theoretical and computational aspects of magnetic molecules: phd thesis

Theoretical and Computational Aspects of Magnetic Molecules

Thesis

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Submitted by

Md. Ehesan Ali (02403001)

Under the Guidance of Prof. S. N. Datta

Department of Chemistry Indian Institute of Technology, Bombay

January, 2007

Dedicated to My Parents

Approval Sheet

Thesis entitled “Theoretical and Computational Aspects of Magnetic

Molecules” by Md. Ehesan Ali is approved for the degree of Doctor of

Philosophy

Examiners

_______________________

_______________________

_______________________

Supervisor

_______________________

_______________________

_______________________

Chairman

________________________

________________________

________________________

Date: ___________

Place: IIT–Bombay

Certificate of course work

This is to certify that Mr. Md. Ehesan Ali was admitted to the Ph.D. program on July 2002. He successfully completed all the courses required for Ph.D. program. The details of the course work are given below.

Sr. No. Course No. Course name Credit 1 CH 521 Interpretative Molecular Spectroscopy 6.00 2 CH 559 Solid State Chemistry and its Applications 6.00 3 CH 821 Topics in Chemistry I 6.00

4 CH 842 Elements of Advanced Molecular Quantum Mechanics 6.00

5 CH 831 Advanced Laboratory Techniques 8.00 6 CHS 802 Seminar 4.00

Total credit 36.00

Place: IIT–Bombay Deputy Registrar (Academic)

Contents

Chapter 1: Magnetic Molecules 1.1. Introduction 1

1.2. Ferromagnetic Molecules 1 1.3. Insights from Literature Survey 2 1.4. Diradicals 3

1.4.1. Stable Diradicals 6 1.4.2. Triradicals and Polyradicals 7

1.5. Interesting Phenomenon related to Magnetic Molecules 8 1.5.1. Single Molecule Magnets 8 1.5.2. Photomagnetism 9 1.5.3. Spintronics 10

1.6. Scope of Molecular Magnetism 10 1.7. Objectives and Organization of the Thesis 11 1.8. References 12

Chapter 2: Theoretical Background

2.1. Introduction 17 2.2. Theoretical Background 19

2.2.1 Single Determinant Approach 19 2.2.2 Two-Determinant Configuration 21 2.2.3 Orbital Perturbation Theory 22 2.2.4 SCF energy 24

2.3. Spin Hamiltonian Treatment 26 2.3.1 Base line 27

2.3.2. Spin Hamiltonian 27 2.3.3. Expectation Values 29

2.4. Coupling Constants 30 2.5. Discussion 33

2.5.1. Factors Influencing Accuracy 33 2.5.2. Numerical Tests 33

2.6. Conclusions 36 2.7. References 37

Chapter 3: Organic Fused–Ring Diradicals

3.1. Introduction 40 3.2. Methodology 42

3.3. Energy Differences 43 3.3.1. 4-oxy-2-naphthalenyl methyl 44

i

3.3.2. 1,8-naphthalenediylbis(methyl) 45 3.3.3. 1-imino-1-naphthalenyl methyl 45 3.3.4. 1,8-naphthalenediylbis(amidogen) 46 3.3.5. 8-methyl-1-naphtyl carbine 47 3.3.6. 8-methyl-1-naphthalenyl imidogen 48 3.3.7. 8-methyl-1-naphthyl diazomethane 49

3.4. Conclusions 50 3.5. Reference 52 Chapter 4: Bis–Nitronyl Nitroxide Diradicals: Influence of Length and Aromaticity of Couplers

4.1. Introduction 54 4.2. Theoretical Background 54 4.3. Computational Strategy 54 4.4. Results and Discussion 55

4.4.1. Rationalization 61 4.4.2. SOMO-SOMO Energy Level Splitting 63 4.4.3. Isotropic Hyperfine Coupling Constant 65

4.5. Diphenylene Acetylene Coupler 65 4.6. Conclusions 70 4.7. Reference 72

Chapter 5: Influence of Aromaticity in Intramolecular Magnetic Coupling

5.1. Introduction 74 5.2. Computational Methodology 74 5.3. Results and Discussion 76

5.3.1. Bond Order and Dihedral Angles 77 5.3.2. Nuclear Independent Chemical Shift 78 5.3.3. SOMO-SOMO Energy Splitting 81 5.3.4. Isotropic Hyperfine Coupling Constant 81

5.4. The m-Phenylene Couplers 82 5.4.1. Calculations 83

5.5. Conclusions 89 5.6. Reference 91

Chapter 6: Photomagnetism 6.1. Introduction 92

6.2. Technical Details and Results 93 6.3. Conclusion 98 6.4. References 98

ii

Chapter 7: Dinuclear Copper Complex

7.1. Introduction 99 7.2. Computational Methodology 102

7.3. Choice of Magnetic Orbitals 102 7.4. Results and Discussion 104 7.5. Conclusions 110 7.6. References 110

Conclusions 112 Summary i Acknowledgment

iii

Chapter 1

Magnetic Molecules This chapter describes a general introduction to magnetic molecules. A detailed literature

survey is also presented. The scope of molecular magnetism and the objectives of the Thesis

are discussed.

Chapter 1 Magnetic Molecules

1

1.1. Introduction

Magnetism has played a vital role in human civilization. The study of ferromagnetism

has been traditionally concerned exclusively with the study of transition elements (like Fe, Co

and Ni), alloys and metal oxides. This field of study has provided numerous technological

rewards based on the exploitation of such materials. In recent years, the focus of research on

magnetism has turned towards molecular systems and crystals. The reasons for this trend are

tunability of magnetic properties as a result of alterable chemical structures, bio-activity of

organic molecules, photo-control of chemical structure, and structure-property relationships.

In other words, the molecule can be tailored to exhibit desired magnetic properties.

Invariably, the building block of such magnetic materials are open-shell molecules

such as organic monoradicals, diradicals in which nonbonding molecular orbitals contain

unpaired electrons, and transition metal complexes with unpaired d-electrons. A large number

of high-spin magnetic molecules have been recently synthesized and investigated.1

1.2. Ferromagnetic Molecules

Molecular ferromagnetism results when the electronic spins in a single molecule are

coupled in a parallel orientation. In organic diradicals, two unpaired electrons at two different

non-bonded molecular orbitals (NBMOs) are coupled through the spacer in either parallel

(S=1) or anti-parallel (S=0) fashion, resulting in ferromagnetic or antiferromagnetic

interaction respectively. There are several rules to qualitatively predict the ground spin state

of such diradicals as discussed below.

According to Longuest-Higgins2, the number of NBMOs can be calculated as n = (N

− 2T) where N is the total number of the carbon atoms and T is the maximum number of

possible double bonds. This simple rule predicts the ground spin state of para-

benzoquinodimethan (1) (Figure 1.1). Four double bonds are possible and the number of

NBMOs is zero. Hence an antiferromagnetic interaction is observed. For meta-

benzoquinodimethan (2) (Figure 1.1), there are two NBMOs and according to the Hund’s

rule, a triplet (S=1) ground state is expected.3

From valence-bond formalism, Ovchinnikov4 suggested that the ground state spin S

can be determined as S = (n* −n)/2 for n* > n, where n* and n are the numbers of starred and

unstarred alternate carbon atoms. This rule predicts the meta- and para- benzoquinodimethan

to be in triplet and singlet ground spin states respectively (Figure 1.2).


1S = 0 S = 1

2

Figure 1.1. para- and meta- isomers of benzoquinodimethan.

* ***

* ** *

1 2

S = 0 S = 1

*

Figure 1.2. Predictions of ground state spin in starred/unstarred model and in the spin polarization

model.

The starred/unstarred model is very closely related to the concept of spin polarization

that can also predict the nature of the magnetic exchange interaction.5 Large positive spin

densities on atoms in conjugated systems induce small negative spin densities on neighboring

atoms. This spin polarization can be rationalized by considering the formation of the

chemical bond. The quantum chemical exchange favors a parallel orientation of the spins of

the electrons in σ- and π-orbital on the same atom over an antiparallel orientation.

Spin alternation rule that has been explained and demonestrated in UHF treatment can

also predict the ferro- and anti-ferromagnetic exchange interaction.6

1.3. Insights from Literature Survey

The first molecular ferromagnet, Fe(Cl)[S2CN(C2H5)2]2, was reported in 1972 and its

crystal was found to have ferromagnetic ordering at 2.43 K.7 Subsequently, Miller et al

synthesized a charge-transfer salt, composed a ferrocene derivative and tetracyanoethylene

with a Curie temperature of TC of 4.8 K.7

The first pure organic magnet, β crystalline phase of p-nitrophenyl nitronylnitroxide

(3) that orders magnetically at 0.65 K,8 is one of the major successes in modern research. In

1991, Wudl and coworkers discovered the organic molecular ferromagnet,

tetrakis(dimethylamino)ethylenefullerene[60] (TDAE-C60, 4) with Curie temperature of 16.1

2


K.9 The origin of the ferromagnetism in TDAE-C60 has been the subject of various studies.

Its synthesis initially raised the hope that the higher TC values would soon be observed by

using other donors. But TDAE-C60 remained as the organic material with the highest TC until

1998 when Mihailovic reported 3-aminophenyl-methano-fullerene[60]-cobaltocene with a

slightly higher TC of 19 K.10 Rassat et al. have synthesized 1,3,5,7-tetramethyl-2-6-

diazaadamantane N, N'-dioxyl (5) (TC = 1.48 K).11 In 5, the two electron spins align in the

parallel fashion through intramolecular interaction, while the intermolecular interaction is

also ferromagnetic in nature. This phenomenon results a three-dimensional ferromagnetic

order below the Curie temperature.

N

NN

O

OO

ON

NO

O

3 5

NMe2NMe2Me2N

Me2N

4 Figure 1.3. Organic ferromagnetic molecules.

1.4. Diradicals

All the above examples of ferromagnetism arise from the intermolecular

ferromagnetic interactions. The intramolecular ferromagnetic order occurs in organic

diradicals and in dinuclear transition metal complexes. The simplest example of

intramolecular ferromagnetic interaction is trimethylenemethane (TMM, 6). TMM is a

diradical that has been widely investigated in different areas of chemistry. It was isolated in

matrix by Dowd in 1966. The significant and diverse impact of TMM has resulted from the

synthesis and studies of TMM derivatives that are stable and can be tailored to desired

molecular properties. TMM is the triplet ground state diradical with a very large singlet-

triplet energy gap (ΔE ). The estimated energy gap by photoelectron spectroscopy is in the

range of 13−16 Kcal mol . This range is confirmed by different computational studies.

The large ΔE indicates a very strong ferromagnetic interaction between the two radical

centers. As a result, TMM is a very attractive building block in molecular magnetism.

12

ST

−1 13 14

ST

3


C Ar

ArC Ar

Ar

Ar Ar

..

O

O O t-Bu

t-Bu

t-Bu

t-Bu

t-But-Bu

NNt-Bu

O t-Bu

O

N+

t-Bu O

6 7

8 9

TMM

Figures 1.4. TMM and its analogous diradicals.

Rajca et al. synthesized a new stable diradical 7, which is a 3-fold symmetric analogue

of TMM with no hetero atom substitution. This is similar but superior to 8 (Young’s

diradical) and 9. All these diradicals are stable and have triplet ground states. But only 7

can be extended to form polyradicals with a strong ferromagnetic coupling whereas the other

two are restricted due to their geometrical features. Recently, Shultz and coworkers have

synthesized stable diradicals 10-13 and studied their magnetic properties. These radicals are

TMM analogues. The authors noticed the structure-magnetic property relationship, that is, the

correlation of the

14 15

16

17

exchange parameters with the phenyl-ring torsion angles (φ).

The strong ferromagnetic interaction can also be achieved via a benzene moiety (2).

The ground state of ortho- and para- substituted diradicals are singlet. Spectroscopic studies

of m-benzoquinodimethane (2) by Berson and their coworkers, Migirdicyan and Platz suggest

a triplet ground state18 and ab initio calculations predict a 10 Kcal mol−1 singlet-triplet energy

gap.19

4


t-Bu t-Bu

NNt-Bu

O O

t-Bu

NN

O

t-Bu t-Bu

O

NN

t-Bu

O O

t-Bu

Nt-Bu

O

N

O

t-Bu

10 11

12 13

Figure 1.5. Recently synthesized TMM analog diradicals.

The exchange coupling constant (J) is consistent with ferromagnetic (JF) as well as

antiferromagnetic (JA) counterparts, that is, J = JF + JA. Antiferromagnetic coupling is

generally found to be more effective than the ferromagnetic interaction.20 Borden and

Davidson observed that the presence of non-disjoint MO (different orthogonal NBMO arising

from the common atom contributions) leads to a greater ferromagnetic coupling.21

14 15 16

Figure 1.6. Diradicals with two parallel couplers.

5


6

The spectroscopic study of dimethylenecyclobutadiene (14), which is an example of

ferromagnetic coupling via two parallel coupling units, suggests a triplet ground state. The ab

initio calculations predict ΔE = 17.7 kcal mol . The spectroscopic studies of the matrix-

isolated species 15 established its singlet ground state with ΔE > −1 kcal mol . Ab initio

calculations suggest an energy gap of −5 kcal mol . For diradical 16, spin alternation rule

predicts the singlet ground state.

ST−1 22

ST−1 23

−1 24

25

1.4.1. Stable Diradicals

The first stable diradical (17) was synthesized by Schlenk almost a century ago, in

1915.26 It was identified as a ground state triplet by EPR measurement in matrix.27 A lot of

efforts has been applied to increase the stability and the magnetic exchange interaction in this

species.28 In fact, all the compounds 18-22 in Figure 1.7 show intense ESR spectra.29

The heteroatoms as spin sites can be attached to a strong ferromagnetic coupler. For

example, 10−13 are stable and ferromagnetically coupled diradicals. Stable, yet weakly

coupled diradicals are known.29−30 The stable diradicals with nitronyl nitroxide fragments are

fascinating species in modern research on molecular magnetism. A large portion of this thesis

will describe the computational studies on different Nitronyl Nitroxide systems. Till now a

large number of nitronyl nitroxide (NN) based diradicals has been experimentally

investigated.31

The intramolecular magnetic exchange coupling constant, as well as the

intermolecular interaction that depends upon the structure and the nature of a molecular

crystal, control the magnetic properties of a molecule-based magnetic material. An estimate

of the intramolecular exchange coupling constant is necessary prior to synthesizing a

successful magnetic material based on organic diradicals or transition metal complexes. The

recent development of computational techniques and theoretical methodologies has enabled

the prediction of magnetic properties of the precursors.32 Here we report the results of the

study of a series of nitronyl nitroxide based diradicals with different conjugated magnetic

couplers.

The magnetic couplings are generally found to arise from spin polarization and spin

delocalization.33 Lahti et al.5 investigated a large number of π-conjugated couplers. They

noticed that most of the spin density is localized on the two-singly occupied σ orbitals


(SOMOs) centered on the radical atoms. The large spin population polarizes the π electrons

near the radical centre. The total π spin density sums to zero over all sites in the singlet state,

but the individual sites may be polarized to have positive or negative spin densities. The spin

polarization effect plays a major role in controlling the nature of the coupling. The presence

of non-bonding molecular orbitals (NBMOs) in organic diradicals makes it difficult to

properly evaluate the energy difference between the lowest states of different spin. The

expected ground state spin may be predicted either by molecular orbital (MO) calculation or

by a valence bond (VB) treatment. In the simple MO model, Hund’s multiplicity rules are

often applied to molecules having degenerate or nondegenerate NBMOs, with the prediction

of a triplet ground state. However, in a variety of conjugated systems the Hund’s criterion

does not necessarily follow, and a singlet ground state results. TME and its derivatives are

the simplest examples of such system. The low-spin nature of TME and the related disjoint

systems was explained by a VB-type electronic exchange. A number of derivations were

made to model the intramolecular exchange in connectivity-conjugated systems by

Ovchinikov,4 Klein,34 Borden and Davidson35 and Sinanŏglu.36 In all these cases the

simplistic MO theory and the Hund’s rule do not follow in a proper way. A large number of

computational studies have been performed on this issue.37−39 It is observed that the spin

polarization argument is more useful to understand the spin density distribution in an open

shell system. t-Bu

t-Bu

t-Bu t-Bu

X

RR

17 R=H, X=H 18R=Me, X=H 19R=i-Pr, X=H 20R=Me, X=Me 21R=CF3, X=H 22

Figure 1.7. The stable Schlenk diradicals (17) and its derivatives. 1.4.2. Triradicals and Polyradicals

Triradicals are relatively unusual. The nature of magnetic interaction between the

radical centers can be divided into three categories: (1) Two 1,3 connected benzene rings in a

7


‘linear’ arrangement, (2) 1,3,5 substituted benzene ring, and (3) three 1,3,5 connected benzene

rings in a “closed loop” arrangement as shown in Figure 1.8. The expression for the magnetic

exchange coupling constant is J = ΔEQD, and 3J = ΔEQD, and 3J = ΔEQD for the three

categories respectively.

Δ EQD=J Δ EQD=3J Δ EQD=3J

Figure 1.8. Triradicals of three types.

Polyradicals (23) are most promising very-high-spin molecular systems. Rajca and

co-workers have synthesized decaradicals, which posses S = 5 high-spin state.40 Due to the

presence of spin defects in single ferromagnetic pathway, the experimental magnetization is

always less than the predicted value. To overcome this problem, Rajca introduced

calix(4)arene in the centre while triarylmethyl radicals are linked to it, and this provides

multiple pathways for the ferromagnetic interaction (24). 41

1.5. Interesting Phenomena Related to Magnetic molecules

Several interesting areas of research have been emerged from molecular magnetism.

Currently a large number of physicists and chemists are involved in these exciting areas. A

few of these are described below.

1.5.1. Single Molecule Magnet (SMM)

The single molecule magnet is an assembly of individual magnetic molecules. To be a

single-molecule magnet, the object must show a net magnetic spin and have no magnetic

interaction between molecules. Caneschi et al. reported the first magnetic molecules

[Mn12O12-(O2CMe)16(H2O)4] with a ground-state spin of S =10 in 1991.42 The SMM term

was first used by Hendrickson in 1996.43 The Mn12O12 complex and its analogous complexes

shows SMM behaviors but in the low temperature range with very small the spin barrier (∼50

8


cm−1). To increase the critical temperature of SMM, (above which it behaves as

n

nn

n

n

n

Ar

Ar

ArAr

ArAr

Ar

Ar

Ar

Ar

Ar Ar Ar

Ar

Ar

Ar

Ar

ArAr

Ar

23 24

Figure 1.9. Magnetic plastics S=10 for 23 and S∼5000 for 24.

Ferromagnetic, antiferromagnetic or paramagnetic) the total spin quantum number S should

be very high and there must be a highly negative zero-field splitting parameters.

A number of theoreticians and experimentalists are making efforts to increase the

critical temperature of SMM to use them in molecular devices.44 Recently Soler et al. have

synthesized Mn12O12 based SMM.45 Davidson et al. calculated the magnetic properties of

these substances using local spin model.46

1.5.2. Photomagnetism

The extension of molecular photochromism results in the possibility of photo

switching of magnetic properties. Matsuda et al. have synthesized a large number of

photomagnetic molecules.47 The number of coordination compounds is limited to some

dithienylethylene derivatives,48 tetracyanoethylene organomettalic compounds, spin crossover

and valence-tautomeric complexes. From these perspectives, the Prussian blue analogue

complexes are the most promising. Recently Dei have synthesized a very interesting

9


10

photomagnetic [{Cu(tren)-NC}6Mo(CN)2]8+ complex.49 A larger number of theoretical and

experimental work has been performed on this issue.50

1.5.3. Spintronics

Spintronics is the "spin-based electronics" and also known as magnetoelectronics.

This refers to the control of electric current through a manipulation of the spin of the

electrons. It has been extensively investigated using layers of ferromagnetic materials from

both fundamental and device application points of view, as in magnetic tunneling junctions

(MTJ).51-53

1.6. Scope of Molecular Magnetism

The materials which display cooperative magnetic phenomena yet are based on

molecular building blocks have added advantages over conventional magnets such as low

density, transparency, electrical insulation, low temperature fabrication as well as offering the

possibility of combining magnetic behavior (either cooperative, or isolated, as in “spin-

crossover compounds”) with other properties such as photo- or thermal responsiveness. In

addition, there are often useful processing advantages such as the ability to deposit the

materials as films, to functionalise them for attachment to substrates and possible

biocompatibility.

Magnetic properties of these materials can have important practical applications for

use in domestic appliances as well as in high-tech sciences. For example, in the rapidly

evolving world of micro-electronics, electronic circuits and storage devices are decreasing in

size and will eventually reach molecular dimensions. Thus, in future conventional semi-

conductor technology could be abandoned in favour of new materials with molecular

magnetic properties (“spintronics”) to build a computer. In addition to their small size,

magnetic materials consisting of molecular entities have the advantage that the molecular

precursors can be prepared under mild conditions in a directed synthesis and are therefore

easily integrated into materials with well-defined magnetic, magneto-optic or magneto-

electric properties. Spintronic devices are used in the field of mass-storage. Recently (in

2002) IBM scientists announced that they could compress massive amounts of data into a

small area, at approximately one trillion bits per square inch (1.5 Gbit/mm²) or roughly 1 TB


11

on a single sided 3.5" diameter disc. Although some applications have already arisen from this

newly emerging field, it is still at an early stage of development. The ultimate rewards of

producing devices based on magnetic systems are increased processing power and

information storage, but this lie some way in the future. At present, the major requirements

for the further successful development of this highly promising field of materials science are

the production and investigation of a range of new systems and developing the basic research

into the underlying principles, namely a detailed description of molecular magnetism.

Molecular magnetism is a field of research where the investigation of the magnetic

properties of isolated molecules as well as of assemblies of molecules is undertaken. These

molecules may contain one or more magnetic centers. The assemblies of molecules occurring

in the solid state may be characterized by very weak interactions between the molecular

entities, thereby displaying magnetic behavior very similar to that of the isolated molecules.

They may consist of extended systems, in which strong magnetic interactions between the

molecular entities are responsible for bulk magnetic properties.

1.7. Objectives and Organization of the Thesis

The main aim of my PhD work has been to investigate the phenomenon called

molecular magnetism. A quantitative measure of magnetism in molecules is available in the

form of the intramolecular magnetic exchange coupling constant J. A knowledge of J helps

in predicting the magnetic moment and EPR frequency. Therefore, the objectives of this

thesis turn out as

1. To theoretically investigate J for diradicals and compare the derived formula with

those obtained by others;

2. To computationally investigate J for organic diradicals by using different ab initio

methodology;

3. To explore the possibility of photo-activation of magnetic properties;

4. To briefly investigate the characteristics of a transition metal complex diraical, and

understand the differences from organic diradicals.

The present thesis has been arranged according to the stated objectives. The arrangement is

as follows.


12

A theoretical formalism for diradicals with nondegenerate HOMOs is given in

Chapter 2, where we present an N-electron interpretation of the spin Hamiltonian and

subsequently show that the Yamaguchi and GND formula arise from special types of

approximations.

In Chapter 3, fused-ring organic diradicals are investigated so as to determine the

ground spin states. The singlet-triplet energy gaps are calculated by using HF, post-HF and

DFT methodologies. A good correlation between the experimental and calculated results is

observed when the S−T energy difference is large. However, these molecules are less

important from the molecular magnetism point of view as these are very unstable.

The broken symmetry (BS) method of calculations has been adopted for the work

described in Chapter 4−7. Chapter 4 deals with a series of nitronyl nitroxide diradicals with

different linear and cyclic couplers. Effects of structural features, like chain length, dihedral

angles, etc., on J are investigated. We also predict the nature of magnetic exchange

interaction in a few new molecules.

The effect of aromaticity of benzene and polyacene couplers on the intramolecular J is

discussed in Chapter 5.

The investigation of the ground state magnetic properties of a few photochromic

molecules is described in Chapter 6. Some of these photochromic molecules can act as

photomagnetic switches.

The investigation of magnetic properties of a dinuclear copper complex is discussed in

Chapter 7.

Summary of the thesis and the conclusions are given subsequently.

1.8. References 1. (a) Gatteschi, D.; Khan, O.; Miller, J. S.; Palacio, F. (Eds.), Magnetic Molecular Materials, Kluwer

Academic Publishers, Dordrecht, 1991. (b) Kahn, O. (Ed.), Magnetism: A Supramolecular Function,

Kluwer Academic Publishers, Dordrecht, 1996. (c) Lahti, P. M. (Ed.) Magnetic Properties of Organic

Materials, Marcel Dekker, Inc., New York, 1999. (d) Miller, J. S.; Epstein.A. J. Angew. Chem., Int. Ed.

Engl. 1994, 33, 385. (e) Rajca, A. Chem. Rev. 1994, 94, 871. (f) Miller,J. S.; Drillon, M. (Eds.)

Magnetism: Molecules to Materials, Models and Experiments, Wiley-VCH, Weinheim, 2001. (g)

Miller, J. S.; Drillon, M. (Eds.) Magnetism: Molecules to Materials II,Molecule-Based Materials,

Wiley-VCH, Weinheim, 2001. (h) Miller, J. S.; Drillon, M. (Eds.) Magnetism: Molecules to Materials

III, Nanosized Magnetic Materials, Wiley-VCH, Weinheim, 2001. (i) Miller, J. S.; Drillon, M. (Eds.)


13

Magnetism: Molecules to Materials IV, Molecule-based Materials (2), Wiley-VCH, Weinheim, 2002.

(j) van Meurs, P.J. High-Spin Molecules of p-Phenylenediamine Radical Cations, PhD Thesis,

Technische Universiteit Eindhoven, 2002. (k)

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Chapter 2

Theoretical Background This chapter describes the theoretical background of the determination of magnetic

exchange coupling constant from first principle calculations. An N-electron spin

Hamiltonian is formulated for diradicals having non-degenerate highest occupied molecular

orbitals. At first, energy expressions are obtained for singlet, broken-symmetry and triplet

single-determinant wave functions of unrestricted Hartree-Fock treatment. Total energy

values for the two-determinant singlet and triplet configurations that can be obtained from a

self-consistent-field treatment are determined next by using the orbital perturbation theory.

This leads to an energy ordering, which is expected to be valid also in an unrestricted

Hartree-Fock Kohn-Sham treatment. The spin Hamiltonian is based on this ordering.

Using the spin Hamiltonian, we obtain an expression for the energy differences from which

the Yamaguchi and GND formula for J can be easily obtained.

Chapter 2 Theoretical Background

2.1. Introduction

The interaction between two magnetic sites A and B in a diradical species is usually

expressed by the Heisenberg two-spin Hamiltonian

A B20H=E JS S− ⋅ (2.1)

where and are the respective spin angular momentum operators for the monoradical

fragments. A positive sign of J indicates a ferromagnetic interaction, whereas a negative sign

indicates an antiferromagnetic interaction. The eigenfunctions of the Heisenberg Hamiltonian

are eigenfunctions of S

AS BS

2 and Sz where S is the total spin angular momentum, and J is directly

related to the energy difference between the eigenstates. For a diradical,

1 0 2 .E(S= ) E(S= )= J− − The two-spin description of magnetic interaction has been

extensively correlated with the electronic structure of diradical systems.1 Recently a large

number ab initio calculations have been performed to evaluate J.2 A proper mapping between

the Heisenberg spin eigenstates and the electronic states is necessitated for the above

procedure. This is true in principle but computationally very expensive.

An alternative approach has been proposed by Noodleman so as to reliably compute

the magnetic exchange coupling constant by density functional theory with less computational

effort.3 The spin polarized, unrestricted formalism and a broken-symmetry (BS) solution for

the lowest spin-state are required in this method. The BS state is not an eigenstate of H . It is

an equal mixture of singlet and triplet states. The coupling constant can be written as

BS T2

(1 Sab

E EJ )′−=

+ (2.2)

where is the overlap integral between the spatial part of α and β orbitals in the BS

solution. Eq. (2.2) is valid for the S=1/2 interacting spins. The quantity E

Sab

BS is the energy of

the BS solution and is the energy of the triplet state in the unrestricted formalism using

the BS orbitals. In a single-determinant approach,

TE ′

TE ′ can be approximated by the energy of

the true triplet state ( ≈ ) because of the very less spin contamination in the high spin

state.

TE ′ TE

17

It is observed in literature that Eq. (2.2) is used in the strongly localized or orthogonal

limit where S → 0 as well as the strongly delocalized limit where S → 1. The current

literature is full of controversy regarding the choice of limit. Generally, in density functional

ab ab


(DFT) based calculations the magnetic orbitals are more delocalized than those obtained from

the unrestricted Hartree-Fock (UHF) calculations, and some of the authors have

recommended the use of Eq. (2.2) also in the strongly delocalized limit.4 However, Bencini et

al. have argued that Eq. (2.2) is to be restricted to the strongly localized limit.5 It has also

been concluded that the limit should be chosen on the basis of the proximity of calculated and

experimental values, rather than a consideration of rigorous theoretical complications.6

Illas et al.7 have shown that the most often-quoted trend concerning the much larger

degree of delocalization of magnetic orbitals obtained from DFT as opposed to UHF is not

fully justified. They have recommended the use of the strongly localized limit for the general

cases. In the strongly delocalized limit, Equation (2.2) becomes

BS T.2J E E≈ − (2.3)

In this situation singlet becomes degenerate with the broken-symmetry state which does not

have any scientific evidence. Despite these problems and several other deficiencies in DFT as

recently mentioned by several authors, Eq. (2.3) produced very impressive numerical results

for some systems by using the so-called B3LYP exchange correlation functional treatment.9

A large literature is found on this issue.10

The following spin projected equations are commonly used in the investigation of J in

different circumstances:

(i) the GND equation11,12

(1) BS T2

max

( E EDFT DFTJ

S−

=) (2.4)

(ii) the Bencini-Ruiz equation13,14

(2) LS T

max max

( E E( 1)

DFT DFTJ

S S−

=+

) (2.5)

(iii) the Yamaguchi equation15

(3) BS T2 2

T B

( E EDFT DFTJ

S S−

=< > − < > S

) . (2.6)

To settle the controversy in the choice of the correct expression for J, Neese has recently

analyzed the two-orbital system in CI and BS languages.16 He has advocated the use of the

corresponding orbital transformation (COT) due to Amos and Hall17 to determine the non-

18


orthogonal, ‘valence-bond’-like magnetic orbital pairs.

It is evident that BS is not the only way to approach the problem of magnetic exchange

coupling at DFT level. Recently Filatov et al.9a-b proposed a methodology based on the

Restricted Ensemble Khon-Sham formalism which deals directly with spin eigenfunctions.

The procedure suffers from the strong dependence of exchange interaction with the exchange

correlation potential.

We find that there are two theoretical aspects involved here. First, the existing literature is

based on the interpretation of the N-electron spin in the two-electron spin picture, that is, from

an attempt to provide an interpretation of the calculated results for an N-electron system in

terms of the Heisenberg two-spin Hamiltonian given by Eq. (2.1). The second aspect deals

with the nature of the wave function. An unrestricted procedure basically relies on a single-

determinantal wave function as the ground state configuration. The so-called BS approach is

based on the single determinant. Therefore, any analysis of the results computed by the BS

method must be based on such wave functions. Complications would still arise when the

highest occupied molecular orbitals (HOMO) are degenerate. This work is on diradicals with

non-degenerate HOMOs.

2.2. Theoretical background

The Heisenberg spin Hamiltonian is an effective Hamiltonian. It is normally written

with a base line that equals the energy of the lowest spin state. For a diradical, the lowest spin

state is a singlet. Nevertheless, a diradical is generally based on a pair of non-bonding

orbitals which implies that the highest occupied molecular orbitals (HOMO) would be either

non-degenerate (the single determinant representation) or degenerate. From two electrons

occupying four spin orbitals corresponding to a pair of HOMOs, it is possible to build a set of

triply degenerate configurations (triplet) and three singlet configurations. Therefore, one

needs to identify the singlet that forms the base line.

2.2.1. Single determinant approach

The single determinants for the singlet (S), broken-symmetry (B) and triplet (T) states

in the unrestricted formalism are written as

ΨS1=||η1(r1)α(s1) η/1(r2)β(s2) ... ηN/2(rN−1)α(sN−1) η/

N/2(rN)β(sN)||,

ΨS2=||η~ 1(r1)α(s1) η~ /1(r2)β(s2) ... η~ N/2+1(rN−1)α(sN−1) η~ /

N/2+1(rN)β(sN)||,

19


ΨB1=||ζ1(r1)α(s1) ζ/1(r2)β(s2) ... ζN/2(rN−1)α(sN−1) ζ/

N/2+1(rN)β(sN)||, (2.7)

ΨB2=||ζ/1(r1)α(s1) ζ1(r2)β(s2) ... ζ/

N/2+1(rN−1)α(sN−1) ζN/2(rN)β(sN)||,

ΨT1=||ξ1(r1)α(s1) ξ/1(r2)β(s2) ... ξN/2(rN−1)α(sN−1) ξN/2+1(rN)α(sN)||,

ΨT2=||ξ/1(r1)α(s1) ξ1(r2)β(s2) ... ξN/2(rN−1)β(sN−1) ξN/2+1(rN)β(sN)||.

In this work, we do not explicitly consider the density functional treatment as was

done by Ginsberg, Noodleman and Davidson. Instead, we put forward a new spin

Hamiltonian that is valid for N electrons. It holds so long as the single determinant picture is

retained, the HOMOs are nondegenerate, and the energy ordering for the lowest six states,

three singlets and a triplet, remains intact. The spatial orbitals η’s are mutually orthogonal and

similarly η/’s are mutually orthogonal. The orbitals η’s are not necessarily orthogonal to η/’s.

These sets {η} and {η/} are strictly determined from the unrestricted calculation. Similarly,

ζ’s are mutually orthogonal, ζ/’s are also mutually orthogonal, but ζ’s need not be orthogonal

to ζ/’s, and the sets {ζ} and {ζ/} are obtained from the BS calculations. The same situation is

valid for the triplet orbitals belonging to the sets {ξ} and {ξ/}. The sets {η}, {η~ }, {ζ} and

{ξ} are in general somewhat different in the valence domain. Similarly, the sets {η/}, {η~ /},

{ζ/}and {ξ/} differ from each other in the valence sector.

The single determinant energy expectation values are 1

1 ( / 2) , ( / 2) ( / 2) , ( / 2)

/ 2 1

, ( / 2) , ( / 2) , ( / 2)1

, ( / 2) , ( / 2) , ( / 2)

( / 2) ,

[ ]

[{ }

{ }]

SS core N N N N

N

a N a N a Na

a N a N a N

N

E E h h

J K J

J K J

J

−

=

= + +

+ − +

+ − +

+

∑

α α β β

α α α α β α

β β β β α β

α ( / 2)N β

(2.8)

where / 2 1

1, ,

1

/ 2 1 / 2 1

, , , , , , 1 1

( )

1 ( ). 2

NScore a a a a

a

N N

a b a b a b a b a b a ba b

E h h

J K J J K J

−

=

− −

= =

= +

+ − + + − +

∑

∑ ∑

α α β β

α α α α α β β β β β β α

(2.9)

We similarly write

20


22 ( / 2 1) , ( / 2 1) ( / 2 1) , ( / 2 1)

/ 2 1

, ( / 2 1) , ( / 2 1) , ( / 2 1)1

, ( / 2 1) , ( / 2 1) , ( / 2 1)

[ ]

[{ }

{ }

SS core N N N N

N

a N a N a Na

a N a N a N

E E h h

J K J

J K J

+ + + +

−

+ +=

+ +

= + +

+ − +

+ − +

∑

α α β β

α α α α β α

β β β β α β

( / 2 1) , ( / 2 1)

]

,N NJ + ++ α β

+

+

+

+

(2.10)

(2.11)

1 21

( / 2) , ( / 2) ( / 2 1) , ( / 2 1)

/ 2 1

, ( / 2) , ( / 2) , ( / 2)1

, ( / 2 1) , ( / 2 1) , ( / 2 1)

[ ]

[{ }

{

B B BBcore N N N N

N

a N a N a Na

a N a N a N

E E E

E h h

J K J

J K J

+ +

−

=

+ +

= =

′ ′= + +

′ ′ ′+ − +

′ ′ ′+ − +

∑

α α β β

α α α α β α

β β β β α

/( / 2) , ( / 2 1)

}]

, N NJ ++β

α β

and (2.12)

1 21

( / 2) , ( / 2) ( / 2) , ( / 2)

/ 2 1

, ( / 2) , ( / 2) , ( / 2)1

, ( / 2 1) , ( / 2 1) , ( / 2 1)

[ ]

[{ }

{

T T TTcore N N N N

N

a N a N a Na

a N a N a N

E E E

E h h

J K J

J K J

−

=

+ +

= =

′′ ′′= + +

′′ ′′ ′′+ − +

′′ ′′ ′′+ − +

∑

α α β β

α α α α β α

α α α α β α

// //( / 2) , ( / 2 1) ( / 2) , ( / 2 1)

}]

( )N N N NJ K+ ++ −α α α α

where the tilde signs and the primes have been used to indicate that the integrals over the S1,

S2, B and T orbitals are in general different from each other.

2.2.2. Two-determinant configurations

A linear combination of the BS determinants produces an approximation to the Ms = 0

component of triplet state

ΨT3′ = 2−1/2(ΨB1+ΨB2) (2.13)

with MS = 0 and similarly an approximation to the third singlet state

ΨS3′ = 2−1/2 (ΨB1−ΨB2). (2.14)

The corresponding energy values are

//

3 BTE E K= − (2.15)

and

21


//

3 BSE E K= + (2.16)

where the quantity K/ is written as /

( / 2),( / 2 1)

( / 2),( / 2 1) / 2 / 2 1 / 2 1 / 2

1 | 2 ,

| core core N N

N N N N N N

K B B K

K ς ς ς ς′+

′+ + +

=< >

′ ′=< > . (2.17)

In a two-configuration self-consistent-field (TCSCF) process, the orbitals in ΨT3′ (and ΨS3′)

would undergo relaxation to some extent to form configuration ΨT3 (and ΨS3) such that ET3′

changes into ET3 that is equal to ET (and ES3′ changes to ES3). One would like to obtain an

approximately correct measure of this change, which is discussed in the following.

2.2.3. Orbital perturbation theory

The Fock operator for state B1 is written as 2 2 1

1

1 12

N / N /B

a a a aa a

( N / )

ˆ ˆ ˆ ˆF h ( J K ) ( J K )ς α ς α ς β ς β

+

′ ′= =

≠

= + − + −∑ ∑ . (2.18)

From the energy expression (2.15), one can determine the Fock operators relevant to the state

T3 in unrestricted formalism. In doing so, we make use of the “frozen core” like assumption

that 1 2core coreB B remains unchanged. The spatial parts of the orbitals, ς and ς ′ , are

varied subject to the orthogonality constraints a b abς ς δ= and a b abς ς δ′ ′ = . We

consider 3 0TEδ = for arbitrary variations of ς ’s and ς ′ ’s. For the valence orbitals, we

need the additional constraint / 2 / 2 1 0.N Nς ς +′ = These variations are then coupled together

by Lagrange’s undetermined multiplier technique, and the multiplier matrix can be easily

shown to be hermitean. By diagonalizing the multiplier matrix, one finds Hartree-Fock

equations for the spatial functions ς and ς ′ . For the B1' component of T3, the Fock

operators are found as

/T3 B11BF F≈ (19a)

for all orbitals iς α and iς β′ , except for the orbitals / 2Nς α and ( / 2 1)Nς β+′ for which

22


/3 1 /

1

/

/ 2 / 2 1

, ˆ ˆ1 | 2 ( )

T BB

core core N N

F F F

F B B K Kς β ς α′ +

≈ +

= − < > + . (19b)

The approximately equal to sign (≈) indicates that the expression for B1F holds with the

possibility of minor changes in the spatial functions from ς and ς ′ to ς and ς ′ . The

perturbation F/ changes the spatial functions that can be used to form the broken symmetry

determinant B1' (and B2' ) in ΨT3. The first order energy correction is non-vanishing only for

the HOMOs. Thus the perturbed orbital energy values can be written as B1 (2)

B1 (2)

...,

...i i i

i i i

ς α ς α ς α

ς β ς β ς β

ε ε ε

ε ε ε′ ′ ′

= + +

= + + (2.20)

for i =1, ..., N/2 –1, and B1 (1) (2)

/ 2 /2 /2 /2

B1 (1) (2)

/ 2 1 /2 1 /2 1 /2 1

... ,

... N N N N

N N N N

ς α ς α ς α ς α

ς β ς β ς β ς β

ε ε ε ε

ε ε ε ε′ ′ ′ ′+ + + +

= + + +

= + + + (2.21)

where

/(1) (1) /

( / 2),( / 2 1)/ 2 ( / 2 1)1 | 2core core N NN N

B B Kς α ς βε ε ′ ++= = − < > = K− . (2.22)

The second order corrections to the orbital energies are given by

/

/

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

B1 B1

/ 2

2/ 2 1(2) 2 / 2

B1 B1

/ 2 1

| | | | | 1 | 2 |

| | | | | 1 | 2 |

i NNcore corei

i N

i NNcore corei

i N

KB B

KB B

ς α

ς ας α ς α

ς βς β

ς β ς β

ς α ς αε

ε ε

ς β ς βε

ε ε

+

+

′′ ′ +

,< >

= < >−

′ ′< >= < >

−

(2.23)

for i =1, …, N/2 –1, and for the orbitals / 2Nς α and / 2 1Nς β+′ ,

/

/

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

B1 B1/ 2 1/ 2( / 2)

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

B1 B1/ 2 1 1/ 2 1

| | | | 1 | 2 |

| | || 1 | 2 | .

N N jNcore coreN j

jNN

N N jNcore coreN j

N j

KB B

KB B

ς α

ς ας α ς α

ς βς β

ς β ς β

ς α ς αε

ε ε

ς β ς βε

ε ε

++

=≠

+

′ + = ′ ′+

|,

|

< >= < >

−

′ ′< >= < >

−

∑

∑

(2.24)

Equation (2.24) is only applicable to the non-degenerate HOMO case.

23


2.2.4. SCF energy

The component B1' (of ΨT3) is an eigenfunction of the operator

/(0) 3

1 1( )

( ).N

T

Bi

electrons

H F i=

= ∑ (2.25a)

This is comparable to the so-called Hartree-Fock Hamiltonian in the usual cases. The

corresponding many-body perturbation is

(0)fullH H H′ = − (2.25b)

where Hfull is the full Hamiltonian in coordinate space.

The zeroth-order energy for the B1' component (in T3) can be written as the sum of the

perturbed orbital energies, that is, / 2 N/2+1

(0) 1 11

1 1( / 2)

(1) (1)

/ 2 / 2 1

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

/ 2 ( / 2 1)1

[ ]

[ ]

[ ( ) ] ...

a a

NB B

Ba a

N

N N

N

N Na aa

E ′ ′′= =

≠

′ +

−

′ ′ +=

= +

+ +

+ + + +

∑ ∑

∑

ς α ς β

ς α ς β

ς α ς β ς α ς β

ε ε

ε ε

ε ε ε ε +

(2.26)

The zeroth order sum in this expression equals EB1(0) while the first order sum equals –2K. A

large number of terms in the second-order sum cancel each other, thereby leaving a residual

sum of only two terms,

/

/

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

/ 2 ( / 2 1)1

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

B1 B1

/ 2 / 2 1

/ 2 1 / 2/ 2

[ ( ) ]

| | | | | 1 | 2 |

| | |

[

N

N Na aa

N NNcore core

N N

N NN

KB B

K

−

′ ′ +=

++

+

+

+ + + =

< >< >

−

′ ′< >+

∑ ς α ς β ς α ς β

ς α

ς α ς α

ς β

ε ε ε ε

ς α ς α

ε ε

ς β ς β 2

B1 B1

/ 2 1 / 2

|.]

N N′ ′+−ς β ς βε ε

(2.27)

The denominators involved are of opposite signs. Therefore, the residual sum is quite small

in magnitude.

24


The first order correction is given by (1) (1)

1 1 2B BE E′ = + K , (2.28)

where (1)1BE is the first-order correction for the original determinant B1. As the SCF energy

is determined by energy up to the first order in many-body perturbation, we get

/

/

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

1 1 B1 B1

/ 2 / 2 1

2/ 2 1 / 2/ 2

B1 B1

/ 2 1 / 2

| | | | | 1 | 2 |

| | | | ...

[

]

N NNB B core core

N N

N NN

N N

KE E B B

K

ς α

ς α ς α

ς β

ς β ς β

ς α ς α

ε ε

ς β ς β

ε ε

++

′

+

+

′ ′+

< >= + < >

−

′ ′< >+ +

−

(2.29)

Comparing (2.15), we find that the energy of T3 is given by

/

/

/

2/ 2 / 2 12 ( / 2 1)

3 B1 B13

/ 2 / 2 1

2/ 2 1 / 2/ 2

B1 B1

/ 2 1 / 2

| | | |1 | 2

| | | | .

[

] ..

N NNT core coreT

N N

N NN

N N

KE E B B

K

ς α

ς α ς α

ς β

ς β ς β

ς α ς α

ε ε

ς β ς β

ε ε

++

+

+

′ ′+

< >= + < >

−

′ ′< >+

−+

(2.30)

Similarly, the S3 singlet energy can be written as

/

/

/

2/ 2 / 2 12 ( / 2 1)

3 B1 B13

/ 2 / 2 1

2/ 2 1 / 2/ 2

B1 B1

/ 2 1 / 2

| | | |1 | 2

| | | |

[

] ...

N NNS core coreS

N N

N NN

N N

KE E B B

K

ς α

ς α ς α

ς β

ς β ς β

ς α ς α

ε ε

ς β ς β

ε ε

++

+

+

′ ′+

< >= + < >

−

′ ′< >+

−+

(2.31)

These formulae apply in the case of non-degenerate HOMOs. The situation that arises in the

degenerate HOMO case is not addressed theoretically in this thesis, although in Chapter 7 we

investigate a system that has the characteristic feature.

A linear combination of S3 and T3 yields a pair of degenerate BS functions ΨB1' and

ΨB2' with energy EB' = EB1' = EB2'. The energy EB' is not directly obtainable from a quantum

chemical computation as the functions B1' and B2' are not self-consistent. As Eqs. (2.28) and

(2.29) show, the formula

25


3 3 2S TE E K= + (2.32)

is approximately valid.

The main purpose of the derivation given in this section is to decide the energy

ordering, as illustrated in Figure 2.1. It is reasonable to expect the same energy ordering from

a DFT calculation. The DFT calculations are generally carried out by Hartree-Fock-Kohn-

Sham equation of the form18

, (2.33) 1 C XCˆ[ ( ) ( )] ( ) ( )i i ih V V φ ε φ+ + =r r r r

that involves the exchange-correlation terms in VXC(r). The conventional Hartree-Fock

method can be viewed as a limiting case where the correlation contribution is completely

neglected.18 In turn, the DFT formalism can be viewed as a treatment where the effect of the

correlation contribution leads to a modification of the J and K integrals from their Hartree-

Fock counterparts plus some additional corrective terms. While the calculated total energy

can vary largely from UHF to UB3LYP because of the accommodation of the correlation

energy, the relative energy for different configurations generally changes by a lesser amount,

thereby leaving the energy ordering of the lowest-lying configurations intact in most cases.

T1,T2

S1

S2

T3/

S3/T1,T2

S1

B1,B2

S2

T3/

S3/

( K/ > 0) ( K/

< 0)

B1,B2K/

K/

S3

S3

T3

T3

Figure 2.1. Schematic illustration of the energetics in the case of nondegenerate HOMO’s. The wave

functions S3 and T3 are multiconfigurational. The UHF (UB3LYP) singlet calculations

generally lead to S1.

2.3. Spin-Hamiltonian Treatment

26


2.3.1. Base line

As S1 and S2 involve stronger repulsions ( and βα )2/( ,)2/(J NN βα )12/( ,)12/(J~ ++ NN are greater

than J/(N/2), (N/2+1)/+ ) and S2 has population in a higher orbital compared to S1, one

obtains the energy ordering E

//1)/2( /2),(

K+NN

S2 > ES1 > ES3. It is assumed here that the difference in one-

electron energies hN/2, N/2 and hN/2+1, N/2+1 does not alter the energy ordering of S1 and S3. If

the energy ordering changes, the BS method would break down as it works on the assumption

that the BS wave function has energy midway between the energies of the singlet and the

triplet. Arbitrary breakdowns from the systematics have not been observed, except when the

Bencini-Ruiz formula (2.5) applies. But ΨS3 is a two-determinant configuration. In other

words, the single determinant picture gives a poor representation of the singlet ground state of

the diradical. In practice, one computes from UHF or related techniques like UB3LYP a

singlet energy that is quite high. Following the energy ordering in Figure 1, the base line for

the spin Hamiltonian is determined by 03,SE the energy of the pure state that is basically

formed from the configuration Ψ

0S3Ψ

S3.

2.3.2. Spin Hamiltonian

We write the N-electron spin-projected form for the full Hamiltonian of the diradical

as

(2.34a) 0 0ˆspin fullH H= P 0P

where

, (2.34b) 0 0 0 0SI TJ QK

I J K

ˆ ˆ ˆ ˆ ...∑ ∑ ∑P = P + P + P +

the symbols S, T, Q, etc. indicating singlet, triplet, quintet, etc. states. The operator is the

projector for the Ith singlet eigenstate

0SIP

0SIΨ of the full coordinate-space Hamiltonian Hfull, and

so on. The eigenstates can be built up from the configurations S1, S2, etc. in an explicitly

carried out many-body treatment. We note the energy eigenvalue orderings

0 0 0S3 S1 S2 ...E E E< < < (2.35a)

etc., besides the equality

0 0 0 0T1 T2 T3 S3 2E E E E= = = − J . (2.35b)

27


A little rearrangement gives rise to the expression

0 0 2 0 0 0 0S3 SI S3 SI

0 0 0TJ S3 TJ

0 0 0QK S3 QK

ˆ ˆ( ) ( )I( 3)

ˆ ( 2 )J>3

ˆ ( 6 ) ...K

spinH E JS E E

E E J

E E J

= − + −≠

+ − +

+ − + +

∑

∑

∑

P P

P

P

(2.36)

where S2 is the squared spin angular momentum operator. The effective spin Hamiltonian is

obtained by replacing the projector in Eq. (2.36) by the unit operator: 0P

/ 0 00 SI

0 0 0TJ S3 TJ

0 0 0QK S3 QK

ˆ( )1 1 I( 3)

ˆ ( 2 )J>3

ˆ ( 6 ) ...K

spin

N NH E J S S E Ei ji j

E E J

E E J

′= − ⋅ + −= = ≠

+ − +

0S3 SI

+ − + +

∑ ∑ ∑

∑

∑

P

P

P

(2.37a)

where is the spin operator for the ith electron, the prime over the second sum indicates j ≠

i, and the base E'

iS

0 is given by

00 S3

34NE E′ = − J . (2.37b)

The Hamiltonian Hspin yields the eigenvalues for the respective eigenstates (that may be found

from an explicit many-body treatment using Hfull). We emphasize that N is the total number

of electrons involved in the calculation of energy and other molecular characteristics. The

retention of the same coupling constant J for all pairs of electrons is related to the

indistinguishability of the electrons.

Equation (2.37a) is to be distinguished from the Heisenberg spin Hamiltonian for

ferromagnetic and antiferromagnetic solids. The latter operator is written as

// 0

1 1

n n

FM AFM pq p qp q

H E J s= =

′′ s= − ⋅∑∑ (2.38)

28


where p and q are indices for the lattice sites, and ps and qs are the associated spins. Each

site has the same spin, say, s. Yamaguchi et al.15 used this Hamiltonian and considered s = ½

so that the maximum number of half spin is n with (n+1)-fold degeneracy. Thus the treatment

by Yamaguchi et al. has been an n-electron treatment where n is the number of monoradical

fragments.

For a diradical, there are only two sites, and in general no lattice is formed. For two sites,

Eq. (2.38) reduces to Eq. (2.1) and 0E ′′ equals E0. In the two-electron two-orbital model, Eq.

(2.37a) reduces to Eq. (2.1) where the need for the projectors are normally overlooked, A and

B represent two magnetically active electrons, and E0' becomes equal to E0. It is important to

note that by an appropriate partitioning of the core and valence spaces ascribed to each site,

the first two terms in Hspin of Eq. (2.37a) for any radical can be reduced to the ferromagnetic

and anti-ferromagnetic lattice Hamiltonian HFM/AFM in Eq. (2.38). Thus the first two terms in

Hspin together represent a general operator.

2.3.3. Expectation values

The total spin is written as ∑=

=N

SS1i

i such that

j

N

1i

N

1ji

/N

1i

22 SSSS i ⋅+= ∑∑∑= ==

. (2.39)

To evaluate the effect of the ji SS ⋅ terms in (2.34) and (2.37), one makes use of the equality

= (SiS S⋅ j i+Sj

− + Si−Sj

+)/2 + SizSjz. We define the overlap integrals as

>=< /S1 |S lkkl ηη ,

>=< /S2 ~|~S lkkl ηη ,

>=< /B1 |S lkkl ζζ , (2.40)

>=< /T1 |S lkkl ξξ ,

and find the following interesting expectation values of <S2>:

22

1

S12

1S1

2 |S2 ∑∑

= =

−=><N/

kkl

N/

l |NS ,

29


212

)2/(1

S212

)2/(1

S22 |S

2 ∑ ∑+

≠=

+

≠=

−=><N/

Nkk

kl

N/

Nll

|NS ,

/ /

2 2 1 2 1 22 B 2

S31 1 1 1

( / 2) ( / 2)

B B

/ 2,( / 2) / 2 1,( / 2 1)

1 1S S2 2 2

1 | 2 S S ,

N/ N/ N/ N/

kl lkk l k l

l N k N

core core N N N N

NS | |

B B

+ +′ ′

= = = =≠ ≠

′ ′

+ +

< > = − −

−

∑ ∑ ∑ ∑ B 2| |

22

1

B12

21

B12 S

2||NS

N/

kkl

N/

)N/ (l l

∑ ∑=

+

≠=

−=>< , (2.41)

212

1

T12

1T1

2 S 12

||NS N/

kkl

-N/

l∑ ∑

+

= =

−+=>< ,

/ /

2 2 1 2 1 22 B 2

T31 1 1 1

( / 2) ( / 2)

B B

/ 2,( / 2) / 2 1,( / 2 1)

1 1S S2 2 2

+ 1 | 2 S S .

N/ N/ N/ N/

kl lkk l k l

l N k N

core core N N N N

NS | |

B B

+ +′ ′

= = = =≠ ≠

′ ′

+ +

< > = − −∑ ∑ ∑ ∑ B 2| |

The condition SklS/B/T = δkl implies <S2>S1,2,3 = 0, <S2>B = 1 and <S >B

B

2T = 2. These ideal

values are rarely obtained from unrestricted calculations on the BS and T states.

Strictly speaking, the S2 operator is not well-defined in DFT. The assumption that

generally gives the difference <STS Skl kl≈ 2>T1 − <S2>B1 – 1 = |SBN/2,N/2+1| 2. The commonly

perceived difference 1−<S2>B1 = |SBN/2, N/2+1| 2 evolves from the very restrictive conditions

TSkl kl=δ . These assumptions do not necessarily hold in every case. In fact, as Eqs. (2.41)

show, it is possible for the calculated <S2>B1 and <S2>T1 to be greater than 1 and 2

respectively, and their difference can be less than 1. Such results are often obtained from

calculations.

2.4. Coupling Constant

The SCF energy values can be obtained as the expectation values of the spin

Hamiltonian. These are generally written as

30


2 0 0 00 SI S3

0 0 0TJ S3 TJ

0 0 0QK S3 QK

3 ˆ( )4 I( 3)

ˆ ( 2 )J>3

ˆ ( 6 ) ...K

NE E J S E E

E E J

E E J

⎛ ⎞′= + − < > + − < >⎜ ⎟⎝ ⎠ ≠

+ − + < >

SI

+ − + < > +

∑

∑

∑

P

P

P

(2.42)

An equivalent expression is

0 0 00 T SI S3 SI

0 0 0TJ S3 TJ

0 0 0QK S3 QK

3 ˆ ˆ2 ( )4 I( 3)

ˆ ( )J>3

ˆ ( ) ...K

NE E J E E

E E

E E

⎛ ⎞′= + − < > + − < >⎜ ⎟⎝ ⎠ ≠

+ − < >

0

+ − < > +

∑

∑

∑

P P

P

P

(2.43)

where . 0 0 0T T1 T2

ˆ ˆ ˆ ˆ=P P + P + P0T3

When the basis is large enough, the expectation values of S2 are nearly equal to the

ideal values 0, 1 and 2 for singlet, BS and triplet configurations. The addition of the sums on

the right side of (2.43) remains more or less the same for S3 and T3 (as the contribution from

the higher energy states is nearly equal in the two cases), and therefore for B1 and B2. One

also expects more or less the same sum for T1 and T2 on a similar ground. This happens

especially in the DFT calculations where the total of the sums is small. We write

0 0 0 0 0 00 SI S3 SI TJ S3

0 0 0QK S3 QK

3 ˆ ˆ( ) ( )4 I( 3) J>3

ˆ ( ) ... K

c

NJE E E E E TJ

E E E

′ + + − < > + − <≠

>

+ − < > + =

∑ ∑

∑

P P

P (2.44)

where Ec remains practically same for the S3, T1 (T2), T3 and B1 (B2) configurations, but

changes with the basis set. The quantity Ec can be interpreted as ES3. In the limit of an

infinitely large basis where basis set truncation error is negligibly small, Ec approaches the

ES3 value in the DF (Hartree-Fock Kohn-Sham) limit.

It is possible to rewrite Eqs. (2.42) and (2.43) in terms of Ec as

31


2 0 0TJ QK

ˆ ˆ2 6J>3 K

cE E J S J J= − < > + < > + < > +...∑ ∑P P , (2.45a)

and equivalently 0T

ˆ2cE E J= − < >P (2.45b)

for a particular basis set. Equation (2.45a) shows that a fair estimate of J can be obtained as

Low spin High spin2 2

High spin Low spin

( )( )

E EJ

S S−

=< > − < >

(2.46)

that is the same as Eq. (6) derived by Yamaguchi from the energy expression for the two-

electron model of a diradical [and the n-electron model of a n-radical]. The high spin state is

defined here as the state that correspond to S = S1+S2. This formula, though approximate,

holds irrespective of the limit of the two-orbital overlap and irrespective of the number of

magnetically active orbitals, and remains valid as long as the single determinantal picture is

retained for B1 (B2) and T1 (T2), and the two-determinant configuration S3 (T3) is valid.

A better relation is obtained from Eq. (45b),

Low spin High spin

0 0T High spin T Low spin

( )ˆ ˆ2( )

E EJ

−=

< > − < >P P, (2.47)

but it will be difficult to use this relation in the absence of the knowledge of the exact states

. This relation is in reality a spin projection formula. When the

difference between

0 0T1 T2 T3and, Ψ Ψ Ψ0

10

T Tˆ< >P and 0

T Bˆ

1< >P equals 1/2, one obtains the GND equation, (2.4).

If one uses the condition SklS/B/T = δkl that is reminiscent of the restricted scheme, one

obtains from Eq. (2.45b), ES3 = Ec, EB1 ≈ Ec−J, and ET1 ≈ Ec−2J such that ES3− ET1 ≈ 2J and

EB1 − ET1 ≈ J. Considering SN/2, (N/2+1) ≡ Sab to be a small but finite quantity, one immediately

gets the Noodleman equation (2.2). Actually, the Noodleman equation requires only one

assumption, that is, while SBN/2, N/2+1 may or may not equal zero, the sum of the squares of the

rest of the overlap integrals for B1 in (2.41) is more or less equal to the sum of the overlap

integral squares for T1 in the same equations. It does not require the restricted Hartree-Fock

type constraints.

The deviation of J from (2.46) mainly occurs due to the spin correlation effects as

32


shown in Eq. (2.45a). For a smaller basis set, the quantity Ec may vary for different

configurations. In such a case, both (2.46) and (2.47) would give rise to a large fractional

deviation in the calculated value of J. Most of the organic diradicals have a small coupling

constant, and the deviation calculated by using Yamaguchi or GND expressions and a smaller

basis set may not be noticeably large, though it is there. As Eq. (2.44) shows, the lack of

constancy of Ec is a general correlation effect.

2.5. Discussion

A spin Hamiltonian can also be derived from effective Hamiltonian theory of Illas et

al. 7 Here, the spin Hamiltonian is obtained for N electrons.

2.5.1. Factors Influencing Accuracy

A number of factors affect the accuracy of a calculation of the magnetic exchange

coupling constant within the BS approach. First of all, the DF methodology is to be

employed. A simple UHF calculation may yield different Ec values for S3, T3, B1 (B2), and

T1 (T2). Second, the basis must be large in size and must contain polarization and diffuse

functions so as to reduce the basis set truncation error. Third, the computed values of <S2>B1

and <S2>T1 need to be as close to 1.00 and 2.00 respectively as possible. A large deviation of

the S2 expectation value indicates the build-up of inaccuracy in the computed energy values,

and then the use of Eqs. (2.46) and (2.47) becomes suspect. Fourth, correction terms can be

added to the spin Hamiltonian defined in (2.37a). Spin biquadratic correction terms have

been considered by Noodleman et al.3c However, as these terms arise only in an indirect way

and not from a direct spin dipole−spin dipole interaction, their contribution to J is expected to

be rather small. Higher order terms in the spin Hamiltonian have been important both

experimentally as theoretically for superconducting couplers.7g

2.5.2. Numerical Tests

The S2-weighted projection

(2.48)

0 0TJ QK

J 3 K 3

2 0T

ˆ ˆ2 < 6 < ..

ˆ 2

W

S> >

= > + >

= < > − < >

∑ ∑P P

P

.+

33


is a crucial quantity. The J value calculated from Eq. (2.46) differs from that calculated by

Eq. (2.47) by a factor of 1/[1−(WB1 – WT1)]. A large ΔW indicates that the Yamaguchi

expression is in error. When the basis is sufficiently large, a measure of W can be obtained

from the approximations 0T

ˆ< P > B1 ≈0.5 and 0T

ˆ< >P T1≈1.0 that leads to Eq. (2.4). These

yield

B1 T1(B1, T1) = 2cE E E− (2.49)

that can be visualized as the average value of ES3 estimable from the computed total energies.

We consider two typical nitronyl nitroxide diradicals, namely, (i) D-NIT2 and (ii) 2,2′-

(1,2-ethynediylid-4,1 phenylene) bis [4,4,5,5–tetramethyl–4,5-dihydro-1H-imidozolyl-oxyl].

These are shown in Figure 2.2. The basic data have been taken from our previous work19,20

using the UB3LYP methodology on Gaussian98 software.21 The quantities W, J and Ec are

calculated here. The results are given in Tables 2.1 and 2.2 respectively. The observed J

values are 349.6 cm−1 for D-NIT2 in solid22 and –3.37 cm−1 for species (ii) in solution.23 The

calculations presented in this section are only to illustrate the performance of the present

methodology rather than to provide accurate values.

Table 2.1. Spin-weighted projection, coupling constant and estimated energy of the singlet

configuration from single-point UB3LYP calculations on D-NIT2.a

W b J in cm−1

c Ec in a.u.b

Basis B1 T1 Eq. (46) Eq. (47) b Eq. (49)

6−31+G** 0.1273 0.0620 −375.1 −350.6 −1145.0839

6−311+G** 0.1286 0.0629 −375.2 −350.5 −1145.3303

6−311++G** 0.1285 0.0629 −375.0 −350.4 −1145.3305

a Geometry optimization at ROHF/6-311G** level, ref. 19. b Assuming

1T BP = 0.5,

1T BP = 1.0.

c Jobs=350.2 cm−1, ref . 22.

34

For species (i), the molecular geometry was optimized by ROHF method using 6-

311G** basis set. Both singlet and triplet optimized geometries are close to each other


and also close to the crystal geometry. The WB1 and WT1 values remain more or less

unchanged while the basis set is changed, but they differ from each other (Table 2.1).

Consequently, the value calculated by the Yamaguchi formula is almost 7% larger than the J

value calculated from (2.47). The bases here are large enough to yield a near constancy of the

calculated J values as the basis size increases. Equation (2.47) obviously yields almost the

exact experimental J. The approach of Ec to a limiting value is manifest in this table.

The second molecule is quite large, and its molecular geometry was optimized at the

UHF level using the 6-31G(d) basis set. Single-point calculations were performed with

higher basis sets. In this case we find almost the same values for WB1 and WT1 such that ΔW≈

0 (Table 2.2). Thus we have more or less the same J value calculated from Eq. (2.46) and Eq.

(2.47). This fortuitous result presumably arises from the linear geometry enforced by the

acetylenic bond and the p-phenylene couplers. The coupling constant is antiferromagnetic

and very small in magnitude. The fractional variation of J with the basis size is significantly

large, and it is obvious that a large basis is required to calculate a reasonably accurate J value.

The largest basis set used is 6-311G(d,p), and the correspondingly calculated J value becomes

(i)

(ii)

Figure 2.2. The diradicals (i) D-NIT2 and (ii) 2,2′-(1,2-ethynediylid-4,1 -phenylene) bis [4,4,5,5 –tetramethyl -4,5-dihydro-1 H-imidozolyl-oxyl].

35


somewhat larger than the observed one. This has happened because the geometry was

optimized at a lower level. In any case, the approach of Ec towards a limiting value is

apparent though the limiting value itself is not manifest.

2.6. Conclusions

An analysis is made of the single determinant approach for a diradical in the non-

degenerate HOMO case. In general, one computes the determinants S1, B1 (B2) and T1 (T2).

Another determinant S2 is in existence. The determinants B1 and B2 can linearly combine to

produce the configurations S3/ and T3/. A two-configuration SCF will lead to S3 and T3 that

are slightly perturbed version of S3/ and T3/. The total energy for S3 (T3) differs from that

for S3/ (T3/) by a second order correction. The energy ordering is expected to S3 < S1 < S2.

Table 2.2. Spin-weighted projections, coupling constant and estimated energy of the singlet configuration from single-point UB3LYP calculations on Species (ii).a

W b J in cm−1

c Ec in a.u. b

Basis B1 T1 Eq. (46) Eq. (47) b Eq. (49)

6−31G** 0.0228 0.0225 −1.52 −1.51 −1455.5768

6−31+G** 0.0247 0.0241 −1.87 −1.87 −1455.6214

6−311G* 0.0229 0.0221 −2.85 −2.86 −1455.8490

6−311G** 0.0226 0.0227 −3.60 −3.60 −1455.8949

a Geometry optimization at UHF/6-31** level, ref. 20. b Assuming

1T BP = 0.5,

1T BP = 1.0.

c Jobs= −3.37 cm−1, ref. 23.

An N-electron effective spin Hamiltonian is formulated. The base line for the spin

Hamiltonian is determined by 0S3E that is the energy of the state 0

S3Ψ , an eigenstate of the full

Hamiltonian in coordinate space. The relationship with the Heisenberg two-spin Hamiltonian

and the Yamaguchi Hamiltonian can be established. An expression for the energy expectation

values is easily obtained. From this, we derive two expressions (2.46) and (2.47) for J, when

36


37

the DFT methodology is adopted and the basis set is large. In general, Eq. (2.46) is

approximate in nature. It is identical with the expression due to Yamaguchi. Equation (2.47)

is more correct. When the basis is very large, (2.47) reduces to the so-called GND formula.

An average estimate of ES3 is also obtained. A smaller basis produces a large fractional

deviation for J.

These expressions are investigated by considering two nitronyl nitroxide diradicals. In

one case, the Yamaguchi approximation differs from the spin projection formula, both

yielding almost unvarying J values for different bases, and a limiting value of ES3 is observed.

In the other case, the Yamaguchi and spin projection give the same J, but the J value

improves with an increasing basis size. For both the species, the spin projection formula leads

to the observed value of J.

2.7. References 1 (a) Nesbet, R. K. Ann. Phys. 1958, 4, 87. (b) Nesbet, R. K. Phys. Rev, 1960, 119, 658. (c) Anderson, P.

W. Phys. Rev. 1959, 115, 5745. (d) Anderson, P. W. Solid Sate Phys. 1963, 14, 99. (e) Herring, C.

Magnetism, Rado, G. T., Shul, H., Eds.; Academic Press : New York, 1965; Vol. 2B. (f) Maynau, D.;

Durand, Ph.; Daudey, J. P.; Malrieu, J. P. Phys. Rev. A 1983, 28, 3193.

2 (a) de Loth, Ph.; Cassoux, P.; Daudey, J. P.; Malrieu, J. P. J. Am. Chem. Soc. 1981, 103, 4007. (b)

Miralles, J.; Castell, O.; Caballol, R. Chem. Phys. 1994, 179, 377. (c) Wang, C.; Fink, K,; Staemmler,

V. Chem. Phys. 1995, 192, 25. (d) Calzado, C. J.; Cabrero, J.; Malrieu, J. P.; Caballol, R. J. Chem.

Phys. 2002, 116, 3985. (e) Calzado, C. J.; Cabrero, J.; Malrieu, J. P.; Caballol, R. J. Chem. Phys. 2002,

116, 2728. (f) Ciofini, I.; Daul, C. A. Coord. Chem. Rev. 2003, 238-239, 187.

3 (a) Noodleman, L. J. Chem. Phys. 1981, 74, 5737. (b) Noodleman, L.; Baerends, E. J. J. Am. Chem.

Soc. 1984, 106, 2316. (c) Noodleman, L.; Davidson, E. R. Chem. Phys. 1986, 109, 131. (d) Noodleman,

L.; Peng, C. Y.; Case, D. A.; Mouesca, J.-M. Coord. Chem. Rev. 1995, 144, 199.

4 Ruiz, E.; Alemany, P.; Alvarez, S.; Cano. J.; J. Am. Chem. Soc. 1997, 119, 1297.

5 Bencini, A.; Gatteschi, D.; Totti, F.; Sanz, D.N.; Mc Clevrty, J. A.; Ward, M. D. J. Phys. Chem. A 1998,

102, 10545.

6 Ross, P. K.; Solomon, E. I. J. Am. Chem. Soc. 1991, 113, 3246.

7 (a) Martin, R. L.; Illas, F. Phys. Rev. Lett. 1997, 79, 1539. (b) Caballol, R.; Castell, O.; Illas, F.;

Moreira, I. di P. R.; Malrieu, J. P. J. Phys. Chem. A 1997, 101, 7860. (c) Barone, V.; Matteo, A. di;

Mele, F.; Moreira, I. di P. R.; Illas, F. Chem. Phys. Lett. 1999, 302, 240. (d) Illas, F.; Moreira, I. di P.

R.; Graaf, C. De; Barone, V. Theor. Chem. Acc. 2000, 104, 265. (e) Graaf, C. de; Sousa, C.; Moreira, I.

di P. R.; Illas, F. J. Phys. Chem. A 2001, 105, 11371. (f) Moreira, I. de P. R.; Calzado, C. J.; Malrieu ,


38

J. P.; Illas, F. Phys. Rev. Lett, 2006, 97, 087003. (g) Moreira, I. de P.R.; Suaud, N.; Guihéry, N.;

Malrieu, J.P.; Caballol, R.; Bofill, J.M.; Illas, F. Phys. Rev. B 2002, 66, 134430.

8 Rodriguez-Fortea, A.; Alemany, P.; Alvarez, S.; Ruiz, E. Inorg. Chem. 2002, 41, 3769.

9 (a) Illas, F.; Moreira, I. di P. R.; Bofill, J. M.; Filatov, M. Phys. Rev. B 2004, 70, 132414. (b) Illas, F.

Moreira, I. de P. R.; Bofill, J. M.; Filatov, M. Theoret. Chem. Acc., 2006, 115, 587. (c) Dai, D.;

Whangbo, M-H. J. Chem. Phys. 2003, 118, 29.

10 (a) Ruiz, E.; Alemany, P.; Alvarez, S.; Cano. J. Inorg. Chem. 1997, 36, 3683. (b) Ruiz, E.; Cano. J.;

Alvarez, S.; Alemany, P. J. Am. Chem. Soc. 120, 11122. (c) Ruiz, E.; Alvarez, S.; Alemany, P. Chem.

Com. 1998, 2762. (d) Ruiz, E.; Cano. J.; Alvarez, S.; Alemany, P. J. Comput. Chem. 1999, 20, 1391. (e)

Yamaguchi, K.; Takahara, Y.; Fueno, T.; Nasu, K. Jpn. J. Appl. Phys. 1987, 26, L1362. (f) Onishi, T.;

Soda, T.; Kitagawa, Y.; Takano, Y.; Daisuke, Y.; Takamizawa, S.; Yoshioka, Y. Yamaguchi, K. Mol.

Cryst. Liq. Cryst. 2000, 143, 133.

11 Ginsberg, A. P. J. Am. Chem. Soc. 1980, 102, 111.

12 Noodleman, L.; Davidson, E. R. Chem. Phys. 1986, 109, 131.

13 Ruiz, E.; Cano. J.; Alvarez, S.; Alemany, P. J. Comput. Chem. 1999, 20, 1391.

14 Bencini, A.; Totti, F.; Daul, C. A.; Doclo, K.; Fantucci, P.; Barone, V. Inorg. Chem. 1997, 36, 5022.

15 (a) Yamaguchi, K.; Fukui, H.; Fueno, T. Chem. Lett. 1986, 625. (b) Yamaguchi, K.; Takahara, Y.;

Fueno, T.; Nasu, K. Jpn. J. Appl. Phys. 1987, 26, L1362. (c) Yamaguchi, K.; Jensen, F.; Dorigo, A.;

Houk, K. N. Chem. Phys. Lett. 1988, 149, 537. (d) Yamaguchi, K.; Takahara, Y.; Fueno, T.; Houk, K.

N. Theo. Chim. Acta 1988, 73, 337.

16 Neese, F. J. Phys. Chem. Solid 2004, 65, 781.

17 Amos, A.T. Hall, G.G. Proc. R. Soc. Ser. A. 1961, 263, 483.

18 Ruiz, E.; Alvarez, S.; Cano, J.; Polo, V. Chem. Phys. 2005, 123, 164110.

19 Vyas, S.; Ali, Md. E.; Hossain, E.; Patwardhan, S.; Datta, S. N. J. Phys. Chem. A 2005, 109, 4213.

20 Ali, Md. E.; Vyas, S.; Datta, S. N. J. Phys. Chem. A 2005, 109, 6272.

21 Gaussian 03, Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J.

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39

Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J.

A. Pople, Gaussian, Inc., Wallingford CT, 2004.

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23 Wautelet, P.; Moigne, J. Le ; Videva, V.; Turek, P.; J. Org. Chem. 2003, 68, 8025.

Chapter 3

Organic Fused─Ring Diradicals

This Chapter describes the investigation of ground state spins of seven diradicals belonging to the

fused ring system by ab initio restricted and unrestricted formalisms. In this work a variety of basis sets

is used. The UHF calculations yield an unrealistically large Singlet−Triplet (S−T) splitting. To avoid

spin contamination completely, we have repeated computations in the restricted (open-shell) Hartree-Fock

framework. The R(O)B3LYP/6-311G(d,p) optimized geometry yields the best total energy for each spin

state and hence the most reliable S−T energy difference. The calculated results are in agreement with the

available experimental findings. Molecules 3 and 7 have widely different geometries in the singlet and

triplet states. The UHF spin density plots obtained from the 4-31G optimized geometries manifest the

phenomenon of spin alternation in the ground state.

Chapter 3 Organic Diradicals

40

3.1. Introduction Non-Kekulé hydrocarbons are known to be diradicals and highly reactive.1 The

presence of degenerate nonbonding molecular orbitals (NBMOs) is responsible for their

extraordinary reactivity.2 A singlet ground state results when the degeneracy is spoiled. In

fact, Hoffmann has shown that when the NBMOs differ by less than 1.5 eV, the ground state

is a triplet.3 It is also well-known that a change of the symmetry of the molecule or a

variation of the electronegativity of the diradical termini can be used to control the spin

multiplicity in the ground state.4 These multiplicities can be reliably predicted by ab-initio

post-Hartree-Fock treatments using large basis sets.5

In a previous work Datta et. al.6 discussed the spin nature of some chain and

monocyclic diradicals, and found the UCCSD(T) methodology with split-valence basis sets to

be a dependable approach to the calculation of the S−T energy difference for diradicals. In

this work we use ab-initio methods to characterize the ground state spin multiplicity of seven

diradicals sharing the naphthalene skeleton (1-7). Among these molecules, one is a 1,3-

substituted naphthalene (1) and the rest are 1,8-substituted naphthalene derivatives (2-7).

These molecules are shown in Figure 1.1. Out of these, 2 and 5 are derivatives with

homonuclear substituents, and 1, 3, 4, 6 and 7 have heteroatom substituents. These non-

Kekulé diradicals exhibit very high reactivity.4 The species 4-oxy-2- naphthalenyl methyl (1)

was observed by ESR.7 While plausible zwitterionic singlet structures may be drawn for

species 1, a Curie law analysis showed that it has a triplet ground state.4 Molecule 1,8-

naphthalenediylbis(methyl) (2) was observed by Pagni et al.8 using the triplet ESR spectrum.

The molecule was postulated to be a ground state triplet.4 INDO calculations for the planar,

anti-conformation of 8-imino-1-naphthalenyl methyl (3) have been carried out by Platz et al.4

Molecule 1,8- naphthalenediylbis(amidogen) (4) was observed by Platz et al.4 from ESR and

subsequent Curie law plot. The species 8-methyl-1-naphthyl carbene (5) was prepared by the

photochemical reaction of 8-methyl-1-naphthyl diazomethane at 4K. Platz et al.4 predicted

molecule 5 to be a ground state triplet based on the observation by Trozzolo et al.9 that 1-

naphthyl carbene is a ground state triplet. The species 8-methyl-1-naphthalenyl imidogen (6)

was matrix isolated by Platz and Burns.10

An accurate calculation of the multiplet splittings in non-Kekulé systems is a

challenging task. Based on the calculations on trimethylene methane (TMM), Borden,

Davidson and Feller11 had initially shown that the restricted (open-shell) Hartree-Fock


Figure 3.1. Species investigated in this work: (1) 4-oxy-2-naphthalenyl methyl, (2) 1,8-

naphthalenediylbis(methyl) or 1,8-naphthaquinodimethane (1,8-NQM), (3) 8-imino-1-naphthalenyl methyl, (4) 1,8- naphthalenediylbis(amidogen), (5) 8-methyl-1-naphthyl carbene, (6) 8-methyl-1-naphthalenyl imidogen and (7) 8-methyl-1-naphthyl diazomethane.

[R(O)HF] and two-configuration self-consistent-field (TCSCF) calculations generally fail to

produce the correct relative energies and geometries although they may provide qualitatively

correct molecular orbitals for the two open−shell electrons in a diradical. This is a result of

the so-called doublet instability problem in RHF which is most severe when the basis set is

small.

41

Hence, these authors advocated the use of the unrestricted Hartree-Fock (UHF)

methods for a reasonably correct description of triplet and open-shell singlet geometries. In

this area, the most detailed investigations were carried out by Cramer and Smith,12a Nachtigall

and Jordan,13c and Mitani et al.14h on the molecules TMM, tetramethylene ethane (TME) and

m-xylylene respectively. They all concluded that the Singlet−Triplet energy differences are

highly sensitive to the methodology and the rigor (basis set) employed in the calculations.


42

The most widely studied diradical systems are TMM,12a-k TME,13a-j m-xylylene,14 and

polycyclic π-conjugated hydrocarbon polymers.15

Species 1-4 and 6 in Figure 1 were shown (and 5 was predicted) to be triplet in the

ground state. 4, 7-10 Species 7 is an exception. The traditional view of the chemist would be to

put one lone electron on the –CH2 substituent and the other on the two nitrogen

atoms of the substituent –CH2N2 in 7. This would indicate, by the rule of spin alternation,

that molecule 7 should be a ground state singlet, or at best a ground state triplet with very

little Singlet-Triplet energy gap. The main objective of this work is to confirm these

observations and generate quantitative data for the S−T energy gap.

3.2. Methodology

In this work we deal with diradical systems that are fairly large in size, and a complete

geometry optimization is required for each species in each spin state at both UHF and

R(O)HF levels. The STO-3G, 4-31G, 6-311G(d) and 6-311G(d,p) basis sets have been

employed in these calculations using the software Gaussian-98, but only the 6-311G(d) and 6-

311G(d,p) results are explicitly shown here. Density functional (DFT) calculations have been

performed by both UB3LYP and R(O)B3LYP methods while using some of the above-

mentioned basis sets. For the post-Hartree-Fock calculations at the unrestricted coupled-

cluster UCCSD and UCCSD(T) levels as well as the unrestricted density functional treatment

at UB3LYP level, the UHF-optimized geometry has been used for each species in each spin

state. The coupled cluster calculations get the sign right and never overestimate the gap.

Sometimes the gap is seriously underestimated. The spin-squared expectation values (which

should be 2.0) fall in a narrow range 3.2 to 3.5. Therefore, the coupled-cluster results are not

shown here. A complete geometry optimization was carried out at the DFT level only in the

restricted formalism using the R(O)B3LYP method.

A correct description of low-lying singlet and triplet states requires the proper

treatment of both static and dynamic correlation energy. The choice of UCCSD and

UCCSD(T) post-Hartree-Fock treatments for the present set of calculations is quite reasonable

from this standpoint. Using m-xylylene, Mitani et al.14h showed that the triplet state tends to

overstabilise relative to the singlet in a simple UHF calculation, whereas Møller-Plesset (MP)

perturbation calculations result in the singlet state being much more stable than the triplet.

Due to the near degeneracy of more than one UHF wave function, the MP perturbation theory


43

fails to yield correct results for diradical species. The zeroth-order UHF Hamiltonian is not

properly represented in such cases, thereby decreasing the credibility of the ensuing

perturbation-theoretic expansion. Coupled-cluster (CC),14h multiconfiguration self-consistent-

field (MCSCF)12a,13c and complete active space perturbation theory (CASPT2N)12a

methodologies treat correlation in a more sensible way, and by employing these calculational

procedures with progressively higher level of sophistication, one can overcome the problem

of one state being relatively more stabilized than the other. The importance of these

methodologies can be visualized from Table 2 in ref. 12a (for MCSCF and CASPT2N), Table

3 in ref. 13c (for MCSCF) and Table 3 in ref. 14h (for CC). Using TMM, Cramer and

Smith12a also demonstrated the restrictive nature of the density functional treatment (DFT).

The unrestricted methods like UHF, UCCSD, UCCSD(T), etc. introduce some bias

due to spin contaminations. Large deviations in the <S2> value raise questions on the

reliability of predictions made using the highly spin-contaminated geometries. There are

special methods like multi-reference coupled-cluster or non-standard version of DFT, which

are able to treat such systems reliably, concerning the interplay of static and dynamic

correlation effects.

The reason to perform computations in the restricted formalism is to correct the effect

of the spin-contaminated geometries in predicting the S−T gaps. Geometry optimization was

carried out at the STO-3G, 6-311G(d) and 6-311G(d,p) level using the R(O)HF methodology

and at the 6-311G(d,p) level using the R(O)B3LYP methodology.

3.3. Energy Differences Ab-initio calculations on species 1-7 were performed at various levels, namely, UHF,

R(O)HF, UB3LYP, R(O)B3LYP, UCCSD and UCCSD(T), by employing STO-3G, split-

valence as well as a few polarized basis sets. In every case, the Singlet−Triplet energy gap

varies with the rigor of calculation as well as the basis set, and this observation is in

agreement with the trends noticed earlier in refs.7, 8, 9 and 15. The UHF methodology

generally yields the spin-contaminated geometry, and fails to give the correct Singlet−Triplet

energy gap. As the level of calculation increases, this gap generally converges. For reasons

discussed in the previous section, results at the MP level and the UCCSD(T) calculations have

not been shown here.


44

Cramer and Smith have demonstrated that the DFT methodology cannot adequately

account for the static correlation effects in closed-shell singlets in the limit of degenerate

frontier molecular orbitals, where one would expect multi-configurational behavior. The DFT

breakdown is expected for molecules 2, 4 and 6.

3.3.1. 4-oxy-2-naphthalenyl methyl

The molecule 4-oxy-2-naphthalenyl methyl (1) is planar with Cs symmetry in both

singlet and triplet states. Table 3.1 shows the energy values computed by the restricted and

the unrestricted formalisms, the <S2> value in the triplet state, and the point group. In all

calculations on this species, except the UB3LYP/6-311G(d) one, the triplet is found to be the

ground state. The best energy gaps obtained by our calculations, 9.7 kcal mol−1 at the

R(O)B3LYP/6-311G(d,p) level are in strong agreement with the CASPT2N/6-31G* energy

gap (11.6 kcal mol−1) reported by Hrovat et al.16

Table 3.1. The ab-initio total energy and the optimized geometry for the spin states of 4-oxy-2-naphthalenyl methyl (1) in the unrestricted Hartree-Fock formalism. S and T indicate singlet and triplet respectively.

Method Basis sets Optimization ET(a.u.) <S2>TES− ET

(kcal mol−1)

UHF 6-311G(d) O -496.1265 3.1744 39.1 UB3LYP 6-311G(d) SP -499.2106 2.0414 –14.7

ROHF 6-311G(d) O −496.0855 2.0000 13.3 ROHF 6-311G(d,p) O −496.1270 2.0000 30.3

ROB3LYP 6-311G(d,p) O −499.2665 2.0000 9.7

CASSCF 6-31G(d) O −496.1703 2.0000 14.2a

CASPT2N 6-31G(d) O −497.6266 2.0000 11.6a

a Ref. 16.

The R(O)B3LYP calculations using a larger basis set including polarization functions

yields a more realistic S−T energy gap. Goodman and Kahn17 estimated, by using

photoacoustic calorimetry, that the energy difference is about 18.5 kcal mol−1. However, the

calculations by Hrovat et al.16 and those presented here indicate that this number is likely to

be an overestimation. For the other molecules too, the computed results have been given in


45

the same fashion. The UHF spin density is plotted using Hyperchem.18 All other calculations

are performed using Gaussian 98 software.19 Table 3.2. The ab-initio total energy and the optimized geometry for the spin states of

1,8−naphthalenediylbis(methyl) (2).


(kcal mol−1)

UHF 6-311G(d) O –460.2974 3.1408 37.8 UB3LYP 6-311G(d) SP –463.3069 2.0938 16.1

ROHF 6-311G(d) O –460.2328 2.0000 37.8 ROHF 6-311G(d,p) O –460.2516 2.0000 37.8

ROB3LYP 6-311G(d,p) O –463.3178 2.0000 9.4

3.3.2. 1,8−naphthalenediylbis(methyl)

The molecule 1,8−naphthalenediylbis(methyl) (2) has C2v symmetry in both singlet

and triplet states. In all the calculations performed here the triplet has emerged as the ground

electronic state (Table 3.2), which agrees with the observation of Platz et al.4 and Pagni et al.8

The molecule seems to be a prime candidate for the DFT breakdown, but its NBMOs take

part in π−orbital formation, thereby making the system simultaneously planar and stable. The

R(O)B3LYP method yields the best result for the Singlet−Triplet splitting (9.4 kcal mol−1).

3.3.3. 8−imino−1−naphthalenyl methyl

The species 8−imino−1−naphthalenyl methyl (3) is a planar molecule with Cs point

group in both singlet and triplet states. Table 3.3 shows the computed Singlet−Triplet energy

gap by different methodologies. The molecule is found to be a ground state triplet in

accordance with the observation of Platz et al.4 who relied on INDO calculations. The INDO

method, however, is grossly inadequate to give rise to the correct S−T splitting. The splitting

calculated by Platz et al.4 was 60 kcal mol−1. The S−T energy gap follow the trends

mentioned earlier and the best value calculated in the present work is 16.4 kcal mol−1

[R(O)B3LYP/6-311G(d,p)].

The two substituents on the naphthalene ring (−CH2 and −NH) are of entirely different

nature. The methylene NBMO is mixed with the π orbitals. The −NH group has the lone


electron in a σ orbital. The most remarkable point about the optimized geometries is that the

singlet has −NH2 and −CH substituents whereas the triplet has −NH and −CH2 substituents, as

shown in Figure 3.2.

H

H

HH

H

H

NH H

21

20

12

3

4

56

7

8

19

9

10

11

12

13

1415

1617

18H

H

NH

H

H H

H

H

HH21

20

12

3

45 6

7

98

10

12

11

1314

16

1718

1915

S T

Figure 3.2. Optimized singlet (S) and triplet (T) structure for 3.

Observing the optimized geometry, one may notice that the nitrogen is in the

state of sp2 hybridization in singlet and sp3 hybridization in triplet whereas the

substituent carbon is in sp3 hybridization in singlet and sp2 hybridization in triplet.

Thus, the optimized singlet and triplet geometries are in reality tautomeric forms. This

observation can be made from the geometry optimization at UHF/6-311G(d),

ROHF/6-311G(d), and ROB3LYP/6-311G(d,p) levels.

Table 3.3. The ab-initio total energy and the optimized geometry for the spin states of 8-

imino-1-naphthalenyl methyl (3).


(kcal mol−1)

UHF 6-311G(d) O –476.3133 3.1359 63.3

UB3LYP 6-311G(d) SP –479.3636 2.0931 24.3

SROHF 6-311G(d) O −476.2455 2.0000 20.8 ROHF 6-311G(d,p) O −476.2638 2.0000 19.9

ROB3LYP 6-311G(d,p) O −479.3741 2.0000 16.4

3.3.4. 1,8−naphthalenediylbis(amidogen)

The molecule 1,8−naphthalenediylbis(amidogen) (4) is a symmetric molecule with C2v

optimized geometry in each spin state. The triplet state is found to be the overall ground state

at each level of calculation. However when dealing with such systems one cannot rely fully

46


47

on nonpolarized bases. The molecule appears to be a prime candidate for DFT breakdown.

The nitrogen atoms of the diradical have three available electrons out of which two electrons

take part in π−bond formation while one electron is still left in a nonbonding orbital. These

NBMOs are degenerate, leading to difficulties in obtaining a consistent energy gap by B3LYP

or other DFT methods in the unrestricted formalism. DFT breakdown does not occur in the

restricted (open-shell) calculations. The best S−T value is 8.7 kcal mol−1 that has been

obtained from the R(O)B3LYP/6-311G(d,p) calculation [Table 3.4].

Table 3.4. The ab-initio total energy and the optimized geometry for the spin states of

1,8- naphthalenediylbis(amidogen) (4) in the unrestricted Hartree-Fock formalism.


(kcal mol−1)

UHF 6-311G(d) O −492.3102 3.1644 80.8

UB3LYP 6-311G(d) SP –495.3982 2.1019 17.3

ROHF 6-311G(d) O –492.2395 2.0000 36.4 ROHF 6-311G(d,p) O –492.2577 2.0000 36.2

ROB3LYP 6-311G(d,p) O –495.4077 2.0000 8.7

3.3.5. 8-methyl-1-naphthyl carbene

The species 8-methyl-1-naphthyl carbene (5) has the point group C1 but the

framework has Cs symmetry. The species can be derived from molecule 2 by [1,8] migration

of a hydrogen atom so that it can be viewed as a slightly higher-energy isomer of molecule 2.

It is found to have a triplet ground state, in agreement with the prediction of Platz et al.4

Unlike the earlier species, the diradical center lies on a single atom. Table 3.5 shows the

computed energy gap between Singlet and Triplet states. The best value for the energy

difference is 7.4 kcal mol−1 [R(O)B3LYP/6-311G(d,p)]. The post-Hartree-Fock CC

calculations involving the split-valence bases were performed with the orbitals 27-124 active

in the CC expansion. However, for the minimal basis, the CC expansion apparently stabilizes

the singlet to a greater extent than the triplet, thus reducing the S-T energy gap drastically.

This happens whenever the diradical is centered on a single atom, that is, also for molecule 6.


48

In such cases, the CC calculation on the triplet using the minimal basis set is not at par with

the Singlet state calculation. The DFT calculations exhibit a systematic trend.

The S−T gap for this molecule is found to be generally higher than that for molecule 2

in the unrestricted formalism and lower in the restricted (open-shell) calculations. The

NBMOs are nondegenerate, and DFT gives rise to fairly good energy differences. Table 3.5. The ab-initio total energy and the optimized geometry for the spin states of

8−methyl−1−naphthyl carbene (5) in the unrestricted Hartree-Fock formalism.


(kcal mol−1)

UHF 6-311G(d) O −460.2737 3.1921 45.2 UB3LYP 6-311G(d) SP −463.2758 2.0385 8.0

ROHF 6-311G(d) O −460.2330 2.0000 22.3 ROHF 6-311G(d,p) O –460.2504 2.0000 19.8

ROB3LYP 6-311G(d,p) O –463.2940 2.0000 7.4


methyl-1-naphthalenyl imidogen (6) in the unrestricted Hartree-Fock formalism. S and T indicate singlet and triplet respectively.


(kcal mol−1)

UHF 6-311G(d) O –476.3248 3.2048 70.5 UB3LYP 6-311G(d) SP −479.3681 2.0641 32.2

ROHF 6-311G(d) O −476.2719 2.0000 37.4 ROHF 6-311G(d,p) O −476.2923 2.0000 40.7

ROB3LYP 6-311G(d,p) O −479.3773 2.0000 27.2

3.3.6. 8−methyl−1−naphthalenyl imidogen

The molecule 8−methyl−1−naphthalenyl imidogen (6) has a planar framework in both

singlet and triplet states with symmetry Cs, the overall point group being C1. Table 3.6 shows

the computed energy gap between singlet and triplet states. From all levels of calculation the

triplet state emerges as the ground state. A single atom is the diradical centre, and again we

observe that the computed S−T energy gaps from CC calculations are unreliable while using

small bases. The species can be viewed as an analog of molecule 5, but the computed energy


gaps are a lot different from those of 5 because of the presence of a heteroatom. It can also be

considered as a higher-energy isomer of molecule 3. The best calculated S−T gap is 27.2 kcal

mol−1 [R(O)B3LYP/6-311G(d,p)].

Like molecule 5, species 6 has almost the same singlet and triplet geometries. The

singlet state has slightly elongated bonds compared to the triplet state, with the differences

varying upto 0.15Å. The bond angles hardly vary in the two spin states.

3.3.7. 8−methyl−1−naphthyl diazomethane

The molecule 8−methyl−1−naphthyl diazomethane (7) shows a large difference in its

singlet and triplet optimized structures. The singlet has Cs symmetry while the point group

for the triplet is C1. See Table 3.7. The NBMOs in this case are somewhat degenerate

leading to a slight DFT breakdown. Because of the presence of the bonds between the

heteroatoms, the UHF calculation using the STO-3G basis very confusingly indicates the

triplet as the ground state. But this is corrected by using the split-valence basis which shows

the Singlet and Triplet to have almost the same energy. The post−Hartree−Fock calculations

invariably point out the singlet to be the more stable species. Here again, the density

functional treatment leads to a systematic trend in the S−T energy difference. Calculations

were also performed using the polarized basis sets to yield the best S−T gap as −21.9 kcal

mol−1 at the R(O)B3LYP/6-311G(d,p) level.

49

0

S T

Figure 3.3. Optimized Singlet (S) and Triplet (T) structures for molecule 7.

1

2

46

89

1112

1314

17

1618

19

2

21

22 N

N

H

HH

H

H

HH

H

H H24

15

3

0

5

N

H HH

H

HH

H

H

HN

H

1

24

68

911

13

15

16

1817

20

22

2324

3

5

710

12

1419

2123

7 1


50

In the singlet state, the molecule takes up a very interesting geometry. One of the

nitrogen atoms becomes equidistant from the two CH2 groups with the length of the C−N bond

of the same order as that of a carbon−nitrogen single bond (Figure 3.3). This points out the

formation of a stable six membered ring by one nitrogen and five carbon atoms. This happen

in all the calculations, both restricted and unrestricted. The nitrogen atoms, however, remain

out of the plane of the carbon atoms so that the point group is only Cs. This situation does not

hold for the triplet case where the nitrogen atom points away from the carbon atom C16 to

which it is not directly bonded.


methyl-1- naphthyl diazomethane (7) in the unrestricted Hartree-Fock formalism.


(kcal mol−1)

UHF 6-311G(d) O −569.1994 3.3101 12.9 UB3LYP 6-311G(d) SP −572.8201 2.0563 −17.9

ROHF 6-311G(d) O −569.1501 2.0000 −18.0 ROHF 6-311G(d,p) O −569.1676 2.0000 −17.3

ROB3LYP 6-311G(d,p) O −572.8337 2.0000 −21.9

3.4. Conclusions

The diradicals TMM, TME and m-xylylene were investigated earlier by a large

number of researchers in great detail.12−14 In a previous occasion, Datta et al.6 discussed the

spin nature of some chain and monocyclic diradicals, and found the UCCSD(T) methodology

with split-valence basis sets to be a dependable approach to the calculation of the S−T energy

difference for diradicals. Here we have explored the ground electronic spin state of seven

organic diradicals belonging to the condensed ring system.

Though the UHF method gives a more or less correct, optimization of the molecular

geometry in each spin state, the relative energies calculated by the UHF method are not

reliable. Hence the method often yields significantly wrong S−T energy differences. The

calculated S−T splitting vastly improves by the application of coupled-cluster methods on the

UHF optimized geometry. The other alternative, Møller-Plesset perturbation theory,


generally yields misleading results for the S−T gap. This is also in general agreement with

the observations of Mitani et al.14g,h Results from the MP calculations are not shown in this

report. The S−T energy gap calculated with small basis sets like STO−3G and 4−31G at the

UCCSD(T) level is not very realistic. One has to use larger basis sets, especially those with

polarization functions. This imposes a limit on the computing ability using coupled-cluster

methods. So, one can resort to the density functional treatment as a workable solution.

There is another aspect of the problem. The unrestricted formalism gives rise to a

very high spin contamination as can be seen from the S2 expectation value computed for the

triplet state. The post-Hartree-Fock methods do not significantly rectify this error. But the

density functional treatment such as UB3LYP reduces spin-contamination and yields <S2>T of

the order of 2.1. The spin-contamination effect is best avoided by the restricted formalism.

The RHF formalism suffers from the difficulty that a much larger basis set is needed to obtain

the correct triplet geometry. This is why we carried out single point UB3LYP calculations

using 6-311G(d) basis sets whereas for each restricted calculations the geometry was

explicitly optimized.

Tg S Figure 3.4. The PM3 spin density contours for molecules 1 in Singlet (S) and Triplet (T) states.

The superscript g indicates the calculated ground state. Plots for 2-7 can be found in ref 20.

The DFT methodology does not always work in the unrestricted formalism. In fact,

we have noticed more or less a systematic trend in the UB3LYP calculations only for

molecules 5-7. Our best results are, therefore, from the R(O)B3LYP/6-311G(d,p)

51


52

calculations. The experimental gaps, once they are measured, are predicted to be found

within a few kcal mol−1 of the values calculated here. The calculated spin ordering in the

ground states are in excellent agreement with the experimental observations discussed in

section 1.4,7-10,12−14

All the molecules except species 7 have triplet ground states. Molecule 7 is the only

species investigated here in which one of the diradical centers is not directly attached to the

ring. This indicates a very low S−T gap or even a ground state singlet. This prediction is

borne out by all the calculations except the UHF ones.

The optimized structures in the singlet and triplet states vary from each other.

Symmetry breaking has been found to be essential in obtaining a correct estimate of the S−T

splitting that is usually of the order of only a few kcal mol−1. The variation has been found to

be the largest for molecules 3 and 7. The structure of molecule 3, as shown in Figure 3.2, is

representative of the stable, that is, the triplet state. The stable singlet is a tautomeric form

with substituents −NH2 and −CH in lieu of −NH and −CH2. The singlet of molecule 7 has a

three-fused-ring non-planar structure that has been evidenced by geometry optimization by all

the methods indicated in Table 3.7. See Figure 3.3. Finally, the rule of spin alternation in

UHF is again found to be robust. It can be used to identify the correct spin nature of the

ground state without fail for the diradical systems as shown in Figure 3.4.

3.5. References

1. (a) Coulson, C. A. J. Chim. Phys. 1948, 45, 243. (b) Moffitt, W. E. J. Chem. Soc., Faraday Trans. 1949,

45, 373.

2. Borden, W. T.; Davidson, E. R. J. Am. Chem. Soc. 1977, 99, 4587.

3. Hoffmann, R. J. Am. Chem. Soc. 1968, 90, 1475.

4. Platz, M. S.; Carrol, G.; Pierrat, F.; Zayas, J.; Auster, S. Tetrahedron, 1982, 38, 777.

5. (a) Conrad, M. P.; Pitzer, R. M.; Schaefer III, H. F. J. Am. Chem. Soc. 1979, 101, 2245. (b) Borden, W.

T.; Davidson, E. Annu. Rev. Phys. Chem. 1979, 30, 125.

6. Datta, S. N.; Mukherjee, P; Jha, P. P. J. Phys. Chem. A. 2003, 107, 5049.

7. Rule, M.; Matlin, A. R.; Hilinski, E. F.; Dougherty, D. A.; Berson, J. A. J. Am. Chem. Soc. 1979, 101,

5098.

8. Pagni, R. M.; Burnett, M. N.; Dodd, J. R. J. Am. Chem. Soc. 1977, 99, 1972.

9. Trozzolo, A. M.; Wasserman, E.; Yager, W. A. J. Am. Chem. Soc. 1965, 87, 129.

10. Platz, M. S.; Burns, J. R. J. Am. Chem. Soc. 1979, 101, 4425.


53

11. Borden, W. T.; Davidson, E. R.; Feller, D. Tetrahedron 1982, 38, 737.

12. (a) Cramer, C. J.; Smith, B. A. J. Phys. Chem. 1996, 100, 9664. (b) Feller, D.; Davidson, E. R.; Borden,

W. T. Isr. J. Chem. 1983, 23, 105. (c) Dietz, F.; Schleitzer, A.; Vogel, H.; Tyutyulkov, N. Z. Phys.

Chem.(Munich), 1999, 209, 67. (d) Gisin, M.; Wirz, J. Helv. Chim. Acta. 1983, 66, 1556. (e) Lahti, P.

M.; Rossi, A. R.; Berson, J. A. J. Am. Chem. Soc. 1985, 107, 2273. (f) Prasad, B. L. V.; Radhakrishnan,

T. P. J. Phys. Chem. 1992, 96, 9232. (g) Li, S.; Ma, J.; Jiang, Y. J. Phys. Chem. A. 1997, 101, 5587. (h)

Li, X.; Paldus, J. Chem. Phys. 1996, 204, 447. (i) Li, S.; Ma, J.; Jiang, Y. J. Phys. Chem. A. 1997, 101,

5567. (j) Pranata, J. J. Am. Chem. Soc. 1992, 114, 10537. (k) Shen, M.; Sinanoglu, O. Stud. Phys.

Theor. Chem. 1987, 51, 373.

13. (a) Hashimoto, K.; Fukutome, H. Bull. Chem. Soc. Jpn. 1981, 54, 3651. (b) Du, P.; Borden, W. T. J.

Am. Chem. Soc. 1987, 109, 930. (c) Nachtigall, P.; Jordan, K. D. J. Am. Chem. Soc. 1992, 114, 4743.

(d) Nachtigall, P.; Jordan, K. D. Ibid. 1993, 115, 270. (e) Rodriguez, E.; Reguero, M.; Caballol, R. J.

Phys. Chem. A, 2000, 104, 6253. (f) Filatov, M.; Shaik, S. Ibid. 1999, 103, 8885. (g) Chakrabarti, A.;

Albert, I. D. L.; Ramasesha, S.; Lalitha, S.; Chandrasekhar, J. Proc. Ind. Acad. Sci. 1993, 105, 53. (h)

Prasad, B. L. V.; Radhakrishnan, T. P. THEOCHEM, 1996, 361, 175. (i) Mahlmann, J.; Kleissinger, M.

Int. J. Quantum Chem. 2000, 77, 446. (j) Pittner, J.; Nachtigall, P.; Carsky, P.; Hubac, I. J. Phys. Chem.

A, 2001, 105, 1354.

14. (a) Kato, S.; Morokuma, K.; Feller, D.; Davidson, E. R.; Borden, W. T. J. Am. Chem. Soc. 1983, 105,

1791. (b) Karafiloglou, P. Croat. Chem. Acta, 1983, 56, 389. (c) Karafiloglou, P. Int. J. Quantum Chem.

1984, 25, 293. (d) Fort, Jr. R. C.; Getty, S. J.; Hrovat, D. A.; Lahti, P. M.; Borden, W. T. J. Am. Chem.

Soc. 1992, 114, 7549. (e) Fang, S.; Lee, M. S.; Hrovat, D. A.; Borden, W. T. J. Am. Chem. Soc. 1995,

117, 6727. (f) Baumgarten, M.; Zhang, J.; Okada, K.; Tyutyulkov, N. Mol. Cryst. Liq. Sci. Technol.

Sect. A. 1997, 305, 509. (g) Mitani, M.; Mori, H.; Takano, Y.; Yamaki, D.; Yoshioka, Y.; Yamaguchi,

K. J. Chem. Phys. 2000, 113, 4035. (h) Mitani, M.; Yamaki, D.; Takano, Y.; Kitagawa, Y.; Yoshioka,

Y.; Yamaguchi, K. J. Chem. Phys. 2000, 113, 10486. (i) Lejeune, V.; Berthier, G.; Despres, A.;

Migirdicyan, E. J. Phys. Chem. 1991, 95, 3895. (j) Sandberg, K. A.; Shultz, D. A. J. Phys. Org. Chem.

1998, 11, 819. (k) Havlas, Z.; Michl, J. J. Chem. Soc. Perkin Trans. 2, 1999, 11, 2299.

15. (a) Klein, D. J.; March, N. H. Int. J. Quantum Chem. 2001, 85, 327. (b) Ivanciuc, O.; Bytautas, L.;

Klein, D. J. J. Chem. Phys. 2002, 116, 4735. (c) Ivanciuc, O.; Klein, D. J.; Bytautas, L. Carbon 40,

2002, 2063.

16. Hrovat, D. A.; Murcko, M. A.; Lahti, P. M.; Borden, W. T. J. Chem. Soc., Perkin Trans. 2, 1998, 5,

1037.

17. Kahn, M. I.; Goodman, J. L. J. Am. Chem.. Soc. 1994, 116, 10342.

18. HyperChem Professional Release 7 for Windows, (Hypercube Inc., Gaimesville, 2002).

19. Frisch, M. J.; et al. Gaussian 98; Gaussian, Inc.: Pittsburgh, PA, 1998. Gaussian 98 for Windows,

(Gaussian Inc., Pittsburgh, 2002).

20. Jha, P.P.; Ali, Md. E.; Datta, S. N. J. Phys. Chem. A 2004, 108, 4084.

Chapter 4

Bis-Nitronyl Nitroxide Diradicals: Influence of Length and Aromaticity of Couplers

A series of Nitronyl Nitroxide (NN) diradicals with linear conjugated couplers and another

series with aromatic couplers have been investigated by broken-symmetry (BS) DFT

approach. The overlap integral between the magnetically active orbitals in the BS state has

been explicitly computed and used for the evaluation of the magnetic exchange coupling

constant (J). The calculated J values are in very good agreement with the observed values in

literature. The magnitude of J depends on the length of the coupler as well as the

conformation of the radical units. The aromaticity of the spacer decreases the strength of the

exchange coupling constant. The SOMO-SOMO energy splitting analysis where SOMO

stands for the singly-occupied molecular orbital, and the calculation of electron paramagnetic

resonance (EPR) parameters have also been carried out. The computed hyperfine coupling

constants support the intramolecular magnetic interactions. The nature of magnetic exchange

coupling constant can also be predicted from the shape of the SOMOs as well as the spin

alternation rule in the unrestricted Hartree-Fock (UHF) treatment. It is found that π-

conjugation along with the spin-polarization plays the major role in controlling the magnitude

and sign of the coupling constant.

Chapter 4 Nitronyl Nitroxide

4.1. Introduction

Nitronyl nitroxide is found to be one of the most promising radicals in molecular

magnetism due to its exceptional stability, facile method of preparation, versatility in

coordination, and ability to generate cooperative magnetic properties. Here we report the

results of the study of a series of nitronyl nitroxide based diradicals with different conjugated

magnetic couplers. In the present work we establish that the strength of the magnetic

interaction decreases with the increase in size of the conjugated coupler in a quantitative way,

and also with the extent of aromaticity of the ring coupler. With this aim, we have studied a

series of NN diradicals with different magnetic couplers: No coupler (1), the ethylenic coupler

(2), 1,4 butadienic coupler (3), 1,6-hexatrienic coupler (4), p-phenylene coupler (5), 2,6-

pyridine coupler (6), m-phenyelene coupler (7), 2,5-furanic coupler (8), 2,5-pyrrolic coupler

(9), and 2,5 m-thiopheneic coupler (10) . All the couplers are π-conjugated molecules.

Three recently synthesized diradicals, (11-13) with larger linear diphenylene acetylene

couplers are also investigated to study the dependence of J on basis sets.

4.2. Theoretical Background

In this work we have explicitly computed the overlap integral . The α-HOMO and

β-HOMO in the BS state have been considered as the magnetic orbitals. It is observed that

the overlap between the magnetic orbitals is very low for all the diradicals except 1a, 1c and

1d. We have further noticed that the <S

abS

2> value for all the calculated BS states deviate very

little from 1.00, and, in particular, the difference ( 2 2TS S BS< > − < > ) is nearly equal to

unity for these systems. Therefore, the magnetic exchange constants have been calculated

here by using both Eq. (2.2) and Eq. (2.4). Only in the moderately large region, for 1a,

1c, 1d and 2, Eq.(2.2) is estimated to yield better result.

abS

4.3. Computational Strategy

The molecular structures of all the diradicals 1-10 (Figure 4.1) have been fully

optimized at ROHF/6-31G(d,p) level. The optimized dihedral angle of diradical 1 between

the two planes of the NN moiety has been found to be 78° in the isolated molecule. But the

crystallographic data suggests that the dihedral angle is 55°.1 So we have taken several values

54


of the dihedral angle between the two NN moieties while keeping the rest of the optimized

molecule intact, and computed the exchange coupling constant for each of these geometries.

The angles considered are 0°, 55°, 78° and 90°.

Single point calculations have been performed on the optimized geometry at the

UB3LYP level with 6-311G(d,p) and 6-311+G(d,p) basis sets. To obtain the broken-

symmetry states, single-point UB3LYP calculations have been carried out using the accurate

guess values of molecular orbitals, which are in turn retrieved from the proper ROHF

calculations. These calculations have been done by using Gaussian 98 quantum chemical

package. The visualization software Molden2 and Molekel3 have also been used. The overlap

integral between the two magnetic orbitals in the BS state has been calculated by a program of

our own. This program utilises the MO coefficients and basis set information at 6-311+G(d,p)

level from the Gaussian 98 log files.

To further support the magnetic properties, the hyperfine coupling constants (hfcc)

have been calculated at B3LYP level by using EPR-II and EPR-III basis sets.4 The diradical

10 contains one S atom, but this atom is not included in the EPR basis set of Gaussian 98.

Therefore, during the calculation of hfcc we have used 6-311G(d,p) basis set for the S atom,

while EPR-II and EPR-III basis sets have been used for the rest of the atoms.

4.4. Results and Discussion

First of all, to make the discussion clear, the computed overlap integrals ( ) are

given in Table 4.1. The moderately large overlap region is manifest for 1a, 1c, 1d and 2. For

these species, neither Eq. (2.4) nor Eq. (2.5) can be used with accuracy. Therefore, Eq. (2.2)

gives a better estimate of J value. For 1b and 9, Eq. (2.2) would make a deviation of about

3% and 2% respectively from the J value calculated from Eq. (2.4). For all others, Eq. (2.4)

represents a better choice.

abS

The calculation of the intramolecular exchange coupling constant between the two NN

monoradicals without any coupler (in species 1) is shown in Table 4.2. The J values for the

planar diradicals with no coupler and π-conjugated linear couplers are tabulated in Table 4.3.

The values for the six member and five member conjugated aromatic couplers are given in

Table 4.4 and Table 4.5 respectively.

55


Table 4.2 shows that the magnitude of J drastically decreases with the increase in the

dihedral angle. The highest J is −923 cm−1 for the planar configuration and lowest value is

N

ΟN

+ N

ΟN

+Ο

Ο

Ο

Ο N

ΟN

+

N

Ο

N+

N

Ο

N+

Ο

Ο N

ΟN

+12

34

5

6

78

91011

12

13

14

N

ΟN

+ N

Ο

N+

Ο

Ο

1

2 3

4

5

6

78

910

11

1213

14

1516

N

ΟN

+NH N

Ο

N+

Ο

Ο

1

2 3

4

5

67

8

910

11

12 1314

15

N

ΟN

+O

N

Ο

N+

Ο

Ο

1

2 3

4

5

6

78

910

11

12 1314

15

N

ΟN

+S

N

Ο

N+

Ο

Ο

1

2 3

4

5

6

78

910

1112 13

14

15

N

ΟN

+ N

ΟN

+Ο

Ο

1

23

4

5

6

78

910

1112 13

14

1516

Ο

Ο N

ΟN

+N

Ο

N+

N

ΟN

+N

N

Ο

N+

Ο

Ο

1 2

34

5 6

7 8

9

1

12

23

344

55

66

7 78 8

9 91010

11

12

12

34

5

6

78

910

10

1

2 3

4

5

6

78

910

11

1213

14

1516

11

12

13

14

15

16

Figure 4.1. The systems under investigation with (1) no coupler, (2) ethylenic coupler, (3) 1,4-

butadienic coupler, (4) 1,6-hexatrienic coupler, (5) p-phenylene coupler, (6) 2,6-

56


57

pyridinic coupler, (7) m-phenylene coupler, (8) 2,5-furanic coupler, (9) 2,5-pyrrolic coupler, and (10) 2,5-thiophenic coupler between the two nitronyl nitroxide monoradicals.

Table 4.1. The computed overlap integral between the two magnetically active orbitals in Broken-symmetry state. The computed results are for the 6-311+G(d,p) basis sets.

Sl. No Coupler Sab

1a 0° dihedral angle −0.494041 1b 55° dihedral angle −0.178056 1c 78° dihedral angle −0.791540 1d 90° dihedral angle 0.569932

2 ethelenic 0.361410 3 1,4-butadienic 0.039911 4 1,6-hexatrinenic −0.072483

5 p-phenylene −0.014067 6 m-pyridinic 0.044348 7 m- phenylene −0.051216

8 2,5-furanic −0.006970 9 2,5-pyrrolic 0.134857

10 2,5-thiophenic 0.036758

−29 cm−1 for the 90° rotated species. This is due to the maximum overlap between the two p-

orbitals in bridging carbon atoms when the dihedral angle is 0°, and the minimum conjugation

when the two p-orbitals in bridging atoms are orthogonal. In crystal structure of 1 it is

observed that the dihedral angle is 55°. The J value calculated for 1b by using Eq. (2.4)

excellently matches with the observed J in molecular crystals. The trend in Table 4.2 makes

it amply clear that the delocalization of the π-electrons plays the major role in controlling the

exchange coupling constant. The larger dihedral angle inhibits conjugation of the π-electrons.

Nevertheless, a weak antiferromagnetic interaction exists even when the two p-orbitals are

orthogonal to each other. In this case, there is a strong localization of the SOMOs. The spin

of the unpaired electron in one of the π-orbitals polarizes the spin of the paired electrons in

the orthogonal σ-orbital. The residual spin polarization is the sole reason for a very weak

antiferromagnetic coupling constant in 1d.

It is observed that the exchange coupling constant decreases with the increase of the

length of the coupler (Table 4.3). In this Table, Eq. (2.2) is a better description for 1a while

Eq. (2.4) is more appropriate for 25a, 3 and 4.6 This is a very normal trend. It is observed that


58

2 has the highest J value. Our theoretical calculations have also supported this finding.6 The

main reason for it is that the steric effects force the dihedral angle of 1 to be 55° in molecular

crystal, which causes loss of delocalization. The rule of spin alternation in the UHF treatment7

can also predict the proper ground spin state for all the cases in Table 4.2 and Table 4.3

(Figure 4.1).

Table 4.2. Single-point energies and calculated intramolecular exchange coupling constants for

the Nitronyl Nitroxide (NN) diradicals without any coupler. The coupling constant J is calculated for different dihedral angles. All the single-point calculations are performed with the UB3LYP methodology for the broken-symmetry state as well as the triplet state.

Energy (a.u.)

<S2> J

(cm−1) Calculated

Dihedral angle

Basis sets

BS T Eq. (2.2) Eq. (2.4)

Exptl.

6-311G** −1067.7727349

1.125262 −1067.7675028

2.046992 −923 −1148

0° 1a 6-311+G** −1067.7904005

1.111334 −1067.7851668

2.046176 −923 −1148

NAa

6-311G** −1067.8615782

1.077337 −1067.8602569

2.060998 −281 −290

55° b

1b 6-311+G** −1067.8806805 1.07538

−1067.8793779 2.06042

−277 −286

−311c

6-311G** −1067.864504

1.073739 −1067.8642504

2.066934 −34 −56

78° d

1c 6-311+G** −1067.8848812 1.070904

−1067.8845819 2.064865

−41 −66

NAa

6-311G** −1067.8637813

1.072735 −1067.8636461

2.068682 −23 −30

90° 1d 6-311+G** −1067.884201

1.070115 −1067.8840684

2.066156 −22 −29

NAa

a Not available in literature; b Rotating the N-C-C-N dihedral angle of fully optimized geometry to 55° so as to get a structure similar to the crystallographic one; c Ref. 1; d Fully optimized geometry at ROHF/6-311G(d,p) level.

The calculated J values are in very good agreement with the observed values for 5-7 in

Table 4.4. Here, we find hardly any difference between Eqs. (2.2) and (2.4). The length of

the coupler in 5 is similar to the butadienic coupler in 3. However, the magnetic exchange

coupling constant is found to be less than that for the linear conjugated coupler. In general,


all the conjugated aromatic couplers are weaker than the liner couplers. The spin alternation

rule for the prediction of the ground state spin is also supported by the experimental results on

1,4 phenylene (5),8 2,6-pyiridinic coupler (6)9 and 2,6 phenylene coupler (7),10 with singlet,

triplet and triplet ground states respectively.

N

ΟN

+

Ο

Ο N

ΟN

+ N

ΟN

+ N

ΟN

+Ο

Ο

N

ΟN

+X

N

Ο

N+

Ο

Ο

N

ΟN

+X

N

Ο

N+

Ο

Ο

(CH=CH )n

1, 2, 3, 4 for n = 0, 1, 2, 3 Antiferromagnetic Antiferromagnetic

6, 7 for X = N, C Ferromagnetic

8, 9, 10 for X = O, NH, S Antiferromagnetic

5

Figure 4.2. Prediction of ground spin states and hence the nature of the magnetic exchange

coupling constants are shown according to the spin alternation rule.

Results for 8, 9 and 10 are given in Table 4.5. Here, again, the GND expression (2.4)

gives a more reliable estimate of J in every case. The data for 5 has been included in this

table for the reason of making a facile comparison. The calculated J is in good agreement

with the observed value for 10.11 Experimental values are lacking for 8 and 9, and the J values

−148 cm-1 and −164 cm−1 are predicted estimates. Again, the spin alternation rule identifies

the proper ground state for 10 as a singlet. The identified ground states for 8 and 9 are both

singlet, in agreement with the computed J values.

The sign of J depends on the parity of the number of bonds in the coupling pathway

through the coupler. When the number of bonds is odd, J is negative like in 1a, 2, 3 and 4 (1,

3, 5 and 7 bonds). In 5, there are two five-bond coupling pathways (odd number) and the

resulting J value is negative. In 6 and 7, there are two even coupling pathways (four- and six-

59


bond couplings), and J is positive. These observations represent a mere restatement of the so-

called spin alternation rule (Figure 4.2). In all three cases 8, 9, and 10, there are one even

Table 4.3. Single-point energies and calculated intramolecular exchange coupling constants for π-conjugated linear couplers. All the single-point calculations are performed with the UB3LYP methodology for the broken-symmetry state as well as the triplet state.

Energy (a.u.)

<S2> J

(cm−1) BS T Calculated Exptl

.

Diradical

Basis sets

Eq. (2.2)

Eq. (2.4)

6-311G** −1067.7727349

1.125262 −1067.7675028

2.046992 −923 −1148

NN NN1a 6-311+G** −1067.7904005

1.111334 −1067.7851668

2.046176 −923 −1148

NAb

6-311G** −1145.3113872

1.139187 −1145.3096214

2.066011 −343 −388

NNNN

2

6-311+G** –1145.3287469 1.1286

–1145.3271496 2.0629

−310 −350

–350c

6-311G** –1222.7385572

1.144146 –1222.7374101

2.084033 −251 −251

NNNN3

6-311+G** –1222.7589119 1.134768

–1222.7578636 2.080213

−230 −230

NAb

6-311G** −1300.1675127

1.130756 −1300.1668185

2.083696 −151 −152

NNNN

4 6-311+G** −1300.1870287 1.120468

−1300.1864071 2.078546

−135 −136

−66d

a Rotating the N-C-C-N dihedral angle of fully optimized geometry to 0° so as to get a planar structure like 2 and 3; b Not available in literature; c Ref. 5(a); d Ref. 5(b); for 1,6 dimethyl derivative.

and one odd pathway. At a first glance, one would think that there is a competition between

the two pathways. In reality, the odd (five-bond) route is supported by the even (four-bond)

path through the heteroatom as the latter contributes two π electrons. The J values for 8, 9,

and 10 are all negative. In magnitude, these are actually larger than the J value for 5 (Table

4.5). This behavior is similar to that known for the Fermi Contact contribution to nuclear

spin-spin couplings transmitted through the π-electronic system in conjugated compounds,

and can be viewed as an extension of the spin alternation rule to the case of heteronuclear

60


aromatic couplers.


aromatic couplers. All the single-point calculations are performed with the UB3LYP methodology for the broken-symmetry state as well as the triplet state.

Diradicals Basis

sets Energy (a.u.)

<S2> J

(cm−1) BS T Calculated Exptl.

Eq. (2.2) Eq. (2.4)

6-311G** −1298.9901966 1.090979

−1298.9897665 2.073425

−94 −94

NNNN

5

6-311+G** −1299.0106241 1.086835

−1299.0102268 2.070655

−87 −87

−72a

6-311G** −1315.0163354

1.069793 −1315.0164212

2.074852 19 19

NNN NN 6

6-311+G** −1315.0289297 1 .083623

−1315.0290336 2.090128

23 23

7b

6-311G** −1298.9864545

1.073124 −1298.9865576

2.078921 23 23

NNNN 7

6-311+G** −1299.0066208 1.070268

−1299.00 6716 2.075584

21 21

20c

aRef 8, bRef 9, cRef 10.

4.4.1. Rationalization

The spin density distribution in all the species investigated here is more or less

(pairwise) symmetric for rotation by 180° around the principal axis (C2). An understanding of

the trend of the J values in each series can be obtained by writing

ij i ji j

J J ρ ρ=

= ∑ (4.1)

where iρ is the spin density on the ith atom in the triplet state, and Jij is the exchange integral

between the π-orbitals of the atoms i and j. The integral Jij is strongest for atoms i and j being

61

nearest neighbors. For a conjugated coupler of N atoms, there are (N+1) nearest neighbors.

But the absolute magnitude of the atomic spin density approximately varies as 1/(N+1).

Therefore, as N increases, the absolute magnitude of J decreases approximately as 1/(N+1).

This is a general trend, and Table 4.3 bears a glowing testimony to it. The trend is clearly set


with |J| exhibiting the order 1a > 2 > 3> 4 in the approximate ratio 1:1/3:1/5:1/7, and the

longer the coupler is, the less antiferromagnetic interaction is there.


five-member aromatic couplers. The p-phenylene diradical is included here for the

purpose of making a comparison possible. All the single-point calculations are

performed with the UB3LYP methodology for the broken-symmetry state as well as

the triplet state.

Diradicals Basis

sets Energy (a.u.)

<S2> J

(cm−1) BS T Calculated Exptl.

Eq.(2.2) Eq.(2.4)

6-311G** −1298.9901966 1.090979

−1298.9897665 2.073425

−94 −94

NN NN

5 6-311+G** −1299.0106241

1.086835 −1299.0102268

2.070655 −87 −87

−72a

6-311G** −1296.7615506

1.098239 −1296.760815

2.065873 −161 −161

ONN NN 8

6-311+G** −1296.7837161 1.092527

−1296.7830414 2.062898

−148 −148

NAb

6-311G** −1276.9299062

1.100927 −1276.9291011

2.07081 −174 −177

NH

NN NN

9

6-311+G** −1276.9503153 1.095613

−1276.9495684 2.067769

−161 −164

NAb

6-311G** –1619.7607667

1.108292 −1619.7599146

2.07297 −187 −187

SNN NN 10

6-311+G** −1619.7811119 1.101501

−1619.7803369 2.069521

−170 −170

−157c

aRef. 8; bNot available in literature; cRef. 11.

In the case of 6-membered ring aromatic couplers, the rule of spin alternation indicates

that an antiferromagnetic coupling exists for o-phenylene and p-phenylene or their

derivatives, and a ferromagnetic coupling exists for m-phenylene. For 5-membered ring

heteronuclear aromatic couplers, the 2,3 and 3,4 species are to be treated as the o-couplers,

and 2,5 species is a p-coupler while the 2,4 one acts as m-coupler, because the hetero atom in

62


63

position 1 provides two π-electrons.

The chain rule here suggests that J ∝ 2(N+1)/(2N+1)2 where 2N is the number of

conjugated atoms in the coupler. Thus, J ∝8/49 for 6-membered p-couplers whereas J ∝ 1/5

for the butadienic coupler. Therefore, the magnitude of the J value decreases by ring

formation. The atomic spin densities in the coupler decrease further due to resonance. So a

6-membered π-coupler has a considerably reduced |J | compared to the value for a linear chain

of 4 carbon atoms. This is turned out by the calculated values for the butadienic coupler 3

(−230 cm−1) in Table 4.3 and the p-phenylenic coupler 5 (−87 cm−1) in Table 4.4. For the 6-

membered m-couplers like 6 and 7, as aromaticity increases, the J value increases (Table 4.4).

Therefore, aromaticity favors the ferromagnetic trend.

The heteronuclear couplers are less aromatic. Therefore, by counting all the six π

electrons, the para coupling with heteronuclear aromatic spacers would entail, and J ∝ 1/5.

But the resonance decreases the atomic spin densities. These two factors lead to a J value that

is almost midway between the J for 3 (−230 cm−1) and the J for 5 (−87cm−1). See Table 4.5.

Here again, the decrease in aromaticity is accompanied by an increase in antiferromagnetic

coupling. This is evidenced from the trend 5 < 8 < 9 < 10 for the absolute magnitude of the

calculated J values given in the same table.

4.4.2. SOMO-SOMO Energy Level Splitting

Hoffmann12 provided a criterion based on the extended Hückel calculations on

benzyne and diradicals, which suggests that if the energy difference between the two SOMOs

(∆ESS) is less than 1.5 eV, the two nonbonding electrons will occupy different degenerate

orbitals with a parallel-spin configuration so as to minimize their electrostatic repulsion and

thereby leading to a triplet ground state. Constantinides et al.13 have investigated a series of

4nπ antiaromatic linear and angular poly-heteroacene molecules by B3LYP/6-31G(d) method

and found that singlet ground states result when ∆ESS > 1.3 eV. Zhang et al.14 have

calculated a series of m-phenylene-bridged diradicals to investigate the effect of substitution

on the S-T energy gaps and ground state multiplicity. They have calculated ∆ESS at

ROB3LYP/6-31G(d) level. The low spin ground state results even when ∆ESS is found to be

0.19 eV. Our calculation of ∆ESS for all the diradicals 1-10 by ROB3LYP/6-311+G(d,p)

method in Table 4.6 does not reveal much information about the ground state spin.


64

For the diradicals 1a-1d the SOMO-SOMO energy gap decreases as the dihedral angle

increases. Thus the magnitude of the J value decreases with the decrease of the SOMO-

SOMO energy gap. This trend is in agreement with the Hay-Thibeault-Hoffmann (HTH)

formula for the triplet-singlet energy difference15 in a dinuclear complex containing two

weakly interacting metal atoms. For species 1, the SOMOs are not degenerate even when the

p-orbitals are orthogonal to each other. A very weak antiferromagnetic interaction is observed

in species 1d. However, as Table 4.6 shows, the same formula does not hold for other

diradicals examined here: species 6 and 7 with relatively large SOMO-SOMO energy gaps are

known to have ferromagnetic coupling and our calculations also support this fact (Table 4.4),

while the others have much smaller gaps but are antiferromagnetically coupled.

The shape of all the SOMOs at ROB3LYP for diradicals 1−10 are given in Figure 4.3.

In general, two types of SOMOs are found, namely, disjoint (where no atoms are common)

and nondisjoint (with common atoms). All the diradicals except 6 and 7 are nondisjoint in

nature. The SOMOs of 2 and 9 seem to be apparently disjoint, but these are in reality

nondisjoint as observed from the molecular orbital coefficients. We find that for the type of

organic diradicals studied here, the ferromagnetic interaction arises when the shapes of the

SOMOs are disjoint in nature as in 6 and 7 (Figure 4.3).

Table 4.6. The energy levels of two SOMOs and their energy differences (∆ESS) at ROB3LYP/6-

311+G(d,p) level for the diradicals 1-10.

Diradicals

ES(1) (a.u.)

ES(2) (a.u.)

∆ESS(eV)

1a -0.08364 -0.11228 0.7793 1b -0.09236 -0.09764 0.1437 1c -0.09701 -0.09850 0.0405 1d -0.09818 -0.09837 0.0052 2 -0.09405 -0.09550 0.0395 3 -0.09599 -0.09625 0.0071 4 -0.09647 -0.09662 0.0041 5 -0.09393 -0.09410 0.0046 6 -0.09189 -0.09590 0.1091 7 -0.09272 -0.09661 0.1059 8 -0.09656 -0.09680 0.0065 9 -0.09670 -0.09681 0.0030

10 -0.09709 -0.09764 0.0150

A similar point of interest arises. Borden and Davidson had argued that if Hückel


65

NBMO’s are not localized to disjoint groups of atoms, the triplet would lie below the

corresponding open-shell singlet at the SCF level. Our results contradict this observation, but

are in good agreement not only with experiment (Table 4.4) but also with the prediction from

the rule of spin alternation.

4.4.3. Isotropic Hyperfine Coupling Constant

From the experimental work it is observed that the hfcc of the two equivalent nitrogen

atoms in Nitronyl Nitroxide monoradicals with different substitutions at α-carbon atoms is in

the range of 7.00−7.81 G.16 The hfcc does not strongly depend on the nature of the

substitution at the α-position, but solvents play a dominant role. Hfcc values for diradicals

with conjugated couplers decrease to half of the values for the corresponding monoradicals.

The experimental values lie in the range of 3-4.5 G for diradicals with different couplers.17

Cirujeda et al.18 calculated the hfcc for several α-nitronyl aminoxyl radicals by

B3LYP method using EPR-II basis sets. They found similar hfcc for the monoradicals with

similar steric constraints between the two rings. This fact also supports that the spin density

distribution in the phenyl ring is not strongly dependent on the nature and position of

substituents. In our publications, the detailed discussions were given on this issue.6, 19

Table 4.7 shows that although the computed hfccs are different for the four nitrogen

atoms in diradicals 1-10, the average hfcc is reliably generated. Similar discrepancy for

different N atoms was also found by other authors. From the spin density distribution in the

triplet state it is observed that the calculated spin density is not homogeneously distributed

through the O-N-C-N-O bond, though the two N atoms are chemically equivalent. This fact

results in different values of hfcc, and is supported by the SOMO’s in Figure 4.3.

4.5. Diphenylene acetylene couplers: A study of size and basis sets effects

We have also investigated NN and imino nitroxide (IN) diradicals with extended

couplers. The following three spin couplers have been chosen (1) 2,2'-(1,2-ethynediyldi-4,1-

phenylene)bis[4,4,5,5-tetramethyl-4,5-dihydro-1 H-imidozolyl-oxyl] (IN-2p-IN, 11), (2) 2,2'-

(1,2-ethynediyldi-4,1 3,1-phenylene)bis[4,4,5,5-tetramethyl-4,5-dihydro-1 H-imidozolyl-

oxyl] (IN-pm-IN, 12) and (3) 2,2'-(1,2-ethynediyldi-4,1 3,1-phenylene)bis[4,4,5,5-

tetramethyl-4,5-dihydro-1 H-imidazole-1-oxyl-3-oxide] (NN-pm-NN, 13).


66

The molecular geometries are optimized at ROHF/6-31G(d) level and J is calculated

Table 4.7. The highperfine coupling constants (hfcc) calculated at B3LYP level with EPR-II and

EPR-III basis sets.

Diradicals Basis sets aN2(G)

aN4 (G)

aN7 (G)

aN9 (G)

Observed aN

1a EPR-II 4.11025 1.04162 1.03287 4.16915

EPR-III 4.23427 1.15317 1.14087 4.29395

1b EPR-II 4.31866 1.06099 1.05223 4.38357 EPR-III 4.47133 1.21081 1.20199 4.53641

1c EPR-II 4.46136 1.06774 1.07244 4.39365 EPR-III 4.67802 1.12935 1.13714 4.60911

1d EPR-II 4.59041 1.02385 1.03317 4.52136 EPR-III 4.73748 1.17743 1.18690 4.66801

2 EPR-II 4.52596 1.50349 1.34520 4.44834 EPR-III 4.17437 1.77937 1.49466 4.23153

3 EPR-II 4.35821 1.71624 1.71544 4.35549 EPR-III 4.43620 1.73844 1.73774 4.43366

4 EPR-II 4.54130 1.72502 1.72495 4.54584 EPR-III 4.50506 1.80138 1.80096 4.50906

5 EPR-II 3.38184 1.65462 1.65528 3.38509 EPR-III 3.50442 1.74658 1.74706 3.50721

6 EPR-II 4.48825 1.66194 1.66113 4.44737 3.3a

EPR-III 4.59373 1.66533 1.66506 4.55285

7 EPR-II 3.51368 1.49883 1.44803 4.62565 EPR-III 3.76404 1.84418 1.80126 4.88496

8 EPR-II 4.83269 1.62270 1.62222 4.83306 EPR-III 4.94187 1.65987 1.65902 4.94245

9 EPR-II 3.78055 1.48492 1.46878 4.86759 EPR-III 3.92236 1.66391 1.64693 5.01510

10 EPR-II 4.33239 1.50318 1.50422 4.30763 3.7b

EPR-III 4.51469 1.59648 1.59748 4.48928

aRef. 9, bRef. 11. according to the procedure described in theoretical background section but using Eq. (2.4).

The ground spin state is also determined using CASSCF methodology.


Figure 4.3. Triplet SOMOs for all the diradicals 4−7, plotted at ROB3LYP/6-311+G(d,p) level.

The UB3LYP total energies for the BS and triplet states of IN-2p-IN (11) are given in

Table 4.8. The stability consistently increases with the basis size. We have considered up to

6-311G(d,p) basis set, as the triplet geometry was optimized at the 6-31G(d) level and a larger

basis would not necessarily generate a good value of the energy difference (EBS−ET). Both 67


the BS and T wave functions suffer from spin contamination effects, but the difference in

<S2> remains more or less equal to 1.0. The intramolecular magnetic exchange coupling

constant J, calculated from Eq. (2.4), shows a smooth trend for all the basis sets. Our best

result, computed with the 6-311G(d,p) basis, is J = −3.60 cm−1 that corresponds to the largest

basis in Table 4.8 and a very small (<S2>T − <S2>BS −1), 1.36×10−4. The experimental J is

−3.37 cm−1. The difference may be attributed to the solvent effect which is not considered in

our computations, the slight difference of (<S2>T − <S2>BS) from 1, and the constraint of

geometry optimization at the HF/6-31G(d) level. The magnetic coupling is manifestly

antiferromagnetic, as predicted by the simple spin alternation rule (Figure 4.4). This is also in

agreement with the observation that linear diradical derivatives of IN with other couplers have

singlet ground states.21

H

H

H

HH

H

H

H

N

N

O

N

N

O

33

..

N

N

N

N

N

N

N

NO

O

H

H

H

H

H

H

H H

..

68

N

NO

ON

NO

ON

NO

H

H

H

H

HH

H

H

.

.

+

−+

−

11 12

13

Figure 4.4. Spin alternation in the diradicals. IN-2p-IN (11) shows a singlet ground state whereas IN-pm-IN (12) and NN-pm-NN (13) have triplet ground states.

Results form UB3LYP single-point calculations on IN-pm-IN (12) and NN-pm-NN

(13) are given in Table 4.9 and Table 4.10 respectively. The coupling between the radical

sites is manifestly ferromagnetic in each case, with positive J values. The calculated J values,

however, vary erratically with basis sets. This is unlike the antiferromagnetic case in Table

4.8. Therefore, the calculated J values are plotted against (Δ<S2> −1), where Δ<S2> = <S2>T

Table 4.8. Total energy from UB3LYP single point calculations on IN-2p-IN (11) in both broken-symmetry (BS) and triplet (T) states using different basis sets.


69

Basis EBS(a.u.)

<S2> ET(a.u.)

<S2> EBS−ET

(cal mol-1) (Δ<S2> −1)

x104J (cm–1)a

6-31G(d) −1455.5299574

1.022834 −1455.5299586

2.023165 0.75 3.31 0.26

6-31G(d,p) −1455.5768206 1.022828

−1455.5768137 2.022475

−4.33 −3.53 −1.52

6-31+G(d,p) −1455.6213633 1.024742

−1455.6213548 2.024146

−5.33 −5.96 −1.87

6-311G(d) −1455.84900831.022876

−1455.8489953 2.022066

–8.16 −8.10 −2.85

6-311G(d,p) −1455.8948799 1.022596

−1455.8948635 2.022732

−10.29 1.36 −3.60

Observedb −3.37

a From Eq. (2.4). b Ref. 20. Table 4.9. Total energy from UB3LYP single point calculations on IN-pm-IN (12) in both broken-

symmetry (BS) and triplet (T) states using different basis sets.

Basis EBS(a.u.) <S2>

ET(a.u.) <S2>

EBS−ET(cal mol-1)

(Δ<S2> −1) x104

J (cm–1)a

6-31G(d) −1455.5310178

1.023946 −1455.5310264

2.02445 5.40 5.04 1.89

6-31G(d,p) −1455.5778669 1.024046

−1455.5778774 2.024745

6.60 6.99 2.32

6-31+G(d,p) −1455.6228021 1.026287

−1455.6228052 2.026526

1.95 2.39 0.68

6-311G(d) −1455.8502143 1.024004

−1455.8502274 2.024968

8.22 9.64 2.87

Extrapolatedb 0.16

a From Eq. (2.4). b Fig. 4.5 (a). − <S2>BS. These plots turn out to be surprisingly linear, and the best straight lines are shown

in Fig. 4.5(a) and Fig. 4.5(b) respectively. The extrapolated values of J, for Δ<S2> =1, are

0.16 cm−1 for 12 and 0.67 cm−1 for 13. From experiment, Wautelet et al20 concluded that the J

values for 12 and 13 are larger than the hfcc but extremely small in magnitude, less than 1 K.

The J values extrapolated here are in agreement with the experimental observations.

In principle, the magnetic exchange coupling constant J can also be determined by the

CASSCF methodology by using J = ES − ET. In practice, however, an explicitly detailed

CASSCF calculation can be performed only on very small species. For larger species such as

the diradicals 11−13, one can carry out a CASSCF calculation with only a handful of active


70

electrons in a handful of active orbitals. This limitation invariably results in a large value of

the calculated energy difference between the spin states, and the J values cannot be accurately

Table 4.10. Total energy from UB3LYP single point calculations on NN-pm-NN (13) in both

broken-symmetry (BS) and triplet (T) states using different basis sets.

Basis EBS(a.u.) <S2>

ET(a.u.) <S2>

EBS−ET(cal mol-1)

(Δ<S2> −1) x104

J (cm–1)a

6-31G(d) −1605.8625392

1.078432 −1605.8625728

2.082158 21.08 37.26 7.40

6-31G(d,p) −1605.9099065 1.082187

−1605.9099236 2.084283

10.73 20.96 3.76

6-31+G(d,p) −1605.9613842 1.075813

−1605.9613969 2.076811

7.97 9.98 2.8

6-311G(d) −1606.2261714 1.077158

−1606.2262023 2.080576

19.39 34.18 6.79

Extrapolatedb 0.67

a From Eq. (2.4). b Fig. 4.5 (b).

determined. In fact, Table 4.11 shows the results from the CASSCF calculations on

diradicals 11−13 by using 10 active electrons in 10 active orbitals. The spin state energy

difference (ET−ES) comes out to be very large in every case, of the order of a few kcal mol−1.

Nevertheless, the CASSCF results definitely identify the ground state spin. Table 4.11 clearly

shows IN-2p-IN (11) to have a singlet ground state whereas IN-pm-IN (12) and NN-pm-NN

(13) have triplet ground states.

4.6. Conclusions

A series of bis-nitrotronyl nitroxide diradicals with ten different conjugated couplers

have been investigated by broken-symmetry density functional treatment. The computed

magnetic exchange coupling constants are in very good agreement with the reported values.

Moreover, J values for 3, 4, 8 and 9 are predicted here (−230 cm-1, −136 cm-1, −148 cm-1 and

−161 cm-1, respectively). Sometimes it becomes necessary to explicitly compute the overlap

integral between the two magnetically active orbitals to calculate the exchange coupling

constant accurately by the broken-symmetry approach. The α-HOMO and β-HOMO in the

BS state are generally found to be magnetic orbitals.


71

0 2 4 6 8 10

1

2

3

0

J

(cm

−1)

(Δ<S2>−1)x104

0 10 20 30 40

2

4

6

8

0 (Δ<S2>−1)x104

J (c

m−1

)

4.5(a) 4.5.(b) Figure 4.5. The best straight line plot of the computed J values against (Δ<S2>−1). The final J is

obtained by extrapolating the straight line to Δ<S2> = 1. We get (a) IN-pm-IN, J= 0.16 cm−1, and (b) NN-pm-NN, J=0.67 cm−1. The standard deviations are (a) 0.25 and (b) 0.52 in cm−1.

Table 4.11. Results from CASSCF (10,10) calculations with different basis sets for the diradicals

11−13.

Molecule Basis set Energy (a.u.) ES −ET

(kcal mol−1) J (cm−1)

Single (S) Triplet (T)

6-31G −1445.7448 −1445.7237 −13.24 −2323 IN-2p-IN (11) 6-311G(d) –1446.6345 −1446.6331 −0.88 −154.1

6-31G −1445.7012 −1445.7788 48.69 8542 IN-pm-IN

(12) 6-311G(d) –1446.5921 –1446.6342 26.42 4632

6-31G −1595.2477 −1595.2859 23.97 4205 NN-pm-NN (13) 6-311G(d) Convergence failure

In conjugated systems, the magnetic interaction is mainly transmitted through the π-

electron conjugation. The strength of antiferromagnetic interaction decreases with the increase

in the length of conjugated couplers. Conjugated linear couplers are more efficient

antiferromagnetic couplers than the aromatic ones of similar length. As the aromaticity of the

spacer decreases, the magnitude of the antiferromagnetic coupling constants increases. In

general, aromaticity favors the ferromagnetic trend. The diradicals with m-couplers are

undoubtedly ferromagnetic. The shape of the SOMOs as well as the rule of spin alternation in


72

the UHF emerge as two robust guidelines for the prediction of the qualitative nature of the

intramolecular magnetic interaction in bis-nitronyl nitroxide diradicals.

The intra-molecular magnetic exchange interaction between the radical units is

antiferromagnetic with J = −3.60 cm−1 for the diradical 11 and ferromagnetic for the diradical

12 and 13 with coupling constants 0.16 cm−1 and 0.67 cm−1 respectively. These calculated

values are in excellent agreement with the experimental results. The calculated J is also

dependent on the basis set so that it requires higher basis sets to get reliable results, especially

when the interaction is very weak.

References 1. Alies, F.; Luneau, D.; Laugier, J.; Rey, P. J. Phys. Chem. 1993, 97, 2922.

2. Schaftenaar, G.; Noordik, J. H. "Molden: a pre- and post-processing program for molecular and

electronic structures", J. Comput.-Aided Mol. Design, 2000, 14, 123.

3. Flükiger, P.; Lüthi, H. P.; Portann, S.; Weber, J. MOLEKEL, v.4.3; Scientific Computing: Manno,

Switzerland, 2002-2002. Portman, S.; Lüthi, H. P. CHIMIA 2000, 54, 766.

4. Barone, V. Recent Advances in Density Functional Methods; Part I; Ed. D. P. Chong; World Scientific

Publ. Co., Singapore, 1996.

5. (a) Ziessel, R.; Stroh, C.; Heise, H.; Köehler, F. K.; Turek, P.; Claiser, N.; Souhassou, M.; Lecomte, C.

J. Am. Chem. Soc. 2004, 126, 12604. (b) Stroh, C.; Ziessel, R.; Raudaschl-Sieber, G.; Köehler, F.;

Turek, P. J. Mater. Chem. 2005, 15, 850.

6. Ali, Md. E.; Vyas, S.; Datta, S. N. J. Phys. Chem. A 2005, 109, 6272.

7. (a) Trindle, C.; Datta, S. N. Int. J. Quantum Chem. 1996, 57, 781. (b)Trindle, C.; Datta, S. N.; Mallik, B.

J. Am. Chem. Soc. 1997, 119, 12947.

8. Caneschi, A.; Chiesi, P.; David, L.; Ferraro, F.; Gatteschi, D.; Sessoli, R. Inorg. Chem. 1993, 32, 1445.

9. Ziessel, R.; Ulrich, G.; Lawson, R. C.; Echegoyen, L.; J. Mater. Chem. 1999, 9, 1435.

10. Shiomi, D.; Tamura, M.; Sawa, H.; Kato, K.; Kinoshita, H. Syn. Metals 1993, 56, 3279.

11. Mitsumori, T.; Inoue, K.; Koga, N.; Iwamura, H.; J. Am. Chem. Soc. 1995, 117, 2467.

12. Hoffmann, R.; Zeiss, G. D.; Van Dine, G. W. J. Am. Chem. Soc. 1968, 90, 1485.

13. Constantinides, C. P.; Koutentis, P. A.; Schatz, J. J. Am. Chem. Soc. 2004, 126, 16232.

14. Zhang, G.; Li, S.; Jiang, Y. J. Phys. Chem. A 2003, 107, 5373.

15. Hay, P. J.; Thibeault, C. J.; Hoffmann, R. J. Am. Chem. Soc. 1975, 97, 4884.

16. (a) D’Anna, J. A. ; Wharton, J. H. J. Chem. Phys. 1970, 53, 4047. (b) Jurgens, O. ; Cirujeda, J. ; Mas,

M. ; Mata, I. ; Cabrero, A. ; Vidal-Gancedo, J. ; Rovira, C.; Molins, E. ; Veciana, J. J. Mater. Chem.

1997, 7, 1723 . (c) Zeissel, R. ; Ulrich, G. ; Lawson, R. C. ; Echegoyen, L. J. Mater. Chem. 1999, 9,

1435 . (d) Shiomi, D. ; Sato, K. ; Takui, T. ; Itoh, K. ; Tamura, M. ; Nishio, Y. ; Kajita, K. ; Nakagawa,

M. ; Ishida, T. ; Nogami, T. Mol. Cryst. Liq. Cryst. 1999, 335, 359.


73

17. (a) Luckhurst, G. R. In Spin Labeling. Theory and applications ; Berliner, J. L., Ed. ; Academic Press:

New York, 1976; p 133 ff. (b) Luckhurst, G. R. ; Pedulli, G. F. J Am. Chem. Soc. 1970, 92, 4738 ; (c)

Dulog, L. ; Kim, J. S. Makromol. Chemie 1989, 190, 2609.

18. Cirujeda, J.; Vidal-Gancedo, J.; Jürgens, O.; Mota, F.; Novoa, J. J. ; Rovira, C.; Veciana, J. J. Am. Chem.

Soc. 2000, 122, 11393.

19. Ali, Md. E.; Datta, S. N. J. Phys. Chem. A 2006, 110, 2776.

20. Wautelet, P.; Le Moigne, J.; Videva, V.; Turek, P. J. Org. Chem. 2003, 68, 8025.

Chapter 5

Influence of Aromaticity in Intramolecular Magnetic Coupling

In this Chapter we discuss the prediction of the intramolecular magnetic exchange coupling

constant (J) for nitronyl nitroxide (NN) diradicals with different linear and angular polyacene

couplers. For the linear acene couplers, J initially decreases with increase in the number of

fused rings. But from anthracene coupler onwards, the J value increases with the number of

benzenoid rings due to an increasing diradical character of the coupler moiety. The J value

for the diradical with a fused bent coupler is always found to be smaller than that for a

diradical with a linear coupler of the same size. Nuclear independent chemical shift (NICS) is

calculated and it is observed that the average of the NICS values per benzenoid ring in the

diradical is less than that in the normal polyacene molecule. An empirical formula for the

magnetic exchange coupling constant of a NN diradical with an aromatic spacer is obtained

by combining the Wiberg bond order (BO), the angle of twist (φ) of the monoradical (NN)

plane from the plane of the coupler, and the NICS values. A comparison of the formula with

the computed values reveals that from tetracene onwards, the diradical nature of the linear

acene couplers becomes prominent thereby leading to an increase in the ferromagnetic

coupling constant. Isotropic hyperfine coupling constants are calculated by using polarized

continuum model for the diradicals in different solvents and in vacuum. In the last section of

this Chapter we have discussed the substituent effects on the m-phenylene couplers and the

verification of the empirical relationship is also described.

Chapter 5 Influence of Aromaticity

74

5.1. Introduction

Nitronyl nitroxide (NN) radicals have become the natural choice in molecular

magnetism since they are stable at ordinary conditions of temperature and pressure and also

have cooperative magnetic properties.1 These radicals are well characterized from structural

and spectroscopic viewpoints. The strong localization of the unpaired electron of NO makes

the NN radicals as ideal ferromagnetic precursors.2 In Chapter 4, we have noticed that the

π-conjugated linear spacers are, as couplers, stronger than the aromatic ones. We have also

noticed that the aromaticity of the coupler plays a major role in controlling the strength of

magnetic interaction. The m-phenylene species is known to be one of the best ferromagnetic

couplers. In the present work, we investigate the magnetic properties of eleven

ferromagnetically coupled NN diradicals with linear and angular polyacene couplers (Figure

5.1). The polyacenes are aromatic hydrocarbons with benzenoid rings. They have been

extensively investigated for their electronic properties, molecular structure, and aromaticity.

Pentacene has attracted a special attention as an active organic semiconductor molecule.3 The

larger polyacenes are predicted to be conductors with nearly zero band gap.4 The objective

of this work is to investigate the intramolecular ferromagnetic interaction mediated by

polyacene spacers.

In the latter part of this chapter we investigate 27 substituted m-phenylene based

nitronyl nitroxide diradicals (Figure 5.2). The object of this portion is finding out the effect of

subtitutent of m-phenylene in intramolecular ferromagnetic coupling.

5.2. Computational Methodology

Molecular geometries are optimized at the ROHF level using 6-311G(d,p) basis sets.

The magnetic exchange coupling constants are calculated using the spin-polarized

unrestricted DFT methodology. The magnetic exchange coupling constants are calculated by

using UB3LYP/6-311+G(d,p) methodology and following equations Eq. (2.4) and Eq. (2.6).

To study the effect of aromaticity of the coupler on the magnetic exchange interaction,

the nucleus-independent chemical shift (NICS) are calculated by B3LYP/GIAO methodology

for all the aromatic rings in each diradical. The NICS values are calculated at the centre of

the rings [NICS(0)]. But the σ framework of C−C and C−H affects the π electrons and hence


NICS is also calculated at a point 1 Å above the ring center [NICS(1)] where the π-electron

density is known to be maximum.

75

Ο Ο

N

Ο

N+N

ΟN

+

Ο N

Ο

N+

N

ΟN

+

Ο

Ο

N

Ο

N+

N

ΟN

+

Ο

Ο

N

Ο

N+

N

ΟN

+

Ο

Ο

Ο

N

ΟN

+ N

Ο

N+

Ο

N

Ο

N+

N

ΟN

+

Ο

Ο

Ο

N

ΟN+

N

Ο

N+

Ο

N

ΟN

+

N

Ο

N+

Ο

Ο

Ο

N

ΟN+

N

Ο

N+

Ο

N

ΟN

+ N

Ο

N+

Ο

A

Ο

Ο

N

ΟN

+N

Ο

N+

C5

1

2 3

45 10

1

2

45

10

6

78

9

6

78

910

1

2 3

4 510

6

78

910

1

2 3

45

6

78

9

67

8

910

1

2 3

4 5

1

2

3

4

5

6

78

910

6

78

910

Figure 5.1. Diradicals under investigation. The benzene rings are identified by alphabets A, B, C, etc. for the shake of convenience of discussion.

1

2 3

45

1

2

3

4

5

6

78

910

3

1 2

3 4

A BA

6

7 8

9

6

78

910

1

2 3

45 10

H H

1

23

45

6

789

10

11

C DA B A B C

BA

A A

A A

A

BB

BB

B

C

C

C

C C

C

DD

D

D

D

EF

10

1

2 3

45

6

78

9

5

EDA B C


76

The isotropic hyperfine coupling constant (hfcc), which is essential in experiment to

characterize the radical systems and to predict the intramolecular exchange interaction, is also

calculated at B3LYP/EPR-II/C-PCM level. The hfcc values are first determined for the

diradical in vacuum. EPR parameters are strongly solvent dependent. To account the solvent

effect, hfcc’s are also calculated using conductor-like polarizable continuum model (CPCM).

Three solvents have been considered. These are the nonpolar solvent benzene (ε=2.25), the

moderately polar and aprotic solvent acetonitrile (ε=36.64), and the polar and protic solvent

water (ε=78.39).


Table 5.1 shows the calculation of J from Eqs. (2.4) and (2.6). The decreasing order

of the J from 1 to 3 is in agreement with our general observation that magnetic exchange

interaction in NN diradical with linear conjugated couplers decreases with the increase in the

length of the coupler.5 The reason for the deviation of 3-5 is that the larger oligoacenes

possess open-shell singlet ground states,6a that is, these acenes are diradicals with disjoint

nature.

The J value decreases remarkably for the bent couplers. The coupler in 6 has 3 fused

rings. Those in 7-9 have 4 fused rings. The coupler in 10 has 4 fused aromatic and one fused

non-aromatic rings, while that in 11 has 7 fused aromatic rings. One consequence of being

bent is that the coupler fragments lose the disjoint diradical character. Also, they become

stronger aromatics as discussed later. Diradicals 3 and 6 are similar, but J is much smaller for

phenanthrene coupler (6) than the anthracene one (3). The J value further decreases for the

1,8 and 1,7 substituted pyrene couplers (7 and 8). The couplers of diradicals 2 and 7 are

similar but the value of J for 2 is more than twice that for 7. Again, 6 and 8 are similar in

length except for an additional ring in 8, but J is larger in 6 than in 8. The same trend, that is,

the decrease of the J value with the increase of conjugation in the bent aromatic coupler is

observed in the case of 9-11. Nevertheless, conjugation within the coupler is not the only

factor that determines the strength of the intramolecular exchange interaction.


77

5.3.1. Bond Order and Dihedral Angles

Wiberg bond index (order)7 is calculated by natural bond orbital (NBO) analysis

(implemented in Gaussian 03) at B3LYP/6-311+G(d,p) level. The calculated bond orders are

given in Table 5.2 along with the angle of rotation of the NN plane from the coupler plane (φ). Table 5.1. Results from single-point broken-symmetry calculations at UB3LYP/6-311+G(d,p) levels

and the calculated J values. The triplet geometry is optimized. Legends: J(2) for GND equation (2.4) with Smax=1, and J(4) from Yamaguchi equation (2.6).

Diradical Energy (a.u.) (<S2>)

J(cm−1)

BS T Eq. (2.4) Eq. (2.6)

1a −1299.0066208 1.0703

−1299.00 6716 2.0756 20.89 20.78

2 −1452.68288294 1.0706

−1452.68295337 2.0754 15.46 15.39

3 −1606.35279897 1.0720

−1606.35286055 2.0775 13.52 13.44

4 −1760.02026569 1.0742

−1760.02032912 2.0821 13.92 13.81

5 −1913.68945590 1.0767

−1913.68953368 2.0910 17.07 16.73

6 −1606.32389842 1.0946

−1606.32393844 2.0974 8.78 8.76

7 −1682.60289023 1.0706

−1682.60291783 2.0726 6.06 6.05

8 −1682.60180740 1.0683

−1682.60183056 2.0701 5.08 5.07

9 −1760.03222796 1.0825

−1760.03224969 2.0842 4.77 4.76

10 −1798.16513473 1.0707

−1798.16515225 2.0722 3.85 3.84

11 −1988.76716320 1.0660

−1988.76716712 2.0663 0.86 0.86

a These values are reported in reference Ref. 5, and the observed J value is 20 cm−1.

The average bond order (BO) for the linear acene couplers (1-5) increases with the

increase of the number of phenyl rings in the coupler. A larger bond order generally favors a

greater conjugation with the radical centers, and hence a larger magnetic exchange coupling


constant. The rotation of the NN plane from the plane of the coupler (φ) has an opposite

effect, that is, if φ increases, J decreases because of the lesser conjugation. In 1-3, J decreases

although BO increases and φ decreases whereas for 3-5, J increases with the size of the

coupler along with the increase in φ. Table 5.2. The calculated Wiberg bond order at B3LYP/6-311+G(d,p) level for the NN–coupler bond

and the average dihedral angle between the NN and coupler planes.

Diradical Bond Ordera Dihedral Angleb

r1 r2 Average φ1 φ2 Average

1 1.05 1.05 1.05 34.74 32.33 33.54 2 1.07 1.07 1.07 25.76 25.78 25.77 3 1.08 1.07 1.08 26.45 22.87 24.66 4 1.07 1.08 1.08 26.87 22.71 24.79 5 1.07 1.08 1.08 27.06 22.61 24.84

6 1.07 1.05 1.06 5.72 43.65 24.68 7 1.07 1.02 1.05 23.01 54.22 38.61 8 1.07 1.03 1.09 22.90 53.87 38.38 9 1.07 1.07 1.07 24.22 23.16 23.69

10 1.07 1.07 1.07 23.96 25.22 24.59 11 1.02 1.02 1.02 55.32 55.31 55.31

a r1 and r2 are bond lengths between the benzenoid ring and the two NN radicals. bφ1 and φ2 are the angles of twist of the two NN moieties from the plane of the coupler.

The planes of the two NN moieties are asymmetrically twisted for the angular

diradicals 6-8. One of the NN planes undergoes a large twist, and this fact is also reflected in

BO. The BO and φ are consistent with the trend in calculated J values for the diradicals 6-8,

and a similar trend is observed for 9-11. For the highly planar and conjugated coupler

coronene (11), the very less J value (0.86 cm−1) is due to the extremely large angle of twist

(φ=55.31°), basically a stereo-electronic effect.

5.3.2. Nuclear Independent Chemical Shift (NICS)

NICS(0) and NICS(1) are calculated at GIAO-B3LYP/6-311+G(d,p) level for

different six membered rings in each coupler. NICS is an accepted measure of aromaticity.

The benzenoid rings are denoted as A, B, C, etc. in Figure 5.1 and the corresponding values of

NICS(0) and NICS(1) are given in Table 5.3.

78


79

Table 5.3. The calculated NICS values at the center of the aromatic rings for diradicals 1-11.

Diradical NICS A B C D

NICS(0) −7.51

1 NICS(1)

−9.24

NICS(0) −7.60 −7.57

2 NICS(1)

−9.62 −9.68

NICS(0) −6.47 −10.3 −6.28

3 NICS(1)

−8.61 −12.22 −8.41

NICS(0) −5.15 −10.5 −10.38 −5.04

4 NICS(1)

−7.51 −12.4 −12.29 −7.35

NICS(0) −3.87 −9.60 −11.40 −9.61

5a

NICS(1) −6.62 −11.62 − 13.64 −

11.51

NICS(0) −7.95 −4.66 −8.06

6 NICS(1)

−9.96 −7.72 −10.16


80

NICS(0) −10.41 −2.86 −10.85 −2.72

7 NICS(1)

−12.19 −6.29 −12.58 −5.95

NICS(0) −10.68 −2.82 −10.32 −2.93

8 NICS(1)

−12.42 −6.38 −12.13 −6.33

NICS(0) −8.04 −5.63 −5.61 −8.20

9 NICS(1)

−8.85 −8.06 −8.81 −11.12

NICS(0) −7.36 −5.21 −5.17 −7.49

10 NICS(1)

−9.37 −8.06 −8.07 −9.48

NICS(0) − 8.65 − 8.37 − 8.50 −8.95

11b NICS(1)

−10.87 −10.85 −10.92 −

10.67

a NICS(0) and NICS(1) for ring E in species 5 are −3.87 and −6.34 respectively; b NICS(0) and NICS(1) for ring

E in species 11 are −8.38 and −10.83 respectively and for ring F are −8.54 and −10.93.

The linear polyacene molecules have already been investigated by Schleyer et al.8

using the same methodology and basis sets. These authors observed that the terminal rings

have less benzenoid character as the size of the linear acene increases. For the angular

acenes, however, the central rings have a reduced benzenoid character except in the


thoroughly aromatic molecule 11. These trends are exactly preserved for the acene couplers

in the diradicals under investigation (Table 5.3).

Table 5.4 shows that the average NICS(1) for a coupler is always less than that for the

normal acene molecule. The difference between NICS(1) of a NN diradical and that of the

corresponding acene molecule is written as ΔNICS. The loss of aromaticity in the coupler

moiety is due to the participation of the conjugated π-electrons in the magnetic exchange

phenomenon. We notice that J is proportional to the fractional change of NICS(1) from the

parent acene, that is, ΔNICS : |NICS(1)| for acene. It is also generally proportional to the

Wiberg bond order BO, cos φ1 and cos φ2. As we discussed in Chapter 4 for linear aliphatic

couplers, the absolute magnitude of atomic spin density approximately varies as 1/(N+1)

where N is the number of conjugated atoms in the coupler, and J is approximately

proportional to 1/(N+1). Similarly, here, J will be further proportional to a factor of 1/(n+1)

where n is the number of benzenoid rings in the ployacene coupler. These proportionalities

can be coupled together to write the qualitative expression

1 2ΔNICS

NICS)

( )( )

( 1)(

BO cos cosJ A

n

φ φ= ×

+ (5.1)

where NICS in the denominator is the absolute magnitude of NICS(1) for the parent acene.

The proportionality constant A is found by considering the experimental value J = 20 cm−1 for

the m-phenylene coupler (with n=1).9 We get A = 426.5 cm−1. The J values calculated from

Eq. (5.1) are given in Table 5.4. It is seen that Eq. (5.1) produces a rough estimate of J, but

for the linear polyacenes, the estimate grows progressively worse from 3 to 5.

For the liner acenes, the average NICS(1) per benzenoid ring increases with the size of

the coupler (Table 5.4). It is also evident that the diradical character increases with the

coupler size.6a Introducing the effective value (1-χd)NICS in place of NICS in the

denominator of Eq. (5.1) for the diradicals with linear couplers, and using the scaling of the

calculated J values by the multiplicative factor 0.9625 (=20.0/20.78), we find the deviation

parameter χd as 0.0, 0.12, 0.17, 0.38 and 0.56 respectively for n equal to 1–5. The deviation

parameter reflects the trend of the increasing diradical character.

81


82

Table 5.4. The calculated NICS(1) values for the diradicals and the corresponding acene molecules, ΔNICS(1) and the J value calculated from Eq. (6) and estimated from Eq. (5.1).

Diradical Average NICS(1) ΔNICS(1) JGND J

Couplersa Acenesb Eq.

(2.5

)

Eq.

(5.1

)

1 −9.24 −10.60c 1.36 20.78 20.00

2 −9.56 −10.80c 1.15 15.39 13.10

3 −9.75 −11.00c 1.25 13.44 10.72

4 −9.89 −11.10c 1.21 13.81 8.22

5 −9.94 −11.20c 1.25 16.73 7.01

6 −9.28 −9.94c 0.66 8.76 5.42

7 −9.25 −10.62d 1.37 6.05 6.19

8 −9.32 −10.62d 1.30 5.07 5.95

9 −9.21 −9.59d 0.38 4.76 3.02

10 −8.75 −9.27d 0.52 3.84 4.23

11 −10.83 −12.17d 1.34 0.86 2.22

aIn NN diradical; bAcene molecule without any NN radical as substituent; cSchleyer et al. Ref. 8; dCalculated at GIAO-B3LYP/6-311+G(d,p) level, our work.


83

The deviation cannot be straight-forwardly applied to the bent couplers where the

central rings are less aromatic. Also, the variation of NICS(1) is not smooth like that in linear

couplers because of the zwitterionic contributions in bent acenes.6b The J values estimated

by Eq. (5.1) in this case are generally in better agreement with the calculated values.

5.3.3. SOMO-SOMO Energy Splitting

The energies of the SOMOs are calculated at UB3LYP/6-311+G(d,p) level. The

(ε1SOMO−ε2

SOMO) energies are very low except for 6. The difference (ε1SOMO−ε2

SOMO)

decreases with increase in length of the linear acenes (1-5). The degeneracy of the SOMOs

for 2, 10 and 11 arises accidentally. The molecular point group is C2 (see Table 5.5) that is

abelian. The rest of the diradicals undergo a distortion from this symmetry. That all the

diradicals have SOMO-SOMO energy difference less than 1.5 eV and all have ferromagnetic

ground states are in agreement with the empirical rule proposed by Hoffman.

5.3.4. Isotropic Hyperfine Coupling Constant (hfcc)

The polarized continuum model (PCM) has been successfully applied to the

investigation of isotropic hyperfine coupling constant (aN) of organic radicals in solution.

The solute-solvent interaction can change aN values by modifying the local spin density. In

this work, we have calculated aN values for all four equivalent N-atoms in each diradical.

Table 5.5. The SOMO-SOMO energy splitting at UB3LYP/6-311+G(d,p) level for the triplet state.

Diradical E(SOMO1) (a.u.)

E(a.u.)

Δ(a.u.)

(SOMO2) E(SOMO)

1 −0.196 −0.191 .2 −0.19836 −0.19831 0.0001 3 −0.199 8 −0.198 6 0.001 4 −0.19920 −0.19421 0.0050 5 −0.198 −0.185 .0131 6 −0 −0 0.7 −0.19963 −0.18928 0.0104 8 −0 −0 0.9 −0.19819 −0.18973 0.

10 −0.198 −0.197 .0006 11 −0 −0 0.

97 83 0 0051

5 0 5

95 .22288

85 0.16391 0590

.19970 .19223 0075 0085

37 .19359

78 0.19358 0000


The experime l values li ange o for diradicals with different

couplers.10 In this work, we have calculated the hfcc for the diradical in gas phase as well as

in three different solvents. The calculated average hfcc for the N-atoms are given in Table

5.6. The average values for gen at in good ent with the

experimental values for general nitronyl nitroxide diradicals. The calculated values indicate

that there is a preference for the spin density to localize on one of the N-atoms in each NN

moiety.

Solvent plays an important role in hf all the species, the hfcc values for N-atoms

increase with the increase in dielectric constant. For linear acenes (1-5), the average hfcc

value increases as the coupler size i aN

values with the calculated J remains missing.

enylene Couplers

igure 5.2). The role of aromaticity of the coupler unit is studied

explici

coupling constant (hfcc) for nitrogen atoms of the

se Benzene Acetonitrile Water

nta e in the r f 3-4.5 G

aN the nitro oms are agreem

cc. In

ncrease. However, a straightforward correlation of the

5.4. The m-Ph

In this section we will further check the validity of Eq. (5.1) and the substituent effects

on phenyl ring of the m-phenylene coupler, which is know as the best ferromagnetic spacer.

Here we have investigated the magnetic properties of 27 substituted m-phenylene based

Nitronyl Nitroxide diradicals. We have considered nine different constituents on the m-

phenylene ring, namely, −COOH, −F, −Cl, −NO2, −Br, −OH, −NH2, −Ph and −CH3, in order

of decreasing −I effect (electron withdrawing power). Each substituent occupies three unique

positions on the ring (F

tly. All molecules are new, with the exception of 15b which has been recently

synthesized.11

Table 5.6. Calculated average isotropic hyperfine

diradical in different environments.

Diradicals Gas pha

84

1 2.78 2.96 2.95 2.97 2 2.90 3.15 3.23 3.25 3 3.26 3.25 3.37 3.35 4 3.26 3.23 3.38 3.40


85

5 3.07 3.29 3.37 3.40 6 2.76 3.15 3.23 3.38

2 3.12 3.14

10 3.09 3.19 3.33 3.29

11

7 2.82 2.90 2.95 3.04 8 2.96 3.0

9 2.78 2.80 2.94 2.95

2.79 2.86

2.92

2.94

5.4.1. Calculations

The calculated JGND a for all the species are given in Table 5.7. The coupling

constant is found to follow the order a < b n every The maximum variation in J is

found for the species with substitution at the common ortho position for b ents (a

isomers). The calc lated J is in the range of 3.59−8.34 cm , except for 15a, 16a and 19a. In

the last three ca s, the g state single the m c interaction is

antiferromagnetic in nature. e ortho b) spec varies from 7.07 cm−1 to 14.87

cm−1 except for 17 nd 19b. , J is 20.78 cm−1 and for 19b, it is 1.95 cm-1. The meta

substitution (c) always yields a −1, except for 16c where

it is 27.11 cm−1.

The decre g trend effect reveals no cle ma r any substituent

osition, as electron delocalization is the major factor for determining the coupling constant.

substituents F, Cl and Br (species 13, 14 and 16), however, a pattern

delocalization, hydrogen bonding, steric effect, etc.

nd JY

< c i case.

oth NN fragm

u −1

se round is a t, and agneti

For th -para ( ies, J

b a For 17b

J value in the range of 20.11-24.05 cm

asin of −I ar-cut syste tics fo

p

For the single atom

clearly emerges. When the substituents are in ortho positions (a and b), the coupling constant

decreases with the decreasing −I effect, whereas for meta position (c), J clearly increases.

This can be understood from spin alternation. An increase in electron density at the ortho

positions leads to a destructive interference. This explanation would fail in the case of a

substituent with more than one atom, as the bond(s) of the substituent would be involved in


86

The angles of rotation (φ1 and φ2 as in Figure 5.3) of NN from the plane of the m-

phenylene coupler are given in Table 5.8 which clearly indicates that J decreases as φ

increases. This is because the conjugation between the NN and coupler fragment decreases

ouble bond character and hence a greater

conjug

with the increase in φ,12 and is in agreement with the work in previous Chapter. The angle φ1

and φ2 are constant in almost every case except 16c and 17c for the substitution at the

common meta position (c) and vary between 26°-33°. The larger values for 1c are due to the

steric effect of the bulkier –COOH group. The reason for the very small and unsymmetrical φ

values for 16c and 17c remains unclear. The larger dihedral angles for a and b isomers are

mainly due to stereo-electronic effects. We find that the calculated J is approximately

proportional to cosφ1 cosφ2.

Wiberg bond order (BO) for NN-Ph for all the cases is greater than 1.0 except for 15a

and 18a. The larger BO (>1.0) indicates a larger d

ation, thus a larger J. It is fair to estimate that J varies linerly with BO. The BO for

15a and 18a are discussed later. An estimate of J can be obtained from Eq. (5.1). This

equation is valid only if steric and other constraints are absent. We have recently shown that


87

N

NN

NO

O

OF

13a

N

NN

N

O

O

O

O

13b

N

NN

N

O

O

O

O

13c

O F

F

N

NN

NO

O

OCl

14a

N

NN

N

O

O

O

O

14b

N

NN

N

O

O

O

O

14c

OCl

Cl

N

NN

NO

O

OBr

16a

N

NN

N

O

O

O

O

16b

N

NN

N

O

O

O

O

16c

OBr

Br

N

NN

NO

O

OCO2H

12a

N

NN

N

O

O

O

ON

NN

N

O

O

O

O

12c

O CO2H

CO2H

N

NN

NO

O

ONO2

15a

N

NN

N

O

O

O

O

15b

N

NN

N

O

O

O

O

15c

ONO2

NO2

12b

N

NN

NO

O

OOH

17a

N

NN

N

O

O

O

O

17b

N

NN

N

O

O

O

O

17c

O OH

OH


N

NN

NO

O

ONH2

18a

N

NN

N

O

O

O

O

18b

N

NN

N

O

O

O

O

18c

O NH2

NH2

N

NN

NO

O

OPh

19a

N

NN

N

O

O

O

O

19b

N

NN

N

O

O

O

O

19c

OPh

Ph

N

NN

NO

O

OCH3

20a

N

NN

N

O

O

O

O

20b

N

NN

N

O

O

O

O

20c

OCH3

CH3

Figure 5.2. The diradicals under investigation.

O

O

N

ON

+ N

O

N+1

23

45

6

7

8

9

10

11

12

1314

15

16

φ1 φ2

Figure 5.3. Schematic representation of m-phenylene-coupled nitronyl nitroxide diradical.

this equation gives an estimate of J for polyacenes.12 For the present systems, we find A to

be about 426.5 cm−1. The calculated coupling constant (Jcal) using equation (5.1) matches

well with calculated values of J for the c group species, except for 18c and 19c where the

calculated ΔNICS(1) is too small and large respectively. For the ortho species, steric

constraint, hydrogen bnding, etc. leads to prominent deviations. The J values estimated from

Eq. (5.1) are given in Table 5.8 along with all the parameters involved in the equation. The

88


89

above equation produces the absolute magnitude of J and does not determine the nature of the

magnetic exchange interaction.

Table 5.7. Calculated exchange coupling constant (J) at UB3LYP/6-311G(d,p) level.

Diradicals Energy(a.u.) <S2>

JGND JY

BS T (cm−1) (cm−1)

12a −1487.6248083 1.0587

−1487.6248263 2.0595

3.94 3.94

12b −1487.6306294 1.069612

−1487.6306716 2.072114

9.24 9.26

12c −1487.6362267 1.0675

−1487.6363185 2.0727

20.11 20.01

13a −1398.2643806

1.0698 −1398.2644187

2.0719 8.34 8.32

13b −1398.2711268 1.071

−1398.2711833 2.0737

12.37 12.35

13c −1398.272202 1.0676

−1398.2723105 2.0740

23.76 23.61

14a −1758.6141617

1.0701 −1758.6141781

2.0710 3.59 3.59

14b −1758.6220774 1.0683

−1758.6221191 2.0705

9.13 9.11

14c −1758.6273742 1.0677

−1758.627484 2.0743

24.05 23.89

15a −1503.45099658

1.0331 −1503.45083192

2.0516 −36.14 −35.35

15b −1503.55431367 1.0723

−1503.55437103 2.0758

12.59 12.55

15c −1503.56186522 1.0678

−1503.56196921 2.0740

22.82 22.68

16a1 −1311.5478844

1.0705 −1311.5478825

2.0709 −0.40 −0.40

16b1 −1311.55683448 1.0710

−1311.55686668 2.0727

7.07 7.06

16c1 −1311.55832363 1.0646

−1311.55844713 2.0719

27.11 26.91

17a −1374.2588476

1.0578 −1374.2588652

2.0587 3.85 3.85

17b −1374.26158508 1.0727

−1374.26167978 2.0777

20.78 20.68

17c −1374.2448564 1.0559

−1374.2449509 2.0616

20.70 20.58

18a −1354.3834627

1.0634 −1354.3835035

2.0656 8.93 8.37


90

18b −1354.3929839 1.0624

−1354.3930518 2.0661

14.87 14.81

18c −1354.3814049 1.0663

−1354.3815049 2.0721

21.9 21.77

19a −1530.1020791

1.0639 −1530.1020719

2.0637 −1.58 −1.58

19b −1530.1105927 1.0664

−1530.1106016 2.0669

1.95 1.95

19c −1530.1158724 1.0668

−1530.1159792 2.0731

23.39 23.12

20a −1338.3213797

1.0658 −1338.3214042

2.0671 5.4 5.4

20b −1338.3316078 1.0649

−1338.3316635 2.0679

12.2 12.16

20c −1338.3336551 1.0663

−1338.3337603 2.0725

23.04 22.89

1Using lanl2dz basis set for Br. The 6-311+G(d,p) basis set for Br yields a JGND value of −0.53 cm−1 for 16a, 7.05 cm−1 for 16b and 27.07cm−1 for 16c. Table 5.8. Calculated NICS(1), bond order (B.O.), angle of twist (φ), and the J value estimated from

Eq. (5) (Jcal ) and calculated from the BS-DFT approach (JGND).

NICS(1) ΔNICS(1) BO φ1 φ 2 Jcal JGND

Diradicalsa Moleculesb (cm-1) (cm-1)

12a −9.71 0.44 1.01 47.48 63.50 5.63 3.94 12b −9.52 −10.15 0.63 1.03 28.82 55.11 13.66 9.24 12c −9.42 0.73 1.04 32.35 32.96 22.61 20.11

13a −9.54 0.85 1.02 46.78 55.81 13.70 8.34 13b −9.63 −10.39 0.76 1.04 23.23 54.91 17.14 12.37 13c −9.68 0.71 1.06 25.62 25.62 25.12 23.76

14a −9.40 0.65 1.01 60.80 61.12 6.56 3.59 14b −9.39 −10.05 0.66 1.04 21.69 61.55 12.89 9.13 14c −9.35 0.70 1.05 25.84 25.84 25.27 24.05

15a −9.08 1.22 0.65 51.25 38.37 16.11 -36.14 15b −9.80 −10.30 0.50 1.06 15.94 54.64 12.21 12.59 15c −9.46 0.84 1.05 26.85 28.54 28.62 22.82

16a −9.32 0.56 1.01 60.23 60.08 6.05 -0.40 16b −9.32 −9.88 0.56 1.03 15.19 69.89 8.26 7.07 16c −9.08 0.80 1.06 28.33 15.53 31.05 27.11


91

17a −8.76 1.06 1.03 68.94 38.55 13.33 3.85 17b −8.66 −9.82 1.16 1.05 25.93 40.65 36.09 20.78 17c −9.23 0.59 1.06 2.33 35.39 22.13 20.70

18a −7.97 0.96 1.04 49.50 47.34 20.99 8.93 18b −7.89 −8.93 1.04 1.06 22.60 44.43 34.71 14.87 18c −8.69 0.24 1.05 28.56 27.04 9.42 21.90

19a −9.44 0.19 0.99 82.88 79.97 0.18 −1.58 19b −8.90 −9.63 0.73 1.02 22.40 95.08 -2.70 1.95 19c −8.74 0.89 1.05 26.34 26.34 33.24 23.39

20a −9.42 0.65 1.01 55.79 56.62 8.60 5.40 20b −9.62 −10.07 0.45 1.04 22.57 54.08 10.74 12.20 20c −9.42 0.73 1.04 32.35 32.96 22.79 20.11

a This work; b Ref. 8. Wiberg bond index for phenyl−NN is small for 15a, 16a and 19a. Each B.O. reported

in Table 5.8 is the average of two phenyl-NN bond orders. The low values are indicative of

the occupation of σ* orbitals by the non-bonding electrons. This favors the Heitler-London

spin pairing, hence the antiferromagnetic coupling, especially in 15a. The formation of a

unique O−O bond in 15a is an interesting phenomenon (Figure 5.4) that is also responsible for

the large antiferromagnetic interaction. Hydrogen-bonding produces mostly structural effects

such as a reduction of the twist angle φ as in 17a and 17b (Figure 5.4), and also a small

amount of spin delocalization effect. The bulky phenyl group leads to an increase in the twist

angle in its neighborhood, (both φ1 and φ2 in 19a and only φ2 in 19b). Thus 19a becomes

antiferromagnetically coupled and 19b is only faintly ferromagnetic. The explanation of the

antiferromagnetic coupling in 16a is a conundrum. It varies from a combination of different

factors, and can be best understood from a comparison of the different factors involved in Eq.

(5.1).

5.5. Conclusions

The magnetic exchange coupling constants are calculated for eleven diradicals by

broken-symmetry density functional method. The coupling constant J is found to decrease for

the linear acene couplers from one to three benzenoid rings, but it increases from three to five


benzenoid rings. The α-HOMO and β-HOMO are not the only magnetically active orbitals

for 3-5. This happens due to the increase of the diradical character of the acene couplers.

The diradical character is lost in the bent couplers. The NICS value at the central rings of the

linear acene is high, while the terminal rings lose

Figure 5.4. The unique O-O bond formation observed for 15a. No such bond is formed in 15b.

Intramolecular H-bonding is found in 17a and 17b. (Similar hydrogen bonds are also

observed for 12a, 18a and 18b).

the benzenoid character. The J value increases with BO, and decreases with the increase in

the angle of twist of the NN mono-radicals from the coupler plane. The qualitatively

proposed equation (5.1) can give a fair estimate of J. Reliable aN values are obtained for the

diradicals in solution.

The magnetic exchange coupling in 24 out 27 studied m-phenylene diradicals are

predicted as ferromagnetic at UB3LYP/6-311+G(d,p) level. Species 15a, 16a and 19a are

92


93

antiferromagnetically coupled. The substitution at the meta position of m-phenylene NN

diradicals has little steric and hydrogen bonding effects. In the case of a larger twist of the

planes of the spin sources from the plane of the coupler, the nature of the interaction changes

from ferromagnetic to antiferromagnetic. In these cases, the B.O. becomes small (even less

than 1.0) due to a partial population of the σ* orbital. The NICS value is found to decrease

from the corresponding mono-substituted phenyl derivatives. The coupling constant can be

estimated from Eq. (5.1) provided that complications arising from sterio-electronic repulsion,

hydrogen bonding, ring formation, etc., are absent. This is found to be generally valid for NN

diradicals with m-phenylene spacer that has substitutions at the common meta position.

5.6. References (1) (a) Blundell, S. J.; Pratt, F. L. J. Phys.: Condens. Matter 2004, 16, R771. (b) Luneau, D.; Rey, P.

Coord. Chem. Rev. 2005, 249, 2591.

(2) Blundell, S.J.; Pratt, F.L. J. Phys.: Condens. Matter 2004, 16, R771.

(3) (a) Hegmann, F. A.; Tykwinski, R. R.; Lui, K. P. H.; Bullock, J. E.; Anthony, J. E. Phys. Rev. Lett. 2002,

89, 227403. (b) Meng, H.; Bendikov, M.; Mitchell, G.; Helgeson, R.; Wudl, F.; Bao, Z.; Siegrist, T.;

Kloc, C.; Chen, C.-H. Adv. Mater. 2003, 15, 1090.

(4) Raghu, C.; Patil, Y. A.; Ramasesha, S. Phys. Rev. B 2002, 65, 155204.

(5) Ali, Md. E.; Datta, S. N. J. Phys. Chem. A 2006, 110, 2776.

(6) (a) Bendikov, M.; Duong, H. M.; Starkey, K.; Houk, K. N.; Carter, E.A., and Wudl, F. J. Am. Chem. Soc.

2004, 126, 7416. (b) Constantinides, C.P.; Koutentis, P.A.; Schatz, J. J. Am. Chem. Soc., 2004, 126,

16232.

(7) Wiberg, K. Tetrahedron 1968, 24, 1083.

(8) Schleyer, P.v.R.; Maerker, C.; Dransfeld, A.; Jiao, H.; Hommes, N.J.R.v.E.

J. Am. Chem. Soc. 1996, 118, 6317 . (b) Chen, Z.; Wannere, C.S.; Corminboeuf, C.; Puchta, R.; and

Schleyer, P.v.R. Chem. Rev. 2005, 105, 3842. (c) Schleyer, P.v.R.; Manoharan, M.; Jiao, H.; Stahl, F.

Org. Lett. 2001, 3, 3643.

(9) Shiomi, D.; Tamura, M.; Sawa, H.; Kato, K.; Kinoshita, H. Syn. Metals 1993, 56, 3279.

(10) Hoffmann, R.; Zeiss, G. D.; Van Dine, G. W. J. Am. Chem. Soc. 1968, 90, 1485.

(11) (a) Luckhurst, G. R. In Spin Labeling. Theory and applications ; Berliner, J. L., Ed. ; Academic Press:

New York, 1976; p 133 ff. (b) Luckhurst, G. R. ; Pedulli, G. F. J Am. Chem. Soc. 1970, 92, 4738 ; (c)

Dulog, L. ; Kim, J. S. Makromol. Chemie 1989, 190, 2609. (d) Catala, L.; Le Moigne, J.; Kyritsakas, N.

Rey, P.; Novoa, J. J.; Turek, P. Chem. Eur. J 2001, 7, 2466.

(12) Ali, Md. E.; Datta, S. N. J. Phys. Chem. A 2006, 110, 13232.

Chapter 6

Photomagnetism

In this Chapter, we investigate another aspect of molecular magnetism. We predict the

photo-switching magnetic properties of four substituted dihydropyrenes from density

functional broken-symmetry calculations. The magnetic exchange coupling constants differ

up to 17 cm−1 between the photo switched isomers. The intramolecular exchange interactions

are ferromagnetic in nature. The calculated coupling constants are much larger than those

reported earlier for photomagnetic organic molecules. We also report the calculated values of

J for a few synthesized diradicals.

Chapter 6 Photomagnetism

6.1. Introduction

Photochromism is the reversible photon-induced transition of a chemical species

between two different forms having different absorption spectra. Photochromic materials

change their geometries and physical properties with irradiation. They are useful in potential

photoswitching. If a photoswitchable molecule is used as a spin coupler between two

magnetic units, the magnetism of the resulting species can change upon irradiation.1

Perfluorocyclopentene is one of the widely studied photochromic spin couplers.

Matsuda et al. have synthesized a large number of nitronyl nitroxide diradicals with

perfluorocyclopentene.1-4 In these diradicals, the intramolecular exchange interaction is very

weak, and the coupling constant J is of the order of the hyperfine coupling constant (hfcc).

The J value differs nearly 150-fold between the open- and closed-ring isomers. Its absolute

magnitude is generally found to be < 10−3 cm−1 for open ring isomers and ∼ 10−2 cm−1 for

closed ring isomers except for NN diradicals with 1,2-bis(2-methyl-1-benzothiophene-3-

yl)perfluorocyclopentene where J equals to −0.76 cm−1 and −4.03 cm−1 respectively.1b,1d,1g As

|J| is very small in the ground state, the photomagnetic properties of these molecules are not

expected to find a great usage. This has led to the investigation of photo-excited states of

diradicals. Teki et al. have investigated the magnetic properties of excited states of nitronyl

nitroxide diradicals with diphenylanthracene coupler.5 Huai et al. have also investigated

similar excited states by theoretical means.6

CPD DDP

hν1, Δ

hν2

Figure 6.1. Conversion of CPD to DDP.

The substituted pyrene molecule exists in two different forms, namely,

cyclophanediene (CPD) and dihydropyrene (DDP) as shown in Figure 6.1.7 The restricted life

time of CPD limits the utility of these molecules. The thermal return of CPD to DDP belongs

to the category of Woodward-Hoffman (W-H) orbital symmetry forbidden processes.

92


R2

R2R1

R1R2

R2R1

R1

a b

R1 R2

1a, 1b -NN -CH3 2a, 2b -NN -CF33a, 3b -IN -CH34a, 4b -IN -CF3

N

N+O

ON

NO

-NN -IN

hν1, Δ

hν2

Figure 6.2. Diradicals under investigation.

Nevertheless, the barrier created from the correlation of the occupied reactant orbitals

with the virtual product orbitals and vice versa is not too high for CPD→DDP conversion.

Recently, Williams et al. have found that proper substitutions can increase the activation

barrier to hinder the thermal conversion.8 In this work, we have investigated the ground-state

photomagnetic properties of nitronyl nitroxide diradicals and imino-nitroxide diradicals with

substituted pyrene couplers. The four sets of diradicals are illustrated in Figure 6.2. One

novelty of this work is in the choice of the coupler. To our knowledge, these molecules have

not been synthesized so far. The J values of the isomers differ by 4.7-9.6 times for each pair.

The magnitude of J for the closed ring isomers is significantly large, which constitutes the

second novelty. Besides, the points of attachment of the NN and IN groups are decided from

the rule of spin alternation9 such that the resulting diradicals are ferromagnetic in nature.

6.2. Technical Details and Results

The theoretical evaluation of the magnetic exchange coupling constant have been

performed using broken-symmetry (BS) density functional (DFT) methodology proposed by

Noodleman. The J value is evaluated using the so called GND [Eq.(2.5)] and Yamaguchi

[Eq.(2.7)] equations.

The molecular geometries of all the eight species (1-4 a and b) are optimized at

ROHF/6-31G(d,p) level using Gaussian 03 software. The optimized molecular geometry for

1a and 1b are shown in Figure 6.3. The magnetic exchange coupling constants, which are

93


calculated at UB3LYP/ 6-311+G(d,p) level, are given in Table 6.1 for all the species.

1a

1b

Figure 6.3. Optimized geometries for 1 in two different states.

We show only JGND and JY in Table 6.1. These are almost equal to each other in every

case. It is observed that NN radicals are much more strongly coupled to each other than the

IN radicals. This is due to the larger spin density on the carbon atoms of the O- N-C-N-O

fragments in NN diradicals. The spin density on the carbon atoms of N-C-N-O fragments in

IN-diradicals is much less.

The J value is greater for the –CF3 substituents than that for the −CH3 groups in the

closed form, and smaller in the open form. This is due to the bulkier group restricting the

angle of rotation (Φ) of the nitronyl nitroxide rings from the coupler plane. The average Φ

follows the orders 1 > 2 and 3 > 4. A smaller Φ gives a greater conjugation. The intra-ring

C−C distances are more or less same in the four closed species. Therefore, J exhibits the

reverse orders, 1 < 2 and 3 < 4. The calculated intra-ring C−C distance and dihedral angles

are given in Table 6.2.

94


95

Table 6.1. Calculated exchange coupling constants (J) and total energies at UB3LYP/6-311+G(d,p)

level. The J values are calculated using the GND and Yamaguchi equations.

Species EB (a.u.) B

<S2>

ET (a.u.)

<S2>

JGND

(cm−1)

JY

(cm−1)

Eq.(2.5) Eq.(2.7) 1a −1762.3338519

1.0670 −1762.3338619

2.0679 2.20 2.20

1b −1762.3569120 1.0750

−1762.3569596 2.0805

10.43 10.37

2a −2357.9438295

1.0700 −2357.9438367

2.0707 1.58 1.58

2b −2357.9639160 1.0784

−2357.9639662 2.0847

11.02 10.95

3a −1611.9511192

1.0727 −1611.9511208

2.0235 0.34 0.34

3b −1611.9723107 1.0242

−1611.9723196 2.0253

1.95 1.95

4a −2207.5623297

1.0243 −2207.5623308

2.0245 0.22 0.22

4b −2207.5831239 1.0250

−2207.5831336 2.0263

2.13 2.13

The opposite effect is found for the open form. The reason is that the substitution of a

bulkier group increases the intra-ring C−C distance by about 0.065 Å, and the bridging C−C

bond lengths also increase. This causes the phenyl rings that are no longer coplanar in CPD

to move further away from each other, thereby weakening the magnetic interaction.

The difference between the magnetic properties of a and b species are not due to the

angle Φ, as the average value of Φ always follows the order b > a. The stronger magnetism in

the b species is evidently an outcome of the shorter route for the transmission of magnetic

interaction and the planarity of the coupler.

The total energy difference between the a and b species in the triplet state are nearly

the same for 1-4. The ratio J b/J a is largest in case of 4, but species 2 is undoubtedly the best

photomagnetic molecule. The (Jb−Ja) for substituted dihydropyrenes is clearly much larger

than those for the diradicals based on perfluorocyclopentene.


96

Table 6.2. Dihedral angles of all the species.

C-C Φ1 Φ2 Φ3 Φ4 Aveg.

1a 2.75976 20.611 26.604 42.221 49.334 34.69 1b 1.54620 22.861 25.359 53.361 53.361 38.74 2a 2.82490 15.687 22.062 39.431 47.673 31.21 2b 1.55642 20.849 23.193 52.864 53.218 37.53 3a 2.75685 14.143 22.947 46.774 53.872 34.43 3b 1.54673 28.965 33.980 51.331 56.277 42.64 4a 2.82046 13.367 22.806 41.856 50.216 32.06 4b 1.55761 26.887 31.878 51.676 56.243 41.67

Table 6.3. Calculated magnetic exchange interactions at UB3LYP/6-311+G(d,p) level.

Species EB (a.u.) B

<S2> ET (a.u.)

<S2> JGND

(cm−1)

Eq.(2.4) 5a -3347.46082819

1.0714 -3347.4608282

2.0714 2.2 x10−3

5b -3347.43784677 1.0785

-3347.43780447 2.0745

-9.28

6a -4451.22202588

1.0832 -4451.22202588

2.0832 0.00

6b -4451.20213889 1.0844

-4451.20213591 2.0841

-0.65

7a -4566.63412348

1.0867 -4566.63410879

2.0854 -3.22

7b -4566.58452280 1.0867

-4566.58452235 2.0867

-0.09

8a -2281.81941755

1.0687 -2281.81941770

2.0687 0.033

8b -2281.73263186 1.0780

-2281.73271138 2.0904

17.45

9a -3292.34823490

1.0713 -3292.34823520

2.0713 0.07

9b -3292.31447756 1.0722

-3292.31437285 2.0597

-22.98


N

N

S

N

N S N

N

S

N

N S

N

N

S

N

N S

SSN

N

S

N

N S

SS

O

O

O

OUV

Vis

O

O

O

O

F2 F2F2F2

5a 5b

UV

Vis

F2F2

6a

O

O

O

O

F2F2

6b

O

O

O

O

F2F2

+_

+

_

_ +

_

+

+_

F2 F2

+_

_

+

+

_

S SS

N

N N

NS

S SS

N

N N

NS

NN N

NN

NN

N

NN N

N NN N

N

O

O O

O

F2 F2

F2

UV

Vis

O

O O

O

F2 F2

F2

7a 7b

O

O

O

O

UV

VisO

O

O

O

8a 8b

O

O

O

O

UV

Vis

9a

NCNC

CN

CF3

F3C

CN

_+

+_

+

_

+

_

+_ +

_

+_

+

_

_

+

O

O

O

O

9a

NCNC

CN

CF3

F3C

CN

+

_

_

+

Figure 6.4. A few recently synthesized (5-7) and a few predicted photomagnetic molecules (8-9).

A few recently synthesized photomagnetic molecules (5-7)2-4 and additional one (8-9)

are also investigated (Figure 6.4). These molecules are quite large and their molecular

geometries are optimized at ROHF/6-31G(d,p) level. The calculated J values at UB3LYP/6-

311+G(d,p) level and given in Table 6.3.

97


98

The diradicals 5 shows a very interesting phenomenon. The open ring structure

shows a very weak ferromagnetic interaction whereas the closed ring structure (5b) shows

quite high antiferromagnetic interaction.

The spacer in species 6 is quite large. This is why the unpaired electrons in 6a do not

interact at al. In close ring species 6b, the interaction is weakly antiferromagnetic in nature.

The switching behavior of 7 is different than the rest of species studied in this Chapter.

The open ring is the “ON” state here and the closed one is “OFF” state, whereas the reverse is

true for the rest of the cases. This is because the conjugation is affected due to the Diels-alder

reaction. The species 5-7 were synthesized and the magnetic properties were studies by EPR

only. The photomagnetic properties of 8 and 9 are predicted here. 6.3. Conclusions

In conclusion, we predict that species 1, 2, 8 and 9 would be good photomagnetic

molecules with J varying by a few cm−1 upon irradiation. Besides, these species are all

ferromagnetically coupled. The a forms (CPD) have very small singlet-triplet energy

differences and would be faintly magnetic. The b isomers (DDP), however, would retain a

fairly considerable magnetic character at a low temperature, and possibly also in an inert

matrix. 6.4. References

1. (a)Tanifuji, N.; Matsuda, K.; Irie, M. Polyhedron 2005, 24, 2484. (b) Matsuda, K.; Irie, M. Polyhedron

2005, 24, 2477. (c) Matsuda, K. Bull. Chem. Soc. Jap. 2005, 78, 383. (d) Tanifuji, N.; Matsuda, K.;

Irie, M. Org. Lett. 2005, 7, 3777. (e) Matsuda, K.; Irie, M. J. Photochem. Photobio. C: Photochem.

Rev. 2004, 5, 69.

T

2. Tanifuji, N.; Irie, M.; Matsuda, K. J. Am. Chem. Soc. 2005, 127, 13344.

3. Matsuda, K.; Matsuo, M.; Irie, M. J. Org. Chem. 2001, 66, 8799.

4. Matsuda, K.; Irie, M. Chem. Lett . 2000, 16.

5. Teki, Y.; Toichi, T.; Nakajima, S. Chem. Eur. J. 2006, 12, 2329.

6. Huai, P.; Shimoi, Y.; Abe ; S. Phys. Rev. B 2005, 72, 094413.

7. Mitchell, R. H.; Ward, T. R.; Chen, Y.; Wang, Y.; Weerawarna, S. A.; Dibble, P. W.; Marsella, M. J.;

Almutairi, A.; Wang, Z.-Q. J. Am. Chem. Soc. 2003, 125, 2974.

8. Williams, R. V.; Edwards, W. D.; Mitchell, R. H.; Robinson, S. G. J. Am. Chem. Soc. 2005, 127, 16207.

9. Trindle, C.; Datta, S. N. Int. J. Quantum Chem. 1996, 57, 781. (b)Trindle, C.; Datta, S. N.; Mallik, B. J.

Am. Chem. Soc. 1997, 119, 12947.

http://www.scopus.com/scopus/search/submit/author.url?author=Teki%2c+Y.&authorId=7004690730&origin=resultslist&src=s

http://www.scopus.com/scopus/search/submit/author.url?author=Toichi%2c+T.&authorId=12766742000&origin=resultslist&src=s

http://www.scopus.com/scopus/search/submit/author.url?author=Nakajima%2c+S.&authorId=7403109481&origin=resultslist&src=s

Chapter 7

Dinuclear Copper Complex

This Chapter describes the investigation of the magnetic properties of a recently

synthesized dinuclear complex, [Cu2(μ−OAc)4(MeNHpy)2]. We have explicitly calculated

the overlap integral Sab between the two magnetic orbitals, and found a value of 0.8589.

Deviating from the common practice of replacing Sab by 1, the computed value of the

integral has been used in calculating the magnetic exchange coupling constant (J). The

calculated J is −290 cm−1, in excellent agreement with the observed value of −285 cm−1.

Also, the calculated J value is a weakly varying function of the Cu-Cu distance.

Furthermore, we have shown that the onset of intramolecular hydrogen bonding reduces the

spin density on the bridging atoms and consequently the magnitude of J. This explains why

the complex under investigation has a J value smaller than that of [Cu2(μ−OAc)4(H2O)2]

(−299 cm−1). While establishing this trend, we predict that the complex [Cu2(μ−OAc)4(py)2]

would have a higher J value, about −300 cm−1.

Chapter 7 Dinuclear Copper Complexes

99

7.1. Introduction

The study of intramolecular magnetic coupling between two metal magnetic centers

within a molecule is a fascinating subject.1 Intramolecular and intermolecular magnetic

interactions play the major role in controlling the magnetic properties of molecular crystals.

The intramolecular magnetic coupling in dinuclear transition metal complexes is controlled by

the number of bridging ligands, the angle M(metal)−L(bridging-ligand)−M(Metal), the M−M

distance, and the nature of bridging ligands and other ligands. A large body of theoretical and

experimental work has been performed to explain the magnetic properties.2 A number of

compounds with the same basic structure Cu2(μ-OAc)4 have been synthesized and the

cooperative magnetic interactions in these have been investigated, thereby establishing that

the through-ligand superexchange leads to the rather strong intramolecular interaction in these

complexes.3−6 Recently, Barquín et al.7 have synthesized a similar compound by introducing

2-methyl imino pyridine in the axial position as shown in Figure 7.1. This has the important

characteristic of having two intramolecular hydrogen bonds as shown by the dotted lines,

while the corresponding analogs with H2O and NH3 molecules as axial ligands do not have

intramolecular hydrogen bonds.

Complexes with the Cu2(μ-OAc)4 basic structure have quite high intramolecular magnetic

exchange coupling constants.8 The coupling constant J is –299 cm–1 for [Cu2(μ-

OAc)4(H2O)2] and −285 cm−1 for [Cu2(μ-OAc)4(NH3)2]. The strong antiferromagnetic

interaction in these compounds was initially thought to be explained by the δ-overlap of the

dx2−y2 orbitals of the two copper atoms, which contain the unpaired electron.9 It is now

theoretically accepted that the through-space interaction has much less contribution to the

overall J in comparison to the through-bridge exchange contribution.10

Ruiz et al. have extensively carried out theoretical calculations by varying the substituents

of the bridging carbon atoms.10 These substitutents have been found to play a very important

role in controlling the value of J. As the electronegativity of the atom or the group attached to

the bridging carbon atoms increases, the J value generally decreases. In the case of very

strongly electronegative groups like −CF3 and −CCl3, however, the opposite effect is

observed. Ruiz et al.10a have also noticed that the changes in axial ligands introduce only a

little variation of J. The qualitative model of Hay-Thibeut-Hoffmann (HTH)11 cannot explain

the observed variation of J in acetate-bridged compounds. This model is mainly based on the


Figure 7.1. The dinuclear copper complex [Cu2(μ−OAc)4(MeNHpy)2].

100

(a) (b)

Figure 7.2. Structure of (a) the pyridine substituted complex, and (b) a conformer

with the MeNHpy ligands in the axial positions rotated through 45°. energy differences between the singly occupied molecular orbitals (SOMOs). The

participation of the axial ligands in the SOMOs is forbidden by symmetry. It is also generally

known that the coupling constant depends on the number of bridging ligands n and can be

expressed as J = JF + nJAF where JF and JAF are respectively the ferromagnetic and

antiferromagnetic contributions. In addition to these controlling factors, the spin density of

the bridging atoms must be ultimately responsible for the J value. Any physical phenomenon


101

that can affect the spin density on the bridging atoms can control the intramolecular coupling

constant.

Table 7.1. The computed overlap integral between the spatial parts of alpha and beta occupied orbitals in the BS state of the dinuclear copper complex.

MO Orbital Energy (a.u.) Sab

a

α β

HOMO −0.20278 −0.20273 0.9997 HOMO-1 −0.20294 −0.20299 0.9997 HOMO-2 −0.23769 −0.23753 −0.2876 HOMO-3 −0.24290 −0.24310 −0.4977 HOMO-4 −0.25369 −0.25370 0.7405 HOMO-5 −0.26021 −0.26016 0.7478 HOMO-6 −0.26385 −0.26384 0.9913 HOMO-7 −0.26912 −0.26914 −0.8126 HOMO-8 −0.27105 −0.27098 0.7020 HOMO-9 −0.27914 −0.27915 −0.9844 HOMO-10 −0.28208 −0.28206 0.9554

aOverlap integral between the magnetic orbital not between the metal atoms.

The main objective of this work is to carry out a theoretical investigation and evaluate the

magnetic exchange coupling constant by broken-symmetry approach while avoiding any

approximation to the overlap integral between the magnetically active orbitals. We have

calculated the intramolecular magnetic exchange coupling constant for the complex

synthesized by Barquín et al.7 The second aim is to reinvestigate the mechanism of the

magnetic exchange interaction of the same complex. This has been done in two ways. First,

we have shown that the contribution of the overlap between the orbitals of the two copper

atoms to the overlap of the two magnetic orbitals is truly small. Second, we have performed

computations on structures with varying Cu-Cu distance while the rest of the structure is kept

intact, and analyzed the spin distribution in each case. The third goal here is to demonstrate

that the J value undergoes reduction with the onset of intramolecular hydrogen bonding. This

has been investigated by comparing the spin density distribution in three systems, namely, the

original complex, a complex with pyridine substitution in the axial position (a possible new

compound) and a complex with the axial ligands in a different conformation. The latter

structures are shown in Figure 7.2.

Table 7.2. Single-point calculations in UB3LYP method using 6-311G(d,p) basis sets for atoms

other than the Cu atoms.


Cu basis set Energy (a.u.)

<S2> J a

(cm−1) BS T

EBS−ET (cm−1)

6-311G(d,p) −4881.0236208 0.987599

−4881.0225427 2.002953

−237 −272

Lanl2dz −4881.0934244 0.983606

−4881.0922747 2.003598

−252 −290

Cep-121G −4881.1188212 0.987053

−4881.1176725 2.003776

−252 −290

Using S = 0.859 in Eq. (2.2).a

ab

7.2. Computational Methodology

The molecular geometry of the complex is obtained from the crystal structure reported

by Barquín et al. in Ref 7. The geometry was not theoretically optimized, as we want to keep

up the effects of the surrounding molecules on the geometry of the selected molecule in the

real crystal. To obtain the broken-symmetry states, all the single-point UB3LYP calculations

have been performed with the accurate guess values of molecular orbitals, which are in turn

retrieved from the proper ROHF calculations. In all the calculations, 6-311G(d,p) basis sets

are used for the lighter atoms. For the copper atoms, we have used both LANL2DZ and CEP-

121G basis sets. We have ourselves written a program to compute the overlap integral S

between the two magnetic orbitals, and to evaluate the contribution of the overlap between the

orbitals of the two copper atoms to S . This program uses the basis set information and

molecular orbital coefficients from the log files of Gaussian 98 software.

ab

ab

7.3. Choice of Magnetic orbitals

The interaction between two magnetic centres A and B is similar to the existence of

very weak chemical bond between them. In the present case, both the singlet and triplet states

are very close to each other in energy, and both can be thermally populated. When the

interaction vanishes, the dynamics of the two active electrons (one from A and another from

B) become uncorrelated. The Heitler-London approach in this case results in two

semilocalized orbitals a and b which can be identified as the magnetic orbitals.

102


103

Orthogonalized magnetic orbitals and natural magnetic orbitals are oft-quoted types of

magnetic orbitals.12 The computational procedure adopted for the BS calculations here

necessitates the choice of the natural magnetic orbitals.

Table 7.3. Single-point calculations by UB3LYP method using 6-311G(d,p) basis sets for

atoms other than the Cu atoms. The Cu-Cu distance is varied.

Cu basis set

Cu-Cu Distance(Å)

Energy (a.u.) <S2>

J a

(cm−1) BS T

EBS−ET (cm−1)

2.62 −4881.061151

0.981953 −4881.0601861

2.003702 −212 −244

2.67 −4881.083968 0.983567

−4881.0828751 2.003693

−240 −276

Lanl2dz 2.72 −4881.0934244 0.983606

−4881.0922747 2.003598

−252 −290

2.77 −4881.09279690.982964

−4881.0916344 2.003604

−255 −294

2.82 −4881.0844034 0.982456

−4881.083226 2.00372

−258 −297

2.87 −4881.065495 0.983909

−4881.0643415 2.003576

−253 −291

2.62 −4881.0870304

0.987054 −4881.0859311

2.003776 −241 −278

2.67 −4881.1097856 0.987016

−4881.1086601 2.003789

−247 −284

Cep-121g 2.72 −4881.1188212 0.987053

−4881.1176725 2.003776

−252 −290

2.77 −4881.1198784 0.986935

−4881.1187121 2.003775

−256 −295

2.82 −4881.1114843 0.98607

−4881.110294 2.003824

−261 −301

2.87 −4881.0924631 0.987672

−4881.0913333 2.003771

−248 −285

a Using S = 0.859 in Eq. (2.2).ab

The overlap integral between the α-HOMO and β-HOMO in the BS state is nearly

unity (0.9993) (Table 7.1). The calculated expectation value of S2 for the BS states are

approximately equals to 0.98 (Table 7.2), indicating that these HOMOs cannot be considered

as the magnetic orbitals. It is well known that the magnetic properties of transition metal

complexes evolve from the nature of the orbitals and their electronic population in the

transition metal ions. Therefore, it is reasonable to expect that the magnetic orbitals are to be

determined by the copper atoms.


To obtain the natural magnetic orbital a for the present complex

[Cu(7)(μ−OAc)4Cu(13)(MeNHpy)2], single-point calculations have been performed by

treating Cu(13) as dummy to prevent any orbital interaction between Cu(13) and the rest of

the molecule. The resulting singly-occupied HOMO is chosen as orbital a. Similarly, the

magnetic orbital b has been obtained by treating atom Cu(7) as dummy. These calculations

have been carried out at the UB3LYP/6-311G(d,p) level. The computed overlap integral

between the selected magnetic orbitals is 0.8589(49).

2.600 2.925

-4881.12

-4881.06

Ener

gy (a

.u.)

Cu-Cu distance in Angstrom

Figure 7.3. The plot of the calculated total energy against the Cu-Cu distance. The total energy

was calculated by UB3LYP method with 6-311G(d,p) basis sets for atoms other than the Cu atoms. The dotted lines are for the triplet states and bold lines are for the broken-symmetry states. The upper set of curves is for the lanl2dz basis set used for the copper atoms and the lower one is for the CEP-121G basis set.


The intramolecular magnetic coupling constants are calculated using Eq. (2.2). The

computed value of the overlap integral Sab remains more or less unchanged through three

digits for the different basis sets and the varying Cu-Cu distance investigated here. Therefore,

104


105

Sab= 0.859 has been used throughout in these calculations to generate a three digit accuracy

for the

Table 7.4. The variation of average spin density square and the difference of average spin density

square between the Broken-Symmetry and Triple state for Cu(7) and Cu(13).

Cu-Cu distance (Å)

2.62 2.67 2.72 2.77 2.82 2.87

PP

2HS(Cu) 0.4533 0.4568 0.4585 0.4637 0.4643 0.4703

PP

2BS(Cu) 0.4497 0.4526 0.4536 0.4566 0.4577 0.4641

-ΔPP

2(Cu) −0.0037 −0.0043 −0.0049 −0.0071 −0.0066 −0.0063

J (cm-1) −278 −284 −290 −295 −301 −285

calculated J. The calculated J values are given in Table 7.2. The metal atoms are known to

control the magnetic properties of transition metal complexes, and it is necessary to treat these

atoms with effective core potentials. In fact, when we use the 6-311G(d,p) basis set for the

valence orbitals of copper atoms, we obtain a J value of –272 cm−1 from Eq. (2.2), whereas

both CEP-121G and LANL2DZ bases give a coupling constant of −290 cm−1. The latter J

values excellently match the observed value –285 cm−1 that was reported by Barquín et al.7

The better agreement for LANL2DZ basis set is largely fortuitous, as the LANL2DZ basis set

is not necessarily better than the 6-311G(d,p) basis. But the improved value obtained from

the use of CEP-121G basis is notable.

The overlap of the d-orbitals of the two copper atoms (mainly dx2−y2) in the magnetic

orbitals is indicative of δ-bonding between the two Cu atoms, that is, the strength of the

through-space interaction. The computed value of this overlap integral is around 0.000011 in

absolute magnitude. This is not surprising as the two copper atoms are considerably away

from each other, but it confirms that the direct exchange would be negligibly small. In each

complex with one copper atom as dummy, the gross d-orbital population is 8.75 and the total

electronic population is 27.32 for the remaining copper atom. To test the absence of direct

exchange further, the distance between the two copper atoms is varied from 2.62 Å to 2.87 Å

with an interval of 0.05 Å. The rest of the structure is kept intact at the crystallographic

geometry. The observed Cu-Cu distance in the crystallized complex is 2.72 Å. Table 7.3

shows a very systematic variation of the single-point total energy with the Cu-Cu distance.


Figure 7.3 illustrates this trend. The CEP-121G basis set consistently yields a lower energy

for each state. Both the bases yield a more stable BS configuration. In the case of the

Table 7.5. Single-point calculations by UB3LYP method using 6-311G(d,p) basis sets for the

atoms other than Cu atoms by replacing the axial ligands with pyridine and by rotating the axial ligands

through 45º.

Cu basis

set Condition Energy (a.u.)

<S2> J a

(cm−1) BS T

EBS−ET (cm−1)

Original −4881.0934244

0.983606 −4881.0922747

2.003598 −252 −290

Lanl2dz Pyridine subst.

−4691.7378766 0.982556

−4691.7366646 2.003666

−266 −306

45º rotation −4881.0903817 0.985857

−4881.08917932.003708

−264 −303

Original −4881.1188212

0.987053 −4881.1176725

2.003776 −252 −290

Cep-121G Pyridine subst.

−4691.7654575 0.986742

−4691.764268 2.003831

−261 −300

45º rotation −4881.1147885 0.986682

−4881.1136113 2.00381

−258 −297

a Using S = 0.859 in Eq. (2.2).ab

LANL2DZ basis set, the minimum of the energy curve is around 2.74 Å. For CEP-121G basis

set it is at about 2.75 Å. These values are in good agreement with the observed Cu-Cu

distance of 2.72 Å in crystal. We notice from Table 7.3 that the calculated J value is only

weakly dependent on the Cu-Cu distance. The absolute magnitude of the calculated coupling

constant increases as the distance increases up to about 2.82 Å and then it decreases (Figure

7.4). This is only possible if there is no δ-bonding. The through-space interaction contributes

very little to the J value, and the through-bridge exchange is the dominant contribution. The

slight variation of the J value arises from the change in the overlap and bonding with the

atoms of the bridging ligands.

The reason for the absence of direct exchange can be understood from Figure 7.5. The

orbitals of the copper atoms as shown in Figure 7.5(b) form the singly occupied 2 2x yd −

106


HOMO’s whereas the orbitals in Figure 7.5(e) are fully occupied. As the 2zd 2 2x yd − point

towards the bridging ligands while the orbitals are along the Cu-Cu axis, the through-

bridge interaction determines the magnetic properties. The overlap between the

2zd

2zd orbitals

is negligibly small.

Table 7.6. The calculated spin densities on different atoms from the UB3LYP/6-311G(d,p) + Cep-

121G results for the broken-symmetry sate.

Cu(7) O(8) C(9) O(31) Cu(13)

Original −0.6766 −0.0795 0.0004 0.0797 0.6705 Pyridine −0.6691 −0.0778 −0.0001 0.0787 0.6685

45° rotated −0.6736 −0.0765 0.0003 0.0766 0.6687 Cu(7) O(30) C(20) O(28) Cu(13)

Original −0.6766 −0.0785 0.0002 0.0805 0.6705 Pyridine −0.6691 −0.0785 0.0001 0.0779 0.6685

45° rotated −0.6736 −0.0758 0.0002 0.0775 0.6687 Cu(7) O(34)a C(26) O(25) Cu(13)

Original −0.6766 −0.0650 0.0660 0.0804 0.6705 Pyridine −0.6691 −0.0721 -0.0011 0.0802 0.6685

45° rotated −0.6736 −0.0707 -0.0010 0.0829 0.6687 Cu(7) O(24) C(23) O(22)a Cu(13)

Original −0.6765 −0.0791 0.0018 0.0660 0.6705

Pyridine −0.6691 −0.0803 0.0011 0.0719 0.6685 45° rotated −0.6736 −0.0823 0.0014 0.0718 0.6687

a Hydrogen bonded oxygen atoms in original compound.

The average squared spin populations of the Cu atoms are correlated with the

computed J values in Table 7.4. The spin densities are obtained from the calculation at

UB3LYP/6-311G(d,p)/CEP-121G level. The variation of −ΔPP

2(Cu) [defined as

ΔP2P (Cu)=P2(Cu)HS−PP

2(Cu) ] with the Cu-Cu distance (Table 7.4) is in general agreement

with the nature of the plot of the calculated J versus the Cu-Cu distance (Figure 7.4). But for

the O atoms, it is observed that only the triplet spin population matches with the nature of

variation of J.

BS

The intramolecular hydrogen bonding has a minor influence on the magnetic exchange

coupling constant. Table 7.5 shows that the absolute magnitude of J increases by about 10

cm–1 for the pyridine substituted species and for the 45º-rotated conformation. To find a

107


reason for this behavior, we have investigated the spin density on the bridging oxygen atoms.

The spin density distribution is shown in Table 7.6. The spin density changes progressively

along each Cu-O-C-O-Cu chain, manifesting an antiferromagnetic trend. The spin densities

on the hydrogen-bonded O(22) and O(34) atoms in the original complex are lower than those

on the non-hydrogen-bonded oxygen atoms, and also much reduced compared to the spin

densities in the pyridine substituted complex and the 45º–rotated conformer. The spin

densities on all other oxygen atoms remain almost unchanged in the latter two species. Thus

the intramolecular hydrogen bonding reduces the spin density on the hydrogen bonded

oxygen atoms. This reduction leads to a diminished extent of the through-bridge magnetic

interaction, thereby lowering the absolute magnitude of the J value.

-320

-220

2.6 2.7 2.8 2.9

Cu-Cu distance in Angstrom

J (c

m-1

)

Figure 7.4. Variation of the calculated J values with the Cu-Cu distance for the lanl2dz (dotted

line) and CEP-121G (bold line) copper basis sets.

The complex [Cu2(μ−OAc)4(py)2] is seen to have a J value of about −300 cm−1 (Table 7.5)

that is as much as that for the water-substituted complex.

In molecular magnetism, the influence of hydrogen bonding on the spin-spin interaction

and spin migration is a very common phenomenon.13 Recently, Desplanches et al.14 have

reported a computational study on dinuclear Cu(II) complexes with two monomeric units

linked by O−H···O to form a dimer. These authors noted that the hydrogen-bonded H atom

does not have a major contribution to the SOMOs but it takes part in spin density

108


delocalization between the two Cu atoms. This leads to an intramolecular magnetic

109

(a).

(b)

(d)

(c)

(e)

Figure 7.5. View of molecular orbitals: (a) LUMO, (b) HOMO (c) HOMO-1 (d) HOMO-2, and (e) HOMO-5. The electronic population of HOMO and HOMO-1 is 1.0. The MOs are obtained from the ROHF/6-311G(d,p) /Lanl2dz calculation for the triplet.


coupling. In the present case, we notice the opposite phenomenon, that is, the spin density

delocalization in the superexchange pathway is reduced by the formation of a hydrogen bond.

The difference is that the monomeric units are directly linked here and hydrogen bonding

reduces the spin distribution in this linkage.

7.5. Conclusions

The magnetic exchange coupling constant of the recently synthesized dinuclear copper

complex [Cu2(μ-OAc)4(MeNHpy)2] has been calculated by using broken-symmetry density

functional methodology, and in doing so we have explicitly computed the overlap integral

between the two magnetic orbitals. The α-HOMO and β-HOMO in BS state are not magnetic

orbitals. The magnetic orbitals are obtained by using the concept of natural magnetic orbitals.

The calculated magnetic exchange coupling constant −290 cm−1 is in good agreement with the

observed value of −285 cm−1.

The direct exchange between the two copper atoms is negligibly small, and the

superexchange interaction is predominant. This conclusion is made after determining the

contribution of the overlap between the orbitals of the two copper atoms to Sab, and also by

studying the spin density distribution while the Cu-Cu distance is varied. There is a lack of δ-

bonding between the orbitals that carry the unpaired electrons. Intramolecular

hydrogen bonding reduces the spin density of the oxygen atoms, and leads to a lower absolute

magnitude of J as compared to the complex that contains H

2 2x yd −

2O instead of MeNHpy. The

magnitude of the J value for [Cu2(μ−OAc)4(py)2] will be as high as that for the complex with

water as axial ligands.

7.6. Reference 1 (a) Kahn, O. Molecular Magnetism; VCH: New York, 1993. (b) Goodenough J. B. Magnetism and the

Chemical Bond; Interscience: New York, 1963. (c) Coronado, E.; Delhaè, P.; Gatteschi, D.; Miller, J. S.

Molecular Magnetism: From Molecular Assemblies to the Devices, Eds.; Nato ASI Series E, Applied

Sciences, Kluwer Academic Publisher: Dordrecht, Netherland, 1996; Vol. 321 (d) Benelli, C.; Gatteschi,

D. Chem. Rev. 2002, 102, 2369. (e) Millar, J. S.; Drillon, M. Magnetism: Molecules to Materials

Nanosized Magnetic Materials, Wiley-VCH, Weinheim, 2002.

2 (a) Master, P. De; Fletcher, S. R.; Skapski, A.C. J. Chem. Soc. Dalton Trans. 1973, 2575. (b) Catterick. J.;

Thornton, P. Adv. Inorg. Chem. Radiochem. 1977, 20, 291. (c) Doednes, R. J. Prog. Inorg. Chem. 1976,

110


111

21, 209. (d) Rao, V. M.; Sathyanarayana, D. N.; Manohar, H. J. Chem. Soc. Dalton Trans. 1983, 2167. (e)

Nakagwa, M.; Inomata, Y.; Howell, F.S. Inorg. Chim. Acta 1999, 295, 121. (d) Ruiz, E.; Rodríguez-

Fortea, A.; Cano, J.; Alvarez, S.; Alemany, P. J. Comput. Chem. 2003, 24, 982. (e) Ruiz, E.; Rodríguez-

Fortea, A.; Alvarez, S. Inorg. Chem. 2003, 42, 4881.

3 Guha, B. Proc. R. Soc. 1952, 206, 353.

4 Bleaney, B.; Bowers, K. D. Proc. R. Soc. London, A, 1952, 214, 451.

5 Niekerk, J. N. van; Schoening, F. R. L. Acta. Crystallogr. 1953, 6, 227.

6 (a) Jotham, R. W.; Kettle, S. F. A.; Marks, J. A.; J. Chem. Soc. Dalton Trans. 1972, 428. (b) Melníc, M.

Coord. Chem. Rev. 1981, 36, 1. (c) Muto, Y.; Nakashima, M.; Tokii, T.; Kato, M.; Suzuki, I. Bull. Chem.

Soc. Jpn. 1987, 60, 2849. (d) Steward, O. M.; McAfee, R. C.; Chang, S.-C.; Piskor, S. R.; Schreiber, W. J.;

Jury, C. F.; Taylor, C. E.; Pletcher, J. F.; Chen. C.-S. Inorg. Chem. 1986, 25, 771.

7 Barquín, M; Garmendia, M. J. G.; Pacheo, S.; Pinilla, E.; Quintela, S.; Seco, J. M., Torres, M. R. Inorg.

Chim. Acta 2004, 357, 3230

8 (a) Sesco, J. M.; González Garmendia, M. J.; Pinilla, E.; Torres, M. R. Polyhedron 2002, 21, 457. (b)

Jotham, R. W.; Kettle, Sidney F. A. Chem. Comm. 1969, 6, 258. (c) Sesto, R. E. D.; Deakin, L; Miller, J.

S. Synthetic Metals 2001, 122, 543. (d) Figgis, B. N.; Martin, R. L. J. Chem. Soc. 1956, 3837. (e)

Yablokov, Yu. V.; Mosina, L. V.; Simonov, Yu. A.; Milkova, L. N.; Ablov, A. V.; Ivanov, V. I. Zh.

Strukt. Chim. 1978, 19, 42.

9 (a) Datta, R. L.; Syamal, A. Elements of Magnetochemistry; Affiliated Ease-West Press Pvt. Ltd.: New

Delhi, 1993. (b) MaxDougall, J. J.; Nathan, L. C.; Nelson, J. H. Inorg. Chim. Acta 1976, 17, 243.

10 (a) Rodríguez-Fortea, A.; Alemany, P.; Alvarez, S.; Ruiz, E. Chem. Eur. J. 2001, 7, 627 (b) Rodríguez-

Fortea, A.; Alemany, P.; Alvarez, S.; Ruiz, E. Inorg. Chem. 2002, 41, 3769. (c) Ruiz, E.; Llunell, M.;

Alemany, P. Sold. Stat. Phys. 2003, 176, 400.

11 Hay, P. J.; Thibeault, C. J.; Hoffmann, R. J. Am. Chem. Soc. 1975, 97, 4884.

12 R. D. Willett, D. Gatteschi and O. Kahn, Magneto-Structural Correlations in Exchange Coupled

Systems, NATO ASI Series, Reidel, Dordrecht, 1985.

13 (a) Romero, F. M.; Ziessel, R.; Bonnet, M.; Pontillon, Y.; Ressouche, E.; Schweizer, J.; Delley, B.; Grand,

A.; Paulsen, C. J. Am. Chem. Soc. 2000, 122, 1298. (b) Hicks, R. G; Lemaire, M. T.; Öhrström, L.;

Richardson, J. F.; Thompson, L. K.; Xu, Z. J. Am. Chem. Soc. 2001, 123, 7154.

14 Desplanches, C.; Ruiz, E.; Rodríguez-Fortea, A.; Alvarez, S., J. Am. Chem. Soc. 2002, 124, 5197.

Conclusions

112

Conclusions

The magnetic exchange coupling constant J, that is related to the difference between

the energy of singlet (ES) and triplet (ET) states, is theoretically examined. A theoretical

formalism on the N-electron spin Hamiltonian is given. From this treatment, a theoretical

expression for J is easily derived, which can be reduced to the popular spin projection formula

due to Giensberg, Noodleman and Davidsion. A few initial calculations are done by

employing post-Hartree-Fock methods. The rest of the calculations are based on the density

functional, broken symmetry methodology. The broken-symmetry calculations are easy to

perfom even on larger molecules using moderate computational facility.

The coupling constant J is very small, of the order of 10-5 a.u., whereas the energy

calculated for each spin state can differ from the accurate energy by a large amount. The

calculation of J relies on the fact that while both ES and ET (or ES and ET) can be in errors,

their difference is correctly reproduced by dependable methodology and a large basis set.

The effect of molecular vibration on J is not studied here. This is because the

vibrational frequencies are more or less the same in the two spin states in most cases. The

difference in vibrational energy would be large when the singlet and triplet geometries highly

differ from each other. The BS calculations are performed using the triplet geometries.

Calculations on a total 79 molecules are presented in this thesis. Each species is

investigated in two different spin state by a number of methodologies and using different

basis sets. A plethora of optimized bond angles, bond lengths and MO coefficients is

available from the respective journal sites where the parts of the work reported herein have

been published.

Fused ring diradicals are investigated in HF and DFT methodologies. The unrestricted

calculations produced very high spin contamination. The rule of spin alternation can predict

the ground spin state. The tautomeric conversion of H atom between singlet and triplet

optimized geometries is observed in a few diradicals.

Series of bis-nitrotronyl nitroxide diradicals with different conjugated couplers have

been investigated. The computed magnetic exchange coupling constants are in very good

agreement with the reported values. The α-HOMO and β-HOMO in the BS state are

Conclusions

113

generally found to be magnetic orbitals. In conjugated systems, the magnetic interaction

is mainly transmitted through the π-electron conjugation. The strength of antiferromagnetic

interaction decreases with the increase in the length of conjugated couplers. Conjugated

linear couplers are more efficient antiferromagnetic couplers than the aromatic ones of similar

length. The diradicals with m-couplers are undoubtedly ferromagnetic. The shape of the

SOMOs as well as the rule of spin alternation in the UHF emerge as two robust guidelines for

the prediction of the qualitative nature of the intramolecular magnetic interaction in bis-

nitronyl nitroxide diradicals. The calculated J also depends on the basis sets.

The coupling constant J is found to decrease for the linear acene couplers from one to

three benzenoid rings, but it increases from three to five benzenoid rings. The α-HOMO and

β-HOMO are not the only magnetically active orbitals for the molecules with 3-5 rings. This

happens due to the increase of the diradical character of the acene couplers. The diradical

character is lost in the bent couplers. The NICS value at the central rings of the linear acene

is high, while the terminal rings lose some of the benzenoid character. The J value increases

with the bond order, and decreases with the increase in the angle of twist of the NN mono-

radicals from the coupler plane. The qualitatively proposed equation (5.1) can give a fair

estimate of J for these molecules. Reliable aN values are obtained for the diradicals in

solution.

We predict a few species that would be good photomagnetic switch molecules with J

varying by a few cm−1 upon irradiation. The predicted species are all ferromagnetically

coupled.

The magnetic exchange coupling constant of the recently synthesized dinuclear copper

complex [Cu2(μ-OAc)4(MeNHpy)2] has been calculated by using broken-symmetry density

functional methodology, and in doing so we have explicitly computed the overlap integral

between the two magnetic orbitals. The complex has several pairs of magnetic orbitals

although HOMO and HOMO-1 pairs are not magnetic orbitals. Overlap integrals are

computed by using the concept of natural magnetic orbitals. The calculated magnetic coupling

constant −290 cm−1 is in good agreement with the observed value of −285 cm−1. The direct

exchange between the two copper atoms is negligibly small, and the superexchange

interaction is predominant. This conclusion is made after determining the contribution of the

overlap between the orbitals of the two copper atoms to Sab, and also by studying the spin

Conclusions

density distribution while the Cu-Cu distance is varied. There is a lack of δ-bonding between

the orbitals that carry the unpaired electrons. The Bencini-Ruiz formula is applicable

to such systems due to the presence of highly degenerate HOMOs. Intramolecular hydrogen

bonding reduces the spin density of the oxygen atoms, and leads to a lower absolute

magnitude of J as compared to the complex that contains H

2 2x yd −

2O instead of MeNHpy. The

magnitude of the J value for [Cu2(μ−OAc)4(py)2] is predicted to be as high as that for the

complex with water as axial ligands, nearly −300 cm−1.

114

Summary

i

Summary of the Thesis

The main aim of the thesis is to study the magnetic properties of diradicals, mostly of

organic origin. Molecule-based magnetic materials have several advantages over traditional

magnets. These are transparent, insulating, photoactive, thermally controllable and bio-active.

Molecular magnetism arises from the spin of the unpaired electron. Chapter 1 gives a general

introduction to magnetic molecules. The origin of magnetic phenomena in molecules and the

empirical ways of determination of the nature of magnetic exchange interaction are discussed.

The roles played by molecular magnetism in modern science and technology, such as

photomagnetic effect, single molecule magnets (SMM), spintronics etc., are also reviewed. The

objectives are clearly stated and a chapter-wise arrangement of the thesis is mentioned.

The theoretical background for the determination of magnetic exchange coupling constant

by broken symmetry (BS) calculations1 is investigated in Chapter 2. In this chapter we formulate

an N-electron spin Hamiltonian for diradicals having non-degenerate highest occupied molecular

orbitals. At first, energy expressions are obtained for singlet, broken-symmetry and triplet single-

determinant wave functions of unrestricted Hartree-Fock treatment. Total energy values for the

two-determinant singlet and triplet configurations that can be obtained from a self-consistent-

field treatment are determined next by using the orbital perturbation theory. This leads to an

energy ordering, which is expected to be valid also in an unrestricted Hartree-Fock Kohn-Sham

treatment. The base line of the spin Hamiltonian is determined from this ordering, and the spin

Hamiltonian is formulated. The spin Hamiltonian reduces to the Heisenberg effective spin

Hamiltonian operator in the two-center two-electron case. Using the spin Hamiltonian, we obtain

expressions for the average value of energy for the broken-symmetry and triplet determinants. It

is shown that the Yamaguchi expression2 for the magnetic exchange coupling constant J is

approximately valid. A more correct expression for J is based on spin projection. The latter

reduces to the Ginsberg-Noodleman-Davidson formula3 when the spin projection difference

equals ½. It normally happens in a density functional calculation using a large basis set. In the

latter case, a good estimate of the energy of the two-determinant singlet is also obtained. These

Summary

ii

observations are briefly discussed by using examples of two nitronyl nitroxide diradicals, one

with a large, and the other with a very small, value of J.

In Chapter 3, we discuss our investigation of the ground state spins of seven diradicals

belonging to the fused ring system by traditional ab-initio methodologies. The systems under

study are (1) 4-oxy-2-naphthalenyl methyl, (2) 1,8-naphthalenediylbis(methyl), (3) 8-imino-1-

naphthalenyl methyl, (4) 1,8- naphthalenediylbis(amidogen), (5) 8-methyl-1-naphthyl carbene,

(6) 8-methyl-1-naphthalenyl imidogen and (7) 8-methyl-1-naphthyl diazomethane (Figure 3.1).

Out of the seven molecules, only 1 was theoretically investigated earlier. To our knowledge, for

2-7, this work represents the first ab-initio investigation. A variety of basis sets has been

employed in these calculations. For each spin state, the molecular geometry has been fully

optimized at the unrestricted Hartree-Fock (UHF) level using the STO-3G, 4-31G, 6-311G(d) and

6-311G(d,p) basis sets. The UHF optimized geometries have been used for Møller-Plesset (MP)

and coupled cluster (CC) calculations as well as the density functional (UB3LYP) treatment.

Results in the unrestricted formalism have been given only at UHF and UB3LYP levels for the 6-

311G(d) basis. The UHF calculations yield an unrealistically large Singlet−Triplet (S−T)

splitting. Splittings calculated with different bases disagree seriously. The S−T gap is smaller in

the split-valence bases. The basis set truncation error can be considerably overcome by

considering electron correlation. Møller-Plesset perturbation theory and UCCSD(T) does not

produce realistic S−T energy gap unless the basis set used is very large which imposes the limit

of computing ability.

For these diradicals, any meaningful result would require larger bases with polarization

functions. Apart form this difficulty, the optimized molecular geometries turned out to be highly

spin-contaminated. The spin-contamination can be significantly reduced by the density

functional UB3LYP treatment. Nevertheless, for most of the diradicals, the UB3LYP method did

not yield a systematic trend. To avoid spin contamination completely, we have repeated

computations in the restricted (open-shell) Hartree-Fock framework. Geometry optimizations

were carried out using STO-3G, 6-311G(d), and 6-311G(d,p) bases at R(O)HF level and 6-

311G(d,p) basis at R(O)B3LYP level for each spin state. The R(O)B3LYP/6-311G(d,p)

optimized geometry yields the best total energy for each spin state and hence the most reliable

Summary

iii

S−T energy difference. Molecules 1-6 are found as ground state triplets. The calculated results

are in agreement with the available experimental findings. Molecules 3 and 7 have widely

different geometries in the singlet and triplet states. The calculations using 6-311G(d) and 6-

311G(d,p) basis sets show that in molecule 3, the substituents of naphthalene are −NH2 and −CH

in singlet but −NH and −CH2 in triplet. The two optimized geometries are tautomeric forms.

Molecule 7 is expected to be either a ground state triplet with a very little S−T gap or a ground

state singlet. This prediction is borne out by the computed results. The R(O)B3LYP/6-

311G(d,p) calculation yields a S−T splitting of −21.9 kcal mol-1. The singlet state becomes

stabilized by forming an additional condensed ring. The UHF spin density plots obtained from

the 4-31G optimized geometries manifest the phenomenon of spin alternation in the ground state.

A series of Nitronyl Nitroxide (NN) diradicals with linear conjugated couplers and

another series with aromatic couplers have been investigated by broken-symmetry (BS) DFT

approach. These are shown in Figure 4.1. The results are discussed in Chapter 4. First we show

that an ethylenic coupler provides a very strong intramolecular magnetic interaction. A recently

synthesized nitronyl nitroxide derivative, D-NIT2, is investigated by ab initio quantum chemical

methods. The broken symmetry approach yields a coupling constant −541 K that is in good

agreement with the observed value in solid state. The overlap integral between the magnetically

active orbitals in the BS state has been explicitly computed and used for the evaluation of the

magnetic exchange coupling constant (J). The calculated J values are in good agreement with the

observed values in literature. The magnitude of J depends on the length of the coupler as well as

the conformation of the radical units. The aromaticity of the spacer decreases the strength of the

exchange coupling constant. The SOMO-SOMO energy splitting analysis where SOMO stands

for the singly-occupied molecular orbital, and the calculation of electron paramagnetic resonance

(EPR) parameters have also been carried out. The computed hyperfine coupling constants

support the intramolecular magnetic interactions. The nature of magnetic exchange coupling

constant can also be predicted from the shape of the SOMOs as well as the spin alternation rule in

the unrestricted Hartree-Fock (UHF) treatment. It is found that π-conjugation along with spin-

polarization play the major role in controlling the magnitude and sign of the coupling constant.

Moreover, the magnetic properties of the diradicals (2,2'-(1,2-ethynediyldi-4,1-

Summary

iv

phenylene)bis[4,4,5,5-tetramethyl-4,5-dihydro-1 H-imidozolyl-oxyl] (IN-2p-IN), 2,2'-(1,2-

ethynediyldi-4,1 3,1-phenylene)bis[4,4,5,5-tetramethyl-4,5-dihydro-1 H-imidozolyl-oxyl] (IN-

pm-IN) and 2,2'-(1,2-ethynediyldi-4,1 3,1-phenylene)bis[4,4,5,5-tetramethyl-4,5-dihydro-1 H-

imidazole-1-oxyl-3-oxide] (NN-pm-NN) are also investigated. The rule of spin alternation in the

UHF clearly shows that the radical sites are antiferromagnetically coupled in IN-2p-IN and

ferromagnetically coupled in NN-2p-NN and NN-pm-NN, in agreement with a previously

carried out experiment.

The molecular geometries are optimized at Hartree-Fock levels. This is followed by

single-point calculations using the density functional (UB3LYP) treatment and the

multiconfigurational (CASSCF) methodology. Magnetic exchange coupling constants are

determined from the broken symmetry approach. The calculated J values, −3.60 cm−1 for IN-2p-

IN, 0.16 cm−1 for NN-2p-NN and 0.67 cm−1 for NN-pm-NN, are in excellent agreement with the

observed values.

Chapter 5 describes our prediction of the intramolecular magnetic exchange coupling

constant (J) for eleven nitronyl nitroxide diradicals (NN) with different linear and angular

polyacene couplers. These are shown in Figure 5.1. For the linear acene couplers, J initially

decreases with increase in the number of fused rings. But from anthracene coupler onwards, the

J value increases with the number of benzenoid rings due to an increasing diradical character of

the coupler moiety. The J value for the diradical with a fused bent coupler is always found to be

smaller than that for a diradical with a linear coupler of the same size. Nuclear independent

chemical shift (NICS) is calculated and it is observed that the average of the NICS values per

benzenoid ring in the diradical is less than that in the normal polyacene molecule. An empirical

formula for the magnetic exchange coupling constant of a NN diradical with an aromatic spacer

is obtained by combining the Wiberg bond order (BO), the angle of twist (φ) of the monoradical

(NN) plane from the plane of the coupler, and the NICS values. A comparison of the formula

with computed values reveals that from tetracene onwards, the diradical nature of the linear acene

couplers becomes prominent thereby leading to an increase in the ferromagnetic coupling

constant. Isotropic hyperfine coupling constants are calculated by using the polarized continuum

model for the diradicals in different solvents, and also for the species in vacuum.

Summary

v

Chapter 5 also describes the prediction of the magnetic exchange coupling constant (J) of

m-phenylene based nitronyl nitroxide (NN) diradicals with nine different substituents in three

unique (common ortho, ortho-meta and common meta) positions on the coupler unit by from the

broken-symmetry density functional (BS-DFT) calculations. The molecules are shown in Figure

5.3. Substitutions at common ortho position always have greater angles of twist between the spin

source and the coupler units. When the twist angle is very large, a change of the intramolecular

interaction from ferromagnetic to antiferromagnetic is observed. In these cases, the coupler-NN

bond order (BO) becomes small due to a partial population of the σ* orbital. Substitution at the

common meta position of m-phenylene in the diradical has little steric and hydrogen bonding

effects. The effect of electron withdrawing power of the subtituent does not reveal any clear cut

expression except for the single atom substitution. In the latter case, an ortho substitution leads

to a decrease of J and a meta substitution increases J with a decreasing −I effect. The nucleus-

independent chemical shift (NICS) value is found to decrease from the corresponding mono-

substituted phenyl derivatives. The exchange coupling constant can be estimated from an

empirical equation when there is hardly any stereo-electronic or hydrogen bonding effects that

may change the twist angles.

The interesting magnetic phenomenon of photomagnetism is explored in Chapter 6. In

this Chapter, we predict the photo-switching magnetic properties of four substituted

dihydropyrenes from broken-symmetry calculations. (Figure 6.2 and Figure 6.4) The magnetic

exchange coupling constants differ up to 17 cm−1. The intramolecular exchange interactions are

ferromagnetic in nature. The calculated coupling constants are much larger than those reported

earlier for photomagnetic organic molecules.

In Chapter 7 of this thesis, we present our results of investigation of the magnetic

properties of a recently synthesized dinuclear complex, [Cu2(μ−OAc)4(MeNHpy)2], by broken-

symmetry (BS) density functional (DFT) methodology. While for the molecules investigated in

the previous chapters the Noodleman,1 Yamaguchi,2 and Ginsberg-Noodleman-Davidson formula

(GND)3 have been applicable, the Bencini-Ruiz formula gives a better description of J in

transition metal complex diradicals.4 The complex has several pairs of magnetic orbitals.

Therefore, we have explicitly calculated the overlap integral Sab between the two natural

Summary

vi

magnetic orbitals, and found a value of 0.8589. Deviating from the common practice of

approximating Sab by 1 for the strongly delocalized systems, the computed value has been used in

calculating the magnetic exchange coupling constant (J) from the two electron-two orbital BS

model. The calculated J is −290 cm−1, in excellent agreement with the observed value of −285

cm−1. The contribution of the overlap between the orbitals of the two copper atoms to Sab is

negligibly small. Also, the calculated J value is a weakly varying function of the Cu-Cu distance.

The last two observations confirm that the through-ligand superexchange phenomenon is

responsible for the high magnetic exchange interaction in the Cu2(μ−OAc)4 complex(es).

Furthermore, we have shown that the onset of intramolecular hydrogen bonding reduces the spin

density on the bridging atoms and consequently the magnitude of J. This explains why the

complex under investigation has a J value smaller than that of [Cu2(μ−OAc)4(H2O)2] (−299

cm−1). While establishing this trend, we have predicted that the complex [Cu2(μ−OAc)4(py)2]

would have a higher J value, about −300 cm−1.

List of Publications

Published

1. Ali, Md. E.; Datta, S. N. “Polyacene Spacers in Intramolecular Magnetic Coupling” J. Phys. Chem. A 2006, 110, 13232.

2. Ali, Md. E.; Datta, S. N. “Density Functional Theory of Prediction of Enhanced Photomagnetic Properties

of Nitronyl Nitroxide and Imino Nitroxide Diradicals with Substituted Dihydropyrene Couplers,” J. Phys. Chem. A, 2006, 110, 10525.

3. Ali, Md. E.; Datta, S. N. “Theoretical Investigation of Magnetic Properties of a dinuclear copper complex

[Cu2(μ-OAc)4(MeNHpy)2],” J. Mol. Struc. :THEOCHEM, 2006, 775, 17. 4. Ali, Md. E.; Datta, S. N. “Broken-Symmetry DFT Investigation on bis-Nitronyl Nitroxide Diradicals:

Influence of Length and Aromaticity of Couplers” J. Phys. Chem. A 2006, 110, 2776. 5. Ali, Md. E.; Vyas, S.; Datta, S. N. “Ab Initio Quantum Chemical Investigation of Spin States of Some

Mono and Diradical Derivatives of Imino Nitroxide and Nitronyl Nitroxide” J. Phys. Chem. A 2005, 109, 6272.

6. Vyas, S.; Ali, Md. E.; Hossain, E.; Patwardhan, S.; Datta, S. N. “Theoretical Investigation of Intramolecular

Magnetic Interaction Through An Ethylenic Coupler” J. Phys. Chem. A 2005, 109, 4213. 7. Datta, S. N.; Jha, P. P.; Ali, Md. E. “Ab-initio Quantum Chemical Investigation of the Spin States of Some

Fused Ring Systems” J. Phys. Chem. A 2004, 108, 4087. Submitted and ‘in preparation’

1. Ali, Md. E.; Misra, A.; Datta, S. N. “N-electron Spin Interpretation of Magnetic Exchange Interaction in Broken-Symmetry approach”.

2. Ali, Md. E.; Hossain, E.; Datta, S. N. “Theoretical Investigation of Substituted m-Phenylene Spacers as

Ferromagnetic Couplers in Nitronyl Nitroxide Diradicals”.

3. Ali, Md. E.; Singharoy, A.; Datta, S. N. “Theoretical Investigation of Magnetic Properties of Trimethylenemethane-Type Nitroxide Diradicals”.

4. Ali, Md. E.; Datta, S. N. “Theoretical Investigation of Magnetic Properties of Photomagnetic Molecules”.

Acknowledgment

I express my deep sense of gratitude, respect and admiration to my guide, Prof. S. N.

Datta, for his constant support and guidance, and for encouraging me to work independently

whenever it was required. I am greatly indebted to him for his priceless help, genial

behavior, moral boost, good wishes, kindness, ebullience, criticism, enormous patience,

useful suggestions and keen interests throughout the continuation of my work. His systematic

approaches to scientific problems and affirmative outlook have always enlivened me. The

affection, love and friendliness of ‘Kakima’ and Gargi deserve humbling admiration.

I also convey deepest gratitude and respect to Prof. Y. U. Sasidhar and Prof. Anindya

Datta for painstakingly evaluating my annual progress reports and bestowing their valuable

suggestions and comments throughout the PhD program.

I am indebted to Prof. Illas of University of Barcelona and Dr. Ciofini of CNRS Paris.

I learned certain scientific techniques from them. This work would have remained largely

incomplete without their help.

Susmit Basu has been associated in my work and is a considerate friend. I would also

like to thank my co-workers Anirban Panda, Nital Mehta, Ekram Hossain, Abhishek

Singharoy, Aritro Sinha Roy, Suvrajit Sengupta, Shubham Vyas, Sameer Patwardhan, Praket

Jha, Anshu Pandey, Prasuan Mukherjee and Dr. Anirban Misra.

I am eternally grateful to my “Abba” and “Maa” (parents), who in spite of great

hardships, have always encouraged me to pursue higher education and finally the PhD work.

Their blessings have caused the real motivation for me to join the program at IIT-Bombay. I

dedicate this thesis to them. I am also indebted to Amina, Rahmat and Barjahan for

continuous moral support. Little “Atiya” has been a constant source of joy. Last, but not the

least of all, Anjum Ismail has been a constant source of inspiration.

Md. Ehesan Ali

January 25th, 2007

theoretical and computational aspects of magnetic molecules: phd thesis

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