )γ 1 2r r

d dxc

AB

( )γ 1 2r r

= −2 r r,1 2(4)

The DI accounts for the electrons delocalized or shared be-tween atomic basins A and B (24, 25). On the other hand,the localization index (LI) accounts for the electrons local-ized in the region A, λ(A), and is defined as:

r r,( )1 2 d d1 2r rAA A

xcAA

2( ) = − =

( )λ γ

δ ,(5)

Its value is always less than or equal to the average numberof electrons or the electron population, N(A), in that region,which results from the density integrated over the region:

dN A( ) = ( )ρ r r1 1

A(6)

It is worth noting that the AIM topological analysis providesan exhaustive partition of molecular space, so LIs and DIsobey the following sum rule:

A B A N AB A

( ) + ( ) = ( )∑ ≠

δ λ1

2, (7)

Substituting eqs 1 and 3 into eqs 4 and 5, we obtain

δ A B N A N B S A S Bik jl,( ) = ( ) ( ) − ( ) ( )2 2i,j,k,l∑ ij kl2( )( |Γ )) (8)

λ A N A S A S Aik jl( ) = ( ) − ( ) ( )2

i,j,k,l∑ ij kl2( )( |Γ )) (9)

where overlaps in terms of spinorbitals have appeared in theright-hand side of the above equations, and the notation isas follows:

r1 r1 r1dS Aij i j

A

( ) = ( ) ( )χ χ*

(10)

Hereafter we will identify δ(A,B ) with mutual electron shar-ing between these two regions, and λ(A) with electron local-ization in A; nothing else is needed to understand subsequentexplanations.

Some Examples To Better Understand the Meaningof LI and DI Indices

It is worth mentioning that for molecular bonds withequally shared pairs such as H2 or N2, a simple relationshipbetween the DI and the number of Lewis bonded pairs (12)(bond order) is generally found. Table 1 lists several examplesof LIs and DIs for a series of diatomic molecules calculatedat the HF/6-311++G(2d,2p) level (24). H2 is a representa-tive example of a molecule with an equally shared pair of elec-trons. In this molecule the LI is 12 for each atom and thedelocalization contribution is 1, which completely agrees withthe prediction of the Lewis model (12) for the electronicstructure of H2. Let us now consider the N2 molecule. Ac-cording to the Lewis model (12), the LI should have the con-tribution of 2 from the 1s2 core electrons, 2 from the lonepair, and 1.5 from the three shared pairs, giving λ(N) = 5.5.Moreover, these three equally shared pairs should give a DIof 3, achieving a total contribution of 14, the number of elec-trons. If we compare these values with the ones in Table 1,we realize that they are quite similar. The fact that δ(N,N´)> 3 means that there is a slight delocalization of the non-bonded pair of a N atom onto the basin of the other N atom.

However, with the exception of equally shared pairs, theDI does not reproduce the results of the Lewis model (24).In molecules AB where there is an important charge trans-fer, the electron pair is not equally shared but partly local-ized on the more electronegative atom A, and as aconsequence, λ(A) increases at expenses of δ(A, B) and λ(B).Thus, LiF presents a δ(Li, F) of 0.18 and a λ(F) equal to9.85 electrons. It is important to note that the DI of LiFdoes not imply a Lewis bond formed from 0.18 pairs of elec-trons, instead it means that in LiF there is a bonded pair thatis very unequally shared. This can be further illustrated withthe isoelectronic sequence involving N2, NO+, CN−, and CO,

(snoitalupoPcimotA.1elbaT N (noitazilacoL,) λλλλλ ,)(noitazilacoleDdna δδδδδ secidnI)

seluceloMcimotaiDfoseireSarof

eluceloM motA N )A( λ )A( δ )B,A(

H2 H 000.1 005.0 000.1N2 N 000.7 974.5 240.3F2 F 000.9 853.8 382.1

FiL iL 060.2 179.1 871.0F 049.9 158.9 ---

OC C 746.4 068.3 475.1O 453.9 765.8 ---

NC − C 722.5 121.4 012.2N 377.8 866.7 ---

ON + N 525.5 323.4 504.2O 574.8 372.7 ---

N ETO : fermorfataD 42 folevelkcoF–eertraHehttadetupmocatad;snmuloc3tsalehtniseulaV.tessisab)p2,d2(G++113_6ehthtiwyroeht

.snortceleforebmunehtotdnopserroc

55





all triply bonded molecules whose DI drops in the sequence3.04, 2.41, 2.21, and 1.57 electrons, respectively. The DIsdecrease with the increased electronegativity difference(greater charge transfer) of the atoms involved in the bond.

The HF Homolytic Dissociation of Molecular Hydrogen

We focus now on the main subject of this contribution,which is the usefulness of DI and LI indices to understandthe incorrect behavior of the HF method in the dissociationof the hydrogen molecule. For the sake of simplicity, we willwork with the simplest basis set, that is, a STO-3G minimalbasis set, consisting of 1sA and 1sB functions both comingfrom the contraction of three primitive Gaussian-type func-tions. The system is studied first at the HF level of theoryand finally in terms of the CI method.

A molecular orbital (MO) scheme of the hydrogen mol-ecule obtained at the HF level is drawn in Scheme I. In thisScheme, one can see how the two 1s atomic orbitals com-bine to form the σg and σu* bonding and antibonding MOs,having the following expressions

gS

sAB

A=+( )

σ1

2 11 ++( )1sB

=−( )

−( )1

2 11 1

Ss s

ABA Buσ

*(11)

where SAB is the overlap between the atomic orbitals 1sA and1sB. See, for instance, refs 2, 26, and 27.

At the restricted HF level, we enforce the restriction ofspinorbitals to be occupied by a couple of electrons. Thus,the restricted HF ground-state wavefunction with a pair ofelectrons occupying the lowest energy MO is

≡ o gσΨ σσg

=+( )

+ +(( ))1

2 11 1 1 1

Ss s s s

ABA B A B

(12)

while the excited-state wavefunction corresponding to twoelectrons occupying the antibonding orbital is

σ σD u u≡

=−

( )11

2 1 SA

Ψ * *

BA B A Bs s s s

( )− −( )1 1 1

(13)

The HF method give us the molecular orbitals σg andσu* to construct the Slater determinant of minimum energy.The CI method constructs all possible Slater determinantsusing the HF wavefunction as follows (3). First arranging then electrons of the HF determinant in the m spinorbitals cho-sen as a basis in all possible manners; that generates a set ofdeterminants based in our HF one. Afterwards a CI wave-function is generated with the coefficients that expand theset of determinants. In our case, there is no need to use anyother determinants than those corresponding to thebiexcitation and the HF one, because the monoexcitacionshave different symmetry, hence they do not mix with theother determinants (3). So that we can write the CI wave-function as follow

Dc c c c= + = +CI o o D D o g g uΨ Ψ Ψ σ σ σ ** *σu (14)

where c0 and cD are the expansion coefficients to be deter-mined in the CI calculation. After application of the varia-tional principle, the above equation gives us the exact resultfor this minimal basis set within the Born–Oppenheimer ap-proximation when neglecting relativistic effects. At infiniteseparation of the hydrogen atoms, it is found that (3):

CI HFElim E s s s sr

A A A A− = −→ ∞

1

21 1 1 1 (15)

The difference between the CI and HF results at infinite in-ternuclear distance given by eq 15 is exactly the repulsionboth electrons feel because of occupying the same spatial partof the spinorbital. Once eq 15 has been deduced (3), the HFdissociation problem turns out evident: both electrons willalways feel each other because they are restricted to share thesame spatial orbital. If the other contributing configurationgiven in eq 13 is used (or we work within the unrestrictedformalism) the dissociation problem is overcome, since thesystem is then described as a pondered contribution of dif-ferent electronic situations, eq 14, leading to a better—actu-ally exact-approach. Figure 1 shows the energy profilescalculated at CI and HF levels of theory; the error of the HFapproach is especially important for large interatomic dis-tances (3).

H 1sB

σu*

σg

1sA H

Scheme I. Molecular orbitals scheme corresponding to molecularhydrogen.

Figure 1. Energy profile for the homolytic bond dissociation of mo-lecular hydrogen at the HF and CI levels of theory.

r(H−H′) / Å

E /

Eh

-0.5

-0.7

-0.9

-1.1

-1.30 1 2 3 4

HF

CI

56





Localizability as a Tool To Analyzethe Homolytic Bond Dissociation Problem

In this section, we will use the definitions of LI and DIto provide an alternative and easy way to understand the ho-molytic bond dissociation problem in terms of localizability.This approach does not need of any MO interpretation andmoreover all quantities required can be calculated with handysoftware, such as the Gaussian 98 (28) and AIMPAC (29)packages.

The following approximations for LI and DI, definedin eqs 4 and 5,

λ H( ) ≈ 11

2− c co D (16)

1 2( ) ≈ + c co DH, Hδ ′ (17)

can be easily derived from the expression of the DM2 (seethe Appendix in the Supplemental MaterialW). As said be-fore, λ(H) gives the number of electrons localized in basinof a H atom, while δ(H, H´) is a measure of the electronsdelocalized or shared between the two atomic basins.

Equations 16 and 17 enable the calculation of the indi-ces with just the need of the contribution of both configura-tions at each interatomic distance; no overlap or densitymatrix is needed. Using this approximation or the exact ex-pression for the DI, the interatomic distance dependence on

the DI—absent in monodeterminantal wavefunctions andimplicit on the coefficient values—clearly come out, as shownin Figure 2 and in the values of Table 2.

Now, the only task remaining is to compute the CI wave-function of the hydrogen molecule at different interatomicdistances. Some results are collected in Table 2. The indices’tendency is plotted in Figure 3. The graphic shows the ex-pected trend: mutual shared electrons (DI) diminish whileatoms separate reaching the down bound zero, indicative ofno electron sharing when atoms are infinitely separated. Be-

Figure 2. Delocalization indices (electrons) at different interatomicdistances. The upper curve is the exact result, while the lower oneis calculated using the approximation given by eq 17.

r (H−H′) / Å

(H− H

′ )

0.0

0.2

0.4

0.6

0.8

0.7 1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.1 3.4 3.7

1 + 2cocD

exact

)seulaVtcaxE(secidnInoitazilacoLdnanoitazilacoleD,palrevOsOM,stneiciffeoCIC,ygrenEFH,ygrenEIC.2elbaTteSsisaBG3-OTSehtgnisusecnatsiDcimotaretnItnereffiDta

r Å/)H–H( E /)IC( Eh E /)FH( Eh co cD S 21 )H( δ )´H,H( λ )H(

1537.0 3731.1 0711.1 3499.0 4601.0 8544.0 7328.0 1885.00538.0 3031.1 3501.1 1199.0 1331.0 3944.0 1787.0 5606.00530.1 6390.1 1650.1 7289.0 5581.0 0754.0 5596.0 2256.00532.1 0940.1 9399.0 9769.0 5152.0 2564.0 6875.0 7017.00534.1 1900.1 6039.0 3449.0 0923.0 8274.0 4444.0 8777.00536.1 0979.0 9178.0 1219.0 0014.0 4974.0 5213.0 7348.00538.1 0959.0 4028.0 6478.0 8484.0 6484.0 3302.0 4898.00530.2 0749.0 7677.0 5738.0 5645.0 8884.0 4521.0 3739.00532.2 3049.0 6047.0 8408.0 6395.0 9194.0 3570.0 4269.00533.2 3839.0 1527.0 7097.0 2216.0 2394.0 1850.0 9079.00534.2 8639.0 2117.0 3877.0 9726.0 3494.0 9440.0 6779.00536.2 0539.0 6786.0 8757.0 5256.0 0694.0 8620.0 6689.00538.2 0439.0 7866.0 5247.0 8966.0 3794.0 0610.0 0299.00530.3 6339.0 7356.0 4137.0 9186.0 2894.0 6900.0 2599.00532.3 4339.0 8146.0 5327.0 3096.0 8894.0 8500.0 1799.00534.3 2339.0 4236.0 0817.0 1696.0 2994.0 5300.0 3899.00536.3 2339.0 0526.0 2417.0 9996.0 5994.0 1200.0 0999.00538.3 2339.0 0916.0 7117.0 5207.0 7994.0 2100.0 4999.00530.4 2339.0 1416.0 0017.0 2407.0 8994.0 7000.0 6999.00532.4 2339.0 0016.0 9807.0 3507.0 9994.0 4000.0 8999.00534.4 2339.0 6606.0 2807.0 0607.0 9994.0 2000.0 9999.0

... ... ... ... ... ... ... ...

∞ E )H( E )H( + )4/1( s1 A s1 A s1| A s1 A /1 √2 /1 √2 2/1 0 1

N ETO : .snortceleforebmunehtotdnopserrocsnmuloc3tsalehtniseulaV

57





side, the electrons localize one on each atom when distanceincreases; LI comes by the upper bound of one electron peratom when no interaction holds. In contrast the HF LI andDI values remain constant at 0.5 and 1 electrons, respectively.From these results, it becomes evident to the students thatthe failure of the restricted HF method to correctly dissoci-ate molecular hydrogen in atomic hydrogen is due to its in-ability to localize the electrons in each hydrogen atom at largeinteratomic distances. This lack of localization in the HFmethod results in the artificial excess of electron repulsionfound in eq 15.

Finally, it is worth noting that any attempt to evaluatethe electron localization must use two-electron quantities, thepair density being the simplest of such quantities. The analysisof localization versus delocalization during the dissociationprocess cannot be performed using atomic charges derivedfrom the electron density only. Any method of computingatomic charges, such as the Mulliken population (30), givesan identical charge of 1 electron to each H, irrespectively ofthe HH bond length considered or the method of calcula-tion used. Thus, this educational example has the additionalvalue of clearly illustrating to students how the pair densitycan be used to get a deeper insight into the nature of mo-lecular electronic structure.

Conclusions

The aim of the above discussion has been to show howa detailed quantitative treatment of localizability of electronsin the hydrogen molecule may be presented to the studentto get further understanding of the homolytic bond disso-ciation problem in the restricted HF method. We have shownthat the untenable necessity of molecular spinorbitals to beoccupied simultaneous by a couple of electrons is responsiblefor localization to hold on the same value while dissociationis occurring. When the CI method is used, the localizabilityof the electrons in the system turns into the intuitive schemeexpected for homolytic dissociation: atomic electronlocalizability increases with interatomic distance until oneelectron localizes in each atom, while mutual shared electronsdecreases to reach no sharing in the limit of noninteractingfragments. In our opinion, this approach represents an alter-

native and instructive view that helps the students to betterunderstand the homolytic bond dissociation problem in re-stricted HF wavefunctions.

Acknowledgments

Financial help has been furnished by the Spanish MCyTProjects No. BQU2002-0412-C02-02 and BQU2002-03334. MS is indebted to the Departament d’Universitats,Recerca i Societat de la Informació (DURSI) of theGeneralitat de Catalunya for financial support through theDistinguished University Research Promotion, 2001. EMthanks the Secretaría de Estado de Educación y Universidadesof the MECD for the doctoral fellowship no. AP2002-0581.We also thank the Centre de Supercomputació de Catalunya(CESCA) for partial funding of computer time.

WSupplemental Material

The derivations for the LI and DI approximations givenin eqs 16 and 17 are available in this issue of JCE Online.

Literature Cited

1. Cramer, C. J. Essentials of Computational Chemistry: Theoriesand Models; John Wiley & Sons: New York, 2002.

2. Pilar, F. L. Elementary Quantum Chemistry; McGraw–Hill:New York, 1990.

3. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Intro-duction to Advanced Electronic Structure Theory; MacmillanPublishing Co.: New York, 1982.

4. Bader, R. F. W. Acc. Chem. Res. 1985, 18, 9–15.5. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clar-

endon: Oxford, 1990.6. Chamorro, E.; Fuentealba, P.; Savin, A. J. Comput. Chem.

2003, 24, 496–504.7. Ponec, R.; Carbó-Dorca, R. Int. J. Quantum Chem. 1999, 72,

85–91.8. Ponec, R.; Yuzhakov, G.; Cooper, D. L. J. Phys. Chem. A 2003,

107, 2100–2105.9. Ángyán, J. G.; Rosta, E.; Surján, P. R. Chem. Phys. Lett. 1999,

299, 1–8. (b) Scemama, A. J. Chem. Theory Comput. 2005,4, 397–409. (c) Fulton, R. J. Phys. Chem. 1993, 97, 7516–7529.

10. Modl, M.; Dolg, M.; Fulde, P.; Stoll, H. J. Chem. Phys. 1996,105, 2353–2363.

11. Fradera, X.; Solà, M. J. Comput. Chem. 2002, 23, 1347–1356.12. Lewis,G. N. J. Am. Chem. Soc. 1916, 38, 762–785.13. Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular

Geometry; Allyn and Bacon: Boston, 1991.14. Gillespie, R. J.; Popelier, P. L. A. Chemical Bonding and Mo-

lecular Geometries: From Lewis to Electron Densities; OxfordUniversity Press: New York, 2001.

15. Gillespie, R. J.; Robinson, E. A. Angew. Chem., Int. Ed. Engl.1996, 35, 495–514.

16. Matta, C. F.; Gillespie, R. J. J. Chem. Educ. 2002, 79, 1141–1152.

17. Bader, R. F. W. Chem. Rev. 1991, 91, 893–928.18. Bader, R. F. W. Can. J. Chem. 1998, 76, 973.19. Becke, A. D.; Edgecombe, K. E. J. Chem. Phys. 1990, 92,

5397–5403.

Figure 3. Localizability along the interatomic distance. The ascend-ing curve is the localization index calculated on a hydrogen atom,the descending curve represents the delocalization index betweenboth hydrogens.

0.0

0.2

0.4

0.6

0.8

1.0

0.7 1.1 1.5 1.9 2.3 2.7 3.1 3.5 3.9

r (H−H′) / Å

λ(H

) or

δ(H− H

′ )λ(H)

δ(H−H′)

58




http://www.jce.divched.org/Journal/Issues/2002/Sep/abs1141.html

http://www.jce.divched.org/Journal/Issues/2002/Sep/abs1141.html


20. Savin, A.; Becke, A. D.; Flad, J.; Nesper, R.; Preuss, H.;Vonschnering, H. G. Angew. Chem., Int. Ed. Engl. 1991, 30,409–412.

21. Savin, A.; Nesper, R.;Wengert, S.; Fassler, T. F. Angew. Chem.,Int. Ed. Engl. 1997, 36, 1809–1832.

22. Bader, R. F. W.; Stephens, M. E. J. Am. Chem. Soc. 1975, 97,7391–7399.

23. Lowdin, P. O. Phys. Rev. 1955, 97, 1474–1489.24. Fradera, X.; Austen, M. A.; Bader, R. F. W. J. Phys. Chem. A

1999, 103, 304–314.25. Fradera, X.; Poater, J.; Simon, S.; Duran, M.; Solà, M. Theor.

Chem. Acc. 2002, 108, 214–224.26. Willis, C. J. J. Chem. Educ. 1988, 65, 418–422.27. Willis, C. J. J. Chem. Educ. 1991, 68, 743–747.28. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B; Scuseria, G. E.;

Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Mont-gomery, J. A.; Stratmann, R. E.; Burant, J. C.;. Dapprich, S;Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.;

Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.;Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski,J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Sal-vador, P.; Dannenberg, J. J.; Malick, D. K.; Rabuck, A. D.;Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.;Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz,P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.;Keith, T.; Al-Laham, M.; Peng, C.; Nanayakkara, A.;Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen, W.;Wong, M. W.; Andres, J. L.; Gonzalez, R.; Head-Gordon, M.;Replogle, E. S.; Pople, J. A. Gaussian 98, Gaussian 98, rev.A11: Pittsburgh, PA, 1998.

29. Biegler–König, F. W.; Bader, R. F. W.; Tang, T.–H. J. Comput.Chem. 1982, 3, 317–328.

30. Mulliken, R. S. J. Chem. Phys. 1955, 23, 1833–1840.31. Platts, J. A.; Laidig, K. E. J. Phys. Chem. 1996, 100, 13455–

13461.

59




Appendix 1. Calculation of λ(H)

First of all, we must determine which region of space (atomic basin) is assigned to

each atom. Partitioning of molecular space must be consistent with the symmetry of the

system. In particular, for a homonuclear diatomic molecule with D∞h symmetry, the

molecular space is exactly split into two regions according to the plane perpendicular to

the bond. Consequenttly, atomic basins in this case are independent, first, of the method

used to partition the space, second, of the method employed to compute the

wavefunction, and, third, the partition is valid irrespective of whether we are considering

a stationary point or not. The Bader’s partition is somewhat controversial for non-

stationary points. In particular, it is important in these points to check that atomic

populations and energies sum the correct molecular value [Platts, 1996 #34]. However, in

this case it is not necessary because only atomic populations are used and the average

number of electrons per atomic basin is obtained directly by construction: it must be 1

electron per region, since both regions are equal and its populations must sum two

electrons. With this information and labeling σg and σ*u MOs as 1 and 2, the LI

calculated from Eq. 9 turns into:

( )( )( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( )

Γ+Γ

+Γ+Γ−=

HSHSHSHS

HSHSHSHSH

22222

21212

12122

11112

22221122

221111111λ (1.1)

The DM2 is symmetric and its diagonal sums to the total number of electrons, that is 2;

the overlap matrix is also symmetric, and summed for all their contributions must give

the identity matrix, so that the follow properties are easily derived:

60

( ) ( ) ( )( ) ( )( ) 222221111 222 =Γ=Γ+Γ Tr

( ) ( ) ( )( )11222211 22 Γ=Γ

( ) ( ) ( ) ( ) 2122221111 ==== BABA HSHSHSHS

( ) ( ) ( ) ( )BBAA HSHSHSHS 21122112 −=−==

(1.2)

Applying these relationships to Eq. 1.1, the following result for the LI is reached:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( ) ( )212

2212

22

2212

22

221122122112

411

222241221121111

411

HSHSTr

HSH

Γ−=

Γ+Γ−=

=

Γ+Γ+Γ−=λ

(1.3)

The above equation holds for any level of calculation when using a minimal basis and it

is just the form of the DM2 matrix that changes the value of ( )Hλ . For any

monodeterminantal wavefunction -like the HF one- the DM2 is diagonal (that is to say

Γ(2)(11|22)=0), so its LI is simply 0.5 electrons whatever the distance between the

hydrogen atoms; in the same way it can be found that the DI is always 1.

The DM2 crossed-element for the CI wavefunction, Γ(2)(11|22), can be obtained

by substituting the wavefunction given by Eq. 14 into Eq. 2 followed by spin-integration:

61

( ) ( ) ( ) ( )

)2()1()2()1(2)2()1()2()1(2

)2()1()2()1(2)2()1()2()1(2

22112211,

11*2

*222

*1

*1

22*2

*2

211

*1

*1

2

*

212

χχχχχχχχ

χχχχχχχχ

γ

oDDo

Do

DoDo

cccc

cc

ccccrr

+

++=

=++=

(1.4)

Where integration of the third electron coordinates is not necessary since we are dealing

with a two-electron problem. In Eq. 1.4 we have assumed that the coefficients of the CI

expansion are real numbers. From the above equation and comparing with Eq. 3, it is

easily recognized that:

( ) ( ) Docc222112 =Γ (1.5)

Thus, the LI for a CI wavefunction is:

( ) ( )212421 HSccH Do−=λ (1.6)

On the other hand, it can be seen from the results in Table 2 that the overlap between σg

and σ*u MOs in an atomic basin (S12(H)) is almost invariant during the bond dissociation

process, its squared value being always restricted to the interval:

( ) 25.02.0 212 ≤≤ HS (1.7)

62

Hence, we can focus on the leading term of the LI and approach the squared overlap to

0.25, so that the LI formula for hydrogen atom becomes:

( ) DoccH −≈21λ (1.8)

Derivation for δ(H,H') can be done in an analogous way.

63

4.3.- The Aromatic Fluctuation Index. (FLU): A New

Aromaticity Index Based on Electron Delocalization.

65

THE JOURNAL OF CHEMICAL PHYSICS122, 014109 ~2005!

The aromatic fluctuation index „FLU…: A new aromaticity indexbased on electron delocalization

Eduard Matito, Miquel Duran, and Miquel Solaa)

Institut de Quı´mica Computacional and Departament de Quı´mica, Universitat de Girona, 17071 Girona,Catalonia, Spain

~Received 6 August 2004; accepted 5 October 2004; published online 14 December 2004!

In this work, the aromatic fluctuation index~FLU! that describes the fluctuation of electronic chargebetween adjacent atoms in a given ring is introduced as a new aromaticity measure. This newelectronic criterion of aromaticity is based on the fact that aromaticity is related to the cyclicdelocalized circulation ofp electrons. It is defined not only considering the amount of electronsharing between contiguous atoms, which should be substantial in aromatic molecules, but alsotaking into account the similarity of electron sharing between adjacent atoms. For a series of ringsin 15 planar polycyclic aromatic hydrocarbons, we have found that, in general, FLU is stronglycorrelated with other widely used indicators of local aromaticity, such as the harmonic-oscillatormodel of aromaticity, the nucleus independent chemical shift, and the para-delocalization index~PDI!. In contrast to PDI, the FLU index can be applied to study the aromaticity of rings with anynumber of members and it can be used to analyze both the local and global aromatic character ofrings and molecules. ©2005 American Institute of Physics.@DOI: 10.1063/1.1824895#

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I. INTRODUCTION

There is no doubt about the fundamental role that amaticity plays in current chemistry.1 Thus, the stability ofcertain molecular structures as well as many successefailures of chemical reactions and changes in activationreaction enthalpies are attributed to a gain or loss of aroticity. Despite the lack of a clear and unambiguous definitof aromaticity, this chemical concept is deeply rooted inchemistry community.1

Since aromaticity is not an observable and, therefore,a directly measurable quantity, several definitions of aromticity have been put forward. The most accepted definitioof aromaticity found in standard chemistry textbooks referthe electronic cyclic delocalization present in all aromaspecies. In this sense, Sondheimer considers aromaticecules as those that have a ‘‘measurable degree of cdelocalization of ap-electron system.’’2 Similarly, Schleyerand Jiao think of aromaticity as ‘‘associated with cyclic arays of mobile electrons with favorable symmetries’’ wh‘‘the unfavorable symmetry properties of antiaromatic stems leads to localized rather than to delocalized electrostructures.’’3 This cyclic mobility of electrons is translateinto characteristic aromatic manifestations such as bolength equalization, abnormal chemical shifts, magneanisotropies, and energetic stabilization.

These particular manifestations of aromaticity have btaken into account to define different indices of aromaticthat evaluate the aromatic character of molecules basetheir structural, magnetic, and energetic properties.4–9 Theharmonic-oscillator model of aromaticity~HOMA! index,

a!Author to whom correspondence should be addressed.Fax: ~134!-972-41-8356. Electronic mail: [email protected]

122, 0140021-9606/2005/122(1)/014109/8/$22.50

Downloaded 24 Dec 2004 to 130.206.124.122. Redistribution subject to A67

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defined by Kruszewski and Krygowski10 and by Krygowskias11

HOMA512a

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n

~Ropt2Ri !2, ~1!

is found to be among the most widespread and effectiv1,4

indices of aromaticity founded on structural criteria. In E~1!, n is the number of bonds considered, anda is an empiri-cal constant~for C–C and C–N bondsa5257.7 and 93.52,respectively! chosen in such a way that HOMA50 for amodel nonaromatic system, and HOMA51 for a system withall bonds equal to an optimal valueRopt ~1.388 and 1.334 Åfor C–C and C–N bonds, respectively!11 assumed to beachieved for fully aromatic systems.Ri stands for a runningbond length.

Magnetic indices of aromaticity are based on tp-electron ring current that is induced when the systemexposed to external magnetic fields. Probably the mwidely used among this group is the nucleus independchemical shift ~NICS!, proposed by Schleyer anco-workers.3,12 It is defined as the negative value of the asolute shielding computed at a ring center or at some ointeresting point of the system. Rings with large negatNICS values are considered aromatic. The more negativeNICS values, the more aromatic the rings are.

Finally, indices of aromaticity based on energetic proerties make use of the fact that conjugated cyclicp-electroncompounds are more stable than their chain analogs.8,13 Themost commonly used energetic measure of aromaticity isaromatic stabilization energy, calculated as the reactionergy of a homodesmotic reaction.14

Less common is the use of measures of aromatifounded on electronic properties. Among them, we can mtion the highest-occupied-molecular-orbital–lowest-occ

109-1 © 2005 American Institute of Physics

IP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

http://dx.doi.org/10.1063/1.1824895

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014109-2 Matito, Duran, and Sola J. Chem. Phys. 122, 014109 (2005)

pied-molecular-orbital gap, the absolute and relative haness, the electrostatic potential, and the polarizability.5 Inaddition, descriptors of aromaticity using electronic paraeters derived from the charge density have been also exined. Following this line of research, Jug defined a criterbased on bond orders15 and Howard and Krygowski16 pro-posed the density at the ring critical point interalia chardensity descriptors.

More recently, several measures of delocalizationrived from natural bond orbital analysis of the charge denhave been used to quantify aromaticity in five-membeheteroaromatic compounds.17 In addition, the degree op-electron delocalization in an aromatic compound obtainfrom an electron localization function18,19~ELF! analysis hasbeen proven to be a good aromaticity indicator,20 although noquantitative measures competing with existing indices hbeen given.

Lately, our group have employed electron delocalizatmeasures derived from the second-order density21,22as a newdescriptor of local aromaticity. In particular, we have usthe para-delocalization index~PDI!,23 which is an average oall delocalization indices~DIs! of para-related carbon atomin a given six-membered ring~6-MR!. The DI value betweenatomsA and B, d(A,B), is obtained by the double integration of the exchange-correlation density over the basinsatomsA and B in the framework of the atoms in the moecules ~AIM ! theory of Bader ~vide infra!.24 Previousworks23,25 have shown that for a series of planar and curvpolycyclic aromatic hydrocarbons~PAHs!, there is a satisfactory correlation between NICS, HOMA, and PDI. In generlarger PDIs go with larger absolute values of NICS alarger HOMA values.

The main inconvenience of PDI is that it cannot accofor the aromaticity of rings having a number of membedifferent from six. Moreover, it can only be used to analythe local aromaticity of a given ring and not to describeglobal aromatic character of a molecule having several furings. To surpass these shortcomings that limit the scopthe PDI measure, we have elaborated a new index also bon electron delocalization measures that can be applieany ring or group of rings. In this sense, the new indexfined here opens up entirely new possibilities for very dtailed aromaticity studies.

Due to the multidimensional character of aromticity,4,6,26,27it is usually recommended to employ more thone aromaticity parameter for comparisons restricted to sregions or groups of relatively similar compounds.26 So far,the most widely used indices of aromaticity are basedstructural, magnetic, and energetic measures. In this conwe believe that the definitions of this new aromaticity indand itsp counterpart derived from alternative indicationsaromaticity, such as the electron delocalization, are veryevant.

II. METHODOLOGY

In this work we have used the negative of the exchancorrelation density@GXC(rW1 ,rW2)# as a function that accountfor electron sharing between two certain regions of the spalthough, in principle, other functions that characterize el


-

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tron sharing could be used as well. Besides, we haveployed the AIM theory24 to obtain an exhaustive physicallsound subdivision of molecular space by means of a dentopological analysis. Other partitions, such as the ELF dsion of space18,28 or the Mulliken-type partitioning in theHilbert space spanned by the basis functions are alsosible. Integration of the exchange-correlation~XC! densityover two atomic basins24 yields atomic localization indicesand DIs defined as21,22

l~A!52EAdrW1E

AdrW2GXC~rW1 ,rW2!, ~2!

and

d~A,B!522EAdrW1E

BdrW2GXC~rW1 ,rW2!, ~3!

respectively.The AIM topological analysis provides an exhausti

partition of molecular space, so these indices obey thelowing sum rule:

1

2 (AÞB

d~A,B!1l~A!5N~A!, ~4!

whereN(A) is the average population of an atomA definedas follows:

^N&A5N~A!5EAr~rW !drW. ~5!

The DI measure,d(A,B), provides a quantitative idea of thnumber of electrons delocalized or shared between atomAand B.22,29 It is worth noting that these indices are closerelated to the fluctuation~or variance! in the average population of the basin of a given atomA, s2, defined by30

s2@N~A!#5^N2&A2^N&A2. ~6!

s2@N(A)# represents the quantum-mechanical uncertaintyN(A) and it is related to the localization and delocalizatiindices through the following equations:30,31

s2@N~A!#5N~A!2l~A!, ~7!

s2@N~A!#51

2 (BÞA

d~A,B!. ~8!

We can quantify the importance of theA–B electron sharing/fluctuation with respect to the total electron sharinfluctuation in atomA using the following quantity:

Flu~A→B!5d~A,B!

(BÞAd~A,B!5

d~A,B!

2@N~A!2l~A!#. ~9!

If the molecule is aromatic, one can expect the transferelectrons to be almost equal fromA to B than fromB to A, sowe can calculate the ratio of the electron exchange betwA andB as

Exc~A2B!5FFlu~A→B!

Flu~B→A!Gd

. ~10!


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to, wo

t ayex

a

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tinodl imoreLeitit

fi-rolato

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ee

athfo

a

ges

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014109-3 New aromaticity index J. Chem. Phys. 122, 014109 (2005)

The delta above is a simple function to make sure thatexchange term is always greater than or equal to 1, so it tathe values

d5H 1 Flu~A→B!.Flu~B→A!

21 Flu~A→B!<Flu~B→A!.~11!

This Exc(A–B) weighting factor gives an idea of the eletron sharing similarity betweenA and B adjacent atoms. Itwill be close to 1 in aromatic species and particularly largethe case of molecules having atomsA or B with highly lo-calized atomic charge. When multiplying the exchange facabove by the quantity of electrons shared by both atomsget a measure ofA–B net mobile charge. It is possible tfind nonaromatic molecules having Exc(A–B) weightingfactors close to 1~for instance, cyclohexane! due to verysimilar or even equal electron sharing between adjacenoms in the analyzed ring~s!. For this reason, to correctlaccount for aromaticity, it is necessary to include in thepression a new factor$@d(A,B)2d ref(A,B)#/d ref(A,B)% thatcompares theA–B electron sharing with that found intypical aromatic molecule,d ref(A,B). By squaring the prod-uct of these two factors for each connected pair of atomthe ring and taking the average for all the pair of adjacatoms in the ring, we obtain the following expression for taromatic fluctuation index:

FLU51

n (A2B

ring HFFlu~A→B!

Flu~B→A!GdFd~A,B!2d ref~A,B!

d ref~A,B! G J 2

, ~12!

which should give a number close to zero for any aromamolecule. Abbreviated hereafter as FLU, this quantityonly analyzes the amount of electron sharing between acent atoms in a given ring, which should be substantiaaromatic molecules, but it also takes into account the silarity of electron sharing between adjacent atoms. In shthis index measures weighted electron delocalization divgences with respect to typical aromatic molecules. The Findex as given by Eq.~12! requires the use of referencparameters. We anticipate here that for planar systemspossible to define a FLU index that is free from this requis~vide infra!. It is worth repeating that although the FLU denition is unique, the quantity chosen to account for electsharing or the topological approach to divide molecuspace, i.e., molecular space portions assigned to each amay be different. Finally, it is also important to remark thlike the HOMA index, this dimensionless FLU quantity cabe applied to analyze both local and global aromaticity.

III. COMPUTATIONAL DETAILS

Molecular geometries of systems analyzed have bfully optimized with the Hartree–Fock~HF! method usingthe 6-31G(d) basis set32 by means of theGAUSSIAN 98

program.33 The calculation of the electronic~FLU and PDI!,geometric~HOMA!, and magnetic~NICS! aromaticity crite-ria has been carried out at the same level of theory. To vdate the level of accuracy of the results obtained with6-31G(d) basis set, we have performed HF calculationsselected systems using the 6-31111G(d,p) basis set.34 Theresults, given as supporting information, show that incre


ees

n

re

t-

-

int

ct

ja-ni-t,r-U

ise

nrm,

,

n

li-er

s-

ing the quality of the basis set only produces minor chanthat do not significantly affect the final conclusions.

The gauge-including-atomic-orbital method35 has beenused to perform calculations of NICS. NICS~0! values havebeen computed at the ring centers determined by the nweighted mean of the heavy atoms coordinates and atring critical point ~RCP!,24 the position of minimal chargedensity in the ring plane, as suggested by Cossioco-workers.36 The results at the RCP reproduce those valcalculated at the geometrical ring center, and for this reasthey are reported as supporting information. We have acomputed NICS~1! values at 1 Å above the molecular planewhere ring current contribution is larger and hence more rresentative of aromatic character.37 For those molecules withnonplanar structures we have fitted the best plane38 and havefound the projection of the ring geometrical center to tfitted plane. Following the plane’s normal vector directiona 1-Å distance from this site, we have reached the poincalculate the NICS~1! for nonplanar species. The NICS ouof-plane component corresponding to the principal axis ppendicular to the ring plane, NICSzz, which has been re-cently recommended as a more direct measure of induring current densities,39 has been also calculated.

Integrations of DIs were performed by the use of tAIMPAC collection of programs40 through the following ex-pression:

d~A,B!54(i , j

N/2

Si j ~A!Si j ~B!. ~13!

The summations in Eq.~13! run over all the occupied molecular orbitals.Si j (A) is the overlap of the molecular orbitals i and j within the basin of atomA. The numerical accu-racy of the AIM calculations has been assessed usingcriteria: ~i! the integration of the Laplacian of the electrodensity @¹2r(r )# within an atomic basin must be closezero and~ii ! the number of electrons in a molecule mustequal to the sum of all the electron populations of a mecule, and also equal to the sum of all the localization indiand half of the delocalization indices in the molecule@Eq.~4!#.29 For all atomic calculations, integrated absolute valuof ¹2r(r ) were always less than 0.001 a.u. For all moecules, errors in the calculated number of electrons wereways less than 0.01 a.u.

All systems studied in the present work contain onC–C and C–N bonds in the ring structure. As a referencearomatic electron sharing to compute the FLU index@Eq.~12!#, we will take the value ofd ref~C,C) in benzene andd ref~C,N) in pyridine. At the HF/6-31G(d) level of theory,these values are 1.4e and 1.2e, respectively.

IV. RESULTS AND DISCUSSION

This section is organized as follows. First, we analythe performance of the FLU index for the assessment of loaromaticities in the series of PAHs depicted in Fig. 1. Tfigure contains also the labels given to each PAH and rstudied. The analysis will be carried out by comparing tFLU results with other widely used and different propert


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based indicators of aromaticity such as the HOMA, NICand PDI indices. Second, we discuss the capability of thdifferent indices to reproduce the expected order of aroticity for the series cyclohexane, cyclohexene, cyclohe1,4-diene, cyclohexa-1,3-diene, and benzene. Finally, fornar PAHs, we define the FLUp index by considering only thesharing/fluctuation ofp electrons and we comment on ibenefits and drawbacks.

FIG. 1. Labels for the molecules and rings studied.


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a-

A. The FLU index: Comparison with HOMA, NICS,and PDI

For the different rings of the systems studied, Tablelists the HOMA indices together with its energetic~EN! andgeometrical~GEO! components@Eq. ~14! defines these quantities, vide infra#, the NICS~0! and NICS~1! values and theirout-of-plane NICSzz components, the PDI of 6-MRs, and thresults obtained with the herein proposed FLU and FLp

indicators of local aromaticity.

1. HOMA results

Table I shows that there is, in most cases, a good cospondence between the different indices, compounds wmore negative NICS values having also larger HOMA aPDI measures and lower FLU indices. In the case of HOMthis can be clearly seen in Fig. 2, which plots FLU versHOMA for the rings ofM1 to M15 species. In particularelectronic FLU and structural HOMA indices give almost tsame order of aromaticity for the different rings in the PAHanalyzed, with few noteworthy exceptions. One of this cresponds to triazine~M14! that HOMA classifies as stronglyaromatic, while it is found only moderately aromatic accoring to FLU, PDI, and NICS indices. Besides, with respectHOMA, FLU also underestimates the aromaticity of ringBin quinoline~M15! and overestimates that of ringB in triph-enylene~M6!.

In general, however, there is a good correlation betwHOMA and FLU indices despite being two indices basedtotally different aspects of aromaticity. In part, this good corelation can be attributed to the fact that FLU is construcfollowing to some extent the HOMA philosophy, i.e., witthe purpose of measuring the aromaticity by compariswith the values of a specific aromatic manifestation in indputable aromatic systems such as benzene or pyridine. Mover, both indices concentrate on differences between ctiguous atoms around the ring. The formula of FLresembles that of the HOMA index except for the weightifactor on the fluctuation parameter, Eq.~10!, and for thefeature that the differenced(A,B)2d ref(A,B) in Eq. ~12! isdivided by the reference value. The main difference, hoever, comes from the fact that, whereas FLU measures cyelectron fluctuation differences, HOMA compares bolengths.

To examine further the discrepancies found in tripenylene~M6!, triazine ~M14!, and quinoline~M15! in theHOMA and FLU orderings of local aromatic character ftheM1 to M15 systems, we have split the HOMA index inttwo completely different terms that according to Krygowsand Cyran´ski41 are linearly independent variables accountifor EN and GEO manifestations of aromaticity in a certaring structure. Then, HOMA can be written as follows:

HOMA512EN2GEO

512a~r opt2 r !22a( i~ r 2r i !

2

n

512a@~r opt2 r !22s2~r !#. ~14!


6

7

7


TABLE I. HF/6-31G(d) calculated HOMA, NICS~ppm!, PDI ~electrons!, and FLU values for the studied systems.

EN GEO HOMA NICS~0! NICSzz(0) NICS~1! NICSzz(1) PDI FLU FLUp

Benzene M1 20.001 0.000 1.001 211.5 217.2 212.9 232.5 0.105 0.000 0.000Naphthalene M2 0.022 0.199 0.779 210.9 213.8 212.6 230.6 0.073 0.012 0.114Anthracene M3-A 0.067 0.416 0.517 28.7 26.9 210.7 224.8 0.059 0.024 0.251

M3-B 0.045 0.072 0.884 214.2 221.9 215.5 238.4 0.070 0.007 0.024Naphthacene M4-A 0.110 0.565 0.325 26.7 20.9 29.0 219.6 0.051 0.031 0.350

M4-B 0.073 0.153 0.774 213.8 220.4 215.2 237.2 0.063 0.011 0.071Chrysene M5-A 0.011 0.130 0.859 211.1 213.7 212.8 230.7 0.079 0.008 0.067

M5-B 0.122 0.325 0.553 28.2 23.6 210.6 223.3 0.052 0.019 0.182Triphenylene M6-A 0.003 0.068 0.930 210.6 211.8 212.3 229.3 0.086 0.003 0.026

M6-B 0.609 0.324 0.067 22.6 13.9 26.2 29.4 0.025 0.027 0.181Pyracylene M7-A 0.000 0.329 0.671 24.9 6.9 27.2 213.3 0.067 0.014 0.128

M7-B 0.406 0.923 20.328 10.1 56.4 4.8 24.2 a 0.050 0.67Phenanthrene M8-A 0.007 0.091 0.902 211.4 214.7 213.0 231.5 0.082 0.005 0.045

M8-B 0.189 0.409 0.402 26.8 20.1 29.4 220.0 0.053 0.025 0.255Acenaphthylene M9-A 0.011 0.192 0.797 29.6 28.7 211.4 226.4 0.070 0.013 0.114

M9-B 0.337 0.703 20.039 2.2 33.7 21.8 4.6 a 0.045 0.578Biphenylene M10-A 0.000 0.193 0.807 26.7 20.7 28.1 217.3 0.088 0.008 0.066

M10-B 1.360 0.571 20.930 17.4 87.0 7.5 35.7 a 0.048 0.29Banzocyclobutadiene M11-A 0.001 0.501 0.497 24.0 8.4 25.4 29.5 0.080 0.022 0.191

M11-B 0.910 1.526 21.437 20.2 96.2 10.7 44.1 a 0.071 1.04Pyridine M12 20.006 0.001 1.005 29.5 215.2 212.5 231.6 0.097 0.001 0.001Pyrimidine M13 0.015 0.000 0.985 27.5 211.8 211.7 229.8 0.089 0.005 0.003Triazine M14 0.023 0.000 0.977 25.3 26.9 210.8 227.0 0.075 0.013 0.000Quinoline M15-A 0.018 0.190 0.792 211.0 214.7 212.6 230.7 0.072 0.015 0.125

M15-B 0.008 0.161 0.830 29.1 211.5 212.1 229.5 0.071 0.017 0.121Cyclohexane M16 5.340 0.000 24.340 22.1 23.5 22.0 3.1 0.007 0.091 bCyclohexane M17 2.955 1.647 23.601 21.6 18.1 23.6 21.8 0.019 0.089 bCyclohexa-1,4-diene M18 0.779 1.984 21.763 1.5 25.4 20.8 2.9 0.014 0.084 bCyclohexa-1,3-diene M19 0.931 2.207 22.138 3.2 30.0 0.8 7.3 0.031 0.078 b

aPDI cannot be computed for non-6-MRs.bNonplanar molecules that prevent easy and exacts–p separation.

o

an

om

mresp4

ered

ostd aA

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-

ar-

The second term of the right-hand side of Eq.~14! is whatKrygowski and Cyran´ski call dearomatization. It has twcontributions: the first contribution~the EN term! arises fromthe difference between the average distance in the ringthat calculated from the minimization of the deformation eergy due to the extension of the single bonds and the cpression of the double ones; the second contribution~theGEO term! is the bond-lengths variance of the bonds foring the ring. Equation~14! cannot be used directly as it is fothe heterocyclic systems since the averaging procedurmeaningless for heterogenic data. Thus, for heterocycliccies, we have followed the procedure described in Ref.As can be seen in Fig. 3, the GEO component is the twhich better correlates with FLU. This is not unexpect

FIG. 2. Plot of FLU vs HOMA for the series of planar aromatic hydrocbonsM1–M15 ~correlation coefficientr 520.957).


nd--

-

ise-1.m

because in both the FLU and GEO expressions the mrelevant component is a variance contribution. We also finmoderate correlation between the EN component of HOMand FLU (r 520.789). The lower correlation of the ENterm with FLU is in line with the fact that the main differences between FLU and HOMA found in the evaluationthe local aromatic character of ringB in triphenylene~M6!and quinoline~M15! arises from the EN term, which is particularly large forM6-B and surprisingly low forM15-B. Onthe other hand, for triazine~M14!, both the EN and GEOterms are responsible for the HOMA overestimation~as com-pared to the rest of analyzed indices! of the local aromatic

FIG. 3. Plot of FLU vs GEO (r 50.952) and EN (r 50.788) with FLU forthe series of planar aromatic hydrocarbonsM1–M15.


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character ofM14. We note in passing that for systems haviHOMA values larger than 1, it is possible to have negatvalues for the EN term.4,41,42

2. NICS results

Table I collects the results of NICS~0!, NICSzz(0),NICS~1!, and NICSzz(1). The differences between NICS~0!and NICSzz(0) give a quantitative idea of the spurious infomation arising from the electron flow perpendicular to tmolecular plane.43 The rings showing a larger departure frothe linear regression of NICS~0! versus NICSzz(0) are thosecorresponding to compounds having heteroatoms anfused rings. Interestingly, these deviations almost disappwhen plotting NICS~1! against NICSzz(1). These resultssuggest that ring currents in adjacent rings and/or the ngen in-plane electron cloud may mask the out-of-plainduced magnetic field and that these effects are less noous when considering only out-of-plane NICSzz values. Ingeneral, there is a good linear relationship between thedifferent NICS values given in Table I and the FLU inde(r 50.90– 0.92). Relevant differences between the variplots of FLU versus NICS are not observed, althoughNICSzz(1) values yield the best-fitted line~Fig. 4!. It isworth noting that the local aromaticity of the inner ringsacenes such as ringsB in anthracene and naphthacene aespecially overestimated by all NICS measures as compto the rest of aromaticity descriptors analyzed. Indeed, ablute NICS values are larger for the inner rings of anthracand naphthacene than for benzene itself, whereas benzethe most aromatic 6-MR of the series according to the FLHOMA, and PDI indices. These results give support toclaimed overestimation by NICS of the local aromaticitythe central rings in polyacenes.9,44

3. PDI results

The last index to be analyzed here is the PDI descripof local aromaticity.23 Due to its definition, this index canonly be applied to analyze the local aromaticity of 6-MRFigure 5 displays the correlation between PDI and FLUall 6-MRs appearing along theM1–M15 series. In generalthere is a moderately good agreement (r 520.840) on theorderings yielded by these two criteria, the main exceptbeing the 6-MR of benzocyclobutadiene, to which PDItributes a high aromatic character, while HOMA~0.497! and

FIG. 4. Plot of NICSzz(1) vs FLU values for the moleculesM1–M15 (r50.918).


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se

eedo-ee is,

e

r

.r

n-

NICS~0! @not only NICS~0! but also NICSzz(0), NICS~1!,and NICSzz(1) values show that the 6-MR of benzocycloutadiene has a low or moderate aromatic character# ~24.0!indicators attributed it to a low or moderate aromaticity. Dspite being reasonable, the correlation between FLU andis lower than that shown by FLU with HOMA or NICS. Thiis surprising if one considers that the FLU and PDI defitions are based on the same manifestation of aromatinamely, the electronic delocalization in aromatic molecul

B. The aromaticity of the M16–M19 systems

In this section, the ordering of aromatic character in tM16–M19 and M1 series given by the different indicesanalyzed.M16 andM17 are nonplanar species, whileM18,M19, andM1 systems possess a completely planar molelar structure. Due to the increase in unsaturated charaalong this series, an intensification of aromaticity is expecfrom M16 to M19 and especially toM1. Indeed, all indicesshow the predictable increase of aromaticity from planM19 to M1. However, the only indices that more or lereproduce the anticipated trend along the full seriesHOMA and FLU. Nonetheless, these two indices disagreedeciding the aromaticity of cyclohexa-1,4-diene~M18! andcyclohexa-1,3-diene~M19!. Although both indices assignsimilar low aromaticity for these compounds, HOMA givesgreater aromaticity toM18, while FLU asserts thatM19 hasa higher aromatic character. Thus, in contrast to HOMA,FLU index succeeds in assigning a larger aromatic charato M19, which has two consecutivep-electron bonds, asopposed toM18, which has thep-electron systems disconnected.

On the other hand, when splitting the HOMA index oits GEO and EN contributions, we can see the expected trwith increasing aromaticity~see Fig. 6!, i.e., stemming en-ergy ~EN term! when getting closer to the unsaturated decalized picture of benzene, while symmetry~GEO term! onthe structure achieves its minima on benzene and cyclohane, and intermediate values for the nonsymmetric molecM17–M19.

For M16–M19, NICS~0! provides a trend completelyopposed to the expected one. The out-of-plane componethe NICS values gives somewhat better results. This notwstanding, NICSzz(0) and NICSzz(1) values fail in predicting

FIG. 5. Plot of PDI vs FLU values for the six-membered rings of molecuM1–M7-A , M8, M9-A , M10-A, M11-A, M12–M15 (r 520.840).


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of

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then

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cyclohexene~M17! as more aromatic than cyclohexadieneand cyclohexa-1,3-diene as less aromatic than cyclohex~M16!. Moreover, the out-of-plane component is not easrelated to thep electrons for nonplanar molecules. In a prvious work,37 Schleyeret al. showed that the NICSp ~NICSvalues obtained considering only the contribution ofp orbit-als! reproduce the expected trend in aromaticity for theM18,M19, andM1 species. Since this NICSp cannot be applied tononplanar molecules,M16 andM17 were not analyzed usingthis index.37 Finally, PDI values fail when predicting a lowearomatic character for cyclohexa-1,4-diene~M18! as com-pared to cyclohexene. Apparently, the aromaticity in cychexene is somewhat overestimated by PDI because theplanar nature of this species makes the basins of carbonpara-position spatially closer than in 1,4-cyclohexadieThis causes an increase of DIs in cyclohexene as compto cyclohexa-1,4-diene.

C. The FLUp index of aromaticity

The main shortcoming of the FLU index, as it is for thHOMA descriptor of aromaticity, is the need for referenparameters. At variance with PDI, NICS, and other indicFLU and HOMA require archetypical aromatic moleculcontaining specific bonds as a reference. The original detion of the FLU index, based on the DIs that account forsharing between a certain pair of atoms, enables the pobility of splitting this index in terms of orbital contributionsGathering thep contribution treated the way reported in SeII opens the possibility of calculating a FLU index freereference parameters. Named hereafter as aromaticp fluc-tuation index (FLUp), this aromaticity criterion is calculateas follows:

FLUp51

n (A2B

ring HFFlup~A→B!

Flup~B→A!Gd dp~A,B!2dav~A,B!

dav~A,B! J 2

,

~15!

where dav is the average value of thedp , that is, thepcomponent of the DI, and Flux(A→B) is calculated as in Eq~9! using also only thep component of the DIs. Correlatioof FLUp with FLU is excellent (r 50.973, see Fig. 7!, thusproving the similarity between FLU and FLUp approaches.Correlations of FLUp with HOMA, NICS, and PDI are also

FIG. 6. Plot of GEO and EN vs HOMA values for the compoundsM16–M19 andM1.


,ne

-

-n-on.ed

,

i-esi-

.

found to be reasonable. An advantage of FLUp with respectto FLU, apart from the fact that no reference parametersrequired, is that the former gives values spread in wiranges. For these reasons, we suggest the use of the Fp

index for evaluation of aromaticity in planar species sincereference systems are necessary, while the FLU index isferred when analyzing nonplanar systems, for which an exs–p separation is not possible. Finally, we recommend usthese FLU indices in conjunction with other differently basdescriptors of aromaticity~HOMA, NICS, PDI,...! to accountfor the multidimensional character of aromaticity.4,6,26,27

V. CONCLUSIONS

While aromaticity has always been associated withcyclic circulation of p electrons, few attempts have beedone until now to quantify this phenomenon by analyzielectron delocalization. To this end, we have defined in twork the aromatic fluctuation index~FLU!, which not onlyanalyzes the amount of electron sharing between contiguatoms, which should be substantial in aromatic molecubut also takes into account the similarity of electron sharbetween adjacent atoms. We have demonstrated that theindex and itsp analog, the FLUp descriptor, are simple andefficient probes for aromaticity. We have shown that, foseries of planar PAHs, FLU and FLUp correlate well, withfew exceptions, with other already existing independent loaromaticity parameters, such as the HOMA descriptor baon structural criteria, the NICS parameter founded on mnetic properties, and the PDI index derived from the analyof electron delocalization, which are of common use nowdays. It remains to be seen whether these indices canapplied to discuss the aromaticity of novel inorganic copounds recently synthesized.45 Research in this direction iscurrently under way in our laboratory.

Note added in proof. It is worth noting that, recently, anew HOMA-like aromaticity indexu, which bears some resemblance to the present FLU index, has been definedreplacing the bond distances in Eq.~1! by the correspondingdelocalization indices@C. F. Matta and J. Herna´ndez-Trujillo,J. Phys. Chem. A107, 7496~2003!#.

FIG. 7. FLUp vs FLU representation for the series of planar aromatic hdrocarbonsM1–M15 (r 50.973).


y3thate

o.p

n

s-

va

ocr,.

. E

.n,

,

.

on-

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ber,

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i-.

re,

the11ison.


ACKNOWLEDGMENTS

Financial help has been furnished by the Spanish MCProject Nos. BQU2002-0412-C02-02 and BQU2002-033and by the DURSI Project No. 2001SGR-00290. One ofauthors~M.S.! is indebted to the DURSI of the Generalitde Catalunya for financial support through the DistinguishUniversity Research Promotion, 2001. Another author~E.M.!thanks the Secretarı´a de Estado de Educacio´n y Univer-sidades of the MECD for the Doctoral Fellowship NAP2002-0581. The authors also thank the Center de Sucomputacio´ de Catalunya~CESCA! for partial funding ofcomputer time. See Ref. 46 for supporting informatioTables SI-1 with HF/6-31G(d) NICS at the RCP and SI-2with HOMA, NICS, FLU, and PDI values for selected sytems at the HF/6-31111G(d,p) level of calculation.

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46See EPAPS Document No. E-JCPSA6-121-310448 for the NICS atRCP and the HOMA, NICS, FLU, and PDI calculated at the HF/6-311G(d,p) level of theory for selected systems. A direct link to thdocument may be found in the online article’s HTML reference sectiThe document may also be reached via the EPAPS homepage~http://www.aip.org/pubservs/epaps.html! or from ftp.aip.org in the directory/epaps/. See the EPAPS homepage for more information.


Table SI-1. HF/6-31G(d) NICS values (ppm) at the RCP for the studied systems.

NICS(0) NICSzz(0) NICS(1) NICSzz(1) M1 -11,5400 -17,2902 -12,8144 -32,1626 M2 -10,9011 -13,7930 -12,5846 -30,5674

M3-A -8,6275 -6,8559 -10,7020 -24,7125 M3-B -14,2434 -21,8590 -15,5305 -38,3889 M4-A -6,5326 -0,5601 -8,9276 -19,3208 M4-B -13,7550 -20,2344 -15,1524 -37,0651 M5-A -11,0749 -13,6854 -12,7420 -30,7023 M5-B -8,1667 -3,5396 -10,5738 -23,2367 M6-A -10,5550 -11,7311 -12,3076 -29,2991 M6-B -2,5768 13,9036 -6,2226 -9,4436 M7-A -4,9285 6,9373 -7,1632 -13,2388 M7-B 10,0954 56,3759 4,7715 24,2917 M8-A -11,3700 -14,6615 -12,9647 -31,4954 M8-B -6,7977 -0,0343 -9,4274 -20,0003 M9-A -9,6209 -8,7228 -11,3530 -26,4247 M9-B 2,2313 33,6825 -1,7503 4,5701 M11-A -6,6495 -0,6385 -8,0647 -17,2797 M11-B 17,4027 86,9912 7,5108 35,7370 M10-A -3,9529 8,4063 -5,4054 -9,4202 M10-B 20,3273 96,2380 10,7727 44,1165 M12 -9,5598 -15,3451 -12,4646 -31,6312 M13 -7,5062 -11,8972 -11,7289 -29,7378 M14 -5,2915 -6,9050 -10,7911 -27,0489

M15-A -10,9314 -14,6562 -12,5549 -30,6694 M15-B -9,1044 -11,5684 -12,0773 -29,4148 M16 -2,0904 23,4933 -1,9705 3,0572 M17 -1,7199 18,0620 -3,3935 -2,0275 M18 1,4968 25,4089 -0,7811 2,8698 M19 3,2114 29,9439 0,8035 7,1926

75

Table SI-2. HF/6-311++G(d,p) calculated HOMA, NICS (ppm), PDI (electrons), FLU values for selected systems.

EN GEO HOMA NICS(0) NICSzz(0) PDI FLU Benzene M1 1,0008 -0,0008 0,0000 -9,4881 -16,8435 0,0989 0,0001

Naphthalene M2 0,7819 0,0204 0,1977 -9,2320 -13,5729 0,0721 0,0118 Anthracene M3-A 0,5213 0,0643 0,4144 -7,1319 -6,8869 0,0579 0,0231

M3-B 0,8914 0,0401 0,0685 -12,8249 -21,5207 0,0684 0,0068 Phenanthrene M8-A 0,9032 0,0061 0,0906 -9,5614 -13,8712 0,0810 0,0055

M8-B 0,4147 0,1779 0,4074 -5,5191 0,0997 0,0434 0,0241 Acenaphthylene M9-A 0,7976 0,0094 0,1929 -8,1481 -8,7441 0,0688 0,0123

M9-B -0,0357 0,3341 0,7016 3,0679 31,5063 (a) 0,0442 Biphenylene M10-A 0,8114 0,0002 0,1883 -4,8902 -0,2752 0,0873 0,0078

M10-B -0,9229 1,3434 0,5795 18,1312 85,7870 (a) 0,0471 Benzocyclobutadiene M11-A 0,4917 0,0016 0,5067 -2,0692 8,7134 0,0787 0,0216

M11-B -1,4261 0,9268 1,4993 21,4710 94,6376 (a) 0,0681 Pyridine M12 0,9917 0,0070 0,0013 -8,0579 -15,0806 0,0961 0,0010

Pyrimidine M13 0,9828 0,0167 0,0005 -6,3940 -11,2403 0,0887 0,0027 Triazine M14 0,9738 0,0262 0,0000 -4,5436 -6,0419 0,0765 0,0065

Quinoline M15-A 0,7903 0,0169 0,1929 -9,4331 -14,3686 0,0707 0,0149 M15-B 0,8317 0,0069 0,1614 -7,8628 -11,4866 0,0705 0,0166

Cyclohexane M16 -4,2938 5,2938 0,0000 -2,5429 23,5219 0,0072 0,0963 Cyclohexene M17 -3,5623 2,9366 1,6257 -1,8167 17,6673 0,0194 0,0910

Cyclohexa-1,4-diene M18 -1,7310 0,7733 1,9577 2,0326 25,1470 0,0147 0,0826 Cyclohexa-1,3-diene M19 -2,1135 0,9257 2,1878 4,1378 29,6917 0,0304 0,0772

(a) PDI can not be computed for non 6-MRs.

76

4.4.- An Analysis of the Changes in Aromaticity and

Planarity Along the Reaction Path of the Simplest Diels-

Alder Reaction. Exploring the Validity of Different Indicators

of Aromaticity.

77

An analysis of the changes in aromaticity and planarity along

the reaction path of the simplest Diels–Alder reaction.

Exploring the validity of different indicators of aromaticity

Eduard Matitoa, Jordi Poaterb, Miquel Durana, Miquel Solaa,*

aInstitut de Quımica Computacional and Departament de Quımica, Universitat de Girona, 17071 Girona, Catalonia, SpainbAfdeling Theoretische Chemie, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands

Received 30 November 2004; accepted 13 December 2004

Available online 27 June 2005

Abstract

In this work, we analyze the changes in aromaticity and planarity along the reaction path of the Diels–Alder reaction between ethene and

1,3-butadiene. To this end, a new index that quantifies the planarity of a given ring is defined. As expected, the planarity of the ring being

formed in the Diels–Alder cycloaddition increases along the reaction path from reactants to product. On the other hand, the aromaticity of the

ring formed is measured using several well-established indices of aromaticity such as the nucleus independent chemical shift (NICS), the

harmonic oscillator model of aromaticity (HOMA), and the para-delocalization index (PDI), as well as a recently defined descriptor of

aromaticity: the aromatic fluctuation index (FLU). The results given by the NICS and PDI indices, at variance with those obtained by means

of the HOMA and FLU indicators of aromaticity, confirm the existence of an aromatic transition state for this reaction. The reasons for the

failure of some of the descriptors of aromaticity employed are discussed. The results support the multidimensional character of aromaticity.

q 2005 Elsevier B.V. All rights reserved.

Keywords: Aromaticity; Nucleus independent chemical shift (NICS); Para-delocalization index (PDI); Harmonic oscillator model of aromaticity (HOMA);

Aromatic fluctuation index (FLU); Planarity; Atoms in Molecules theory (AIM); Diels–Alder reaction

1. Introduction

The well-known Diels–Alder (DA) [1–3] reaction

between ethene and 1,3-butadiene to yield cyclohexene is

the prototype of a thermally allowed 4sC2s cycloaddition.

This reaction has been extensively investigated using

different theoretical methods. It is now well-recognized

that this reaction takes place via a synchronous and

concerted mechanism through an aromatic boatlike tran-

sition state (TS) [2,4]. By 1938 [5], the analogy between the

p electrons of benzene and the six delocalized electrons in

the cyclic TS of the DA reaction was already recognized.

The aromatic nature of this TS has been later confirmed

theoretically using magnetic-based indices such as

0166-1280/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2005.02.020

* Corresponding author. Tel.: C34 972 418 912; fax: C34 972 418 356.

E-mail address: [email protected] (M. Sola).

79

the nucleus independent chemical shifts (NICS) and the

magnetic susceptibility exaltations [3,6].

Aromaticity is a concept of central importance in

physical organic chemistry [7–10]. It has been very useful

in the rationalization of the structure, stability, and

reactivity of many molecules. Even though this concept

was introduced in 1865 by Kekule [11], it has no precise and

collectively assumed definition yet. Probably the most

widely accepted description of aromaticity was formulated

by Schleyer and Jiao in 1996 [9]. These authors defined

aromatic systems as conjugated cyclic p-electron com-

pounds that exhibit a cyclic electron delocalization leading

to bond length equalization, abnormal chemical shifts and

magnetic anisotropies, as well as energetic stabilization.

According to this definition, aromaticity manifests itself

through a variety of phenomena, which can be quantified to

have a measure of aromaticity. Thus, the evaluation of

aromaticity is usually performed by analyzing its

manifestations and this leads to the classical structural,

magnetic, energetic, and reactivity-based measures of

Journal of Molecular Structure: THEOCHEM 727 (2005) 165–171

www.elsevier.com/locate/theochem

http://www.elsevier.com/locate/theochem

E. Matito et al. / Journal of Molecular Structure: THEOCHEM 727 (2005) 165–171166

aromaticity [10,12]. At this point, we must note that, as

found by Katritzky-Krygowski and co-workers by means of

principal component analyses, aromaticity is a multidimen-

sional property and, as a consequence, aromatic compounds

cannot be fully characterized using a single index [12–16].

In fact, different studies have confirmed that different

indices of aromaticity can afford divergent answers [17,18].

Consequently, to make reliable comparisons restricted

to groups of relatively similar compounds it is

usually recommended to employ a set of aromaticity

descriptors [16–18].

Aromaticity in DA reactions has been until now analyzed

using only magnetic-based indices. Because of the multi-

dimensional character of this phenomenon, we believe that

it is convenient to discuss the aromaticity in DA cycloaddi-

tions using descriptors based on different properties. Since

not all aromaticity indices give the same results, it is worth

analyzing certain particular cases in which the behavior of

aromaticity is well-established, such as in the DA reaction,

to detect possible limitations and failures of commonly used

indicators of aromaticity. Thus, the goal of the present work

is to quantify the aromaticity along the reaction path of the

simplest DA reaction between 1,3-butadiene and ethene

(see Scheme 1) using several indicators of aromaticity, to

discuss their behavior for this particular reaction.

As a structure-based measure, we have made use of the

harmonic oscillator model of aromaticity (HOMA) index,

defined by Kruszewski and Krygowski as [19]

HOMA Z 1 Ka

n

Xn

iZ1

ðRopt KRiÞ2; (1)

where n is the number of bonds considered, and a is an

empirical constant (for C–C bonds aZ257.7) fixed to give

HOMAZ0 for a model non-aromatic system, and

HOMAZ1 for a system with all bonds equal to an optimal

value Ropt (1.388 A for C–C bonds), assumed to be achieved

for fully aromatic systems. Ri stands for a running bond

length. This index has been found to be one of the most

effective structural indicators of aromaticity [7,13].

Magnetic indices of aromaticity are based on the

p-electron ring current that is induced when the system is

exposed to external magnetic fields. In this work, we have

Scheme 1. Schematic representation of reactants, transition state and product of t

the symbol (#) corresponds approximately to the position where NICS(1) has bee

80

used the NICS, proposed by Schleyer and co-workers

[9,20], as a magnetic descriptor of aromaticity. This is one

of the most widely employed indicators of aromaticity. It is

defined as the negative value of the absolute shielding

computed at a ring center or at some other interesting point

of the system. Rings with large negative NICS values are

considered aromatic. The more negative the NICS value, the

more aromatic the ring is.

As an aromaticity criterion based on electron delocaliza-

tion, we have employed the para-delocalization index (PDI)

[21], which is obtained using the delocalization index (DI)

[22,23] as defined in the framework of the Atoms in

Molecules (AIM) theory of Bader [24]. The PDI is an

average of all DI of para-related carbon atoms in a given

six-membered ring. The DI value between atoms A and B,

d(A,B), is obtained by double integration of the exchange–

correlation density ðGXCð~r1; ~r2ÞÞ over the basins of atoms

A and B, which are defined from the condition of zero-flux

gradient in the one-electron density, rð~rÞ [24]:

dðA;BÞZK

ðA

ðB

GXCð~r1;~r2Þd~r1 d~r2 K

ðB

ðA

GXCð~r1;~r2Þd~r1 d~r2

ZK2

ðA

ðB

GXCð~r1;~r2Þd~r1 d~r2: ð2Þ

d(A,B) provides a quantitative idea of the number of

electrons delocalized or shared between atoms A and B.

Therefore, the PDI is clearly related to the idea of electron

delocalization so often found in textbook definitions of

aromaticity. Previous works [21] have shown that for a

series of planar and curved polycyclic aromatic hydro-

carbons there is a satisfactory correlation between NICS,

HOMA, and PDI. In general, larger PDIs go with larger

absolute values of NICS and larger HOMA values.

Another electronically based criterion of aromaticity has

been recently defined by Matito et al. [25], the aromatic

fluctuation index (FLU), which describes the fluctuation of

electronic charge between adjacent atoms in a given ring.

The FLU index is based on the fact that aromaticity is

related to the cyclic delocalized circulation of p electrons,

and it is constructed not only considering the amount of

electron sharing between contiguous atoms, which should

he Diels–Alder reaction between ethene and 1,3-butadiene. The position of

n calculated; (*) for NICS(0); and (&) for NICS(K1) (see text for details).

E. Matito et al. / Journal of Molecular Structure: THEOCHEM 727 (2005) 165–171 167

be substantial in aromatic molecules, but also taking into

account the similarity of electron sharing between adjacent

atoms. It is defined as

FLU Z1

n

XRING

AKB

FluðA/BÞ

FluðB/AÞ

d dðA;BÞKdrefðA;BÞ

drefðA;BÞ

" #2

;

(3)

with the sum running over all adjacent pairs of atoms around

the ring, n being equal to the number of members in the

ring, dref(C,C)Z1.4 (the d(C,C) value in benzene at the

HF/6-31G(d) level [25]), and the fluctuation from atom A to

atom B reading as follows

FluðA/BÞ ZdðA;BÞP

BsA dðA;BÞZ

dðA;BÞ

2ðNðAÞKlðAÞÞ; (4)

where d(A,B) is the DI between basins of atoms A and B,

N(A) is the population of basin A, and l(A) is the number of

electrons localized in basin A. FLU is close to 0 in aromatic

species, and differing from it in non-aromatic ones.

Finally, another characteristic of an aromatic species is

its planarity that facilitates the delocalization of the

p-electrons around the ring. This is the reason why we

have also tried to evaluate the aromaticity through an

analysis of the planarity of the system along the reaction

path. To this end, we have devised a procedure that finds the

best fitted plane p to a given ring. The methodology

followed is detailed in the next section.

2. Obtention of best fitted plane p

Our purpose is to find the best fitted plane p to a certain

cloud of points PhPiiZ1,m. We use the well-known

Eq. (5) that gives the distance between a given point Pi to a

plane p.

dðp;PiÞ Z jaTPi Ca0j: (5)

In this equation, a is the normalized plane’s normal

vector. This restriction on the modulus of the vector

a makes no loss of generalization. The best fitted plane is

obtained by minimizing the error function F(a,P) that

measures the sum of the square distances d(p,Pi) from the

set of points PhPiiZ1,m to the plane p.

Fða;PÞ ZXm

i

dðp;PiÞ2 Z

Xm

i

ðaTPi Ca0Þ2

ZXm

i

½a1xi1 C/Canxi

n Ca02: (6)

81

First, let us find the minimum according to a0:

vFða;PÞ

va0

Z 0 (7)

2Xm

iZ1

½aTPi Ca0 Z 0;

aTXm

iZ1

½Pi

mZ aT P ZKa0: (8)

Thus, now we can write F(a,P) as

Fða;PÞ ZXm

i

ðaTðPi K PÞÞ2

ZXm

i

½a1ðxi1 K x1ÞC/Canðx

in K xnÞ

2: (9)

Forcing the a vector to have modulus 1 as stated on

Eq. (5), we look for the minima of the above expression

under this normalization restriction that is included in

the minimization through a Lagrange multiplier in the

following way:

Lða; lÞ Z Fða;PÞKlðjjajj2 K1Þ

ZXm

i

ðaT ðPi K PÞÞ2 KlðaT a K1Þ

vLða; lÞ

vaj

Z 0:

(10)

We obtain the following set of equations:

Xm

iZ1

½aTðPi K PÞðxij K xjÞ Z laj: (11)

On the other hand, the covariance matrix reads:

S Z sij Z

Xm

kZ1

ðxki K xiÞðx

kj K xjÞ

m

8>>><>>>:

9>>>=>>>;: (12)

Hence, we can write the equation above in terms of the

covariance matrix elements:

a1s1j C/Cajðsnj Kl0ÞC/Cansnj Z 0 0% j%n;

(13)

where l 0Zl/m. The last set of equations can be easily

recognized as the following secular formula:

SA Z AL: (14)

A is the matrix that collects eigenvectors (i.e. normal

vectors of different planes) of covariance matrix S, and L is

a diagonal matrix with the eigenvalues (corresponding to the

F(a,P) values). The lowest eigenvector is the vector

perpendicular to the best fitted plane in the way stated

above. The corresponding eigenvalue is an unambiguous

measure of the planarity for the set of points P, since its


value quantify in an unequivocal way how far the points are

from the best fitted plane. A similar derivation and an

alternative one can be found in Ref. [26].

The best fitted plane is usually mistaken as the plane

obtained in a typical multiregression method. Our best fitted

plane is found in the basis of minimizing the distances of P

to p. On the other hand, the multiregression methods lead to

the best plane in order to do predictions on a certain variable

X, i.e. the minimization is over a function resulting of the

difference between the X values and its expectation values

according to p.

The quantity F(a,P) obtained for the best fitted plane p,

from each set of points P—corresponding to a molecular

geometry determined by the nuclear positions of the atoms

in the ring—will be given as a measure of molecular

planarity; hereafter these quantities will be referred to as the

Root-Summed-Square (RSS) values.

3. Computational details

All calculations have been performed with the GAUSSIAN

98 [27] and AIMPAC [28] packages of programs, at the

B3LYP level of theory [29] with the 6-31G* basis set [30].

The intrinsic reaction path (IRP) [31] for the DA reaction

has been computed with the GAUSSIAN 98 package [27],

going downhill from the TS in mass-weighted coordinates

using the algorithm by Gonzalez and Schlegel [32]. The TS

of the DA reaction was characterized by the existence of a

unique imaginary frequency corresponding to the C–C bond

formation and breaking. All aromaticity criteria have also

been evaluated at the same B3LYP/6-31G* level of theory.

The GIAO method [33] has been used to perform

calculations of NICS at 1 A (NICS(1)) above the ring center,

which is determined by the non-weighted mean of the heavy

atoms coordinates, following the direction given by the

normal vector corresponding to the best fitted plane.

Integrations of DIs were performed by use of the AIMPAC

[28] collection of programs. Calculation of these DIs with

the density functional theory (DFT) cannot be performed

exactly because the electron-pair density is not available at

this level of theory [34]. As an approximation, we have used

the Kohn–Sham orbitals obtained from a DFT calculation to

compute HF-like DIs through the expression:

dðA;BÞ Z 4XN=2

i;j

SijðAÞSijðBÞ: (15)

The summations in Eq. (15) run over all the N/2 occupied

molecular orbitals. Sij(A) is the overlap of the molecular

orbitals i and j within the basin of atom A. Eq. (15) does not

account for electron correlation effects. In practice, the

values of the DIs obtained using this approximation are

generally closer to the HF values than correlated DIs

obtained with a configuration interaction method [34].

The numerical accuracy of the AIM calculations has been

82

assessed using two criteria: (i) the integration of the

Laplacian of the electron density ðV2rð~rÞÞ within an atomic

basin must be close to zero; (ii) the number of electrons in

a molecule must be equal to the sum of all the electron

populations of the molecule, and also equal to the sum of all

the localization indices and half of the delocalization indices

in the molecule. For all atomic calculations, integrated

absolute values of ðV2rð~rÞÞ were always less than 0.001 a.u.

For all molecules, errors in the calculated number of

electrons were always less than 0.01 a.u.

4. Results and discussion

The present work analyzes the aromaticity along the DA

reaction between 1,3-butadiene and ethene to yield

cyclohexene (boat conformation), which is often taken as

a prototype of a pericyclic concerted reaction. As said in

Section 1, this reaction is characterized by an aromatic TS,

thus along the reaction path we expect a peak of aromaticity

around the TS, which, in principle, should be shown by the

different aromaticity criteria.

Table 1 lists all values obtained from the different

aromaticity criteria applied to the analysis of the DA

reaction, while Fig. 1 depicts the NICS(1), PDI, FLU,

HOMA, and RSS values along the reaction path. From

Table 1, it is seen that only the magnetic NICS(1) and the

electronic PDI criteria find the most aromatic point along

the reaction path around the TS of the reaction. Because of

the lack of a symmetry plane, the NICS(1) computed at 1 A

above the ring center determined by the non-weighted mean

of the heavy atoms coordinates, following the positive

direction of the normal vector corresponding to the best

fitted plane does not coincide with the NICS computed at

1 A below the ring center (NICS(K1)) (see Scheme 1).

However, all definitions of NICS, and in particular NICS(0)

and NICS(K1), also find that a structure close to the TS is

the most aromatic species along the reaction path of this DA

reaction. For this reason, only NICS(1) values are discussed

in this work. On the other hand, both the HOMA and the

electronic FLU indices consider the cyclohexene molecule

(product), as the most aromatic species in the reaction. The

RSS measure of the planarity shows that the cyclohexene

species is the flattest system along the DA reaction path. In

principle, the flatter a structure, the easier the p-electron

delocalization. Therefore, likewise the HOMA and FLU

indices, the RSS measure also fails in considering the

cyclohexene as the most aromatic species along the DA

reaction. We see in this example that the study of the

planarity along a reaction path may fail to account for the

aromaticity of the species involved, since the most planar

structures are not necessarily those having the most

extended p-electron delocalization. Thus, in this case, an

aromaticity analysis only based on geometrical indices such

as HOMA and RSS gives a wrong answer. However, it is

also true that these indices can be very useful when dealing

Table 1

Reaction coordinate (IRP in amu1/2 bohr), RSS, HOMA, NICS(1) (ppm), PDI (electrons), and FLU for different points along the reaction path of the DA

reaction between 1,3-butadiene and ethene

IRP RSS HOMA NICS(1) PDI FLU

K3.491 0.865 K170.116 K2.335 0.055 0.339

K2.892 0.848 K150.941 K2.967 0.058 0.327

K2.293 0.829 K132.591 K3.803 0.063 0.313

K1.694 0.807 K115.002 K4.951 0.069 0.293

K1.094 0.782 K98.287 K6.534 0.077 0.266

K0.496 0.758 K82.619 K8.669 0.086 0.231

K0.296 0.750 K77.671 K9.489 0.089 0.218

K0.099 0.742 K72.925 K10.293 0.091 0.204

0.000 0.732 K66.285 K11.416 0.094 0.183

0.099 0.721 K59.981 K12.251 0.096 0.163

0.397 0.711 K53.819 K16.975 0.095 0.144

0.697 0.700 K47.992 K16.449 0.092 0.128

1.297 0.678 K37.455 K13.830 0.081 0.106

1.897 0.654 K28.388 K10.812 0.067 0.095

2.497 0.625 K20.773 K8.421 0.053 0.091

3.097 0.593 K14.589 K6.755 0.043 0.089

3.397 0.575 K12.021 K6.140 0.039 0.089

3.597 0.563 K10.497 K5.792 0.037 0.088

Negative values of the IRP correspond to the reactants side of the reaction path, positive values to the product side, and IRPZ0.000 corresponds to the TS of the

DA cycloaddition.

–4 –2 0 2 4

IRP

–20

–16

–12

–8

–4

0

NIC

S(1

), H

OM

A/1

0

0

0.1

0.2

0.3

0.4

0.5

PD

I, F

LU, R

SS

/2

NICS(1)PDIFLUHOMA/10RSS/2

Fig. 1. Plot of NICS(1) (ppm), PDI (electrons), FLU, HOMA (values

divided by 10), and RSS (values divided by 2) versus the reaction

coordinate (IRP in amu1/2 bohr). Negative values of the IRP correspond to

the reactants side of the reaction path, positive values to the product side,

and IRPZ0.000 corresponds to the TS of the DA cycloaddition.


with the aromaticity in structurally similar compounds.

Finally, it is worth noting that the species with the highest

slope of RSS (see Table 1 and Fig. 1) along the IRC

corresponds to the most aromatic structure on the reaction

path, the TS.

HOMA and FLU values measure variances of the

structural and electronic patterns, respectively, around the

ring. Therefore, HOMA and FLU might fail if they are not

applied to stable species because, while reactions are

occurring, structural and electronic parameters suffer

major changes. The larger the difference between the

standard bond lengths for aromatic species and the average

bond lengths, the lower aromaticity predicted with HOMA

(lower HOMA values). And, the higher the differences

between the standard in aromatic systems and the actual

electronic sharing, the lower aromaticity predicted by FLU

(higher FLU values). For this reason, in general, HOMA

and FLU can not be used to compare the aromaticity

between species which undergo large structural or

electronic changes.

The present study has shown that FLU and HOMA

indices are not suitable for the study of aromaticity along a

reaction path because they give wrong aromaticity differ-

ences when applied to species that are structurally or

electronically very different. In principle, we expect that any

criterion to quantify aromaticity that is defined using a

reference model of aromaticity will have problems to find

the TS of the DA reaction as the most aromatic species

along the reaction path. However, FLU and HOMA indices

have proven to behave properly in aromaticity studies of

systems having the usual aromatic structures [13,25].

In these circumstances, changes in aromaticity of the

different species analyzed are less dramatic and are well

83

quantified by variance-like indices. Although both FLU

and HOMA present the same wrong behavior in this DA

reaction, this result cannot be generalized for all cases.

For instance, in a recent work, Matito et al. [25] have

studied the change of aromaticity along the

series: cyclohexane, cyclohexene, cycohexa-1,4-diene,

cyclohexa-1,3-diene, and benzene, a set of molecules with

increasing p-conjugated character from non-aromatic

cyclohexane to aromatic benzene. These authors found


that while FLU correctly reproduces the predicted steady

increase of the aromatic character along the series,

the HOMA index fails to account for the expected

continuous intensification in aromaticity.

Finally, although NICS behaves correctly for the DA

reaction, it has been recently shown that it fails to recognize

the decrease of aromaticity that takes place when going

from planar to pyramidalized pyracylene [18], at variance

with PDI and HOMA values that correctly account for this

reduction of aromaticity. NICS is also known to over-

estimate the local aromaticity of inner rings in linear

polyacenes [10,35]. On the other hand, HOMA and PDI fail

to correctly account for the increase of aromaticity [36]

when going from C60 to CC1060 [37]. Indeed, there is no single

descriptor of aromaticity that behaves correctly for all

situations. This fact reinforces the idea of aromaticity as

being a multidimensional property [12–14,16]. For this

reason, we also strongly recommend the use of more than a

single parameter for aromaticity studies.

5. Conclusions

In the present study, the paradigmatic DA reaction

between 1,3-butadiene and ethene, presenting an aromatic

TS, has been analyzed to show that some aromaticity indices

may fail to describe aromaticity in chemical reactions. The

NICS and PDI indicators of aromaticity correctly predict

that a structure close to the TS is the most aromatic species

along the reaction path. On the contrary, we have found that

HOMA and FLU indices are unsuccessful to account for the

aromaticity of the TS. The same is true for the new defined

geometric RSS index, which quantifies the planarity of a

given structure. This index shows that the most planar

species along the reaction path of the DA reaction between

1,3-butadiene and ethene is cyclohexene, the final product.

Inclusion of electron correlation effects will obviously

change the quantitative values of the indices, but we do not

expect that it may alter in a significant way the qualitative

conclusions of this work. In particular, with respect to the

PDI and FLU results, it has been found that the Hartree–

Fock (HF) values of DIs represent upper bounds to the

number of electron pairs shared between atoms [22,34,38].

Consequently, inclusion of correlation energy will reduce

the values of the PDI index (the effect on the FLU indices is

less clear). However, we think that conclusions obtained

from the comparison of the different values will remain

unchanged. This is the case, for instance, of the relation

between the DIs of meta and para related carbon atoms in

benzene. These values are 0.068 and 0.106 e at the HF/3-

21G* level, respectively, and 0.048 and 0.071 e at the

CISD/3-21G* level, respectively [39]. Thus, although the

values of DIs are quite different, both the HF and the CISD

levels of calculation qualitatively agree in assigning a larger

DI for the carbon atoms in para position.

84

Finally, the failure of some indices to detect the

aromaticity of the TS in the simplest DA cycloaddition

reinforces the idea of the multidimensional character of

aromaticity and the need for using several criteria to

quantify the aromatic character of a given species. The

results presented in this paper clearly confirm the necessity

of finding new indices of aromaticity that can be success-

fully applied to a series of well-defined situations, as far as

the aromatic behavior is concerned, such as the DA reaction.

More research is underway in our laboratory concerning this

particular issue.

Acknowledgements

Financial help has been furnished by the Spanish MCyT

projects No. BQU2002-0412-C02-02 and BQU2002-03334.

M.S. is indebted to the Departament d’Universitats, Recerca

i Societat de la Informacio (DURSI) of the Generalitat de

Catalunya for financial support through the Distinguished

University Research Promotion 2001. E.M. thanks the

Secretarıa de Estado de Educacion y Universidades of the

MECD for the doctoral fellowship no. AP2002-0581. J.P.

also thanks the DURSI for the postdoctoral fellowship

2004BE00028. We are also grateful to the Centre de

Supercomputacio de Catalunya (CESCA) for partial fund-

ing of computer time.

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4.5.- Aromaticity Measures from Fuzzy-Atom Bond Orders

(FBO). The Aromatic Fluctuation (FLU) and the para-

Delocalization (PDI) indexes.

87

Aromaticity Measures from Fuzzy-Atom Bond Orders (FBO). The Aromatic Fluctuation(FLU) and the para-Delocalization (PDI) Indexes

Eduard Matito, Pedro Salvador, Miquel Duran, and Miquel Sola*Institut de Quı´mica Computacional and Departament de Quı´mica, UniVersitat de Girona, 17071 Girona,Catalonia, Spain

ReceiVed: December 19, 2005; In Final Form: February 4, 2006

In the past few years, there has been a growing interest for aromaticity measures based on electron densitydescriptors, the para-delocalization (PDI) and the aromatic fluctuation (FLU) indexes being two recent examples.These aromaticity indexes have been applied successfully to describe the aromaticity of carbon skeletonmolecules. Although the results obtained are encouraging, because they follow the trends of other existingaromaticity measures, their calculation is rather expensive because they are based on electron delocalizationindexes (DI) that involve cumbersome atomic integrations. However, cheaper electron-sharing indexes (ESIs),which in principle could play the same role as the DI in such aromaticity calculations, can be found in theliterature. In this letter we show that PDI and FLU can be calculated using fuzzy-atom bond order (FBO)measures instead of DIs with an important saving of computing time. In addition, a basis-set-dependencestudy is performed to assess the reliability of these measures. FLU and PDI based on FBO are shown to beboth good aromaticity indexes and almost basis-set-independent measures. This result opens up a wide rangeof possibilities for PDI and FLU to also be calculated on large organic systems. As an example, the DI andFBO-based FLU and PDI indexes have also been calculated and compared for the C60 molecule.

Introduction

The quest for new aromaticity indexes1 has recently putforward two new aromaticity indicators based on the electronicstructure of molecules: the para-delocalization (PDI)2 and thearomatic fluctuation (FLU)3 indexes. These quantities have beencalculated using the so-called delocalization index (DI),4,5 anelectron-sharing index (ESI) defined in the framework of theatoms-in-molecules (AIM) theory. In this approach, the zero-flux condition over the electron density provides a 3D exhaustivepartitioning of the molecular space into atomic domains, mostoften containing an atomic center. The integration of theexchange-correlation density over two different atomic domainsleads to the DI between the corresponding atoms. Despite theirwidespread use throughout the literature, these indexes have aserious drawback: the numerical integrations over the atomicdomains are often cumbersome because of the complicated andsharp shape of these atomic domains. As a result, the numericalprecision gathered can sometimes be rather poor,6 especiallyfor calculations including electron correlation, where the numberof elements of the second-order reduced density matrix is larger(for monodeterminantal wave functions the density matrices arediagonal regardless of their order).

Recently, we have shown that, together with the DI, anothertwo ESIs, namely, the Mayer-Wiberg index (MWI)7 and thefuzzy-atom bond orders (FBO),8 give similar quantitive trendsin their values for a large set of molecules.9 In particular, thecorrelation of these three ESIs for carbon-carbon bonds isexcellent. The Hilbert-space-based MWI index is determinedeasily. However, it is strongly basis-set-dependent, and un-physical negative values can be obtained for nonbondedinteractions, such as the ones used to construct the PDI index.On the contrary, the FBO values are less sensitive to basis-set

variations and the computational cost of the required numericalintegrations8 is reduced enormously with respect to the DI.

With this in mind, the aim of this letter is twofold. On oneside, we will explore the use of FBO for the determination ofFLU and PDI indexes for a set of aromatic and nonaromaticmolecules and compare the values with those of our previouswork,3 where DIs were used, instead. Second, we will assessthe dependence of the aromaticity indexes upon the basis setused. The validity of very modest basis sets such as STO-3Gand 3-21G is also addressed with the aim of being able to furtherextend the present study to much larger organic systems.

Fuzzy Bond Order and Delocalization Index.The ESIs wewill use in this letter are based upon the exhaustive partitioningof the molecular space into atomic domains. This allows theintegration of any function restricted to a particular atomicdomain to provide the respective atomic contribution to its totalvalue. Such partitioning of the space can generally be expressedin terms of some nonnegative weight factors,wA(r ), defined ateach pointr of the space for every atom A, in such a way thatthe following condition is fulfilled:

where the summation runs for all atoms of the molecule.Alternatively, both the DI and FBO for monodeterminantal

wave functions arise from the integration of the so-calledexchange density.4 For a closed-shell single determinant wavefunction, it can be obtained easily through the nondiagonal partof the spinless first-order reduced density matrix,γ1(r ,r ′), as

The double integration over the whole space of the exchange

∑A

wA(r ) ) 1 (1)

γx(r1,r2) ) 12

γ1(r1,r2) γ1(r2,r1) ) 12

|γ1(r1,r2)|2 (2)

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10.1021/jp057387i CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 03/28/2006

89

density provides the number of electrons of the system

Now, to obtain the atomic and atom pair contributions of thisintegral, one simply inserts the identity (eq 1) into the integration(eq 3) for each electron coordinate and interchanges thesummation and integration symbols to get

whereγA1(r1,r2) can be seen as the atomic nondiagonal first-

order density matrix of atom A.The DI and FBO between two atoms A and B are defined

simply as

whereas the localization index of atom A,λ(A), stands for theintegration of both electron coordinates over the same atomicbasin

By substitution of the definitions 5 and 6 into 4

the relationship between the electron population of a given atom,N(A), and its localization and delocalization indexes arisenaturally

Also, some authors7 define the valence of an atom A,V(A), asthe sum of its bond orders

Finally, the only difference between the DI and FBO is givenby the definition of the weight functions (eq 1). Hence, for theDI we have

whereΩA represents the spatial atomic domain of atom A. Thatis, wA(r ) ) 1 if the pointr belongs to the atomic basin of theatom A and is zero otherwise.

On the contrary, the weight factors could be allowed to varybetween 0 and 1 in a continuous way, thus reflecting to whichextent every point of the physical space belongs to each atom.In this way the atoms are fuzzy entities that overlap andpenetrate into each other. There are many ways to define fuzzyatoms, depending on the shape and nature of the atomic weightfactors (e.g., Hirshfeld,10 Davidson,11 etc.). In this letter we haveused the original definition of the FBO, in which the weightfactors are simply taken as those derived by Becke12 for hismulticenter integration scheme, tuned with the set of covalentatomic radii of Koga.13

Aromaticity Indexes

PDI. Despite the fact that electron delocalization is the keypoint to the aromaticity, there have been just few attempts2,3,14-20

to give quantitative aromaticity measures from ESIs. Becausethese indexes measure the electron sharing between pairs ofatoms, one would expect to obtain from them a reliable measureof the degree of delocalization in a given ring. The PDI wasone of the first local measures of aromaticity proposed basedon ESIs.2 The PDI is calculated as the average of the three DIsbetween pairs of atoms in para position in a six-membered ring(6-MR). Although quite simple and resolutive, the PDI has aserious disadvantage: it can be computed only for 6-MRmolecules. An alternative definition of the PDI for 5-MR hasbeen suggested (∆DI)2, but unfortunately the results are lesssuccessful than those obtained from the original 6-MR PDI.

FLU. The electron fluctuation index (FLU) is an appealingproposal to calculate the aromaticity from electron-density-baseddescriptors.3 Unlike PDI, FLU can cope with different ring sizesand has been proven to reproduce the trends of other aromaticityindexes in the literature. It is worth noticing that some of ushave recently pointed out that the FLU and HOMA indexesshould be applied with care to the study of chemical reactivitybecause they fail to recognize those cases where the aromaticityis enhanced upon deviation from the equilibrium geometry, asis the case of the Diels-Alder reaction.21 Fortunately, thesystems with such behavior are scarce.

The FLU index was constructed following the HOMAphilosophy, that is, measuring divergences (electron-sharingdifferences in the FLU framework) from typical aromaticmolecules. To this end, the following formula is given for FLU

where the summation runs over all adjacent pairs of atomsaround the ring,n is equal to the number of atoms of the ring,V(A) is the valence (see eq 9) of atom A,δ(A,B) andδref(A,B)are the ESI values for the atomic pair A and B and its referencevalue, respectively, and

Note that the ratio between the valences of the atoms hasreplaced the fluctuation measures in the original definition ofFLU.3 It can be shown easily that both definitions are equivalent.The second factor in eq 11 measures the relative divergenceswith respect to a typical aromatic system, and the first factor ineq 11 penalizes the highly localized electron systems.

The weakness of FLU, as is also the case of HOMA, is theneed for typical aromatic systems as a reference. From the latter,the corresponding ESI for a given bond is used as a reference

∫∫γx(r1,r2) dr1 dr2 ) N (3)

∑A

∑B∫∫wA(r1) wB(r2) γx(r1,r2) dr1 dr2 )

1

2∑A

∑B∫∫wA(r1) γ1(r1,r2) wB(r2) γ1(r2,r1) dr1 dr2 )

1

2∑A

∑B∫∫γA

1(r1,r2) γB1(r2,r1) dr1 dr2 ) N (4)

δ(A,B) ) ∫∫wA(r1) wB(r2) γx(r1,r2) dr1 dr2 +

∫∫wB(r1) wA(r2) γx(r1,r2) dr1 dr2 )

∫∫γA1(r1,r2) γB

1(r2,r1) dr1 dr2 (5)

λ(A) ) ∫∫wA(r1) wA(r2) γx(r1,r2) dr1 dr2 )

12∫∫|γA

1(r1,r2)|2 dr1 dr2 (6)

N ) ∑A

λ(A) + ∑A

∑B>A

δ(A,B) ) ∑A

λ(A) +1

2∑A

∑B*A

δ(A,B)

(7)

N(A) ) λ(A) +1

2∑B*A

δ(A,B) (8)

V(A) ) ∑B*A

δ(A,B) ) 2[N(A) - λ(A)] (9)

wA(r ) ) 1 r ∈ ΩA

0 otherwise(10)

FLU )1

n∑A-B

RING[(V(B)

V(A))R(δ(A,B) - δref (A,B)

δref (A,B) )]2

(11)

R ) 1 V(B) > V(A)-1 V(B) e V(A)

(12)

Aromaticity Measures from Fuzzy-Atom Bond Orders J. Phys. Chem. A, Vol. 110, No. 15, 20065109

90

for the calculation of the aromaticity of other systems. Therefore,the main shortcoming for this kind of aromaticity measures isthe somewhat arbitrary choice of the reference parameters. In aprevious work,3 the reference DI values for C-C and C-Nbonds were obtained from benzene and pyridine, respectively.In a forthcoming work we will also give the reference parameterfor B-N bonds.22

FLUπ. Because the lack of parameters is the main drawbackof FLU, we explored the possibility of aπ counterpart of it,FLUπ, which would measure the degree of global delocalizationof theπ system. In this way no reference parameters would beneeded. The FLUπ measure can only be exactly determined forplanar molecules, where the corresponding ESIs can be exactlydecomposed into itsσ andπ contributions. However, appropriateorbital localization schemes could be applied to obtain ap-proximateπ contributions of the ESI measures for nonplanarmolecules. Research in this direction is under way in ourlaboratory.

The FLUπ formula is indeed very similar to eq 11

where nowδav is the average value of theπ component of thecorresponding ESI, and only theπ component of the ESI andvalence is needed. The correlation between FLUπ and FLU hasbeen shown to be excellent for a series of organic compounds.3

However, one should not expect it to always be like this. Indeed,FLUπ is measuring the degree of homogeneous delocalizationin a π system, whereas FLU is measuring the degree ofsimilarity with respect to reference aromatic molecules. In thissense, when the aromaticity of the system comes from thedelocalization of theπ system, as is the case in organic species,both indexes should speak in the same voice. However, somedifferences can arise when the aromaticity of the system is given,for instance, by the delocalization of theσ electrons, as it couldbe the case of inorganic species.

Recently, Bultinck et al. have computed PDI values by usingthe Mulliken-like partition in the Hilbert space spanned by thebasis functions instead of the AIM atomic basins for theintegration of the exchange-correlation density. Somehow, theyhave decided to rename the index as an average two-center index(ATI).14 The authors show that there is a good agreementbetween ATI and PDI for a large family of benzenoid rings ofpolycyclic aromatic hydrocarbons. In the same spirit the bondorder index of aromaticity (BOIA) measure was also introduced,which, in our opinion, represents a simplified version of ouralready defined FLU index. Finally, it is worth mentioning inthis context that aσ-π separation for the ELF function hasbeen proposed recently to analyze aromaticity.19-20

Computational Details

All calculations were performed with Gaussian 98,23 AIM-PAC24 packages, and the FUZZY25 and other software of ourown for the FLU and PDI calculations. AIMPAC and FUZZYare freeware, and the FLU program is available upon requestto the authors. The integration into the fuzzy atom domains havebeen carried out using our implementation of Becke’s multi-center numerical integration. Particularly, a Gauss-Legendregrid of 30 radial points combined with a Lebedev quadratureof 110 angular points has been used throughout. The accuracyobtained with such a modest grid is comparable to that achievedby AIM integrations, though a much more extensive grid isnecessary for the latter.

We have studied the same set of molecules as in ref 3 (seeScheme 1), calculated at the Hartree-Fock level of theorywith the STO-3G, 3-21G, 6-31G*, 6-311G**, and 6-311++G-(3df,2pd) basis sets.

The reference FBO parameters for FLU calculations are 1.40e for C-C and 1.58 e for C-N. They correspond to the FBOvalues of the C-C and C-N bonds obtained for the referencebenzene and pyridine molecules at the HF/6-31G* level oftheory. Note that the latter is slightly different than the suggestedvalue of ref 3. Although the same general trends have beenobserved for both DI and FBO, the actual values for a givenbond do differ and, in some exceptional cases,6 the descriptionof the bonding can, indeed, be very different.

To illustrate the computational saving in the numericalintegration consider the following results: the calculation ofbenzene at the HF/STO-3G level of theory took about 3 s forFBO, 1 h for DI with the default PROAIM integration algorithm,and 6 h with the PROMEGA26 one in a simple 1.6 GHz LinuxPC. Although most of calculations can be performed with thesimplest PROAIM algorithm, unpredictable numerical instabili-ties may appear eventually, forcing the use of the more robustPROMEGA algorithm to obtain reliable measures. This isespecially true for inorganic compounds. For instance, for theMg3Na2 system at the B3LYP/6-311+G* level of theory, about10 h were needed for a PROMEGA integration, whereasFUZZY ran for just 20 s in the same PC.

Results

The values of the PDI, FLU, and FLUπ indexes calculated atthe HF/6-31G* level of theory using the FBOs for the set of

SCHEME 1: Set of Molecules and Aromatic RingsStudied

FLUπ )1

n∑A-B

RING[(Vπ(B)

Vπ(A))R(δπ(A,B) - δav

δav)]2

(13)


91

molecules in ref 3 are gathered in Table S1 of the SupportingInformation. The corresponding results using DI from ref 3 arealso included for comparison in this Table. Figures 1-3 showthe correlation between values obtained using DIs and thoseemploying FBOs. PDI was computed only for 6-MRs; likewise,FLUπ was calculated only for planar molecules.

The PDI results obtained from DIs and FBOs are strikinglysimilar, especially for molecules containing only C-C bonds.The correlation depicted in Figure 1 is almost perfect, also withall points close to they ) x line. The only exceptions are thosecorresponding to triazine and pyrimidine molecules, for whichthe FBO between the C and N atoms in the para position is

larger than the corresponding DI. This can be explained on thebasis of a larger polarization of the C-N bonds induced by theAIM partitioning (the AIM and fuzzy atom partial charges onthe C atom of the reference pyridine are+0, 750, and-0.052electrons) that is translated into a lesser availability of theelectrons of the C atoms for ring delocalization.

The correlation corresponding to the FLU and FLUπ resultsare displayed in Figures 2 and 3. In both cases all points arelocated below they ) x line, which clearly indicates that theFBO-based values are systematically smaller than those obtainedby using DI. The points corresponding to the genuinelynonaromatic molecules slightly leave out of the general trend,but it is well known that nonaromatic species are difficult tocorrelate, much more if they are included in the set of aromaticones. Nevertheless, the correlation is again excellent.

The main discrepancy in the FLU correlation comes fromthe triazine species. In this case, the value of FLU is lower withFBO (nearly zero) than using DI. The reason is simple: theelectron sharing of C-N when going from pyridine (taken as avalue of reference) to triazine decreases from the DI point ofview (leading to higher FLU and lower aromaticity), but standswith a similar magnitude from the FBO perspective (thus,leading to a quite aromatic molecule). Because the FLUπ isbased on averages rather than differences, this discrepancy isnot observed. However, for the same reason, both pyridine andtriazine are predicted to have the same degree of aromaticity.

In general, it seems that for aromatic rings the FBO betweenthe same pair of bonded atoms are not too dependent on theactual molecule, so the reference and average values areexpected not to be too different. Therefore, FLU and FLUπdescribe aromaticity in a very similar fashion.

Alternatively, the dispersion on the corresponding DI valuesis larger, which makes the FLU values larger, and thus theycover a wider range than the FBO-based ones. However, thiseffect is not translated into the FLUπ results because nonoticeable discrepancies are observed for the genuinely aromaticmolecules of the set. Hence, this seems to indicate that the natureof the dispersion on the DI between bonded atoms is ofσ origin.In short, FLU and FLUπ are more prone to differ in thequantitative prediction of the aromaticity if using DI asdescriptors.

To assess the consistency of the method, we have also carriedout a systematic analysis of the basis-set effects on the valuesof the aromaticity indexes. In particular, we have used, inaddition to the 6-31G* basis set, the STO-3G, 3-21G, 6-311G**,and 6-311++G(3df,2pd) basis sets. Each molecule has beenfully optimized at the given level of theory, and the aromaticityindexes have been computed with the corresponding electrondensity using FBOs. In Figure 4 we have represented ther 2

Figure 1. DI- vs FBO-based PDI measures for the set of molecules atthe HF/6-31G* level of theory.

Figure 2. DI- vs FBO-based FLU measures for the set of moleculesat the HF/6-31G* level of theory.

Figure 3. DI- vs FBO-based FLUπ measures for the planar moleculesof the set at the HF/6-31G* level of theory.

Figure 4. Correlation coefficientsr 2 for the pairwise basis-setcomparison of FBO-based PDI, FLU and FLUπ indexes for the wholeset of molecules studied.


92

coefficients for the linear correlation between each pair of basissets and for each aromaticity index. The correlations are, onceagain, excellent. It is striking to see that, even for the PDI values,which are based on nonbonded interactions, ther 2 coefficientbetween the minimal STO-3G and the extended 6-311++G-(3df,2pd) basis sets exceeds 0.99. The least correlated valuesare obviously those from smaller basis sets, and concerningmainly the FLU index, whose values for the STO-3G basis setlay between 0.98 and 0.99. However, one must bear in mindthat theδref(A,B) value has not been changed from basis to basis,in order to use auniVersal DI of reference regardless of thebasis set employed. It is worth noticing that despite the use ofthis universal reference the correlations coefficients are alwaysabove 0.98.

The validity of the results obtained with such minimal basissets opens up a wide range of possibilities for PDI and FLU tobe calculated on much larger organic systems, for which thecalculation of other aromaticity indexes such as NICS could beprohibitive.

Finally, to show the applicability of the method to largesystems, we have also determined the FBO-based FLU and PDIvalues for fullerene C60 and compared them with previous DI-based results from ref 27 (see Table 1). The calculations havebeen performed at the HF/6-31G*//AM1 level of theory andthe atomic numeration together with the layer’s nomenclatureare also taken from ref 27. One can see clearly that results fromFBO mimic those already reported with DI. Consequently, thesame chemical picture of the aromaticity features of the systemis obtained with a substantial reduction of the computationalcost.

Conclusions

PDI, FLU, and itsπ counterpart, FLUπ, measures have beencalculated for a series of aromatic and nonaromatic moleculesreplacing the DI from the AIM theory by FBO. The resultsobtained have been compared with the those from ref 3 obtainedusing DIs. The corresponding values for each aromaticity indexare in excellent agreement, particularly for the moleculescontaining only C-C bonds. The correlation for the PDI indexis surprisingly good, taking into account that it is based on theelectron sharing between nonbonded atoms.

The slight deviations between the FBO and DI between C-Cand C-N bonded atoms are translated into some differencesbetween the FLU and FLUπ indexes, as the correspondingelectron-sharing reference and average values change, respec-tively. Nevertheless, the correlation between the FBO- and DI-based FLU and FLUπ is still excellent.

In addition, a basis-set-dependence study has been performedto assess the reliability of the FBO-based indexes. The indexesare strongly insensitive to the basis set, even for the modestSTO-3G basis set. This result opens up a wide range ofpossibilities for PDI and FLU to also be calculated on largeorganic systems.

Acknowledgment. Financial help has been furnished by theSpanish MEC project nos. CTQ2005-08797-C02-01/BQU andCTQ2005-02698/BQU. We are also indebted to the Departamentd′Universitats, Recerca i Societat de la Informacio´ (DURSI) ofthe Generalitat de Catalunya for financial support through projectno. 2005SGR-00238. E.M. thanks the Secretarı´a de Estado deEducacio´n y Universidades of the MECD for doctoral fellowshipno. AP2002-0581. We also thank the Centre de Supercomputa-cio de Catalunya (CESCA) for partial funding of computer time.

Supporting Information Available: The values of the PDI,FLU, and FLUπ indexes calculated at the HF/6-31G* level oftheory using the FBOs for the set of molecules in ref 3 aregathered in Table S1. The corresponding results using DI fromref 3 are also included for comparison. This material is availablefree of charge via the Internet at http://pubs.acs.org.


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TABLE 1: Buckminsterfullerene DI and FBO (in Electrons)and C-C Distances (in Å) for Symmetrically Different C-CPairsa

atom B δFBO(1,B) δAIM(1,B)b r 1,B layer

4 0.069 0.053 2.369 15 1.135 1.118 1.464 1 (ortho)11 0.003 0.002 4.525 212 0.013 0.012 3.583 213 0.073 0.056 2.467 1 (meta)14 0.004 0.003 4.124 215 0.008 0.007 3.705 216 1.423 1.415 1.385 1 (ortho)17 0.046 0.046 2.849 1 (para)18 0.074 0.056 2.467 1 (meta)23 0.013 0.012 3.583 124 0.004 0.003 4.124 225 0.003 0.002 4.525 226 0.000 0.000 5.217 327 0.001 0.000 5.419 334 0.000 0.000 6.146 435 0.001 0.000 5.797 336 0.001 0.000 5.489 337 0.007 0.008 4.836 238 0.003 0.003 4.609 242 0.000 0.000 7.114 444 0.000 0.000 6.708 445 0.000 0.000 6.961 451 0.001 0.000 5.489 252 0.000 0.000 6.146 353 0.000 0.000 6.672 454 0.001 0.000 5.797 355 0.002 0.002 6.073 356 0.000 0.000 6.978 457 0.000 0.000 6.518 458 0.000 0.000 6.672 4

PDI 0.046 0.046FLU (5-MR) 0.036 0.041FLU (6-MR) 0.018 0.020

a Atom 1 has been chosen as a reference.b From ref 27.


93

(19) Santos, J. C.; Andres, J. L.; Aizman, A.; Fuentealba, P.J. Chem.Theory Comput.2005, 1, 83.

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(22) Matxain, J.; Eriksson L. A.; Mercero J. M.; Lopez X.; Ugalde J.M.; Poater, J.; Matito, E.; M.; Sola`, M., to be submitted for publication,2006.

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94

Supporting Information Table S1. Results for FLU, PDI and FLU-π from FBO and DI measures (in electrons)

for the set of molecules studied at the HF/6-31G* level of theory.

DI values extracted from Ref. [3]

FLU PDI FLU-π DI FBO DI FBO DI FBO

Benzene M1 0.0000 0.0009 0.1047 0.0988 0.0000 0.0000 Naphthalene M2 0.0121 0.0110 0.0730 0.0723 0.1142 0.1014 Anthracene A M3-A 0.0237 0.0213 0.0588 0.0590 0.2507 0.2168 Anthracene B M3-B 0.0067 0.0051 0.0695 0.0680 0.0238 0.0222

Naphthacene A M4-A 0.0314 0.0282 0.0509 0.0516 0.3500 0.3025 Naphthacene B M4-B 0.0112 0.0092 0.0634 0.0623 0.0715 0.0648

Chrysene A M5-A 0.0077 0.0070 0.0793 0.0778 0.0673 0.0599 Chrysene B M5-B 0.0190 0.0166 0.0524 0.0524 0.1824 0.1595

Triphenylene A M6-A 0.0034 0.0033 0.0864 0.0845 0.0259 0.0239 Triphenylene B M6-B 0.0267 0.0226 0.0255 0.0268 0.1812 0.1492 Pyracylene A M7-A 0.0143 0.0128 0.0674 0.0666 0.1275 0.1116 Pyracylene B M7-B 0.0499 0.0443 - - 0.6756 0.5676

Phenanthrene A M8-A 0.0055 0.0050 0.0824 0.0808 0.0446 0.0403 Phenanthrene B M8-B 0.0248 0.0216 0.0434 0.0441 0.2545 0.2197

Acenaphthylene A M9-A 0.0126 0.0111 0.0698 0.0691 0.1140 0.1003 Acenaphthylene B M9-B 0.0454 0.0399 - - 0.5782 0.4838

Biphenylene A M11-A 0.0081 0.0084 0.0884 0.0866 0.0664 0.0588 Biphenylene B M11-B 0.0480 0.0427 - - 0.2970 0.2295

Benzocyclobutadiene A M10-A 0.0223 0.0217 0.0800 0.0791 0.1911 0.1658 Benzocyclobutadiene B M10-B 0.0707 0.0632 - - 1.0467 0.8713

Pyridine M12 0.0009 0.0005 0.0970 0.0998 0.0011 0.0005 Pyrimidine M13 0.0045 0.0004 0.0886 0.0982 0.0025 0.0004

Triazine M14 0.0129 0.0005 0.0750 0.0946 0.0000 0.0000 Quinoline A M15-A 0.0154 0.0113 0.0717 0.0726 0.1248 0.1046 Quinoline B M15-B 0.0173 0.0128 0.0707 0.0722 0.1213 0.1026

Cyclohexane M16 0.0909 0.0577 0.0072 0.0108 - - Cyclohexene M17 0.0893 0.0660 0.0195 0.0258 1.2757 0.9073

Cyclohexa-1.4-diene M18 0.0836 0.0690 0.0137 0.0186 1.3018 0.8735 Cyclohexa-1.3-diene M19 0.0781 0.0641 0.0308 0.0337 - -

95

4.6.- Electron Sharing Indexes at Correlated Level.

Application to Aromaticity Calculations.

97

Electron sharing indexes at the correlated level.

Application to aromaticity calculations.

Eduard Matito,* Miquel Solà, Pedro Salvador, Miquel Duran

Institut de Química Computacional and Departament de Química, Universitat de Girona, 17071 Girona, Catalonia, Spain

Electron sharing indexes (ESI) have been applied to numerous bonding situations

to grasp inside in the nature of the molecular electronic structures. Some of the most popular ESI given in the literature, namely the delocalization index (DI), defined in the context of the quantum theory of atoms in molecules (QTAIM), and the Fuzzy-Atom bond order (FBO), are here calculated at correlated level for a wide set of molecules. Both approaches are based on the same quantity, the exchange-correlation density, to recover the electron sharing extent, and their differences lay on the definition of an atom in a molecule. In addition, while FBO atomic regions enable accurate and fast integrations, QTAIM definition of an atom leads to atomic domains that occasionally make the integration over these ones rather cumbersome. Besides, when working with a many-body wavefunction one can decide whether to calculate the ESI from first-order density matrices, or from second-order ones. The former way is usually preferred, since it avoids the calculation of the second-order density matrix, which is difficult to handle. Results from both definitions are discussed.

Although these indexes are quite similar in their definition and give similar descriptions, when analyzed in greater detail, they reproduce different features of the bonding. In this manuscript DI is shown to explain certain bonding situations that FBO fail to cope with.

Finally, these indexes are applied to the description of the aromaticity, through the aromatic fluctuation (FLU) and the para-DI (PDI) indexes. While FLU and PDI indexes have been successfully applied using the DI measures, other ESI based on other partition such as Fuzzy-Atom can be used. The results provided in this manuscript for carbon skeleton molecules encourage the use of FBO within FLU and PDI indexes even at correlated level.

e-mail: [email protected]

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Introduction

The analysis of electronic distribution in molecules is best understood with the

concept of bond order (BO). Its definition is strongly related with the seminal work of

Coulson,1 who provided the first attempt to quantify the extent of electronic sharing

between two atoms by means of quantum mechanics calculations. It is worth noticing that

in this work Coulson had the intention of using these bonds of fractional order –as he

stated– to study some polyenes and aromatic molecules. It is thus of remarkable

importance that, already from the very beginning, the concepts of aromaticity and bond

order were tightly connected.

From then on, the study of bond orders followed its own way, gaining a prominent

situation in the descriptors used for the analysis of electronic structure of molecules. In

order to avoid controversy, since not all indexes that account for the electron sharing of a

couple of atoms can be regarded as bond orders, we prefer the term electron sharing index

(ESI) to comprise them all. Several ESI have been put forward in the literature since

Coulson’s work. It is not our intention to make here an extensive quotation of all papers

treating chemical bonding by the use of ESI, and therefore we will just comment on the

relevant works for our present purposes. In this sense, it is worth noticing that for a long

time Mulliken’s analysis2 (and its extension to bond orders and valences)3-7 has been one

of the preferred for its simplicity and inexpensive computational cost. On the other hand,

since the work of Wiberg8 there has been a growing interest on ESI based on the

exchange-correlation density,9 being the major contributions those reported for Atoms-in-

Molecules10 and Fuzzy-Atom11 partitions.

Aromaticity was discovered as a multifold property whose characterization was

done by means of other criteria such as typical magnetic manifestations,12 bond length

equalization,13 aromatic stabilization energies14 or more occasionally by characteristic

electron delocalization.15 It is widely known that all these manifestations individually are

insufficient for the characterization of the aromaticity of a given species. Since aromaticity

is said to be multifold, several aromaticity manifestations need to be taken into account

simultaneously to finally assess the aromaticity of a certain compound.16 Furthermore, the

need for continuous examination of existing aromaticity indexes to recover situations

whose aromatic character is perfectly known can provide further evidences of lacks and

100

weak points of current aromaticity measures to ensure a correct description of aromaticity

in molecular systems.

The goal of this work is thus twofold. First of all, we will review the concept of an

ESI from Fuzzy-Atom and QTAIM partitions, starting from the calculation of correlated

first- and second-order density matrices; while its adequacy to describe electronic structure

in molecules is discussed. And secondly, the use of ESI for the definition of aromaticity

indexes is revised, while it is tested for correlated calculations. The present manuscript is

devoted to open a discussion on ESI and aromaticity in a conference named after the man,

Michael Faraday, who discovered in 1825 the first aromatic molecule, benzene.17

Electron Sharing Indexes (ESI)

The concept of bond order, or electron sharing index (ESI)18 according to Fulton’s

terminology, has gained a prominent role in the quest for the best tool to characterize the

electronic structure and chemical bonding of molecular systems. Since the seminal work

of Wiberg,8 different indexes have been put forward in the literature and, despite there is

no unique definition, many of them have a common pattern: they measure the extent the

electrons are shared by two (or more) entities, usually atoms. In this way, well-known

quantities such as the delocalization index (DI),19, 20 fuzzy atom bond order (FBO),11

Mayer Bond Order4-7 or Wiberg Index8 (MWI), among others, may be included in this

definition and can reproduce to some extent the Lewis structures of ninety years ago.21

These ESI can be classified according to the way they define an atom within the

molecule. In the pioneer works,2, 3, 8, 22 the atoms were formally defined by the Hilbert

space of the basis functions assigned to them. Obviously, this definition is restricted to the

LCAO-MO approximation and may lead to severe difficulties if basis functions lacking

atomic character are included, such as diffuse or bond functions. Another avenue is to

partition the 3D physical space into regions assigned in one way or another to each atom.

These atomic regions or domains have usually sharp boundaries, like the Voronoi cells,23

the muffin-tin spheres,24 the Wigner-Seitz cells,25 the Daudel loges,26 the ELF basins,27 or

Bader’s atomic basins;10 however they can also “exhibit a continuous transition from one

101

to another”,11 like the fuzzy atoms. The 3D space atomic definition is more general, in the

sense that it can be potentially applied beyond the LCAO, contrary to the Hilbert-space

case. Since Hilbert space approaches have been widely studied, in the present work, we

will only focus on the family of ESI defined on the 3D-space.

Since the advent of the QTAIM10 (quantum theory of atoms in molecules) the

preferred way of most researchers to define an atom in a molecule has been the Bader

atomic basin: a region of the space containing an attractor surrounded by a zero-flux

surface or by infinity. Such atomic regions are non-interprenetrating10 and can often

exhibit complicated shapes, which make the integration over its domain rather

cumbersome.

In contrast, a fuzzy atom partitioning of the space provides atomic domains that

can overlap and hence part of the physical space is assumed to be shared to some extent by

two or more atoms. There are many ways to define the fuzzy atomic domains. All of them

are described by a non-negative weight function wA(r) defined for each atom A at every

point of the space r in such a way that wA(r) is close to one in the vicinity of the nucleus of

atom A and gradually tends to zero as the distance to the given nucleus increases.

However, the numerical integrations to be carried out within the fuzzy atom

framework are noticeable less expensive and straightforward than those involving the AIM

atomic basins. On the other hand, there is indeed some arbitrariness concerning the

definition of the weight factors that characterize the fuzzy atoms. In previous studies11, 28,

29 we have used Becke’s30 method for multicentric integration, where the weight factors

have been obtained through an algebraic function for each atom that is equal to one in the

vicinity of the respective nucleus and progressively decreases to zero with the distance.

This function depends upon some set of atomic radii that control the size of the Voronoi

cells11 and a upon stiffness parameter that controls the “shape” of the atom.

We have also tested other weight functions, particularly adjusting the ratio of

atomic radii using the position of the extremum of the density along the line connecting

the atoms or even those derived from Hirshfeld’s original idea of promolecular densities.

Numerical experience 11, 29 seems to indicate that reasonable results can be obtained using

the simplest Becke scheme and a balanced set of covalent atomic radii, like those from

102

Koga31 or Slater.32 The results for covalent unpolarized bonds are very similar to those

obtained using a QTAIM partitioning of the space. For ionic systems or bonds involving

B, Be or Al atoms the deviation is more important, basically because the relative atomic

sizes predicted by the QTAIM partitioning are very different from those used to defined

the fuzzy atoms. As a result the chemical bonds are apparently strongly polarized in

QTAIM,33 presenting exaggerated partial atomic charges but also the side effect of a much

lower bond order, according with the predicted ionicity of the bond.

In the present work we have also explored the possibility of using the topology of

the electron density to determine the set of atomic radii for each molecule to be used in

Becke’s integration scheme. As it will be further discussed, typical features of the

QTAIM-based ESI can be easily recovered using this mixed scheme with much lower

computational cost.

Even though there is a huge amount of literature behind the concept of ESI, the

most accepted approaches to the calculation of such quantity are somehow based on the

Ruedenberg’s generalized exchange density term.9 Originally introduced as a

decomposition of the pair density, this term is more generally known as the exchange-

correlation density (XCD). The physical meaning of the XCD becomes evident if such

function is defined as the difference between the pair density of the system and a fictitious

pair distribution constructed by two independent one-particle distributions,

( ) ( ) ( ) ( ) ( ) ( ) ( )212

21

11

21 ,, xxxxxxxcrrrrrr γγγγ −= (1)

where ( ) ( )11 xrγ and ( ) ( )21

2 , xx rrγ are the diagonal parts of the first- (1-RDM) and the

second-order (2-RDM) reduced density matrices, respectively. They are commonly known

simply as the density and the pair density.

In the MO approximation, both the pair density and the density can be written in

terms of the molecular spin orbitals as

103

( ) ( ) ∑∫ ∫ Γ=ΨΨ−=

lkji

lkklijji

spin Rn xxxxxdxdNNxx

,,

212*

1*

3*

212 )()()()()1(,

3

rrrrrK

rrr χχχχγ , (2)

and

( ) ∑∫ ∫ Γ=ΨΨ=ji

jj

iispin R

n xxxdxdNx,

11*

2*

1 )()(3

rrrK

rr χχρ , (3)

where Γklij and Γj

i are the second- and first-order density matrices; hereafter DM2 and

DM1, respectively.

For wavefunctions expanded in terms of Slater determinants, the DM1 and DM2

are computed from the coefficients of the wave function expansion. For

monodeterminantal wave functions (HF or DFT within the Kohn-Sham formalism) these

density matrices are diagonal at any order, and therefore the 2-RDM can be expressed in

terms of the 1-RDM; that is, the XCD can be written entirely in terms of the 1-RDM:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2

211

121

211

21 ;;;, xxxxxxxxxcrrrrrrrr γγγγ == (4)

This formula can be used for correlated wavefunctions too, even though, as an

approximation to the true XCD. It will be referred hereafter as Hartree-Fock like

exchange-correlation density (HF-XCD).

The electron density contains information about the position of a single electron

regardless the position of the other N-1 electrons. In the same way, the pair density is a

two-electron quantity that accounts for the distribution of an electron pair, irrespective of

the position of the remaining N-2 ones. Hence, the XCD carries the information about the

motion of a pair of electrons contained in the pair density, while it avoids that of the

individual distribution of the electrons inherent in the one-particle density. Such a function

is expected to be close to zero for two points distant in the space and thus it is not

surprising that it integrates to N, the number of auto-interactions artificially contained in

the independent electron pair distribution,

104

( ) Nxdxdxxxc =∫ 2121, rrrrγ (5)

It is easy to understand why the XCD has become so popular in the definition of

ESIs. The integration of the XCD is decomposed over all pairs of atomic domains

providing a measure of the extent of correlation (interaction) between a given pair of

regions. Indeed, back in 1975, Bader and Stephens19 pointed out that the integration of the

XCD upon two different regions of the space A and B leads to a measure of the correlative

interactions between the electrons in these two moieties:

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫∫∫ ∫ −⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

B ABAB Axc xdxdxxxdxxdxxdxdxx 2121

211

111

12121 ,, rrrrrrrrrrrr γγγγ , (6)

which at the same time is a measure of the extent to which electrons are delocalized over

these regions. However, it was not Bader the first to propose such a quantity as an ESI.

ESI based on the HF-XCD and XCD

The first ESI based on the QTAIM space partitioning was introduced by

Cioslowski et al in 1991.34 He defined the so-called covalent bond order, which was based

on localized orbitals obtained by an isopycnic transformation,35, 36 and whose formula

reads:

( ) ∑=i

iiiiiC BSASBA )()(2, 2λδ , (7)

where the Sii(A) is the diagonal element of the atomic overlap matrix (AOM) of the

localized spin orbitals integrated over the domain of atom A so that maximize

∑i

iii AS 22 )(λ , and λi is the corresponding occupancy. These localized spin orbitals must

be related to the original natural spin orbitals through a unitary transformation, which

guarantees that the same density is reproduced from both (isos pyknos).

105

Two years later, Fulton18 proposed another ESI in the framework of the QTAIM by

using the 1-RDM in following manner:

( ) ( ) ( )[ ] ( ) ( )[ ]

( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫∫ ∫

∫ ∫=+

+=

A BB A

A B

F

xdxdxxxxxdxdxxxx

xdxdxxxxBA

212/1

1212/1

211

212/1

1212/1

211

212/1

1212/1

211

;;2;;

;;,

rrrrrrrrrrrr

rrrrrr

γγγγ

γγδ, (8)

whereupon the pseudo-square root of the 1-RDM reads

( ) ( )[ ] )()(; 21*2/12/1

211 xxxx kk

kk

rrrr ηηλγ ∑= , (9)

λk and ηk(x) being the natural occupancies and natural spin orbitals, respectively. It is

more convenient to express Eq. (8) in following manner:

( ) ∑=ij

ijijjiF BSASBA )()(2, 2/12/1 λλδ , (10)

where the Sij(A) are the elements of the overlap matrix of the natural spin orbitals

integrated over the domain of atom A.

One year later,37 Ángyán et al. proposed a mapping between the Mayer’s Hilbert-

space based bond order and the QTAIM formalism that leads yet to a different ESI, which

was using again the HF-XCD, and tuned out to coincide with Bader’s definition for a

single-determinant wave function:

( ) ( ) ( ) ( ) ( ) ∑∫ ∫ ==ij

ijijjiA B

A BSASxdxdxxxxBA )()(2;;2, 21121

211 λλγγδ rrrrrr (11)

As a curiosity, it is worth to point out that Fulton originally proposed Eq. (11), but

he refused it afterwards because with this formulation the sum of all electron sharing

106

indexes for a given atom A is not proportional to its population, N(A). It is easy to prove

this relationship is fulfilled in Eq. (8).

Finally in 1999 Fradera et al.20 retook Bader’s 1975 original work based on the

XCD19 to use it explicitly as an ESI, what it is nowadays known as the delocalization

index (DI):

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫∫∫ ∫ −==B ABAB A

xc xdxdxxxdxxdxxdxdxxBA 21212

111

111

2121 ,22,2, rrrrrrrrrrrr γγγγδ (12)

where the 1-RDM and 2-RDM are integrated over the atomic domains to provide the

atomic and atomic pair populations, respectively.

More recently, Bochicchio et al.38 have studied in detail Fradera’s DI and have

showed that in the case of a correlated wave function it can be expressed as the sum of two

terms, one of each being Ángyán’s ESI37, 39 (which they call it exchange term) and the

other coming entirely from the second order density matrix (cumulant term):

( ) ( ) ( )BABABA cumA ,,, δδδ += (13)

The cumulant term measures the difference between the true XCD and the HF-

XCD one; hence for monodeterminantal wave functions ( ) 0, =BAcumδ . In most cases the

magnitude of the cumulant contribution is rather small. However, in a recent work by

Bochicchio et al.38 high cumulant contributions have been reported, which unfortunately

could not be reproduced in the present work (vide infra).

It can be easily shown that Fulton’s, Ángyán’s and Fradera’s definitions coincide

for single-determinant wave functions. It is only interesting to see how they perform at the

correlated level and what is the effect of using the true XCD instead of the HF-XCD

approximated one.

On the other hand, all those ESIs have been always computed in the framework of

the QTAIM theory. However, any other 3D-space partitioning of the space could in

107

principle be applied in all cases. In the present work we explore the possibility of

computing such ESI at the correlated level in the framework of the fuzzy atoms, for which

the necessary numerical integrations are much more efficient from a computational point

of view.

Statistical interpretation of the ESI

Another interesting property of the ESI from equations (8) to (13) is that half of its

same-atom contribution

( ) ( )2, AAA δλ = (14)

can be regarded as the number of electrons localized in the atomic basin; according to

Fradera et al. this quantity is called localization index (LI).20

Taking into account that the expected number of electrons in the basin of atom A, the

population of A, is calculated as:

( ) ( ) ( )ANrdrA

=∫ 111 rrγ , (15)

and its variance reads,

( )[ ] ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )2

,

1

2121

22

AAANxdxdxxAN

ANANANAN

Axc

δλγ

σ

=−=−=

=−−=

∫ rrrr , (16)

where one must bear in mind that the pair distribution does not account for the self-

pairing, and calculate the variance accordingly.

108

Thus, the LI for an atom is the expected number of electrons minus its uncertainty

(variance) in its atomic basin. On the other hand, calling the formula for the covariance of

different atoms population,

( ) ( )[ ] ( ) ( ) ( ) ( )

( ) ( )2,,

·,

2121BAxdxdxx

BNANBNANBNANV

B Axc

δγ −=−=

=−=

∫ ∫rrrr (17)

one can easily see that ESI between two atoms A and B is proportional to the covariance of

their electron populations.

Finally, the sum of all ESI values involving a given atom,

( ) ( ) ( )[ ] ( )[ ] ( )AANBNANVBABABA

δσδ === ∑∑≠≠

2

21,

21, , (18)

may be regarded as a valence of the atom, or the number of electrons of atom A shared

with the rest of the atoms in the molecule. This statistical interpretation holds for any ESI

given in Eqs. (8) to (13) calculated through a monodeterminantal wave function. In

addition, for ESI in Eq. (13) this interpretation is also valid for many-body wave functions,

since the latter is based on the true XCD, and therefore on the true pair density.

From the previous analysis it can be deduced that all discussed ESI in this work are

non-negative quantities. From Eqs. (4) and (12), it is clear that the ESI thus defined cannot

be negative for a single determinant wavefunctions. For correlated wavefunctions the

same is true for Cioslowski’s, Fulton’s and Ángyán’s because they come also from the

square of a real value. Finally, the ESI as defined in Eq. (12) for a given pair of atoms A

and B must be also non-negative because, when the population of one atom N(A) increases

it is necessarily at expenses of the decrease of the population on the other one N(B). Hence

the covariance40 of both quantities is negative. To put it in a nutshell, all ESI reported here

are non-negative defined.

109

ESI Spin cases

If the ESI is constructed entirely from the DM1 it can only be decomposed

according to the two spin cases the DM1 is built of: alpha and beta. These two

contributions will only differ for an open-shell calculation. On the contrary, when the ESI

is calculated from the DM2, it can be decomposed into three contributions,

( ) ( ) ( ) ( )BABABABA ,2,,, ''' σσσσσσ δδδδ ++= (19)

( ) ( )∫ ∫=B A

xc rdrdrrBA 2121 ,2, rrrrσσσσ γδ and ( ) ( )∫ ∫=B A

xc rdrdrrBA 2121'' ,2, rrrrσσσσ γδ (20)

accounting for the three spin cases (alpha, alpha), (beta, beta) and (alpha, beta) that are

included in the true XCD. The first two are equivalent for a restricted closed-shell

calculation. Therefore, it can be interesting to decompose the ESI into the same spin,

( ) ( )BABA ,, ''σσσσ δδ + , and different spin, ( )BA,'σσδ , contributions. The natural names for

these quantities are Fermi-ESI and Coulomb-ESI, respectively, named after Fermi and

Coulomb pair densities. Although the ESI is a strictly positive quantity, it is worth

noticing that ESIσσ’ can be negative, since the Coulomb pair density is not necessarily

positive definite.

Once introduced the spin decomposition, let us focus on the sum rules fulfilled by

each aforementioned ESI. The most general sum rule is as follows:

( ) ( ) ( ) σσλδ NNNANAAAi

ii

iij

ji +===⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ ∑∑ ∑

≠

',21 , (21)

where an exhaustive partition of molecular space Aii=1...n has been supposed. It is easily

shown that Eq. (21) must be fulfilled by all indexes from (8) to (13), except for the

Ángyán’s index at correlated level. Indeed, if the ( )BA,δ integrates to N –as stated in Eq.

(5)– for both correlated and single determinant calculations, it is pretty evident that neither

( )BAA ,δ nor the cumulant part of ( )BA,δ can integrate to N (cf. Eq. (13)). This is one of

the shortcomings of the ( )BAA ,δ index: for many-body calculations it does not provide a

110

decomposition of the N electrons in the system. As it will be shown later, the number of

electrons missing to count up to N corresponds roughly to the unlike spin pairs.

In order to avoid this shortcoming, Bochicchio et al.38 introduced the unshared

population. The unshared populations correspond to those LI which would accomplish Eq.

(21) when using ( )BAA ,δ instead of ( )BA,δ , say:

( ) ( ) ( ) σσλδ NNNANAAAi

ii

iA

ijji

A +===⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ ∑∑ ∑

≠

',21 . (22)

It is worth noticing that these ( )AAλ or unshared populations no longer fulfil Eq. (14).

Additionally, some sum rules are obeyed by the spin cases in which ( )BA,δ is

decomposed according to Eq. (19):

( ) ( ) ( ) σσσσσσ λδ ∑∑ ∑ ==⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛

≠ ii

ii

ijji NANAAA ,

21 (23)

( ) ( ) 0,21 '' =⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛∑ ∑

≠ii

ijji AAA σσσσ λδ (24)

Therefore, one can expect ( )BA,'σσδ to vary from negative to positive values,

while ( )BA,σσδ is always non-negative.

111

The density relaxation

Any property can be defined as either an expectation value of the corresponding

operator or as a response of the molecule with respect to a perturbation (a derivative). The

Hellmann-Feynman theorem ensures that both methods lead to the same result. This is true

for exact wavefunctions, and some approximate theoretical methods, such as self-

consistent field approach. Unhopefully, Hellmann-Feynman theorem does not hold for

most many-body methods, with the exception of CASSCF and Full-CI calculations, being

the latter prohibitive for most molecular systems. This fact leads to the ambiguity of

whether calculating a given property from the corresponding energy derivative or from its

expectation value. In this sense, the density matrices can be determined either as the

expectation value of the proper Dirac’s delta based operator or as a wave function

response. The density matrix obtained in the latter case is known as response density or

relaxed density, since it includes all effects due to orbital relaxation within the coupled-

perturbed Hartree-Fock (CPHF) calculations.

One of the epistemological problems around the relaxed density (in practice, the

wave function which reproduces such density) is that the density obtained in this way is

not N-representable. As a consequence, the population of the natural orbitals can be larger

than one or negative. Such a situation is especially uncomfortable for Fulton’s ESI, where

the square root of the natural occupancies appears. With this aim, to avoid this physically

unsound situation, we believe one should always use the unrelaxed density for the

calculations of ESI.

The relaxed second order reduced density matrices have also been defined,41 but

unfortunately they are generally not available in standard computational packages. On the

other hand, the second order reduced density matrices for methods such as MP2 and

CCSD are not straightforward computed, since N-electron wavefunctions cannot be

produced from these methods. Although energy-derivative second-order reduced density

matrices can be obtained, its use is discouraged because such DM2 is not N-representable

inasmuch as it does not sum up to the number of electron pairs.42 Therefore, exact ESI as

defined in equation (12) are only obtainable from HF, CISD and CASSCF calculations.

This fact, together with the computational expenses needed for CASSCF calculations, has

led us to restrict our study to the CISD results presented here, whereas CASSCF

112

calculations have been only calculated for the systems where such level of theory is

needed to properly describe it (vide infra). It is worth mentioning in this context that DM2

have been only calculated for CISD calculations.

Aromaticity Indexes from ESI

Despite the electron delocalization is one of the key features of aromatic

compounds, there have been few attempts to give quantitative aromaticity measures from

ESI. Since these indexes measure the electron sharing between pairs of atoms, one would

expect to obtain from them a reliable measure of the degree of delocalization in a given

ring. Recently, two aromaticity measures based upon ESI have been put forward, the para-

DI (PDI),43 and the fluctuation aromatic index (FLU).44

The para-delocalization index (PDI)

Bader and coworkers45 reported that for QTAIM-ESI the extent of electron sharing

in benzene is greater for para-related carbons (para-ESI) than for meta-related ones

(meta-ESI); it has been shown that it also holds for Fuzzy-Atom ESI.46 Poater et al.43

undertook an analysis of the QTAIM para-ESI in a series of polycyclic aromatic

hydrocarbons (PAH) with six membered rings. They concluded a close relationship

between PDI and other measures of aromaticity such as ASE, HOMA or NICS, thus

validating PDI as a reliable measure of aromaticity in six membered rings.

The aromatic fluctuation index (FLU)

The FLU index44 is constructed following HOMA47 philosophy, i.e., measuring

divergences from certain aromatic molecules chosen as references. The formula given for

FLU reads:

( ) ( )( )∑

− ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

RING

BA ref

ref

BABABA

AVBV

nFLU

2

,,,

)()(1

δδδα

, (25)

113

where the summation runs over all adjacent pairs of atoms around the ring, n is equal to

the number of atoms of the ring and ( )BAref ,δ is the ESI for the atomic pair A and B from

the aromatic molecule chosen as a reference, and α is a constant to ensure the quotient of

atomic valences is greater than one,

⎩⎨⎧

≤−>

=)()(1)()(1

AVBVAVBV

α (26)

and therefore penalizes species with high electron localization, thus leading to a lesser

delocalization in the ring. The reference ESI (QTAIM,44 Fuzzy-Atom46) values for C-C

and C-N bonds were obtained from benzene (1.4e, 1.4e) and pyridine (1.2e, 1.6e),

respectively at the HF/6-31G* level of theory. The larger the difference, the larger FLU;

ergo, low FLU values correspond to aromatic molecules.

It is fair to say that Bird48 has compared Gordy bond orders of all bonded pairs in a

given ring to define a measure of aromaticity that has some resemblance to our FLU

index. In this context it is also worth to mention the work of Matta et al.49, 50 who also

attempted to construct an HOMA-like47 index from QTAIM by substituting the bond

length by the total electron delocalization. Although Bird’s and Matta’s indexes resemble

FLU, and actually should be recovering the same aromaticity characteristics, one should

notice the three indexes are indeed different. The differential feature of the FLU index is

that it weights the divergence of electron sharing for each pair of bonded atoms with a

quantity that accounts for electron localization -cf. Eq. (25)-.

114

Computational details:

A wide set of molecules (H2, N2, F2, LiF, CO, CN-, NO+, He2, Ar2, LiH, HF, NaH,

HCl, NH3, CH4, H2O, B2H6, Mg32-, and C6H6) have been studied to analyze the effect of

the inclusion of electron correlation and the differences between the different indexes in

equations (8)-(13). All geometry optimizations and correlated wave function calculations

were obtained with Gaussian0351 at the CISD/6-311++G(2d,2p) level of theory using

Cartesian d and f orbitals (6d 10f). Only the valence electrons were correlated (frozen

core). In the case of the benzene, a smaller basis set, 6-311G(d,p), has been used. For the

sake of clarity, the nomenclature of the atoms for B2H6 has been given in Scheme1.

Unrelaxed density matrices may be obtained from Gaussian03 package with the

keywords density=rhoci and use=l916. A program designed in our laboratory52 was used

to generate the DM2 and DM1 matrices from the coefficients of the CISD expansion (the

matrices thus computed are always unrelaxed). In the present work, only coefficients with

values larger than 10-15 were considered, and the elements of DM2 greater than 10-12 were

used for subsequent calculations.53

The atomic overlap matrices for AIM and Fuzzy-Atom domains were generated

with AIMPAC54 and FUZZY55 programs, respectively. For the latter, the Becke’s

multicenter integration algorithm with Chebyshev and Lebedev radial and angular

quadratures has been used. A grid of 46 radial by 140 angular points per atom has been

used in all cases. We have employed the recommended stiffness parameter k = 3 and

adjusted the size of the Voronoi cells with the set of atomic radii determined by Koga,31

except otherwise indicated. Finally, other software of our group56 was used to generate the

different ESI from the atomic overlap matrices and the corresponding density matrices.

The aromatic molecules taken into study consist of a subset of that already studied

in references 44 and 46. Due to the computational cost of CISD calculations we have

limited ourselves to the molecules of maximum of three rings, that is, those of Scheme 2.

The calculation of PDI and FLU indexes have been performed at the CISD//6-31G(d) level

of theory. For such systems the calculation of the second order density matrix is

prohibitive, and therefore only indexes using the HF-XCD can be determined. The

reported results have been obtained using Ángyán’s ( )BAA ,δ index.

115

Results and discussion

ESI at correlated level

Since the effects of the inclusion of electron correlation in the topology of the

density have already been addressed in the literature,57-59 we will not discuss on this topic.

We will rather focus on the influence that the Fermi and Coulomb correlation have on the

ESI under study. In order to purely evaluate the effects of inclusion of electron correlation

on the wave function, reference Hartree-Fock and B3LYP wavefunction calculations have

been performed also at the CISD/6-311++G(2d,2p) geometry.

Tables 1a and 1b contain the calculations for the ESI with QTAIM molecular space

decomposition, whereas Tables 2a and 2b contain the Fuzzy-Atom counterpart.

Comparison of the QTAIM- and Fuzzy-Atom-ESI shows that for equally-shared covalent

compounds, such as H2, N2, F2, and Mg32- and for compounds with bonds closed to this

situation, like C-H bonds in CH4, ESI obtained from the two procedures are almost the

same for the different methods of calculation used. Differences become more important

for compounds with polar covalent bonds (such as CO, CN-, NO+, CO2, NH3, and H2O)

for which the Fuzzy-Atom-ESI is larger than that derived from the QTAIM approach, HCl

being an exception to this general trend. Interestingly, for the isoelectronic series N2, NO+,

CN-, and CO, the differences in δ(A,B) computed from Fuzzy-Atom and QTAIM

partitions widen with the increase in the electronegativity differences between the two

bonded atoms (from 0.081 e in N2 to 0.879 e in CO). The differences between the two ESI

are even larger for species ionically bonded like LiF, LiH or NaH. For instance, for LiF,

the δ(Li,F) is 0.186 e according to the QTAIM partition while the Fuzzy-Atom-ESI is

1.981 e. Obviously, the observed differences must be ascribed to the different partition

used, which is almost equivalent for equally-shared bonds and very different for ionic

species. Some authors have provided support to the idea that the QTAIM partitioning

exaggerates electron densities at electronegative atoms,33 although others have opposed to

this view.60, 61 Our results indicate that, as compared to the Fuzzy-Atom approach,

QTAIM overestimates the charge separation in polar and ionic bonds. However, being

both charges and bond orders non observable quantities, one can not tell whether QTAIM

results are better from a chemical point of view than Fuzzy-Atom approach ones to

partition the space or vice versa.

116

Let us focus now on the results for single-determinant wave function methods. As

already reported for AIM,62 in spite of the correlation included within B3LYP,

( )BALYPB ,3δ results are much closer to ( )BAHF ,δ than to CISD ones, which makes us

notice that in general Coulomb correlation is not fully reflected on these ( )BALYPB ,3δ

calculations, for both AIM and fuzzy partitions. If were about to decide whether

( )BAHF ,δ or ( )BALYPB ,3δ is closer to CISD one, based on a least squares fitting

procedure, one concludes that HF is slightly closer to CISD than B3LYP for QTAIM, and

just the opposite for Fuzzy-Atom. The reader may see how ( )BALYPB ,3δ values for

QTAIM partition are surprisingly larger for LiH, NaH, H2O or CO2, whereas CISD and

HF values almost coincide. Such a finding is not reproduced by Fuzzy-Atom partition,

whose B3LYP results are actually much closer to CISD than HF ones.

Nevertheless, in spite of the partition used, both ( )BAHF ,δ or ( )BALYPB ,3δ results

are much closer to ( )BA,σσδ or ( )BAA ,δ , rather than to ( )BA,δ . In particular, the main

differences between HF (or B3LYP) and ( )BA,δ are found in those cases where the

bonding is given between two open-shell moieties (N2, NO+, CN-), whereas situations of

non-bonding or closed-shell interactions are hardly affected by the inclusion of electron

correlation (see for instance noble gases or ionic compounds in Tables 1a and 1b).

Providing ( )BAA ,δ does not reflect the inclusion of electron correlation as

( )BA,δ does, it seems to indicate that the correlation in ESI is introduced through the 2-

RDM (only explicitly contained in the true XCD) rather than 1-RDM. However, when we

now turn our sight to ( )BA,σσδ , and compare these values with ( )BAA ,δ , we see they

follow the same trend, and thus we shall conclude that what makes the differences in the

calculation of ( )BA,δ is definitely the Coulomb correlation included in DM2.

Let us make it clear by reviewing the formulas already given. In Eq. (4) as a result

of HF-XCD being approximated by the 1-RDM, the Coulomb contribution is not

contemplated. Indeed, if we separate spin cases already in Eq. (4), it is pretty clear it must

be 0),( =21 rrαβγ xc , since the 1-RDM does not contain cross-spin terms. The same holds for

ESI based on HF-XCD, namely, ( )BAHF ,δ , ( )BAA ,δ and ( )BALYPB ,3δ , since for the

117

latter exact 2-RDM is not available within Kohn-Sham formalism, and XCD is also

approximated as a HF-XCD. On the other hand, when XCD is calculated exactly from a

correlated 2-RDM the Coulomb contribution arises,

),()()(),( )2()1()1(212121 rrrrrr αββααβ γγγγ −=xc , in particular ( )BA,δ for correlated

calculation contains such contribution.

A different case is that of Fulton index, as noticed by Wang and Werstiuk,63

( )BAF ,δ can be regarded as an approximation to ( )BA,δ . And its origin comes from the

work of Buijse and Baerends64, 65 who proposed the square of Eq. (9) as an approximation

to actual XCD for correlated calculations. Hence, unlike HF-XCD we may expect

( )BAF ,δ to partially include the Coulomb information of the 2-RDM, at the time it

supposes a different and closer65 approach to the actual XCD.

As a result, when using a HF-XCD (even if we perform a correlated calculation)

we are a) not explicitly considering the Coulomb interaction and b) approximating parallel

spin XCD by formula given in Eq. (4). In order to measure the first effect we simply must

check ( )BA,'σσδ , which will be equally zero for any ESI calculated from HF-XCD. In

order to judge the approximation of HF-XCD to recover the actual XCD for correlated

calculations, as already mentioned, we just need to focus on the cumulant term. But since

we want to isolate the effect of Coulomb correlation from the actual approximation given

by HF-XCD we have collected the difference ( )BA,σσδ - ( )BAA ,δ in the last column of

Tables 1 and 2; it simply measures the differences between XCD and HF-XCD for like

spins. The reader can check that Aδδ σσ − is in general higher than ( )BA,'σσδ for systems

with merely electrostatic or non-bonding interaction, and the opposite is found for

covalently bonded species.

In summary, as expected, the inclusion of electron correlation in the calculation of

ESI plays a marginal role in non-bonded or weakly bonded species, has a light influence in

ionic species, and a prominent role in equally-shared or polar covalent bonding. Such is

the case for the isoelectronic series N2, NO+, CN-, and CO, where the inclusion of

correlation by means of CISD drastically changes the value of the ESI.

118

Desirable properties of an ESI

Providing a universal unique definition of an ESI does not (and maybe cannot)

exist, one can only use the knowledge gathered on his chemical grounds to impose or try

to define an ESI so that a certain set of properties are fulfilled. For instance, reproducing

the growing tendency with lower saturation, as in the series: ethane, ethene, ethyne; an

intermediate ESI value for C-C in benzene between formal single and double bond,

asymptotically tending to zero as the distance between the two atoms increase to infinity,

are among the desirable properties one may impose to an ESI in order to give truthful

results. For instance, some of us66 recently discussed the dissociation H2 molecule with

QTAIM-ESI.

Another desired property of an ESI is being always non-negative, and to tend to

zero for a no electron sharing situation. If the index can achieve negative values, probably

the index can hardly be called sharing index, since a negative sharing is rather unphysical.

One should take into account that this is the case of Wiberg-Mayer Bond Order, because it

bases on Mulliken’s partition, whereas, as already noticed, ESI from Fuzzy-Atom and

QTAIM is always positive.

A recent study of Ponec67 brought our attention to another interesting property for

an ESI to be fulfilled. This is the case of the dissociation of ionic molecules in neutral

species.68 Such molecules should have a small number of electrons shared in their

equilibrium geometry, however, when going to larger atomic distances one of the

electrons should go from the negative charged atom to the positive one. Therefore, at some

given distance, there must be a maximum of electron sharing, since we start from rather

electrostatic interaction to move towards a more covalent situation. Ponec showed that

indeed, QTAIM-ESI –or simply the delocalization index (DI)19, 20– exhibits such a

maximum, whereas a Mulliken-like scheme clearly fails to reproduce it.

Since Ponec did not study this dissociation process with the fuzzy atom approach,

we decided to recalculate the ( )BAF ,δ and ( )BAA ,δ indexes with both QTAIM and our

fuzzy atom partition for the LiH dissociation by using the same method, CAS(4,4) with a

rather large basis set. Results are reported in Figure 1. One can easily see how, as already

reported by Ponec, QTAIM ESI detects a maximum for both ESI, whereas the simplest

119

Fuzzy-Atom approach based on Becke’s partitioning fails to reproduce this maximum.

The fact that the ratio between the atomic radius of the Li and H atom is kept fixed during

the dissociation is clearly responsible for this situation. Hence, we have searched for a way

of dynamically tuning the radii used in the Fuzzy-Atom approach11 to reproduce the

maximum of electron sharing which in principle one would predict for the dissociation of

this ionic species. Because of success of QTAIM approach, based on the topology of the

electron density, a natural way to define the radii of an atom in a molecule may be its

distance to the bond critical point at which it is connected. Whereas for polyatomic

molecules this definition can be controversial (a given nucleus can be connected through

different gradient paths to several critical points), such definition for diatomic molecules is

quite simple. The results obtained for such Fuzzy-Atom approach are striking, since a

noticeable improvement in the results is gathered (cf. Figure1). With these radii the Fuzzy

Atom predicts a slight maximum near the region where QTAIM-ESI displays it.

This finding encourages the possibility of using the information of the topology of

the density as a way to dynamically adjust the ratio of atomic radii for a fuzzy atom

calculation. Needless to say, these atomic basins, unlike QTAIM’s, would not possess the

important features related with the fulfilment of the atomic virial theorem and other

interesting properties. However, the numerical values of the ESIs will be quantitatively

very similar to the true QTAIM ones. This opens the possibility of using the Fuzzy-Atom

approach when the calculation of a QTAIM ESI is prohibitive from a computational point

of view, or even to avoid the presence of some spurious non-nuclear attractors.69, 70

Research in this line is underway in our laboratory.

120

Aromaticity indexes at correlated level. PDI and FLU

So far, the FLU and PDI aromaticity indexes have been reported for

monodeterminantal wave functions (either HF or Kohn-Sham), but no attempt to explore

these indexes for many-body methods has been done. Since the calculation of DM2 is

prohibitive for these systems, here we propose to use the ( )BAF ,δ that needs only of the

DM1 to construct these three aromaticity indexes, and have been already shown the

closest ESI to ( )BA,δ fully correlated value.

Let us notice here an important feature of the results for benzene reported in Table

1b and 2b. Contrary to what is found with QTAIM partition, regardless the ESI used

Fuzzy-Atom approach shows that the electron sharing is smaller for para-related atoms

than for meta-related atoms in benzene. Therefore, one would anticipate Fuzzy-Atom PDI

to fail to reproduce this aromaticity trend found by QTAIM. However, as already

demonstrated,46 Fuzzy-Atom do not suffer of this drawback.

The results reported in Table 3 suggest that in general the inclusion of the electron

correlation does not drastically change the values already obtained. However, as compared

to a previous work where the effect of the basis set was discussed,46 the effects due to the

inclusion of electron correlation are more notorious.

Since the interactions are mainly covalent ones, the FLU index is affected by a

lower ESI value when correlation is taken into account. Although PDI account for the

para-related atoms interaction (a non-bonding interaction) its effect is as noticeable as that

of bonded pairs in the ring. Nonetheless, the values follow the expected trends, and

therefore there is no reason why these measures should not be valid at correlated level of

theory.

121

24

Conclusions

The effect of the inclusion correlation has been analyzed for the ESI from different

partition schemes (QTAIM and Fuzzy-Atom), based on both the first- and the second-

order density matrices. It has been shown that for weakly bonded molecules most indexes

coincide and the effect of the electron correlation is minor; whereas covalent and polar

covalent molecules the electron correlation effects are notorious, and ESI based on first-

order density matrices do not wholly reproduce the trends shown by the ESI based on

second-order one.

Fuzzy-atom approach for the set of radii determined by Koga has been shown to

not reproduce a maximum of electron sharing in the dissociation of LiH species in neutral

atoms. Nevertheless, an improvement in the description is gathered if the radius

corresponding to the distance from the attractor to the bond critical point is used.

Finally, aromaticity indexes have been studied for first time for correlated

calculations, leading to a general trend similar to that already reported for

monodeterminantal methods. Nevertheless, it is worth mentioning that these indexes

depend more on the methodology employed than in the basis set used. PDI and FLU for

correlated calculations can be calculated yielding reasonable results.

Acknowledgements

Financial help has been furnished by the Spanish MEC Projects No. CTQ2005-

08797-C02-01/BQU and CTQ2005-02698/BQU. We are also indebted to the Departament

d’Universitats, Recerca i Societat de la Informació (DURSI) of the Generalitat de

Catalunya for financial support through the Project No. 2005SGR-00238. We also thank

the Centre de Supercomputació de Catalunya (CESCA) for partial funding of computer

time.

122

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125

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Rev. Lett., 1999, 83, 1930.

126

Scheme 1

B B

H H

H H

Hb

Hb

127

Scheme2

128

Figure 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7

r(Li···H)

ESI

),( HLiAδ),( HLiFδ

),( HLiAδ


),( HLiFδ

QTAIMQTAIMfuzzy Beckefuzzy Beckefuzzy-rhofuzzy-rho


),( HLiAδ


),( HLiFδ


129

Table1-a. AIM-ESI for the different diatomic molecules taken into study. When the ESI refers to the same atom, half of the value is given instead

Pair ( )BAHF ,δ ( )BALYPB ,3δ ( )BA,δ ( )BA,σσδ ( )BA,'σσδ ( )BAA ,δ ( )BAF ,δ Cum. Aδδ σσ −

H2 H-H 0.500 0.500 0.575 0.500 0.075 0.490 0.578 0.085 0.010 H-H' 1.000 1.000 0.849 1.000 -0.151 0.950 0.844 -0.100 0.050

N2 N-N 5.480 5.477 5.896 5.564 0.332 5.408 5.796 0.488 0.156 N-N' 3.041 3.046 2.209 2.872 -0.664 2.833 2.407 -0.625 0.039

F2 F-F 8.372 8.356 8.501 8.375 0.126 8.262 8.520 0.239 0.113 F-F 1.256 1.289 0.997 1.250 -0.252 1.152 0.960 -0.155 0.097

LiF Li-Li 1.972 1.969 1.974 1.972 0.002 1.972 1.975 0.002 0.000 Li-F 0.175 0.219 0.186 0.189 -0.003 0.181 0.184 0.005 0.008 F-F 9.855 9.812 9.840 9.839 0.002 9.668 9.841 0.173 0.171

CO C-C 3.870 3.944 4.004 3.932 0.072 3.860 4.052 0.144 0.072 C-O 1.569 1.813 1.377 1.620 -0.244 1.617 1.466 -0.241 0.003 O-O 8.561 8.244 8.620 8.448 0.172 8.196 8.481 0.424 0.252

CN- C-C 4.142 4.285 4.496 4.307 0.190 4.179 4.454 0.318 0.128 C-N 2.228 2.450 1.883 2.263 -0.379 2.214 1.969 -0.331 0.049 N-N 7.630 7.266 7.620 7.431 0.190 7.231 7.578 0.389 0.200

NO+ N-N 4.381 4.518 4.846 4.577 0.269 4.445 4.770 0.401 0.132 N-O 2.471 2.646 1.927 2.466 -0.538 2.425 2.078 -0.498 0.041 O-O 7.148 6.836 7.227 6.958 0.269 6.794 7.152 0.433 0.164

He2 He-He 1.999 1.998 1.999 1.999 0.000 1.968 1.999 0.031 0.031 He-He' 0.003 0.004 0.003 0.003 0.000 0.003 0.003 0.000 0.000

Ar2 Ar-Ar 17.994 17.992 17.994 17.994 0.000 17.805 17.994 0.189 0.189 Ar-Ar' 0.012 0.015 0.013 0.013 0.000 0.012 0.013 0.001 0.000

LiH Li-Li 1.990 1.991 1.994 1.991 0.003 1.991 1.995 0.003 0.000 Li-H 0.197 0.242 0.205 0.211 -0.006 0.195 0.203 0.009 0.015 H-H 1.814 1.769 1.797 1.794 0.003 1.695 1.798 0.101 0.098

HF H-H 0.023 0.033 0.045 0.032 0.014 0.030 0.050 0.015 0.001 H-F 0.403 0.484 0.436 0.463 -0.027 0.442 0.426 -0.006 0.021 F-F 9.574 9.483 9.519 9.506 0.014 9.357 9.524 0.162 0.149

NaH Na-Na 9.997 10.020 10.023 10.006 0.017 10.005 10.028 0.018 0.001 Na-H 0.380 0.535 0.401 0.436 -0.034 0.390 0.390 0.012 0.046 H-H 1.623 1.446 1.576 1.559 0.017 1.466 1.582 0.110 0.093

HCl H-H 0.253 0.258 0.333 0.273 0.059 0.261 0.342 0.071 0.012 H-Cl 1.046 1.062 0.933 1.052 -0.119 1.011 0.915 -0.078 0.041 Cl-Cl 16.702 16.680 16.734 16.675 0.059 16.461 16.744 0.273 0.214

130

Table1-b. AIM-ESI for the different polyatomic molecules taken into study. When the ESI refers to the same atom, half of the value is given instead

Pair ( )BAHF ,δ ( )BALYPB ,3δ ( )BA,δ ( )BA,σσδ ( )BA,'σσδ ( )BAA ,δ ( )BAF ,δ Cum. Aδδ σσ − NH3 N-N 6.687 6.581 6.815 6.645 0.170 6.515 6.793 0.300 0.130

N-H 0.897 0.919 0.784 0.897 -0.113 0.870 0.798 -0.086 0.027 H-H 0.188 0.199 0.244 0.199 0.045 0.192 0.253 0.052 0.007 H-H' 0.019 0.021 0.034 0.022 0.011 0.019 0.018 0.015 0.004

CH4 C-C 3.868 3.982 4.312 4.022 0.290 3.948 4.238 0.364 0.074 C-H 0.982 0.984 0.821 0.966 -0.145 0.942 0.858 -0.120 0.025 H-H 0.485 0.455 0.520 0.463 0.057 0.449 0.530 0.071 0.014 H-H' 0.044 0.044 0.054 0.044 0.010 0.039 0.035 0.015 0.004

H2O O-O 8.609 8.418 8.538 8.476 0.061 8.321 8.541 0.217 0.155 O-H 0.628 0.703 0.615 0.676 -0.061 0.651 0.612 -0.037 0.025 H-H 0.064 0.083 0.107 0.080 0.027 0.077 0.114 0.030 0.003 H-H' 0.007 0.010 0.019 0.011 0.008 0.008 0.008 0.011 0.003

CO2 C-C 2.234 2.408 2.496 2.368 0.128 2.321 2.467 0.175 0.047 C-O 1.152 1.408 1.125 1.254 -0.128 1.237 1.155 -0.112 0.016 O-O 8.566 8.198 8.484 8.404 0.081 8.245 8.486 0.239 0.159 O-O' 0.330 0.381 0.285 0.318 -0.033 0.313 0.252 -0.028 0.005

B2H6 B-B 2.113 2.208 2.234 2.169 0.065 2.149 2.216 0.084 0.019 H-H 1.243 1.107 1.218 1.192 0.027 1.158 1.229 0.061 0.034 Hb-Hb 1.095 0.962 1.084 1.048 0.036 1.011 1.090 0.073 0.038 B-H 0.490 0.602 0.497 0.536 -0.039 0.520 0.504 -0.023 0.016 B-Hb 0.274 0.349 0.295 0.307 -0.012 0.296 0.294 -0.001 0.012 Hb-Hb’ 0.222 0.184 0.171 0.196 -0.025 0.197 0.179 -0.026 0.000 B-B' 0.052 0.100 0.041 0.061 -0.020 0.066 0.054 -0.025 -0.005 H-H’ 0.128 0.110 0.112 0.117 -0.005 0.113 0.103 -0.002 0.004 H-Hb 0.112 0.096 0.096 0.102 -0.005 0.100 0.092 -0.003 0.002 H-H'cis 0.018 0.025 0.020 0.014 0.005 0.010 0.009 0.010 0.005 H-H'trans 0.014 0.017 0.017 0.017 0.000 0.014 0.012 0.003 0.003 B-H'vic 0.010 0.010 0.011 0.016 -0.004 0.019 0.016 -0.008 -0.004

C6H6 C-C 3.931 3.952 4.256 4.017 0.238 3.923 4.190 0.333 0.094 H-H 0.457 0.438 0.487 0.448 0.039 0.440 0.501 0.047 0.008 C-H 0.974 0.964 0.852 0.954 -0.102 0.946 0.879 -0.093 0.009 C-Co 1.390 1.391 1.175 1.360 -0.185 1.345 1.230 -0.170 0.015 C-Cm 0.073 0.073 0.054 0.044 0.010 0.070 0.058 -0.016 -0.026 C-Cp 0.099 0.103 0.071 0.102 -0.031 0.093 0.075 -0.022 0.010 H-Ho’ 0.005 0.006 0.016 0.011 0.005 0.005 0.005 0.011 0.005 H-Hm’ 0.001 0.002 0.006 0.004 0.002 0.001 0.001 0.004 0.002 H-Hp’ 0.000 0.000 0.004 0.002 0.002 0.000 0.000 0.004 0.002 C-Ho 0.048 0.049 0.046 0.043 0.003 0.046 0.040 0.001 -0.003 C-Hm 0.008 0.010 0.009 0.009 0.000 0.008 0.007 0.001 0.001 C-Hp 0.005 0.005 0.007 0.006 0.002 0.005 0.004 0.003 0.001

Mg32- Mg-Mg 11.563 11.533 * * * * * * *

Mg-Mg' 0.437 0.467 * * * * * * * * Inaccurate PROAIM integration (ΔN = 1.1e)

131

Table2-a. Fuzzy atom-ESI for the different diatomic molecules taken into study. When the ESI refers to the same atom, half of the value is given instead.

Pair ( )BAHF ,δ ( )BALYPB ,3δ ( )BA,δ ( )BA,σσδ ( )BA,'σσδ ( )BAA ,δ ( )BAF ,δ Cum. Aδδ σσ − H2 H-H 0.500 0.500 0.567 0.500 0.067 0.489 0.570 0.078 0.011 H-H' 1.000 1.000 0.865 1.000 -0.135 0.951 0.860 -0.086 0.049

N2 N-N 5.441 5.447 5.805 5.520 0.285 5.369 5.717 0.436 0.152 N-N' 3.118 3.107 2.390 2.960 -0.569 2.912 2.566 -0.521 0.048

F2 F-F 8.201 8.221 8.345 8.226 0.120 8.108 8.350 0.237 0.117 F-F 1.599 1.558 1.310 1.549 -0.239 1.460 1.300 -0.150 0.089

LiF Li-Li 2.453 2.435 2.566 2.479 0.087 2.439 2.530 0.127 0.040 Li-F 2.245 2.181 1.981 2.154 -0.173 2.145 2.053 -0.164 0.009 F-F 7.303 7.384 7.453 7.367 0.087 7.237 7.417 0.216 0.130

CO C-C 4.707 4.678 4.993 4.781 0.212 4.630 4.929 0.364 0.152 C-O 2.832 2.764 2.256 2.680 -0.424 2.660 2.385 -0.403 0.020 O-O 6.461 6.557 6.750 6.539 0.212 6.384 6.686 0.367 0.155

CN- C-C 5.047 5.032 5.372 5.124 0.248 4.962 5.300 0.409 0.161 C-N 3.076 3.048 2.426 2.921 -0.496 2.874 2.570 -0.448 0.048 N-N 5.878 5.921 6.203 5.955 0.248 5.787 6.131 0.415 0.167

NO+ N-N 5.024 4.997 5.396 5.111 0.285 4.963 5.306 0.434 0.149 N-O 3.027 2.993 2.293 2.863 -0.571 2.824 2.474 -0.531 0.040 O-O 5.949 6.010 6.311 6.026 0.285 5.878 6.220 0.433 0.148

He2 He-He 1.926 1.930 1.929 1.929 0.001 1.899 1.929 0.030 0.029 He-He' 0.143 0.134 0.142 0.143 -0.001 0.139 0.141 0.002 0.003

Ar2 Ar-Ar 17.585 17.589 17.593 17.588 0.004 17.410 17.592 0.183 0.179 Ar-Ar' 0.877 0.873 0.814 0.823 -0.009 0.802 0.815 0.013 0.021

LiH Li-Li 2.338 2.321 2.383 2.329 0.054 2.313 2.388 0.070 0.016 Li-H 0.969 0.961 0.856 0.965 -0.109 0.897 0.848 -0.041 0.068 H-H 0.693 0.718 0.760 0.705 0.055 0.676 0.764 0.085 0.030

HF H-H 0.292 0.286 0.373 0.309 0.065 0.288 0.364 0.085 0.020 H-F 1.410 1.382 1.246 1.375 -0.129 1.352 1.264 -0.106 0.023 F-F 8.298 8.332 8.381 8.317 0.065 8.189 8.372 0.192 0.128

NaH Na-Na 10.296 10.255 10.339 10.274 0.066 10.258 10.345 0.081 0.016 Na-H 0.952 0.922 0.805 0.936 -0.131 0.849 0.793 -0.044 0.087 H-H 0.753 0.823 0.856 0.790 0.066 0.753 0.862 0.103 0.037

HCl H-H 0.274 0.274 0.344 0.287 0.057 0.273 0.349 0.071 0.014 H-Cl 1.150 1.139 1.017 1.131 -0.113 1.091 1.006 -0.074 0.040 Cl-Cl 16.576 16.588 16.639 16.583 0.057 16.370 16.645 0.270 0.213

132

Table2-b. Fuzzy-atom-ESI for the different polyatomic molecules taken into study. When the ESI refers to the same atom, half of the value is given instead

Pair ( )BAHF ,δ ( )BALYPB ,3δ ( )BA,δ ( )BA,σσδ ( )BA,'σσδ ( )BAA ,δ ( )BAF ,δ Cum. Aδδ σσ − NH3 N-N 5.559 5.547 5.791 5.588 0.204 5.469 5.736 0.323 0.119

N-H 1.084 1.090 0.926 1.062 -0.136 1.042 0.963 -0.116 0.020 H-H 0.338 0.334 0.400 0.346 0.054 0.331 0.404 0.069 0.015 H-H' 0.059 0.061 0.077 0.064 0.014 0.057 0.055 0.020 0.007

CH4 C-C 4.160 4.189 4.472 4.215 0.256 4.134 4.399 0.338 0.082 C-H 0.985 0.984 0.839 0.967 -0.128 0.944 0.875 -0.105 0.023 H-H 0.410 0.399 0.460 0.409 0.050 0.396 0.468 0.064 0.013 H-H' 0.044 0.046 0.056 0.047 0.009 0.042 0.038 0.014 0.005

H2O O-O 6.984 6.950 7.120 6.984 0.136 6.852 7.089 0.268 0.132 O-H 1.184 1.198 1.029 1.166 -0.136 1.146 1.061 -0.116 0.020 H-H 0.288 0.290 0.361 0.303 0.058 0.287 0.359 0.074 0.016 H-H' 0.072 0.074 0.100 0.080 0.021 0.070 0.071 0.030 0.010

CO2 C-C 3.576 3.678 4.029 3.744 0.286 3.613 3.925 0.417 0.131 C-O 2.375 2.382 1.993 2.279 -0.286 2.290 2.098 -0.297 -0.011 O-O 6.736 6.636 6.895 6.736 0.159 6.597 6.857 0.298 0.139 O-O' 0.202 0.285 0.195 0.227 -0.032 0.213 0.167 -0.017 0.015

B2H6 B-B 3.540 3.570 3.798 3.598 0.200 3.523 3.747 0.275 0.075 H-H 0.406 0.394 0.420 0.456 -0.036 0.441 0.423 -0.021 0.015 Hb-Hb 0.268 0.253 0.296 0.263 0.033 0.255 0.305 0.041 0.007 B-H 0.943 0.935 0.826 0.929 -0.103 0.905 0.847 -0.080 0.023 B-Hb 0.461 0.458 0.444 0.402 0.042 0.393 0.455 0.051 0.009 Hb-Hb’ 0.018 0.020 0.025 0.033 -0.008 0.032 0.026 -0.007 0.002 B-B' 0.840 0.873 0.686 0.808 -0.122 0.816 0.750 -0.129 -0.007 H-H’ 0.034 0.035 0.028 0.021 0.006 0.018 0.015 0.010 0.004 H-Hb 0.011 0.012 0.015 0.011 0.003 0.010 0.009 0.004 0.001 H-H'cis 0.001 0.002 0.010 0.006 0.004 0.001 0.001 0.009 0.004 H-H'trans 0.006 0.008 0.010 0.009 0.001 0.006 0.005 0.004 0.004 B-H'vic 0.034 0.038 0.033 0.032 0.000 0.033 0.028 0.000 0.000

C6H6 C-C 3.955 3.958 4.220 4.021 0.199 3.930 4.159 0.290 0.091 H-H 0.378 0.372 0.413 0.379 0.034 0.372 0.424 0.041 0.007 C-H 0.939 0.932 0.837 0.924 -0.087 0.914 0.860 -0.077 0.010 C-Co 1.437 1.437 1.265 1.411 -0.146 1.393 1.314 -0.128 0.018 C-Cm 0.105 0.105 0.078 0.077 0.001 0.100 0.086 -0.022 -0.023 C-Cp 0.098 0.102 0.072 0.099 -0.027 0.092 0.075 -0.020 0.007 H-Ho’ 0.004 0.006 0.014 0.009 0.004 0.005 0.004 0.009 0.005 H-Hm’ 0.001 0.002 0.005 0.003 0.002 0.001 0.001 0.004 0.002 H-Hp’ 0.000 0.000 0.003 0.002 0.002 0.000 0.000 0.003 0.002 C-Ho 0.055 0.057 0.053 0.052 0.001 0.054 0.048 -0.001 -0.002 C-Hm 0.008 0.010 0.009 0.009 0.000 0.008 0.007 0.001 0.001 C-Hp 0.005 0.005 0.007 0.005 0.002 0.005 0.004 0.002 0.001

Mg32- Mg-Mg 11.461 11.438 11.920 11.835 0.085 11.666 11.935 0.254 0.169

Mg-Mg' 0.540 0.563 0.747 0.832 -0.085 0.774 0.732 -0.027 0.059

133

Table3. CISD/6-31G(d) results for FLU and PDI aromaticity indexes from QTAIM and Fuzzy-Atom partitions.

FLU PDI QTAIM Fuzzy-Atom QTAIM Fuzzy-Atom HFa CISD HFa CISD HFa CISD HFa CISD

M1 0.000 0.000 0.001 0.000 0.105 0.094 0.099 0.092 M2 0.012 0.011 0.011 0.011 0.073 0.070 0.072 0.069

M3-A 0.024 0.021 0.021 0.020 0.059 0.058 0.059 0.058 M3-B 0.007 0.006 0.005 0.008 0.070 0.066 0.068 0.065 M8-A 0.005 0.005 0.005 0.006 0.082 0.079 0.081 0.078 M8-B 0.025 0.023 0.022 0.023 0.043 0.042 0.044 0.043 M9-A 0.013 0.012 0.011 0.012 0.070 0.067 0.069 0.066 M9-B 0.045 0.043 0.040 0.041 - - - -

M11-A 0.008 0.007 0.008 0.007 0.088 0.085 0.087 0.083 M11-B 0.048 0.046 0.043 0.049 - - - M10-A 0.022 0.019 0.022 0.017 0.080 0.078 0.079 0.076 M10-B 0.071 0.067 0.063 0.061 - - - - M12 0.001 0.001 0.001 0.003 0.097 0.091 0.100 0.093 M13 0.005 0.007 0.000 0.006 0.089 0.085 0.098 0.093 M14 0.013 0.021 0.001 0.009 0.075 0.075 0.095 0.090

M15-A 0.015 0.015 0.011 0.011 0.072 0.069 0.073 0.069 M15-B 0.017 0.020 0.013 0.016 0.071 0.068 0.072 0.069 M16 0.091 0.088 0.058 0.067 0.007 0.007 0.011 0.011 M17 0.089 0.087 0.066 0.069 0.019 0.020 0.026 0.027 M18 0.084 0.082 0.069 0.065 0.014 0.014 0.019 0.018 M19 0.078 0.075 0.064 0.060 0.031 0.076 0.034 0.033

aFrom reference 46

134

4.7.- Aromaticity Analyses by Means of the Quantum

Theory of Atoms in Molecules.

135

Aromaticity Analyses by Means of the

Quantum Theory of Atoms in Molecules

Eduard Matito,ϒ Jordi Poater,*,§ and Miquel Solà*,ϒ

Institut de Química Computacional and Departament de Química, Universitat de Girona, Campus de Montilivi, 17071 Girona, Catalonia, Spain, Afdeling Theoretische

Chemie, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands

[email protected] [email protected]

An overview of recent aromaticity indicators defined in the framework of the

Quantum Theory of Atoms in Molecules (QTAIM) is presented. Two new indexes

based on the calculation of the QTAIM delocalization indexes have been defined: the

para-delocalization index (PDI) and the aromatic fluctuation index (FLU). We give a

short review of recent calculations of local aromaticity in a series of polycyclic

aromatic hydrocarbons employing these new defined indexes with special emphasis

on the strengths and weaknesses of these novel as well as previously defined

descriptors of aromaticity.

* To whom correspondence should be addressed. Phone: +34-972.418.912. Fax: +34-972.418.356. ϒ Institut de Química Computacional and Departament de Química, Universitat de Girona. § Afdeling Theoretische Chemie, Vrije Universiteit.

137

1 Introduction

Aromaticity is a concept formulated to account for the unusual properties of an

important class of organic and inorganic molecules: the aromatic compounds.1,2

Aromaticity is currently enjoying a resurgence of interest as its scope of application

has expanded from the organic to the inorganic realm of chemistry.3 The new class of

all-metal and inorganic aromatic compounds synthesized has prompted new

definitions of aromaticity applicable to both the classic and the novel aromatic

molecules. According to the most recent definition of Schleyer and coworkers,4

aromaticity is a manifestation of electron delocalization in closed circuits, either in

two or three dimensions, which results in energy lowering, often quite substantial, and

a variety of unusual chemical and physical properties. These include a tendency

toward bond length equalization, unusual reactivity, and characteristic spectroscopic

features as well as distinctive magnetic properties related to strong induced ring

currents in aromatic systems.5

Despite aromaticity being a not directly observable quantity, its importance as

a central concept in chemistry has not diminished since the discovery of benzene by

Michael Faraday in 1825.6 Yet its quantification has proved remarkably elusive.

Nearly everyone agrees that there is not a well-established method to quantify the

aromatic character of molecules yet. The evaluation of aromaticity is usually done

indirectly by measuring some physicochemical property that reflects the aromatic

character of molecules. Thus, most aromaticity indicators are based on the classical

aromaticity criteria, namely, structural, magnetic, energetic, and reactivity-based

measures,2 although more recently, new ways to quantify the aromaticity based on

electronic properties of the molecules have been devised (for a recent review see

138

reference 7). Although the existence of many aromaticity descriptors complicates

matters, it is also true that the use of differently-based aromaticity criteria is

recommended for aromaticity analysis due to its multidimensional character.8,9

The structure-based measures of aromaticity rely on the idea that important

manifestations of aromaticity are equalization of bond lengths and symmetry. Among

the most common structural-based indices of aromaticity, one of the most effective is

the harmonic oscillator model of aromaticity (HOMA) index,10 defined by

Kruszewski and Krygowski as:

( )∑=

−−=n

iiopt RR

nHOMA

1

21 α , (1)

where n is the number of bonds considered and α is an empirical constant fixed to

give HOMA = 0 for a model nonaromatic system and HOMA = 1 for a system with

all bonds equal to an optimal value Ropt, assumed to be achieved for fully aromatic

systems. Ri stands for a running bond length. The HOMA value can be splitted into

the energetic (EN) and geometric (GEO) contributions according to the relation:11

( ) ( )∑ −−−−=−−=i

iopt RRn

RRGEOENHOMA 2211 αα . (2)

The GEO contribution measures the decrease/increase in bond length alternation

while the EN term takes into account the lengthening/shortening of the mean bond

lengths of the ring.

Magnetic indices of aromaticity are based on the π-electron ring current that is

induced when the system is exposed to external magnetic fields. Probably the most

139

widely used magnetic-based indicator of aromaticity is the nucleus independent

chemical shift (NICS), proposed by Schleyer and co-workers.4,12 It is defined as the

negative value of the absolute shielding computed at a ring center or at some other

interesting point of the system. Rings with large negative NICS values are considered

aromatic. The more negative the NICS values, the more aromatic the rings are.

Finally, energetic-based indices of aromaticity make use of the fact that

conjugated cyclic π-electron compounds are more stable than their chain analogues.

The most common used energetic measure of aromaticity is the aromatic stabilization

energy (ASE), calculated as the reaction energy of a homodesmotic reaction.13

Although aromaticity plays a prominent role in current chemistry, aromaticity

measures based upon electronic descriptors are scarce. Among them, we can mention

the HOMO-LUMO gap, the absolute and relative hardness, the electrostatic potential,

and the polarizability.14 None of them refer directly to the electronic delocalization in

aromatic species. If we take into account that electron delocalization is the main

responsible for aromaticity manifestations, this fact is surprising. The electron

structure of the molecules is obviously strongly connected with aromaticity, and

QTAIM provides a wide set of concepts to characterize electronic structure: atomic

populations and charges, electron localization and delocalization (through the so-

called LIs and DIs, vide infra), BCP, RCP (one should bear in mind that aromaticity

manifests on molecules possessing ring structures), among others.15 In this sense, the

QTAIM provides a formidable scheme for the construction of aromaticity measures.

140

In the last five years, we have concentrated our research efforts to quantify

aromaticity by measuring the degree of electron delocalization in molecules using the

set of tools that the QTAIM theory provides for this task.7 In this chapter, we briefly

review the new aromaticity criteria defined by us that make use of the QTAIM

concepts. The present chapter is organized as follows. Section 2 covers the definition

of electron delocalization in the QTAIM theory; section 3 analyzes the electron

delocalization in benzene, the quintaessential aromatic molecule; section 4 gives the

definition of the new aromatic indexes based on the analysis of electron

delocalization; section 5 discusses several applications of the new defined indexes to

the analysis of aromaticity; and finally, in section 6, we summarize the main

conclusions.

2 The Fermi hole and the Delocalization Index

Most methods used to determine the degree of electronic

localization/delocalization in molecules employ the two-electron density, ( )21 , rr rrΓ ,

also named second-order density or pair density.16 This is the simplest quantity that

describes the pair behavior and it is usually interpreted as the probability density of

finding two electrons simultaneously at positions 1rr and 2r

r , regardless the position of

the other N-2 electrons. This function can be splitted into an uncorrelated pair density

and a part that gathers all exchange and correlation effects:

( ) ( ) ( ) ( )212121 ,, rrrrrr xcrrrrrr

Γ+=Γ ρρ . (3)

The uncorrelated component of the pair density, given by the product )()( 21 rr rr ρρ ,

provides the probability of finding simultaneously two independent electrons at

141

positions 1rr and 2r

r . The difference between ),( 21 rr rrΓ and )()( 21 rr rr ρρ is known as the

exchange-correlation density,17 ),( 21 rrXCrr

Γ , which is a measure of the degree to which

density is excluded at 2rr because of the presence of an electron at 1r

r . Integration of

the exchange-correlation density of a given molecule through all space yields the

negative of the total number of electrons in this molecule. The Fermi hole density is

another two-electron function directly connected to the exchange-correlation density.

It is defined as:

( ) ( )( )1

2121

,,

rrr

rr xcxc r

rrrr

ρρ

Γ= , (4)

and it represents the decrease/increase, with respect to the uncorrelated probability

density, in the probability of finding an electron in position 2rr when a reference

electron is fixed at position 1rr . In two landmark works,18,19 Bader and Stephens

showed that the extent of localization or delocalization of an electron at 1rr is

determined by the corresponding localization/delocalization of its Fermi hole. A

localized Fermi hole indicates the presence of localized electronic charge in the

position of the electron of reference and vice versa. For this reason, Fermi density

hole maps have been widely used to analyze the electronic

localization/delocalization.20

Eq. (4) shows that the exchange-correlation density is nothing but the Fermi

hole density weighted by the density of the reference electron. Bader and

coworkers19,21 used this quantity to define the localization and delocalization indexes

(LIs and DIs) from the double integration of the exchange-correlation density over the

atomic basins defined within the QTAIM theory:

142

( ) ( )∫ ∫ Γ−=A A

xc rdrdrrA 2121 , rrrrλ (5)

( ) ( )∫ ∫ Γ−=B A

xc rdrdrrBA 2121 ,2, rrrrδ . (6)

The term δ(A,B) gives a quantitative measure of the number of electrons delocalized

or shared between atomic basins A and B, while λ(A) is a measure of the average

number of electrons localized on basin A. The following sum rule can be easily

demonstrated:

∑≠

+=AB

BAAAN ),(21)()( δλ . (7)

Eq. (7) proves that the total number of electrons belonging to a given basin can be

exactly partitioned into its localized ( )(Aλ ) and delocalized ( ∑≠AB

BA ),(21 δ ) parts. In

addition, one can define the global delocalization or valence of atomic basin A as:

[ ])()(2),()( AANBAAVAB

λδ −== ∑≠

. (8)

As shown in the next sections of this chapter, we have employed these quantities to

define new electronically-based aromaticity descriptors.

3 Electron delocalization in aromatic systems

In 1996, Bader and coworkers22 in a seminal work analyzed the electron

delocalization in benzene, the archetypal aromatic molecule. They observed by means

of contour maps of the Fermi-hole density in conjugated species that the interatomic

delocalization of the π-electrons on a given atom, in general, decreases with the

distance of the second atom from the one in question. However, for benzene, there is

143

significantly greater delocalization of the π density into the basins of the para-related

as compared to the meta-related carbons, despite the distance being shorter in the

latter. These data are in concordance with the energy ordering of the principal

resonance structures of benzene: the two Kekulé structures are the most important,

followed by the Dewar structure connecting para-related carbon atoms that is, in turn,

more relevant than those connecting meta-related ones. In order to corroborate what

happens in benzene, Bader and coworkers22 studied the effect of geometrical

distortions on the delocalization of the π electrons in benzene by considering a

symmetrical distortion (S), in which each equilibrium C-C bond length of 1.42 Å was

increased by 0.06 Å in an a1g stretching mode, and a b2u unsymmetrical one (U)

obtained by alternately increasing and decreasing the bond lengths to 1.54 and 1.34 Å,

respectively. The results indicated that there is no significant change in the

delocalization of the π electrons for S, however for U it is seen how the delocalization

between para carbons largely decreases. The values of the DIs for benzene also

confirm the larger delocalization in para than in meta. In particular, Bader and co-

workers obtained the following DIs at the HF/6-31G* level of theory: δ(C,C’)para =

0.101 e and δ(C,C’)meta = 0.070 e. At the same level of theory, it is important to

recognize that Fulton and Mixon reported some years before almost identical values

for δ(C,C’)para and δ(C,C’)meta.23 Interestingly, these authors showed that the

δ(C,C’)para has a large π component (0.09 e), at variance with δ(C,C’)meta. The larger

DI found between para-related carbon atoms than between meta-related carbons has

been corroborated using larger basis sets and higher levels of calculation. For

instance, the CISD/6-311++G** value of δ(C,C’)para and δ(C,C’)meta in benzene are

xxx and yyy e, respectively. Therefore, this result is not an artifact of the method used

144

and, consequently, it has sound physical foundation for being the basis for the

definition of our PDI index of aromaticity (vide infra).

4 Aromaticity electronic criteria based on QTAIM

Sondheimer defined as aromatic those molecules showing a "measurable

degree of cyclic delocalization of a π-electron system".24 Likewise, Schleyer and

coworkers25 considered the aromaticity "associated with cyclic arrays of mobile

electrons with favorable symmetries” while “the unfavorable symmetry properties of

antiaromatic systems lead to localized, rather than to delocalized electronic

structures". These definitions stress the existence of a direct connection between

aromaticity and electron delocalization. As said in the introduction section,

unfortunately, neither the delocalization nor the aromaticity has a quantitative

descriptor universally accepted. This is the reason for the continuous search for new

aromaticity indexes, and the exhaustive revision of the existing ones in this quest for a

less ambiguous index which, at the same time, agrees with the most elementary

chemical grounds on aromaticity. Recently, two aromaticity measures based upon DIs

have been put forward, the para-DI (PDI) and the fluctuation aromatic index (FLU)

measures. In this context it is worth to mention, first, the work of Matta et al.26 who

also attempted to construct an HOMA-like index from QTAIM by substituting the

bond length by the total electron delocalization, and second, the use by Giambiagi et

al.27 and Bultink et al.28 of the n-center electron DIs as descriptors of aromaticity.

145

4.1 The para-delocalization index (PDI)

As said before, Bader and coworkers reported that the delocalization in

benzene is greater for para-related carbons (para-DI) than for meta-related ones

(meta-DI). With this idea in mind, Poater et al.29 decided to undertake a study to

validate the averaged para-DI in 6-MRs as a measure of local aromaticity. The PDI is

a specific measure of aromaticity for 6-MR, where three para-related positions exist,

namely (1,4), (2,5) and (3,6):

( ) ( ) ( )3

6,35,24,1 δδδ ++=PDI , (9)

where δ(A,B), the DI, is defined as in equation (6).

The correlation between PDI and other measures of aromaticity (HOMA,

magnetic susceptibility and NICS) for the series of polycyclic aromatic hydrocarbons

(PAHs) given in Scheme 1 (M1-M11) served to validate this measure as a reliable

index of aromaticity.29

Scheme 1 here

The main shortcoming of PDI is that it can only be applied to 6-MR. This

disadvantage was somehow overcome by the introduction of ΔDI.29 ΔDI bases on the

intuitive idea of comparing the electron delocalization between formal single and

double bonds. For compounds with a formal Lewis structure, the difference between

the DI for double and single bonds in a given ring was suggested as a measure of

aromaticity. The lower the difference (the lower ΔDI) the closer electron

146

delocalization in double and single bounds; an indication that both DI for single and

double bonds exhibit typical delocalization of aromatic compounds. On the other

hand, the higher the difference, the closer the structure to the Lewis one, thus showing

a situation with quite localized electrons, which is known to prevent aromaticity. This

relationship was especially useful for 5-MR species, namely the series C4H4-X

(X=CH-, NH, S, O, SiH-, PH, CH2, AlH, SiH+, BH, CH+), for which a reasonable

agreement between ΔDI with NICS values was found. Nonetheless, ΔDI needs of a

clear Lewis structure to compare double and single bounds, and its application to

different sized rings is less clear. In this sense, the quest for a new aromaticity index

based on DIs, powerful enough to deal with rings of any size, led to the aromatic

fluctuation index (FLU).

4.2 The aromatic fluctuation index (FLU)

Sondheimer’s definition24 suggests using cyclic electron delocalization as a

measure of aromaticity. While PDI focused in para-related carbon atoms, and ΔDI in

a couple of bonds in the ring, there was no attempt to construct an aromaticity index

by examining the DI of all bonded pairs in a given ring. It is worth mentioning,

however, that Bird30 has compared Gordy bond orders of all bonded pairs in a given

ring to define a measure of aromaticity that has some resemblance to our FLU index.

The FLU index was constructed following the HOMA philosophy, i.e., measuring

divergences (DI differences for each single pair bonded) from aromatic molecules

chosen as a reference; cf. Eq. (1). The following formula was given for FLU,31

( ) ( )( )∑

− ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

RING

BA ref

ref

BABABA

AVBV

nFLU

2

,,,

)()(1

δδδα

, (10)

147

where the summation runs over all adjacent pairs of atoms around the ring, n is equal

to the number of atoms of the ring, V(A) is the global delocalization of atom A given

in Eq. (8), ( )BA,δ and ( )BAref ,δ are the DI values for the atomic pairs A and B and

its reference value, respectively, and

⎩⎨⎧

≤−>

=)()(1)()(1

AVBVAVBV

α . (11)

The second factor in Eq. (10) measures the relative divergences with respect to a

typical aromatic system, and the first factor in Eq. (10) penalizes those with highly

localized electrons. The reference DI values for C-C and C-N bonds were obtained

from benzene (1.4 e) and pyridine (1.2 e), respectively at the HF/6-31G* level of

theory. In a forthcoming work we will also give the reference parameter for B-N

bonds as that obtained from borazine (0.77 e).32

As one can easily see, unlike PDI, FLU can cope with rings of different sizes

and it provides global and local measures of aromaticity. However, the weakness of

FLU, as it is also the case of HOMA, is the need for typical aromatic systems as

reference. In addition, one must be aware that FLU is actually measuring the relative

electronic divergence of a given ring in a molecule with respect to a molecule chosen

as a reference. Therefore, it is worth noticing that FLU (the same holds for HOMA)

must be applied with care to the study of chemical reactivity, since it fails to

recognize those cases where aromaticity is enhanced upon deviation from the

equilibrium geometry (ref. 33 and vide infra).

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4.3 The π-fluctuation aromatic index (FLUπ)

In order to overcome the need for reference parameters which prevents the

FLU to be straightforwardly applied to any molecule, another index based on QTAIM

was designed, the FLUπ; this measures the divergence of π-delocalization with respect

to its average:31

( )∑− ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

RING

BA av

avBAAVBV

nFLU

2

,)()(1

δδδπ

α

π

ππ , (12)

with avδ being the average value of the π-DI for the bonded pairs in the ring, and the

other quantities refer to the aforementioned ones calculated by only using π-orbitals.

Therefore, FLUπ can only be exactly calculated for planar molecules, where DI can be

(exactly) decomposed into its σ and π contributions. Nevertheless, orbital localization

schemes can always be used to obtain approximate π-DI for non-planar molecules.

The correlation between FLUπ and FLU is shown to be excellent for a series of

organic compounds (cf. Table 1 and Scheme 1). There is no reason to expect it to be

always like this. Indeed, FLUπ is measuring the degree of homogeneous

delocalization in a π-system, whereas FLU is measuring the degree of similarity with

respect to reference aromatic molecules. Hence, for organic species where aromaticity

comes from the delocalization of the π-system, one expects both indexes to reproduce

the same trends, while some differences can arise for inorganic species, where the

aromaticity of the system is not exclusively driven by the π-electron delocalization.

149

Table 1 here

In general, the correlation between PDI, FLU, and FLUπ with other

aromaticity indexes is reasonably good (cf. Table 2) for the series of compounds

given in Scheme 1, perhaps with the exception of NICS values, whose correlation is

lesser as one would expect. Therefore, one can see how QTAIM provides a

formidable scheme to provide aromaticity measures. It is also worth noticing that

some other molecular decompositions, like Fuzzy-Atom34 can provide aromaticity

measures in excellent agreement with those derived from QTAIM DIs.35,36

Table 2 here

5 Applications of QTAIM to Aromaticity Analysis

In the present section we present some applications of the QTAIM to the

analysis of aromaticity in buckybowls, fullerenes, substituted and unsubstituted

PAHs, and in the simplest Diels-Alder reaction.

5.1 Aromaticity of buckybowls and fullerenes

Different studies have attributed an ambiguous aromatic character to

fullerenes,37 which are considered aromatic by some criteria and nonaromatic by

others.38-40 For instance, magnetic properties show extensive cyclic π-electron

delocalization and substantial ring currents; however, against their aromaticity, these

systems are very reactive. We have analyzed the local aromaticity of fullerenes and

150

buckybowls,41,42 which also display fullerene-like physicochemical properties, in

order to clarify their aromatic character, a property closely connected to their

reactivity. Table 3 encloses the HF/6-31G*//AM1 values of NICS, PDI, and HOMA

aromaticity measures, together with the average of the pyramidalization angles, for a

series of planar and bowl-shaped PAHs and fullerenes, i.e. from benzene to

buckminsterfullerene (C60), represented in Scheme 2.

Table 3 and Scheme 2 here

The three local aromaticity criteria almost give the same trend for the different

rings of the PAHs studied. A clear aromatic character is assigned to the hexagonal

rings of benzene, naphthalene, and C20H10 and to the outer 6-MRs of C26H12 (C) and

C30H12 (C and D), whereas the inner 6-MRs of C26H12 (A), C30H12 (A), and C60 are

found to be moderately aromatic. In contrast to the significant local aromaticities of 6-

MRs, the 5MRs present antiaromatic character.29 It is also worth noticing a certain

convergence in the local aromaticity of the inner 6-MRs when going from the most

aromatic benzene to the partially aromatic C60. Unexpectedly, for the bowl-shaped

PAHs, the most pyramidalized outer rings possess the largest local aromaticities.

Another interesting fullerene is C70,41 whose structure arises from the insertion

of an equatorial belt of five 6-MRs to C60 (see Scheme 3), thus causing it to present

different reactivities and local aromaticities.43 Experimentally, the pole is more

reactive than the equatorial belt, which should imply a lower aromaticity of the

former. The HF/6-31G*//AM1 aromaticity criteria in Table 3 confirm that ring E, as

expected, presents the largest aromaticity, followed by ring B, located in the pole and,

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unexpectedly, by ring D, even though this is located in the equatorial belt. By

comparison with C60 results, rings B and E of C70 are more aromatic than 6-MRs of

C60, in line with the accepted greater aromaticity of C70,38 and despite it is more

reactive.39 On the other hand, NICS values of C70 are surprisingly high compared to

C60, even though they present similar geometric environments and pyramidalization

angles.

Scheme 3 here

5.2 Substituent effect on aromaticity

Benzene is considered to be the archetype of aromaticity, fulfilling all criteria

attributed to this property.2 This molecule has been used as the reference for the

proposal of quantitative descriptors of the substituent effects, that is, the Hammett

substituent constants.44 We have tried to establish a relationship between the

substituent effect and the aromaticity of a series of monosubstituted derivatives of

benzene.45 Table 4 contains the HF/6-31+G*//B3LYP/6-311+G** NICS, and the

B3LYP/6-311G**//B3LYP/6-311+G** HOMA and PDI aromaticity measures

together with different substituent constants (see reference 45 for the corresponding

explanation). It is seen that, even though the nature of the substituents varies in a

considerable extent along the series (σp varying from –0.66 for a strongly electron-

donating NH2 substituent to 1.91 for a strongly electron-accepting NN+ one), no

apparent aromaticity changes are observed. This proves the high resistance of the π-

electron structure of benzene, in agreement with its preference for substitution

reactions instead of addition ones. In addition, PDI is the only index that gives a direct

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correlation between aromaticity and the substituent constants (see Figure 1), proving

to be a good descriptor for the changes of π-electron delocalization in substituted

benzenes.45

Table 4 and Figure 1 here

The next study analyzes the substituent effect in a series of carbazole

derivatives (see Scheme 4) and tries to find quantitative predictions for the reactivity

of these systems as a function of the substituent by measuring different local

aromaticity criteria.46 As it can be seen in Figure 2 for the substituted ring, the results

for the three aromaticity criteria are scattered over a narrow range of values. In

addition, there is a clear divergence in the ordering of the different systems by local

aromaticity values given by the B3LYP/6-31++G** values of NICS, HOMA, and

PDI. Therefore, at variance with previous aromaticity analyses that presented a

relatively good agreement among the different aromaticity criteria, in this case one

must be very cautious with the results, not being possible to have a definite answer

about the relative aromaticity of these rings. As in the previous case, substituents

slightly affect the π-electron structure of the aromatic ring. Finally, it is worth saying

that the different trends found by these criteria are not completely unexpected, since

we are considering descriptors based on different physical properties.

Scheme 4 and Figure 2 here

The last study on this subject has consisted on analyzing how metal cations

and ionization affect the aromaticity of the Watson-Crick guanine-cytosine base pair

(GC).47 Table 5 contains the B3LYP aromaticity results (HOMA, PDI, NICS, and

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FLU) for the 6- and 5MRs studied (see Figure 3). The H-bond formation in GC

implies a certain loss of π-charge in N1 and a gain in O6, respectively, thus increasing

the relevance of the resonance structure 2 (see Figure 3), which favors the

intensification of the aromatic character of the guanine 6-MR. The increase of the

aromaticity of the guanine and cytosine 6-MRs due to the interactions with Cu+ and

Ca2+ is also attributed to the strengthening of H-bonding in the GC pair, which

stabilizes the charge separation resonance structure 2.48 This effect is stronger for the

divalent Ca2+ metal cation than for the monovalent Cu+. Meanwhile, the observed

reduction of aromaticity in the 5- and 6-MRs of guanine due to ionization or the

interaction with Cu2+ is caused by the oxidation process that removes one electron π,

disrupting the π-electron distribution.

Table 5 and Figure 3 here

5.3 Assessment of Clar’s aromatic π-sextet rule

The introduction of Hückel’s 4n + 2 rule allowed a better comprehension of

aromaticity,49 although it could only be applied to monocyclic conjugated systems.

This was solved later on by means of Clar’s model of the extra stability of 6n π-

electron benzenoid species.50 According to Clar’s rule, the Kekulé resonance structure

with the largest number of disjoint aromatic π-sextets, i.e. benzene-like moieties, is

the most important for the characterization of the properties of polycyclic aromatic

hydrocarbons (PAHs). Aromatic π-sextets are defined as six π-electrons localized in a

single benzene-like ring separated from adjacent rings by formal C-C single bonds.

For instance, the application of this rule to phenathrene indicates that the resonance

154

structure 2 (Scheme 5) is more important than the resonance structure 1; which is

translated into a major local aromaticity of outer rings than inner ones. The Clar

structure of a given PAH is the resonance structure having the maximum number of

isolated and localized aromatic π-sextets, with a minimum number of localized double

bonds. A PAH with more π-sextets is kinetically more stable than its isomer with

fewer. In addition, π-sextets are considered to be the most aromatic rings of PAHs.

Some PAHs present a single Clar structure (i.e. phenanthrene), whereas others present

several ones. For these latter, Clar’s rule cannot differentiate which of the

corresponding resonance structures is mainly responsible for the aromaticity of the

system. In this study we pretend to investigate whether three local aromaticity criteria,

PDI, HOMA, and NICS, give results consistent with the qualitative original Clar’s π-

sextet rule.51

Scheme 5 here

The PAHs taken under study are represented in Scheme 6, with Mol. 1 – 5

having a single Clar structure and Mol. 6 – 10 having several Clar valence structures,

also represented. The corresponding local aromaticity values, calculated at the

B3LYP/6-31G* level, can be found in Table 6. First, for systems presenting a single

Clar structure (Mol. 1 – 5), π-sextet rings present higher PDI values, higher HOMA,

and more negative NICS than non-π-sextet rings. Hence, all three aromaticity criteria

applied perfectly agree with the qualitative description given by Clar’s rule. Second,

for systems presenting several Clar structures, it is seen how the overall aromaticity of

the system given by PDI and HOMA agrees with the superposition of all possible Clar

structures. For instance, for Mol. 7, PDI and HOMA attribute a very similar

155

aromaticity to rings A and B, which proves the non-localizability of the π-sextet.

However, NICS attributes a much larger aromatic character to ring B than A, although

this is due to the claimed overestimation by NICS of the local aromaticity of inner

rings of PAHs, which will be commented later on.

Scheme 6 and Table 6 here

5.4 Aromaticity along the Diels-Alder reaction. The failure of some aromaticity

indexes.

This work33 analyzes the aromaticity along the Diels-Alder reaction between

1,3-butadiene and ethane to yield cyclohexene (see Scheme 7),52,53 which is often

taken as a prototype of a pericyclic concerted reaction. This reaction is characterized

by an aromatic TS,53,54 thus along the reaction path a peak of aromaticity around the

TS is expected. The trends of aromaticity along the path for the different criteria

applied, at the B3LYP/6-31G* level, can be found in Figure 4. Only the magnetic

NICS(1) and the electronic PDI criteria find the most aromatic point along the

reaction path around the TS of the reaction. On the other hand, both geometric

HOMA and electronic FLU consider cyclohexene as the most aromatic species in this

reaction. This latter trend is also given by the RSS (Root Summed Squares) of the best

fitted plane for atoms in the ring, which is an unambiguous measure of the molecular

planarity.33 It shows the product as the most planar species, thus in principle implying

a higher π-electron delocalization. The failure of RSS proves that not necessarily the

flatter a structure, the more aromatic. On the other hand, HOMA and FLU measure

variances of structural and electronic patterns around the ring, and might fail if they

156

are not applied in stable species, as it is the case of a reaction with major structural

and electronic changes. Therefore, the failure of some indices to detect the aromaticity

of the TS in the simplest Diels-Alder cycloaddition reinforces the idea of the

multidimensional character of aromaticity8 and the need for using several criteria to

quantify it.

Scheme 7 and Figure 4 here

5.5 Problematic cases for the NICS indicator of aromaticity

Pyracylene was previously observed to present a divergent aromatic character

depending on the measure used.29 Hence, whereas NICS (−0.1 ppm) found 6-MRs to

be nonaromatic, HOMA (0.755) and PDI (0.067 e) attributed a medium aromaticity to

them. By comparison to naphthalene and acenaphthylene aromaticities, we considered

that this divergence was due to an underestimation of NICS, and to prove this

hypothesis we analyzed the change of aromaticity of the 6-MRs by increasing the

pyramidalization of the pyracylene structure.55 The B3LYP/6-31G* PDI, FLU,

HOMA, and HOMO-LUMO gap in Table 7 show a slight decrease of aromaticity

with the increase of pyramidalization (FLU is invariable), as experimentally

expected.56 On the other hand, unexpectedly, both NICS(0) and magnetizability

measures show an increase of aromaticity with a higher distortion. In order to try to

understand the reason for the apparent failure of magnetic-based indices for this

system, the ring current density maps (at 0.9 Å above the ring plane) were analyzed.57

As can be seen in Figure 5 for planar pyracylene, dominant paratropic ring currents

circulating on pentagons are observed, more intense than the diatropic ring current

157

flowing on the naphthalene perimeter. However, with the increase of the distortion

(see Figure 6), whereas the diatropic ring current remains constant, the paratropic in

pentagons decreases up to a breaking point for θ = 40°. Therefore, magnetic-based

NICS(0) criterion fails for the 6-MRs of pyracyclene because of the strong paratropic

ring currents on pentagons. Finally, NICS at 1 Å above and below the center of the

ring, considered to better reflect the π-electron effects,58 were also calculated;

however also opposite tendencies were obtained.

Table 7, Figure 5, and Figure 6 here

The last study has consisted of analyzing the local aromaticity of three series

of benzenoid compounds, namely, the [n]acenes, [n]phenacenes, and [n]helicenes for

n = 1 – 9.59 Values are not included but the tendencies of B3LYP/6-31G* NICS, PDI,

and HOMA for each ring along the different series to see the changes when going

from linear acenes to angular phenacenes, and from these to non-planar angular

helicenes, are given in Figures 7 to 9. It is seen in Figure 7 that for [n]acenes, inner

rings are more aromatic than outer rings, even though the former are more reactive.

For [n]phenacenes (see Figure 8), all indicators show that the external rings are the

most aromatic; and from the external to the central rings, the local aromaticity varies

in a damped alternate way. Finally, HOMA and PDI indices indicate a small change

in aromaticity when going from [n]phenacenes to [n]helicenes (see Figure 9), despite

the loss of planarity in the latter species. NICS results show the same behavior for

systems with n ≤ 5; however, for larger systems NICS considers the terminal rings of

[n]helicenes clearly more aromatic than those of [n]phenacenes. This increase of local

aromaticity in terminal rings of [n]helicenes (n = 6 – 9) according to NICS has to be

attributed to magnetic couplings with neighboring rings placed above that alter the

158

expected trend in the local aromaticities. It is worth pointing out that for [n]acenes

with n = 6–9 it exists a controversy about their corresponding lowest-lying singlet

state.60 Thus, in a recent work, we also analyzed the aromaticity of this series of

[n]acenes at their diradical singlet state.35 It was found that, while inner rings have the

largest aromatic character in closed-shell singlet states, the outer rings become the

most aromatic for the diradical singlet states.

Figure 7, Figure 8, and Figure 9 here

6 Conclusions

One of the key aspects of aromatic compounds is the π-electronic

delocalization (but also σ- and even δ- in all-metal and inorganic aromatic species)

present in these molecules. In this Chapter, we have defined two new aromaticity

indexes founded on the evaluation of the electron delocalization in the framework of

the QTAIM, namely, the para-delocalization (PDI) and the aromatic fluctuation

(FLU) indexes. We have shown that theoretical studies of electron delocalization

using QTAIM-based tools have significantly improved our understanding of

aromaticity in fullerenes, substituted benzene derivatives, polycyclic aromatic

hydrocarbons, and chemical reactivity. We have also proved the usefulness of these

indexes when applied to the analysis of the particular behavior of NICS in systems

where this indicator of aromaticity fails. Indeed, the lack of a universally accepted

measure of aromaticity, its multidimensional character, and the limitations of almost

all descriptors of aromaticity stress the need for new aromaticity criteria as those

defined in this Chapter. In a given study over a series of compounds, one can safely

reach a definite conclusion about their aromaticity only when differently-based

159

indicators of aromaticity lead to the same results. For this reason, in our opinion, a

careful analysis of aromaticity in a set of molecules must be performed using

electronically-based descriptors like the present PDI or FLU indexes together with

geometrical-based indicators such as the HOMA index, magnetically-based measures

like NICS, and energetically-based descriptors such as ASEs.

Acknowledgments

The authors are grateful to Prof. Dr. Miquel Duran and Dr. Xavier Fradera for helpful

discussions. Financial help has been furnished by the Spanish Ministerio de

Educación y Ciencia (MEC) Project No. CTQ2005-08797-C02-01/BQU. E.M. and

J.P. thank the MEC for the doctoral fellowship no. AP2002-0581 and the DURSI for

the postdoctoral fellowship 2005BE00282, respectively. Excellent service by the

Centre de Supercomputació de Catalunya (CESCA) is gratefully acknowledged.

160

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166

Table 1. HF/6-31G(d) calculated HOMA, NICS (ppm), PDI (electrons), FLU values for the series of PAH in Scheme 1.(a)

EN GEO HOMA NICS(0) NICSzz(0) NICS(1) NICSzz(1) PDI FLU FLUπ Benzene M1 -0.001 0.000 1.001 -11.5 -17.2 -12.9 -32.5 0.105 0,000 0.000

Naphthalene M2 0.022 0.199 0.779 -10.9 -13.8 -12.6 -30.6 0.073 0,012 0.114 Anthracene M3-A 0.067 0.416 0.517 -8.7 -6.9 -10.7 -24.8 0.059 0,024 0.251

M3-B 0.045 0.072 0.884 -14.2 -21.9 -15.5 -38.4 0.070 0,007 0.024 Naphthacene M4-A 0.110 0.565 0.325 -6.7 -0.9 -9.0 -19.6 0.051 0,031 0.350

M4-B 0.073 0.153 0.774 -13.8 -20.4 -15.2 -37.2 0.063 0,011 0.071 Chrysene M5-A 0.011 0.130 0.859 -11.1 -13.7 -12.8 -30.7 0.079 0,008 0.067

M5-B 0.122 0.325 0.553 -8.2 -3.6 -10.6 -23.3 0.052 0,019 0.182 Triphenylene M6-A 0.003 0.068 0.930 -10.6 -11.8 -12.3 -29.3 0.086 0,003 0.026

M6-B 0.609 0.324 0.067 -2.6 13.9 -6.2 -9.4 0.025 0,027 0.181 Pyracylene M7-A 0.000 0.329 0.671 -4.9 6.9 -7.2 -13.3 0.067 0,014 0.128

M7-B 0.406 0.923 -0.328 10.1 56.4 4.8 24.2 (b) 0,050 0.676 Phenanthrene M8-A 0.007 0.091 0.902 -11.4 -14.7 -13.0 -31.5 0.082 0,005 0.045

M8-B 0.189 0.409 0.402 -6.8 -0.1 -9.4 -20.0 0.053 0,025 0.255 Acenaphthylene M9-A 0.011 0.192 0.797 -9.6 -8.7 -11.4 -26.4 0.070 0,013 0.114

M9-B 0.337 0.703 -0.039 2.2 33.7 -1.8 4.6 (b) 0,045 0.578 Biphenylene M10-A 0.000 0.193 0.807 -6.7 -0.7 -8.1 -17.3 0.088 0,008 0.066

M10-B 1.360 0.571 -0.930 17.4 87.0 7.5 35.7 (b) 0,048 0.297 Benzocyclobutadiene M11-A 0.001 0.501 0.497 -4.0 8.4 -5.4 -9.5 0.080 0,022 0.191

M11-B 0.910 1.526 -1.437 20.2 96.2 10.7 44.1 (b) 0,071 1.047 Pyridine M12 -0.006 0.001 1.005 -9.5 -15.2 -12.5 -31.6 0.097 0,001 0.001

Pyrimidine M13 0.015 0.000 0.985 -7.5 -11.8 -11.7 -29.8 0.089 0,005 0.003 Triazine M14 0.023 0.000 0.977 -5.3 -6.9 -10.8 -27.0 0.075 0,013 0.000

Quinoline M15-A 0.018 0.190 0.792 -11.0 -14.7 -12.6 -30.7 0,072 0,015 0.125 M15-B 0.008 0.161 0.830 -9.1 -11.5 -12.1 -29.5 0,071 0,017 0.121

Cyclohexane M16 5.340 0.000 -4.340 -2.1 23.5 -2.0 3.1 0.007 0,091 (c) Cyclohexene M17 2.955 1.647 -3.601 -1.6 18.1 -3.6 -1.8 0.019 0,089 (c)

Cyclohexa-1,4-diene M18 0.779 1.984 -1.763 1.5 25.4 -0.8 2.9 0.014 0,084 (c) Cyclohexa-1,3-diene M19 0.931 2.207 -2.138 3.2 30.0 0.8 7.3 0.031 0,078 (c) (a) Reprinted with permission from ref 31. Copyright 2005 American Institute of Physics. (b) PDI can not be computed for non 5-MRs. (c) Non planar molecules that prevent easy and exact σ-π separation.

167

Table 2. Pearson coefficient (r2) of correlation between different aromaticity indexes

for the series of molecules in Table 1 at the HF/6-31G(d) level of theory.

HOMA NICS(0) NICS(0)zz NICS(1) NICS(1)zz PDI FLU FLUπ

HOMA 1.00 0.30 0.33 0.43 0.41 0.68 0.90 0.86 NICS(0) 1.00 0.98 0.96 0.97 0.50 0.47 0.40

NICS(0)zz 1.00 0.96 0.98 0.64 0.49 0.41 NICS(1) 1.00 0.99 0.55 0.60 0.50

NICS(1)zz 1.00 0.60 0.58 0.48 PDI 1.00 0.79 0.62 FLU 1.00 0.95 FLUπ 1.00

168

Table 3. HF/6-31G*//AM1 calculated values of NICS (ppm), HOMA, para-

delocalization (PDI) (electrons) indices, average pyramidalization angles for the

carbon atoms present in a given ring (Pyr) (degrees) for a series of aromatic

compounds.(a)

Molecule ring NICS HOMA PDI PyrC6H6 6A -11.7 0.987 0.101 0.0 C10H8 6A -11.3 0.807 0.074 0.0 C14H8 6A -2.7 0.603 0.067 0.0 5B 13.1 -0.205 0.0 C20H10 6A -8.6 0.652 0.058 4.6 5B 7.6 0.357 9.1 C26H12 6A -5.6 0.474 0.037 6.9 5B 3.9 -0.142 6.3 6C -10.0 0.746 0.078 2.8 C30H12 6A -6.5 0.390 0.043 9.2 5B 6.8 0.113 10.1 6C -9.4 0.652 0.061 5.1 6D -8.1 0.614 0.057 4.6 C60 6A -6.8 0.256 0.046 11.6 5B 6.3 -0.485 11.6C70 5A 2.8 -0.481 11.9 6B -11.5 0.294 0.046 11.8 5C -1.3 -0.301 11.0 6D -8.8 0.141 0.028 10.1 6E -17.3 0.697 0.059 9.6

(a) Adapted with permission from ref 41. Copyright 2003 Wiley-VCH.

169

Table 4. GIAO/HF/6-31+G* NICS, B3LYP/6-311+G** HOMA, and B3LYP/6-

311G** PDI aromaticity indices, calculated at the B3LYP/6-311+G** geometry for

the different substituted benzene (C6H5X) structures. Substituent constants: σ+, σ-, σm,

σp, R+ and R-.(a)

-X NICS HOMA PDI σ+ /σ- σm σp R+/R-

-NN+ -10.6 0.96 0.080 3.43 1.76 1.91 1.85 -NO -9.8 0.98 0.091 1.63 0.62 0.91 1.14 -NO2 -10.9 0.99 0.096 1.27 0.71 0.78 0.62 -CN -10.3 0.98 0.096 1 0.56 0.66 0.49 -COCl -9.9 0.98 0.095 1.24 0.51 0.61 0.78 -COCH3 -9.7 0.98 0.097 0.84 0.38 0.5 0.51 -COOCH3 -9.8 0.98 0.097 0.75 0.37 0.45 0.14 -COOH -9.7 0.98 0.097 0.77 0.37 0.45 0.43 -CHO -9.6 0.97 0.095 1.03 0.35 0.42 0.70 -CONH2 -9.9 0.98 0.098 0.61 0.28 0.36 0.35 -CCH -10.1 0.97 0.096 0.53 0.21 0.23 0.31 -Cl -10.7 0.99 0.099 0.19 0.37 0.23 -0.31 -F -11.7 0.99 0.098 -0.03 0.34 0.06 -0.52 -H -9.7 0.99 0.103 0 0 0 0 -Ph -9.3 0.98 0.098 -0.18 0.06 -0.01 -0.30 -CH3 -9.7 0.98 0.100 -0.31 -0.07 -0.17 -0.32 -OCH3 -10.8 0.98 0.094 -0.78 0.12 -0.27 -1.07 -NH2 -9.8 0.98 0.093 -1.3 -0.16 -0.66 -1.38 -OH -10.8 0.99 0.095 -0.92 0.12 -0.37 -1.25

(a) Reprinted with permission from ref 45. Copyright 2004 American Chemical Society.

170

Table 5. NICS, PDI, HOMA, and FLU aromaticity measure of the five- and six-

membered rings of guanine (G) and six-membered ring of cytosine (C) computed with

the B3LYP method. NICS in ppm and PDI in electrons.(a)

NICS PDI HOMA FLU System G-5 G-6 C-6 G-6 C-6 G-5 G-6 C-6 G-5 G-6 C-6 GC -11.94 -4.10 -1.86 0.036 0.040 0.848 0.795 0.703 0.025 0.033 0.035[GC]·+ -5.41 -0.31 -2.49 0.023 0.042 0.829 0.550 0.773 0.028 0.048 0.031Ca2+-GC -10.67 -4.76 -2.53 0.044 0.045 0.843 0.886 0.797 0.023 0.024 0.029Cu+-GC -10.64 -4.59 -2.25 0.040 0.043 0.869 0.898 0.761 0.021 0.027 0.032Cu2+-GC -7.37 -2.00 -3.07 0.022 0.040 0.915 0.760 0.822 0.033 0.058 0.039(a) Adapted with permission from ref 47. Copyright 2005 Taylor and Francis. Details about the basis set used can be found in ref 47.

171

Table 6. PDI (in electrons), HOMA, and NICS (in ppm) values for the series of PAHs

taken into study. See the corresponding numbering in Scheme 6.(a)

Molecule Param. Ring 1 2 3 4 5 6 7 8 9 10

A 0.080 0.069 0.086 0.084 0.083 0.076 0.066 0.080 0.069 0.079B 0.047 0.043 0.026 0.034 0.041 0.066 0.053 0.066 0.057C 0.044 0.068 0.038 0.031D 0.073 0.084 0.085PD

I

E 0.085A 0.856 0.834 0.889 0.811 0.872 0.769 0.619 0.829 0.697 0.749B 0.435 0.553 -0.030 0.383 0.356 0.696 0.542 0.730 0.305C 0.518 0.788 0.266 -0.097D 0.838 0.883 0.820H

OM

A

E 0.883A -10.06 -12.74 -8.63 -9.44 -9.93 -9.98 -8.84 -9.94 -9.30 -10.19 B -6.82 -5.07 -1.18 -4.13 -5.38 -12.60 -7.69 -11.69 -7.68 C -5.47 -11.27 -4.58 -3.91 D -11.58 -9.81 -9.55 N

ICS

E -8.99 (a) Adapted with permission from ref 51. Copyright 2005 Wiley-VCH.

172

Table 7. θ angle (degrees), PDI (electrons), FLU and HOMA indices at hexagons,

HOMO-LUMO energy difference (eV), NICS(0) values at hexagons (ppm), and

average magnetizabilities (ppm-au) for the different distorted pyracylene molecules

optimized at the B3LYP/6-31G** level of theory.(a)

θ PDI FLU HOMA ΔEHOMO-LUMO NICS(0) NICS(1)in NICS(1)out χ 0. 0.068 0.011 0.755 2.84 −0.1 -2.8 -2.8 −875 10. 0.067 0.011 0.761 2.83 −0.1 -3.8 -2.1 −917 20. 0.067 0.011 0.771 2.82 −0.6 -5.5 -1.6 −989 30. 0.066 0.011 0.744 2.80 −1.7 -7.6 -1.2 −107640. 0.065 0.011 0.572 2.79 −3.7 -9.9 -0.8 −1057

(a) Adapted with permission from ref 55. Copyright 2004 American Chemical Society.

173

Figure Captions

Scheme 1. Reprinted with permission from ref 31. Copyright 2005 American Institute

of Physics.

Scheme 2. Reprinted with permission from ref 41. Copyright 2003 Wiley-VCH.


Scheme 4. Reprinted with permission from ref 46. Copyright 2004 Royal Society of

Chemistry.


Scheme 6. Adapted with permission from ref 51. Copyright 2005 Wiley-VCH.

Scheme 7. Reprinted with permission from ref 33. Copyright 2005 Elsevier.

Figure 1. The plot of PDI vs. σm (a) and σp (b). The correlation coefficients are equal

to −0.83 and −0.91, respectively. Reprinted with permission from ref 45. Copyright

2004 American Chemical Society.

Figure 2. Comparative plot of HOMA, NICS, and PDI for the substituted ring (Subs)

for the series of carbazole derivatives studied. The x-plot numeration can be found in

ref 46. Reprinted with permission from ref 46. Copyright 2004 Royal Society of

Chemistry.

Figure 3. Schematic representation of the charge transfer in GC base pair. Reprinted

with permission from ref 47. Copyright 2005 Taylor and Francis.

Figure 4. Plot of NICS(1) (ppm), PDI (electrons), FLU, HOMA (values divided by

10), and RSS (values divided by 2) versus the reaction coordinate (IRP in amu1/2

bohr). Negative values of the IRP correspond to the reactants side of the reaction path,

positive values to the product side, and IRP=0.000 corresponds to the TS of the DA

cycloaddition. Reprinted with permission from ref 33. Copyright 2005 Elsevier.

174

Figure 5. First-order π-electron current density map in planar pyracylene calculated

with the 6-31G** basis set using the CTOCD-DZ2 approach at the B3LYP/6-31G**

optimized geometry. The plot plane lies at 0.9 a.u. above the molecular plane. The

unitary inducing magnetic field is perpendicular and pointing outward so that

diatropic/paratropic circulations are clockwise/counter clockwise. Reprinted with

permission from ref 55. Copyright 2004 American Chemical Society.

Figure 6. The same as in Figure 5 for the distorted θ = 40º pyracylene. The plot plane

is parallel to: a) a 6-MR ring (in blue); b) a 5-MR (in blue). The green portion of the

molecular frame lies above the plot plane. Reprinted with permission from ref 55.

Copyright 2004 American Chemical Society.

Figure 7. Schematic representation of trends in [n]acenes (n = 1 – 9) according to

PDI, HOMA, and NICS results. Reprinted with permission from ref 59. Copyright


Figure 8. Schematic representation of trends in [n]phenacenes (n = 1 – 9) according

to PDI, HOMA, and NICS results. Reprinted with permission from ref 59. Copyright


Figure 9. Schematic representation of trends in [n]helicenes (n = 1 – 9) according to

PDI, HOMA, and NICS results. Reprinted with permission from ref 59. Copyright


175

Scheme 1.

B

B

A

A

A

B

BB

AA

A

B

B

A

A B BA

M1: Benzene M2: Naphthalene M3: Anthracene

M6: Triphenylene

M8: Phenanthrene

M4: Naphthacene M5: Chrysene

M9: Acenaphthylene

M10: Biphenylene

M7: Pyracylene

M11: Benzocyclobutadiene

176

Scheme 1. cont.

N

N

M12: Pyridine

M15: Quinoline

N

N

M13: Pyrimidine M14: Triazine

N

N

N

M17: Cyclohexene M18: Cyclohexa-1,4-diene M19: Cyclohexa-1,3-diene

M16: Cyclohexane

BA

177

Scheme 2.

A A A

B

A

BC

BA

A

BC

D

A

B

C6H6C10H8

C14H8

C20H10C26H12

C30H12 C60

178

Scheme 3.

A

B

C

D

Eree

rde

rddrcd

rbc

raarab

rcc

179

Scheme 4.

N2

C1

C5C4

C3C6

C7

C8C9

C10

C11

C12C13

H

H

H

HH

H

H

X

X = CH3, OH, Br, CN, CH3CO, COOHY = H, CH3

Y

180

Scheme 5.

1

2 Clar structure

181

Scheme 6.

A

A

A

A A

BB B

B B

C

C

D

Mol. 1Mol. 2 Mol. 3

Mol. 4

Mol. 5

Mol. 6

Mol. 7

Mol. 8

a a

a

b b

b

c

c

A A

A

B

B

Mol. 9 Mol. 10

a

ab

b

A

A

B

B

C

C

D

D

E

182

Scheme 7.

+ *# &

183

Figure 1.

0.06

0.07

0.08

0.09

0.1

0.11

0 0.5 1 1.5

σm

PDI

0.06

0.07

0.08

0.09

0.1

0.11

0 0.5 1 1.5

σp

PDI

(a) (b)

184

Figure 2.

1 2 3 4 5 6 7 8 9Compound

-14

-13.6

-13.2

-12.8

-12.4

-12N

ICS

0.068

0.072

0.076

0.08

0.084

PDI

0.8

0.84

0.88

0.92

0.96

HO

MA

NICS

HOMA

PDI

185

Figure 3.

NH

N7

N

N1

O6

H

HN2 H

N3

NH

N4HH

O2

NH

N7

N

N1+

O6-

H

HN2 H

N3

NH

N4HH

O2

1 2

186

Figure 4.

187

Figure 5.

188

Figure 6.

189

Figure 7.

A3 A4 A5 A6 A7 A8 A9

PDI

HOMA

NICS

190

Figure 8.

P3 P4 P5 P6 P7 P8 P9

PDI

HOMA

NICS

191

Figure 9.

H3 H4 H5 H6 H7 H8 H9

PDI

HOMA

NICS

192

5.- Applications II: The Electron

Localization Function

193

5.1.- The Electron Localization Function at Correlated

Level.

195

Electron Localization Function at the Correlated Level

Eduard Matito(a), Bernard Silvi(b)*, Miquel Duran(a) and Miquel Solà(a)*

(a) Institut de Química Computacional and Departament de Química, Universitat de

Girona, 17071 Girona, Catalonia, Spain

(b) Laboratoire de Chimie Théorique (UMR-CNRS 7616), Université Pierre et Marie

Curie-Paris 6, 4 Place Jussieu 75252-Paris Cédex, France

Abstract

The electron localization function (ELF) has been proved so far a valuable tool

to determine the location of electron pairs. Because of that, the ELF has been widely

used to understand the nature of the chemical bonding and to discuss the mechanism

of chemical reactions. Up to now, most applications of the ELF have been performed

with monodeterminantal methods and only few attempts to calculate this function for

correlated wavefunctions have been carried out. Here, a formulation of ELF valid for

mono- and multiconfigurational wavefunctions is given and compared with previous

recently reported approaches. The method described does not require the use of the

homogeneous electron gas to define the ELF, at variance with the ELF definition

given by Becke. The effect of the electron correlation in the ELF, introduced by

means of CISD calculations, is discussed in the light of the results derived from a set

of atomic and molecular systems.

Keywords: ELF (Electron Localization Function), correlated methods, second-order

density.

197

1. Introduction

The first to recognize the importance of the electron pair in chemical bonding

was G. N. Lewis ninety years ago.1 His idea that chemical bonds are formed by shared

electron pairs has been one of the most fruitful concepts in modern chemistry.2,3 The

electron pair plays a dominant role in the present day chemistry, being a powerful

concept to rationalize and understand the molecular structure (through the VSEPR

theory4), bonding, and chemical reactivity. Because of their importance, quantum

chemists have pursued the development of procedures to determine the spatial

domains where local electron pairing takes place in molecules. The first attempts in

this direction were based on the idea of the so-called localized orbitals.5 The

Laplacian of the one-electron density is another function used to denote electronic

localization.6,7 Afterwards, with the intention to calculate the electron pair regions

with greater precision in the molecular space, theoretical chemists turned its interest

to two-electron quantities.8 The simplest of such quantities that describes the pair

behavior is the so-called pair density. Other related functions employed to locate the

electron pairs have their origin in the seminal works of Artman9 and Lennard-Jones.10

Among them we can mention the exchange-correlation density,11,12 the Fermi hole,13

the Lennard-Jones function,14 the Luken and Culberson function15 and the conditional

pair density.15,16

Nevertheless, nowadays the most widely used procedure to analyze electronic

localization is probably the electron localization function (ELF) introduced by Becke

and Edgecombe from the leading term in the Taylor series expansion of the

spherically averaged same-spin conditional pair probability.17-20 The definition of the

198

conditional probability (CP) i.e., the probability density of finding electron 2 nearby

r2 when electron 1 is at r1, reads:

( )( ) ( )

( )1

212

21,,

rrrrrP r

rrrr

ργ

= , (1)

with ( )( )212 , rr rrγ and ( )1r

rρ being the pair density and the density, respectively. Both the

pair density and the CP contain all necessary information about the correlation of

electrons. However, a good advantage of the CP function is that it is actually

discounting irrelevant information concerning the position of the reference electron.

Hence, it is not surprising that Becke and Edgecombe17 chose the CP of same

spin electrons as a measure of electron localization. In particular, they used the

spherical average of the CP expanded by series of Taylor around the position of the

reference electron:21

( ) ( ) ( ) ( )00

1

1

2

0

· ,sinh,41, rrrr

rr rrrrrrrrr==

−

∇∇ +∇

∇=+⎟⎟

⎠

⎞⎜⎜⎝

⎛=+ ∫ ∫ ss

ss srrPs

ssrrPddesrrPe σσσσπ

θσσ φθπ

, (2)

that after using the Taylor expansion of sinh around the reference electron (s=0)

generates an expression whose leading term reads:

( ) ( ) ( )0

22

0

22· ,61,

611, rrr

rr rrrrrrK

rrr==

∇ +∇≈+⎟⎠⎞

⎜⎝⎛ +∇+=+

sssss srrPssrrPssrrPe σσσσσσ , (3)

199

The r.h.s. of Eq. (3) holds because the Pauli principle forces the same spin pair

density at the coalescence point to be zero, γσσ(r,r)=0. Thus, the leading term of the

Taylor expansion of the spherically averaged CP is proportional to:

( )0

2 ,21

rr

rrr=

+∇=ss srrPD σσ

σ , (4)

Hence, Becke and Edgecombe17 used the relative ratio of Dσ with respect to the same

quantity for the homogenous electron gas (HEG) assuming a monodeterminantal

situation where the pair density can be calculated from the density:22

( ) ( ) ( ) 3535322353220 31036

53 ρρπρπ σ

σ FcD === , (5)

with ( ) 3223103 π=Fc being the Fermi constant.23 Hence the relative ratio reads:

( ) ( )( ) ( )[ ]

( ) ( )( )[ ] 3/8

0

22

0

3/53/22

2·

0 2

,

653

3,)(

rc

srr

r

ssrrPe

DD

rDF

ss

s

s

r

rrr

r

rrrr rr

r

rr

σ

σσ

σ

σσ

σ

σ

ρ

γ

ρπ=

=

∇ +∇=

+== (6)

The (conditional) probability of finding an electron with spin σ when there is

already another electron with the same spin nearby is lower when the former is

localized. Therefore, the ratio in Eq. (6) accounts for electron localization; the higher

200

it is, the lower the localization. In order to range the electron localization measure in

the interval [0,1], Becke and Edgecombe chose the following scaling to define the

ELF:24

( )202 11

)(11

σσ

ηDDrD

ELF+

=+

=≡ r , (7)

thus, ELF=1 corresponds to a completely localized situation, and 0 to its opposite.

ELF=0.5 within an HEG.

The ELF helps in clarifying the structure of molecules,25 the shell structure of

atoms,26,27 and its topological interpretation gives rise to the molecular space splitting

in terms of chemical meaningful regions such as bonds, lone pairs, and core regions.19

The ELF has also been calculated separately for alpha and beta orbitals with the aim

to study different spin contributions on some radical systems.28 The ELF has been

used to gain inside on aromaticity20,29 and to analyze the changes in electronic

structure along a given reaction coordinate30 and to distinguish between similar

mechanisms of reaction.31

Becke17 developed the ELF formula in Eq. (7) for a monodeterminantal case

where the two-bodies quantities can be expressed in terms of one-body ones. The fact

that Becke based his ELF on the CP (a two-body quantity) and later on worked out his

expression for the monodeterminantal case had the disadvantage of the loss of

generalization, since only monodeterminantal cases were contemplated. However, he

overcame an epistemological problem: the natural extension of the ELF at correlated

201

level should also take into account the correlated expression of its reference, the spin-

unpolarized HEG. To the best of our knowledge the exact analytical expression of the

pair density for the HEG is unknown.

The investigations of the ELF at correlated level are scarce, mainly due to the

considerable computational effort needed for the calculation of the pair density. The

purpose of this paper is to investigate the ELF at correlated level of theory. Thus, we

firstly explore the definitions of the ELF already given for correlated calculations to

show they are indeed equivalent. Secondly, we apply the ELF to the calculation of

some atomic systems up to krypton, to analyze its ability to recover shell structure and

shells populations at the CISD method. And finally, we take a series of isoelectronic

molecules (CO, CN-, N2, and NO+) and another series of molecules known to be

problematic for monodeterminantal approaches (F2, FCl- and H2O2) to show the effect

of the inclusion of electron correlation in the population and variance of ELF basins.

202

2. Methods

This section is organized as follows. First, we analyze the previous definitions

of correlated ELF given by different authors and we prove that all of them are

equivalent; second, definitions of different populations and their variances are

provided; and finally, we discuss the computational tools used in this study.

2.1 Correlated electron localization function

A) An alternative ELF with no need to arbitrary references to HEG

The ELF thus defined, Eqs. (6)-(7), has advocated several criticisms because

of the arbitrariness when choosing the HEG as a reference. However, recently one of

us32 derived a quantity to account for the electron localization that in turn was

identical to the ELF, and thus served as an alternative derivation to ELF without the

arbitrariness of measuring with respect to the HEG. A similar process was followed

by Kohout33,34 to achieve identical results. Here we demonstrate the equivalence of

both formulations, which lead to the same ELF definition. (vide supra)

The interpretation of the ELF in terms of the second derivative of the pair

density is given in the papers of Dobson.35 The ELF is indeed the angularly averaged

(spherical averaged) measure of the Fermi hole curvature, and in turn the Fermi hole

curvature can be also understood as the density of the kinetic energy of relative

motion pairs of the same spin electrons centered at r. Dobson suggests the integration

of the CP (Eq. (3)) over a small sphere of radius R:

203

( ) ( )0

25

2

00

22 ,15

24,61

rrrr

rrrrrr==

+∇≈+∇∫ ss

R

ss srrPRdsssrrPs σσσσ ππ (8)

When R is close to zero the charge inside the sphere can be approximated as the

density of its central point times its volume, and therefore the density is proportional

to the inverse of the third power of R. In this case we are left with:

( )( ) ( )

( )[ ] 3/80

22

0

25

2

,,

r

srrsrrPR ss

ss r

rrrrrr rr

rrσ

σσσσ

ρ

γ=

=

+∇∝+∇ (9)

where we have supposed a closed-shell situation for the sake of simplicity. The reader

should note that excepting for constant factors we have reached the expression already

given by Eq. (6) in terms of two-body quantities without using the arbitrary reference

of the HEG, or in other words, one can give another interpretation to Eq. (6) used to

define the ELF. Hence the ELF can be understood as the number of electrons with

spin σ nearby r. It is worth noticing that the ELF is actually an approximation to this

quantity.

The expression in the numerator of Eq. (9) is a measure of the spin

composition of the system32 and is usually referred as the Fermi hole curvature. It is

not the real Fermi hole curvature, since the γαα it is not a Fermi hole, but a pair

density of the same spin electrons. However, one can easily demonstrate that the

curvature of the Fermi hole,

204

( ) ( ) ( ) ( )( )1

2121)2(

21,,

rrrrrrr r

rrrrrr

σ

σσσσσσ

ρρργ −

=Δ , (10)

is indeed very similar to the pair density curvature. Hence:

( )( )

( )[ ] ( )( )

( )[ ]r

srrr

r

srrsrr ssss

ss r

rrrr

r

rrrrrr rrrr

rr σ

σσσ

σ

σσσσ

ρ

γρ

ρ

γ0

)2(220

)2(2

0

2,,

, ==

=

+∇≈∇−

+∇=+Δ∇ , (11)

The r.h.s. of the latter equation holds for volumes where the density is small enough

to be considered constant. Here, the advantage of using the curvature instead of the

function or its gradient clearly shows: the former is the only one that does not vanish

when r1=r2.36 Therefore, one could get an alternative ELF definition in terms of the

Fermi hole curvature, which would be equal to the current definition excepting for the

exponent of the density.

B) On the equivalence of already proposed derivations of correlated ELF

One of us recently32 proved that:

( ) )(;);( 3/2 rcqrNqrD iirrr

π≈ , (12)

where D is the size-dependent pair composition of the system, -our Eq. (6) considered

in small sphere centered at r with charge qi-, N the population of the volume

205

enclosed in the sphere centered at r, and cπ the size-independent spin composition

given by ( ) ( )13/5

2122 3, rrrP rrr ρσσ∇ . This way, an empirical proof for the

approximation from Eq. (8) to the current definition of the ELF was given. In ref. 32

it is shown how the approximation works fine for small volumes, differing slightly for

large volumes where no electron homogeneity is granted; in Eq. (30) of ref. 32 a

generalization of ELF for open-shell cases is provided:

( ) ( ))(2

,,)( 3/8

0

)2(2

0

)2(2

rc

srrsrrrD

F

ssss

ρ

γγ ααββ

==+∇++∇

=rr

rrrrrrr , (13)

where we have taken proper normalization in order to reproduce our equation (6) for

closed-shell species. On his side, Kohout34 decided to split the molecular space into

regions of fixed charge:11

( ) ( )∫Ω

==Ωiq

rdrqrN iq

,

1,; rrrσσσ ρ (14)

And taking the same spin pair density as given by Eq. (10) we have:

( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫∫ ∫Ω ΩΩ ΩΩ Ω

Δ+=iq iqiq iqiq iq

rdrdrrrrdrdrrrdrdrr, ,, ,, ,

21211212121212 ,, rrrrrrrrrrrrr σσσσσσσ ρρργ (15)

In the situation of fixed charge (Eq. (14)) we get:

206

( ) ( )iqiq FqD ,2

,)2( Ω+=Ω σσ

σσσ (16)

So, in this context Eq. (16) is measuring not only the number of expected pairs of

electrons in Ω, but provided the population in these regions is constant, we are indeed

measuring F, which is to say -as previously demonstrated by Bader3,7- a quantity that

accounts for the electron localization. This qσ-restricted pair population is what

Kohout suggests (after proper scaling) as a measure of electron localizability. In fact,

is not the quantity itself, but the first non-vanishing term in the expansion of Eq. (16)

around position r1:

( ) ( ) ( )( ) ( ) ( )∫ ∫Ω Ω =

∇−=Ωiq iq

rdrdrrrrDrr

iq

, ,21

212122

212,2 ,·

21 rrrrrrr

rr

σσσσ γ (17)

Expression (17) is indeed equally to the latter expression (8) excepting for a factor

proportional to the density, which in turn is constant in this context. Then, Kohout

uses an arbitrary factor to normalize the latter expression and fit the one already

given by Becke and Edgecombe.17 It is worth noting that Kohout and coworkers did

some preliminary CASSCF calculations using Eq. (17)33 as an expression for the

correlated ELF. Moreover, Kohout et al. have recently proposed an antiparallel

version of the localization function.37 Finally, the paper of Scemama et al.38 should

also be mentioned in this context because of the definition of the EPLF (electron pair

localization function) constructed by analogy with the ELF in order to compute this

function from quantum Monte Carlo data.

207

2.2 Variance and covariance basin population

The partition of the molecular space enables basin-related properties to be

calculated by integrating the density of a certain property over the volume of the

basins. Thus, for a basin labeled Ωi, one can define the average population as:

( ) ( )∫Ω

=Ωi

rdrN irrρ , (18)

and its variance:

( ) ( ) ( )( ) ( ) ( ) ( )[ ]∫Ω

−ΩΩ−=Ω−Ω=Ωi

iiiii NNrdrdrrNN 1, 2121)2(22 rrrrγσ (19)

One can also define likewise the pair population:

( ) ( )∫ ∫Ω Ω

=ΩΩ≡ΩΩi j

rdrdrrNNN jiji 21212 ),()()·(, rrrrγ , (20)

The variance is a measure of the quantum mechanical uncertainty of the basin

population, which can be interpreted as a consequence of the electron delocalization.

The variance, ( )iΩ2σ , can also be split in terms of contribution from other basins,

208

( ) ∑≠

ΩΩ−=Ωij

jii V ),(2σ , (21)

where ),( jiV ΩΩ is the covariance, and is calculated as follows:

( )( )∫ ∫Ω Ω

−=

=ΩΩ−ΩΩ=ΩΩ

i j

rdrdrrrr

NNNNV jijiji

2121212 )()(),(

)()()()·(),(rrrrrr ρργ (22)

Finally, the integrated spin density may also be important for open-shell systems:

[ ]∫Ω

−=Ωi

rdrrS iz 111 )()()( rrr βα ρρ (23)

2.3 Computational details

Since coupled cluster and MP2 methods do not fulfill the Hellmann-Feynman

theorem, neither the density nor the pair density can be obtained as an expectation

value. Therefore, the pair density and the density thus obtained are relaxed and, in

general, not N-representable39 (i.e. derivable from a N-electron wavefunction). For

this reason, here we have preferred to deal with the configuration interaction with

singles and doubles (CISD) unrelaxed density and pair density that have been

employed in Eq. (13) to obtained correlated ELFs. The complete active space (CAS)

unrelaxed density and pair density could also be used, but in this work we will focus

our study on CISD results. The DFT calculations in this paper are performed with

Kohn-Sham formalism, and building up an approximate HF-like pair density,40 thus

reducing the ELF expression to that originally given by Becke and Edgecombe.17

209

In order to compute the ELF from Eqs. (13) and (7), one must calculate the

curvature of the pair density, that in terms of molecular orbitals reads:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∇∇+

+∇+∇Γ=∇ ∑= )()(2

)()()()()()(,

*

2**2*

122

11 rr

rrrrrrrr

lj

ljjl

ijklki

klijrr rr

rrrrrrrr

rr

χχ

χχχχχχγ σσ , (24)

In case we are dealing with a linear combination of gaussian primitives, the latter

expression turns into derivatives of gaussian functions, whose analytical expressions

are straightforward. In this work, we have worked with such functions, obtained with

the Gaussian03 program.41 From the same program the coefficients of the expansion

were obtained and used by an own set of programs to construct the Γ matrix in Eq.

(24). The wavefunction together with the Γ matrix were read by our own modification

of the ToPMoD package of programs42 to calculate the curvature of the pair density

and finally, the ELF.

In this study we have focused on configuration interaction (CI) calculations to

introduce the electron correlation. The electron correlation of parallel spins was

introduced by mixing the Hartree-Fock (HF) determinant with other mono-excited

and bi-excited determinants, what it is known as a configuration interaction with

singles and doubles (CISD).43 Unlike atoms, CISD calculations for molecules were

performed within the restriction of frozen core. All molecules were optimized with

the CISD, HF, B3LYP, and BP86 levels of theory with Pople’s 6-311+G(2df,2p)

basis set. Additional single point calculations were performed for all methods at the

210

HF geometry with the aim to study the purely electronic effects in the inclusion of

electron correlation.

The coefficients of the CI expansion have been obtained from energy

calculations performed with the Gaussian03 program;41 only coefficients with values

above 10-6 were taken. The density and the pair density thus obtained are unrelaxed. It

is well known that relaxed CISD densities are better for the calculation of one-

electron response properties,44 but this is not the goal of the present study, and since

they are not N-representable,39 the unrelaxed ones have been preferred for this study.

Only values of density and pair density above 10-12 au have been considered.

3. Results and discussion

With the goal to briefly explore the influence of the electron correlation in the

ELF expression, we have limited ourselves to calculate some atomic systems (atoms

from Li to Kr), four diatomic isoelectronic molecules (CO, N2, NO+, and CN-), and a

series of molecules which are expected to be more affected by the inclusion of the

electron correlation (F2, FCl-, and H2O2).

3.1 Atomic calculations

The ELF has been previously demonstrated as a good tool to discern the shell

structure of atoms.26,27,45 Unlike the Laplacian of the density, which is unable to

recover quantitatively the number of electrons per atomic shell46 and even the shell

structure for heavy atoms,47 the ELF reproduces the shell structure and gives a quite

211

good approach (0.1-0.2 electrons of deviation)26 to the number of electrons per shell.

Recently Kohout demonstrated that the one-electron potential also reveals with a

reasonable accuracy the electron numbers for the occupation of shells.48

In order to check whether ELF provides better shell numbers with the

improvement of the wavefunction,26 ELF calculations for the ground state of atoms

from Li to Kr with CISD/6-311+G(2df,2p) level of theory have been performed. The

basins boundaries have been assumed to be spherical and the integration of the atomic

electron density has been carried out analytically (see appendix). Table 1 contains the

data from ELF calculated with CISD and the 6-311+G(2df,2p) basis set. The radius of

the K-shell is the result of the competition between the attractive (electron-nucleus)

and repulsive forces (electron-electron). Since the attractive potential increases with

Z, the net effect is the reduction of the shell radii, and the increase of the shell number

with Z. For the same reason, atoms from Li to Sc, L-shell populations lay below the

expected on the chemical grounds, decreasing shell population with Z. Because of the

screening effect of d electrons on the L-shell, the opposite situation is found for atoms

from Ti to Kr.

Table 1, here

Shell radii and population of atoms with electrons in 3d orbitals are quite

unpredictable, especially concerning the M and N-shell numbers; their configuration

is no longer ruled exclusively by the aufbau procedure and the Hund’s rule plays a

prominent role. One should also take into account that real orbitals p and d do not

212

guarantee the spherical symmetry excepting for empty, half-occupied and full

occupied configurations.

3.2 The isoelectronic series: CO, CN-, N2, and NO+

The four molecular systems CO, CN-, N2, and NO+ are isoelectronic but

present different bonding situations. Although the four systems are covalent, one

would advance nitrogen molecule to have non-polar covalent character, while NO+,

CN- and CO are somewhat polarized. In the formal Lewis sense one would expect a

triple bond for all species and some lone pairs around the nuclei. This kind of

molecules with large electron charge in the bonding regions are especially susceptible

to the inclusion of electron correlation in the calculations.49

Table 2 contains the basins population and its variance for the isoelectronic

series with the four methods studied at the optimized geometry of each method and

also at the HF geometry. Focusing either in the results derived from the calculations at

different geometries (which gather simultaneously the electronic and geometric

effects of electron correlation) or in the results with HF geometry, we can generally

assess that the population of monosynaptic valence basins increases at the expense of

populations in disynaptic bonding basins, which is consistent with the general

expectance of smaller force constants and larger internuclear distances at the CI or

DFT levels of theory. At the optimal geometry of each method, the calculated

decrease of the V(A,B) basin population with respect to the HF value is larger for the

BP86 functional than for the B3LYP and CISD approaches, the latter yielding the

smallest change. The net charge transfers towards the lone pair due to electron

213

correlation are often correlated with the bond lengthening: this is always the case for

the BP86 and B3LYP results, and most of CISD results, whereas CISD value for

V(N) of CN- show an unexpected opposite behavior.

Table 2, here

The loge theory, as introduced by Daudel et al.50, suggests the minimization of

electron fluctuation in individual domains as the way to reach the most likely

decomposition of molecular space: the smaller the lack of localization, the better the

decomposition is. It suggests the idea of taking the sum of variances (Σσ2 in Table1)

of basin populations as a measure of the best partitioning achieved.51 In this context,

Gallegos et al.52 explicitly calculated the maximal probability domains for N2 and CO.

Since the best partition found by Gallegos et al. for N2 and CO yields smaller

populations for the bonding regions than the ELF partition at the HF level, one would

expect the populations of bonding regions for N2 and CO to decrease with respect to

the values given by HF with the inclusion of electron correlation. This is indeed what

we have found in our DFT and CISD calculations. Furthermore, in all cases the CISD

partitioning yields the smallest sum of variances. It is worth noting that for DFT

calculation the variance calculation is not exact since an approximate expression of

the two particle density is used instead of the (unknown) exact one.

These results seem to indicate that further correlated ELF partitioning can lead

to better agreement with loge theory. However, one should take into account that only

a real space exploring of maximal probability spaces against ELF partitioning will

lead to the definite answer about the similarity of these two approaches.

214

3.3 The series F2, FCl- and H2O2

Finally, the results for F2, FCl-, and H2O2 are given in Table 3. These

molecules are supposed to be more affected by the inclusion of electron correlation.53

Indeed, F2 exhibits two symmetric V(F,F) basins for correlated methods, instead of a

single one as in the HF case, holding less electrons than in the single V(F,F) situation;

this number is specially small for DFT methods. H2O2 is partially well described by

the HF method, yielding lower V(O,O) populations and variances for CISD and DFT

calculations; for the latter this basin is even split into two symmetric ones.

Table 3, here

The FCl- is a special case, where the inclusion of electron correlation leads to

notorious change in both the structure and the electron distribution. First, one must

notice that HF equilibrium distance is about 1 Å larger than that obtained with more

accurate methods; DFT methods approach better this distance, while our CISD result

agrees with that of 2.111Å given by the EOMEA-CCSD/aug-cc-PV5Z method.54

Especially it is worth to comment that the spin density integrated in each basin, cf. Eq.

(23), is zero for Cl basin within HF. This is an indication that the HF picture of the

molecule is a closed-shell fragment given by the anion Cl- interacting with a neutral

fluorine atom. Therefore, the ELF results reported for the HF wave function differ

considerably from those calculated with DFT or CISD methods. In addition, it is

noticeable the differences of the populations and variances between CISD and DFT.

The inclusion of the electron correlation leads to not only shorter bond distances, but

215

also to partial electron transfer from the chlorine atom towards the fluorine one; an

electron transfer slightly exaggerated by the DFT methods, as one can deduce from

both the populations and the spin integrated densities.

Conclusions

In this paper we review the ELF definition for correlated wavefunctions,

showing the equivalence of the most recently proposed definitions of ELF. This ELF

definition reproduces Becke and Edgecombe’s one for monodeterminantal wave-

functions. We give the first results for CISD calculations of ELF, together with the

first populations and variance analysis for a correlated partitioning according to the

ELF function. Our analysis shows a clear reduction of the bonding region populations

as expected from previous results derived from the loge theory. It is worth noting that

there is a qualitative agreement between the CISD and DFT results in all the systems

investigated here. Although more molecular systems need to be explored to assess the

effect of the electron correlation in the ELF and its consequences for the description

of the molecular structure, results for F2, ClF, and HOOH indicate that correlated ELF

can differ dramatically in some cases from the uncorrelated one. Since the description

of some chemical reactions may also be very sensitive to the inclusion of electron

correlation, we are currently investigating the ability of correlated ELF to deal with

such chemical processes. We hope to be able to report the results on chemical

reactions analyzed using correlated ELF in due time.

216

Acknowledgements

Financial help has been furnished by the Spanish MEC Projects No. CTQ2005-08797-

C02-01/BQU and CTQ2005-02698/BQU We are also indebted to the Departament

d’Universitats, Recerca i Societat de la Informació (DURSI) of the Generalitat de

Catalunya for financial support through the Project No. 2005SGR-00238. E.M. thanks

the Secretaría de Estado de Educación y Universidades of the MECD for the doctoral

fellowship no. AP2002-0581. We also thank the Centre de Supercomputació de

Catalunya (CESCA) for partial funding of computer time.

217

Appendix: Integration of atomic populations

In order to compute the populations given in Table 1 one must find the ELF

minima along the radial direction, since they indicate the separation between shells.

Afterwards, one must integrate the electron density over a spherical basin of radius ra,

say, to compute integrals of the form:

( ) ( ) ( ) drddrrGrGrIar

a θφθπ π

νμμν2

2

0 0 0

sin∫ ∫ ∫=rr , (25)

with the cartesian gaussian function reading:

( ) 2rnml ezyxNrG μμμμ αμμ

−= , (26)

thus ( )arIμν can be written as a:

( ) ( )aa rRNNrI μνμνμννμμν ΘΦ= , (27)

with:

( ) ( )∫ +−++++++=ar

rnnllmma drerrR

0

2 2νμνμνμνμ αα

μν , (28)

218

∫ +++++=Θπ

μν θθθ νμνμνμ

0

1sincos dllmmnn , ∫ ++=Φπ

μν φφφ νμνμ

2

0

sincos dmmll (29)

The angular integrals have the following general form:

( ) ∫=Φπ

φφφn

ml dml0

sincos, (30)

whose result is as follows:

( ) ( ) ( )

( ) ( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=Φ

∑ ∑∫

∫∑∑+

= −=

−++

−−+

= =+

km

kMl

dei

dekm

jl

iml

lm

M

mM

lMk

kn

Mmlimml

nkjmli

l

j

m

k

kmml

0

),min(

0,max0

2

0

22

0 0

12

1

12

1,

φ

φ

πφ

πφ

(31)

Likewise, the radial part can be depicted as:

( ) ∫ −=ar

rlal drerrR

0

2α (32)

and easily evaluated through the following recursion formulas:

219

( ) ( )al

la

al rRler

rR ar2

1

21

22

−−

− −+=

ααα (33)

with:

( ) aa rerfrR ααπ

20 = (34)

( ) ( )2

121

1arerR a

α

α−−= (35)

220

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223

Table 1. Shell radii (rX) and shell numbers populations (NX) for the ground state of

atoms from Li to Kr. All quantities in atomic units.

NK rK NL rL NM rN NN N Li 2.017 1.616 0.983 3.000

Be 2.017 1.027 1.983 4.000

B 2.083 0.787 2.917 5.000

C 2.174 0.646 3.826 6.000

N 2.121 0.482 4.879 7.000

O 2.058 0.376 5.942 8.000

F 2.239 0.364 6.761 9.000

Ne 2.140 0.295 7.860 10.000

Na 2.171 0.263 7.871 2.249 0.958 11.000

Mg 2.179 0.236 7.851 1.677 1.970 12.000

Al 2.187 0.214 7.910 1.445 2.903 13.000

Si 2.194 0.195 7.970 1.256 3.836 14.000

P 2.198 0.179 7.867 1.034 4.935 15.000

S 2.203 0.166 7.926 0.933 5.870 15.999

Cl 2.198 0.154 7.809 0.800 6.994 17.001

Ar 2.207 0.144 7.856 0.736 7.937 18.000

K 2.201 0.134 7.868 0.671 8.028 3.331 0.902 18.999

Ca 2.204 0.127 7.856 0.615 7.965 2.513 1.975 20.000

Sc 2.212 0.120 7.934 0.575 9.324 2.318 1.529 20.999

Ti 2.215 0.114 8.138 0.548 10.898 2.264 0.749 22.000

V 2.219 0.108 8.176 0.517 11.061 2.083 1.544 23.000

Cr 2.223 0.103 8.049 0.474 12.873 3.007 0.855 24.000

Mn 2.229 0.099 8.156 0.447 13.908 2.255 0.708 25.001

Fe 2.232 0.095 8.281 0.427 14.026 1.938 1.462 26.001

Co 2.238 0.091 8.448 0.407 15.572 1.891 0.742 27.000

Ni 2.242 0.088 8.496 0.392 16.703 2.294 0.560 28.001

Cu 2.236 0.084 8.279 0.362 17.694 2.745 0.791 29.000

Zn 2.242 0.081 8.311 0.344 16.944 1.702 2.503 30.000

Ga 2.233 0.078 8.377 0.329 16.894 1.485 3.496 31.000

Ge 2.236 0.075 8.419 0.315 17.341 1.491 4.005 32.001

As 2.236 0.073 8.459 0.302 17.098 1.284 5.208 33.001

Se 2.237 0.070 8.487 0.289 16.973 1.157 6.304 34.001

Br 2.236 0.068 8.546 0.279 16.880 1.065 7.338 35.000

Kr 2.239 0.066 8.542 0.268 17.026 1.012 8.193 36.000

224

Table 2. Basin populations and its variance for CO, CN-, N2 and NO+ calculated at

the HF, B3LYP, BP86 and CISD levels, and the (HF) columns correspond to single

point’s calculation at a given method with the HF geometry. Bond lengths in Å and

populations and variances in electrons. Since core basins do not significantly change

from one method to the other, only valence basins (V) are included.

HF B3LYP (HF) BP86 (HF) CISD (HF) CO Bond lengtha 1.103 1.125 1.103 1.137 1.103 1.121 1.103

V(C) 2.37 2.53 2.52 2.58 2.56 2.39 2.40 σ2 0.67 0.77 0.77 0.80 0.80 0.81 0.75 V(C,O) 3.32 3.08 3.19 2.99 3.16 2.99 3.07 σ2 1.45 1.41 1.43 1.39 1.43 1.42 1.40 V(O) 4.11 4.19 4.10 4.23 4.08 4.43 4.33 σ2 1.37 1.40 1.39 1.40 1.39 0.95 1.03 Σσ2 4.05 4.16 4.17 4.17 4.20 3.36 3.76

CN- Bond length 1.152 1.171 1.152 1.183 1.152 1.169 1.152 V(C) 2.66 2.84 2.82 2.90 2.87 2.73 2.68 σ2 0.88 0.97 0.97 1.00 0.99 0.86 0.90 V(C,N) 3.81 3.48 3.54 3.39 3.48 3.81 3.78 σ2 1.51 1.26 1.49 1.46 1.48 1.36 1.37 V(N) 3.34 3.48 3.44 3.52 3.46 3.29 3.35 σ2 1.20 1.26 1.24 1.27 1.25 1.19 1.12 Σσ2 4.12 4.25 4.25 4.28 4.27 3.94 3.94

N2 Bond lengthb 1.066 1.091 1.066 1.103 1.066 1.088 1.066 V(N) 2.97 3.16 3.12 3.21 3.15 3.08 3.05 σ2 1.05 1.12 1.11 1.14 1.12 1.03 1.01 V(N,N) 3.85 3.49 3.58 3.37 3.52 3.64 3.66 σ2 1.52 1.49 1.50 1.47 1.50 1.43 1.42 Σσ2 4.19 4.32 4.29 4.36 4.32 4.07 4.04

NO+ Bond length 1.026 1.057 1.026 1.071 1.026 1.051 1.026 V(N) 2.64 2.82 2.81 2.89 2.85 2.79 2.70 σ2 0.86 0.96 0.95 1.00 0.98 0.85 0.95 V(N,O) 3.58 3.18 3.32 3.06 3.27 3.24 3.13 σ2 1.49 1.43 1.46 1.41 1.45 1.36 1.37 V(O) 3.60 3.79 3.66 3.84 3.69 3.78 3.99 σ2 1.28 1.34 1.31 1.36 1.31 1.25 1.04 Σσ2 4.22 4.34 4.33 4.39 4.35 4.06 3.98

a Experimental bond length is 1.128 Å.55 b Experimental bond length is 1.098 Å.55

225

Table 3. Basin populations and its variance for H2O2, F2 and FCl- calculated at the

HF, B3LYP, BP86 and CISD levels within the corresponding geometry. Distances in

Å, angles in degrees, and populations and variances in electrons. Since core basins do

not significantly change from one method to the other, only valence basins (V) are

included.

HF B3LYP BP86 CISD

H2O2a rOH 0.942 0.966 0.976 0.954

rOO 1.386 1.454 1.476 1.421 ∠HOO 102.7 100.1 99.2 100.8 (HOOH 116.3 120.2 119.8 118.0 V(O, H) 1.77 1.71 1.68 1.74 σ2 0.78 0.77 0.77 0.76 V(O,O) 0.77 2x0.29 2x0.25 0.69 σ2 0.58 0.51 0.40 0.53 V(O) 2.36 2.45 2.48 2.41 σ2 1.02 1.03 1.07 1.02

FCl- Bond length 3.173 2.291 2.322 2.120 V(F) 6.90 7.37 7.32 7.23 σ2 0.40 0.73 0.73 1.36 Sz(F) 0.47 0.22 0.21 0.29 V(Cl) 7.91 7.42 7.41 7.56 σ2 0.59 0.90 0.90 0.28 Sz(Cl) 0.01 0.26 0.25 0.18

F2 Bond lengthb 1.331 1.399 1.418 1.381 V(F) 6.59 6.73 6.74 6.66 σ2 0.94 0.96 0.96 0.93 V(F,F) 0.55 2x0.16 2x0.13 2x0.23 σ2 0.45 0.15 0.12 0.21

a Experimental values are rOH = 0.950 Å, rOO = 1.475 Å, and ∠HOO = 94.8º55 b Experimental bond length is 1.417 Å.55

226

5.2.- Bonding in Methylalkalimetal (CH3M)n (M = Li – K; n

= 1, 4). Agreement and Divergences between AIM

and ELF Analyses.

227

Bonding in Methylalkalimetals (CH3M)n (M ) Li, Na, K; n ) 1, 4). Agreement andDivergences between AIM and ELF Analyses†

Eduard Matito, ‡ Jordi Poater,*,§ F. Matthias Bickelhaupt,§ and Miquel Sola* ,‡

Institut de Quı´mica Computacional and Departament de Quı´mica, UniVersitat de Girona, 17071 Girona,Catalonia, Spain, and Afdeling Theoretische Chemie, Scheikundig Laboratorium der Vrije UniVersiteit,De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands

ReceiVed: December 26, 2005; In Final Form: February 11, 2006

The chemical bonding in methylalkalimetals (CH3M)n (M ) Li-K; n ) 1, 4) has been investigated by makinguse of topological analyses grounded in the theory of atoms in molecules (AIM) and in the electron localizationfunction (ELF). Both analyses describe the C-M bond as an ionic interaction. However, while AIM diagnosesa decrease of ionicity with tetramerization, ELF considers tetramers more ionic. Divergences emerge alsowhen dealing with the bonding topology given by each technique. For the methylalkalimetal tetramers, theELF analysis shows that each methyl carbon atom interacts through a bond pair with each of the three hydrogenatoms belonging to the same methyl group and through an ionic bond with the triangular face of the tetrahedralmetal cluster in front of which the methyl group is located. On the other hand, the AIM topological descriptionescapes from the traditional bonding schemes, presenting hypervalent carbon and alkalimetal atoms. Ourresults illustrate that fundamental concepts, such as that of the chemical bond, have a different, even collidingmeaning in AIM and ELF theories.

1. Introduction

Organometallic species containing polar metal-carbon bonds,1

such as organolithium, organomagnesium, and organozinccompounds, are important in organic synthesis. They combinein a unique way high reactivity and ease of access. Besides theirsynthetic utility, they are also of interest because of their unusualgeometries that challenge conventional bonding considerations.2

Methyl derivatives of organoalkalimetal compounds constitutethe simplest organometallic compounds containing the archetypecarbon-metal bond, and because of that, they have beenanalyzed in numerous theoretical3-9 and experimental10-15

studies. The high polarity of the alkalimetal-carbon bondpresent in these compounds results in a partially negativelycharged carbon atom, which to an important extent determinesthe behavior of these compounds as carbon nucleophiles andbases. The carbon-alkalimetal bond is commonly viewedpredominantly ionic, with simple ion-pair Coulombic attractionbeing much more important for the description of the bond thanthe covalent bonding. This picture has emerged from advancedpopulation analysis methods such as the natural populationanalysis (NPA), the integrated projected population (IPP)method, or the atoms in molecules (AIM) approach that yieldlithium atomic charges between+0.75 to+0.90 e.3,6,16-18 AIMtopological analysis6,17,19and modern valence-bond description20

of the carbon-alkalimetal bond provide further support for theionic character of this bond. Finally, Streitwieser, Bushby, andSteel have shown that a simple electrostatic model is able toreproduce the ratio of carbon-carbon and lithium-lithiumdistances in the methyllithium tetramer.4,21

As a result, the established idea about this bond is that theelectron of the metal is nearly completely transferred to methyl,and covalent interactions play only a marginal role. Nevertheless,there is also some experimental and theoretical evidence of arelevant covalent contribution to the alkalimetal-carbon bond.For instance, the large carbon-lithium NMR coupling constantsof up to 17 Hz observed for organolithium aggregates22 areindicative of the importance of the covalent character of thecarbon-lithium bond. Besides, the solubility of simple organo-lithium compounds in nonpolar solvents has been also consid-ered as a manifestation of the covalent nature of the C-Libond.23 Moreover, Streitwieser’s aforementioned ideal distanceratio is not found in other similar lithium tetramers such as(LiH)4, (LiOH)4, and (LiF)4, and for the two latter examples anelectrostatic model wrongly predicts a planar eight-memberedring structure to be more stable than the tetrahedral structure.24

In addition, Hirshfeld25 and Voronoi deformation density16,26

charge population analyses in methyllithium yield a Li atomiccharge of+0.50 and+0.39 e, respectively, pointing out thatthe charge transfer from Li to the methyl group is far from beingcomplete, i.e., opposite to what is expected for an ionic carbon-lithium bond.26 Finally, we have recently shown by means ofquantitative Kohn-Sham orbital mixing and energy decomposi-tion analyses that the trend in homolytic M-CH3 bond strengthof the MCH3 species along the series M) Li, Na, K, and Rbis governed by the overlap between the singly occupiedmolecular orbitals (SOMOs) of the methyl and alkalimetalradicals.27 These results provide evidence of the importance ofthe covalent contributions in the C-M bond.

The studies mentioned above show that the nature of thecarbon-metal bond in methyl derivatives is interesting andimportant, but also controversial. In the present work, wereinvestigate the carbon-metal bond by performing atoms inmolecules (AIM)28 and electron localization function (ELF)29

topological analyses of the (MCH3)n species (M) Li, Na, and

† Dedicated to Professor Ramon Carbo´-Dorca on the occasion of his 65thbirthday.

* To whom correspondence should be addressed. E-mail: [email protected].

‡ Universitat de Girona.§ Scheikundig Laboratorium der Vrije Universiteit.

7189J. Phys. Chem. B2006,110,7189-7198

10.1021/jp057517n CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 03/15/2006

229

K; n ) 1, 4) depicted in Figure 1. Our goal is to shed lightfrom yet other perspectives onto the nature of the carbon-metalbond using the complementary approaches of AIM and ELFanalyses. Although the AIM study of the electron density forM ) Li was already carried out by Ritchie and Bachrach someyears ago,17 and Vidal et al. studied the KCH3 species with AIMand ELF,19 in this paper we extend and complement theseprevious analyses for the full M) Li, Na, and K,n ) 1, 4series to discuss the effect of going from Li to the heavieralkalimetals and also the effect of tetramerization. Our AIMresults include the localization and delocalization indices30,31

derived from the second-order density. Moreover, we willprovide a comparison between the AIM and ELF descriptionsof the bonding in these species. Here we anticipate that, notunexpectedly, both AIM and ELF yield a picture of the carbon-alkalimetal bond that is highly polar. Interestingly, however,the two models disagree regarding a number of fundamentalbonding descriptors. Implications of these findings are discussed.

2. Methodology

2.1. AIM and ELF Topological Theories.The partitioningof the molecular space into basins of attractors allows thecalculation of several properties by integration over these basins.In the atoms in molecules (AIM)28,32-34 theory, the basins aredefined as a region in the Euclidean space bounded by a zero-flux surface in the gradient vectors of the one-electron density,F(r ), or by infinity. In this way, a molecule is split into itsconstituent atoms (atomic basins) and, if present, nonnuclearattractors, using only the one-electron density distribution. Sucha division of the topological space is exhaustive, so that manymolecular properties, e.g. energy, dipole moments, electronpopulations, etc., can be written as the sum of atomic contribu-tions. A different partitioning of the space was proposed in 1990by Becke and Edgecombe35 using a local scalar function, theelectron localization function (ELF) denoted byη(r ), which isrelated to the Fermi hole curvature. As shown by Savin et al.,29

the ELF measures the excess of kinetic energy density due tothe Pauli repulsion. In the region of space where the Paulirepulsion is strong (single electron or opposite spin-pairbehavior) the ELF is close to one, whereas where the probabilityof finding the same-spin electrons close together is high, theELF tends to zero. For anN-electron single determinantalclosed-shell wave function built from Hartree-Fock (HF) or

Kohn-Sham orbitals,φj, the ELF function is given by:36

where

and whereN is the number of electrons andΓ(2)σσ(rb1,rb2) is oneof the same-spin contributions to the second-order density,Γ(2)-(rb1,rb2). As the ELF is a scalar function, the analysis of its gradientfield can be carried out in order to locate its attractors (localmaxima) and the corresponding basins. There are basically twochemical types of basins, core (C) and valence (V), that arecharacterized by their synaptic order, which is the number ofcore basins with which they share a common boundary.37

Graphical representations of the bonding are obtained by plottingisosurfaces of the ELF which delimit volumes within whichthe Pauli repulsion is rather weak. The localization domainsare called irreducible when they contain only one attractor andare called reducible otherwise. The reduction of reducibledomains is another criterion of discrimination between basins,and the reductions occur at a critical value of the bondingisosurface. The domains are ordered with respect to the ELFcritical values, yielding bifurcations (tree diagrams).

The partitioning of the molecular space enables basin-relatedproperties to be calculated by integrating the density of a certainproperty over the volume of the basins. Thus, for a basin labeledΩi, one can define the average population as:

and its variance:

The variance is a measure of the quantum mechanicaluncertainty of the basin population, which can be interpretedas a consequence of the electron delocalization. Equation 6 canbe written in terms of the exchange-correlation density,ΓXC-(rb1,rb2), as:

whereλ(Ωi) is the so-called localization index (LI), so that wecan see the varianceσ2(Ωi) is nothing but a measure of totalelectron delocalization into the basinΩi.38 Besides, the relativefluctuation parameter introduced by Bader30,39 indicates the

Figure 1. Geometry of the methylalkalimetal monomer (a), themethylalkalimetal tetramer in the staggered conformation (b), and themethylalkalimetal tetramer in the eclipsed conformation (c).

η ) 1

1 + ( DDh

)2(1)

D )1

F∇2

2Γ(2)σσ( rb1, rb2) )

1

2∑j)1

N

|∇φj|2 -1

8

|∇F|2

F(2)

Dh ) 310

(3π2)2/3F5/3 (3)

F ) ∑j)1

N

|φj|2 (4)

Nh (Ωi) ) ∫ΩiF( rb)drb (5)

σ2(Ωi) ) ⟨(N(Ωi) - Nh (Ωi))2⟩ ) ∫Ωi

Γ(2)( rb1, rb2)drb1drb2 -

Nh (Ωi)[Nh (Ωi) - 1] (6)

σ2(Ωi) ) ∫ΩiΓXC( rb1, rb2)drb1drb2 + Nh (Ωi) )

Nh (Ωi) - λ(Ωi) (7)

7190 J. Phys. Chem. B, Vol. 110, No. 14, 2006 Matito et al.

230

relative electronic delocalization of a particular atomic basin:

which is positive and expected to be less than one in most cases.It is important noticing the difference between this quantity andthe LI in eq 7. WhileλF(Ωi) is a fluctuation parameter that standsfor the ratio of electrons delocalized, the latter is a measure ofthe number of electrons localized into the basinΩi. It is alwaysless than the corresponding atomic population, N(Ωi), exceptfor totally isolated atoms, where there is no exchange orcorrelation with electrons in other atoms.

The variance,σ2(Ωi), can also be spread in terms ofcontributions from other basins, the covariance,V(Ωi,Ωj), whichhas a clear relationship with the so-called delocalization index(DI), δ(Ωi,Ωj):

The DI, δ(Ωi,Ωj), accounts for the electrons delocalized orshared between basinsΩi andΩj.31 As the total variance in acertain basin can be written in terms of covariance, we have:

From this quantity above one can do the usual contributionanalysis (CA), listed on the tables as a percentage:

The contribution analysis gives us the main contributionarising from other basins to the variance, i.e., the delocalizedelectrons of basinΩj on basinΩi, giving us a measure ofelectron pair sharing between two regions of the molecularspace.

It is worth noting that for closed-shell molecules and for singledeterminant wave functions, LIs and DIs can be written in termsof the integralsSkl(Ωi), i.e., the overlap between molecular spin-orbitalsk and l in a basinΩi as:31

where the summations run over all the occupied molecular spin-orbitals. The accuracy of eqs 12 and 13 for computing LIs andDIs at the HF level has been discussed in a previous work.40

Finally, using eqs 7-10 it can be proved that the following

relations between LIs and DIs and variance, covariance, andaverage populations exist:

Quantities such as DIs, variances and covariances have beenused to discuss the degree of ionicity/covalency of a givenbond,31,41 and they will be applied to our systems. Moreover,within the framework of the AIM theory, the magnitude of thedensity (F(rbcp)), of the Laplacian of the density (∇2F(rbcp)), andof the ratio between the perpendicular and the parallel curvaturesat the bond critical point (|λ1|/λ3) have been previously appliedin order to classify an interaction as either ionic or covalent, orhaving an intermediate character.28,32,33,42,43Thus, a clearly ionicinteraction (i.e. LiF) would be characterized by the aboveparameters as:|λ1|/λ3 < 1, F(rbcp) small, and∇2F(rbcp) > 0;and a clearly covalent interaction (i.e. N2) as: |λ1|/λ3 > 1,F(rbcp) large, and∇2F(rbcp) < 0. Finally, the energy density atthe bond critical point,H(rbcp), which is equal to the sum ofthe potential and kinetic energy densities, has been also usedto characterize chemical bonds:33 a negative value ofH(rbcp)indicates that the potential energy dominates, thus denoting aconcentration of charge and a covalent bond, while for positivevalues the kinetic energy is more important, implying a closed-shell interaction.

2.2. Computational Details.Molecular geometries for allthe methylalkalimetals have been fully optimized at the HF/6-311G** level of theory by means of the Gaussian 98 program.44

For comparison purposes, monomers have also been calculatedat the CISD/6-311G** level of theory. All stationary pointsfound have been characterized as either minima ornth-ordersaddle points by computing the vibrational harmonic frequen-cies;nth-order saddle points haven imaginary frequencies whilefor minima all frequencies are real. The wave functions andelectron densities required to carry out the AIM and ELFtopological analyses have also been obtained at the same levelof theory using the Gaussian 98 program. Although DensityFunctional Theory (DFT) performs somehow better for thesesystems,27 the HF method has been chosen because within DFTthe calculation of LIs and DIs is problematic. The reason isthat the second-order density is not available at the DFT leveland must be approximated using a HF-like expression.40

However, HF reproduces qualitative features of trends ingeometry and energy correctly. For each molecule, a topologicalanalysis ofF(r ) has been performed and electron populations(N) have been obtained for each atom, using the AIMPACpackage of programs.45 For the monomers calculated with CISD/6-311G** method, DIs and LIs have been computed at the samelevel of theory using the approximation suggested by Wang andWerstiuk.46 The numerical accuracy of the AIM calculationshas been assessed using two criteria: (i) The integration of∇2F-(r ) within an atomic basin must be close to zero; (ii) The numberof electrons in a molecule must be equal to the sum of all atompopulations of a molecule and also equal to the sum of all theLIs and DIs in the molecule according to eq 14. For all theatomic calculations, integrated values of∇2F(r ) were alwaysless than 0.001 au. For all the molecules, errors in calculatednumber of electrons were always less than 0.01 au. In addition,a visual analysis of the critical points has been carried out bymeans of the AIM2000 software.47 The Gaussian 9844 wavefunction output was also treated with the TopMod package48 in

λF(Ωi) )σ2(Ωi)

Nh (Ωi)(8)

V(Ωi,Ωj) ) ⟨N(Ωi)‚N(Ωj)⟩ - ⟨N(Ωi)⟩⟨N(Ωj)⟩ )

∫Ωi∫Ωj

(Γ(2)( rb1, rb2) - F( rb1)F( rb2))drb1drb2 )

∫Ωi∫Ωj

ΓXC( rb1, rb2)drb1drb2 ) -δ(Ωi,Ωj)

2(9)

σ2(Ωi) ) - ∑j*i

V(Ωi,Ωj) ) ∑j*i

δ(Ωi,Ωj)

2(10)

CA(Ωi|Ωj) )V(Ωi,Ωj)

∑k*j

V(Ωj,Ωk)

× 100) -V(Ωi,Ωj)

σ2(Ωj)‚100 (11)

λ(Ωi) ) ∑k,l

(Skl(Ωi))2 (12)

δ(Ωi,Ωj) ) 2∑k,l

Skl(Ωi)Skl(Ωj) (13)

N ) ∑i

λ(Ωi) +1

2∑

i

(∑j*i

δ(Ωi,Ωj)) ) ∑i

(N(Ωi) -

σ2(Ωi)) - ∑i

(∑j*i

V(Ωi,Ωj)) ) ∑i

N(Ωi) (14)

Bonding in Methylalkalimetals J. Phys. Chem. B, Vol. 110, No. 14, 20067191

231

order to perform the ELF analysis. The Vis5d program49 wasused for the visualization of the ELF.

3. Results and Discussion

The present section is divided into two subsections. The firstone takes into account the methylalkalimetal monomers, andthe second one considers the tetramers. For each section, themolecular structures and the AIM and ELF analyses will bediscussed with special emphasis on the question to what extentthe AIM and ELF topological descriptions agree and where theyyield colliding or differing pictures.

3.1. Methylalkalimetal Monomers. Structures.Table 1summarizes the structural parameters obtained at the HF/6-311G** and CISD/6-311G** levels of theory for the threemethylalkalimetal monomers studied: CH3Li, CH3Na, andCH3K. As it can be seen, the HF/6-311G** C-M bond distanceincreases when descending the periodic table from 1.985 (Li),to 2.341 (Na), and to 2.718 Å (K). The C-H bond distance isquite invariant and amounts to ca. 1.10 Å throughout the series.Meanwhile, the∠MCH angles are also kept quite constant forthe three monomers (111-112°), not varying the pyramidal-ization of theseC3V symmetric structures when going down theperiodic table. This is also observed for the∠HCH angles, whichare almost constant (106-107°).

The above HF/6-311G** geometrical parameters reasonablyagree with CISD/6-311G** results and previous theoreticalstudies7-9,16,27as well as microwave experiments,10,12which alsoyield a monotonic increase of the C-M bond along CH3Li, CH3-Na, and CH3K. Actually, the experimental C-M bond lengths(1.959,12 2.299,12 and 2.633 Å10 for Li, Na, and K, respectively)are systematically shorter, by 2-3%, than our theoretical values.Thus, just focusing on this series of molecules, the optimizationat the HF/6-311G** level of theory can be considered satisfac-tory, achieving structural data quite similar to that yielded bymore expensive methods, like the CISD or the CCSD(T) levelsof theory.8 Metal fluoride distances are also well reproduced atthe HF level.50

AIM Analysis.Table 2 contains the most important propertiesobtained from the AIM analysis. It is worth noting that an AIManalysis of CH3Li species at the HF/6-311G** level has beenalready reported by Ponec and co-workers.6 Here, we comple-ment their analysis by comparing the results obtained for CH3-Li with those of CH3Na and CH3K. The metal atomic chargesfor the three monomers are:+0.914 (Li), +0.799 (Na), and+0.844 (K) electrons, suggesting the picture of a metal cationbound to a carbanion. The charge separation across the C-Mbond in methylalkalimetal monomers decreases from Li to Naand increases from Na to K. This behavior has been already

previously observed and traced to the interplay of two effects:the increasing polarity of the C-M bond when going from Lito Na is overcompensated by the reduction of the participationof the alkalimetal npσ atomic orbital.27 Thus, from the AIMmetal charges, we could conclude that these monomers arehighly (80-90%) ionic systems. However, as pointed outearlier,26,27 atomic charges are very dependent on the schemeof calculation employed, and, as a consequence, they cannotbe used as absolute bond polarity indicators.

Table 3 contains additional parameters derived from the first-order density from a Bader’s theory point of view. From theresults in Table 3, it is seen that for all three cases the parametersobtained are|λ1|/λ3 < 1, F(rbcp) small,∇2F(rbcp) > 0, andH(rbcp)> 0. These values suggest that the C-M interaction inmethylalkalimetal monomers is a typical closed-shell (electro-static) interaction, and, indeed, they are not far from those foundfor the alkalimetal fluorides (see Table 3). In contrast, for apolar covalent bond such as the C-F bond in CH3F, the valuesof F(rbcp) are relatively large andH(rbcp) < 0, as it can be seenin Table 3. According to theF(rbcp) and∇2F(rbcp) values, theionicity of the C-M bond increases when going from Li to K,as expected from the metal electronegative differences. Finally,Table 2 gathers LIs and DIs extracted from the second-orderdensity, together with the localization percentages (%λ), whichfor comparison along the series Li to K are even more valuablethan the LIs. The large values of the localization percentagesin the alkalimetal are also expected for a closed-shell interaction.Descending along the first column of the Periodic Table, themetal %λ increases from 95.2% (Li) to 99.0% (K), in line withthe C-M bond increase of ionicity. The lowδ(C,M) delocal-ization indices are also consistent with a predominantly ionicC-M bond. However, in contrast toF(rbcp), ∇2F(rbcp), and %λ,DIs increase when going from Li (0.177 e) to Na (0.360 e),and slightly decrease from Na to K (0.328 e). This tendency isalso observed if we take into account the methyl group as alonely entity, as it can be seen from theδ(CH3,M) values (Li0.198 e; Na 0.411 e; K 0.376 e), and also if we fix the same

TABLE 1: HF/6-311G** (first row), CISD/6-311G** (secondrow in parentheses), and mm-wave Gas Phase (third row inbrackets) Geometrical Parameters of the MethylalkalimetalMonomers. Distances Are Given in Angstroms (Å) andAngles in Degrees

molecule R(C-M) R(C-H) ∠MCH ∠HCH

CH3Li 1.985 1.095 112.8 105.9(1.981) (1.100) (112.8) (105.9)[1.959]a [1.111]a [106.2]a

CH3Na 2.341 1.092 111.5 107.4(2.335) (1.098) (111.5) (107.3)[2.299]a [1.091]a [106.0]a

CH3K 2.718 1.097 112.8 106.0(2.674) (1.099) (112.9) (105.9)[2.633]b [1.135]b [107.0]b

a Results from ref 12.b Results from ref 10.

TABLE 2: HF/6-311G** (CISD/6-311G** given inparentheses) Atomic Populations (N), Atomic Charges (q),Localization Indices (λ), Localization Percentages (%λ), andDelocalization Indices (δ) Obtained from the AIM Analysisfor the Methylalkalimetal Monomers. Units Are Electrons

molecule atom N q λ %λ pair δ(A,B)

CH3Lia C 6.578 -0.578 4.900 74.5 C,Li 0.177(6.691) (-0.691) (5.222) (78.1) (0.180)

Li 2.086 0.914 1.986 95.2 C,H 1.060(2.091) (0.909) (1.990) (95.2) (0.919)

H 1.112 -0.112 0.521 46.9 H,H′ 0.057(1.073) (-0.073) (0.565) (52.7) (0.044)

CH3,Li 0.198(0.202)

CH3Na C 6.466 -0.466 4.700 72.7 C,Na 0.360(6.560) (-0.560) (5.013) (76.4) (0.352)

Na 10.201 0.799 9.996 98.0 C,H 1.057(10.223) (0.777) (10.021) (98.0) (0.914)

H 1.111 -0.111 0.518 46.6 H,H′ 0.056(1.072) (-0.072) (0.563) (52.5) (0.044)

CH3,Na 0.411(0.352)

CH3K C 6.429 -0.429 4.677 72.8 C,K 0.328(6.491) (-0.491) (4.924) (75.9) (0.353)

K 18.156 0.844 17.968 99.0 C,H 1.058(18.191) (0.809) (17.988) (98.9) (0.928)

H 1.138 -0.138 0.540 47.5 H,H′ 0.061(1.106) (-0.106) (0.584) (52.8) (0.049)

CH3,K 0.376(0.408)

a Some of these results can be found in ref 6.


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∠MCH pyramidalization angle for the three methylalkalimetalmonomers. Thus, instead ofδ(C,M) decreasing when descendingthe Periodic Table due to the C-M bond being more ionic, weobserve the opposite trend found for the atomic charges, i.e.,δ(C,Li) < δ(C,Na) > δ(C,K). In fact, the trend in DIs isgoverned by the atomic charges, as a consequence of thefollowing relationship derived from eq 14:

According to eq 15, a lower positive atomic charge on themetal corresponds to larger atomic populations, and becausethe LIs are almost equal to the number of core electrons for allsystems, this goes with a biggerN(Ωi) - λ(Ωi) difference, thusimplying a largerδ(C,M) value (theδ(H,M) values are similarand close to zero for all MCH3 systems studied). This explainsthe largerδ(C,Na) in NaCH3 as compared toδ(C,Li) in LiCH3.Very recently, Vidal, Melchor, and Dobado19 have studied theCH3K system at the DFT level by means of AIM and ELFtechniques, also attributing a clear ionic character to the C-Kbond. In addition, from Fermi hole analyses, Ponec et al.6 havealso concluded that the C-Li bonds in CH3Li are predominantlyionic. Finally, to demonstrate the validity of the above HFresults, values for AIM analysis at the CISD/6-311G** levelof theory are also shown in Table 2. As one can see, they arepretty close to HF ones, especially those concerning the DIs ofthe polar C-M bonds, stressing once again the adequacy ofthe HF calculations reported. The DIs of the covalent C-Hbonds are somewhat smaller when correlation is included asexpected from previous studies.40,46

ELF Analysis.The ELF method represents an interestingalternative to AIM in order to analyze chemical bonding inmolecules and solids.29,51 The results obtained from the ELFmethod for methyllithium, -sodium, and -potassium are sum-marized in Table 4. In all three cases there are two core

attractors, labeled C(C) and C(M), and corresponding to carbonand metal core electrons, respectively; three protonated disyn-aptic attractors, V(C,Hi), which represent the three C-H bonds,and one monsynaptic attractor, V(C), associated to the valenceelectrons of the carbon atom. As a general trend, for the threemetals, the C(C) basins present populations slightly larger than2 (∼2.05 e); the C(M) basin populations are 2.00, 10.04, and18.06 for Li, Na, and K, respectively, which are the expectedvalues for the core part of each metal atom. With respect to thevalence basins, V(C,Hi) basin populations are slightly lower than2, the typical value for the single bond in a Lewis sense. Thereason for the so marked pairing of electrons is the conditionprovided by the ELF, that seeks for regions on the molecularspace where the probability of finding a couple of electrons ishigher. And finally, the presence of the V(C) basin and the lackof a valence basin associated with both atoms, V(C,M),51

indicates the existence of an unshared-electron interaction29,52

between the metal and the methyl group. This V(C) basin canbe attributed to the nonbonding electron density in the valenceshell of the C, at close distance to the core C(C) basin. Further,there is no monosynaptic V(M) basin, which implies a transferof the electron density of the valence shell of the metal to themethyl group. This view is supported by the V(C) populations,N(V(C)), close to two. Thus, for these three systems, the averagenumber of electrons per basin, obtained by integration of theelectron density function over the basins, corresponds roughlyto the number of electrons expected on chemical grounds foran ionic system. The above classification of basins is verysimilar to that found for the MCCH (M) Li, Na, K) systemsanalyzed by Mierzwicki et al.,53 which also lack the disynapticV(C,M) basin.

To study the electronic delocalization of each basin, aspreviously done in the AIM analysis, it is useful to analyze therelative fluctuation parameter,λF(Ωι), calculated from eq 8. Thecore basins present very low relative fluctuation values,especially the metal core basins,λF(C(C)) ) 0.12 andλF(C(M))

TABLE 3: HF/6-311G** Density Values (G(r bcp)), Laplacian of the Density Values (∇2G(r bcp)), Ratio Values between thePerpendicular and the Parallel Curvatures (|λ1|/λ3), and Local Energy Density (H(r bcp)) at the Bond Critical Point of the C-MBond for the Methylalkalimetal Monomers and Tetramers. For Comparison, the Same Quantities Are Given for the M-FBonds in Alkalimetal Fluorides and the C-F Bond of CH3F. All Quantities Are Given in au

molecule F(rbcp) ∇2F(rbcp) λ1 λ2 λ3 |λ1|/λ3 H(rbcp)

CH3Li 0.0441 0.2180 -0.0634 -0.0634 0.3446 0.1840 0.00143CH3Na 0.0330 0.1500 -0.0336 -0.0336 0.2172 0.1546 0.00235CH3K 0.0293 0.0928 -0.0247 -0.0247 0.1422 0.1736 0.00187(CH3Li) 4 ecl 0.0229 0.1140 -0.0277 -0.0271 0.1688 0.1638 0.00453(CH3Na)4 ecl 0.0155 0.0728 -0.0158 -0.0157 0.1041 0.1521 0.00268(CH3K)4 stg 0.0143 0.0488 -0.0112 -0.0093 0.0692 0.1618 0.00148LiF 0.0701 0.7224 -0.1616 -0.1616 1.0457 0.1545 0.0274NaF 0.0486 0.4300 -0.0767 -0.0767 0.5832 0.1315 0.0156KF 0.0475 0.2844 -0.0577 -0.0577 0.3999 0.1443 0.0059CH3F 0.2351 0.5336 -0.3984 -0.3984 1.3302 0.3000 -0.2993

TABLE 4: HF/6-311G** ELF Values at Attractors ( η), Basin Populations (N(Ωi)), Standard Deviations (σ2(Ωi)), RelativeFluctuations (λF(Ωi)), and Contributions of the Other Basins (%) to σ2(Ωi), Obtained for the Methylalkalimetal Monomers

molecule Ω η N(Ωi) σ2(Ωi) λF(Ωi) contribution analysis

CH3Li C(C) 0.13 2.05 0.24 0.12 28%V(C), 24%V(C,Hi)C(Li) 0.09 2.00 0.07 0.03 71%V(C), 10%V(C,Hi)V(C,Hi) 0.80 1.97 0.68 0.34 36%V(C), 27%V(C,Hi), 9%C(C)V(C) 0.80 2.01 0.85 0.42 28%V(C,Hi), 8%C(Li), 8%C(C)

CH3Na C(C) 0.13 2.05 0.24 0.12 28%V(C), 24%V(C,Hi)C(Na) 0.05 10.04 0.13 0.01 77%V(C), 8%V(C,Hi)V(C,Hi) 0.80 1.99 0.68 0.34 35%V(C), 27%V(C,Hi), 9%C(C)V(C) 0.80 1.92 0.89 0.46 27%V(C,Hi), 11%C(Na), 8%C(C)

CH3K C(C) 0.15 2.07 0.24 0.12 28%V(C), 24%V(C,Hi)C(K) 0.05 18.06 0.16 0.01 79%V(C), 7%V(C,Hi)V(C,Hi) 0.79 1.97 0.67 0.34 36%V(C), 27%V(C,Hi), 9%C(C)V(C) 0.79 1.92 0.90 0.47 27%V(C,Hi), 12%C(K), 8%C(C)

N(Ωi) ) λ(Ωi) +1

2∑j*i

δ(Ωi,Ωj) (15)


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≈ 0.01-0.03, showing the high localization of these electrons,while the protonated disynaptic valence basins present a higherconstant relative fluctuation (0.34) for the three monomers,showing the similar character of this bond. And finally, therelative fluctuation corresponding to the V(C) basin is evenhigher and increases when descending the periodic table from0.42 (Li), to 0.46 (Na), and to 0.47 (K). These fluctuations mustbe complemented with the contribution percentages of otherbasins toσ2(Ωi). If we focus on the V(C) basin, the mostelectronically spread, it is seen how the protonated disynapticbasins account for most of the total population variance of thisbasin (3× 27-28%≈ 80%). The rest of the contribution mainlycomes from the core basin of the carbon atom, which remainsquite constant (8%), and also from the core basin of the metalatom, whose percentage increases with the size of the alkali-metal, as expected. This low contribution from the C(M) to theV(C) reinforces the ionic nature of the C-M bond.

From a more qualitative point of view, Figure 2 contains anELF isosurface plot for methyllithium, where it is possible todistinguish the C(M), C(C), V(C,Hi), and V(C) basins. As theattractor V(C) is close to the core region of the C and it onlycircumscribes the carbon core at low ELFη values (η ≈ 0.1),this bonding situation can be described as ionic.29 The ELFisosurface plots for methylsodium and methylpotassium are notincluded as being visually equivalent. In addition, the bifurcationgraph (see Figure 3 andη values in Table 5) provides a hierarchythat is consistent with the relative fluctuation values. It mustbe mentioned that all three systems present almost the samebifurcation graph with the corresponding ELF values. Thus, theC(M) basin is the first one to be partitioned,η ≈ 0.05-0.09,followed by C(C), withη ≈ 0.13-0.15, and it is not tillη ≈0.80 that the valence basin is split into V(C,Hi) and V(C) basins.It is worth mentioning that if we strictly apply the conditionsgiven by Savin et al.37 to decide whether a valence attractor isconnected to a core attractor or not, the conclusion about thenature of the V(C) basin and the C-M bond would change.The frontier between ionicity and polar covalency was proposedfor bifurcation values ofη ≈ 0.02 for the C(M) basin. Since inour systems the bifurcation occurs atη ≈ 0.05-0.09, thenaccording to this criterion, we would have a V(C,M) basin anda polar covalent C-M bond, especially for CH3Li. However,we consider that the Savin et al. criterion37 is probably too strict(for instance, for LiF the bifurcation occurs at 0.05, and athreshold spectrum ofη ≈ 0-0.1 instead of a sharp cutoff seems

to be more reasonable). Finally, in agreement with AIM analysis,the ELF results (η, σ2, N, λF) also show that the C-Na bondresembles C-K rather than C-Li.

From the relative fluctuation values together with the bifurca-tion graph above, we can attribute an ionic character to the C-Mbond. Thus, both AIM and ELF analyses agree that the C-Mbond is highly polar and associated with ionic bonding.

3.2. Methylalkalimetal Tetramers. Structures. All methyl-alkalimetal tetramers (CH3M)4 have Td symmetry and consistof a tetrahedral cluster of alkalimetal atoms surrounded by fourmethyl groups, one on each M3 face, oriented with respect tothe latter either eclipsed or staggered (see Figure 1). Thestructure looks more or less like a distorted cube, as it can beseen in Figure 4. Depending on the size of the alkalimetal itlooks more like two tetrahedrons, the inner tetrahedron pointingwith its vertexes to the faces of the outer one.

Table 5 summarizes the different geometrical parametersoptimized at the HF/6-311G** level of theory of the distortedcube structures for the three methylalkalimetal tetramers studied,considering the two possible spatial conformations: staggeredand eclipsed for each of them. Tetramerization, i.e., going fromthe monomer to the tetramer, causes the C-M bond to elongatesubstantially by 0.2-0.3 Å, whereas the C-H bond distanceincreases only slightly by 0.01 Å. The distortion of the cubecan be measured by means of theR andâ angles (see Figure 4)and the dihedral angle of a cube side,∠CMCM, which decreaseswhen descending the periodic table, thus getting closer to aregular cube for M) K. There is also a remarkable increase inpyramidalization of the methyl group as follows from the∠HCHangle, which decreases by 3° for lithium and up to 5° for theheavier alkalimetals.

With respect to the eclipsed conformation, the C-M bonddistance increases monotonically when descending the periodictable, from 2.211 (Li), to 2.613 (Na), and to 3.024 Å (K). TheM-M bond distances also increase from Li to Na by about 0.6Å and from Na to K by 0.4 Å. As previously observed for themonomers, the C-H bond distance is kept quite constant at1.10 Å, as well as the extent of pyramidalization of the methylgroups with∠HCH angles of about 102-103°. Compared tothe eclipsed conformation, for the staggered one, the C-H bondsare only marginally shorter (by 0.001 Å), and the methyl groupsslightly less pyramidal (∠HCH angle increases by less than 2°),mainly because of the steric repulsion of the metal on thehydrogen atoms. Our HF/6-311G** molecular structures are ingood agreement with the existing crystal structures (see Table5).

Finally, according to the relative energies obtained for thetetramers studied in both conformations, listed in Table 5, theeclipsed form is the most stable for methyllithium and -sodiumtetramers, whereas the staggered form is the most stable formethylpotassium tetramer. Here, it must be mentioned thatcrystal structures of the three tetramers taken into study alwaysyield the staggered conformation of the methyl groups withrespect to the M3 face to which they are coordinated, whereaswe find the eclipsed orientation to be the lowest-energy structurefor M ) Li and Na.9 Intermolecular interactions and crystalpacking may be responsible for the staggered conformationobserved experimentally for the methyllithium and -sodiumtetramers. The HF/6-311G** frequency analysis shows that thetwo Na conformers and the most stable form for the Li and Ktetramers are minima, whereas the Li tetramer in its staggeredform and the K tetramer in its eclipsed conformation are fourth-order saddle points with a four times degenerate imaginaryfrequency.

Figure 2. ELF isosurface of the methyllithium monomer withη valuesof 0.60 (a) and 0.80 (b). The color scale code used for the localizationdomain is as follows: core, red; valence protonated, green; valencedisynaptic, yellow.

Figure 3. Localization domain reduction tree-diagram of methylal-kalimetal monomers CH3M with M ) Li, Na, and K.


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AIM Analysis. AIM and ELF analyses provide very similarresults for the eclipsed and staggered conformations of themethylalkalimetal tetramers. For this reason, the analysis willbe focused on the most stable conformation for each species(eclipsed for Li and Na, and staggered for K, which are minimain their respective potential energy surfaces). The valuesobtained from the AIM analysis of atomic populations, atomiccharges, LIs, localization percentages, and DIs of these speciesare listed in Table 6. The values corresponding to the least stableconformations can be found in the Supporting Information.

With respect to the monomers, the metal charge is slightlylower by 0.02 e in the methyllithium tetramer, whereas formethylsodium and -potassium tetramers the metal charges are

larger by 0.08 and 0.05 e, respectively. As a result, the threetetramers have almost the same metal charge. Now, the reductionof the npσ AO contribution in the bonding when going from Lito Na is less relevant, as far as the metal charge is concerned,because the npσ admixture to the metal cluster orbitals makesthem point inward into the metal tetrahedron, and the chargecollected by the resulting metal orbitals is almost entirelyaccommodated on the metal cluster.

Table 3 also contains the values for the tetramers of theF-(rbcp), ∇2F(rbcp), H(rbcp), and |λ1|/λ3 parameters. As it can beseen from Table 3, the values ofF(rbcp), ∇2F(rbcp), and|λ1|/λ3

are slightly lower for the tetramers than for the monomers, butthey follow the same tendency:|λ1|/λ3 < 1, F(rbcp) small,∇2F( rbcp) > 0, andH(rbcp) > 0, thus indicating clearly the ionicnature of the C-M bond. As stated for the monomers, the ioniccharacter of this bond regularly increases from Li to K accordingto F(rbcp) and∇2F(rbcp) values. LIs for the metal atom in thetetramers (Table 6) are slightly lower by 0.01 (Li) to 0.05 (K)e in comparison with the corresponding monomers. In line withthis, the localization percentages (%λ) for the tetramers arealmost identical to those of the corresponding monomers andreveal an increase of ionicity from methyllithium to methylpo-tassium.δ(C,M) values for the tetramers are approximatelyreduced to one-third as compared to those found for thecorresponding monomers, because now the metal atom interactswith three equivalent C atoms. These low DIs are characteristicof ionic interactions.43 The increase in DIs when going from Lito K fits with the reduction of LIs in the same direction. Theδ(C,H) values are only slightly inferior to those of the monomers(∼1.04 e). It is also worth noting theδ(C,C′) values, whichdecrease when going from Li (0.056), to Na (0.023), and to K(0.007), especially due to an increase in the C-C′ distance. Itis also interesting to mention theδ(M,M ′) values that are almostimperceptible (0.002-0.004 e), showing that there is noelectronic delocalization between the metal atoms.

On the other hand, Figure 5 depicts the different critical pointsfound for the three tetramers studied in their most stableconformation, including bond, ring, and cage critical points.These representations should help us to characterize the differentbonds present in the tetramers. According to AIM theory, thecriterion for assigning bonding between two atoms is theexistence of a bond critical point connecting them by a gradientpath.54,55Note, however, that this is an unproven premise.56-58

For the methyllithium tetramer, either in the eclipsed or

TABLE 5: HF/6-311G** Relative Energy and Geometrical Parameters of the Eclipsed and Staggered MethylalkalimetalTetramers. Relative Energies Are Given in kcal‚mol-1, Distances in Angstroms (Å), and Angles in Degreesa,b

(CH3Li) 4 (CH3Na)4 (CH3K)4

ecl stg ecl stg ecl stg

∆E 0.00 4.15 0.00 1.79 0.0 -3.05R(C,M) 2.211 2.224 2.613 2.602 3.024 2.978

[2.256(6)]c [2.57-2.68]d [2.947(2), 3.017(4)]e

R(C,H) 1.100 1.098 1.101 1.100 1.104 1.104[1.072(2)]c [1.094]d [1.082(4), 1.103(2)]e

R(M,M) 2.413 2.400 3.015 3.020 3.761 3.746[2.591(9)]c [2.97-3.17]d

R(C,C) 3.641 3.608 4.188 4.162 4.689 4.591[3.621(6)]c

∠HCH 103.02 103.62 102.64 104.26 102.58 103.32[108.2(2)]c [106.2]d [104.8(2), 105.8(2)]e

∠CMCM 23.23 24.06 18.95 18.49 12.69 11.70R 109.36 109.87 106.50 106.18 101.70 100.87â 66.14 65.31 70.46 70.93 76.92 77.95

a R andâ angles correspond to those in Figure 4.b Experimental values in brackets.c Experimental neutron diffraction, 1.5 K.14 d Experimentalneutron+ synchrotron diffraction, 1.5 K.15 e Experimental neutron diffraction, 1.35 K, for the CD3K3 entities in the (CD3K)6 unit cell of themethylpotassium crystal.13

Figure 4. Schematic vision of the methylalkalimetal tetramersgeometry. The anglesR, â and the dihedral∠CMCM are shown.

TABLE 6: HF/6-311G** Atomic Populations (N), AtomicCharges (q), Localization Indices (λ), LocalizationPercentages (%λ), and Delocalization Indices (δ) Obtainedfrom the Atoms in Molecules Analysis for the Most StableMethylalkalimetal Tetramers. Units Are Electrons

molecule atom N q λ %λ pair δ(A,B)

(CH3Li) 4 C 6.533 -0.533 4.763 72.9 C,Li 0.071ecl Li 2.105 0.895 1.972 93.7 C,H 1.043

H 1.121 -0.121 0.527 47.0 H,H′ 0.059C,C′ 0.056Li,Li ′ 0.002

(CH3Na)4 C 6.443 -0.443 4.701 73.0 C,Na 0.088ecl Na 10.122 0.878 9.953 98.3 C,H 1.046

H 1.145 -0.145 0.545 47.7 H,H′ 0.063C,C′ 0.023Na,Na′ 0.003

(CH3K)4 C 6.405 -0.495 4.657 72.7 C,K 0.109stg K 18.116 0.884 17.908 98.9 C,H 1.047

H 1.160 -0.160 0.556 48.0 H,H′ 0.065C,C′ 0.007K,K ′ 0.004


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staggered conformations, it can be seen that each carbon atomis bound to three hydrogen atoms, to the other three carbonatoms (despite the separation of more than 3.5 Å) and at thesame time to the three lithium atoms closest to it, whichconstitute the face of the metal cluster that the methyl group ispointing toward. Thus, taking into consideration the bond criticalpoints, this situation would mean that each carbon is formingnine chemical bonds. A cage critical point in the center of thedistorted cube is also observed, and on each face of the cubewe find two ring critical points, corresponding to C-Li-Ccoupling. This unexpected topological description, observedpreviously at the HF/3-21G level,17 cannot be attributed to anartifact deriving from a too low level of quantum chemicaltheory since it is reproduced using the correlated B3LYP andMP2 methods with the 6-311++G(3df,3pd) basis set. A similarbehavior to that found for the methyllithium tetramer has beenpreviously put forth in the interaction of Mn55 and Ti59 withthe carbons of a cyclopentadienyl ring and in the metallocenesof Al+, Fe, and Ge.34 According to Bader et al.,55 the bondingof a metal atom to an unsaturated ring is not well representedin terms of a set of individual bond paths, but rather by a bondedcone of density. However, there are also a number of authorsthat consider that the conjecture that a bond path between twoatoms in an equilibrium structure implies the presence of achemical bond is not valid.57,60This latter view is supported bythe presence of bond paths between anions in ionic crystals61

and in many H‚‚‚H repulsive interactions in polycyclic aromatichydrocarbons (PAHs) such as kekulene, phenanthrene, chrysene,or benzanthracene56,60,62 and other sterically overcrowdedmolecules.63 In our case, DIs help in the aim of clarifyingwhether there is C-C bonding in these systems. Indeed, thelow δ(C,C′) values clearly point out that there is no direct C-C

bonding.64 Less clear is the situation for the M-M bonding,since metallic bonding is usually associated with low DIs.43

For the methylsodium tetramer, we find that each sodiumatom is bound to the three closest carbon atoms, and each carbonatom is bound to the three hydrogen atoms of the correspondingmethyl group, together to the three closest Na atoms. Howeverthere is no bond critical point between the carbon atoms forthis structure. We also observe a cage critical point, but nowthere is only one ring critical point on each face of the cube,which is less distorted with respect to the Li one. Themethylpotassium tetramer, with the less distorted cube structure,also presents the same critical points. Thus, each carbon atomin Na and K tetramers is connected to six atoms according tothe AIM analysis (cf. Figures 5b and 5c).

ELF Analysis. Table 7 contains the most important parametersobtained from this analysis for the three tetramers in their moststable conformation. The Supporting Information contains theresults for the least stable conformation.

As found in the AIM analysis, ELF values for the monomersand the tetramers do not differ much. First, as it can be seenfrom Table 7, the study has defined the same basins as for themonomers: C(C), C(M), V(C,Hi), and V(C). This proves thatthere is no new interaction when going from the monomer tothe tetramer structure, i.e. new disynaptic attractors as V(C,M),V(C,C′), or V(M,M′).

Focusing on the basin population values for the tetramers,C(C) basins are slightly more populated by 0.02-0.04 e thanin the monomers. The same happens for the population of C(M)basin when M) Li by 0.03 e, whereas for M) Na and K thepopulation decreases by 0.03 and 0.04 e, respectively. Withrespect to the disynaptic basins, the V(C,Hi) populations arealso lower by 0.02-0.03 e in the tetramers than in the

Figure 5. AIM representation of the methylalkalimetal tetramers (CH3M)4 with M ) Li (a), Na (b), and K (c) in eclipsed conformation for Li andNa, and in staggered conformation for K, including the bond (red), ring (yellow), and cage (green) critical points. M is depicted in white, whilehydrogen and carbons are colored in gray and black.

TABLE 7: HF/6-311G** ELF Values at Attractors ( η), Basin Populations (N(Ωi)), Standard Deviations (σ2(Ωi)), RelativeFluctuations (λF(Ωi)), and Contributions of the Other Basins (%) to σ2(Ωi), Obtained for the Most Stable MethylalkalimetalTetramers

molecule Ω η N(Ωi) σ2(Ωi) λF(Ωi) contribution analysis

(CH3Li) 4 C(C) 0.16 2.09 0.25 0.12 28%V(C), 24%V(C,Hi)ecl C(Li) 0.03 2.03 0.10 0.05 22%V(C), 11%V(C,Hi)

V(C,Hi) 0.79 1.95 0.68 0.34 35%V(C), 27%V(C,Hi), 9%C(C)V(C) 0.79 2.03 0.95 0.47 26%V(C,Hi)

(CH3Na)4 C(C) 0.23 2.09 0.25 0.12 28%V(C), 24%V(C,Hi)ecl C(Na) 0.03 10.01 0.14 0.01 25%V(C), 8%V(C,Hi)


(CH3K)4 C(C) 0.16 2.09 0.24 0.12 28%V(C), 24%V(C,Hi)stg C(K) 0.05 18.02 0.19 0.01 19%V(C), 14%V(C,Hi)



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monomers. Meanwhile, V(C) basin populations are those thatundergo the most important changes when forming the tetramer,increasing by 0.02 (Li)- 0.08 (Na, K) e, and at the same timeequalizing their populations (2.00-2.03 e), thus reinforcing theionic character with respect to the monomers. Less importantare the changes on the relative fluctuation parameter (λF(Ωι)),which is kept almost unaltered for the three first basins, whileonly V(C) is increased by 0.05, thus showing a higher electronicdelocalization for this basin. The same happens for the contribu-tion analysis results also enclosed in Table 7, which presentvery similar values to those found in the monomers.

If we look at the ELF isosurface plots depicted in Figure 6for the methyllithium tetramer in its eclipsed conformation, morevaluable information can be obtained. ELF isosurface plots forthe other systems studied have not been enclosed as they arealmost visually equivalent to these ones. First, the abovecommented four different basins can be easily distinguished,especially in the plot with ELF) 0.80. In this case, it is clearlyseen how the monosynaptic V(C) basin does not point to anyof the three closest monosynaptic C(Li) basins, but to the centerof a triangular face of the tetrahedron defined by the Li4 unit.The V(C) basin does not lie in the line connecting the cores,but the line connecting the C(C) and the center of the threeC(Li), pretty close to the core basin of C, and circumscribing it(see Figure 6a). So this bonding situation can also be describedas ionic, like for the monomers. Each methyl group ionicallyinteracts with one face of the tetrahedron created with three Liatoms. This ELF picture is very similar to that found for Li4H4,in which each H localizes one valence electron of the Li atoms.65

The same is true for the other metals and different conformersanalyzed in the present work.

Again, the bifurcation analysis scheme (see Figure 7) providesa hierarchy that is consistent with the relative fluctuation values.For the tetramers, the C(M) basin is partitioned earlier,η ≈0.03-0.05, than for the monomers, indicating a minor increasein ionicity when going from the monomers to the tetramers,but C(C) basin is partitioned later withη ≈ 0.16-0.23. Andagain, it is not untilη ≈ 0.75-0.80 that the valence basin issplit into V(C,Hi) and V(C) basins. So, also for this analysis,the similarity of the results with respect to those obtained bythe monomers makes us attribute a noticeable ionic characterto the C-M interaction in methylalkalimetal tetramers.

4. Conclusions

The AIM and ELF topological approaches partially agree butalso show significant divergences in the description of thebonding in methylalkalimetals (CH3M)n with M ) Li, Na, K, n) 1, 4. They agree that the C-M bond in these compounds ishighly polar. However, whereas AIM indicates that tetramer-ization of CH3M slightly reduces the polarity, ELF suggeststhe opposite.

More importantly, and also more strikingly, AIM yieldsnonavalent carbons in tetramethyllithium. According to AIM,the carbon atoms in this methylalkalimetal have three individualbonds to the three closest hydrogen atoms, three individualbonds to the three closest metal atoms, and three individualbonds to the three closest carbon atoms. At variance, ELF yieldstetravalent carbon atoms that form three covalent bonds withtheir hydrogens plus one polar bond with a triangular face ofthe central metal cluster. This ELF topological result fits wellwith the bonding picture that emerges from quantitative Kohn-Sham molecular orbital analyses.16

Acknowledgment. Financial help has been furnished by theSpanish MEC Project No. CTQ2005-08797-C02-01/BQU, TheNetherlands Organization for Scientific Research (NWO), andthe HPC-Europa as well as the Training and Mobility ofResearchers (TMR) programs of the European Union. E.M. andJ.P. thank the Ministerio de Educacio´n y Ciencia for the doctoralfellowship no. AP2002-0581 and the DURSI for the postdoctoralfellowship 2004BE00028, respectively. Excellent service by theStichting Academisch Rekencentrum Amsterdam (SARA) andthe Centre de Supercomputacio´ de Catalunya (CESCA) isgratefully acknowledged.

Supporting Information Available: Tables containing theAIM and ELF results for the least stable (CH3M)4 (M ) Li,Na, and K) conformers. This material is available free of chargevia the Internet at http://pubs.acs.org.


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Figure 7. Localization domain reduction tree-diagram of methylal-kalimetal tetramers (CH3M)4 with M ) Li, Na, and K.


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238

The Bonding in Methylalkalimetal Oligomers (CH3M)n

(M = Li – K; n = 1, 4). Agreement and divergences between AIM and

ELF Analysesξ

Eduard Matito,1 Jordi Poater,*2 F. Matthias Bickelhaupt2 and Miquel Solà*1

1 Institut de Química Computacional and Departament de Química, Universitat de Girona, 17071 Girona, Catalonia, Spain 2 Afdeling Theoretische Chemie, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands

Supporting Information

Table 6-SI S2

Table 7-SI S3

S 1

239

Table 6-SI. HF/6-311G** atomic populations (N), atomic charges (q), localization

indices (λ), localization percentages (%λ), and delocalization indices (δ) obtained from

the Atoms in Molecules analysis for the least stable methylalkalimetal tetramers. Units

are electrons.

Molecule Atom N q λ %λ Pair δ(A,B)

(CH3Li)4 C 6.600 -0.600 4.830 73.2 C,Li 0.070

stg Li 2.100 0.900 1.974 94.0 C,H 1.045

H 1.100 -0.100 0.507 46.1 H,H’ 0.057

C,C’ 0.054

Li,Li’ 0.002

(CH3Na)4 C 6.500 -0.500 4.750 73.0 C,Na 0.100

stg Na 10.120 0.880 9.960 98.0 C,H 1.060

H 1.130 -0.130 0.530 47.0 H,H’ 0.060

C,C’ 0.026

Na,Na’ 0.003

(CH3K)4 C 6.377 -0.377 4.644 72.8 C,K 0.103

ecl K 18.121 0.879 17.909 98.8 C,H 1.041

H 1.167 -0.167 0.565 48.4 H,H’ 0.066

C,C’ 0.007

K,K’ 0.004

S 2

240

Table 7-SI. HF/6-311G** ELF values at attractors (η), basin populations (N(Ωi)),

standard deviations (σ2(Ωi)), relative fluctuation (λF(Ωi)), and contributions of the other

basins (%) to σ2(Ωi), obtained for the least stable methylalkalimetal tetramers.

Molecule Ω η N(Ωi) σ2(Ωi) λ F(Ωi) Contribution analysis

(CH3Li)4 C(C) 0.16 2.09 0.25 0.12 28%V(C),24%V(C,Hi)

stg C(Li) 0.02 2.03 0.09 0.05 33%V(C)

V(C,Hi) 0.79 1.95 0.68 0.35 36%V(C),27%V(C,Hi),9%C(C)

V(C) 0.79 2.03 0.95 0.47 26%V(C,Hi)

(CH3Na)4 C(C) 0.23 2.09 0.25 0.12 28%V(C),24%V(C,Hi)

stg C(Na) 0.03 10.01 0.12 0.01 25%V(C),8%V(C,Hi)

V(C,Hi) 0.75 1.98 0.68 0.34 36%V(C),27%V(C,Hi),9%C(C)

V(C) 0.75 1.95 0.92 0.47 26%V(C,Hi)

(CH3K)4 C(C) 0.25 2.09 0.24 0.12 28%V(C),24%V(C,Hi)

ecl C(K) 0.05 18.02 0.19 0.01 26%V(C),8%V(C,Hi)

V(C,Hi) 0.79 1.96 0.67 0.34 36%V(C),26%V(C,Hi),12%C(C)

V(C) 0.79 2.00 0.92 0.46 26%V(C,Hi)

S 3

241

5.3.- Comment on the “Nature of Bonding in the Thermal

Cyclization of (Z)-1,2,4,6-Heptatetraene and Its

Heterosubstituted Analogues”.

243

Comment on the “Nature of Bonding in theThermal Cyclization of (Z)-1,2,4,6-Heptatetraeneand Its Heterosubstituted Analogues”

Eduard Matito, Miquel Sola`,* Miquel Duran, andJordi Poater*

Institut de Quı´mica Computacional andDepartament de Quı´mica, UniVersitat de Girona,17071 Girona, Catalonia, Spain

ReceiVed: May 7, 2004; In Final Form: September 24, 2004

In a recent paper by Chamorro and Notario (CN),1 thepericyclic or pseudopericyclic character of the thermal cycliza-tion of (Z)-1,2,4,6-heptatetraene (C) and its heterosubsti-tuted analogues (A and B) (see Scheme 1) was discussed bymeans of a topological electron localization function (ELF)analysis applied to the corresponding transition states (TSs).The aim of this work was to explore whether an ELF canprovide further insight in the context of a recent controversy2-5

concerning the nature of the TSs involved in reactionsA andB. The authors concluded that their ELF results, which weremainly based on the examination of electron density fluctua-tion, provide proof in support of a single disrotatory pericyclicbond interaction (mechanism 1 in Scheme 1) for reactionsA,B, andC.

In our opinion, the analysis of electron density fluctuationused to reach this conclusion is not solid enough to warrantsuch an assertion. To make our point, we have optimized theTS and computed the ELF (at the same B3LYP/6-31+G(d) levelof theory using the HF/6-31G** optimized geometry of the TSand with the same programs and visualization tools used in thework by CN1) for a well-established pseudopericyclic reaction(mechanism 2 in Scheme 1), namely, the cyclization of 5-oxo-2,4-pentadienal to pyran-2-one (reactionD in Scheme 1), thatwas not considered in the study of CN.1 The TS of reactionDwas optimized at the HF/6-31G** level (its geometry can be

found in the Supporting Information) because both the B3LYP/6-31G** (see ref 5) and B3LYP/6-31+G(d) methods give areaction path without an energy barrier. This TS was character-ized by the existence of a unique imaginary frequency corre-sponding to the ring closure, as well as by computing theintrinsic reaction path from the TS going downhill to reactantsand products in mass-weighted Cartesian coordinates. Eventhough the original discussion was focused on the three reactionsstudied by CN,1-4 Rodrıguez-Otero et al.5 introduced this fourthreaction, which is of great importance because it allows thecomparison of the controversialA and B reactions with areaction that has an unquestionable single disrotatory pericyclicTS (reactionC) and another that has an undeniable pseudo-pericyclic nature (reactionD).

CN1 gave basically three reasons to assign a pericyclic natureto reactionsA andB: (i) the lone pair of N in TSA or O in TSB has a low contribution to the whole process, so that it can beconsidered just a mere stabilization factor; (ii) the populationsand fluctuations of the basins resemble those expected inpericyclic reactions; and more importantly, (iii) the measuresof cyclic electron density fluctuations from one basin to itscontiguous basins, following a clockwise or anticlockwisecirculation around the ring reaction center, are essentially thesame for reactionsA, B, andC.

Taking into account the results obtained for reactionD listedin Table 1, these are our remarks to the aforementioned threearguments:

(i) In the pseudopericyclic reactionD, the contribution of thebasin V(C2,C3)6 to the total electron delocalization of the O7lone pair basin facing the C2 atom (basin number10 in Table1) is 3.4% (3.3-3.7%), which is even lower than the contribu-tions of the basin V(C2,C3) for the equivalent electron pairdelocalization in reactionsA, B, and C.1 Therefore, a lowcontribution of the basin V(C2,C3) to the heteroatom electronlone pair delocalization is not a conclusive indication that thelone pair plays a mere stabilization role and that, as aconsequence, the reaction is pericyclic.

SCHEME 1. Suggested Reaction Mechanisms for the Thermal Cyclization of (Z)-1,2,4,6-Heptatetraene and ItsHeterosubstituted Analogues via a Pericyclic Disrotatory Pathway (Mechanism 1) or through PseudopericyclicNucleophilic Addition (Mechanism 2)

7591J. Phys. Chem. B2005,109,7591-7593

10.1021/jp048033e CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 03/30/2005

245

(ii) Comparison of basin populations for reactionD in Table1 with those populations of reactionsA, B, andC in Tables1-3 of ref 1 shows that basin populations in reactionsA andBare closer to either those of reactionsC or D depending on thebasin considered. Therefore, basin populations are not veryinformative for discerning between the pericyclic and pseudo-pericyclic nature of reactionsA andB. In addition, the variancesof the basin populations are not enlightening enough, since thedifferences and similarities of reactionsA andB with respectto C andD lead one to conclude that reactionsA andB havea character intermediate betweenC andD, somewhat closer toC yet not far fromD.

(iii) CN1 based their strongest argument in favor of thepericyclic character of reactionsA andB on a measure of cyclicelectron flow in terms of covariance data. When analyzing thecomplete cyclic fluctuation pattern, CN compared covariancecontributions coming from related basins corresponding todifferent reactions. In our opinion, this is not strictly correct,since the compared covariance contributions came from basinshaving different electronic populations. In this sense, we believethat more consistent results can be obtained by analyzingcovariance contributions arising from the same basin. Inparticular, we recommend comparing covariance contributionsto a given bonding basin coming from adjacent bonding andlone pair basins for the clockwise and counterclockwisedirections to determine the direction in which the electronfluctuation is larger. This procedure is supported by the resultsof a study of cyclic π-conjugated molecules, where thecomparison of electron fluctuation in both directions for eachbasin has been found to be a good indicator of aromaticity.7

Figure 1 depicts the ratios between percentages of covariancecontributions in both directions, with an arrow pointing to thepreferred way of fluctuation.8 When two consecutive bondsfluctuate charge in opposite directions, the atom between suffersfrom electron depletion (no arrows pointing to the atom) orelectron concentration (both arrows pointing toward the atom),whose magnitude can be counted in terms of listed ratios (seethe numbers above the bonds). In reactionC, no electron

accumulation is observed in the ring atoms resembling a typicalaromatic situation, as expected for a pericyclic reaction. Clearelectron depletions are observed in atoms 3 and 6 of reactionD, while an electron accumulation is found on atoms 7 and 5.A very similar trend is observed for reactionB, but with adepletion on atom 4 instead of 3. Finally, reactionA has almostthe same fluctuation pattern asB, with both being far from thearomatic-like behavior expected for a pseudopericyclic reaction,although the covariance contribution ratios are somewhat lowerin A. Thus, in light of these ELF results, we conclude thatBmore resembles a pseudopericyclic reaction than a pericyclic

TABLE 1: B3LYP/6-31+G(d) Basin Populations,Ni, Standard Deviations,σ2(Ni), Relative Fluctuations,λ(Ni), and MainContributions of Other Basins, i(%), to σ2(Ni) for the Transition State of Reaction Da

basin N(Ωi) σ2[N(Ωi)] λ[N(Ωi)] contribution analysis (%)

1 V(H1,C3) 2.06 0.66 0.32 2 (3.0) 5 (13.6) 6 (16.7) 7 (3.0) 11 (24.2)12 (10.6) 13 (10.6) 14 (3.0)

2 V(H2,C4) 2.10 0.61 0.29 1 (3.4) 3 (5.1) 11 (30.5) 14 (39.0) 15 (3.4)3 V(H3,C5) 2.12 0.65 0.31 2 (4.8) 4 (3.2) 14 (48.4) 15 (27.4)4 V(H4,C6) 2.13 0.56 0.27 3 (3.7) 10 (5.6) 14 (5.6) 15 (31.5) 16 (25.9)

17 (13.0)5 V(C2,C3) 1.30 0.84 0.64 1 (11.0) 6 (32.9) 7 (7.3) 8 (6.1) 9 (6.1)

10 (3.7) 11 (9.8) 12 (7.3)6 V(C2,C3) 1.47 0.94 0.64 1 (12.0) 5 (29.3) 7 (6.5) 8 (5.4) 9 (6.5)

10 (3.3) 11 (12.0) 12 (3.3) 13 (8.7)7 V(O1) 4.72 1.42 0.30 5 (4.3) 6 (4.3) 8 (33.3) 9 (33.3)8 V(C2,O1) 1.53 0.99 0.65 5 (5.1) 6 (5.1) 7 (48.0) 9 (24.5) 10 (4.1)9 V(C2,O1) 1.51 0.95 0.63 5 (5.4) 6 (6.5) 7 (50.5) 8 (25.8)10 V(O7) 2.32 1.17 0.51 8 (3.5) 15 (3.5) 16 (28.1) 17 (43.0)11 V(C3,C4) 2.39 1.14 0.48 1 (14.3) 2 (16.1) 5 (7.1) 6 (9.8) 12 (7.1)

13 (7.1) 14 (18.8)12 V(C3) 0.39 0.34 0.86 1 (23.3) 2 (3.3) 5 (20.0) 6 (10.0) 7 (3.3)

11 (26.7) 13 (6.7) 14 (3.3)13 V(C3) 0.39 0.34 0.86 1 (22.6) 2 (3.2) 5 (6.5) 6 (25.8) 7 (3.2)

11 (25.8) 12 (6.5) 14 (3.2)14 V(C4,C5) 3.25 1.42 0.44 2 (16.4) 3 (21.4) 11 (15.0) 15 (23.6)15 V(C5,C6) 2.39 1.11 0.47 3 (15.7) 4 (15.7) 10 (3.7) 14 (30.6) 16 (13.0)

17 (3.7)16 V(C6,O7) 2.20 1.21 0.55 4 (11.9) 10 (27.1) 15 (11.9) 17 (35.6)17 V(O7) 3.04 1.27 0.42 4 (5.6) 10 (39.5) 15 (3.2) 16 (33.9)

a Only percentages above 3% are listed.

Figure 1. Electron fluctuation in terms of contribution analysis. Arrowspoint to the atom where the electron fluctuation of the bond electronpair is larger.

7592 J. Phys. Chem. B, Vol. 109, No. 15, 2005 Comments

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one, whileA shows an intermediate situation, yet closer to thepseudopericyclic character.

The phenomenon described above is further supported by ELFrepresentations. From Figure 1 of CN, at ELF) 0.40,9 it isclearly seen that TSs in reactionsA and B present a highelectronic localization around atoms C3 and C5, while C4 andC6 core basins are easily visualized due to the poor localizationaround them. The same is observed in Figure 2, where the ELF) 0.6 picture for reactionD is drawn.

The closer proximity of reactionsA andB to pseudopericyclicreactionD can be reinforced by observing the absence of a cy-clic loop of interacting orbitals by means of the contributionanalyses (see Tables 1-3 of CN and Table 1 of the presentwork). For reactionsA andB, it is seen how V(C6,X) (X) N(A), O (B)) does not contribute to V(C2,C3) and, conversely,how V(C2,C3) contributes to V(X) but not to V(C6,X). On theother hand, for reactionC (Table 3 of CN), V(C6,C7)contributes to V(C2,C3) and, in turn, this to V(C6,C7). Thus,one can follow the contributions around the ring, achieving acyclic bonding which is fundamental in the definition ofpericyclic reactions. Finally, reactionD (see Table 1) presentsthe same contribution behavior as reactionsA andB.

As a whole, we think that our work proves that CN haveincorrectly assigned a pericyclic nature to reactionsA andB.Our conclusion is achieved by means of a covariance analysisbased on contributions arising from the same basin and bycomparison to the reaction of the cyclization of 5-oxo-2,4-pentadienal to pyran-2-one (D), with an already assignedpseudopericyclic nature.

Acknowledgment. Financial help has been furnished by theSpanish MCyT Projects Nos. BQU2002-0412-C02-02 andBQU2002-03334.M.S.isindebtedtotheDepartamentd’Universitats,

Recerca i Societat de la Informacio´ (DURSI) of the Generalitatde Catalunya, for financial support through the DistinguishedUniversity Research Promotion 2001. E.M. thanks the Secretarıáde Estado de Educacioń y Universidades of the MECD for thedoctoral fellowship no. AP2002-0581. We also thank the Centrede Supercomputacio´ de Catalunya (CESCA) for partial fundingof computer time. The authors are indebted to the Referee forfruitful comments.

Supporting Information Available: Cartesian coordinatescorresponding to the geometry of the transition state of reactionD, calculated at the HF/6-31G** level of theory. This materialis available free of charge via the Internet at http://pubs.acs.org.

References and Notes(1) Chamorro, E. E.; Notario, R.J. Phys. Chem. A2004, 108, 4099.(2) de Lera, A. R.; Alvarez, R.; Lecea, B.; Torrado, A.; Cossıó, F. P.

Angew. Chem., Int. Ed.2001, 40, 557.(3) de Lera, A. R.; Cossıó, F. P.Angew. Chem., Int. Ed.2002, 41,

1150.(4) Rodrıguez-Otero, J.; Cabaleiro-Lago, E. M.Angew. Chem., Int. Ed.

2002, 41, 1147.(5) Rodrıguez-Otero, J.; Cabaleiro-Lago, E. M.Chem.sEur. J.2003,

9, 1837.(6) In reactionD, the V(C2,C3) basin is split into two, but this does

not affect the conclusions.(7) Matito, E.; Duran, M.; Sola`, M. J. Chem. Phys.2005, 122, 014109.(8) For instance, the value 1.50 assigned to the C5-C6 bond in Figure

1 for the TS of reactionD is obtained by dividing the covariance percentageof the V(C5,C6) basin population in the V(C4,C5) basin (counterclockwisedirection) by those of the V(C6,O7) and V(O7) basins (clockwise direction),that is, 30.6/(13.0+ 3.7 + 3.7) ) 1.5. Only percentages above 3%, thoselisted in Table 1, are considered.

(9) The ELF pictures in ref 1 were depicted at ELF) 0.84, ELF)0.40, and ELF) 0.04. We could reproduce the 0.04 and 0.84 pictures, butwe failed in recreating ELF) 0.40. However, the same pictures arise whenan ELF value close to 0.60 is used. Thus, we attribute our failing toreproduce ELF) 0.40 to a possible misprint in ref 1. Bifurcation diagramsin ref 1 support this hypothesis.

Figure 2. Localization domains of the ELF at the transition state of reactionD, depicted for isosurface values of 0.04, 0.60, and 0.84. Core basinsin red, V(C-H) in light blue, V(C-C) in green, and V(C3) and V(O)s in blue.

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5.4.- Electron fluctuation in Pericyclic and Pseudopericyclic

Reactions.

249

DOI: 10.1002/cphc.200500446

Electron Fluctuation in Pericyclicand Pseudopericyclic Reactions

Eduard Matito,[a] Jordi Poater,*[b] Miquel Duran,[a] andMiquel Sol*[a]

Since the very beginning pericyclic reactions have presenteddifficult mechanistic problems. It was not until Woodward andHoffmann set up their theory of orbital symmetry conservationthat the mechanisms of these reactions clearly showed up. Afew years after that, Ross et al.[1] proposed the term “pseudo-pericylic” to describe a set of reactions which broke the lawsgiven by Woodward and Hoffmann. So far, the most noticeablework towards a proper characterization of pseudopericyclic re-actions has been conducted by Birney and co-workers.[2–4]

Zhou and Birney[4] described a pseudopericyclic reaction as areaction with a low or a nonexistent barrier, with planar transi-tion structure and cyclic overlap disconnections. Although thisdefinition is widely accepted by the chemistry community,these criteria become controversial when trying to classify bor-derline cases.[5–11] Some authors have proved the energy barrierto be an ambiguous criterion, since some pseudopericyclic re-actions exhibit significant energy barriers.[6, 8] Planarity is alsofar from being a definitive criterion; some authors reportedquite planar pericyclic reactions[6,8] while other authors foundfar-from-planar pseudopericyclic mechanisms.[4] Another popu-lar descriptor to characterize the reaction mechanism is the ar-omaticity of the transition state (TS). As stated by the Evan’sprinciple,[12] one would expect pericyclic reactions to have aro-matic TSs, unlike pseudopericyclic reactions where the discon-nected cyclic overlap prevents aromaticity. However, one hasto bear in mind that aromaticity is a multidimensional quantity,that is, different indexes can lead to different conclusions.[8–10]

In this Communication we study the electron fluctuation ofthe TS structure for a set of different electrocyclic processes(A–H) by means of the electron localization function (ELF).[13, 14]

The electron distribution in the TS will provide enough proofto discern between both mechanisms. In a pericyclic reactionthe electron charge is spread among the bonds involved inthe rearrangement, and that is the reason for aromatic TSs. Be-sides, pseudopericyclic reactions are characteristic for electronaccumulations and depletions on different atoms, and thus theelectron distribution in the TS is far from being uniform for the

bonds involved in the rearrangement. Since ELF accounts forthe electron distribution we expect connected (delocalized)pictures of bonds in pericyclic reactions, while pseudopericyclicreactions will give rise to disconnected (localized) pictures.The following set of reactions,[15] containing several kinds of

processes, serves our purposes: A, B, C, and D are electrocyclicring closures that have been recently a matter of discus-sion,[5–11] E is a cycloaddition of acetylketene to acetone,[3] F aClaisen rearrangement,[16] G is an electrocyclic ring opening,[2]

and H the butadiene electrocyclic ring closure (reactionschemes are given in the Supporting Information). Reactions C,F, and H are generally accepted to be pericyclic while D, E, andG have been characterized as pseudopericyclic. On the otherhand, A and B are controversial reactions which have beenclassified as pericyclic or pseudopericyclic by different au-thors.[5–11] In particular, these reactions have been recently con-sidered as a borderline case between pericyclic and pseudo-pericyclic.[10] Although the ELF pictures are clear enough to elu-cidate the reaction mechanism (the reader can notice that corebasins are clearly visualized due to poor delocalization in pseu-dopericyclic reactions), fluctuation diagrams are also providedin Figures 1 and 2 to give the preferred electron fluctuation di-rection and an idea of its magnitude. The electron fluctuationis measured as the relative electron sharing between a givenelectron pair bond and the adjacent bonds, showing the pre-ferred direction of electron fluctuation (see ref. [11] for the de-tailed definition). We compare covariance contributions to agiven bonding basin coming from adjacent bonding and thelone pair basins[17] for the clockwise and counterclockwise di-rections to determine the direction in which the electron fluc-tuation is larger. When two consecutive bonds fluctuatecharge in opposite directions, the atom in between suffersfrom electron depletion (no arrows pointing to the atom) orelectron concentration (both arrows pointing towards theatom), whose magnitude can be deduced from the ratiosgiven above the arrows. The arrows on the figures give the di-rection of fluctuation with an estimation of its magnitude cal-culated as the ratio between the fluctuation of the clockwiseand counterclockwise directions. This procedure is supportedby a recent study on cyclic p-conjugated molecules, where therelative ratio of electron fluctuation on both sides of thebonds in the ring was found to be a good indicator of aroma-ticity.[18]

In the TSs of near barrierless reactions, we expect almost noelectron density on the sigma bonds that are to be formed,since the TSs of these reactions resemble the reactants. Hence,we focus on the electron distribution of the p bonds that arebroken and formed in the TS. The electron organization be-comes quite evident from Figures 1 and 2,[19] and thus, onewould characterize reactions A, B, D, E, and G as pseudopericy-clic since they exhibit electron depletions and accumulationsof fluctuation (cf. diagrams on Figures 1 and 2 where severalpairs of arrows point towards or depart from the same atom),breaking the cyclic uniform electron distribution expected in apericyclic process. On the other hand reactions C, F, and H ex-hibit a cyclic electron distribution without electron depletions

[a] E. Matito, Prof. Dr. M. Duran, Prof. Dr. M. SolInstitut de Qumica Computacional and Departament de QumicaUniversitat de Girona, 17071 Girona, Catalonia (Spain)Fax: (+34)972-418-356E-mail : [email protected]

[b] Dr. J. PoaterAfdeling Theoretische Chemie, Scheikundig Laboratoriumder Vrije Universiteit, De Boelelaan 10831081 HV Amsterdam (The Netherlands)Fax: (+31)205-987-629

Supporting information for this article is available on the WWW underhttp://www.chemphyschem.org or from the author.

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or accumulations of fluctuation for bonds involved in the p-electron rearrangement, and therefore they are pericyclic.Various criteria are used to discern between a pericyclic and

pseudopericyclic mechanism, a recent one is also based on anELF analysis,[20] but as yet none of them have been proved de-finitive except for the recent work by LKpez et al. ,[21] whichprovides a way to characterize a restricted group of electrocy-clic processes (those involving ring closures) by means of ellip-ticity calculations along the reaction path. On the other hand,the here-reported electron distribution of TS studied by meansof the ELF provides a clear picture of the electron rearrange-ment, which reveals the mechanism of the reaction for severalelectrocyclic processes with the only need of the TS structure.We believe that the qualitative conclusions reached hereinshould not be altered in a significant way by the use of largerbasis sets and higher-level theoretical methods. In particular,the TS geometry and the energy barrier of reactions A-C havebeen calculated using larger basis sets and more accurate

methods yielding the samequalitative results.[9] Further-more, some authors have dem-onstrated the adequacy ofthese calculations to study peri-cyclic processes by showingthat the B3LYP/6-31G* resultsare similar to those derivedfrom experimental data or moreaccurate theoretical calcula-tions.[22–25] In addition, it isworth noticing that the ELFfunction has been proven quiteinsensible to basis-set changesand only slight changes havebeen observed with exact calcu-lations on small atoms.[26]

Acknowledgments

Financial help has been furnishedby the Spanish MCyT project Nos.BQU2002-0412-C02-02,CTQ2005-08797-C02-01/BQU and BQU2002-03334. We are indebted toProfessor D. M. Birney for fruitfulsuggestions.

Keywords: electrocyclicreactions · electron localizationfunction · pericyclic reactions ·transition states

[1] J. A. Ross, R. P. Seiders, D. M.Lemal, J. Am. Chem. Soc. 1976, 98,4325–4327.

[2] D. M. Birney, J. Am. Chem. Soc.2000, 122, 10917–10925.

[3] W. Shumway, S. Ham, J. Moer, B. R.Whittlesey, D. M. Birney, J. Org. Chem. 2000, 65, 7731–7739.

[4] C. Zhou, D. M. Birney, J. Am. Chem. Soc. 2002, 124, 5231–5241.[5] E. E. Chamorro, R. Notario, J. Phys. Chem. B 2005, 109, 7594–7595.[6] A. R. de Lera, R. Alvarez, B. Lecea, A. Torrado, F. P. CossOo, Angew. Chem.

2001, 113, 570–574; Angew. Chem. Int. Ed. 2001, 40, 557–561.[7] E. E. Chamorro, R. Notario, J. Phys. Chem. A 2004, 108, 4099–4104.[8] A. R. de Lera, F. P. CossOo, Angew. Chem. 2002, 114, 1198–1200; Angew.

Chem. Int. Ed. 2002, 41, 1150–1152.[9] J. RodrOguez-Otero, E. M. Cabaleiro-Lago, Angew. Chem. 2002, 114,1195–1198; Angew. Chem. Int. Ed. 2002, 41, 1147–1150.

[10] J. RodrOguez-Otero, E. M. Cabaleiro-Lago, Chem. Eur. J. 2003, 9, 1837–1843.

[11] E. Matito, M. Sol, M. Duran, J. Poater, J. Phys. Chem. B 2005, 109, 7591–7593.

[12] M. J. S. Dewar, Angew. Chem. 1971, 83, 859–875; Angew. Chem. Int. Ed.Engl. 1971, 10, 761–776.

[13] A. D. Becke, K. E. Edgecombe, J. Chem. Phys. 1990, 92, 5397–5403.[14] A. Savin, R. Nesper, S. Wengert, T. F. FPssler, Angew. Chem. 1997, 109,

1892–1918; Angew. Chem. Int. Ed. Engl. 1997, 36, 1809–1832.[15] All calculations were performed at the B3LYP/6-31G* level of theory

with Gaussian ( M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, R. E.Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N.

Figure 1. Scheme of the transition-state structures together with relative electron fluctuation magnitudes andELF=0.60 pictures for reactions A–D.

Figure 2. Scheme of the transition-state structures together with relative electron fluctuation magnitudes andELF=0.60 pictures for reactions E–H.

112 www.chemphyschem.org G 2006 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim ChemPhysChem 2006, 7, 111 – 113

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www.chemphyschem.org

Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B.Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson,P. Y. Ayala, Q. Cui, K. Morokuma, P. Salvador, J. J. Dannenberg, D. K.Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V.

Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Ko-maromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham,C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. G. Johnson,

W. Chen, M. W. Wong, J. L. Andres, R. Gonzalez, M. Head-Gordon, E. S.Replogle, J. A. Pople,Frisch, Gaussian 98, rev. A11, Pittsburgh, PA; 1998).Frequency calculations enabled TS characterization.

[16] O. Wiest, K. A. Black, K. N. Houk, J. Am. Chem. Soc. 1994, 116, 10336–10337.

[17] Unlike in ref. [9] , the fluctuation of the C2C1 bond in reaction C dueto adjacent CH bonds has also been considered to count the samenumber of formal bonds on each side.

[18] E. Matito, M. Duran, M. Sol, J. Chem. Phys. 2005, 122, 0141091–0141098.

[19] ToPMoD program (S. Noury, X. Krokidis, F. Fuster, B. Silvi, Comput. Chem.1999, 23, 597) was used for the ELF calculations. The cutoff for the ELFpictures was set to 0.60.

[20] E. Chamorro, J. Chem. Phys. 2003, 118, 8687–8698.[21] C. S. LKpez, O. N. Faza, F. P. CossOo, D. M. York, A. R. de Lera, Chem. Eur. J.

2005, 11, 1734–1738.[22] B. R. Beno, K. N. Houk, D. A. Singleton, J. Am. Chem. Soc. 1996, 118,

9984–9985.[23] T. C. Dinadayalane, R. Vijaya, A. Smitha, G. N. Sastry, J. Phys. Chem. A

2002, 106, 1627–1633.[24] H. Isobe, S. Yamanaka, K. Yamaguchi, Int. J. Quantum Chem. 2003, 95,

532–545.[25] O. Wiest, K. N. Houk, Top. Curr. Chem. 1996, 183, 1–24.[26] M. Kohout, A. Savin, J. Comput. Chem. 1997, 18, 1431–1439.

Received: August 4, 2005Revised: September 20, 2005Published online on November 30, 2005

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A

NHNHNH

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(N) 2.11 0.30 0.14 8 (3.4) 12 (24.1) 15 (27.6) 16 (41.4) 17 (3.4) 2 C(C1) 2.09 0.26 0.12 8 (28.0) 12 (4.0) 15 (32.0) 16 (4.0) 17 (28.0) 18 (4.0) 3 C(C2) 2.10 0.26 0.12 9 (29.2) 10 (4.2) 17 (29.2) 18 (37.5) 4 C(C3) 2.09 0.26 0.12 9 (4.0) 10 (28.0) 18 (36.0) 19 (28.0) 20 (4.0) 5 C(C4) 2.09 0.26 0.12 11 (30.4) 19 (30.4) 20 (39.1) 6 C(C5) 2.08 0.25 0.12 11 (4.0) 13 (4.0) 14 (4.0) 19 (4.0) 20 (40.0) 21 (24.0) 22 (20.0)7 C(C6) 2.09 0.26 0.13 13 (26.9) 14 (26.9) 20 (3.8) 21 (19.2) 22 (23.1) 8 V(H1,C1) 2.16 0.65 0.30 2 (11.1) 9 (3.2) 12 (4.8) 15 (33.3) 16 (9.5) 17 (30.2) 18 (4.8) 9 V(H2,C2) 2.12 0.66 0.31 3 (10.9) 8 (3.1) 10 (4.7) 17 (28.1) 18 (43.7) 19 (3.1) 10 V(H3,C3) 2.12 0.65 0.31 4 (10.9) 9 (4.7) 11 (4.7) 17 (3.1) 18 (39.1) 19 (29.7) 20 (4.7) 11 V(H4,C4) 2.08 0.68 0.32 5 (10.4) 10 (4.5) 19 (28.4) 20 (41.8) 12 V(H5,N) 1.98 0.75 0.38 1 (9.6) 8 (4.1) 15 (32.9) 16 (45.2) 13 V(H6,C6) 2.07 0.66 0.32 7 (10.8) 14 (27.7) 20 (4.6) 21 (24.6) 22 (26.2) 14 V(H7,C6) 2.05 0.67 0.33 7 (10.8) 13 (27.7) 20 (4.6) 21 (24.6) 22 (26.2) 15 V(N,C1) 2.69 1.33 0.49 1 (6.1) 2 (6.1) 8 (15.9) 12 (18.2) 16 (32.6) 17 (15.2) 16 V(N) 2.75 1.28 0.47 1 (9.4) 8 (4.7) 12 (25.8) 15 (33.6) 17 (4.7) 20 (7.0) 21 (5.5) 17 V(C1,C2) 2.42 1.16 0.48 2 (6.1) 3 (6.1) 8 (16.5) 9 (15.7) 15 (17.4) 16 (5.2) 18 (25.2)18 V(C2,C3) 3.19 1.44 0.45 3 (6.2) 4 (6.2) 9 (19.4) 10 (17.4) 17 (20.1) 19 (17.4) 19 V(C3,C4) 2.47 1.20 0.48 4 (5.8) 5 (5.8) 10 (15.8) 11 (15.8) 18 (20.8) 20 (25.0) 20 V(C4,C5) 3.41 1.49 0.44 5 (6.0) 6 (6.7) 11 (18.8) 16 (6.0) 19 (20.1) 21 (13.4) 22 (14.1)21 V(C5,C6) 1.88 1.04 0.55 6 (5.7) 7 (4.8) 13 (15.2) 14 (15.2) 16 (6.7) 20 (19.0) 22 (26.7)22 V(C5,C6) 1.92 1.04 0.54 6 (5.0) 7 (5.9) 13 (16.8) 14 (16.8) 20 (20.8) 21 (27.7)

1.54

1.16

1.111.10

1.82

NH5

4

3

2 1

Pseudopericyclic: The ELF picture is discontinuous around

regions involving C1 and C3, whereas there is electron

accumulation over C2 and C4 (cf. fluctuation diagram).

Thereby, it is a pseudopericyclic reaction.

254

B

OOO

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(O) 2.12 0.34 0.16 8 (3.0) 14 (30.3) 15 (21.2) 16 (42.4) 17 (3.0) 2 C(C1) 2.08 0.25 0.12 8 (29.2) 14 (4.2) 15 (25.0) 16 (4.2) 17 (33.3) 18 (4.2) 3 C(C2) 2.09 0.26 0.13 9 (30.4) 17 (34.8) 18 (34.8) 4 C(C3) 2.09 0.26 0.12 10 (29.2) 17 (4.2) 18 (33.3) 19 (29.2) 20 (4.2) 5 C(C4) 2.09 0.26 0.12 11 (29.2) 19 (33.3) 20 (37.5) 6 C(C5) 2.08 0.25 0.12 12 (4.2) 13 (4.2) 19 (4.2) 20 (37.5) 21 (25.0) 22 (25.0) 7 C(C6) 2.09 0.26 0.13 12 (26.9) 13 (26.9) 20 (3.8) 21 (19.2) 22 (23.1) 8 V(H1,C1) 2.18 0.64 0.29 2 (11.5) 9 (3.3) 14 (6.6) 15 (23.0) 16 (13.1) 17 (34.4) 18 (4.9) 9 V(H2,C2) 2.11 0.66 0.31 3 (10.9) 8 (3.1) 10 (4.7) 17 (35.9) 18 (35.9) 19 (3.1)

10 V(H3,C3) 2.12 0.64 0.30 4 (11.3) 9 (4.8) 11 (3.2) 17 (4.8) 18 (37.1) 19 (32.3) 20 (4.8) 11 V(H4,C4) 2.08 0.67 0.32 5 (10.8) 10 (3.1) 18 (3.1) 19 (32.3) 20 (40.0) 22 (3.1) 12 V(H5,C6) 2.05 0.67 0.33 7 (10.6) 13 (27.3) 20 (4.5) 21 (24.2) 22 (25.8) 13 V(H6,C6) 2.07 0.67 0.32 7 (10.6) 12 (27.3) 20 (4.5) 21 (24.2) 22 (25.8) 14 V(O) 2.27 1.19 0.52 1 (8.5) 8 (3.4) 15 (26.3) 16 (39.8) 17 (4.2) 20 (5.9) 21 (4.2) 15 V(O,C1) 2.07 1.18 0.57 1 (6.0) 2 (5.1) 8 (12.0) 14 (26.5) 16 (32.5) 17 (13.7) 16 V(O) 3.03 1.27 0.42 1 (10.9) 8 (6.2) 14 (36.7) 15 (29.7) 17 (4.7) 20 (3.1) 17 V(C1,C2) 2.80 1.32 0.47 2 (6.0) 3 (6.0) 8 (15.8) 9 (17.3) 14 (3.8) 15 (12.0) 16 (4.5) 18 (24.1)18 V(C2,C3) 2.86 1.34 0.47 3 (6.0) 4 (6.0) 9 (17.3) 10 (17.3) 17 (24.1) 19 (18.0) 20 (3.0) 19 V(C3,C4) 2.65 1.27 0.48 4 (5.6) 5 (6.3) 10 (15.9) 11 (16.7) 18 (19.0) 20 (25.4) 20 V(C4,C5) 3.19 1.45 0.45 5 (6.2) 6 (6.2) 11 (17.8) 14 (4.8) 19 (21.9) 21 (12.3) 22 (13.7) 21 V(C5,C6) 1.90 1.04 0.55 6 (5.9) 7 (4.9) 12 (15.7) 13 (15.7) 14 (4.9) 20 (17.6) 22 (28.4) 22 V(C5,C6) 1.93 1.04 0.54 6 (5.8) 7 (5.8) 12 (16.5) 13 (16.5) 20 (19.4) 21 (28.2)

1.27

1.29

1.381.14

3.46

O5

4

3

2 1

Pseudopericyclic: The ELF picture is analogous to the

previous one: disconnections around regions involving C1

and C3, and electron accumulation over C2 and C4. Thus it

is also a pseudopericyclic reaction.

255

C

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(C1) 2.09 0.26 0.12 8 (4.0) 12 (28.0) 15 (28.0) 16 (36.0) 17 (4.0) 2 C(C2) 2.09 0.26 0.12 8 (29.2) 16 (37.5) 17 (29.2) 18 (4.2) 3 C(C3) 2.09 0.26 0.12 9 (30.4) 17 (30.4) 18 (39.1) 4 C(C4) 2.09 0.26 0.12 10 (29.2) 18 (37.5) 19 (29.2) 20 (4.2) 5 C(C5) 2.09 0.26 0.12 11 (29.2) 18 (4.2) 19 (29.2) 20 (37.5) 6 C(C6) 2.09 0.25 0.12 11 (4.0) 13 (4.0) 14 (4.0) 19 (4.0) 20 (44.0) 21 (20.0) 22 (20.0) 7 C(C7) 2.09 0.26 0.12 13 (26.9) 14 (26.9) 20 (3.8) 21 (19.2) 22 (23.1) 8 V(H1,C2) 2.13 0.66 0.31 2 (10.9) 9 (3.1) 12 (3.1) 15 (4.7) 16 (39.1) 17 (31.3) 18 (4.7) 9 V(H2,C3) 2.12 0.65 0.31 3 (11.3) 8 (3.2) 10 (4.8) 16 (3.2) 17 (30.6) 18 (41.9) 19 (3.2)

10 V(H3,C4) 2.12 0.65 0.31 4 (11.3) 9 (4.8) 11 (4.8) 17 (3.2) 18 (40.3) 19 (30.6) 20 (3.2) 11 V(H4,C5) 2.08 0.67 0.32 5 (10.6) 10 (4.5) 18 (3.0) 19 (30.3) 20 (40.9) 22 (3.0) 12 V(H5,C1) 2.11 0.70 0.33 1 (10.4) 15 (28.4) 16 (38.8) 20 (6.0) 21 (4.5) 13 V(H6,C7) 2.07 0.66 0.32 7 (10.8) 14 (29.2) 20 (6.2) 21 (21.5) 22 (26.2) 14 V(H7,C7) 2.09 0.66 0.32 7 (10.4) 13 (28.4) 20 (6.0) 21 (20.9) 22 (25.4) 15 V(H8,C1) 2.12 0.65 0.31 1 (11.1) 8 (4.8) 12 (30.2) 16 (41.3) 17 (3.2) 20 (4.8) 16 V(C1,C2) 3.14 1.45 0.46 1 (6.2) 2 (6.2) 8 (17.1) 12 (17.8) 15 (17.8) 17 (17.8) 20 (6.2) 17 V(C2,C3) 2.51 1.20 0.48 2 (6.0) 3 (6.0) 8 (17.1) 9 (16.2) 16 (22.2) 18 (23.1) 18 V(C3,C4) 3.11 1.43 0.46 3 (6.3) 4 (6.3) 9 (18.3) 10 (17.6) 17 (19.0) 19 (18.3) 20 (3.5) 19 V(C4,C5) 2.51 1.21 0.48 4 (5.8) 5 (5.8) 10 (15.7) 11 (16.5) 18 (21.5) 20 (24.0) 20 V(C5,C6) 3.64 1.58 0.44 5 (5.7) 6 (7.0) 11 (17.1) 16 (5.7) 18 (3.2) 19 (18.4) 21 (13.9) 22 (14.6)21 V(C6,C7) 1.70 0.99 0.58 6 (5.2) 7 (5.2) 12 (3.1) 13 (14.6) 14 (14.6) 16 (3.1) 20 (22.9) 22 (27.1)22 V(C6,C7) 1.90 1.03 0.54 6 (5.0) 7 (5.9) 13 (16.8) 14 (16.8) 20 (22.8) 21 (25.7)

1.51

1.09

1.03

1.03

1.02

6

5

4

3

1

2

Pericyclic: The ELF picture exhibits high delocalization

over the whole ring, and is thus pericyclic. The electron

distribution is extremely symmetric along the whole ring

but in the regions near the closure; as stated in the

Communication: this region resembles the reactants one,

thus one must focus his attention to other bonds

broken/formed, which density has already change.

256

NH

O

NH

HN

O

HN NH

HN

O

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%)

1 C(C1) 2.09 0.26 0.12 9 (32.0) 10 (32.0) 11 (4.0) 14 (28.0) 23 (4.0) 2 C(C2) 2.09 0.26 0.12 13 (30.4) 15 (34.8) 16 (30.4) 24 (4.3) 3 C(C3) 2.09 0.26 0.12 12 (30.4) 15 (34.8) 17 (30.4) 18 (4.3) 4 C(C4) 2.08 0.25 0.12 15 (4.2) 17 (37.5) 19 (8.3) 20 (29.2) 21 (16.7) 22 (4.2) 5 C(N1) 2.11 0.30 0.14 9 (3.4) 10 (3.4) 11 (27.6) 14 (31.0) 21 (20.7) 23 (13.8) 6 C(O) 2.11 0.33 0.16 17 (3.1) 19 (37.5) 20 (21.9) 22 (37.5) 7 C(N2) 2.10 0.29 0.14 8 (23.3) 13 (3.3) 15 (3.3) 16 (26.7) 24 (43.3) 8 V(H1,N2) 1.94 0.77 0.40 7 (9.3) 10 (4.0) 13 (4.0) 15 (4.0) 16 (29.3) 24 (45.3) 9 V(H2,C1) 2.20 0.66 0.30 1 (12.1) 10 (30.3) 11 (4.5) 14 (31.8) 21 (3.0) 23 (4.5) 24 (4.5)10 V(H3,C1) 2.20 0.68 0.31 1 (12.1) 8 (4.5) 9 (30.3) 11 (3.0) 14 (30.3) 21 (3.0) 23 (4.5) 24 (4.5)

11 V(H4,N1) 2.10 0.78 0.37 5 (10.3) 9 (3.8) 14 (33.3) 21 (21.8) 23 (17.9) 12 V(H5,C3) 2.06 0.67 0.32 3 (10.8) 13 (3.1) 15 (33.8) 17 (29.2) 18 (9.2) 13 V(H6,C2) 2.14 0.64 0.30 2 (11.3) 8 (4.8) 12 (3.2) 15 (32.3) 16 (30.6) 17 (3.2) 24 (11.3)14 V(C1,N1) 2.86 1.40 0.49 1 (5.0) 5 (6.5) 9 (15.1) 10 (14.4) 11 (18.7) 21 (15.1) 23 (14.4)15 V(C2,C3) 2.77 1.33 0.48 2 (6.0) 3 (6.0) 12 (16.4) 13 (14.9) 16 (15.7) 17 (18.7) 18 (6.0)

24 (4.5) 16 V(C2,N2) 2.58 1.30 0.50 2 (5.5) 7 (6.3) 8 (17.2) 13 (14.8) 15 (16.4) 24 (33.6) 17 V(C3,C4) 2.79 1.32 0.47 3 (5.4) 4 (7.0) 12 (14.7) 15 (19.4) 18 (6.2) 19 (5.4) 20 (14.0)

21 (11.6) 22 (4.7) 18 V(C3) 0.34 0.30 0.88 3 (3.8) 12 (23.1) 13 (3.8) 15 (30.8) 16 (3.8) 17 (30.8) 20 (3.8)19 V(O) 2.66 1.20 0.45 6 (10.2) 17 (5.9) 20 (33.1) 22 (42.4) 20 V(C4,O) 2.29 1.28 0.56 4 (5.5) 6 (5.5) 17 (14.2) 19 (30.7) 21 (7.9) 22 (31.5) 21 V(C4,N1) 1.83 1.00 0.55 4 (3.9) 5 (5.9) 11 (16.7) 14 (20.6) 17 (14.7) 20 (9.8) 22 (3.9)

23 (13.7) 22 V(O) 2.71 1.22 0.45 6 (10.1) 17 (5.0) 19 (42.0) 20 (33.6) 21 (3.4) 23 V(C4,N1) 0.90 0.65 0.73 5 (6.2) 9 (4.7) 10 (4.7) 11 (21.9) 14 (31.3) 21 (21.9) 24 V(N2) 2.96 1.18 0.40 7 (11.0) 8 (28.8) 13 (5.9) 15 (5.1) 16 (36.4)

3 4

2 2

HN

O

HN1.42

7.20

1.82

1.41

2.67

1

1

Pseudopericyclic: The ELF picture is clearly cut for regions

around C4 and C2 where one would expect continuous

electron distribution (homogenous fluctuation)

D

257

E

O

O O

O

O O

O

OO

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(C1) 2.09 0.26 0.13 12 (26.9) 13 (26.9) 14 (26.9) 21 (19.2) 2 C(C2) 2.09 0.25 0.12 21 (28.6) 22 (33.3) 23 (28.6) 24 (4.8) 31 (4.8) 3 C(C3) 2.09 0.26 0.13 11 (28.0) 22 (28.0) 25 (16.0) 26 (16.0) 27 (8.0) 28 (4.0) 4 C(C4) 2.09 0.25 0.12 25 (22.7) 26 (22.7) 29 (13.6) 30 (40.9) 5 C(O1) 2.12 0.34 0.16 29 (71.9) 30 (28.1) 6 C(O2) 2.13 0.34 0.16 23 (20.6) 24 (38.2) 31 (35.3) 7 C(O3) 2.14 0.34 0.16 32 (30.3) 33 (42.4) 34 (21.2) 35 (3.0) 36 (3.0) 8 C(C5) 2.08 0.25 0.12 32 (4.3) 33 (8.7) 34 (26.1) 35 (30.4) 36 (30.4) 9 C(C6) 2.09 0.26 0.13 18 (26.9) 19 (26.9) 20 (26.9) 36 (19.2)

10 C(C7) 2.09 0.26 0.13 15 (28.0) 16 (28.0) 17 (24.0) 35 (20.0) 11 V(H1,C3) 2.01 0.69 0.34 3 (10.4) 22 (23.9) 25 (14.9) 26 (14.9) 27 (10.4) 28 (10.4) 12 V(H2,C1) 1.99 0.65 0.32 1 (11.3) 13 (30.6) 14 (29.0) 21 (24.2) 13 V(H3,C1) 1.97 0.65 0.33 1 (10.8) 12 (29.2) 14 (29.2) 21 (24.6) 14 V(H4,C1) 2.00 0.65 0.33 1 (10.9) 11 (3.1) 12 (28.1) 13 (29.7) 21 (23.4) 15 V(H5,C7) 1.99 0.65 0.33 10 (10.9) 16 (28.1) 17 (29.7) 35 (23.4) 16 V(H6,C7) 2.00 0.66 0.33 10 (11.1) 15 (28.6) 17 (28.6) 35 (23.8) 17 V(H7,C7) 1.95 0.65 0.34 10 (9.4) 15 (29.7) 16 (28.1) 35 (25.0) 18 V(H8,C6) 1.96 0.66 0.34 9 (10.6) 19 (28.8) 20 (27.3) 36 (24.2) 19 V(H9,C6) 1.97 0.65 0.33 9 (10.8) 18 (29.2) 20 (29.2) 36 (24.6) 20 V(H10,C6) 2.00 0.64 0.32 9 (11.3) 18 (29.0) 19 (30.6) 36 (24.2) 21 V(C1,C2) 2.02 1.01 0.50 1 (5.0) 2 (6.0) 12 (15.0) 13 (16.0) 14 (15.0) 22 (15.0) 23 (13.0) 24 (6.0) 22 V(C2,C3) 2.40 1.16 0.48 2 (6.1) 3 (6.1) 11 (14.0) 21 (13.2) 23 (13.2) 24 (4.4) 25 (8.8) 26 (7.9) 27 (7.9) 28 (7.0) 31 (4.4)

23 V(C2,O2) 2.15 1.23 0.57 2 (4.9) 6 (5.7) 21 (10.7) 22 (12.3) 24 (32.0) 31 (27.9)

24 V(O2) 2.88 1.26 0.44 6 (10.5) 21 (4.8) 22 (4.0) 23 (31.5) 31 (40.3) 32 (4.2) 25 V(C3,C4) 1.56 0.95 0.61 3 (4.2) 4 (5.2) 11 (10.4) 22 (10.4) 26 (24.0) 27 (9.4) 29 (8.3) 30 (13.5)26 V(C3,C4) 1.46 0.90 0.62 3 (4.5) 4 (5.6) 11 (11.2) 22 (10.1) 25 (25.8) 28 (9.0) 29 (7.9) 30 (12.4) 32 (3.4)

27 V(C3) 0.43 0.37 0.85 3 (5.7) 11 (20.0) 22 (25.7) 25 (25.7) 26 (5.7) 28 (5.7) 28 V(C3) 0.40 0.35 0.86 3 (3.1) 11 (21.9) 21 (3.1) 22 (25.0) 23 (3.1) 25 (6.3) 26 (25.0) 27 (6.3) 29 V(O1) 4.91 1.45 0.29 5 (16.1) 25 (5.6) 26 (4.9) 30 (60.8) 29 (3.1) 30 (3.1) 30 V(C4,O1) 2.79 1.44 0.52 4 (6.3) 5 (6.3) 25 (9.2) 26 (7.7) 29 (61.3) 32 (4.2) 31 V(O2) 2.55 1.22 0.48 6 (9.9) 22 (4.1) 23 (28.1) 24 (41.3) 32 V(O3) 2.13 1.15 0.54 7 (8.8) 25 (3.5) 30 (5.3) 33 (39.8) 34 (26.5) 35 (3.5) 33 V(O3) 3.11 1.30 0.42 7 (10.9) 32 (35.2) 34 (32.8) 35 (3.9) 36 (4.7) 34 V(O3,C5) 2.20 1.23 0.56 7 (5.6) 8 (4.8) 32 (24.2) 33 (33.9) 35 (10.5) 36 (10.5) 36 (13.9)35 V(C5,C7) 2.06 1.02 0.49 8 (6.9) 10 (5.0) 15 (14.9) 16 (14.9) 17 (15.8) 32 (4.0) 33 (5.0) 34 (12.9)36 V(C5,C6) 2.06 1.02 0.49 8 (6.9) 9 (5.0) 18 (15.8) 19 (15.8) 20 (14.9) 33 (5.9) 34 (12.9) 35 (13.9)

258

Only percentages above 3% are listed.

O

O O5

3

4

32

2

1.23

1.342.15

13.27

Pseudopericyclic: The ELF picture exhibits high

localization over atoms O2, C3, and O3, while there is

electron depletion (core parts are easily visualized

indicating no valence electron localization near the nucleus)

on atoms C2, C4 and C5. The transition state shows a

typical pseudopericyclic reaction with alternative

localization, far away from the aromatic picture expected in

a pericyclic reaction. Thus, the reaction occurs through a

pseudopericyclic mechanism.

259

F

O OO

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(C1) 2.09 0.26 0.12 10 (29.2) 12 (4.2) 15 (37.5) 16 (29.2) 2 C(C2) 2.09 0.25 0.12 8 (30.4) 9 (30.4) 15 (4.3) 16 (34.8) 3 C(C3) 2.09 0.26 0.12 10 (4.0) 11 (28.0) 12 (28.0) 15 (36.0) 16 (4.0) 4 C(O) 2.13 0.34 0.16 7 (3.1) 17 (40.6) 18 (37.5) 19 (15.6) 20 (3.1) 5 C(C4) 2.09 0.26 0.12 7 (4.2) 13 (29.2) 14 (29.2) 20 (37.5) 6 C(C5) 2.09 0.26 0.12 7 (28.0) 13 (4.0) 14 (4.0) 17 (4.0) 18 (4.0) 19 (20.0) 20 (36.0)7 V(H1,C5) 2.20 0.66 0.30 6 (10.6) 13 (4.5) 14 (4.5) 17 (9.1) 18 (6.1) 19 (18.2) 20 (39.4)8 V(H2,C2) 2.13 0.64 0.30 2 (11.3) 9 (32.3) 10 (3.2) 15 (4.8) 16 (33.9) 17 (4.8) 18 (6.5) 9 V(H3,C2) 2.14 0.67 0.31 2 (10.9) 8 (31.3) 10 (3.1) 15 (4.7) 16 (32.8) 17 (7.8) 18 (4.7) 10 V(H4,C1) 2.12 0.66 0.31 1 (10.8) 8 (3.1) 9 (3.1) 11 (3.1) 12 (4.6) 15 (40.0) 16 (30.8)11 V(H5,C3) 2.13 0.67 0.31 3 (10.8) 10 (3.1) 12 (30.8) 15 (36.9) 16 (3.1) 20 (6.2) 12 V(H6,C3) 2.15 0.66 0.31 3 (10.8) 10 (4.6) 11 (30.8) 15 (38.5) 16 (3.1) 20 (4.6) 13 V(H7,C4) 2.13 0.67 0.32 5 (10.3) 7 (4.4) 14 (29.4) 15 (4.4) 20 (39.7) 14 V(H8,C4) 2.15 0.67 0.31 5 (10.6) 7 (4.5) 13 (30.3) 15 (3.0) 20 (40.9) 15 V(C1,C3) 3.11 1.45 0.47 1 (6.1) 3 (6.1) 10 (17.7) 11 (16.3) 12 (17.0) 16 (20.4) 20 (4.8) 16 V(C1,C2) 2.66 1.29 0.49 1 (5.4) 2 (6.2) 8 (16.2) 9 (16.2) 10 (15.4) 15 (23.1) 17 (3.8)

18 (4.6) 17 V(O) 2.78 1.22 0.44 4 (10.6) 7 (4.9) 9 (4.1) 16 (4.1) 18 (39.8) 19 (24.4) 20 (4.9) 18 V(O) 2.72 1.23 0.45 4 (9.8) 7 (3.3) 8 (3.3) 16 (4.9) 17 (40.2) 19 (24.6) 20 (5.7) 19 V(O,C5) 1.76 1.06 0.60 4 (4.8) 6 (4.8) 7 (11.5) 17 (28.8) 18 (28.8) 20 (15.4) 20 V(C4,C5) 3.24 1.48 0.46 5 (6.1) 6 (6.1) 7 (17.6) 13 (18.2) 14 (18.2) 15 (4.7) 17 (4.1)

18 (4.7) 19 (10.8) Only percentages above 3% are listed.

O3.09

1.16 2.43

1.95

1

2

34

5

Pericyclic: The ELF picture exhibits delocalization over the

atoms bonded, the arrows point towards different atoms and

no electron depletion or accumulation is noticed over these

atoms; the ELF isosurface exhibits a continue electron

distribution (with the exception of the closure/opening

region) indicating a quite aromatic structure.

260

G

O

H

O

H

O

H

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(C1) 2.09 0.26 0.12 7 (32.0) 11 (32.0) 12 (24.0) 13 (4.0) 18 (8.0) 2 C(C2) 2.09 0.26 0.13 8 (29.2) 9 (4.2) 11 (29.2) 13 (37.5) 3 C(C3) 2.08 0.26 0.12 9 (32.0) 11 (4.0) 13 (32.0) 14 (28.0) 15 (4.0) 4 C(C4) 2.08 0.25 0.12 10 (4.0) 14 (32.0) 15 (48.0) 16 (8.0) 17 (4.0) 18 (4.0) 5 C(O) 2.12 0.34 0.16 12 (20.6) 17 (17.6) 18 (52.9) 6 C(C5) 2.08 0.25 0.12 10 (29.2) 14 (4.2) 15 (33.3) 16 (33.3) 7 V(H1,C1) 2.21 0.63 0.29 1 (12.7) 8 (3.2) 11 (33.3) 12 (22.2) 13 (4.8) 15 (3.2) 17 (3.2) 18 (14.3) 8 V(H2,C2) 2.15 0.65 0.30 2 (11.1) 7 (3.2) 9 (3.2) 11 (31.7) 13 (42.9) 14 (3.2) 9 V(H3,C3) 2.14 0.65 0.30 3 (12.7) 8 (3.2) 11 (3.2) 13 (36.5) 14 (31.7) 15 (6.3)

10 V(H4,C5) 1.96 0.62 0.32 6 (11.3) 14 (3.2) 15 (38.7) 16 (38.7) 11 V(C1,C2) 2.63 1.25 0.48 1 (6.5) 2 (5.6) 7 (16.9) 8 (16.1) 12 (12.1) 13 (25.8)

18 (5.6) 12 V(C1,O) 2.05 1.15 0.56 1 (5.2) 5 (6.1) 7 (12.2) 11 (13.0) 17 (14.8) 18 (42.6)13 V(C2,C3) 3.01 1.40 0.47 2 (6.5) 3 (5.8) 8 (19.4) 9 (16.5) 11 (23.0) 14 (17.3)14 V(C3,C4) 2.52 1.19 0.47 3 (5.8) 4 (6.7) 9 (16.7) 13 (20.0) 15 (28.3) 16 (5.8)

17 (5.8) 18 (3.3) 15 V(C4,C5) 3.49 1.53 0.44 4 (7.8) 6 (5.2) 10 (15.6) 14 (22.1) 16 (24.7) 17 (7.1)

18 (5.8) 16 V(C5) 1.97 0.90 0.46 6 (9.1) 10 (27.3) 14 (8.0) 15 (43.2) 17 V(C4,O) 1.43 0.90 0.63 5 (6.7) 12 (18.9) 14 (7.8) 15 (12.2) 18 (43.3) 18 V(O) 3.89 1.46 0.38 5 (12.3) 7 (6.2) 11 (4.8) 12 (33.6) 15 (6.2) 17 (26.7)


O

1.28

1.63

8.83

1.30

3.491

2

3

4

Pseudopericyclic: The ELF picture exhibits high localization

over the oxygen and the other atoms in odd positions. The

electron distribution is not delocalised over the ring, but

localized alternatively unlike a typical pericyclic reaction;

indeed, is a pseudopericyclic reaction.

261

H


1

3

2

4

1.00

1.00

1.00

1.00

Pericyclic: The ELF picture exhibits high delocalization

over the whole ring. The electron distribution is extremely

symmetric and all bonds delocalise the same relative charge

to their neighbour bonds. It is a typical pericyclic reaction.

basin N(Ωi) σ2[N(Ωi)] λ[Ν(Ωi)] Contribution analysis (%) 1 C(C1) 2.08 0.24 0.12 5 (37.5) 6 (12.5) 7 (29.2) 8 (4.2) 9 (4.2) 10 (4.2) 11 (4.2) 12 (4.2) 2 C(C2) 2.07 0.24 0.12 5 (4.2) 6 (12.5) 7 (4.2) 8 (37.5) 9 (29.2) 10 (4.2) 11 (4.2) 12 (4.2) 3 C(C3) 2.07 0.24 0.12 5 (4.3) 7 (30.4) 8 (4.3) 9 (4.3) 10 (39.1) 11 (13.0) 12 (4.3) 4 C(C4) 2.07 0.24 0.12 5 (4.3) 7 (4.3) 8 (4.3) 9 (30.4) 10 (4.3) 11 (13.0) 12 (39.1) 5 V(C1) 2.14 1.00 0.47 1 (9.0) 6 (20.0) 7 (22.0) 8 (16.0) 9 (8.0) 10 (8.0) 11 (5.0) 12 (9.0) 6 V(C1,C2) 1.29 0.87 0.67 1 (3.5) 2 (3.5) 5 (23.5) 7 (15.3) 8 (23.5) 9 (15.3) 10 (5.9) 11 (3.5) 12 (5.9)7 V(C1,C3) 2.27 1.10 0.48 1 (6.3) 3 (6.3) 5 (19.8) 6 (11.7) 8 (7.2) 9 (8.1) 10 (19.8) 11 (11.7) 12 (7.2)8 V(C2) 2.14 1.00 0.47 2 (9.0) 5 (16.0) 6 (20.0) 7 (8.0) 9 (22.0) 10 (9.0) 11 (5.0) 12 (8.0) 9 V(C2,C4) 2.27 1.10 0.48 2 (6.3) 4 (6.3) 5 (7.2) 6 (11.7) 7 (8.1) 8 (19.8) 10 (7.2) 11 (11.7) 12 (19.8)10 V(C3) 2.14 1.00 0.47 3 (9.0) 5 (8.0) 6 (5.0) 7 (22.0) 8 (9.0) 9 (8.0) 11 (20.0) 12 (16.0) 11 V(C3,C4) 1.29 0.87 0.67 3 (3.4) 4 (3.4) 5 (5.7) 6 (3.4) 7 (14.9) 8 (5.7) 9 (14.9) 10 (23.0) 12 (23.0)12 V(C4) 2.14 1.00 0.47 4 (9.0) 5 (9.0) 6 (5.0) 7 (8.0) 8 (8.0) 9 (22.0) 10 (16.0) 11 (20.0)

262

Transition state geometries A N 0.114983 1.404552 0.042479 C 1.378051 1.087222 -0.066239 C 1.827193 -0.238268 0.178968 C 0.963298 -1.312361 0.079940 C -0.426711 -1.258490 -0.197869 C -1.274077 -0.196725 -0.060830 H 2.118604 1.820926 -0.405539 H 2.896630 -0.419740 0.239947 H 1.408529 -2.306919 0.066534 H -0.859431 -2.142603 -0.669212 H -0.113549 2.328720 -0.334681 C -2.508069 0.247594 0.118693 H -2.853163 1.194363 -0.289664 H -3.194484 -0.293143 0.770811 B O -0.027844 -1.389885 -0.163641 C 1.201838 -1.156040 -0.049679 C 1.769586 0.099886 0.285009 C 1.037377 1.260960 0.078505 C -0.333011 1.301511 -0.256438 C -1.208526 0.271588 -0.064524 H 1.893875 -1.970122 -0.318267 H 2.845168 0.153888 0.421980 H 1.580721 2.204107 0.033182 H -0.706459 2.176089 -0.788543 C -2.400456 -0.212313 0.212931 H -3.089348 0.369568 0.821548 H -2.702060 -1.207999 -0.095594 C C -0.141316 1.505122 0.258818 C -1.376059 1.099202 -0.205022 C -1.852200 -0.207495 0.045125 C -0.996072 -1.291637 0.170807 C 0.405852 -1.287938 -0.088462 C 1.268638 -0.237039 -0.133253 H -1.950304 1.742964 -0.872544 H -2.915682 -0.416861 -0.057182 H -1.453563 -2.276957 0.236601 H 0.797189 -2.224636 -0.495132 H 0.192001 1.150300 1.224904 C 2.514321 0.210711 -0.048384 H 3.241663 -0.262579 0.613228

263

H 2.847982 1.099050 -0.579249 H 0.301732 2.443165 -0.068406 D C -0.428955 1.736070 -0.078869 H -1.677159 0.195812 -1.166080 C -1.328181 -1.040745 0.237559 C 0.059570 -1.261204 0.069032 C 1.079170 -0.288769 -0.100777 N 0.574370 1.062371 0.410459 H -0.942967 2.472571 0.529418 H -0.582688 1.751760 -1.147957 H 0.852011 1.232450 1.373299 O 2.272843 -0.391030 -0.300902 H 0.455013 -2.240639 0.329174 H -1.823556 -1.699188 0.961998 N -2.110515 -0.127605 -0.301075 E C 2.855600 -1.290508 -0.033199 C 1.471480 -0.658207 0.021506 C 1.450137 0.781916 0.023188 C 0.421046 1.680379 -0.110045 O 0.083004 2.808092 -0.092083 O 0.486545 -1.422781 0.059639 O -1.037201 0.649978 -0.731478 C -1.529435 -0.270670 -0.047362 H 2.402908 1.291245 0.138662 H 2.849336 -2.236883 0.514842 H 3.101246 -1.509296 -1.080545 H 3.635686 -0.634141 0.365880 C -2.337508 -1.331925 -0.744659 C -1.607922 -0.213218 1.455997 H -1.578510 -1.215768 1.889442 H -0.809773 0.396443 1.882551 H -2.572528 0.245782 1.718426 H -1.975935 -2.319395 -0.440091 H -3.390682 -1.250914 -0.443849 H -2.254558 -1.224861 -1.827557

264

F C 1.397469 0.473924 -0.305475 C 1.378725 -0.872607 0.078293 C 0.621958 1.384696 0.404087 O -0.519463 -1.375927 -0.135471 C -1.484545 0.731312 -0.401245 C -1.329078 -0.448863 0.294122 H -1.640494 -0.494996 1.339342 H 1.766004 -1.635540 -0.579618 H 1.338936 -1.150897 1.127237 H 1.653099 0.711124 -1.336724 H 0.410661 1.215691 1.457431 H 0.529811 2.418490 0.086859 H -1.196522 0.791263 -1.441693 H -2.167468 1.501291 -0.027977 G C -0.467854 -1.401515 0.000000 C 0.932650 -1.221525 0.000000 C 1.168543 0.138839 0.000000 C 0.000000 0.928848 0.000000 O -1.157575 -0.326531 0.000000 C -0.328902 2.201662 0.000000 H -0.995669 -2.356380 0.000000 H 1.667085 -2.015505 0.000000 H 2.141978 0.614958 0.000000 H -1.379417 2.491315 0.000000 H C -0.779680 -0.732758 0.179043 C -0.665359 0.624814 -0.179046 C 0.722518 0.684823 0.179127 C 0.722521 -0.678481 -0.179125 Ref. 12:

Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Salvador, P.; Dannenberg, J. J.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M.; Peng,

265

C.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Gonzalez, R.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98, Gaussian 98, rev. A11: Pittsburgh, PA, 1998.

266

6.- Results and Discussion

267

0.0

0.2

0.4

0.6

0.8

1.0

0.7 1.1 1.5 1.9 2.3 2.7 3.1 3.5r(H-H) A

LI/D

I

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Ener

gy (a

u)

6.- Results and Discussion

The results of the present thesis can be divided into three parts: the

calculation of the ESI, the aromaticity studies and the analyses of the ELF.

Electron Sharing Indexes

All the ESI reviewed in this manuscript are based on the XCD. The adequacy

of this function to recover the extent of electron sharing has been discussed in the

introduction. Its ability to explain the electron distribution of molecules has been

extensively discussed in the literature. Particularly it is interesting noticing the

pedagogic potential of this ESI to explain some difficulties arising in the

computational chemistry, as it is the case of the homolytic dissociation of the

hydrogen molecule in the framework of a restricted calculation with a single

determinant. Chapter 4.2 discusses how the impossibility of occupying two different

orbitals with one electron each, because of the restricted formalism, leads to a

hydrogen-hydrogen electron sharing of one electron all along the dissociation

process; likewise the energy of the dissociation process is overestimated within the

same formalism. The usage of two Slater determinants enables the description of

the process as an average between two configurations: the ground state one (with

two electrons occupying the σ bonding orbital), and the excited state (were the two

electrons have been promoted to the σ∗ antibonding orbital). This way the

dissociation energy decreases, similarly the electron sharing between the hydrogen

atoms decreases with the bond distance to reach zero for two non-interacting

hydrogen atoms; see the picture and the table below:

LI

ESI (DI)

E(CI)

E(HF)

Figure2. The LI and the ESI(DI), the CI and HF energies for H2 dissociation.

269

r(H-H)Å E. CI(au) E. HF(au) δ(H-H) CI

0.7351 -1.1373 -1.1170 0.8237

1.0350 -1.0936 -1.0561 0.6955

1.4350 -1.0091 -0.9306 0.4444

1.8350 -0.9590 -0.8204 0.2033

2.2350 -0.9403 -0.7406 0.0753

2.4350 -0.9368 -0.7112 0.0449

3.0350 -0.9336 -0.6537 0.0096

3.6350 -0.9332 -0.6250 0.0021

4.2350 -0.9332 -0.6100 0.0004

... ... ... ...

∞ E(H) E(H)+K12/2 0

Another interesting property of this system, is that the symmetry itself gives the only

non-penetrating molecular space decomposition into atomic fragments, thus these

results holds for some other partitions of the molecular space such as Voronoi cells.

In this sense, it has also been analyzed how the partition of the molecular

space enables different ESI with different features but a common background. In

chapter 4.1 the differences between the Mulliken, Fuzzy-Atom and the QTAIM

partitions have been compared, to show the same qualitative trends, especially

concerning carbon-carbon bonds. It is of special importance the agreement

between QTAIM and Fuzzy-Atom partitions, since the former are widely used in the

literature, but its computational expense sometimes prevents its usage. The

following graphic shows how for bonded pairs of atoms both partitions agree, with

the exception of certain pairs, as those containing Be, B or Al, or the for P-O bonds:

Table1. The ESI (DI), the CI and HF energies for H2 dissociation.

Figure3. Correlation betweem QTAIM-ESI and Fuzzy-Atom ESI.

QTAIM vs Fuzzy-Atom

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

FBO

QTA

IM

Be,B,Al

P-O

270

On the other hand, the QTAIM partition has been demonstrated as a very suitable

one, since –unlike Fuzzy-Atom approach- it enables the description of the

dissociation of an ionic species into neutral atoms, where a maximum of electron

sharing is expected at a certain distance (see Figure4).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7

r(Li···H)

ESI

However, when using the information given by the topology of the electron density

to define a Fuzzy-Atom, a better description of this dissociation may be gathered, as

it also shown in the picture above (fuzzy-rho). The conclusions hold for the CAS(4,4)

calculation of LiH with two approximations to the true QTAIM-ESI, the Angyan (A) and

the Fulton (F) ones (see chapter 4.6 for further details).

Therefore, one concludes that the QTAIM partition may be replaced by the

less computationally expensive Fuzzy-Atom partition, to reproduce the same

qualitative trends. Nonetheless, it is specially recommendable to use the

information regarding the topology of the density (whose calculation is

computationally inexpensive) to better approach the QTAIM partition, and maybe

achieve quantitative results. In our laboratory we are currently exploring such

possibility for the set of molecules given in chapter 4.1.


),( HLiAδ


),( HLiFδ



),( HLiAδ


),( HLiFδ


Figure4. LiH dissociation in neutral species; Angyan and Fulton ESI for QTAIM, Becke’s Fuzzy-Atom and density based Fuzzy-Atom partitions.

271

Aromaticity and its Quantification

Aromaticity is a cornerstone of current chemistry, and it quantification turns

out to be one of the challenges of the present researchers in the field. Since

aromaticity it is not an observable, several aromaticity descriptors may be put

forward, and thus any proposal of a new aromaticity quantitative descriptor should

be also followed by a proper testing of its ability to reproduce some trends of

aromaticity known from chemical grounds.

In the present thesis we have defined a new aromaticity index, FLU (the short

form of fluctuation aromaticity index), which uses the electron sharing between

adjacent atoms along the ring on its definition; the formula is as follows:

( ) ( )( )∑

− ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

RING

BA ref

ref

BABABA

AVBV

nFLU

2

,,,

)()(1

δδδα

, (28)

where n stands for the number of members in a given ring, ( )BA,δ and ( )BAref ,δ

are the ESI of the corresponding pair and the ESI of the corresponding atoms A-B in

a typical aromatic molecule chosen as a reference respectively; V(A) is the valence

of atom A and reads:

[ ])()(2),()( AANBAAVAB

λδ −== ∑≠

, (29)

and finally α is set so that the ratio of valences in Eq. (28) is greater than 1,

penalizing those cases with highly localized electrons:

⎩⎨⎧

≤−>

=)()(1)()(1

AVBVAVBV

α . (30)

The FLU aromaticity index has been show to give good correlations with other

indexes. In particular, the correlations of FLU with the PDI and HOMA are excellent,

272

and even though NICS correlation with FLU, PDI or HOMA is not so good (see figures

in chapter 4.3), one can easily attribute it to the multidimensional character of

aromaticity.

Although FLU needs a reference value taken from aromatic molecules with

the bond pairs of the molecule we want to take into study; this is not a big deal

because most organic aromatic molecules are composed of carbon-carbon bonds,

and thus benzene can serve as a reference. However, a striking shortcoming of the

FLU index, as it is the case of HOMA index, is its inadequacy to study chemical

reactivity. This is shown in chapter 4.4 by analyzing the value of different

aromaticity indexes along the reaction path of the simplest Diels-Alder reaction; this

graphic illustrates the phenomena:

Here we can see how the most aromatic point in the reaction path (the transition

state) is only stated as such by PDI and NICS values, whereas FLU or HOMA give the

cyclohexene as the most aromatic species. It is due to the fact that the structure

Figure5. NICS, PDI, FLU, HOMA, and Root Sum Square (RSS) of the distances of the atoms to the best fitted plane for the Diels-Alder reaction path.

273

and electron sharing of the reference molecule (benzene) is more similar to the

cyclohexene rather than the transition state, where the bond lengths are unusually

larger and therefore yield lower electron sharing between the corresponding atoms.

As a consequence one may conclude that when aromaticity is enhanced by larger

bond lengthening, HOMA or FLU indexes may fail to reproduce the expected trends.

It is not that these indexes are completely useless for reactivity studies, but at least

it is clear by results of chapter 4.4 that for such analysis one must apply these

indexes with care.

Another aromaticity index, which uses π-electron sharing and avoids

references, is the FLUπ. It has been defined as:

( )∑− ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

RING

BA av

avBAAVBV

nFLU

2

,)()(1

δδδπ

α

π

ππ , (31)

where all quantities correspond to the π counterpart of those given in Eq. (30), and

avδ is the average of all bonded ESI in a given ring. This index needs not for

reference parameters, but it can be only exactly calculated for planar molecules,

where the exact orbital decomposition of the ESI into its σ and π parts is possible.

Usually aromatic molecules are also planar, but if we were about to estimate the

aromaticity of a non-planar species with FLUπ we would need the implementation of

localization scheme to compute it. FLUπ has been shown to reproduce the same

trends of FLU for the series of molecules studied in chapter 4.3.

Finally it is worth noticing that the ESI used in Eqs. (30) and (31) use the XCD

function but with a QTAIM partition. However, there is no reason why we should limit

ourselves to ESI calculated from this partition. In this line, in chapter 4.5 FLU, PDI

and FLUπ have been shown to yield quantitative the same results for QTAIM partition

than for Fuzzy-Atom one (see Figures 1-3 in chapter 4.5). Besides, the basis set

dependence of these indexes for the Fuzzy-Atom partition has been tested in the

same chapter, with the finding of strong independence of the basis set employed.

274

The Electron Localization Function

Defined by Becke and Edgecombe by 1990, the ELF has been one the most

successful approaches to the study of electron localization. While the ELF has been

extensively used for single determinant wave functions (HF or DFT within the Kohn-

Sham formalism), there has been few attempts to address this issue for wave

functions composed of several determinants. In chapter 5.1 we have reviewed the

definitions given in the literature for the ELF to show their equivalence, and that

indeed there is no need of the arbitrary factor originally used by Becke and

Edgecombe to define the ELF. The expression of the ELF at correlated level for both

closed- and open-shell systems reads:

( ) ( )( )[ ]

12

3/80

)2(2

0

)2(2

2

,,1

−

==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +∇++∇+=

rc

srrsrrELF

F

ssssr

rrrrrrrr

ρ

γγ ααββ

. (32)

In the same chapter the ELF is shown to recover the shell structure for atoms Li to

Kr, and to partially reproduce the shell numbers, with lesser accuracy as the atomic

number increases. Finally, chapter 5.1 includes the values for the integration of

certain interesting functions over the ELF basins (the regions of molecular space

which are surrounded by a zero flux surface of a certain function or by infinity; for

ELF basins the function corresponds to the ELF function, whereas for QTAIM basins,

the function is simply the density). For instance, the density and the pair density

lead to the population and pair populations, whose combination enable the

calculation of the covariance and the variance of the basin populations. Unlike

QTAIM, whose basins are uniquely related with an atom in a molecule, the ELF

partition leads to a molecular space partition which is more connected to Lewis

concepts, such as bonding regions, lone pairs or core basins.

The literature is full of examples which show the way to obtain chemically

interesting information from the ELF; we also address this point in chapters 5.2-5.4.

The first of these chapters is devoted to analyze a special case, where two of the

most widely used tools to elucidate the electronic structure of molecules, QTAIM

and ELF, differ on their explanation of the electronic structure of (CH3Li)4 species.

275

For this system the QTAIM reports nine bond paths connecting carbon with the

three closest hydrogen atoms, the other three carbon atoms, and the three closest

lithium atoms, as in the QTAIM picture shown below:

Following Bader,65 each bond path must be associated with a bond, and therefore

from a QTAIM perspective the carbons in this cluster should be nonavalent. On its

side, the ELF picture of the same molecule looks:

Figure6. Electron-density topology of (CH3Li)4 on its eclipsed conformation; the bond critical points are in red and the ring critical

Figure7. The ELF topology of (CH3Li)4 eclipsed conformation; the lithium basins are red, the carbon ones are yellow and the hydrogen ones are blue. Isosurface ELF=0.80.

276

where each carbon basin (in yellow) points towards the plane consisting of three

lithium atoms, as if the bonding interactions were four: three covalent ones with the

hydrogen atoms, plus one polar bond with the center of the three lithium plane.

The last two chapters of this thesis discuss the utility of the ELF to discern

between two mechanisms of reaction which may lead to confusion, the pericyclic

and pseudopericyclic reactions. Pericyclic reactions are widely known and

characterized by the Woodward and Hoffmann rules, whereas the pseudopericyclic

ones were discovered by 1976, when Ross et al.66 put forward reactions which

broke the rules given by Woodward and Hoffmann. Pseudopericyclic reactions are

low or non-existing barrier reactions, with planar transition state and cyclic overlap

disconnections. Since other researchers documented pericyclic reactions with low

barriers or with quite planar transition states, these features of the pseudopericyclic

reactions are sometimes arbitrary, and it is hard to say whether the reaction occurs

through a pericyclic or pseupericyclic mechanism.

We propose the analysis of the transition state of the electrocyclic processes

by means of the ELF, to investigate the electron localizability, in order to assess

whether exists connectivity between the adjacent bonds which are about to be

broken/formed in the concerted reaction. In a pericyclic reaction the electron

charge is spread among the bonds involved in the rearrangement, whereas

pseudopericyclic reactions are characteristic for electron accumulations and

depletions on different atoms, and thus the electron distribution is far from being

uniform for the bonds involved in the rearrangement. Since the ELF accounts for the

electron distribution we expect connected (delocalized) pictures of bonds in

pericyclic reactions, while pseudopericyclic reactions will give rise to disconnected

(localized) pictures. The magnitude of fluctuation between adjacent bonds can be

also used to assess the localized/delocalized character of the transition states.

This way one can help elucidating the mechanisms of reaction for

controversial cases. In Figure8, the ELF picture of the transition state for a series of

electrocyclic processes is depicted, together with the fluctuation diagrams. From

ELF analysis one may conclude that reactions C, F and G occur through a pericyclic

mechanism, while the others are pseudopericyclic reactions.

277

A B

1.54

1.16

1.111.10

1.82

NH5

4

3

2 1

1.27

1.29

1.381.14

3.46

O5

4

3

2 1

C D

1.51

1.09

1.03

1.03

1.02

2

5

4

3

1

2

3 4

2 2

HN

O

HN1.42

7.20

1.82

1.41

2.67

1

1

E F

O

O O5

3

4

32

2

1.23

1.342.15

13.27

O3.09

1.16 2.43

1.951

2

34

5

G H

O

1.28

1.63

8.83

1.30

3.491

2

3

4

1

3

2

4

1.00

1.00

1.00

1.00

Figure8. The ELF=0.60 isosurfaces of the transition state of electrocyclic reactions A-H. The fluctuation diagrams with arrows pointing towards the preferred direction of fluctuation, and its magnitude are also given. Two arrows with significant magnitude pointing towards the same atoms is an indication of electron accumulation, two arrows pointing outwards means electron depletion.

278

7.- Conclusions

279

7.- Conclusions

Applications I: Electron Sharing Indexes and Aromaticity.

1.- The ESI based on QTAIM and Fuzzy-Atom partitions, and the Mayer Bond

Order (MBO), mainly reproduce the same trends in the bonding. This does not

necessarily mean that the three ESI have the same value. The exceptions to this are

mainly due to systems containing aluminum, boron and beryllium. Specially striking

is the relationship found for carbon-carbon bonds; this result suggests

hydrocarbons and other carbon skeleton species, may reproduce the same bonding

patterns from the three ESI, whose computational cost ranges from extremely

cheap (MBO) to considerable expensive (QTAIM).

2.- The localizability of electrons in the hydrogen molecule may be presented

to an undergraduate student to get a deeper understanding of the homolytic bond

dissociation problem in the restricted HF method. The necessity of molecular spin

orbitals to be doubly occupied is the responsible for localization to hold on the same

value while dissociation is happening. To avoid this shortcoming one must use

correlated methods, such as configuration interaction. The QTAIM ESI enables an

easy explanation of this phenomena: for monodeterminantal wave functions the ESI

is equally 1 for the whole reaction path, whereas for CI the atomic electron

localizability increases with interatomic distance until one electron localizes in each

atom, while mutual shared electrons decrease to reach no sharing in the limit of

non-interacting fragments.

3.- We have defined in this thesis the aromatic fluctuation index (FLU), which

analyzes the amount of electron sharing between contiguous atoms, which should

be substantial in aromatic molecules, but it also takes into account the similarity of

electron sharing between adjacent atoms. We have demonstrated that the FLU

index and its π analog, the FLUπ descriptor, are simple and efficient probes for

aromaticity. We have shown that, for a series of planar PAHs, FLU and FLUπ

correlate well, with few exceptions, with other already existing independent local

aromaticity parameters, like the geometry-based descriptor, HOMA, the magnetic-

281

based criterion, NICS, and the electronic-based PDI index, which are of common use

nowadays.

4.- The paradigmatic Diels-Alder reaction between 1,3-butadiene and ethene,

presenting an aromatic TS, has been analyzed to show that some aromaticity

indices may fail to describe aromaticity in chemical reactions. The NICS and PDI

indicators of aromaticity correctly predict that a structure close to the TS is the most

aromatic species along the reaction path. On the contrary, we have found that

HOMA and FLU indices are unsuccessful to account for the aromaticity of the TS.

These indexes show that the most planar species along the reaction path of the DA

reaction between 1,3-butadiene and ethene is cyclohexene, the final product. The

failure of some indexes to detect the aromaticity of the TS in the simplest DA

cycloaddition reinforces the idea of the multidimensional character of aromaticity

and the need for several criteria to quantify the aromatic character of a given

species.

5.- PDI, FLU and FLUπ, measures have been calculated for a series of

aromatic and non-aromatic molecules replacing the QTAIM ESI by the Fuzzy-Atom

one. The corresponding values for each aromaticity index are in excellent

agreement, particularly for the molecules containing only C-C bonds. The correlation

for the PDI index is surprisingly good, taking into account that it is based on the

electron sharing between non-bonded atoms. The slight deviations between the

FBO and DI between C-C and C-N bonded atoms are translated into some

differences between the FLU and FLUπ indexes, as the corresponding electron

sharing reference and average values change, respectively. Nevertheless, the

correlation between the FBO- and DI-based FLU and FLUπ is still excellent. It has

been proved that these indexes are strongly insensitive to the basis set, even for

the modest STO-3G basis set.

6.- The effect of the inclusion of correlation has been analyzed for the ESI from

different partition schemes (QTAIM and Fuzzy-Atom), based on both the first- and

the second-order density matrices. It has been shown that for weakly bonded

molecules most indexes coincide and the effect of the electron correlation is minor;

whereas for covalent and polar covalent molecules the electron correlation effects

282

are notorious, and ESI based on first-order density matrices do not wholly reproduce

the trends shown by the ESI based on second-order one. Fuzzy-Atom approach for

the set of radii determined by Koga has been shown to not reproduce a maximum

of electron sharing in the dissociation of LiH species in neutral atoms. Nevertheless,

an improvement in the description is gathered if the radius corresponding to the

distance from the attractor to the bond critical point is used. Finally, aromaticity

indexes have been studied for first time for correlated calculations, leading to a

general trend similar to that already reported for monodeterminantal methods.

Nevertheless, is worth to mention these indexes are more dependent on the

methodology employed rather than the basis set used.

Applications II: The Electron Localization Function (ELF)

7.- We have reviewed the ELF definition for correlated wave functions,

showing the equivalence of the most recently proposed definitions of the ELF, and

how its monodeterminantal version reproduces Becke and Edgecombe’s definition.

We have computed the ELF for CISD level of theory, providing populations and

variance analysis for a correlated partitioning according to the ELF function. These

results have been discussed with respect to results derived from the loge theory.

8.- The AIM and ELF topological approaches partially agree but also show

significant divergences in the description of the bonding in methylalkalimetals

(CH3M)n with M = Li – K, n = 1, 4. They coincide in indicating that C-M bonds for

these compounds are highly polar. However, whereas AIM indicates that

tetramerization of CH3M slightly reduces the polarity, ELF suggests the opposite.

More importantly, and also more strikingly, AIM yields nonavalent carbons in

tetramethyllithium. According to AIM, the carbon atoms in this methylalkalimetal

have three individual bonds to the three closest hydrogen atoms, three individual

bonds to the three closest metal atoms, and three individual bonds to the three

closest carbon atoms. At variance, ELF yields tetravalent carbon atoms that form

three covalent bonds with their hydrogens plus one polar bond with a triangular

face of the central metal cluster.

283

9.- We have shown that a recent work by Chamorro and Notario67 has

incorrectly assigned a pericyclic nature to reactions A and B in chapter 5.3. Our

conclusion is achieved by means of a covariance analysis based on contributions

arising from the same basin, and by comparison to the reaction of cyclization of 5-

oxo-2,4-pentadienal to pyran-2-one, with an already assigned pseudopericyclic

nature.

10.- We have reported that the electron distribution of TS studied by means

of the ELF provides a clear picture of the electron rearrangement, which reveals the

mechanism of the reaction for several electrocyclic processes with the only need of

the TS structure. This way, a reliable manner to distinguish between pericyclic and

pseudopericyclic mechanisms have been put forward.

284

8.- Complete List of Publications

285

8.- Complete list of publications:

1.- Matito E., Solà M., Duran M., and Poater J.; Comment on “Nature of Bonding in the Thermal Cyclization of (Z)-1,2,4,6-Heptatetraene and its Heterosubstituted Analogues“; J. Phys. Chem. B 109, 7591-7593 (2005) 2.- Matito E., Duran M., and Solà M.; The aromatic fluctuation index (FLU): A new aromaticity index based on electron delocalization; J. Chem. Phys. 122, 014109 (2005) 3.- Matito E., Poater J., Duran M., Solà M.; An analysis of the changes in aromaticity and planarity along the reaction path of the simplest Diels-Alder reaction. Exploring the validity of different indicators of aromaticity; J. Mol. Struct. (THEOCHEM) 727, 165-171 (2005) 4.- Matito E., Poater J., Solà M., Duran M., and Salvador P.; Comparison of the AIM Delocalization Index and the Mayer and Fuzzy Atom Bond Orders; J. Phys. Chem. A 109, 9904-9910 (2005) 5.- Matito E., Poater J., Duran M., and Solà M.; Electron fluctuation in pericyclic and pseudopericyclic reactions; ChemPhysChem 7, 111-113 (2006) 6.- Matito E., Kobus J., and Styszyński J.; Bond centred functions in relativistic and non-relativistic calculations for diatomics; Chem. Phys. 321, 277-284 (2006) 7.- Matito E., Poater J., Bickelhaupt F.M., and Solà M.; Bonding in Methylalkalimetal (CH3M)n (M = Li – K; n = 1, 4). Agreement and Divergences between AIM and ELF Analyses; J. Phys. Chem. B 110, 7189-7198 (2006) 8.- Matito E., Salvador P., Duran M., and Solà M.; Aromaticity measures from Fuzzy-Atom Bond Orders. The aromatic fluctuation (FLU) and the Para-Delocalization (PDI) indexes; J. Phys. Chem. A 110, 5108-5113 (2006) 9.- Jiménez-Halla J.O.C., Matito E., Robles J., and Solà M.; Nucleus-independent chemical shift (NICS) profiles in a series of monocyclic planar inorganic compounds; J. Organometallic Chem. (ASAP) 10.- Matito E., Duran M., and Solà M.; Exploring the Hartree-Fock Dissociation Problem in the Hydrogen Molecule by means of Electron Localization Measures; J. Chem. Educ. (in press, to appear issue July2006) 11.- Matito E., Silvi B., Duran M., and Solà M.; Electron Localization Function at Correlated Level; J. Chem. Phys. (accepted) 12.- Matito E., Poater J. and Solà M.; Aromaticity Analyses by Means of the Quantum Theory of Atoms in Molecules. Recent Advances in the Quantum Theory of Atoms in Molecules; R. J. Boyd and C. F. Matta, Eds., Wiley-VCH, New York, 2006, (accepted)

287

13.- Matito E., Salvador P., Duran M., and Solà M; Electron sharing indexes at correlated level. Application to aromaticity calculations. Faraday Discussions (accepted)

14- Jiménez-Halla J.O.C., Matito E., Robles J., Solà M.; Interaction of Na+ with cyclo-[Mg3]2- triggers an unprecedented change of σ- to π-aromaticity. (submitted)

15.- Güell M., Matito E., Luis J.M., Poater J., and Solà M.; Analysis of Electron Delocalization in Aromatic Systems: Individual Molecular Orbital Contributions to Para-Delocalization Indexes (PDI). PCCP (submitted)

16.- Matxain J. M., Eriksson L. A., Mercero J. M., Lopez X., Ugalde J. M., Poater J., Matito E., Solà M.; Stability and Aromaticity of B12N12 Fullerene Dimer. JCTC (submitted)

288

9.- References

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APENDIX: Technical details

Gaussian Package

For the geometry optimization and single point calculations of the systems

given in this thesis we have only used the commercial software Gaussian, in the two

last versions, Gaussian9868 and Gaussian03.69 Let us give some details about the

calculations performed with Gaussian.

• The wave function

In most works the analysis of the wave function is crucial for the

understanding of the chemical problem. In this sense, PROAIM and ToPMoD

computational packages need the wave function given in specific format, thus the

Gaussian keyword out=wfn will be used in most calculations. This keyword is used

in combination with the keyword density=current to obtain the wave function at the

current level of theory. Otherwise, by default, Gaussian prints the HF wave function

for all post-HF methods (CISD, CASSCF, CCSD, etc.). Furthermore, one must change

the threshold used in link 9999 (l9999) of Gaussian in order to print occupancies

for CISD which are above 10-7. By default an own modified version of l9999 of

Guassian (both 98 and 03) set all occupancies lower than 10-7 to 10-7, this way we

make sure PROAIM program considers also this orbitals to compute the AOM.

In a HF calculation if we want Gaussian to print the virtual orbitals the

keyword we must use is the Iop(99/18=-1), it is needed for the modified version of

ToPMoD program used in chapter 5.1.

• Density Matrices

The calculation of the DM1 and DM2 goes through a program designed in

our laboratory,59 which takes the coefficients of the CI expansion, as indicated

before. The program creates the DM1 and DM2 from these coefficients, in terms of

molecular spin orbitals. However, most programs work either with natural spin

orbitals or with atomic orbitals; our case is the former. Thus, we need to transform

297

all those quantities (in practice the so-called atomic overlap matrices in a given

partition) from natural spin orbitals to molecular spin orbitals. To this aim, we

diagonalize the DM1 to obtain the natural orbitals.

The diagonalization of a given symmetric matrix, as it is the case of DM1, goes

through a unitary transformation. Such a transformation is unique excepting for a

multiplicative factor in each eigenvector given. The only case of the unitary

transformation being undetermined by more than a multiplicative factor is when the

eigenvalues are degenerated; for degenerated eigenvalues any combination of the

corresponding eigenvectors can be used in the unitary transformation. The reason

is simple: any combination of degenerated eigenvectors is also an eigenvector of

the system with the same eigenvalue. It occurs on molecules with degenerated

levels, which is a common situation. Therefore, unless we are dealing with a

molecule with no degenerated levels, for the diagonalization of the DM1 one needs

to read from Gaussian (or any other software used) the specific unitary

transformation used by this program.

By default all gradient calculations for correlated calculation are performed

within link 913. This link computes the relaxed first-order reduced density matrix, as

one checks by simple inspection of natural occupancies (which are above one or

below zero). In order to obtain the unrelaxed first-order reduced density matrix. It is

necessary to force Gaussian to use old link 916, by setting use=l916. Likewise, one

must set density=current to also write the unrelaxed density in the wave function

file. In general it is better optimizing the geometry with the usual link (916) and,

once converged, using the link 913 to obtain the unrelaxed density. Finally it is also

worth noticing that sometimes one needs to set off the symmetry (keyword

nosymm) to achieve the wave function convergence with link 916.

• Input example:

Here the example of NH3 molecule for the calculation of the unrelaxed

density (relaxed density is also obtained) for CISD with 6-311++G(2d,2p) basis set,

with Cartesian d and f functions. The wave function and the formatted checkpoint

(fchk) files are also obtained; gfinput is included to print the basis set.

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%subst l9999 /users/eduard/g03 %chk=NH3.chk %rwf=/tmp/eduard/1.scr,2000mb,/tmp/eduard/2.scr,2000mb,/tmp/eduard/3.scr,2000mb,/tmp/eduard/4.scr,2000mb,/tmp/eduard/5.scr,-1 #CISD/6-311++G(2d,2p) 6d 10f opt out=wfn density=current HF 0 1 n h 1 hc h 1 hc 2 hch h 1 hc 3 hch 2 dih hc=1.00964742 hch=107.08384316 dih=114.57936516 NH3rel.wfn --Link1-- %subst l9999 /users/eduard/g03 %chk=NH3.chk %rwf=/tmp/eduard/1.scr,2000mb,/tmp/eduard/2.scr,2000mb,/tmp/eduard/3.scr,2000mb,/tmp/eduard/4.scr,2000mb,/tmp/eduard/5.scr,-1 #CISD/6-311++G(2d,2p) 6d 10f scf=tight geom=check out=wfn Iop(9/40=7) Iop(9/28=-1) density=rhoci fchk use=l916 nosymm gfinput 0 1 NH3unrel.wfn

299

30/04/2006 I semblava que aquest dia tardaria en arribar... I el cert és que aquests quatre

anys m’han passat sense quasi adonar-me’n. Aquesta tesi representa el meu esforç de

quatre anys i conté gran part dels treballs que he realitzat a l’Institut. Al contrari que la

tesina, que contenia principalment els treballs sobre la teoria i l’algoritme de càlcul de

matrius de densitat, aquesta tesi versa sobre els treballs més pràctics, amb l’excepció

potser d’algun capítol.

Arribats aquí toca agrair a tots aquells que m’han ajudat en aquests quatre anys. Primer

de tot estan els meus directors de tesi, en Miquel Solà i en Miquel Duran (Quel).

Gràcies Miquel pel teu recolzament i per la teva paciència al llarg d’aquests quatre anys,

els teus consells i la teva ajuda han fet d’aquesta tesi el que avui presento. Hem

començat tantes coses que crec que necessitaria dues tesis per donar cabuda a tot plegat.

A tu Quel, et dono les gràcies per la teva visió crítica, la teva capacitat d’estructurar els

treballs, que ha donat més rigor als projectes que ocupen aquestes pàgines. I sobretot per

les xerrades i discussions, sobre Hellmann-Feynman, matrius de densitat d’ordre n, etc.

No dubtis pas que acabarem trobant “a Catalan molecule”. A tots dos us dec també

l’oportunitat de col·laborar amb vària gent de l’estranger, així com poder assistir i

participar en una colla de congressos.

També dec part del que hi ha aquí escrit als meus col·laboradors. A en Jordi, de la mà de

qui em vaig endinsar en el món d’AIM i els anàlisis de l’estructura electrònica; a en

Bernard Silvi per la seva hospitalitat en la meva breu estada a París, arran de la qual

vam començar l’estudi de la funció de localització electrònica a nivell correlacionat; i a

en Matthias Bickelhaupt per la seva ajuda en l’estudi de l’anàlisi electrònica dels

clusters (CH3M)4. Last but not least, en Pedro, amb qui apart dels projectes que dicta la

“chart” que ens hem marcat, he passat hores discutint sobre recerca de les que n’he

après (i em queda per aprendre) molt; siempre quedará la duda de quién de los dos apura

más los deadlines ...

I am also endebted to the people from Poland, to the ones in Torun: Jacek Kobus and

Jacek Karwowski for their kind hospitality. And of course to the ones in Szczecin,

Marcin, Kasia, Jacek and Jerzy; they helped to make my stay in Szczecin a nice

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experience. Specially to the last two, I acknowledge their dedication to the research

projects we have been dealing with.

Gràcies també a tots aquells IQCians que m’han fet més amena la meva estada a

l’Institut: en Juanma, l’Albert, en Torrentxu, en Xunges, la Montse, en Giro, en Quim,

en Pata, en Ferran, la Mireia, l’Òscar, la Sívia, l’Anna, en Samat, en Dani, la Cristina,

en Julian i l’Eugeni. Sense oblidar el sector “senior”: l’Emili, en Lluís, la Sílvia, en

Sergei, en Joan i l’Alexander. I a en Josep Maria, de qui he après molt, sobretot quan

vaig compartir amb ell l’assignatura de Química Física. A la Carme, que ve a jugar el

paper de mare a l’IQC, li estic agraït per la seva infinita comprensió i la seva ajuda amb

l’engorrós món de la paperassa.

Als experimentals per les hores que hem passat plegats: d’excursió al País Vasc, mirant

el futbol, a l’Excalibur, al Cucut i és clar, a les 18:00 (més o menys...) a Can Paco.

També vull dedicar unes línies als meus amics. Als de Girona, en Marc, en Gerard,

l’Albert, l’Ernest, l’Inma i l’Òlbia; als de la Universitat, en Sebes, l’Hèctor, en Bernat,

l’Eva, l’Ester, en Carles, l’Amadeu i la Núria; i als SASA: l’Albert, la Marta, en Pere i

la Gemma.

I per acabar vull agrair als meus pares la seva fe incondicional i la seva paciència amb

aquesta “vida científica” (per no dir bohèmia) que he portat aquests quatre anys. I al

meu germà, en Xevi, per la seva confiança i les llargues converses a mitja nit. I ja

només em queda agrair-li a la que avui per avui és la persona més important a la meva

vida, l’Anna. Gràcies per ajudar-me en el disseny de la caràtula de la tesi, per ser com

ets i per donar-me els millors mesos que he viscut. T’estimo.

A tots plegats, i a tu que tens la paciència de llegir-me,

Moltes gràcies.

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development, implementation and application of …iqc.udg.es/~eduard/recerca/thesis.pdf · girona,...

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