Lecture Notes in Chemistry Edited by: Prof. Dr. Gaston Berthier Universite de Paris

Prof. Dr. Michael J. S. Dewar The University of Texas

Prof. Dr. Hanns Fischer Universitat Zurich

Prof. Dr. Kenichi Fukui Kyoto University

Prof. Dr. George G. 'Hall University of Nottingham

Prof. Dr. Jiirgen Hinze Universitat Bielefeld

Prof. Dr. Hans Jaffe University of Cincinnati

Prof. Dr. Joshua Jortner Tel-Aviv University

Prof. Dr. Werner Kutzelnigg Universitiit Bochum

Prof. Dr. Klaus Ruedenberg Iowa State University

Prof Dr. Jacopo Tomasi Universia di Pis a

56

D. Heidrich W. Kliesch W. Quapp

Properties of Chemically Interesting Potential Energy Surfaces

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors

D. Heidrich Sektion Chemie

w. Kliesch W. Quapp Sektion Mathematik Universitat Leipzig 0-7010 Leipzig

ISBN -13: 978-3-540-54286-5 e-ISBN -13: 978-3-642-93499-5 DOl: 10.1007/978-3-642-93499-5

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted

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© Springer-Verlag Berlin Heidelberg 1991

2151/3140-543210 - Printed on acid-free paper

PREFACE

The exploration, characterization, and representation of potential

energy hypersurfaces (PES) of chemical systems consisting of N

interacting atoms is a task of increasing importance especially as a

basis for modern reactivity theory.

In Chap. 1 of this book, meaning and problems of the potential surface

concept are summarized. The mathematical analysis of the PES is

subject of the Chapters 2 and 3. It covers the calculation and

characterization of chemically interesting points, curves and global

situations on a reaction PES. Since adequate mathematical re­

presentations are of increasing importance for the informative

processes wi thin the chemical communi ty , the presentation of the

mathematical methods in this book also implies educational aspects.

Finally, the dependence of PES properties on those approximations is

considered which occur in the usual application of quantum chemical

theory. This presentation may give the chance to deepen the chemist's

PES intuition concerning the handling of reaction PES as a source for

microscopic and macroscopic information with and without medium

influence (Chap. 4). Thus, we hope to stimulate theoretical under­

standing and research. Considering the importance of proton transfer

reactions in chemistry and biology as well as their advantages in a

theoretical treatment, they form the main source for selected examples

in Chap. 4. For a full account of experimental and theoretical work in

this field, we refer to excellent reviews in the literature.

As soon as Pulay's pioneering work on the problem of quantum chemical

geometry optimization was su,pported by important contributions of

McIver and Komornicki in the early seventies, the utilization of the

gradient of potential energy has revolutionized the a priori

calculation of chemical reactivity. The gradient concept was suitable

as a basis for a systematic study of chemical reactions and the

further development of reaction tqeory. In 1973, one of the authors

(D.H.) published his first quantum chemical program system with some

applications using the gradient of the potential energy in semi­

empirical methods. The main interest was in the structure and

stability of carbocations forming intermediates in important chemical

standard reactions.

Later on, we started a cooperation between the Departments of Che­

mistry and Mathematics at our University to improve the possibilities

of PES analysis. At present our work follows three main directions:

VI

~ development of mathematical methods for the analysis of PES,

~ application of quantum chemical ab initio methods on models of

chemical standard reactions (exploration of reaction mechanisms),

~ the role of vibrations and intramolecular vibrational redistribution

(IVR) during a reaction.

We hope that our experience and results gained in these fields will

enrich this book. Dietmar Heidrich is responsible for Chaps. 1 and 4,

Wolfgang Kliesch is the leading author for Chap. 2 (up to 2.5),

Wolfgang Quapp from 2.6 up to Chap.3.

We would like to thank some colleagues for their help and support to

overcome problems and to finish this book:

Dr.S.Ackermann (Leipzig, Germany),

Dr.J.Pancl~ (Prague, Czechoslovakia),

Dr.J.Reinhold (Leipzig, Germany),

Prof.Dr.P.v.R.Schleyer (Erlangen, Germany),

Prof.Dr.Z.Slanina (Prague, Czechoslovakia),

Prof.Dr.C.Weiss (Leipzig, Germany),

Prof.Dr.M. and Dr.B.Winnewisser (GieSen, Germany),

Prof.Dr.R.Zahradnlk (Prague, Czechoslovakia).

Special thanks should be given to

Doz.Dr.H.-J.Hofmann (Leipzig, Germany),

Prof.Dr.J.Tomasi (Pisa, Italy)

who have substantially contributed to this work.

wolfgang Quapp would like to express his appreciation to the Fonds der

Chemischen Industrie and the Deutsche Forschungsgemeinschaft for

supporting in part this work.

Leipzig, April 1991

D.H., W.K and W.Q.

CONTENTS

1 GUIDELINES IN THE DEVELOPMENT OF THE THEORY OF CHEMICAL REACTIVITY USING THE POTENTIAL ENERGY SURFACE (PES) CONCEPT 1

1.1 The Potential Energy Surface (PES) Concept 2 1.2 The Dimensionality Problem 3 1.3 On the Definition of a Reaction Path (RP) 4 1.4 The Hierarchy and Competition of Reaction Theories 11 1.5 What about the Calculation of Absolute Reaction Rates? 17 1.6 Potential Energy Calculation and Gradient Revolution 19 1.7 The "State of the Art" in Everyday Study of Chemical

Reactivity 23 References 26

2 ANALYSIS OF MULTIDIMENSIONAL POTENTIAL ENERGY SURFACES -STATIONARY AND CRITICAL POINTS 31

2.1 Basic Definitions and Notations 31 2.2 Geometrical Properties of PES 33 2.3 Stationary Points 35 2.4 Location of Stationary Points 38

2.4.1 The Newton Process and its Modifications 41 2.4.2 Update Methods 48 2.4.3 Quasi-Newton Methods 60 2.4.4 Descent Methods 66 2.4.5 A Global Newton-like Method 71

2.5 Testing of Numerical Procedures 76 2.6 Zero Eigenvalues of the Hessian 78

2.6.1 Translational and Rotational Invariance 78 2.6.2 "True" Zero Eigenvalues: Catastrophe Points 86 2.6.3 Flat Bottoms and Double Minimum Potentials 95

References 97

3 ANALYSIS OF MULTIDIMENSIONAL POTENTIAL ENERGY SURFACES - PATHS - 101

3.1 the Simple Valley Floor Line 3.2 Mathematics of Valley Floors

3.2.1 Gradient Extremals (GE) 3.2.2 GE and Bifurcation Points 3.2.3 GE for Higher-Dimensional Cases

3.3 Steepest Descent Paths 3.4 The Independence of Steepest Descent Paths from

Parameterization and Coordinate System 3.4.1 Parameterization 3.4.2 Invariance from Coordinate System 3.4.3 Mass-Weighted Cartesian Coordinates

References

4 QUANTUM CHEMICAL PES CALCULATIONS:

101 107 107 111 121 122

126 126 128 132 136

THE PROTON TRANSFER REACTIONS 138 4.1 The Problem in Visualization of PES Properties 139

4.1.1 RP Energy Profiles and Surfaces Derived from Usual PES Sections 139

4.1.2 Graphical Presentation of Three-center Problems 144

VIII

4.1.3 Interaction Surface of an Attacking Species with a Fixed Valence System 144

4.1.4 Empirically Derived Diagrams of more Complex Reactions PES 147

4.1.5 Energy Profiles from Mathematically Defined RP Calculations 148

4.1.6 Summary 150 References 150

4.2 PES Properties Along the Bimolecular Single Proton Transfer 152 4.2.1 Formulation of the Reaction Mechanisms 152 4.2.2 The Proton Transfer Energy 154 4.2.3 Discussion of most Recent PES Data of Bimolecular

Single Proton Transfer 155 4.2.4 Gas-Phase Results and Medium Influenced

Experimental Data 162 4.2.5 Theoretical Approach to Medium Influence and

the PES Concept 167 4.2.6 Proton Transfer, Transition State Theory, and

Quantum Chemistry 173 References 176

Index 180

1 GUIDELINES IN THE DEvELOPMENT OF THE THEORY OF" CHEMICAL REACTIVITY

USING THE POTENTIAL ENERGY SURF" ACE (PES) CONCEPT

The PES concept forms th~ basis for a variety of theories, models and methods for the study of chemical reactivity. These methods represent a variety of classical, semiclassical, and completely quantum-mechanical methodologies with different degrees of accuracy and applicability to perform calculations of microscopic and macroscopic attributes of chemical reactions. Beyond the investigation of reaction mechanisms, problems which arise from multiphoton excitations in a hypothetical mode-specific chemistry may also be analysed on the basis of the PES concept. Its realization depends on the extent of the intramolecular vibrational redistribution (IVR) which may also be attributed to special PES properties. Frequently, chemistry textbooks present simultaneously different approaches for the interpretation of chemical reactions. Using the transition state theory (TST) , concepts and rules have been derived from the qualitative valence bond as well as from the molecular orbital theory. Both quantum chemical approaches represent useful tools in learning and understanding certain aspects of chemistry by a qualitative consideration of particular electronic configurations and interactions. However, for an explicit treatment of all factors influencing a chemical reaction, it is initially unavoidable to utilize the numerical results of potential energy calculations with respect to the most relevant parts of the PES. Today, the energy requirements of a reaction may be derived by an energy profile over a reaction path (RP) by using the results of quantum chemical methods. On the one hand, the particular meaning of theoretically derived potential energy profiles along a RP consists in the clear separation of all macroscopic effects and influences of the medium from the factors which intrinsically determine the course of a reaction. On the other hand, the potential energy data gained by theoretical gas phase chemistry with the utilization of a fruitful interaction with modern vapour-phase experiments, give the additional possibility of a subsequent use of (or search for) theoretical methods suited for the calculation and interpretation of macroscopic and environmental effects.

2

Quantum chemistry, therefore, powerfully contributes towards extending

and refining chemist's ability to forecast potential energy hyper­

surfaces PES ("PES intuition") and to form ideas concerning the

corresponding free enthalpy surfaces. Considering the growing possibi­

lities of reaction theory guaranteed by the constant development of

sophisticated methods in quantum chemistry, this is at least one

promising process of an ever-improving understanding of the course and

the inherent properties of chemical reactions as well as of the

theoretical simplifications used in quantum theory.

1.1 The Potential Energy Surface (PES) Concept

The PES is one of the most important issues of theoretical chemistry.

One reason consists in the validity of the Born-Oppenheimer approxima­tion (1927) for systems in the ground state. This seems to be con­

firmed by experience which has been gathered until now. The Born­

Oppenheimer approximation makes the molecular structure to the central

dogma of molecular science. considering the molecular structure

controversyl, we feel that a possible revolution in the understanding

of chemical structure will be based on an extensi ve check of the

Born-Oppenheimer approximation and the quantum mechanical theory used

in general at this time. Modern PES analyses in particular show that

isopotential or quasi-isopotential energy domains, which in a

classical sense exclude well-defined structures, are not just

exceptions. 2 symmetric double minimum potentials with low barrier, as

being assumed in the case of NH3 by the existence of two pyramidal

forms of the molecule, are rather fictitious explanations of the

actual state of the system. 1e Both experimental and theoretical

studies of systems with such attributes may show new theoretical lines

for developing the theory along or outside the field of "classical

quantum mechanics".

The Born-Oppenheimer approximation decouples the electronic motion

from that of the nuclei thus allowing to determine the potential

energy for any atomic configuration (we refer to the textbooks and t ' , 11 I' , 3-6 men ~on espec~a y some new pub ~cat~ons ). In other words, a global

PES provides the potential energy as a function of the nuclear

geometry of the system. For N atoms the potential energy

(la,b)

depends on n=3N-6 (for linear systems o~ n=3N-5) independent

3

coordinates q (Eq.1a). If the potential energy is plotted against

these n coordinates we get a hypersurface over the n-dimensional

coordinate space with respect to a given electronic state. In the case

of 3N cartesian coordinates collected in the column vector x (Eq.1b)

we accept 6 (or 5) redundant coordinates which arise from the transla­

tion and rotation of the whole system. These degrees of freedom do not

influence the potential energy.

It should be added that in ~antum chemistry the zero of the potential energy is defined by the isolated electrons and bare nuclei. Thus, the so-called total energy can be represented in terms of the electronic energy Eel and the nuclear repulsion energy En:

(2a)

It is evident that the total energy by definition includes the very large and predominant part of the energy due to the inner-shell electrons which are primarily not responsible for chemical bonding. However, the reference point of the potential energy may be shifted towards the energy of isolated reactants (or any other particular atomic arrangement):

Epot = Etotal(system) - Etotal(reactants). (2b)

In empirical PES calculations 'of interacting molecular species the energy appears directly in this form.

wi th the creation of the transi tion state theory, particle

interactions have been increasingly visualized by geographical

analoga, e.g., by mapping isopotential lines of energy mountains in

subspaces of the coordinates. This is highly visual and has, until

now, decisively, influenced the pattern of thinking in everyday

chemistry. PES contain all the information which in connection with

reaction theories allows one to find most of the entities describing

chemical reactions beyond the proper potential energy data. PES also

form the basis for the study of quantum mechanical effects concerning

the nuclei (wave function of the nuclei) by solving the Schrodinger

equation in which the nuclei are subject to the forces defined by the

energy functional.

1.2 The Dimensionality Problem

It is well known that the PES cannot be given analytically in quantum

chemistry, but can be calculated point by point through iterative

solution of matrix eigenvalue problems arising from the application of

LCAO-MO SCF CI (Linear Combination of Atomic Orbi tals - Molecular

Orbital Self-Consistent-Field Configuration Interaction) methods. If

4

we calculate over a grid of 10 points per coordinate ~e would have to

determine 103N-6

points of the PES. Taking N >3 the number of points is already

overwhelming (cf.for example Refs.7,8). Thus, the decisive problem in

calculating PES is the dimensionality of the PES in the case of

polyatomic systems. The way to circumvent the dimensionality problem

when studying chemical reactions, is to restrict oneself to a reaction

path as shall be introduced in the following.

1.3 On the Definition of a Reaction Path (RP)

Most of the literature that has been accumulated over the years on

this important problem has been collected in several excellent surveys.6,9,25 It is beyond the scope of this section to repeat all

the references; we focus our interest on a number of recent

developments of this field. Further literature can be found in the

following chapters.

until now, a reaction path has in general been described as a

continuous line in a multi-dimensional coordinate space which connects

the minimizers of the energy functional representing reactants and

products, along points of lowest energy with respect to the energy of

nearby-points. This qualitative characterization of· a minimum energy path (MEP) was useful and fruitful for the understanding and discussion of chemical reactivity within the scope of conventional

"0 0.

W

,

Fig.l. Two-dimensional illustration of the relief path which corresponds to the path in the "energy mountains", and the RP which represents the projection of the relief path onto the coordinate space

5

transi tion state theory (TST). 10,11 However, when going behind the

conventional TST and working in a larger coordinate space by advanced

mathematical methods, the question for a physically sui table· and

unique definition of a RP is a much more complex problem than at first

assumed and has to be studied in detail.

Let us first explain a number of terms which are important for the

further considerations (for similar attempts see for instance Ref.13):

The energy profile over such an RP defines a mountain path (relief path12 , Fig .1) which leads over a highest energy point, the saddle. The saddle point (SP) of primary chemical interest has only one

direction of negative curvature in the energy profile (Figs. 1,2).

Therefore, it is called a SP of' index 1 on the PES (cf. the table in

Sect. 2.3; in the literature one can also find the term SP of first

order).

Such a saddle point defines the so-called transition structure which

may be used as the structure of the corresponding transition state

when using the conventional transition state theory. In conventional

TST, this point of the PES defines the atomic configurations of the

transition state and is subject to a statistical treatment, in order

to determine thermodynamic quantities. However, the coordinates of the

transition state may, but need not, agree with this transition

structure. An identification of the transition structure with the

geometry of the transition state is mostly possible or at least a good

initial approximation, but in certain cases (gas phase results) not

sufficient (cf.Sect. 1.4).

The term transition state must be reserved for denoting the energy states as well as the structure of the energy maximum along a free energy (enthalpy) path (SP on a free enthalpy surface) at the actual temperature.

For the RP the term reaction coordinate (RC) is also frequently used.

It originates from the very beginning of the transition state theory10

where it was used to characterize the motion over the saddle point.

Today, the term RC instead of RP is mostly used when the RP is

represented by a straight-line axis in the usual E vs RP diagrams.

A RP of a PES is a static path which neglects all kinetic energy

terms. Therefore, it cannot have a direct physical meaning (in

contrast to trajectories which form solutions of the equations of

motion). This also means that the RP must not be interpreted as

indicating the detailed stereochemical course of a reaction. Never­

theless, it has a great theoretical and chemical importance. An

uniquely defined and computable RP should form a suitable basis for

advanced reaction theories, since it represents a leading line around

6

which the trajectories of the reaction should be mainly oriented. For

a mathematical description of curves on the PES such asa RP, two

basic mathematical quantities are of main interest: the gradient g of

the potential energy (i) and the corresponding Hessian matrix H (ii).

(i): The gradient of the energy functional at some point is a vector

in the vector space of displacements which collects all first

partial derivatives of E with respect to the atomic coordinates. The

negative gradient of a certain atomic configuration can physically be

interpreted as the vector of the forces acting on the atoms of a

system in order to reach the equilibrium geometry, i. e., the next

minimizer of the potential energy. We will illustrate the gradient,

its properties and utilization in Chaps. 2 and 3. Here we only mention

that the gradient always "intersects" the isopotential lines (in

general hypersurfaces) perpendicularly; its negative direction

indicates the direction of steepest descent on a PES. We note that the

steepest descent represents a local mathematical characteristic.

However, it gives no information about whether a point lies on a

valley floor line or not.

(ii) The Hessian matrix represents the second partial derivatives of

the potential energy w.r.t. coordinates (these physically correspond

to "classical" force constants when calculating at minimizers; cf.

also the calculation of the vibrational frequencies at this point).

The Hessian matrix contains important information on the PES curvature

around the given point on the PES. The number of the Hessian

evaluations required in a given procedure determines the effort of the

method when tracing an RP or any other curve.

Now, the question has to be answered: Can we uniquely define an RP as

a basis for further development of reaction theory?

Until now, MEPs are determined by using different mathematical

procedures of path-following, which are supposed to be in agreement

with the chemical requirements. In particular, strategies for MEPs in

ascent are in most cases not identical with those used for the

descent.

In a first step A we analyse the path tracing along a single

(isolated) reaction channel, Le., along an MEP between two minima

passing a saddle point (Fig.2a). In a second step B, the more complex

situation is described when considering the higher dimensions of a

real chemical system (Fig.2b).

7

~ For case A it can easily be shown (cf.Fig.2) that a quantitative

formulation of a RP depends on whether we are moving (a) downhill from

the saddle, or (aa) uphill to the saddle point (SP).

Let us now characterize descent and ascent RP in more detail:

(a) The descent (down-hill) path:

starting at an SP along the descent vector (direction of negative

curvature on the PES, gained from the Hessian matrix) and going

down-hill by (mass-weighted) steepest descent (i.e. using the gradient

only) towards reactants and products, respectively, defines the

so-called intrinsic reaction coordinate14 (see Sects. 3.3 and 3.4). It

is in general not identical with a valley floor path (cf. l.h.s. of

Fig.2a) and, in general, it approaches the reactant minimum from its

weakest ascent (from the lowest frequency mode when using

mass-weighted coordinates), 6,15, 16a i. e., not automaticly from those

directions which reflect the reaction under consideration.

It must be added that in the case of symmetric RP branching the

steepest descent path does not correspond to an MEP in this region

(r.h.s. of Fig.2a) .16 Apart from that fact, the steepest descent

evidently operates very well when considering case A.

(aa) The ascent (uphill) path:

A valley floor line in ascent found by starting at a minimizer of the

energy should follow the stream bed direction of the reaction channel

and describes a path of shallowest ascent in the coordinate space

(Fig. 2). Such a path is in general not identical to the steepest

descent path of the opposite direction, but has to meet it at the

saddle. Considerable effort (e. g., the calculation of the Hessian

matrices or higher derivatives) is needed to determine a path uphill17

(mostly imagined by following valleys uphill).

Note that the reversal of the sign of the PES is no way out (a SP of index 1 would change into a SP of index N-1 1). The reversal of the sign of the gradient direction on the PES fails, too. Thus, a simple reversal of the concept of steepest descent is not possible).

Tracing a path in ascent (as well as in descent) can mathematically be

described by the so-called gradient extremals18 (section 3.2). They

follow extreme values of the gradient norm found along each of the

isopotential lines. In tracing a stream bed path we have to follow the

smallest gradient norm derived in this way. We note that this path

does not necessarily intersect the iso-potential lines of the PES in a

perpendicular way. This approach is still in statu nascendi with

regard to the application in higher-dimensional systems. We mention

that the path-following procedures uphill do not necessarily lead to a

8

Fig. 2a. Approaches to reaction paths (mass-weighted or not) for a valley-SP-valley domain (case A): - steepest descent from the saddle point (solid line): knowledge of the SP geometry is presupposed; the path approaches the minimizer from its weakest ascent; it does not follow an MEP when symmetric bran­ching of the RP occurs, cf. r.h.s. of Fig.2a. - The uphill stream bed path (minimum gradient extremal, dotted line in the l.h.s.): complicated path tracing, e.g. due to dissipation of the original valley floor. Note that along orthogonal RP directions a very different ascent in energy occurs when comparing the uphill and downhill path.

Fig. 2b. The onset-orientation problem of a RP when additional SPs (SP',SP", SPfff) occur in the precursory region of the reaction; the figure shows a symbolic two-dimensional representation of case B outlined in the text (the addi tional processes must generally be described by additional coordinates which cannot be drawn in two dimensions)

saddle because a valley may be flatten anywhere, therefore irritating

the path direction uphill.

In the textbooks and large parts of the literature a minimum energy path is nolens volens identified with the points of a valley floor leading directly up to the saddle of interest and vice versa. However,

9

such a valley floor line only exists in an idealized case (many textbook illustrations are concerned with simple bimolecular reactions such as collinear H+D, which cannot be generalized) and it is better to use the term stream bed path. The latter may suddenly change the original valley floor direction (dissipation of a valley) to reach the actual position of the saddle point (Fig.2a).

From the simplified case A, which we discussed above, it must be con­

cluded that at least two different MEPs can be defined, one for the

ascent and one for the descent (within the same coordinate system).

!!=:J Now, we take up the situation of case B (Fig.2b) by considering

the whole coordinate space of more complex chemical systems.

In a polyatomic system, the chemical process considered may be

accompanied by a number of possible rearrangements and formation of

complexes. These precursory processes should have low-lying saddle

points (cf. Figs.2b and 4). Additionally, other chemical reactions may

occur in competition. We again discuss the path following depending on

moving uphill (b) or downhill (bb).

(b) The RP in ascent can only be understood as a MEP when competing

processes with low SPs are missing.

We have to note that the term MEP has been originally minted when using mathematically not well-defined procedures. An example is the use of two stepwise fixed guiding coordinates in a simple bimolecular reaction and optimizing all other coordinates in order to get energy minima. These relaxed one- and two-dimensional cross-sections realized by the coordinate driving procedures, are successful for obtaining well-approximated MEP in favourable cases (see Sect. 4.2.1.). Thus, the term MEP is related to the real or thought progress of the reaction in question. When we use the term MEP in many-dimensional systems resigning to any coordinate constraints, then the term MEP would "only" define the way to a nearby low energy SP, mostly not representing the course of the desired reaction.

consequently, in order to describe a special chemical reaction we have

to start along eigenvectors (displacements of coordinates) which

belong to eigenvalues of the Hessian at the reactant minimum (or

belong to normal modes calculated from the mass-weighted Hessian) and

which need not, mostly will not be the weakest one. The selection of

one or more eigenvalues of the Hessian, the eigenvectors of which give

a suitable direction for the onset of the reaction path uphill to the

saddle of interest, can mostly not unambiguously be realized, and/or

the eigenvector-following cannot unambiguously be continued, though

well suited as procedure for SP localization algorithms.

A peculiarity of an ascent path is that branching and dissipation points (critical points, see Sects. 2.6 and 3.2) which characterize the regions of branching or dissipation of a valley, may make the path tracing more difficult by the occurrence of zeros of the Hessian matrix. On the other hand, this may promote the development of procedures which allow to locate branching points (cf.Ref.19).

10

(bb) Along steepest descent, the MEP approaches the reactants opposite to the actual course of the chemical reaction. Hence, because the reactants are approached in the direction of weakest ascent, it cannot represent the characteristic atomic movement at the begin of a more complex chemical reaction! Hence, apart from mathematical difficulties in realizing RPs, we observe problems concerning the physical requirements of the paths. -

The term MEP is useful for a general description of the course of RP in chemical reactions. However, from the points A and B we have learned that this term represents no mathematically well-defined path. Therefore, we observe the use of various RP definitions; but they are not yet complete for a general use in theories beyond conventional TST. The chemically and mathematically most difficult syndrome of the RP approach, we describe as general onset-orientation problem of the RP, cf. also Sect. 1.4 (onset at the reactants, irrespective of the direction of RPtracing).

We note that any RP definition may involve further difficulties: A reaction may proceed far away from the course of a static defined RP. It is well known that the preferential path may not only deviate from the RP by tunneling induced corner cutting (for Refs.see 20a, cf. additionally Ref.20b) but also by vibrationally induced corner cutting (Ref.21). Model systems (for example: F+ DBr(v) * FD + Br) demonstrate this important role of reaction dynamics in comparison with static properties of conventionally defined RPs (Ref.20a).

However, although RP versions previously developed do not meet all physical requirements necessary to serve as a general basis for more sophisticated reaction theories, they are extremely useful for a detailed mathematical analysis of PES. The suitability of a certain RP definition also depends on the reaction type considered and from the reaction theory used (if it needs the whole path or only the part near the saddle). There have been interesting attempts to take the steepest descent path as a basis for the first levels in the dynamic approach to chemical reactions. At first sight surprising, this path is among the most difficult to determine with sufficient accuracy. 22 New methods of determining the steepest descent path use higher energy derivatives.· Gonzales and schlege122bde·scribed such a path for cartesian and internal coordinates - with and without mass-weighting -giving thereby a survey of the corresponding literature (see also Chap.3).

For certain quantitative descriptions the introduction of mass­weighting of the RP may be an· urgent problem (the mass-weighting, being formally analogous to that in the Wilson theory of normal modes at minima of a PES (Ref.23), admits kinetic effects through a "back-

11

door", see Sect. 3.4). Additionally, questions of the coordinate invariance of these paths have to be answered (cf. Refs. 24a,25a,9a and Chap.3). Starting with the cartesian coordinate system, one may transform and describe the RP in any other coordinate systems by means of differential geometry. In this manner, a gradient path (e.g. the steepest descent) is invariant with respect to the choice of the coordinate system and may be regarded as a fundamental (mathematical) characteristic of a chemical reaction.

In searching for new and still more complete RP definitions, the

problem consists in findin<J an ascent path along the direction(s)

characterizing the actual atomic movements connected with the reaction

under investigation or a steepest descent path which overcomes the

problems near the reactant minimum. The descent path is the more

advantageous one, first of all because of the inclusion of the SP of

interest by definition (a path in ascent may find any other SP).

One may hope for a new impulse for further progress in this field.

Ideas outlined in the generalized valley approximation26 (GVE:

generalized valley equations) may form the basis for a new running-up

to find a suitable ansatz. However, it seems more realistic to create

individualized descriptions of a given chemical reaction using that RP

definition most promising in the present case. In this manner there

should be a number of possibilities for enhanced reactivity theories

in the grey zone between conventional TST and trajectory calculations

or dynamical quantum mechanics (see Sect. 1. 4). They should allow

certain new insights into the "heart" of common chemical reactions.

Independently of the considerations in this section concerning the advanced definition of a RP, which should allow the study of the environment of the path (orthogonal directions), we repeat that each of the different MEPs (among them descent algorithms which are numerically easier to realize, e.g. quasi-Newton procedures, Sect. 2.4) is able to show whether or not there is a further minimizer between a given saddle point and a given minimizer. This information is sufficient for interpreting conventional TST and elucidating the mechanism of chemical reactions on the basis of a PES analysis.

1.4 The Hierarchy and competition of Reaction Theories

One fundamental development of reaction theory is illustrated in Fig.3

when following the anti-clockwise direction. Beginning with the TST we

have an increase in the level of sophistication as far as the area of

dynamic quantum mechanics, and in the same direction a strong decrease

in the applicability of the methods to polyatomic systems. In the

clockwise direction (r.h.s. of Fig.3) the first methods require

knowledge of the whole, analytically described PES of small chemical

12

systems or closed parts of the PES of larger systems, to allow model calculations on fundamental questions of reaction theory. At the bottom of the cycle one observes a free field for new developments of the theory, coming from both the left as well as from the right side. On the l.h.s. of Fig.3, the theory with the most reduced input of PES information is' the conventional TST, 10,11 . which only requires the exact knowledge of minima (educts, intermediates) and saddle points of index 1 (SP1, transition ~tructure) on a RP. In the conventional TST,

POLYATQMIC SYST SMALL SYSTEMS r "'\

• RP DYNAMICS

VARIATIONAL TST

Fig.3. Cycle of reaction theories based on the PES concept

where the SP defines the position of the transition-state, it is not necessary to know the RP explicitly. In many cases conventional TST provides a sufficient description of the dynamics, especially at low to moderate temperatures, with a tunnelling correction when necessary. The next stage of sophistication may be a suitable and unique definition of the RP (cf. Sects. 1.3 and 2.6) and ,the calculation of the behaviour of the potential transversal to the RP (Fig. 1). This orthogonal characteristic may be given in terms of a fictitious vibrational analysis along points of a RP. with this information two lines of further development of reaction theory have been established:

13

(1) the variational TST (VTST) calculations and

(ii) approaches to formulate a RP Hamiltonian (RPH).

In case (i) we have a variational correction to the conventional TST

where a more general criterion is used for locating the transition

state. In this approach a search is undertaken for the true dynamic

bottlenecks of reactions which are represented by the maximum of the

generalized free energy along the RP. Thus, by estimating the density

of states along the RP, a transition state may be found which is located away from the SP (transition structure) of the PES, or a

transition state is found at all when reactions do not possess a

potential barrier. The approach is applied to bimolecular as well as

unimolecular reactions (for leading references see Refs.27-32).

This is of particular interest for reactions along a RP without aSP,

Le. "reactions on attractive PES" 28 , e.g. potentials of unimolecular

bond fission processes and the reverse bimolecular recombinations, ion

molecule reactions or a large number of proton transfer reactions in

the gas phase. For certain reactions of this kind, the results of the

calculations of Bowers and Chesnavich33 demonstrate "that either zero,

one, or two minimum flux transition states can be present along a

given RP. The number of transition states depends on the particular details of the PES and the amount of nonfixed energy available to the

system." The VTST approach seems to be the most basic level of an actual

dynamic theory, which is capable of outlining the entropy barriers.

We briefly summarize the Truhlar-Garrett canonical variational theory

(CVT) 29b which represents a definite realization of an old proposal

for constructing free-energy surfaces (cf. for instance Ref.llba). It

may be characterized by three steps

A. Location of the SP on the PES

B. Determining (in mass-scaled coordinates) the steepest path of

descent towards both products and reactants, starting from the SP

C. Determining fictitious vibrational frequencies from the projected

force constant matrices (excluding the path direction as well as

rotation and translation of the whole system) at points along that RP

with the aim of calculating partition functions (number of states) and

fictitious thermodynamic entities as a function of the RP. The

resulting free energy curve along the MEP permits the location of the

transition state at the maximum of this curve. Refinements may be

made by including tunneling corrections and anharmonic corrections to

the potential. -

14

It should be noted that the kind of RP used may considerably affect

the quality of the results when the theory therein is directed toward

more than representing an improved transition-state region, as in the

case of the RPH discussed below.

However, successful VTST enables a simple explanation of the

possibility to give representations of free energy profiles over a RP,

i.e~. plots of macroscopic vs. microscopic entities.

In the case of (ii) we find an extension of the TST with the attempt to create a new methodology34-37,22a In order to define a RP­

Hamiltonian (RPH) for a given system, the same entities have to be

calculated as in the application of the CVT characterized above.

However, point C here requires an extension: Apart from the

calculation of the eigenvectors and eigenvalues of a generalized force

constant matrix (or the normal mode characteristic) at the various

points along the RP, one additionally needs curvature coupling

coefficients which involve the derivatives of the normal modes w.r.t.

distance along the RP. This requires energy derivatives of the third

order. They give the RP curvature and include the couplings' of the

normal modes with each other and to the RP.

The procedure in regard to RPH-calculations still remains expensive,

with it in no sense being automatic. The RPH concept has been used to

determine temperature-dependent rate constants for the isomerization

of the methoxy radical. These show broad agreement with the experiment

where previous calculations failed. 38 This study is the most complete

and successful treatment of that type for a system with more than 3

atoms (for a recent application of the RPH method on the gas phase SN2

nucleophilic sUbstitution reaction, see Ref.39).

However, considering Refs. 20a,24,26 and particularly 15b, the

procedure is subject to some criticism in its concept as well as in

some aspects of realization. The picture of the dynamics in this model

is represented by an one-dimensional motion along the reaction path

coupled to harmonic vibrations in the remaining 3N-7 directions. Thus,

the model must critically depend upon the definition of the RP. The

IRe (cf.Sect. 1.3), being the path of steepest descent, finally

reaches the reactant minimum along the normal mode of lowest frequency.6,15,16a However, this must by no means be identical to the

actual normal mode direction of the reactant which mainly determines

the course of a given reaction. Thus, the reaction path curvature,

important in the calculation of the coupling between RP and

vibrational motions in that theory, may become pathological in many

cases, at least near the minimum.

15

This was critically outlined when the authors of the RPH used the

example of malonaldehyde. 15b

* Q"H. '0

I n ~c~c'1l

I H

* .H 0" "0

II I c c ~ ""c/" '11

I H

The reactive process is clearly the motion of essentially only one

hydrogen atom, H*, with the other degrees of freedom playing a modest

role. The problem with the reaction path description here is that the

reaction path always arrives at the reactant and product wells along

the normal mode of lowest frequency (of the appropriate symmetry);

.. , For the above example the lowest frequency is associated with

some floppy skeletal vibrational motion that is quite unrelated to the

motion of atom H*, while the relevant vibration is the O-H* stretch,

which is the highest frequency of the reactant .... Thus, .. . the

reaction is in no sensible way well described as being one-dimensional

motion along this path. 15b

In the meantime new variations15b,40,41 have been proposed, among them

a cartesian reaction-path model with which one most simply deals. The

model is based upon all cartesian coordinates of the N atoms, but

"chemical intuition" is required to select one to three cartesian

coordinates necessary for describing the displacements along a RP

while all the other coordinates should only slightly move within a

harmonic bath. As it is shown, this model does not include an explicit

and unique definition of a RP in order to obtain a possibly more

natural description of the dynamics.

Evidently, we observe a rediscovery of approximated RP of various kind

(cf. also Sect. 4.2.1) as an input in the advanced reaction theories

outlined above. We feel that there may not be a general solution to

define a RP. We repeat our opinion (see Sect. 1. 3) that a possible

conclusion could be to individualize the reactions or reaction types

in order to select the most suited RP definition(s) (including certain

coordinate driving procedures) and/or reaction theories for each of

them.

Even if the proposed methods still have some shortcomings, a great

deal may already be learned from investigating the RP (theoretical

tracing the geometry change along the RP is called reaction

ergodography) 42, viz. curvature, environment and stability of the RP

and free energy properties of the PES regions in question. The RPH

16

method may also be useful for analysing coupling between internal

degrees of freedom and the RP. Methods of this type may form a bridge

toward total dynamic treatment by trajectory calculations which solve

the classical equations of motion for certain initial quantum

conditions, or by quantum mechanical reactive scattering calculations

(presently possible for collinear atomic-diatomic systems).43

Up to now, reaction path calculations have been predominantly related

to nonsolvated species in the ground electronic state. In cases where

the gas phase RP is not expected to be seriously perturbed by solution

effects, it can successfully be used to describe the reaction progress

also in solution (see Sect. 4.3.5). However, the reacting system for

the PES analysis should additionally involve at least a limited number

of solvent molecules when they could play an "active" role along

the RP (limited supermolecule approach to consider active medium

molecules) .

In order to properly simulate the dynamics of a reaction (cf.r.h.s. of

Fig.3), an analytical representation of the PES or of parts of it,

which determine the chemical reaction, must be attempted. Calcula­

tions with respect to the dynamics of chemical reactions could still

be more suitable for a study of the intrinsic mystery of a chemical

reaction (yielding information to the energy transfer between the

various degrees of freedom etc.) in comparison to those described

above. However, the main problem is to limit the necessary PES range

and to find a good analytical PES function for this range. This

function does not reduce but increases inherent inaccuracies in the

quantum chemical PES calculation (cf. next section). In the scientific

literature we find some further progress in generating PES by using

empirical or certain combined methods involving fitting of the PES by

utilizing points found by ab initio computations. Techniques, problems

and results on this field have recently been discussed. 31

In this context we mention the young yet promising field of research

which is represented by the reactive "transition state spectroscopy"

(TSS). Spectra of transition state configurations have been measured

as well as computed. 44 Although restricted to very small reacting

systems, the TSS "could assist in the quest for understanding of the choreography of a chemical reaction,,44b by probing experimentally the

PES and dynamics in the transition state region, for instance by

photodetachment spectra of transition states utilizing metastable

levels of such states known as resonances.

17

1.5 What about the Calculation of Absolute Reaction Rates?

The TST provides an extremely useful conceptual framework in which one may systematically discuss chemical reactions in almost all con­di tions. For a quanti tati ve treatment in conventional TST one must determine the minima and the saddle point which are arranged on a RP. Both structure types belong to the stationary points of a PES and can effectively be localized by" modern optimization techniques, delving into quantum chemistry. The macroscopic entities of TST (A:iH, A:iS, A*c;, k, ••• ) as well as the thermodynamic reaction characteristics (ARH, ARS, ARG, K, ••• ) may be determined by use of statistical thermo­dynamics. However, the first goal of the TST, namely the calculation of absolute reaction rates, could not be attained until now (cf.Ref.l1b). There are two important reasons:

(i) conceptual restrictions (ii) the difficulty in determining accurate potential energy

differences (A*E in Fig.4).

(i): The restrictions due .to the basic assumptions of TST, such as the quasi-equilibrium hypothesis (Boltzmann-partition in the transition­state), the non-recrossing behaviour, or the further suppositions in modern unimolecular TST11 ,45 (in the RRKM [Rice-Ramsberger-Kassel­Marcus)-theory), are well-known. It may be a more suitable approach to consider energy properties along the whole reaction channel on the PES in an extended form of the TST, although the methods which make some allowance for this, may have other conceptual shortcomings (e.g., due to unsolved problems in the definition of the RP, cf. Sect. 1.4). Furthermore, they are laborious for polyatomic systems, as reported above. Finally, a theoretical treatment of reactions in solution is affected by a number of additional problems and, in contrast to gas-phase TST, condensed-phase TST cannot receive extensive testing against classical and quantum dynamics. Regardless of these problems, already from the very beginning of TST11 a thermodynamic formulation has been proposed in order to introduce the solvent effect. The different changes in potential energy contribution and in free enthalpy, respectively, of reactants and transition state produce a change in the potential barrier and in the activation free enthalpy (Eqs.3a,b), respectively, when going from the gas phase to solution. However, in a pioneering paper, Kramers46 referred to the occurrence of barrier recrossing

18

owing to solvent collisions, and hence restricting the validity of TST in solution (cf.also Chapt. 4.2.5). In the scientific literature one finds an intense and interesting scrutiny concerning the applicability of TST versions for solution reactions11e,47-49 (compare also the concept of "imbalanced transition states") 50. In point (ii) it must be stated that there is a fundamental difficulty in calculating the rate constant k, e.g., by the famous equations (A~o: cf. Fig.4)

k= K(kTlh) Q

-*- exp(-A:IE lRT) QA QB 0 (3a)

K(kTlh) exp(A:t:slR -A~lRT) • (3b)

Because of the exponential dependence in the equation, the reaction rate constant is changed by the factor 10 if the energy barrier is changed by only 5.5 kJlmol (room temperature). In spite of all progress which has been made in the last years,

E

I

I I I ... ITS, l~

a b

RC c

Fig.4. Microscopic entities of a chemical reaction (the entities designated for selected positions) a: precursory zone, b: reaction zone, c: product zone AREpot: (Exo)ergicity of the reaction (measured between the most

st.able educt and product arrangements)

Reaction enthalpy at 0 K (AREo = AREpot + AREZpV)

Potential energy barrier of the reaction

Barrier at 0 K (A:lEo= A=lEpot+ A*EZPV)

are

:' Zero point vibrational energies, TS Transition structures

19

it is still difficult to calculate potential energy barriers of

activation within the required accuracy. This is due to approximations

still necessary in MO theory (see Sect. 1.6 and 1.7). Furthermore, the

situation may become worse with the required inclusion of the

zero-point vibrational energy A*:Ezpv (Fig.4) in Eo (Eq.3a). A

computation of EZpv requires the expensive determination of the

vibrational modes of the educts and the transition structure with its

inevitable inaccuracies for larger systems.

We have to add that calculations of the desired accuracy will be

rendered even more difficult with elements of the higher rows in the

periodic table. This is due to problems which are connected with the

calculation of relativistic and spin corrections. For this reason,

theoretical treatments of corresponding heterogeneous catalysis are

rather limited

Finally, it is

a reaction in

till now.

evident that

soluti

the calculation of a potential barrier for

o n is much more difficult. The problem

covers calculations of the reaction barrier by more or less extended

supermolecule approaches, where the specific influence of a restricted

number of solvent molecules is considered (already leading to a

considerable enlargement of the dimension of the system), as well as

the inclusion of further solvent shells which may be realized by using

combinations with simplifying concepts such as continuum models etc.

(cf. Sect. 4.2.5.).

1.6 Potential Energy Calculation and Gradient Revolution

Recent advances in computational techniques enable an application of

rigorous ab initio theoretical methods to study an increasing number

of fundamental chemical problems on the basis of TST type and extended

theories. Present trends in quantum chemical calculations have been

reviewed,51 see also: "An Experimental Chemist's Guide to ab initio

Quantum Chemistry".52 The theoretical methods are capable of supplying

a wealth of information which complements experimental results.

Systems and structures being inaccessible to experimental investi­

gation may be studied by these methods.

The widely used and ever-improving semi-empirical methods of quantum

chemistry provided important results or, at least, prepared the way

for a final theoretical treatment of chemical problems through the

explicit formulation and modelling in connection with processing the

experimental material. The spectrum of data spans from preliminary

results to useful trend predictions to first approaches to sets of

20

stationary points of the PES of chemically interesting systems. This is of considerable importance, in particular, for calculating larger systems which are inaccessible to the application of more sophi­sticated ab initio methods. Moreover, we refer to these kinds of concepts as force field calculations (molecular mechanics) which approximate the potential field (Born-Oppenheimer approximation) by "classical" energy relations and adjustable parameters. These methods have successfully accompanied and completed the ab initio calculations until now. For the literature covering these methods and their results, we refer to other surveys. 53 ,54 Because of the use of analytical potentials, the procedures are not as time-consuming as ab initio methods. However, their importance is placed behind the conceptually stronger ab initio methods, and they are not suited to localize structures between the minimizers on the PES as it is of primary importance for the kinetic characteristic of a chemical reaction.

As discussed above, a fundamental problem in studying chemical reactivity is the quality of the calculated potential energy. Based on the ab initio SCF MO theory, the following quantities are decisive to obtain reliable potential energies:

(i) The s i z e of the basis set of atomic functions forming the MOs of the system: For sufficiently large basis sets we reach the so-called Hartree-Fock limi t ~F' which represents the best energy wi thin the independent electron model of the MO theory.

(ii) The correlation energy Ecorr : This is determined by the equation

Epot = ~F + Ecorr (4)

and calculated by methods based upon the variational principle, ranging from conventional configuration interaction to multicon­figuration SCF(MCSCF) methods, or based on perturbation theory.55 Many body perturbation theory, (MBPT) approximates the true Hamiltonian covering such popular metqods as the nth-order M~ller-Plesset approach MPn. 56 The well-known coupled electron pair approximation (CEPA)57,58 can be regarded as an approximate version of coupled-cluster theory59 which in some aspects includes all orders of perturbation theory.

(iii) The zero-point" vibrational energy, EZPV' may be an important third factor to determine a good approximation to the energy Eo at 0 K

(5)

21

The computation of Ezpv requires the above-mentioned time-consuminq determination of normal modes (cf. also Sect.3.4.3) due to

EZPV = 1/2 h r Vi· i

(6)

Once we have calculated the normal modes, there is no further hindrance to calculate all macroscopic entities of thermodynamics. The types of (Gaussian) basis sets used in ab initio MO SCF theory have been described elsewhere. It may be helpful to list a few references as examples. 51a ,,60 The basis set desiqnation in quantum chemistry appears to be mysterious for non-specialists. However, it frequently has been explained,51a,60b e.q. recently in a succinct form on current levels of theory. 61,62 In order to characterize the theoretical level which has been employed for the PES calculation, a double-slash notation is used, introduced by Pople. It characterizes the method and basis set which are used for calculatinq stationary points of PES ("qeometry optimization") by the information found after the double slashes. The more sOPhisticated level for subsequent sinqle point calculations of the PES is desiqnated by the information precedinq the double slashes. The usual short-hand notation in the followinq example

MP2/6-31G+G(d,p)//HF/6-31G(d) means that a second order Meller-Plesset (correlation) enerqy with the 6-31+G(d,p) basis set (other notation 6-31+G**) has been calculated at points qained by qeometry optimization usinq the 6-31G(d) basis on the Hartree-Fock level.

The sets of polarization functions (d on non-hydroqen atoms) are qiven in parentheses. "+" desiqnates the use of an additional set of diffuse (small. exponent) functions on nonhydroqen atoms ("++":on all atoms, Ref.63). Polarization functions of low exponent and low-exponent diffuse sand p functions are especially important for a correct description of systems exhibitinq larqer concentrations of neqative charqe (e.q., calculations of deprotonation enerqies, description of ion pairs etc.).

compared with the MO theory, the valence bond theory has not been as successful in calculatinq PES sections of reactinq systems. However, we refer to some possible advantaqes of this theory which is capable of producinq more easily the actual chemical information. 64-66 Nonorthoqonal VB techniques are also well-suited for avoidinq65,66 the so-called "basis set superposition error" (BSSE) which occurs in MO theory when comparinq the enerqies between separated reactants and their complexes (or transition structures) when usinq limited basis sets. A valuation of methods (such as the counter-poise method) for correctinq the BSSE in MO theory has recently been published. 67 ,68

22

At the beginning of the seventies, the development of exact and rapid optimization methods to obtain stable structures, transition structures and RP on a reaction PES of polyatomic systems could be regarded as the bottleneck for further progress in the quantum chemical description of chemical reactions. Similar difficulties arose while performing force-field calculations. However, at that time a revolution took place in research on PES by the introduction of the (analytical) gradient of the potential energy as the basic entity in the optimization procedures. pioneering work69 ,70 and Komornicki. 71 ,72

This development was initiated by Pulay's the famous papers of McIver and

Modifications in the gradient-following, the utilization of symmetry conservation,71 the use of the squared gradient norm instead of the energy function itself,72 and the use of higher energy derivatives form the basis for a large number of methods in locating the important nuclear configurations of a PES, as well as tracing chemically interesting curves (paths) of the PES. The gradient methods were conceptually and practically so convincing that most of the new computer programs removed almost all other methodologies, at least in the calculation of PES properties of polyatomic systems. During the span of a few years, this development occurred in semi-empirical quantum chemical methods,73 then after a certain period of time, gradient procedures also began to occupy the field of ab initio methods. 70, 74, 75 Some time later, Popie et al. showed the way to obtain a n a 1 y tic second derivatives of the Hartree-Fock energy with respect to the nuclear coordinates. 76 ,77 Today, algorithms that generate energy derivatives (up to the fourth order) to points on a PES and their computer codes will be available for most Hartree-Fock SCF functions. 22, 78 An analytic expression of second derivatives of the electronic energy for full configuration interaction wave functions has now been obtained in an elegant manner. 79

The expert handling of the energy derivatives opened enormous possibilities for the quantum chemical investigation of reaction PES.

23

1.7 The "State of the Art" in Everyday Study of Chemical Reactivity

In Fig.4 the entities of the "microscopic" interpretation of a complex

chemical reaction have been illustrated. The figure shows a potential

energy profile of a sequence of elementary steps in a reaction. The

precursory reaction phase indicates possible physicochemical processes

which may be of importance in realizing suitable atomic arrangements

and molecule complexes as starting points for the actual reaction.

These precursory processes may exhibit (as in Fig.4) transition

structures, e.g. conformational rearrangements in the educts or educt

complexes, which cannot be qualitatively distinguished from the

usually more energetic SPs of the chemical reaction steps.

What may be presently done when using quantum chemistry in application

to actual problems in chemistry?

Because of the problems outlined above, reactions of the real

chemistry have been discussed in the literature in general in terms of

A ~ t (or A =IE , not in terms of the rate constant, k) which is po 0 frequently sUfficient for the first approach to the reaction

mechanism. Exploring the stationary points along a chemical reaction

path, a mechanism of the reaction (Fig.4) can be proposed. The

mechanism may be confirmed or better understood, at least by a

qualitative estimation of macroscopic (entropic) effects. In case of

the application of the transition state theory, the macroscopic

effects (vapour phase) may be estimated by the ratio of the partition

functions in Eq. (3a). When the energy differences between different

but frequently similar transition structures, representing different

mechanisms, are relatively high, then it is no problem to present a

reaction mechanism. If alternative mechanisms differ only by about

10-30 kJ/mol in the potential barrier or activation energy, the task

will be more complex, but modern MO theory is in most cases still

capable of giving reliable, at least qualitatively correct predictions

concerning the relative stability of the competing structures, in most

cases, by using all the experience of theoretical calculations at our

disposal.

At present, the more accurate calculations were performed for systems

consisting of several to more than ten atoms. Even if there is no

possibility to use more sophi~ticated quantum chemical methods,

questions to the theory may be formulated, so as to allow a discussion

of reI a t i v e changes in the energy terms between suitably

24

related chemical systems, therefore organizing a certain extent of

compensation of errors in t.~o as well as in the derived quantities

(the entropic effects being among them).

Consequently, each nonspecialist should look in particular at the

(i) type and character of the quantum chemical method used (cf.next

section)

(ii) kind of chemical modelling and interpretation of the results.

Theoretical chemists should explicitly demonstrate the possibility or

the need for procedures and models used for a given task. Clear

statements of this kind are not always found in the literature.

From the obtained geometries and potential energies a huge number of

microscopic states can be calculated, which determine any particular

macroscopic entity by forming an appropriate average using the

partition functions of statistical thermodynamics. For instance, the

internal energy U(T) at the temperature T may generally be written by

defining it as the sum of non-interacting rotational, vibrational, and

electronic contributions

(7)

This approximation is the first feasible approach, which is also

supported by empirical experience and mathematical estimations. In the

second step, the coupling between the motions may be added as a

perturbation. The terms on the r.h.s. of Eq. (7) may be obtained by

solving the corresponding Schrodinger equation for these motions.

until now, the calculations have mostly' applied the rigid rotor­

harmonic oscillator (RRHO) approximation. The calculation of the

vibrational frequencies is the most time-consuming procedure, severely

limiting the size of the chemical system. Hence, the calculation of

the normal modes cannot be performed in many cases on the same level

as Epot ' although their exact computation represents a still higher

demand of accuracy. Enthalpy changes may be determined by Eq.(8)

t.H(298) = t.U(298) + pt.V • (8)

For other entities and details of the calculation, we refer to the

textbooks of statistical thermodynamics or to the corresponding

programs. A very useful ab-initio coverage of the theoretical

experience and computational success in calculating quantum chemical

reaction enthalpies has recently been published. 61 The calculation of

the fictitious thermodynamic quantities of TST is realized when using

the corresponding modified relation for the vibrational partition

function where the imaginary normal mode ("decomposition mode") of the

transition structure is excluded.

25

Semi-empirical versions of MO theory such as the MINCO-38 0 , MND081 ,

AM182 and related methods which are significantly less time-consuming as ab initio methods played a pioneering role in calculating the

vibrational frequencies,83 and hence opened the door rather early to

the determination of thermodynamic properties, such as heat capacities

and entropies. 84. Now, ab initio methods occupy this field more and

more (see Ref.51a). All experience made in PES analysis can now be transferred into the

computation of cross-sections of f r e e enthalpy surfaces for such

reactions where this is necessary (cf • the VTST, Sect. 1.4). The

calculation of transition states in solution has now become a more

realistic aim for reactions of real chemical interest (see also Sect. 4.2.5).11e,85,86

Of course, the processes under the influence of solvents, aggregation

or adsorption are frequently of such complexity that the success of

the energy calculations is not guaranteed and the determination of the

macroscopic effects will be questionable. However, if the predicted

behaviour of a reaction in vacuo differs from that observed experi­

mentally, then this may be an important suggestion for, additional

macroscopic and medium effects controlling the reaction. Today, the

theoretical framework based upon the PES concept is a widely used

instrument of organic and inorganic chemistry for interpreting

reaction mechanisms. It is virtuously applied in the higher levels of

quantum chemistry by a number of leading scientists in this field. 51a

26

References (Chapter 1)

1a Cf. for example: Woolley RG (1978) J Amer Chern Soc 100:1073 and (1982) Structure and Bonding 52:1,

b Claverie P, Diner S (1980) Isr J Chern 19:54, c Weiniger SH (1984) J Chern Educ 61:339, d Woolley RG (1985) J Chern Educ 62:1082, e Amann A, Gans W (1989) Angew Chern 101:277(p 284)

and references in these papers 2 cf. for instance: Boldyrev AI, Charkin OP, Bochenko KV, Klimenko NM

Rambidi NG (1979) Russ J Inorg Chern 24:341, Maruani J, Serre J (eds) (1983) Symmetries and Properties of Nonrigid Molecules, Elsevier, Amsterdam; Schleyer PvR, Sawaryn A, Reed AE, Hobza P(1986) J Comput Chern 7:666; Winnewisser BP (1985) in: Molecular Spectroscopy: Modern Research, Academic Press, Orlando, Vol 3, Chap 6, p 321

3 Kutzelnigg W (1978) Einftlhrung in die Theoretische Chemie, Band 2,Verlag Chemie, Weinheim

4 ZUlicke L (1985) Quantenchemie, Band 2, VEB Deutscher Verlag der Wissenschaften, Berlin

5 Hirst DM (1985) Potential Energy Surfaces, Taylor & Francis, London - Philadelphia

6 Mezey PG (1987) Potential Energy Hypersurfaces (studies in Physical and Theoretical Chemistry 53), Elsevier, Amsterdam

7 Miller WH (1983) J Phys Chem 87:3811 8 For a fine illustration cf. MUller K (1980) Angew Chern, Int Ed Engl

19:1 9 cf. for instance:

a Friedrich B, Herman Z, Zahradnik R, Havlas Z (1988) in: Advances in Quantum Chemistry, Vol 19, p 247, Academic Press;

b Slanina Z (1986) Contemporary Theory of Chemical Isomerism, Academia/Reidel

10 Pelzer H, wigner E (1932) Z Phys Chern, Abt B 15:445; Eyring H (1935) J Chern Phys 3:107

11a Glasstone S, Laidler KJ, Eyring H (1941) The Theory of Rate Pro­cesses, McGraw-Hill, New York;

b Laidler KJ (1969) Theories of Chemical Reaction Rates, McGraw­Hill, New York, a) p 78;

c cf.especially: (1983) J Phys Chem 87: number 15, dedicated to Eyring;

d Laidler KJ, King MC (1983) J Phys Chern 87:2657; e Kreevoy MM, Truhlar DG (1986) in: Weissberger A (ed) Techniques of

Chemistry, Vol 6: Bernasconi CF (ed) , part 1, p 13, Wiley, New York, and literature therein

12 Mezey PG (1980) Theoret Chim Acta 54:95 13 Havlas Z, Zahradnik R (1984) Int J Quantum Chern 26:607 14 Fukui K (1970) J Phys Chern 74:4161

27

15a Tachibana A, Fukui K (1979) Theoret Chim Acta 51:189

b Ruf AR, Miller WH (1988) J Chem Soc (Faraday Trans 2) 84:1523

16a Pechukas P (1976) J Chem Phys 64:1516;

b Tachibana A, Okazaki I, Koizumi M, Hori K, Yamabe T (1985) J Am Chem Soc 107:1190;

c Tachibana A, Fueno H, Yamabe T (1986) J Amer Chem Soc 108:4346

17 for instance: a panci~ P (1977) Collect Czech Chem Commun 42:16;

b Banerjee A, Adams N, Simons J (1985) J Phys Chem 89:52;

c Baker J (1986) J Comput Chem 7:385;

d Kliesch W, Schenk K, Heidrich D, Dachsel H (1988) J Comput Chem 9:810;

e Basilevski MV, Shamov AG (1981) Chem Phys 60:347

18a Basilevsky MV (1982) Chem Phys 67:337;

b Hoffman DK, Nord RS, Ruedenberg K (1986) Theor Chim Acta 69:265;

c Quapp W (1989) Theor Chim Acta 75:447

19 Baker J, Gill PMW (1988) J Comput Chem 9:465

20a Hartke B, Manz J (1988) J Amer Chem Soc 110:3063;

b survey for describing tunnelling paths: Schatz'GC (1987) Chem Rev 87:81

21 Parr CA, Polanyi JC, Wong WH (1973), J Chem Phys 58:5

22a Page M, McIver J (1988) J Chem Phys 88:922;

b Gonzales C, Schlegel HB (1990) J Chem Phys 90:2154 and J Phys Chem 94:5523;

c Page M, Doubleday Ch, McIver JW,Jr (1990) J Chem Phys 93:5634

23 Wilson EB, Decius JC, Cross PC (1955) Molecular vibrations, McGraw-Hill, New York

24a Quapp W, Heidrich D (1984) Theoret Chim Acta 66:245;

b Quapp W, Dachsel H, Heidrich D (1990) J Molec Struct 205:245

25 Schlegel HB (1987) in: Lawley KP (ed) Ab initio Methods in Quantum Chemistry-I, Advances in Chemical Physics, Vol 68, Prigogine I, Rice SA (eds) wiley, New York, a) pp 253,278

26 Walet NR, Klein A, Do Dang G (1989) J Chem Phys 91:2848

27 Truhlar DG, Hase WL, Hynes JT (1983) J Phys Chem 87:2664

28 Troe J (1988) Ber Bunsenges Phys Chem 92:243

29a Truhlar DG, Garrett BC (1980) Acc Chem Res 13:440;

b Truhlar DG, Garrett BC (1984) Ann Rev Phys Chem 36:159

30 Marcus RA (1988) JCS Farad Trans 2 84:1237

31 Truhlar DG, Steckler R, Gordon MS (1987) Chem Rev 87:217, and references therein

32 Tucker SC, Truhlar DG (1989) in: Bertran J, Csizmadia (eds) New Theoretical Concepts for Understanding Organic Reactions, Kluwer Acad Publ, Dordrecht, NATO ASI series, Vol 267 (Series C) p 291

33 Song K, Chesnavich WJ (1989) J Chem Phys 91:4664 and references therein

28

34 Marcus RA (1968) J Chem Phys 49:2610; Miller WH, Handy NC, Adams JE (1980) J Chem Phys 72:99; Morokuma K, Kato S (1981) in: Truhlar DG (ed) Potential Energy Surfaces and Dynamics calculations, Plenum, New York

35 Miller WH (1986) in: Clary DC (ed) The Theory of Chemical Reaction Dynamics, Reidel, Dordrecht, p 27

36 Basilevsky MV (1977) Chem Phys 24:81 37 Further references in Refs 31,38 38 Colwell SM (1988) Theor Chim Acta 74:123 39 Ryaboy VM (1989) ChemCPhys Letters 159:371 40 Miller WH, Ruf, BA, Tyng Y (1988) J Chem Phys 89:6298 41 Shida N, Barbara PF, Alml f JE (1989) J Chem Phys 99:4061 42 Kato S, Fukui K (1976) J Amer Chem Soc 98:6395 43 cf. for instance: Nikitin EE, ZUlicke L (1978) Theory of Chemical

Elementary Processes, Lecture Notes in chemistry, Vol 8, Springer­verlag, Berlin, New York; Urena AG (1987) in:Prigogine I, Rice SA (eds) Advances in Chemical Physics, Vol LXVI, wiley, New York

44a Polanyi JC, Prisant MG, wright JS (1987) J Phys Chem 91:4727; b Polanyi JC (1987) Science 236:689; c Brooks PR (1988) Chem Rev 88:407; d Pollak E, Schlier Ch (1989) Acc Chem Res 22:223; e Gomez Lorente JM, Pollak E (1989) J Chem Phys 90:5406; f Metz RB, Kitsopoulos T, Weaver A, Neumark DM (1990) J Phys Chem

94:2240; g Zeweil AH, Bernstein RB (1988) Chem Eng News 66(45):24; h Schatz GC (1990) J Phys Chem 94:6157

45 For further textbooks or surveys see for example: a Pritchard HO (1984) The Quantum Theory of Unimolecular Reactions,

Cambridge University Press, cambridge; b Robinson PJ, Holbrook KA (1972) Unimolecular Reactions, Wiley­

Interscience, London, New York, Sydney, Toronto; cHase WL (1983) Acc Chem Res 16:258

46 Kramers HA (1940) Physica 7:284 47 Hynes JT (1985) in: Baer M (ed) Theory of Chemical Reaction

Dynamics, CRC Press, Boca Raton FL, Vol 4, P 171 48 Bertran J (1989) in: Bertran J, Csizmadia IG (eds)

New Theoretical Concepts for Understanding Organic Reactions, Kluwer Acad Publ, Dordrecht, NATO ASI Series, Vol 267 (Series C), p 231

49 Tucker SC, Truhlar DG (1989) in: ibid, p 331 50 Overview by: Bernasconi CF (1987) Acc Chem Res 20:301 51a cf. Hehre WJ, Radom L, Schleyer PvR, Pople JA (1986) Ab initio MO

Theory, wiley, New York b Kutzelnigg W (1988) in: J Mol Struct (Theochem) Present and Future

Trends in Quantum Chemical Calculations, 181:33-54

29

52 Simons J (1991) J Phys Chem 95:1017

53 Altona C, Faber DH (1974) Topics in CUrrent Chem, Vol 45,

Springer Verlag, Berlin;

Burkert U, Allinger NL (1982) Molecular Mechanics, ACS Monograph 177, Am Chem Soc, Washington;

Clark T (1985) A Handbook of computational Chemistry, wiley­Interscience, New York (Chap. 2 and Refs therein);

Rasmussen K (1985) Potential Energy Functions in Conformational Analysis (Lecture Notes in Chemistry, Vol 37) Springer Verlag, Berlin;

Allinger NL et al. (1989) J Am Chem Soc 111:8551,8566,8576; (1990) 112:8293,8307

54 Ermer 0 (1981) Aspekte von Kraftfeldrechnungen, Wolfgang-Baur-Ver­lag, MUnchen

55 cf. the short and clear survey by Pulay P (1987) in: Lawley KP (ed) Ab initio Methods in Quantum Chemistry-II, Advances in Chemical Physics, Vol 69, Prigogine I, Rice SA (eds), Wiley, New York

56a Bartlett RJ, Silver MD (1975) J Chem Phys 62:325

b Pople JA, Binkley JS, Seeger R (1976) Int J Quantum Chem, Symp 10:1

57 Meyer W (1973) J Chem Phys 58:1017,

58 Ahlrichs R, Lischka H, Staemmler V, Kutzelnigg W (1975)J Chem Phys 62:1225

59 ~izek J (1966) J Chem Phys 45:4256

60a Davidson ER, Feller D (1986) Chem Rev 86:681

b ~arsky P, Urban M (1980) Ab initio Calculations (Lecture Notes in Chemistry, Vol 16), Springer Verlag, Berlin

61 Del Bene JE (1986) in: Liebmann JF, Greenberg A (eds) Molecular Structure and Energetics, Vol 1, VCH Publ., P 319

62 Sauer J (1989) Chem Rev 89:199 (prologue)

63 spitznagel GW, Clark T, Chandrasekhar J, Schleyer PvR (1982) J Comput Chem 3:363

64 Hiberty PC, Lefour JM (1987) J Chim Phys et Phys Chim BioI 84:607

65 Cooper DL, Gerratt J, Raimondi M (1987) in: Lawley KP (ed) Ab ini­tio Methods in Quantum Chemistry-II, Advances in Chemical Physics, Vol 69, Prigogine I, Rice SA (eds) Wiley, New York

66 Van Lenthe JH, van Duijneveldt-van de Rijdt JGCM, van Duijneveveldt FB (1987) ibid.

67 Hobza P, Zahradnik R (1988) Chem Rev 88:871

68 Chalasinski G, Gutowski M (1988) Chem Rev 88:943

69 Pulay P (1970) Dissertation, Stuttgart

70 Pulay P (1977) in: Schaefer HF (ed) Applications of Electronic Structure Theory, Plenum, New York, p 153

71 McIver JW, Komornicki A (1971) Chem Phys Lett 10:303

72 McIver JW, Komornicki,A (1972) J Am Chem Soc 94:2625

30

73 Rinaldi D, Rivail J-L (1972) C R Acad Sci 274:1664; Pulay P, Torok F (1973) Mol Phys 25:1153; pancl~ J (1973) Theor Chim Acta 29:21; Grimmer M, Heidrich D (1973) Z Chem 13:356;

74 Komornicki A, Ishida K, Morokuma K, Ditchfield R, Conrad M (1977) Chem Phys Lett 45:595

75 Schlegel HB (1975) Ph D thesis, Queens University, Kingston, ontario, Canada

76 Pople JA, Krishnan R, Schlegel HB, Binkley JS (1979) Int J Quantum Chem Symp 13:225

77 Pople JA, Schlegel HB, Krishnan R, DeFrees DJ, Binkley JS, Frisch MJ, Whiteside RA, Hout RJ, Hehre WJ (1981) Int J Quantum Chem Symp 15:269

78 For reference see for instance:

Jorgensen P, Simons J (eds) (1986) Geometrical Derivatives of Energy Surfaces and Molecular Properties, Reidel, Dordrecht; Bernardi F, Robb MA (1987) in: Lawley KP (ed) Advances in Chemical Physics, Vol 67/1, wiley-Interscience, p 155; Amos RD, Gaw JF, Handy, NC, Carter S (1988) JCS, Faraday Trans 2, 84:1247; Amos RD, Rice JE (1988) Program CADPAC: The Cambridge Analytic Derivatives Package, University of Cambridge, Cambridge, England; Dupuis M, Watts JD, Villar HO, Hurst GJB (1987) Program HONDO: Version 7.0; Frisch MJ, Head-Gordon M, Schlegel HB, Raghavachari K, Binkley JS, Gonzalez C, DeFrees D, Fox DJ, Whiteside RA, Seeger R, Melius CF, Kahn LR, Stewart JJP, Fluder EM, Topiol S, Pople JA (1988) Program GAUSSIAN 88; Gaussian, Inc: Pittsburgh PA; Ahlrichs R, Bar M, Ehrig M, Haser M, Horn H, Kolmel Ch (1989) Program TURBOMOLE, Version 2.0 Beta, Universitat Karlsruhe (suited for workstations) and other ones, cf. also: Clementi E (ed) (1990) Modern Techniques in ~omputational ~hemist­ry: MOTECC-90, ESCOM, Leiden

79 cf.Osamura (1989) Theoret Chim Acta 76:113, and references therein 80 Bingham RC, Dewar MJS, Lo DH (1975) J Am Chem Soc 97:1285,1294 81 Dewar MJS, Thiel W (1977) J Am Chem Soc 99:4899,4907 82 Dewar MJS, Zoebisch EG, Healy EF, Steward JP (1985) J Am Chem Soc

107:3902 83 see for instance: Dewar MJS, Ford GP (1977) J Am Chem Soc 99:1685 84 see for instance: Dewar MJS, Ford GP (1977) J Am Chem Soc 99:7822 85 Madura JD, Pettitt BM, Mc Cammon JA (1987) Chem Phys Letters

141:83, (1989) Chem Phys 129:185 86 Jorgensen WL (1989) Acc Chem Res 22:184

2 ANALYSIS OF MULTIDIMENSIONAL POTENTIAL ENERGY SURFACES

- STATIONARY AND CRITICAL POINTS -

As outlined in the introducing chapter, the minimizers and the saddle

points of index one of the energy functional are corner-stones of most

reaction theories in chemistry. Thus the applicability of these

theories strongly depends on the availability of mathematical methods,

which compute such points in an effective manner. Therefore the

current chapter is engaged in the presentation of methods, which

enable to compute minim~ and/or saddles of a PES.

Since the scientist will always be in a better position to use

numerical techniques and software effectively if he understands some

of the basic theoretical background, the numerical procedures

considered in this chapter are not treated free of mathematical

context as it is usually done in text books about theoretical

chemistry, but rather they are imbedded in a mathematical framework.

In this sense the chapter is started by an exploration of some

mathematical quantities that describe geometrical features of a PES.

Properties that allow to classify the stationary points (i.e. points

at which the gradient vanishes) of an energy functional are also

given. The chapter is finished by some considerations pertaining to

the critical points (i. e. points at which the determinant of the

Hessian matrix vanishes) of energy functionals.

The mathematical background material presumed for the understanding of

the current chapter corresponds to that of a basic course in

mathematics and can be found in any text book about real analysis1 .

2.1 Basic Definitions and Notations

We are dealing with a molecular system consisting of N atoms which

follows the Born-Oppenheimer approximation. Such a system possesses in

general a well-defined structure and can be completely described by

the cartesian coordinates

i=l(l)N,

of the nuclei of all N atoms. The index i refers to the number of the

nucleus. By collecting all the coordinates of the nuclei, a 3N-tupel

32

(1)

is obtained, which characterizes the molecular system as a whole. All

these n-tupels, n=3N, are elements of the n-dimensional vector space

~n. Since the coordinates ~i' ~i' Ci are assigned to the components

of the vector of Eq.(l) in a unique manner, the system can be uniquely

reconstructed from that vector.

When we choose the n-dimensional Euclidian space En (cf. Ref.2), which

is associated with the vector space ~n, as the configuration space,

then the possible nuclear arrangements of a molecular system may be

identified with the points of En, and the n-tupels of Eq. (1) are

position vectors that describe the points of En with respect to the

chosen Cartesian coordinate system. The forces acting on the nuclei of

the system may be identified with the vectors of ~n. So we have always

to distinguish between position vectors that define a point of En (we

shall call it points) and vectors that describe a force or a

displacement.

The vector space ~n may be equipped with the inner product

n <xly>:= E xiYi = xTy

i=l

(":=" means equal per definition), where xi and Yi denote the i-th

component of the vectors x and y, respectively, and the norm

IIxll :=..; <xix> •

Sometimes it is useful to restrict the freedom of movement of some

nuclei of a molecular system. In such a situation the possible

arrangements of the nuclei correspond to points of a Euclidian space

En with n<3N. Therefore, in the following the number n indicates an

"arbitrary" dimension, which is chosen in accordance with the problem

under consideration.

Since not all points of En correspond to chemically meaningful

arrangements, we shall confine always to domains V of ~n which do not

contain vectors that match "pathological" nuclear arrangements.

The energy functional

E: V ~ ~n .. ~,

which provides the potential energy for the nuclear arrangements of

the molecular system, may always be twice continuously differentiable.

The set

r E := {(x,E(x» I xeV } ~ ~nx ~

is called the graph of E. It is the mathematically correct description

of a PES. Note that a point of a PES is completely described only by

33

the pair (x,E(x». But since for each nuclear arrangement x the value E(x) is uniquely determined, a point of a PES is sufficiently characterized if only the nuclear arrangement is given. We take the following notational conventions: Points of En and vectors of IRn are denoted by small italic and bold-faced Roman letters, respectively. The components of a vector are always consecutively numbered by a lower index and are marked by small Roman letters,

T x = (x1 ,x2 , ••• , X n) .

The upper index T (or T) indicates the transposition. The relation between the (~, 11, C) -coordinates of the nuclei of a molecular system and the components xi of the vector x is evident by Eg.(l). Scalars are marked by small Greek letters. For matrices we shall employ bold-faced Roman capital letters. Identity matrices are always denoted by I.

2.2 Geometrical Properties of PES

Subject of this section are the (local) geometrical features of a PES that are related to the first and second partial derivatives of the energy functional. The vector of the first partial derivatives of an energy functional E at a point xe2) is called the gradient of E at x and is denoted by grad E(x) (or VE(x»,

grad E (x) : = ( a:~:) , BE(x) )T • , aXn •

By assigning to each point xe2) the vector grad E(X), a function

g(x) = grad E(x),

is obtained which is called the gradient field of E (the notion is due to the geometrical representation). Frequently (two-dimensional) PES are visualized by the level surfaces of E, i.e. the sets

~(~) := {xelRn I E(x)=~}, ~elR.

The corresponding gradient field may be an alternative hereto.

THE 0 R E K ~.3 At regular points of E (i. e. the gradient does not vanish at that points), the vector field -grad E(x) is perpendicular to the level surfaces of E.

34

In lEn a straight line through a point x (defined by the position

vector x) is described by the linear function s: ~ ~ ~n,

dE~n and IIdll=l.

The vector d is the directional vector, which determines the straight

line. All functions s which differ only in the position vector x

generate identical or parallel straight lines. For the derivative of

the function E(s(·» with respect to ~, i.e. the derivative of E at x

along d, the relation

(2)

holds by the chain rule. In particular we obtain

The right hand term is the directional derivative of E at x along d,

which we denote by 8E(x)/8d. Higher directional derivatives are

defined in the same manner.

If d is equal to the i -th uni t vector u i of ~n , i. e. i T u =(0,0, ••• ,0,1,0, ••• ,0) (only the i-th component differs from zero),

then

8E(X)/8Ui = 8E(X)/8Xi •

The importance of the directional derivative 8E(x)/8d consists in that

it is a measure for the slope of a PES along d in a small vicinity of

the point (x,E(x». It is easy to see that

8E(x)/8d = <g(x) Id> = IIg(x)1I cos {3,

where (3 is the angle between the normalized gradient vector

grad E(x)

IIgrad E(x) II

and the vector d. Since

18E(X)/8dl ~ IIg(x)1I

(V means "for all") and A

8E(X)/8gx = IIg(X)II, A

the vector gx (-gx) indicates the direction of steepest ascent (descent) of a PES at a point (x,E(x». Therefore, -g(x) is also

called steepest descent vector. Another important quantity for PES analysis the curvature of a PES at a given point is related to the eigenvalues of the Hessian

35

matrix H(x), i.e. the matrix of the second partial derivatives of E,

H(x):=

a2E (x) a2E (x)

axnax1 axn ax2

at a point xeD. The Hessian matrix is always symmetric,l i.e.

H(X)=H(X)T, and the eigenvalues are always real numbers. 4 If H(x) is

positive (negative) definite, i.e. all eigenvalues of H(x) are

positive (negative), then E is convex (concave) at x.

The curvature of E at a point xeD along d is again defined by the

function E(s(·». By deriving Eq.(2) we obtain

d 2E(S("»

Hence,

The second directional derivative of E (left hand term) indicates the

curvature of a PES along d in the vicinity of a point (x,E(x». If the

directional derivative a2E(x)/ad2 is positive (negative), then we say

the functional E is convex (concave) along d (at x). If d is an

eigenvector of H(x), then a2E(x) /ad2 is equal to the corresponding

eigenvalue. In that case d is called main direction of curvature.

Generally it holds

n r Ai<dlei >.

i=1

where e i is the eigenvector pertaining to the eigenvalue Ai of H(x).

2.3 Stationary Points

The chemically most interesting arrangements of nuclei correspond to

the stationary points xst of the energy functional E (see Table 1) ,

which are defined by the condition

grad E(Xst) = o. (3)

Hence, each stationary point of E is a solution of the nonlinear

36

equation

g(x) = O. (4)

When a gradient vector g(x) is decomposed according to Eq.(l) we get

the forces that act on the nuclei of the molecular system. In these

terms, Eq. (3) means that the forces (the "force vector") vanish (es) at

stationary points. The solution set of Eq.(4) is denoted by g-l(O),

g-l(O) := {xeV I g(x)=O}.

This notational conventiop will also be employed for other functions.

A stationary point can be classified by means of the points of its

neighborhoods.

If x* is an arbitrary point of ~n, then the point set

'I1(x* ,e) := {xe~nl IIx-x*lI<e}

is the (e-)neighborhood of x*.

We are able to distinguish three types of stationary points:

.. xst is a (local) minimizer of E if there is an e>O such that

E(Xst) ~ E(x) for all xe'l1(xst ,e).

.. xst is a saddle point of E if for each e>O there are points x, y of

'11 (Xst , e) with E(Xst)~E(X) and E(Xst»E(y) •

.. xst is a (local) maximizer of E if xst is a local minimizer of -E .

THEOREM

Let x st be a

eigenvalues of

2. 5 ,6

stationary H(Xst) .

point of E and let

(i) If xst is a local minimizer of E then Ai~ 0, i=l(l)n.

be

(ii) If all Ai are positive then xst is an isolated minimizer of E.

(iii) If there is at least one negative eigenvalue Aj then xst is a

saddle point of E.

the

since the potential energy of a molecular system is invariant with

respect to translations and/or rotations of the whole system, the

spectrum of the Hessian matrix possesses always six (five in the

linear case) zero eigenvalues at a stationary point. Therefore,

theorem 2 cannot be employed in deciding the type of x st , when all

eigenvalues of H(Xst ) are non-negative.

The saddle points may be classified according to the number of the

negative eigenvalues of the Hessian matrix. A stationary point xst is

37

Table 1. The characteristic of stationary pOintsa of a PES

xst chemical meaning, relevance

MIN STABLE SYSTEM, energy increase with respect to all that

displacements of Rn which do not correspond to a translation

and/or a rotation of the whole system:

Reactants, products, reactive intermediates conformers/isomer~ etc.

and their

SPl INSTABLE SYSTEM, energy decrease with respect to the

displacements parallel to the eigenvector pertaining to the

negative eigenvalue of the Hessian matrix at x SP:

Transition structure in TST and related theories

SP2 INSTABLE SYSTEM, energy decrease with respect to dis­placements of the two-dimensional subspace p2(xsp)~Rn.

Relevance

- for TST and related theories when it can be interpreted as a superposition of two transition structures (see Ref.7); virtual SP2 (V-SP2)

- beyond TST when it can be interpreted as a hilltop point

in the subspace p2 indicating a region with low proba­bility for trajectories

SP3 INSTABLE SYSTEM, energy decrease with respect to the

displacements of the three-dimensional subspace p3(Xsp)~Rn.

Relevance

- for TST and related theories when it occurs as a virtual SP3 (V-SP3) (see Ref.7) etc.

MAX INSTABLE SYSTEM, energy decrease with respect to all that

displacements of Rn which do not correspond to a translation and/or a rotation of the whole system.

a zero eigenvalues of the Hessian (not due to overall rotation and translation) are not considered in this table.

38

called a saddle point of index v if the matrix H(Xst) possesses v

negative eigenvalues, counted according to the multiplicity. The index v is equal to the number of the imaginary frequencies of the corresponding transition structure. The subspace of Rn generated by the eigenvectors pertaining to the v negative eigenvalues of H(XsP) is denoted by pV(xsp). Its dimension is equal to v. We shall use the abbreviations:

MIH for minimizers of E, SP(v) for saddle points (of index v) of E and MAX for maximizers of E.

The meaning of the stationary points for reaction theories and PES studies is outlined in Table 1.

2.4 Location of Stationary Points

This section is concerned with some numerical methods which are of interest when searching for minimizers and/or saddle points of an energy functional. A series of recent papersS- 10 reviews the state of art in computing stationary points. So we want to focus our attention on some well-working Newton-like methods, which also form a basis for further developments, and on a procedure proposed by Zirilli et al. 11 The latter one may be regarded as a global Newton-like method. These methods have been mathematically thoroughly investigated so that substantial assertions can be given. As many Newton-like methods are local procedures, i.e. they are only defined within a certain neighborhood of a stationary point of E, we confine the subsequent considerations to an open and convex subset Do~D (an open subset does not contain its boundary points) so that the following assumptions are fulfilled:

(Al) The energy functional E is twice continuously differentiable on Do.

(A2) There is a stationary point xst belonging, to Do with det H (Xst) .. o.

(A3) There is a constant yalue K such that IIH(X) - H(Xst) II ~ KIIX_Xstll V xeDo •

These assumptions are basic for the procedures considered in th,is section. Since Do is convex, the conditions (Al) and (A2) ensure the

39

uniqueness of xst in D • The condition (A3) (Lipschitz continuity of o st the Hessian matrix at x ) is necessary to ensure some convergence

results. An energy functional defined on R3N (overall translations/rotations of the molecular system are admited) can never satisfy the assumption (A2) because of the invariance of the electronical energy of a molecular system against overall translations and rotations of the system. However, when in the (3 -dimensional) (~ , 11 , l;) -coordinate system one nucleus is fixed at the origin, a further one at the ~-axis and a third one at the (~,lI)-plane, then the invariance effects can be avoided, and the assumption (A2) will be fulfilled in general. But note, the energy functional E is changed to a functional E*: R3N- 6 --+ R (E*: R3N- 5 --+ R in the linear case). However, the stationary points of E* correspond to those of E (see Ref. 12), so that, on principle, no difficulties arise if E* is used instead of E. When the assumption (A2) is not fulfilled, a procedure will not necessarily fail, but a well outcome cannot be guaranteed. We start with a short discussion on the both ways that allow a determination of stationary points. Firstly, stationary points can be determined by solving Eq. (4). But, since the solution set of this equation contains the minimizers, the maximizers as well as the saddle points of E, an additional examination is necessary to determine the type of the solution when an unspecific (Newton-like) method is used, i.e. a method which does not look for a certain type of stationary points in a well-aimed manner. The specific Newton-like methods are able to locate either only minimizers or only saddle points by considering the directions of energy decrease (descent methods) or by preserving the number of the negative eigenvalues of some matrices (see Sect. 2.4.4), respectively. The former class is well investigated whereas the knowledge about the,latter one is very rare. Alternatively, one can use an auxiliary functional, namely the defect

functional

(5)

instead of E to locate the stationary points of E. This way relies on the observation that by Eq.(3) and the positivity of u, i.e.

u(x) ~ 0 V xeD,

each stationary point of E is a (global) minimizer of u (see Fig.1). In particular the computation of saddle points becomes simpler by this way (at least for many cases), because minimizers may be located by descent methods, which· - in contrast to the quasi-Newton methods -

40

even still work when a poor initial guess is only available. But,

there is also a serious disadvantage. The functional u- may possess

minimizers which do not correspond to stationary points of E.

Furthermore, one has no possibility to look in a well-aimed manner for

minimizers of u- that correspond to a certain type of stationary points

of E. Therefore, whenever the functional u- is used to locate a saddle

point (or a minimizer) of E, it is necessary to check subsequently

whether the obtained minimizer is a saddle point (or a minimizer) of E

indeed.

,. I \ I \ I \ ~6' I \

~

I I \ I

X \ ?-g \ ..... ~/

Fig.1. A "u--function" and the underlaying energy functional E (one-dimensional case)

In 1972 the "u--method" has been used by McIver and Komornicki 13 for

the first time to locate saddle points. Today saddle points are

frequently computed in this ·way.

The minimizers of u- can be computed by solving the nonlinear equation

grad u-(x) = o. (6)

For characterizing the solution set of Eq. (6) a further notion is

needed.

The kernel of a matrix H, denoted by ker H, is the set of all vectors

of ~n which are mapped on the null vector by H,

ker H := {xe~n I Hx=O}.

If H is regular, then ker H only contains the null vector. In other

words, only singular matrices have a non-trivial kernel. The dimension

of the kernel of a singular matrix H is equal to the multiplicity of

41

the zero eigenvalue of M.

Proposition 1.

Let x* be a solution of Eq. (6).

(i) If H(x*) is regular, then x* is a stationary point of E.

(ii) If g(x*)*O, then H(x*) is singular, and g(x*)e ker H(x*).

The statements are simple consequences of the equation

grad u(x) = H(x)g(x) , (7)

which is obtained by deriving Eq.(5).

When searching for stationary points of E by the u-method, the

gradient must always be checked if no nucleus of the considered

molecular system has been frozen. This is a consequence of

proposition 1.

The elements of the set

-1 Vinf(E) := ~xeV\g (0) I g(x)eker H(x) ~

are the inflection points of E.

Proposition 2.

(i) Each minimizer x* of u which is not a stationary point of E, i.e. u(x*)*O, is an inflection point of E.

(ii) Each saddle point of u is an inflection point of E.

Proof: statement (i) is a simple consequence of Eq.(7). For a saddle

point x* of u always the relation u(x*»O holds. Hence, g(x*)*O. On

the other hand,

H(x*)g(x*) = grad u(x*) = O.

This means g(x*)e ker H(x*). Consequently, x*eV. f • ~n

2.4.1 The Newton Process and its Modifications

This sUbsection is engaged in a unified approach to the Newton-like

methods.

A functional E can locally be approximated, i.e. on a certain

neighborhood U(y,o) of a point yeV, by means of the model function

'liy(P) <g(y) Ip> + ~ pTH(Y)p, lip 11::50 , (8)

such that

42

Vxe'U(y,6) •

Obviously, tl;le function E; is obtained when E is expanded into a Taylor series on a neighborhood of y and the terms of third and higher order are neglected. The stationary points of E; are determined alone by the model function ~y' The vector p = x-y is a stationary point of ~y if and only if it is a solution of the equation

qrad ~y(p) = g(y) + H(y)p = o. (9)

(Recall, p is a stationary point of ~y if the first derivative of ~y vanishes at p.) Hence, if H(y) is regular, then p = _H(y)-lg(y) is the only stationary point of ~y' If the functional E is approximated by E* sufficiently well in the vicinity of a stationary point xst of E [Le. ye'U(xst ,6», then the point x = y+p can be employed as an estimate to xst. When this procedure is applied to x (instead of y), then an improved estimate to xst may be obtained. This is the basic idea of the classical Newton process:

k+l p

} k=O,1,2, •••

The vector pk is called the Newton vector.

(lOa)

(lOb)

If the vector veRn is different from, zero, then En can be decomposed by the subsets

!Ix (v) := {yeEn <vly-x> = O}

n;(v) := {yeEn <vly-x> > O}

n;(v) := {yeEn <vly-x> < O},

Iv n;(v)

x !I~(V)

n;(V)

Fig. 2. Decomposition of En

43

The arbitrary (but fixed) vector v is perpendicular to the hyperplane + - n .

~ (v). The subsets n (v) and n (v) are half-spaces of E (w1th respect x x x n

to the hyperplane ~ (v», see Fig. 2. Obviously, all vectors of IR x which are affixed to x and point into the half-space n;(v) (n;(v»

form an angle of less than 90 degrees with v (-v).

Proposition 3.

Suppose H(x) is regular and g(x)*O.

(i) If at the point x the energy functional E is convex (concave)

along g(x), then the Newton vector p and the steepest descent vector

point into the same (opposite) half-space(s) vith respect to the

hyperplane ~x(g(x».

(ii) If e

(negative)

is an eigenvector of H(x) pertaining to the positive

eigenvalue A, then the Nevton vector and the steepest

descent vector point into the same (opposite) half-space(s) vith

respect to ~he hyperplane ~x(e).

Proof: (i): This property is a consequence of the relation

<-g(x)lp > = <g(x) IH(X)-lg (X».

(ii): It holds

<elp> = -<eIH(X)-lg (X» -1 -A <elg(x»

Hence,

A<P I e>=<-g(x) Ie>.

By discussing the cases A>O and A<O, statement (ii) is obtained _

Remark: The property (ii) is the mathematically correct formulation of

a statement given by Simons et al. 14 •

As a consequence of proposition 3 we obtain: In the vicinity of a

minimizer of E the Newton vectors and the steepest descent vectors

always point to the minimizer. In the vicinity of a saddle point the

Newton vectors always point to the saddle point whereas a steepest

descent vector point to the saddle point only if E is convex along

that vector. This observation forms the basis for a modified

Newton-like method which looks for stationary points of prescribed

type (see Sect.2.4.3).

An important characteristic of the Newton process is the rate of

convergence. If {Xk} is a sequences of iterates converging to the

limit point xst, the rate of convergence is determined by means of the

44

estimation

11~+l_xstll .~ Lkllxk_xstllw, V k!:ko '

If the sequence {Xk} satisfies the above condition for some w>O and a sequence {Lk} of positive numbers, then its rate of convergence can be taken from Table 2.

Table 2. Rates of convergence

rate condition

linear· w=l,

superlinear w=1,

quadratic w=2,

superquadratic w=2,

sup Lk~~' ~<l kl!:k o

lim Lk=O k.",

An iterative process possesses some rate of convergence if any sequence of iterates generated by this process has this rate at least. The rate of convergence measures the improvement (numerical progress), which may be achieved by one iteration in some small vicinity of a stationary point. It is ~f some interest for practical purposes, since it influences the computational cost (among other quantities). Iterative processes with a superquadratic or higher rate of convergence do ~xist15, but they are of minor importance because of the large numerical effort.

THE 0 R E It 3. 5

If the assumptions (Ai) and (A2) are fulfilled, then there is an open set :Dl s;2)o such that for any xOe:Dl the Newton iterates (lOb) are veIl defined, remain in :Dl and converge superlinerarly to a stationary point xst of E. Itoreover, if additionally assumption (AJ) is true, the sequence of iterates converges quadratically to xst.

Hence, the classical Newton process converges very rapidly if the initial guess belongs to the domain of attraction of a stationary point or if an iterate falls into this domain. Exactly that property

45

has made the Newton process so attractive. Because the total running time primarily depends on the effort that is necessary to compute a new iterate, the numerical cost that is necessary to perform one iteration step is of interest additionally to the rate of convergence for practical purposes. In particular, an iterative process, which converges more slowly than the Newton process, may less time consuming than this one, whenever the iterates can cheaper be computed. Therefore the classical Newton process is primarily regarded as a model procedure. Useful and well-working algorithms are obtained by modifying this basic process in a suitable manner. During the last two decades the Newton process has been essentially improved in two directions:

- reduction of the numerical effort caused by the evaluation of the Hessian matrix and its inverse

- enlarging the domain of attraction.

According to the kind of modif ication we distinquish four types of Newton-like methods:

~ Quasi-Newton methods Quasi-Newton methods aim at the reduction of the numerical cost caused by the evaluation of the Hessian matrix. This is done by approximating the matrices H(xk) (or H(xk ) -1) by estimates ~ (~), which can cheaply be computed. update procedures (a few of them are treated in Sect. 2.4.2) modify a matrix ~-1 by adding a low-rank correction matrix Ck_1 so that the matrix ~ := ~-1 + Ck_1 may be utilized as an estimate to H(Xk). T~e matrix Ck- 1 is built up by quantities that have been used before, so that no sUbstantial cost is incurred additionally by the computation of the matrix ~. Some quasi-Newton methods are considered in detail in Sect. 2.4.3.

~ Descent methods As mentioned above, the Newton process is a local procedure, i. e. convergence occurs only if the initial quess Xo is chosen sufficiently closed to a stationary point (see theorem 3). When a minimizatio.n problem is under consideration, the applicability of the Newton method can essentially be extended if the characteristic of a minimizer is taken into account. If the iterate xk belongs to a domain D on which o E is convex (i.e. H(x) is positive definite for all'xeD ), then the o correction vector pk+1 (see Eq.(10a» indicates always a direction of descent (i.e. the energy decreases along pk+1) , and there is a real number Il so that the point xk+1=xk + Ilpk+1 satisfies the relation

E(Xk+1 ) <E(Xk ). Hence, xk+1 is 't min '" th' p01n x e .uo • In 1S W?y

into a neighborhood of xm1n

46

an improved estimate to a stationary

a sequence of estimates may be driven

on which the Newton method converges.

Descent methods are described in Sect. 2.4.4.

~ Trust region methods Since the model functions ~ (see Eq.(8», yeU(Xst ,8), do not possess

any minimizer if E is not c~nvex on U(Xst ,8) (i.e. the matrices H(y), st yeU(x ,8), possess at l,east one negative eigenvalue), the Newton

process will fail in this case, or it will converge to a saddle point

or maximizer of E. However, if ~y is restricted to the closure of some

neighborhood U(y,8*) of y, then there is a point p* that minimizes ~y

on the closure of U(y,8*). Trust region methods are adapted to this

situation. They utilize the solution of the constrained minimization

problem

minn(E*(Y+P) I II P II:S8*) (11) pelR

as correction vector in Eq. (lOb). In this way a sequence of points

with decreasing energy values is generated. If E is convex on U(y,8*),

then a solution p* of Eq.(9), which satisfies the condition IIp*II<8, is

also a solution to problem (11).5 A disadvantage of this method is

that the minimization problem (11) is harder to solve than the linear

Eq. (9). Furthermore, it is difficult to control the trust region

radius 8* in a numerically effective manner. The reader who is

interested in that methods is recommended to the Refs. 5,6,16-18.

~ Truncated Newton methods Methods of this type especially aim at the reduction of the numerical

cost due to the computation of the correction vector pk. They

determine pk by solving Eq.(9) iteratively within a given tolerance,

which depends on the distance between the estimate xk and the desired

stationary point xst. So, when looking for 'a minimizer, the vector pk

can be computed with low precision outside a certain neighborhood of

the minimizer and more accurate in the vicinity of the minimizer19 ,20.

Schlick and overton21 have given a truncated Newton method for the

energy minimization of molecular systems, which employs an

interesting, physically motivated decomposition of the Hessian matrix.

In practice the Newton-like methods do not occur in "pure" form.

Mostly effective procedures involve elements of several basic methods.

47

For example, descent methods use updates to estimate the Hessian

matrix.

Table 3. Requirements to the geometry of the PES and to the initial

guess XO for some Newton-like minimization procedures.

method requirement to

E XO

quasi-Newton convex on 'D o a x e'D1 0

descent convex on 'Do xOe'Do

trust region xOe'Do

a'D ~'D denotes the set on which the method is contractive. 1 0

Notice that the quasi-Newton methods, descent methods and trust region

methods form a hierarchy with respect to the goodness of the initial

guess XO when searching for a minimizer of E, see Table 3. Whenever XO

(or any iterate) falls into the subset 'D 1 on which the considered

descent method is contractive, then this method works like a (damped)

quasi-Newton method. A trust region method behaves as a descent method

if XO (or any iterate) falls into a domain 'Do on that E is convex.

If the Hessian matrix is approximated, then the model function (8)

must be reset by the more general model function

(12)

If the point x is an element of a sequence (i.e. x = Xk), we write ~k and ~. Model functions ~x, which pertain to different quasi-Newton

methods, only differ in the choice of the matrices Mk (or the method

by that they are computed).

Some of the properties concerning the model function ~x are summarized

in

Proposition 4.

(i) If Mx is a regular matrix, then ~x possesses a unique.ly determined stationary point p* which can be expressed as

-1 p* = -Mx g(x) .

(ii) If Mx is positive definite, then p* is a minimizer of ~x.

The vector p* is called the quasi-Newton vector at the point x. If

Mx=H(X), then p* is the Newton vector. If Mx=I then p* corresponds to

48

the steepest descent vector. We want to discuss both extreme cases in

more detail now. If the minimizer p* of each function iltk is equal to the steepest

descent vector, we obtain the steepest descent method,

xk+1 = xk _ g(Xk ), k=1,2, •••

which neglects the second order information (no use of the Hessian

matrix). The numerical effort required by the gradient method is low,

but the rate of convergence is only linear. On the other hand, when p*

is always the Newton vector, the classical Newton process

xk+1 = xk _ H(Xk)-lg(Xk ), k=1,2, •••

is obtained, which possesses an excellent local behaviour of

convergence (see theorem 3), but the numerical effort is high, because

the evaluation of the Hessian matrix is a very time-consuming

operation. Obviously, there is a strong relation between the rate of

convergence and the utilized amount of the information provided by the

Hessian matrix. since the frequent evaluation of the Hessian matrix is

not acceptable in computational chemistry, we should aspire a

compromise between the numerical effort and the quadratic rate of

convergence. A class of methods based on such a compromise is the

class of the quasi-Newton methods. Before we are able to consider it

in detail, we have to take up the question: How can a Hessian matrix

be approximated in a numerically effective manner?

2.4.2 Update Methods

As mentioned above, update formulae have been developed to avoid a

frequent evaluation of the Hessian matrix. They modify a given matrix

by using quantities which have been employed before in the

procedure (gradient and quasi-Newton vector). If a matrix Mo

(approximating H(Xo » is available, then by an update formula a

sequence of matrices {Mk} may be generated which can be used instead

of the sequence {H(Xk )} in the Newton process given by Eqs.(lO).

Since at each point xk the Hessian matrix H(x) is uniquely defined by

the relation1

lim 119(xk+h)-g(xk ) - H(xk)hll = 0, IIhll,.O

it is reasonable to require that an estimate Mk to H(xk ) satisfies the

secant equation (quasi-Newton condition)

(13)

49

Note that the matrix Mk is not uniquely determined by Eq.(13).

Apart from the quasi-Newton condition we have no further information

about Mk • Thus we should preserve as much as possible of what we

already have. This is done by choosing Mk as the solution of the

cons~rained minimization problem

min {1I~-1 - MIIF I MeA}, (14)

where A - the set of admissible matrices - is a subset of

M(p,q):= {Mef(Rn ) I Mp ='q}

and feRn} is the set of all real (n,n}-matrices. lIollF denotes the

Frobenius norm4 The set A must be specified according to

circumstances.

A formula (procedure), which modifies a given matrix so that the new

matrix is a solution of the constrained minimization problem (14), is

called a least-change update. The term "update" will be clear when we

consider some updates in detail.

For simplicity we employ the notations M+ and Mc (x+ and xc) instead k k-1 of ~ and Mk _1 (x and x ) for the subsequent considerations.

Furthermore, we define

p = x+-xc and q = g(x+)-g(xc }.

The general single-rank update22 , which satisfies the secant

condition, is given by the formula

(q-McP)VT M+= M + ,

c <vip> <v I p> .. O, veRn. (lS)

For all vectors Z perpendicular to v, the equality M+Z = McZ holds.

Notice, the correction matrix (second right-hand term) in Eq.(15) is

generated by the dyadic product of the vectors v and (q-McP). The

dyadic product of two vectors v,weRn is the matrix

The i-th column of the matrix vwT is the vector v multiplied by the

i-th component of the vector w. Hence vwT is a rank-one matrix.

From Eq. (15) two important special cases may be derived. By setting

v=p the well-known Broyden formula 23

(q-McP)pT M = M + --~~--+ c <pip>

is obtained. If v = q-Mcp is

single rank update24 ,25

(16)

set, Eq. (15) results in the symmetric

50

(17)

which is known as Hurtagh-Sargent (HS-) Update26 in the quantum

chemical literature. For long time it has been employed in PES

analysis. Clearly, the MS-update is symmetric only if the matrix Mc is

symmetric.

THE 0 R E H 4. 6 ,22

The matrix M+ may be defined by Eq.(16).

(i) If p*O and A=M(p,q), then the matrix M+ is the unique solution to

the minimization problem (14).

(ii) Let Vcs~n be an open and convex set containing the points Xc and

x+ with xc*Xst If H(x) is Lipschitz continuous at each point xeVc '

i. e.

IIH (x) -H (y) II s -rllx-yll V yeV c'

then

(18)

The inequality (18) means 'that if the approximation M+ to H(X+) gets

worse, then it is done in a controlled way. Only in that sense an

updated matrix can be regarded as an approximation to the Hessian

matrix.

The MS-update (see Eq. (17» is not a least-change update, Le. it

cannot be obtained as a solution of the constrained minimization

problem (14), but it belongs to another important class of updates,

namely to the Broyden's class (see below). A symmetric least-change

update must be at least of rank two.

Since a quasi-Newton procedure requires the inverse of the matrices Mk

(see procedure I below), a matrix computed by one of the update

formulae given above cannot directly be employed in such a procedure.

But the Sherman-Morris on-Woodbury formula (see lemma 1) enables to -1 invert an updated matrix M+ in a simple way if the matrix Mc is

known.

L e m m a 1. 6

Let v,we~n, and assume that the matrix Mc is regular.

regular if and only if

Then M +WVT is c

51

T -1 '1 := 1 + v MC W '* o.

Furthermore,

(M + wvT)-l = M-1 _.1 -1 T-1 c c '1 Mc WV Mc .

A straightforwarq application of lemma 1 shows that if M+ is defined

by Eq. (16), then

(P_M~lq)pTM~l

T -1 P Mc q

(Broyden's second update), and if M+ is defined by Eq.(17), then

-1 -1 T (p-Mc q) (P-Mc q)

provided Mc is symmetric and <P_M~lqlq> '* o.

(19)

(20)

We proceed to the symmetric least-change updates now. In the following

we denote the subset of the symmetric (and positive definite) matrices

of M(p,q) by ~(p,q) (~+(P,q».

THE 0 R E H 5. 27

The matrix Mc may be symmetric and p,*O. Then, if ,4 = ~(p, q), the

unique solution to the minimization problem (14) is given by

The above update formula is known by the name PSB-update (Povell' s symmetric Broyden update28 ). Another symmetric double-rank update is

the DFP-update,

T qq , (21)

which is due to Davidon24 , Fletcher and powell29 • For further

considerations we rewrite it as follows:

qpT ) [ pqT) qqT ----- M 1------ + -----<qlp> c <qlp> <qlp>·

(22)

52

Some important properties of the OFP-update are summarized in

THE 0 R E H 6. 22 ,27

Assume that the matrix Mc is symmetric and regular and that M+ is

defined by Eq.(21).

(i) If Mc is positive definite, then M+ is the unique solution to the

constrained minimization problem

min {IIA-1/2(Mc-M)A-1f2I1F I Me9'+(p,q)},

where Ae9' + (p, q) .

(ii) M+ is regular if and only if <qIM~1q> ~ o.

-1 (iii) If M+ is regular then M+ can be expressed as

<plq>

-1 T-1 Mc qq Mc

<qIM~1q>

(23 )

(24)

(iv) If Mc is positive definite then M+ is positive definite if and

only if

<qlp> > o. (25)

Remark 1. Note that in contrast to problem (14) a weighted Frobenius

norm (weighted by the matrix A-1 / 2 ) is used in problem (23).

Remark 2. Sometimes the update given by Eq. (24) is called

Fletcher-Powell update.

It should be stressed that the OFP-update preserves the positive

definiteness only if condition (25) is fulfilled. Because this fact is

often overlooked, we give some explanations hereto.

L e m m a 2. 30

The energy functional E is convex on the convex subset Vo if and only

if <g(y)-g(x) ly-x>~O V x,yeVo •

Hence, if E is convex on Vo ' x+ and Xc are included in Vo and <qlp>~o,

then the matrix M+ defined by Eq.(21) is positive definite (provided

Mc is positive definite).

If E is not convex on Vo ' then there are points x*,y*eVo such that

<g(y*)-g(x*)ly*-x*> < o.

Therefore, the inner product <qlp> may be smaller than zero. Because

53

M+ may be indefinite even if Mc is positive definite. Thus, under the

assumption that the matrix Mc is positive definite, the positive

definiteness of the matrix M+ is an inherent property of the

OFP-update only if E is convex on a convex subset Vo~Rn. When E is not

convex, it must always be checked whether the condition (25) is

fulfilled (provided the positive definiteness of M+ is of interest).

On the other hand, if Mc is positive definite and <plq> > 0, then the

matrix M+ defined by Eq. (21) is positive definite even if H(X+) is

indefinite. This fact may be an advantage when searching for

minimizers (see Sect. 2.4.4).

The update formulae considered till now yield approximations to the

Hessian matrix H(x) at a point x+. The Newton process, however,

employs the inverse matrices H(X+)-1. Thus it is reasonable to develop

inverse (least-change) updates which provide approximations M+ to the -1 matrix H(X+) • To this end we reset Eq.(13) (quasi-Newton condition)

by the condition

(26)

(approximate matrices to H(x)-1 we shall always mark by a bar) and try

to minimize the "change" (Mk - 1 - Mk ) subject to the condition (26). In

other words, we look for matrices M which solve the constrained

minimization problem

where A is a subset of M(q,p)={MeE(Rn ) I Mq=p} and Ae~+(p,q). Since an inverse update has the same structure as the corresponding

direct update, each inverse update can be derived from its direct update by the transformations

p ~ q, M+ ~ M+, Mc ~ M . (27) c So it is not hard to derive the dual Broyden update,

- T

M+ Me + (p-Mcq)q

<qlq>

Note, this update does not correspond to that given by Eq. (19) •

Generally, an inverse least-change update does not provide the inverse

matrix of the direct update, Le. M+M+"'I! If the matrix Mc is positive

definite, the inverse MS-update corresponds to that defined by

Eq.(20),

updates.

54

. -M-1 M if M-1 = M • I..e. + = + c c This is an exception among the

Two update formulae which may be derived from each other by the

transformation (27) are called dual updates.

At present, the Broyden-Fletcher-Goldfarb-Shanno (BFGS-) update31- 34

<pLq>

<qlp-Mcq> T ----- pp, <plq>"O, (28)

is regarded as the best working inverse update for minimization

procedures (see Sect. 2.4.3). It can be rewritten as

M+ = (I _ pqT )Mc(1 _ qpT) + ppT <qlp> <qlp> <qlp>

<qlp>"O. (29)

THE 0 R E H 7. 22 ,27

Let M+ be the matrix defined by Eq. (28). The matrix Mc may be

symmetric and regular. - --1 (i) The matrix M+ is regular if and only if <plMc p> .. O.

(ii) If Mc is positive definite, then M+ is positive definite if and

only if <qlp> > o.

(iii) The matrix M+ is the unique solution to the minimization problem

min {UA1/2(M - M )A1/2U I Me9'+(q,p)}, c F

where Ae9' + (p, q) .

(iv) If M+ is regular then M~1 can be expressed as

-M-1 + c

<qlp>

--1 T--1 Mc pp Mc

<pIM-1p> (30)

The BFGS- and the DFP-formula are dual updates, i.e. Eq.(28) can be

derived from Eq.(21) by the transformations (27) and vice versa.

If necessary, we shall use the abbreviations BFGS, DFP, MS etc. as

lower indices of a matrix instead of "+" or "c" to indicate that this

matrix is computed by the corresponding update.

There is a close relationship between the DFP- and the BFGS-update. If

Mc and Mc are regular, then

MBFGS = M~;p + yyT and MDFP M;;GS + wwT

55

where

v <qlii q>1/2 __ _ c ( p ii q ) c <plq> <qIMcq>

(31)

and

1/2( q MCP ) w = <PIM P> -- - • c <plq> <pIMcP>

(32)

-1 --1 Notice, the matrices MBFGS and MDFP (MDFP and MBFGS ) differ only by a sinqle-rank matrix. This observation leads to the definition of two

classes of one-parametric updates, T - T-

ii, -1 T - pp Mcqq Mc

+ ,vvT := MDFP +,vv =M +----c <plq> <qIMcq>

(33)

and

qqT T -1 + twwT

Mc: PP Me + twwT Mt := MBFGS Mc + ----

<qlp> <PIMcP> (34)

where the vectors v and ware aqain defined by Eq. (31) and Eq. (32), respectively. A simple calculation shows that the update formulae (33) and (34) satisfy the secant conditions (26) and (13), respectively, for all,. Furthermore one can prove that

ii,= (l-,)M;;p + ,iiBFGS' ,eR,

and --1 Mt = (l-t)MBFGS + tMDFP, teR.

The update Equations (33) and {34) are dual ones with22

t(,)

Therefore, we want to confine the subsequent considerations to the updates· which are defined by Eq. (33) (Broydens's class of updates). The updates of this class which can be written as a convex linear combination (i.e~ te[O,l]) of the BFGS- and DFP-update are of particular interest (Broyden's convex class), because they preserve the positive definiteness and the symmetry under the same assumption as the BFGS- and DFP-update.

If <plq> ~ ~ q Mcq then Eq. (33) can

- - T (p-Mcq) (p-Mcq)

ii", iic + <p-Mcqlq>

where

'" = <plq> - ~<p-iicqlq>

If ",=0, then ii~ with o

<plq>

~o

56

be rewritten as35

vvT

- '" <p-ii qlq> c

(35)

(36)

(37)

corresponds to the MS-update (see Eq.(20»i hence it belongs to the

Broyden's class and the OFP-update of Eq. (24) and the BFGS-update of

Eq.(28) may be regarded as perturbed MS-updates.

Now the question arises: How should the parameter ~ be chosen? When

looking for a minimizer, Oavidon proposed to select the parameter ~

that minimizes among all positive definite matrices M~ the ratio of

the largest to the smallest eigenvalue (condition number) of the

matrix

ii~1/2ii~ii~1/2 (38)

(optimal conditioning). Notice, the eigenvalues of the matrix (38)

correspond to those ones of the generalized eigenvalue problem

Proposition 5. 37

It ~he (n,n)-matrix ii~ is symmetric and positive definite, then the

matrix (38) possesses n-2 eigenvalues equal to one.

Thus the eigenvectors ei, i=1(1)n-2, pertaining to the (generalized)

eigenvalue A=l of ii~ (see Eq.(39» span a subspace sn-2 of ~n which is

not affected by the update, i.e.

ii~z = iicz

We define

where Amax(~) and Amin(~) are the largest and the smallest eigenvalue

of the matrix (38), respectively. Furthermore, we take the following

57

notations

f3 := <plq>,

The ratio 7/ lI p11 2 is nothing else but the curvature of IJ!c (the approximate function (12) of the energy functional E in the vicinity

of xc) at Xc along the quasi-Newton vector p. Since

f3 = <plq> = <plg(xc+p)-g(xc » ~ <pIH(Xc)P>'

f3/lIpIl2 can be regarded as ,an estimate to the curvature of E at Xc along p.

L em m a 3. 37

Let Mc be a symmetric and positive definite matrix. Then the matrix M~ defined by Eq.(33) is positive definite if and only if

~(CX7-f32) f3>0 and 1 + >0.

In other words, if in Eq.(33) the parameter ~ is chosen that

f32 ~ > -

then M~ is positive definite, provided f3>0. The value ~* is called the critical parameter of the Broyden's class. Since

CX7-f32 = IIM1/2qIl2I1M-1/2pIl2 - <ii1/ 2qIM-1/ 2p>2 ~ 0 (40) c c c c

by the Cauchy-Schwarz inequality6, the updates of the Broyden's convex class (~e[0,1]) are always positive definite if f3>0. The updates corresponding to a parameter ~>1 form the preconvex part of the Broyden's class. If the Broyden's class is defined by Eq.(34), then the preconvex part corresponds to the parameters 1<0.

Proposi tion 6.

The matrix Mc may be symmetric and positive definite. Then the HS-update (see Eq.(20» with M~1=Mc

(i) preserves the positive definiteness if f3>7,

(ii) does not belong to the Broyden's convex class,

(iii) belongs to the pre convex part of the Broyden's class if cx<f3.

Proof: (i): Since Mc is positive definite, 7>0. According to lemma 3, the matrix M~ (defined by Eqs.(33) and (37» is positive definite if

o

58

~>o and if ~o satisfies the inequality

~o>~*·

Hence, the following relations must be fulfilled:

~(0:7-~2) > -1

~ > 7.

(ii): Note that ~o=~/(~-O:). Since Mc is positive definite by

assumption, 0:>0. Hence,

if o:>~ then ~ <0 0

and

if o:<~ then ~o>l.

The last relation also proves the statement (iii) •

THE 0 R E H 8. 36 ,37

The matrix Mc may be symmetric and positive definite. If the vectors p

and Mcq are linearly independent, ~ = <plq> > 0, then among all

positive definite matrices M~, defined by Eq.(33), A(M~) is minimized

by M~ , where oc

1

~(7-~) 20:1

0:7-~2 if {3,s

0:+7 ~oc=

{3 20:7

{3-o: if {3> 0:+1

Remark 1: If the vectors p and Mcq are linearly dependent, Le.

Mcq="p, "elR, then the vector v defined by Eq. (31) vanishes and the

update Eq.(33)-provides for all ~ the same matrix. In other words, for

that case there is no optimally conditioned update.

Remark 2: If (3)20:7/(0:+7) then the optimally conditioned update among

the positive definite updates of the Broyden's class is the MS-update!

Now the question arises: Under what conditions are the BFGS- (~=1) and

the DFP-update (~=O) the optimally conditioned updates among the

positive definite updates of the Broyden's class when p is the

quasi-Newton vector?

The assumption of theorem 8 may be fulfilled.

59

If ~oc=o, Le. the DFP-update is the optimally conditioned update,

then

2CX7 ~ s CX+7 and 7=~,

since ~>o by assumption. Thus

--1 <plq-Mc p> = o.

Because of the supposition p = -Mcg(xC ) , we obtain

--i <plg(x+» = <plq>-<pIMc p> = o.

Hence, the DFP-update may be optimally conditioned only if one of the

three conditions

~ x+= Xc

~ P .1 g(x+)

~ g(x+)=o

is fulfilled. But all these conditions correspond to "pathological"

cases, which hardly occur in practice. In other words, the probability

that the DFP-update is optimally conditioned is zero. If x+ belongs to

some neighborhood of a stationary point, then the DFP-update may be

approximately optimally conditioned.

Suppose ~ oc =1, i. e. the BFGS-update is the optimally condi tioned

update, then

and (CX-~)7=O.

Since Mc is positive definite and p~O by assumption, 7=<pIM~lp> > 0 so

that cx=~ and

<qlp-Mcq> = o.

By considering that p = -M g(x) (quasi-Newton vector!), the above c c equation may be rewritten as follows

o <qlp> - <qIMcq>

o <qlp> - <qIMc(g(x+)-g(Xc »>

o <qIMcg(x+»

o <g(x+) IMcg(x+» - <g(xc ) IMcg(x+»

Since Mc is symmetric,

<-g(x+) Ip> = <g(xc ) IMcg(x+».

Thus, the angle ~ between the steepest descent vector and the

60

quasi-Newton vector at x+ is given by the equation

Hg(x+) II COS"

Upll

where g+= g(x+)/lIg(X+)U.

(41)

Notice, the right-hand term is essentially determined by the ratio of the gradient vector and the quasi-Newton vector. We suppose now that the Eq. (41) is fulfilled. Then a=~ (see above). From inequality (40) we ob~ain

~2+a'1 $ 2a'1

~a+~'1 $ 2a'1

~(a+'1) $ 2a'1,

since a=~, and 'oc=l because of theorem 8. Thus the BFGS-update is optimally conditioned only if the angle between the steepest descent vector and the gradient at x+ is related to the ratio of the gradient vector and the quasi-Newton vector in a well defined manner. The last results we summarize in

Proposition 7.

The assumptions of theorem 8 may be fulfilled. If the vector p is equal to the quasi-Newton vector l i.e.

p =. -iicg(xc ) ,

then the BFGS-update (see Eq.(28» is the optimally conditioned update among the positive definite updates of the Broyden1s class if and only if Eq.(41) holds.

2.4.3 Quasi-Newton Methods

As explained at the beginning of this section, a quasi-Newton method is obtained when in the classical Newton process the Hessian matrix is approximated by estimate matrices which are computed by an update formula. Subsequently a few results pertaining to the behaviour of convergence of some quasi-Newton methods are presented. Furthermore, the applicability of these methods for locating minimizers and/or saddle points of energy functionals is discussed. The notations introduced in the preceding sUbsection are maintained. A quasi-Newton method is essentially described by the following steps:

61

Pro c e d u reI (quasi-Newton method)

1. Initialization choose xO (initial guess),

C (truncation error), Mo (initial estimate to the matrix H(XO)-l)

k:=O

2. computation of the quasi-Nevton vector

pk := _~CJ(Xk)

3. computation of the ner estimate

xk+1 := xk+pk

4. Stopping test

if max(lpkn, IE(Xk )-E(Xk+1)1)<C, then stop

5. Updating

~+1 := ~ + Ck

6. Continuation k := k+1, go to step 2

Remark 1: step 5 must still be specified. The matrix Ck 'has been introduced to indicate that the matrix ~ is modified by a (low-rank) correction matrix. When step 5 is reset by a specific update formula, the quasi-Newton method is named after that update (for instance: BFGS-method, DFP-method, Broyden-method, ••• ).

Remark 2: Procedure I can also be started with an estimate No to H(Xo). In that case the matrix ~ must be reset by ..;1 (step 2 and step 5).

Remark 3: In contrast to the classical Newton process, a quasi-Newton procedure is not self-correcting, Le. the errors are accumulated, because a matrix ~ depends on all preceding matrices Mi , i=O(1)k-1. Therefore, the choice of the initial matrix No affects the' entire numerical process.

Convergence properties, comparable with those of the Newton process, cannot be expected for quasi-Newton methods, because a certain charge must be paid for the use of estimates to the Hessian matrix. But super linear convergence, at least, should occur.

L e m m a 4. 38

The assumption (AI). (A2). and (A3) may be fUlfilled. Further suppose

62

that all matrices of the sequence ~~~ are regular, and that for some

xOeV the sequence generated by the iterative process o

Xk+l = xk _ ~lg(Xk), k=I,2,... (42)

remains in Vo' and satisfies xk * xst for all k and lim xk = xst. Then t k*'" st

the sequence ~xk~ converges superlinearly to XS and g(x )=0 holds,

if and only if

lim k*", o.

It has been verified22 that the above condition is equivalent to the

condition

lim IIqkllfllpkll = lim <pk,qk> = I k*", k*",

"k k k "k . k where p :=p flip II and 9 1S again the normalized gradient at x • Thus a quasi-Newton method converges super linearly if and only if the quasi-Newton vector converges in magnitude and direction to the Newton vector. consequently, the gradient method (Le. Mk=I for all k in procedure I) never can superlinearly converge. Quasi-Newton methods are frequently started with the identity matrix, i.e. Mo=r, to save the effort pertaining to the evaluation of the Hessian matrix at the initial guess. By the above lemma, however, such a choice does not promot~ the behaviour of convergence. That is to say, the proportion of the computational cost saved by avoiding the evaluation of H(Xo ) is eventually spent for additional iterations again. This fact should be considered in particular for ab initio calculations.

A Newton-like procedure is locally convergent at a stationary point xst if there is an e>O and a ~>O such that whenever xOeu(xst,e) and IIMo-H(Xst)II<~, the sequence {Xk} is well-defined and converges to xst.

THE 0 R E H 9. 22

Suppose E satisfies the assumptions (AI), (A2), and (A3). Then the Broyden-method is locally and superlinearly convergent at xst.

THE 0 R E H 10. 39

Suppose E satisfies the assumptions (AI), (A2), and (A3). tlurtagh-Sargent method is locally and linearly convergent

Then the at xst.

63

Moreover,

IIxk+1_xstIIS't"IIxk_xstII , k=O, 1, •.. ,

with'rE[O, 1/2].

Remark: If the assumptions of lemma 4 are fulfilled, then the

MS-method also converges superlinearly.

THE 0 REM 11. 27 ,40

The hypotheses (AI), (A2), and (A3) may be fulfilled. Then the

PSB-method is locally and super linearly convergent at xst. If in addition H(Xst ) is positive definite, BFGS-method are locally and superlinearly

then also the

convergent at xst DFP- and

In principle, the Broyden and the PSB-method may be employed to locate

minimizers as well as saddle points via Eq.(4), provided the initial

estimates X O and Mo are chosen in an appropriate manner. But today

saddle points are mostly computed via the defect functional (]" (see

Eq. (5» • For the DFP- and the BFGS-method there are only (mathematically)

proved statements concerning the convergence to minimizers. Since

local convergence occurs only if the initial matrix Mo is a (rough)

estimate to H(Xst) (see the definition above), Mo should be chosen

very carefully. A computer run started with a good initial guess X O

but a bad initial estimate Mo (e.g. Mo=I) may fail. This fact should

be taken into account in particular when searching for saddle points.

At present the Broyden method is regarded as the best working equation

solver6 ,41 whereas the BFGS-method is regarded as the best working solver for minimization problems. 19 ,22,42,43

The MS- and the BFGS-method are the basis for the optimization

procedures incorporated into the program packages GAUSSIAN 8844 and

CADPAC45 •

In recent years many work has been spent to develop and to improve methods that find saddle points. 14, 46-48 Recently a modified

quasi-Newton proposed, 49

method adapted

but without any

Sylvester's law of inertia:

L e m m a 5. 4

to saddle point problems

proofs. The modification

has been

relies on

Two real symmetric matrices M1 and M2 have the same index of inertia if and only if there exists a real nonsingular matrix C with M1=C™2C.

64

Recall, the index in(M)=(i+,io,i_), is

eigenvalues (i+), the

number of the negative

of inertia of a symmetric matrix M,

the triple of the number of, the positive

number of the zero-eigenvalues (io) and the

eigenvalues of M. Thus it holds

in(H(Xmin»=(3N-6,6,0) and in(H(XsPv»=(3N-6-V,6,V).

If an update formula can be written in the form

M = C™ C + C (43)

then all matrices Mk generated by the corresponding quasi-Newton

method possess the same index of inertia as the initial matrix Mo.

This observation is the starting point to derive a quasi-Newton method

which converges to a stationary point with prescribed index of

inertia.

It is easy to verify that the update of Eq. (30) and the (inverse)

MS-update (see Eq. (20» can be expressed in the product form of Eq. (43)49 with

C I -p(v'i q+Mcp) T

<pi McP>

X = <PIMcP>

<plq>

for the update of Eq. (30) --1 (Mc = Mc )'

c = I - PM~l(p-Mcq)(P-Mcq)T,

p and

and

<p-Mcqlg(xc »

<p-Mcqlq>

(44 )

(45)

(46)

for the MS-update. Obviously, the index of inertia of both updates is

remained if X>O.

Recall, the inner product <plq> is an estimate to the curvature of E

along p at Xc and

<PIMcP> = <pl-g(xc »· If at Xc the functional E is convex (concave) along p, then

<pl-g(xc » > ° «0) by proposition 3. Hence, X may be regarded as a

measure for the quality of the approximation M to H(x ). c c A quasi-Newton procedure which looks for stationary points of

prescribed index of inertia is

65

Pro c e d u r e II (modified BFGS-method)

1. Initialization choose XO (initial guess),

C (truncation error), Mo (initial estimate to the Hessian H(XO)-l)

k:=O

2. Computation of the quasi-Newton vector

pk : __ ~g(~)

3. computation of the nev estimate

xk+1 :_ xk+pk

4. computation of the vector qk

qk:_g(Xk+1)_g(Xk )

5. Checking of the control quantity

<pkl_g(~» if < 0 then set pk:=_pk and go to step 3

<pklq~

6. Stopping test

if max(llpkU, IE(Xk)-E(Xk+1)1)<C, then stop

7. Updating

~+l = ~ +

T (pk_~qk)pk + pk(pk+~qk) T

<pklq~

8. Continuation k:=k+l, go to step 2

<qk I pk_~q~ p~kT

<pklq~2

since the matrices ~ are updated by the BFGS-formula (step 7), the inverse matrices ~l=:~, which are estimates to the matrix H(xk), can be computed by Eq.(30). Thus the index of inertia remains if

<pk 1 ~p~ <pk I-g (Xk) > Xk = > O.

<pklq~ <pklq~

If Xk<O then the calculated quasi-Newton vector pk is inconsistent with the estimated curvature of E along pk at xc. This contradiction is solved by stepping in the opposite direction. Although minimizers can also be located by procedure II, it is, above all, a procedure to search for saddle points. The initial guess xO has to be chosen very carefully, because it

66

determines the type of the stationary point for which the procedure

looks. Notice, procedure II will well work only if the restricted functional

E* (see the beginning of Sect. 2.4) is utilized (regularity of the

matrices "k!)'

2.4.4 Descent Methods

Descent methods are specific (quasi-)Newton methods which look for

minimizers only. They differ from the general (quasi-)Newton methods

in the line search step which is added to ensure that the procedure

makes a sufficient progress in the direction to a minimizer,

particularly in the case when the initial guess is far away from a

solution. Line search means that at a point xk the energy functional E

is minimized along the (quasi-)Newton vector pk, i.e. a positive value

~k is determined such that

(47)

(exact line search). This one-dimensional minimization problem may

cause a considerable additional numerical effort. Therefore, the

solution of problem (47) is not exactly calculated in general. Only a

value promoting the energy decrease is determined (inexact line search) . The feasibility of a line search step is guaranteed by

L em m a 6. 22

If there is a vector pe~n so that

<g(x) Ip> < 0 (48)

then there is a positive value ~0>0 such that E(x+~p)<E(X) for a11

~e(O,~o)'

A vector p satisfying the condition (48) is called a descent vector at

x. By the above lemma, each vector pe~n forming an angle of less than

900 with the steepest descent vector (negative gradient) is a descent

vector. So the steepest descent vector (p = -g(x» can be regarded as

the limit case.

An important characterization of descent directions is given by

67

L e m m a 7. S

A vector pelRn is a descent vector at x if and only if there is a

positive definite matrix H such that -1

P =-H g(x). ( 49)

Let e 1 , ••• ,en be the eigenvectors of the Hessian matrix H(x) pertaining to the eigenvalues A1 , ••• ,An . Then

n. . g(x) = r a.e1 , a.= <g(x)le1 >,

i=l 1 1

and the quasi-Newton vector p can be expressed as

-1 P = -H(x) g(x)

Now the equation

n 2 <g(x)lp> = - r (a·/A.)

i=l 1 1

shows that the positive definiteness of H(x) is a sufficient but not a necessary condition for p to be a descent vector. On the other hand, by lemma 7 a positive definite matrix H, which satisfies Eq.(49), may always be constructed if p is a descent vector. Thus H = H (x) holds only if E is convex on some neighborhood of x.

Let Sk(/3,7), /3e(O,O.S) and 7e(/3,1), be the set of steplengths /J.>O which satisfy the conditions

E(Xk+/J.pk) ~E(Xk) + /J./3<g(Xk ) Ipk>, (50a)

<g(Xk+/J.pk) Ip~ 2: 7<9(Xk ) Ipk>. (SOb)

The first condition is to ensure that by means of /J. a certain energy decrease is attained. But /J. may be very small such that the progress in the direction to a minimiz~r can be insignificant. To avoid such a situation the second condition is taken.

L e m m a 8. 6

The energy functional E may be bounded below, and the initial guess XO

may be chosen such that 9 is uniformly continuous on the level set

(51)

Then, if the search vector pk is a descent vector, the sets Sk(/3,7), 0</3<7<1, are non-empty, i.e. there are values /J. satisfying the conditions (50).

68

Lemma 8 ensures the existence of step lengths satisfying the conditions

(50), but it does not say anything about the way by that such

step lengths can be obtained. In Ref. 30 a method is suggested (see

procedure III), which yields step lengths with the desired properties.

First the method generates an interval [Vr , VI] for that the

conditions

are fulfilled, where

and

1

E(xk+J.LP)-E(Xk )

J.L<g(Xk ) Ipk>

<g(Xk+J.Lpk) Ipk>

<g(Xk ) Ipk>

(52)

ifJ.L 1

otherwise

Then the interval is reduced by bisection where the conditions (52)

are kept.

P r o c e du r e III (line search procedure)

1. choose il>o and O<p<l

vr:=J.L

2. if f k (ilk )<f3 then go to step 6

3. if hk(vr)~~ then J.Lk:=vr and stop

else vI:=vr

5. if fk(vr)~f3 then go to step 3

else go to step 9

6. vl:=vrP

7. if fk(vI)~f3 then vr:=vI and go to

8. if hk(v l ) <~ then J.Lk:=v I and stop

step 6

69

10. if fk(Vm)~~ then vl:=vm and go to step 8 else vr:=vm and go to step 9

Under the assumption of lemma 8 the procedure terminates after a finite number of steps with a value ~ which fulfils the inequalities (50)30.

Recently an iterative steplength method50 has been adapted from a more general line search technique which is due to Komornicki51 and tested in conjunction with the DFP- and BFGS-method. The step length Jlk is iteratively calculated by the formula

2 k k Jlk ,i-1<P Ig(x »

Jlk,i= - --------k-------------k~~~-k-------------------k-------k---

2(E(x +Jlk ,i-1P ) - E(x ) - Jlk ,i-1<P Ig(x » (53)

The procedure is started with the initial guess Jlk ,o=l or Jlk ,o=Jlk- 1 • By a simple calculation we obtain

Jlk ,i+1Jlk,i(1 .1 2

( 1 Jlk,i ) k k ~ Jlk,i 1 - 2" ....,.,..-==-- <P Ig(x »

Jlk ,i+1

provided Jlk ,i+1<1. Thus, it is not difficult to modify the iterative process in a manner that the conditions (50) are fulfilled. The main steps of a descent procedure are summarized in

Pro c e d u r e IV (descent method)

1. Initialization choose XO (initial guess),

k:=O

e (truncation error), te[O,l], Mo (initial estimate to the matrix H(Xo»

2. Computation of the quasi-Newton vector k -1 k

P := -~ g(x )

3. Line search Jlk := procedure_III(Jl,p)

4. Computation of the new estimate

xk+1 := xk+Jlkpk

70

5. Stopping test

if max (1lpkll, IE(Xk)_E(xk+l)I, Ig(xk+ln)<e, then stop

6. Updating qk := g(Xk+1)_g(Xk)

qk

~+l := ~ -

7. Continuation

k k T , Mkp P Mk

<pkIMkP~

k:=k+l, go to step 2.

Remark: When exact line searches are performed, the procedure yields

the same iterates for all updates of the Broyden's convex class

(tE [0, 1 J) • 5 2 When inexact I ine searches are performed, the sequences

of iterates may markedly vary for different choices of t.

THE 0 R E H 12. 5 ,22

Let E be bounded below on ~n. Furthermore, for the initial guess XO

the level set (51) may be compact. Consider the iteration

Xk +1= xk + ~kpk, .... k=O, 1, . •. ,

where pk is a descent vector. Then there is a sequence {f..Lk } with

f..LkESk«(X'~) for all k and k Ak

lim <g(x ) Ip > = 0,

Corollary 12.1

Ak P

If the assumption of theorem 12 is fulfilled and {Xk} is a sequence

generated by the steepest descent method, then the sequence {E(Xk )} is

decreasing, and {g(Xk)} converges to zero.

Remark 1: The convergence of the sequence ~g(Xk) ~ does not imply the

convergence of the sequence ~xk~.

Remark 2: There is no guarantee

~xk~ is a minimizer. Wolfe53 has

is achieved by the method of

that the limit point of the sequence

given an example where a saddle point

steepest descent.

71

Corollary 12.222

Under the assumption of theorem 12 the procedure IV converges if {Mk}

is a sequence of symmetric, positive definite matrices with uniformly

bounded condition numbers.

Descent methods converge, in general, only linearly, but it holds

THEOREM 13. 54

Assume that the procedure IV, with te[O,l), is implemented so that ~k

satisfies the inequalities (50) and ~k=l whenever this satisfies the

inequalities (50). If the conditions

(i) there are constants K and p such that IIH(x) _H(Xmin) II S KIIX_XminIlP

(ii) the level set (51) is convex and there exist positive

constants P1 and P2 such that p 1 11Z11 2 S ZTH(X) z S P 2 11Zlll for all ze~n and all xe~(E(xo»

(iii) Mo is positive definite

are fulfilled then the sequence {Xk} converges to xmin superlinearly.

By the theorem, the line search step should be executed only if the

full quasi-Newton step does not lead to a decrease of energy.

2.4.5 A global Newton-like Method

All methods considered in the previous sUbsections rely on successive

local approximations to the energy functional. A stationary point xst

is achieved in general if either the sequence of estimates falls into

the domain of attraction of ~st and remains there, or, at least, the

initial guess belongs to a neighborhood of xst on which the energy

functional is convex. If these conditions are not fulfilled then the

success of a search depends on whether the method generates a seqUence

which is running into such a domains or not. So the question arises:

How can an initial guess be driven in a controlled way into a

neighborhood of a stationary point when, for instance, the descent

methods fail? This task has global character and must just be attacked

by appropriate mathematical means.

The object of this sUbsection consists in solving Eq. (3) by an

associated differential equation. There are several ways to do this.

We describe a method which is due to Zirilli et al. 11 who have

72

considered the second order differential equation

~ = -fx - grad u(x) (54a)

with the initial conditions

x(O)=xo and . 0 x(O)=w , (54b)

where XO is an arbitrary guess and WO is the initial tangent to the

corresponding solution curve of Eq.(54a) at xO. ~ denotes a positive

constant and f a positive real function, i.e.

f: [0, ... ) ,. (0,"')'

The differential Eq.(54a) represents Newton's second

dissipative force) and so the solution curves have

law (with a

a physically

meaning within a semi-classical framework. This question is discussed

in detail in Ref.55.

When searching for minimizers only, the equation

x = -grad E(x) (55)

can be integrated instead of Eq.(54a); cf.Ref.56,57. Equation (55)

describes the (mechanical) motion of a molecular system in a

conservative force field.

It is easy to verify that each stationary point xst of E corresponds

to a constant solution curve X(t)Exst of Eq.(54a), which is an

asymptotically stable solution (the definition is given below). This

property allows to determine the stationary points of E by the

differential Eq.(54a). In this context it is useful to rewrite

Eq. (54a) as a first order system:

x = ~-1/2 v (56a)

v -f/~ v - ~-1/2grad u(x)

with

x(O)=xo and v(O)=wo . (56b)

For a better understanding of the mathematical background of the method, a little excursion to the stability theory58 of differential

equations is useful. To this end we consider the first order differential equation

yet) = F(t,y),

where F is a continuous function defined on the region

'R = ~(t,y)elRxlRn I o~t< ... , lIyll<a~

(57)

with some constant a. 111(·; to'Yo) may denote a solution function of

73

Eq.(57) that satisfies the initial condition ~(to; to'Yo)=yo. Notice, the function ~(t;to'Yo) is only defined for tl!:to (the "past" of the function (t<to ) is out of interest). The solution ~ of Eq.(57) is said to be stable if for every £>0 there exists a ~=~(£,to»O such that whenever h)(to)-yOI<~ the solutions ~(o; to'Yo) exist for tl!:to and satisfy

The solution ~ is said to be"asymptotically stable if it is stable and if there exists a ~o>O such that whenever 1I~(to)-yOIl<~o the solutions ~(o;to'Yo) approaches the solution ~ as t~, i.e.

lim I~(t; to'Yo) - ~(t)n = o. t .. ",

To make it clearer the definition is illustrated by Fig. 3.

b

a

y

-+-11-----I-~----I..--+~_- <pet) fJ-

y 'P(t;to.yO) ._._._._._._;{_._.-

oS -t-I----f---t---f---+---I--<f'(t)

'Ft-

Fig. 3. A stable (a) and an asymptotically stable (b) solution

The solution ~ is stable if each function i}(o;t ,yo) with 1I~(t )_yOIl<~ o 0 oscillates within a certain "neighborhood of ~" as indicated in Fig. 3a. The solution ~ is asymptotically stable if the oscillations ease off and i}(t; to'Yo) approaches ~(t) as t~, see Fig.3b.

74

A minimizer x* of u is nondegenerated if the matrix H{x*) is regular,

i.e. det H{x*)*O.

THE 0 R E H 14. 11b

Let xst be a nondegenerated minimizer of the defect functional u (see Eq.(5)) which may be twice continuously differentiable. If the

function f satisfies the condition

lim f{t)=f:3>O, t,. ..

then the constant solution curves X{t)=xst and v{t)=O are an

asymptotically stable solution of Eq. (54a).

According to this theorem, a solution of Eq. (3) may be computed by

tracing a trajectory of the system of differential equations (56) till

it approaches a constant solution curve that corresponds to a

stationary point of E. since the asymptotical stability is a local

property, a curve emanating from XO will approach an asymptotically

stable solution x=xst of Eq.{55a) with reliability only if XO and V O

belong to the stability domain of that solution function. Thus,

theoretically the situation is similar to that of Newton's method. In

practice, however, most trajectories flow into a stability domain. So

a curve tracing represents a systematic search. This is a considerable

advantage for the computational practice.

By using a special integration technique11b the following system of

linear equations

[ L{Xk) + WkIJpk

xk+l

is obtained, where

L(xk )= H(Xk)TH(xk) + r g.(Xk ) Hi (Xk) i=l ~

(58)

(59)

and Hi (Xk) is

gradient g (x) )

the Hessian matrix of gi (x) (i-th component of the

at xk, 1:>0 is a fixed steplength. Since the second

right-hand-term of Eq. (59) vanishes at stationary points, it can be

neglected.

Equation (58) can be regarded as a Gauss-Newton method30 with memory,

where "memory" means that the current search vector is involved in the

75

computation of the new search vector. The present form of Eq.(58) is

not suitable for quantum chemical purposes, because the Hessian matrix

is to evaluate at each iterate xk. This disadvantage can be overcome

by utilizing updates55 • Th~ method is described roughly by

Pro c e d u r e V (global Newton-like method)

1. Initialization choose XO (initial guess),

WO (initial tangent)

~>O (steplength)

k:=O, Mo:=H(Xo ), pO:=~wo

2. computation of grad u(Xk)

grad U(Xk ) := Mk9(Xk )

3. Building-up of the matrix Lk

Lk := (M~Mk + WkI), where wk is chosen such that det Lk*O

4. Computation of the quasi-Newton vector pk+l := L;l(grad U(Xk) + (~/~2)pk)

5. Computation of the new estimate xk+l := xk + pk+l

6. stopping test

if Ilg(Xk+1 ) II < e, then stop

7. Updating qk g(Xk+1)_g(Xk)

~+1

8. Continuation

T (qk_MkPk)pk

<pklqk>

k := k+l, go to step 2

Remark: In step 7 the Broyden update is used. Of course other updates

can also be employed.

In numerical practice variable steplengths are used. A method for

steplength controlling, which relies on the decrease of the total

mechanical energy along a trajectory, has been suggested by

Aluffi-Pentini et al. 59 (see also Ref.55). Further numerical items are

described in ref.55.

Methods for unconstrained optimization based on the solution of a

system of differential equations are numerically more expensive than

76

quasi-Newton methods. Their effectivity strongly depends on the choice

of the integration method. Some integration techniques appropriate for

solving differential equations like Eq. (54a) or Eq. (55) are

comprehensively discussed in Ref.60.

2.5 Testing of Numerical Procedures

After a series of numerical methods has been investigated in the

previous sections, the questions arise:

Are these methods appropriate to quantum chemical purposes?

Which of them is to be preferred when a certain class of

problems is investigated?

Today both questions can only be decided by experimentation. That is

to say, each procedure has to be tested on a variety of test problems

which has been chosen to represent the different features which might

occur in PES calculations. Clearly experimentation can never give a

guarantee of good performance in the sense of a mathematical proof.

The choice of the test functions may considerably affect the test

result. Therefore, they should be chosen with care. The better a test

problem is adapted to the characteristics of a numerical procedure,

the better the test result will be. Thus, a serious testing should

not only rely on one (artificial, well-adapted) test function, but on

some different molecular systems of different size (number of nuclei)

with different characteristic (deep minima, clear saddles, flat

minima/saddles, etc.).

What effects should be considered in any case when testing a numerical

procedure?

~ The choice of the initial guess

A numerical procedure should always be tested by using different

initial guesses. In particular the robustness, Le. the influence of

small perturbations of the guess to the outcome, should be examined.

Since in particular the descent methods behave like quasi-Newton

methods in the vicinity of a minimizer, differences between them will

become evident only if the initial gUesses are chosen outside of the

domain of attraction. (Recall, descent methods have just been created

for that case!). Therefore, a descent method should also be tested

with initial guesses far away from a minimizer.

~ The number of nuclei

The efficiency of a numerical method strongly depends on the number of

77

floating point multiplications which are necessary to achieve a

certain numerical progress. This number, however, does not

proportionally increase with the dimension of the problem (the product

of two (n,n) -matrices requires n 3 floating-point multiplications, a

matrix/vector product requires n2 such multiplications). Therefore, a

numerical procedure which solves a given problem of low dimension

effectively, must not solve a similar problem of higher dimension in

an effectively way and vice versa.

~ The procedure-inherent parameters, if existing

The second question mentioned above implies the question for a measure

of efficiency that relates the numerical effort and the numerical

progress. The rate of convergence only indicates the numerical

progress per iteration step in the final phase (i.e. when the iterates

fall into a certain vicinity of a stationary point), but it does not

consider the numerical effort (function calls, evaluation of Hessian

matrices, matrix/vector multiplications etc.) which is caused by one

iteration step. Therefore the rate of convergence says nothing about

the efficiency of a numerical procedure. Recently the rate55

(Xst) r proc

(number of efficiency), where

F is the number of function calls,

G is the number of gradient evaluations

0_ st x ~x ,

H is the number of evaluations of the Hessian matrix and

a1 and a 2 are positive constants (see below),

(60)

has been proposed to measure the efficiency of a numerical procedure

with regard to a given problem. xst denotes an estimate to the correct

solution, so that grad E(Xst):l:O in general. In Eq.(60) the numerical

effort is measured by the number of gradient evaluations. The factors

a1 and a2 are to relate the function calls F and the Hessian

evaluations H to G, respectively. If the first and second partial

derivatives are computed by finite differences, then we have to set

a1=1/n and a2=n. The number of efficiency indicates the average numerical progress per

computed gradient for a given problem. Thus it allows to compare test

results obtained by different numerical procedures for a given

problem. But the question whether a procedure works better than

another one cannot generally be settled by it.

78

since experimentation is the only way to get information about the efficiency of a numerical procedure, test reports are a valuable support for the user of a program, provided the results can be reproduced and allow a comparison with other testings. The request for reproducibility requires that the report supplies information about the initial conditions (initial guess, fixed coordinates, parameters, etc.). Furthermore, a test report should involve the following information:

1- the stationary point which has been reached 2. the norm of the gradient at that point 3. the energy at that point 4. the number of function calls 5. the number of gradient and Hessian evaluations

The reader who is interested in more details concerning the preparation of a test report is recommended to the paper by Crowder et al. 61

2.6 Zero Eigenvalues of the Hessian

2.6.1 Translational and Rotational Invariance

A molecular system with N atoms is well described by Eq.(l), but it is obvious that we can translate or rotate the molecule as a whole without any change of the potential energy. The geometrical specification of a translation, as well as that of a rotation, requires three degrees of freedom. Thus, it is to be expected that we are left with (3N-6) degrees of freedom for an "internal" description of a molecular motion (3N-5 for a linear molecule). However, a serious difficulty occurs: Only the three translational degrees of freedom can be linearly separated from 1R3N in a global way, but not the three dimensions of rotation; the latter remain curvilinearly coupled to the vibrational degrees of freedom. In this section we shall explain how this problem can be handled. 12 ,62 Additionally we introduce .the notation Pk with

Pk := (~k'~k,Ck)T,

for points PkEE3 where the index k refers to atoms, not to coordinates, cf. Sect. 2.1. All Pk' k=l, ••• ,N of a molecule form an E3N_point p= (P1 T, ••• , PN T) T • We need this other form rather than

x=(X1 , .•• ,x3N)T as previously used.

79

(i) Translational invariance means

(61)

with one and the same vector tER3 for all atoms. In concise notation we write

E(p) = E(p+t)

with p,tER3N and with the 3N-vector T T T t=(t , ••• ,t) built from 3-vectors t. We rewrite th~ invariance property in the form of a one-dimensional function

f 1 (s) := E(p+t+sx) = E(p+sx), (62)

for an arbitrary but fixed vector XEE3N and for an arbitrary SER1. If P and x are fixed than f 1 (s) is a mapping of R into R ranging over the fixed coordinates of the R3N, i.e., over 3N-vectors the components of which are functions of the real variable s. We take derivatives off1 with respect to s by the chain rule:

df1 (S)/dS f~(S) = g(p+t+sX)TX = g(p+sx)Tx

xTg(p+t+sx) = xTg.(p+SX) Thus

Analogously:

f~(O) = XTH(p+t)X = xTH(P)X •

Becaus!3 XElE3N is arbitrary., 9 and H fulfill the same condition of translational invariance as E, i.e.

g(p+t) g(p)

H(p+t) = H(p)

for all PER3N and t=(t, ••• ,t)T with tER3. If we write Eq.(62) in a form

f 2 (s) := E(p+st) = E(p),

we obtain ,

f 2 (O) T t g(p) = O.

(63)

(64)

(65)

We denote by T the subspace of R3N of all admissible translations:

T := J 3N 3 1 t E R I ~ = t E R for all k=l, ••• ,N ~ •

The scalar product in Eq.(65) is an orthogonality condition. For all pElE3N the gradient vector g(p) lies in the subspace T~cR3N orthogonal

80

to T. We can reformulate Eq.(65) into

and hence we obtain the identity

N r gk« = 0, «-(,~,c

k=l for the components gk« of the gradient vector g(p). We write Eq. (63) in the form

f 3 (S) :- g(p+st) - g(p) ,

and obtain the d~rivative ,

f 3 (0) = R(p)t = O.

(66)

(67)

Each tET is a vector in the kernel of the Hessian matrix R(p). Because dim(T)=3, with three coordinates of the displacement tER3, the translational invariance implies the existence of three zero eigenvalues of the Hessian matrix. Equation (67) also means that the rows (or the columns by symmetry) of R(p) belong to T~.

(ii) In E3 any orthonormal 3*3-matrix A gives a rotation of a point Pk' If we use APk for all the atoms of the molecule, we describe a rotation of the molecule, and the rotational invariance of E means

T T E(Pl, ••• ,PN) - E(AP1,·.·,APN) with AA =1, A A=I,

or concisely

E(p) E(AP)

with a N*N block~diagonal matrix A:

A [A 0 0 1 o A • . . • 0 o 0 A

of N rows and N columns of 3*3-matrices, thus A is 3N*3N. with

f 4 (S) := E(p+sx) = E(A(p+sx» = E(AP + sAx)

3N for p,x E E , S E H, we obtain the relations

f~(O) = xT g(p) (AX)T g(AP) and

(68)

(69)

(70)

81

since XElE3N is again arbitrary we have the rotational invariance

condition for the gradient63

g(AP) = Ag(p) (71)

and for the Hessian matrix

H(AP) = A H(p) AT • (72)

We see that g(p) and R(p) do not satisfy the same invariance

condition as E(p)! Equation 171) shows that the norm of the gradient

of E is still preserved in a rotation, but this is not the case for

its direction in R3N (see Fig.4). Equation (72) means that the Hessian . T -1 .

matrix undergoes a similarity transformation (because A =A ), Wh1Ch

changes its form, but preserves the eigenvalues.

Fig.4. Rotation

of the gradient

We require the subspace of rotations. For a given angle u the matrix A

is determined by Euler's formula64

APk A(U)Pk = Pk cos U + (dxPk) sin U + (PkTd)d (I-cos u)

(73)

We find A(O)Pk=Pk and A(O) TPk=Pk. Differentiation of Eq. (73) with

respect to U gives ,

A(U)Pk (dxPk ) cos U + dx(dxPk ) sin U (74) , , T

with A(O)Pk dXPk and A(O) Pk= - dxPk •

(The latter relation follows with an arbitrary zelE3 from

82

In the next approach, with u in the rotational matrix

f 5 {U) := E{A{u)p) = E{p) ,

we obtain

where we have used Eq.(71), and thus

(A{U)TA(U) p)T g{p) = o.

If u=o we have , , T

f 5 {O) = (A{O)p) g{A{O)p) = O.

, If we describe the vector A{O)p with Eq.(74) in the following form

, T A{O)p = (dxP1, .•• ,dxPN) =: dxp

(75)

(76)

(77)

with p,d e 1R 3N , dk=d e 1R3 , dTd =1, k=1, .•. ,N, we obta:Ln from

Eq.(77) the simple symbolic equation:

( dxp )T g{p) = 0 (78)

Equations (76) and (78) are scalar-products. Since they are zero, the

gradient vector at p is orthogonal to dxp (cf.Eq.65 as well). We still

treat the subspace

3N 3N. ·3 T D{p) = ~ yelR I y = dxp, peE , f~xed, ~=delR , d d=1~ , (79)

which obviously depends in a nonlinear way on point p because the

vector-product contains the sine function. We rewrite Eq.(71) in the

form

f 6 {u) := AT{U) g{A{u)p) = g{p) , (80)

and obtain ,

A(U)T g{A{u)p) + AT{U) H{A{u)p) ,

f 6 {u) A{u)p = 0

, T A{u) A{u)g{p) +AT{u)A{U) H{p) AT{U)A(U)P (81)

Thus

H{p) (AT{U)A(U)P) -(AT{u)A(u»T g{p) (82)

For u=O this identity becomes

H{p) (dxp) dxg (p) . (83)

83

The equation means, if the gradient at p vanishes, that vectors in the

subspace D(p) are vectors of the kernel of H(p), to eigenvalue zero.

Thus, for stationary points (if we choose p"O) , Eq. (83) shows that

D(p) is really the eigenspace of H belonging to all rotations.

It should be noted that in quantum chemical geometry optimization carried out to locate stationary points of the PES, the three rotational modes are only zero exactly at the stationary point. Therefore, their values reached depend on the quality of the geometry optimizations. It has been estimated that for g(x) <E-5au, the "rotational modes" will have.frequencies below 10 wavenumbers.

From Eq.(83) we draw a further important result: If g(p) .. O, the

vectors of the rotational subspace D(p) are not eigenvectors to

eigenvalue zero, or more precisely, there are no zero eigenvalues of

H(p) belo~ging to the rotational invariance of the potential energy.

Of course, there are three degrees of freedom which come from

rotations of the molecule as a whole, but the matrix H(p) has non-zero

eigenvalues also for these degrees of freedom.

Equation (83) illustrates a widely discussed problem, the so-called

"zero"-eigenvalues of a rotation orthogonal to a chemical reaction

path,65 - by falsification of the "folklore" in the field. Computing

molecular paths, three non-zero eigenvalues emerge. The proposed

projector method65 to suppress these values changes the whole force

constant matrix H into (1-P)H(1-P). It gives the intended 6 zero

eigenvalues, but the rema1n1ng (3N-6) eigenvalues are now also

changed. Thus, the projected force constant matrix describes another

PES problem. From a physical point of view, the point is that a

rotate? gradient (see Fig.4), i.e. a rotated force vector in a

non-stationary point, requires a force to turn itself. In general, any

displacement out of a stationary structure results in a gradient which

does not point to the center of mass. An example is the HeN

isomerization (cf. Fig.1 of Sect. 3.1). Hence, we will get

dxg(p)E.O(p), which excludes a solution of the "eigenvector" equation

(83) in this subspace D(p). The vector dxp in the l.h.s. of Eq.(83) is

changed by action of H(p) in another part of the R3N- 3 represented by

dxg(p). The diagonalisation of H(p) in R3N- 3 finds the three

eigenvectors of "rotation" outside the subspace D(p), or in other

words, it couples rotational motion and relaxation of the molecular

structure along the gradient of E (cf. also the computer experiments

of Frederic et al. 66 ).

We summarize the results of the section in a theorem.

84

THE 0 R E N 15. 12 ,62 3 3N Supposing teR , t=(t, ••• ,t)eR , A is an orthonormal 3*3 matrix and

A is the N*N block-diagonal matrix built vith A. We choose an arbitrary point peE3N • (i) The translational and rotational invariance of E means (Eqs.61,68)

E(p+t) = E(p), E(AP) = E(p), and it includes (Eq.63) g(p+t) = g(p), R(p+t) - R(p). But in contrast ve rind Eqs. (71,72) g(AP) = Ag(p) and R(AP) = AR(P)AT•

(ii) Vectors teT are orthogonal to g(p) (Eq.65) and are zero eigenvectors of R(p) (Eq.67). The orthogonal complement T~of T is a (3N-3)-dimensional subspace of R3N containing all molecular configurations independent of translational displacements.

(iii) For any fixed peE3N the subspace of vectors dxpeD(p) (Eq.79) is also orthogonal tog(p) (Eq.78), but there are zero eigenvectors of R(p) only at stationary points pst.

(iv) In pst ve can separate the E3N into a 6-dimensional (5-dim.for linear molecules) Cartesian subspace TeD(pst) and into a trans­lationally and rotationally invariant remainder of dimension (3N-6) or (3N-5).

We give an algorithm for point (iv) of Theorem 15 for the case when p=pst and when the molecule is nonlinear:

Pro c e d u reV

step 1. Generate vectors

t(=l/v'N (t(, ••• ,t()T,

t~=l/v'N (t~, ••• ,t~)T, __ = T

tc-l/v~ (tc,···,tC) ,

T 3 t(=(l,O,O) eJR

T 3 t~=(O,l,O) eJR

T 3 t C=(O,O,l) eJR ,

vhich are an orthonormal basis of T. and vith Pkst=«(k'~k,Ck)T generate vectors

T T 3 Y~(P)=(Y(l'···'Y~N) , Y(k=(O,-Ck'~k) eJR , k=l, ••• ,N

T T 3 Y~(P)=(Y~l'···'Y~N) , Y~k=(Ck,O'-~k) eJR , k=l, ••• ,N

_ T T 3 YC(P)-(YC1 '···'YCN) , YCk=(-~k'(k'O) eJR , k=l, .•• ,N,

vhich are a basis in D(pst).

85

Step 2. Apply the modified Schmidt orthogonalisation67 to

where e i are the axes vectors of the Cartesian (3N-3)

center-of-mass coordinates, and the last three vectors in the

R3N are already seen as linearly dependent on the others for

every p. The orthogonalisation stops when (3N-3) linearly

independent vectors have been found.

Result: An orthonormal basis that splits in the desired way and three

zero vectors which are to be omitted.

NOTE: From a purely kinematical point of view the separation of vibrational and rotational motion is also impossible (Ref.68). If we do not ask for the forces of intramolecular motions, then we can assume the molecular ensemble to be a nonrigid N-body system. Guichardet (cf.Ref.69) made the center-of-mass system into a principal fiber bundle wi th a rotation group as the structure group. It is equipped with a connection (a structure from differential geometry) by the Eckart condition of rotationless constraint. The base manifold of this bundle is called the internal space. The fact that the connection has non-vanishing curvature gives rise to the non-separability of vibration from rotation. According to the connection theory, the rotation, vibration, and internal motion of the molecule are rigorously realized. The vibration is not equal to the internal motion, but rather it induces the internal motion. There exists a moving frame, called the Eckart frame, relative to which the molecule moves without rotation, but the frame depends on a choice of the molecular motion and is not unique for any molecular configuration. For this reason, the Eckart frame is not suitable for describing motions of a nonrigid molecule (Ref.69). The base manifold of internal motion is a Riemannian manifold (Refs. 70, 71). This conformation space of a molecule has some good properties, but one conspicuously missing property is the general local structure. It is rather complicated and not as well-behaved as one might hope. The case of a triatomic is still simple. But it is to be noted that even though the base manifold here is a trivial bundle, the connection has nonvanishing curvature, i.e., it is not flat with respect to this connection (cf. also Refs.72-74). Because we are interested in problems of the PES in this textbook, we do not go any deeper into the problems of kinematics.

86

2.6.2 ItTrue" Zero Eigenvalues: Catastrophe Points

For a function where most points yelEn are non-stationary, there are

the stationary points which organize the global qualitative

topography. Briefly said, you get most information by looking at the

points with striking mathematical behaviour!

True zero eigenvalues of the Hessian occur if the potential energy, E,

is a pure cubic potential function only.75 Normally, in an "ordinary"

potential function with quadratic terms, the third derivatives

characterize the anharmonicity of vibrations within a small

neighborhood of the energy minimum. However, if we do not assume

quadratic terms, a situation arises in which the usual concept that

eigenvalues of the Hessian sufficiently characterize the PES will be

meaningless. In this case the eigenvalues are zero. If we artificially

enforce the local quadratic approximation of the popular GF matrix

technique,76 then we may be led to the erroneous conclusion that no

catastrophe points can exist. But a correct analysis including cubic

and possibly higher degree terms does not exclude the possibility of

bifurcating reaction paths at stationary pOints. 70

We discuss the simplest two cases, namely a potential which is cubic

in one (i) or in two (ii) degrees of freedom.

(i) If H has a single zero eigenvalue then the coordinate axes can be

rotated to coincide with this eigenvector direction of the zero

eigenvalue, so that the new variable Y1 corresponds to the "zero

eigenvector" • The PES should then have a non-vanishing quadratic

approximation in all other variables y i but the first non-vanishing

term in Y1 is at least cubic. The significance of this is more easily

seen by considering the simplest potential, for which

then we find

(85)

Since gl= 3Y12 + C1 ' it is immediately seen that if C1>0 then E has

no stationary points with gl=O along the Y1 axis and the case can be

excluded, whereas if C1<0, there are two stationary points, a maximum

and a minimum (see Fig.5):

(86)

87

1.------------------.---.

o

-1~~~----~----------~

Fig.5. The one-parameter family of functions E(Y1,C1)=Y~+C1Y1 contains members with two isolated stationary points or with no stationary

3 point. They are separated by y 1 with a doubly-degenerate stationary point

with the corresponding energy value

A peculiarity occurs if the parameter C1 varies from negative values to zero: The C1=0 marks the creation of a pair of stationary points destroying the former minimum and maximum of E and building a shoulder of the potential curve, even at Y1=0 where E has a zero-gradient and a zero-eigenvalue." This is Thom's'8 simplest "fold" catastrophe. It is convenient to refer to a stationary point with gl=O which is the coalescence of several stationary points forming a catastrophe point. On the other hand, the catastrophe point Y1=0 of the term Y13 is easy to unfold by the addition of a perturbation term C1Y1; this is the general structure of all possible PES with zero eigenvalues: they are in practical examples unfolded by perturbation terms, see the early papers. '9, 80

88

(ii) If H has two zero eigenvalues, the form of E is then cubic for at

least two variables of the degenerate subspace of Y1 and Y2:

(87)

Equation (87) is already a reduced standard form coming from a

rotation of axes and a suitable scaling of E of any more general form.

E should have also a non-vanishing quadratic approximation for all

other variables Yi. Now ~e look at the 20 germ of form (87) and treat

different cases of the constants a and b. We look for straight lines

(88)

through the origin which are additional solutions of a gradient system

of E (see Sect. 3.3). Thus, we search for curves with an orthogonal

crossing of equipotential lines, and which go through the origin

Y1=Y2=0. The two first gradient equations of the potential of Eq. (87)

are 2 2 -8E(y)/8Y1 = -3(Y1 + aY2 )

(90) -8E(y)/8Y2

We introduce polar coordinates Y1= r cosB and Y2= r sinB and get

Y1Y1 + Y2Y2 r dr dt = -3 (y1 3 + 3ay1y 22 + by 23) (91)

Y1Y2 - Y2Y1 r2 dB

dt = -3Y2(2a-1)y12 + bY1Y2 -ay 22) (92)

using Eq. (88) we obtain the equations

dr r dt

3 -3Y2 f(m) , with f(m) m3 + 3am + b (93)

2 dB 3 r dt = -3Y2 gem) , gem) = (2a-1)m2 + bm - a • (94)

From Eqs.(90), it is immediately apparent that y 2=0 is an orthogonal

gradient path. A value of m with g(m)=O for which dB/dt=O corresponds

to yet another straight gradient path through the origin in addition

to y 2=0, while a value of m with f(m)=Ofor which dr/dt=O implies that

the gradient path is orthogonal to this radius vector. Thus, the roots

of f(m)=o and g(m)=O determine the qualitative form of all paths. f(m)

is already the reduced form of a cubic equation which is solved by the

Cardano formula. f(m)=O will have three real roots if

(95)

and E then is designated as having an elliptic umbilic or monkey

89

saddle (see Fig.6, region III, and Figs.10,ll). If

b 2 + 4a3 > 0

there will be one real root, the hyperbolic umbilic, while if

b 2 + 4a3 =0

(96)

(97)

on the cusp in Fig.6, then two of the three real roots become equal,

resulting in a parabolic umbilic. Similarly, g(m)=O will have two real

roots (region II in Fig.6) if

(98)

and no real roots 'if the inequality is reversed. The set of possible

potentials E depends on a space of parameters a,b of codimension two.

In Fig.6 the regions of this plane satisfy Eqs.(95-98), thus

corresponding to the different umbilics. The three distinct regions

have to be considered separately •

.. ... ... .. ; .......... : .... ...... ~ .. ... .

OM , -- --- •• --- .0..

_1L-~--~------~~--------~------~

Fig. 6. Classification of cubic potentials according to the roots of the help functions f and g. The potentials corresponding to the points indicated by +,~,o,*, are shown in Figs. 7-12.

Region I: hyperbolic umbilic

Here f has one root m1 but g has none. The straight orthogonal line

Y2=0 only goes through the origin which is analogous to case (i). All

other gradient paths are curvilinear. An ascending ridge ends in the

flat catastrophe point Y1=Y2=0 and goes uphill, forming a valley on

the other side of m1 . The PES looks like an armchair (see Fig.9 below) •

90

1

o

-1

-1 o 1

Fig.7. Hyperbolic umbilic. Equipotential lines (thin), straight ortho­gonal lines (arrows), general gradient solution curves (broken arrows), v: valley, r: ridge, hill. The potential in point + in Fig.6 is

3 2 E=Y1 +O. 75Y1Y2

1

o

-1

-1 o 1

Fig.S. Hyperbolic umbilic to region II (lines see Fig. 7). The 3 2 3 potential in point 0 in Fig.6 is E=Y1 +3Y1Y2 +Y2

91

Region II: hyperbolic umbilic also

Here f also has one root, called m3 , where Y1=m3Y2 with

m3= l' (,,15-1)/2' - l' (,,15+1) /2'

is the stra~ght equipotential line through zero which is orthogonally

crossed by gradient paths, and g now has two roots , m1=(V5-1)/2 and

m2=-(V5+1)/2, which, along with Y2=O, give a cross of the three

straight gradient lines. Valley v uphill and ridge r downhill from the

catastrophe point, in region I and Fig. 7, are now accompanied by

corresponding straight ridges or valleys which are indicated by the

gradient paths of m1 and m2 • We also have three ridges and three

valleys meeting at the zero point. The lines m1 and m2 go from a hill

to a valley. But ridge r on the left-hand side of m3 descends and

valley v on the right-hand side of m3 ascends, as well.

This means that two distinct valleys on one side of m3 are balanced by

two hills on the opposite side. The line Y2=O is, on the upper side,

the floor of an ascending valley between the two hills, and, on the

lower side, it is the ridge of a descending hill between two valleys

which descend steeper. The ascending valley attracts all orthogonal

gradient curves from the inner side of the two accompanying hills

between the m1 and m2 lines; thus all of these gradient lines approach

the origin, while on the other side of m3 they recede from the ridge

of the hill. The figure looks like an armchair (see Fig.9).

Fig.9. Hyperbolic umbilic 3 2

to germ E = Y1 + 3Y1Y2 ' giving an "armchair"

92

Region III: elliptic umbilic

Here f has three roots, m3 ,m4 , and ms ' and g also has the two roots

ml ,m2 in addition to Y250. This again is a situation in which three

valleys and three hills meet at O. The star (*) in Fig. 6 marks the

potential a=-l, 0=0, thus the Yl3_3YlY22 form of a three-fold

Fig.lO. Elliptic umbilic, lines see Fig.7. E function as in Fig.ll

Fig.11- "Monkey saddle" 3 2 to germ E=Yl -3Yl Y2

. , I i I I

. • I ' I , ! I

I I I I ~ ,

1'1 , ,

93

symmetric "monkey saddle". Exactly three orthogonal gradient curves pass through the catastrophe point Y1=Y2=O, and they are the straight lines Y2-0 and the two solutions m1 ,m2 of g(m)=o. The difference to Fig.8 is the reversed ascent of the line Y2EO. The roots of f(m)=O are labelled m3 ,m4 ,mS• Obviously the potential vanishes along these lines which divide the hills from the valleys and which are orthogonally crossed by the gradient lines. An example of the elliptic umbilic is the H6non-Heiles potential, whose classical and quantum mechanics are well known both qualitatively and quantitatively.81,82 An example is the H;-PES, in

the equilateral triangle configuration of this molecule. 83 ,84 Another application is given in atomic physics, see Eisenberg and Greiner. 8S

CUsp line between the regions II and III: parabolic umbilic

All the catastrophe points discussed above are characterized by three valleys and three ridges, which meet at the catastrophe point. If, in a limiting case, one of the ridges and its opposing valley are flat along the ridge or floor line, then we arrive at the parabolic umbilic, see Fig.12 (cf. Fig. 12 in Sect. 3.2.2). The other two valleys in the leftlower half of the landscape, in Fig.12, are divided by the flat barrier (the line y=x), which becomes in the right upper half a flat valley floor between the two remaining hills.

1

o

-1

-1"

----­.--

/~~ . ~.-- .--- -----

o 1

3 2 3 Fig. 12. Parabolic umbilic, potential E--Y1 - 3Y1Y2 +2Y2

94

As already outlined in case (i) the catastrophe P9ints can be unfolded by a perturbation, viz by e (y 1 2 +y 22) in the elliptic case (see an example in Ref.86) or by eY1Y2 in the hyperbolic case. More complicated patterns than (i) or (ii) are possible if the cubic terms in E are also omitted, and the potential development starts in some variables with quartic (higher) terms. It is an important mathematical result of the catastrophe theory of Thom78 that for a certain class of functions there are only a few essentially distinct canonical forms where the unfolding of the singularity depends on only five or fewer parameters (see Table '4). The mathematical properties of the elementary catastrophes are standard, or canonical. This means that they need only be studied once. It is for this reason that the classification of the elementary catastrophes and the reduction of each to a standard form has been useful. We have used the unfoldings of some catastrophe points in Sect. 3.2 for the illustration of valley floor paths and their bifurcations. 89

Further illustrations of cusps are given. 90 When catastrophe functions occur in physical systems, a multiplicity of physical phenomena can be

expected. These include modality, inaccessibility, sudden jumps, hysteresis, divergence, divergence of linear response, critical

Table 4. The elementary catastrophes and unfoldings of a potential87 ,88 (for simplicity x=Yl' Y=Y2)

Name Germ

Fold x3

Cusp :!:x4

Swallow-tail x5

Butterfly :!:x6

Hyperbolic umbilic x3+ y3

Elliptic umbilic x3_3xy2

Parabolic umbilic x2y:!:y4

x2y:!:yS

x3+y4

Unfolding

2 (;2x +(;lX

3 2 (;3X +(;2X +(;lX

432 (;4X +(;3X +(;2X +(;lX

(;3XY+(;2Y+(;lX

(;3 (X2+y2)+(;2Y+(;lX

2 2 (;4Y +(;3X +(;2Y+(;lX

322 (;SY +(;4Y +(;3X +(;2Y+(;lX

2 2 (;SXY +(;4Y +(;3XY+(;2Y+(;lX

95

slowing down, and anomalous variance. Not all of these

phenomenological fingerprints may be presented in a system described

by a catastrophe, but a sufficient number is usually present to make

the identification of a catastrophe unmistakable. 91 ,92

2.6.3 Flat Bottoms and Double Minimum Potentials

We assume at the bottom of a potential well a flattening of one degree

of freedom, which simply becomes shallow, for example, for the bending

of a triatomic molecule. The other two possible degrees of freedom

still remain strong pot~ntial bowls, but the third degree acquires a

nearly zero ascent. An example is the molecule LiNC (see literature in

Ref. 93), a long studied candidate for a "poly topic" molecule with

low-lying free rotor states. 94 Schleyer et al. 93 suggested that

lithium isocyanide, LiNC, is a fluxional molecule with practically

free rotation of the Li + around the CN- anion. As the CN distance

varies somewhat during the lithium rotation, so the separation does

between lithium and the other atoms, yet no structure appears to be of

significantly lower energy than another. Hence, the geometry of the

species cannot be described in terms of a fixed angle. The molecule is

nearly structure less , at least in one dimension. A curvilinear

coordinate describing the rotational path of the Li+ will indicate a

valley floor path of nearly equal height. The eigenvalue wb in the

direction of this path is zero throughout.

If the path is not totally flat, as it may be the case looking at some

microwave results,95 and LiNC in the isocyanide linear structure has a

slight minimum, then we arrive at a further example of a "quantum

cusp" for the valley path of the large amplitude bending around the

minimum (cf.Table 4 in Sect. 2.6.2).

The most important member of a cuspoid, locally describing a one­dimensional potential curve, is the "cusp,,96-101

1/4 x4 + 1/2 ax2 + cx (99)

with two parameters, a and c, thus we have a control-parameter co­

dimension of two. If a=c=O we directly have a quartic potential with a

zero eigenvalue at x=O, but it is nevertheless a potential well and a

possible bowl for an angle vibration of a molecular bending mode, see

Gilmore et al. 91 for a calculation of corresponding spectral bands.

The tangent line in x=o touches the curve in a higher order than

two. 102 A potential function is structurally stable if its critical

96

structure does not chanqe despite the addition of a small perturbation. It must be stressed that it is the shape of the potential that must be stable, not the molecule itself. The potential function (99) at a=c=O may lose stability for exceptional values of the parameters. If a and c are not zero, we can treat the control-parameter plane (c,a). The equilibrium surface for potential (99) is

x3 + ax + c = o.

The plane (c,a) is divided by the so called cusp-shaped curve, see Fiq.13, which is a bifurcation set:

(a/3)3 + (C/2)2 = 0 • (100)

Disjoint open reqions divided by the curve of Eq. (100) indicate potential curves (99) with a qualitatively different shape. curves outside the cusp parameterize potentials (99) with a sinqle minimum. The set inside the cusp parameterizes potentials (99) with a doubly deqenerate well. It is also an example of a potential function which describes a symmetry breakinq alonq a reaction path. 75 The case is often observed in chemistry. Examples are the inversion of NH3 and related compounds, the rinq puckerinq of nearly planar rinqs103 or the internal rotation of 'subqroups of atoms in a molecule (see104). Also spectroscopically well-established are quasi linear molecules, 105, 106 where the existence of a sliqht double minimum valley leads to a chanqe in the qualitative numberinq of the l-type doublinq quantum

numbers ql'

a

o

-0.5

-1

-1.5

-1 o 1 c

Fiq.13. Cusp control-parameter plane

97

In general, the direction of freedom of a nearly free motion in a

molecule will be intrinsically curvilinear. Monkenbusch107 gives a

procedure for the automatic choice of "dynamic" coordinates for

semi-rigid molecules. The rigid degrees of freedom are treated as

constraints where the corresponding force constants have non-zero

values. The non-rigid degrees of freedom are given by a set of

corresponding M eigenvectors of the Hessian where the eigenvalues are

zero within numerical accuracy. A curvilinear character of free motion

can be included in a treatment by a nonlinear coordinate transforma­

tion of the cartesian coordinates xi into M curvilinear internal

coordinates 8p

The coefficients b ip are elements of the B-matrix,108 and they are

given by the corresponding M zero eigenvectors. The coefficients c ipq are generated ina numeric way.106 There is no need to predestine the

new 8 p coordinates analytically, at least in a range where the 8 p do

not become too large. There is a derivation of analytical type c ipq coefficients for the ordinarily used internal coordinates, as

distances and angles, by the formula manipulation package REDUCE. 109

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99

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100

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McGraw-Hill, New York 109 Dachsel H, Quapp W (1991) J Math Chem 6:77

3 ANALYSIS OF MULTIDIMENSIONAL POTENTIAL ENERGY SURFACES

- PATHS -

3.1 The Simple Valley Floor Line

We assume a chemical reaction along a valley path crossing a saddle

point (SP1) of the PES. We at first also assume no occurrence of zero

eigenvalues of the Hessian orthogonal on the path, which would exclude

valley path bifurcation. starting from a minimum in the direction of

the eigenvector of the weakest positive eigenvalue of H(x) upwards in

the energy mountains, a simple and deep valley will be traced.

An example is the isomerization of HCN to CNH. The reaction should

exhibit this type of valley on the PES, and an excitation of the

bending mode v 2 and its overtones progress along the valley is 1-3 assumed. A two-dimensional illustration is given in Fig.1.

~~-..

..".. ~ .. -.. ~ ~ . ~-

;'~

-~-SP

I I~' _ I ! i I

~~M==IN~~' ! -L~~~~i~ ________ ~

Fig.1. 20 landscape of a potential valley path

102

The potential energy E(Xl ,x2 ) is defined by taking the CN diatom fixed

on the xl-axis with the zero point in the center of mass, and taking

pure (Xl ,x2 ) cartesian coordinates to describe the migration of the

H-atom around the CN kernel.

Let us choose a point (xl ,x2 ) near the valley floor line of E in the

configuration space of the coordinates, and let us expand the PES up

to the second order

E(Xl+t.Xl , x 2+t.x2 ) E(xl ,X2 ) + ( 8E 8E t.x2) '" 8Xl (xl ,x2 ) t.xl + 8x (xl ,x2 )

2

+ H 82E (t.xl ) 2 + 82E t.xl t.x2 + 82E (t.x2 )2 ) --2 2 8x 2 8Xl 8X18X2 2

Eo + gT t.x + 1 t.xT H t.x 2 (1)

Figure 2 elucidates this formula. At E=E(xl ,x2 ) we define the

tangential plane of the PES. Its slope is given by the partial

derivatives (8E/8Xl )= gl and (8E/8x2 )=g2' which we collect in the

gradient vector g=(gl,g2)T. g(x) is independent of the terms of second

order which form the harmonic part of the surface. This part may be

characterized by the distances over the corners of the tangential

plane sector. We denote the purely harmonic terms with

Fig.2. PES and

tangential plane

E

103

and calculate the principal axes of the harmonic part of the surface

by a rotation of the (Axl ,AX2 ) axial system in the local (Yl'Y2)

system:

AXl Yl cosa - Y2 sina

with a rotational angle a from tan(2a)=2H12/(Hll-H22) for

Hll""H22 •

(2)

The new axes (Yl ,Y2) represent the eigenvectors of the Hessian in

Eq. (1) at the point (xl ,x2). These vectors can be obtained by the

solution of Eq. (2):

Yl = AXl cosa + AX2 sina

Y2 =-Axl sina + AX2 cosa . We further denote

Gl = gl cosa + g2 sina

G2 =-gl sina + g2 cosa

and

All Hll cos2a + 2H12 cosa sina + H22 sin2a

A22 Hll sin2a - 2H12 cosa sina + H22 cos2a

and with Eq.(l) we arrive at A12=O and

122 E(xl +Axl ,X2+Ax2 ) = E(xl,x2)+G1Yl+G2Y2+i(AllYl +A22Y2 )

In fact, with

G = BE = BE dxl + BE dx and 1 BYl BXldYl BX2ay~'

G = BE = 2 BY2

(3)

(4)

we now have .the slopes of the PES in the direction of the new local

principal axes (Yl ,Y2 ) of the Hessian.

In a matrix representation, writing

= [cosa -sina 1 D sina cosa and D DT = E

104

we have the relations

llx D y (2' )

Y DT llx, (3' )

T T A = DT H D. and g D G, G = D g, H = DAD ,

Next, we turn our attention to an equipotential line, for instance, as

a function Y2=Y2(Y1)' which leads through the energy Eo=E(x1 ,x2). We

set E(x1+11x1 ,X2+11X2 ) - Eo 0

and obtain an expression for a second order curve in (Y1 ,Y2):

GTy + .! yTAY = 0 2

t Al1(Y12 2( :~1 )Y1 ( :~1 )2)

G 2 + + 1 + - 2Al1

1 (2 (G2 ) + ( :~2 )2) -

G 2 2

'2 A22 Y2 + 2 --x;;- Y2 2A22 (5)

where the last terms (the quadratic complements) are written down by

assuming Aii"O. Now. we have the curve for a harmonic, equipotential

line through Eo':

(6)

The r.h.s. corresponds to the radius of the curve, and its center is

M=(-G1/A11,-G2/A22)' We note that, away from the stationary points, the higher order terms are more important and the harmonic curve (6)

can give only a crude imagination of the valley structure.

Nevertheless, we can treat some special cases (cf.Fig.3). The point

(Y1'Y2) = (0,0) is the initial point of the curve; other interesting

points are y* at Y1= -G1/A11 with

G2 )112 G2 + 2)

A22 - A22

# following from Eq.(6), and y at Y2=-G2/A22 with

The different influence of the gradient G=(G1 ,G2) T and the (still

diagonal) Hessian terms (A11 ,A22 ) on the equipotential line can now be

observed.

105

y*

M

Fig. 3. Selected points on an equipotential line around the valley

floor path y#

We are interested in the valley floor direction which corresponds in

Fig.3 to the major principal axis of the ellipse. Along the path we

observe the property of a parallel behaviour of gradient and the

"least" eigenvector Yl of the Hessian. 4 Thus, for the y coordinate

system we have to assume the zero point at y# with G2=0 on the

smallest eigenvector. We observe point y# along this direction of

least ascent. In the direction of point y#, the gradient has its

minimal value, and the equipotential line has its maximal curvature.

Now we assume a general path of a PES and trace it under the local

property G2=0 from the minimum to the saddle. Near the minimum we

assume, as usual, a "convex" PES (that is, every two points on an

equipotential line can be connected by a distance lying fully inside

the potential well of lower energy), and we can draw the conclusion

All>O, A22 >0 from Eq.(7)

(7)

Hence, the harmonic, equipotential line is an ellipse having at Y2=0

the radius Gl/All around the center Yl= -Gl/All (Gl/All is the

semi-major axis of the ellipse, thus l/All >1/A22 and All<A22 ).

106

The course of a reaction should lead from this minimum with 0<A11sA22 to a saddle point which is characterized by A11<0, A22 >O. Assuming a

continuous valley we will find an inflection point somewhere on the

path (cf. Proposition in Sect. 2.4 and Sect. 2.6.3), viz, a passage of

All through zero (cf. Fig.4). At this position we find the remainder

of Eq.(5) in the following equation

2 G1 Y 1 + A22 ( y 2) = 0, (8)

(because G2=A11=O), and this corresponds to an opening of the ellipse

to a parabola

2 Y1 = (-A22 /G1 ) ( Y2) •

The slope of the valley floor line, from a global point of view, does

not dramatically change in this point: here, the slope has its maximal

value. The valley goes further uphill, characterized by the gradient

G1 ; a numerical procedure may lead to difficulties when the inverse

Hessian is used. But if we verify the value of A, the problem can be

handled. 5 Another aspect of the curvature of the valley floor is the

existence of more than one inflection point along the path. 5a

Further uphill we have A11<O, and in Eq.(7) we obtain a branch of a

hyperbola for the equipotential line through Eo. Finally, at the

saddle point we have the limiting case G=O, and the hyperbola

degenerates into two straight lines crossing each other at the saddle

point. Because the harmonic approximation (1) should be fairly

accurate near the saddle, the crossing of two equipotential straight

lines is a characteristic feature of the SP. Mezey et al. 6 proposed

the X-form of the two lines as the basis for a localization algorithm

of SPs. A compilation of the path situations mentioned above is given

in Fig.5.

Fig.4. Energy profile along the floor line of a valley

E

x

107

MIN

=0 <0

Fig.5. Equipotential lines along a simplified valley which is harmonically approximated on any energy level in x2

3.2 Mathematics of Valley Floors

3.2.1 Gradient Extremals (GE)

We generalize the two-dimensional "floor path problem" of a valley,

which we have analyzed in Sect.3.1, to the n-dimensional case,

nS(3N-3). In general, if n>2 we cannot visualize the PES in such a

dimension. We will discuss representations of equipotential sections. In the 20 case, a section E(x,y)=const of E with a "shifted" 20

configuration plane gives 10 figures of the section, curves of the

type in Fig.1 of Sect. 3.1 (E(x,y)=O is the section with the x,y­

plane). In dimensions n>2, the equipotential section of E has a

dimension of (n-1). We fistly treat an example starting with the 20

function E(x,y)=x2+y2. In the 30 space of (x,y,E), the graph of E is a

20 paraboloid of rotation. The section E(x,y)=l is the circle of

radius one at the height of the energy axis of one, or, projected into

the configuration plane, as well.

We proceed to the 30 potential E(X,y,Z)=x2+y2+z2. The 30 potential

landscape of the graph of E in the four dimensions of the 40 space

(x,y,z,E) cannot be drawn. However, we can study the section

108

E(x,y,z)=I, which is an ordinary 2D spherical surface x2+y2+z2=1 in

the 3D space of the coordinates (x,y,z). It can be considered either

as being shifted at the height one of the dimension axis of E, or, as

being projected as an equipotential hypersurface down into the

configuration space. In the general n-dimensional configuration space,

the PES is a curvilinear n-dimensional hypersurface in the (n+l)

dimensions of the space of coordinates plus E, and equipotential

sections of E are (n-l)-dimensional hypersurfaces.

In this section we study the course of valley floors in an n­

dimensional landscape. The illustrations are two dimensional examples.

DEFINITION 7-12

The projection of the floor line in the configuration space intersects

every (n-l)-dimensional contour hyPersurface E(x1 , .•• ,xn )=const at

that point (or in those points) where the absolute value of the

gradient vector is smallest compared to other absolute gradient values

on the same contour. The line is a gradient extremal. (If it exists,

and if it is a one-dimensional line, see below for a counter example).

Any gradient vector is always perpendicular to the contour

hypersurface especially to the corresponding tangential hyperplane.

This is to say that the floor line intersects any contour in that

point where the slope of the line is minimal, i. e., where contours

equidistant in energy are spaced farthest apart, Le., where the

valley is "least steep". Basilevsky and Shamov12 call a corresponding

walk mountaineer's algorithm.

We introduce the defect functional13 of the gradient norm (see Fig.l

of Sect. 2.4)

~(x) = 1/2 "grad E(X)"2 1/2 (11)

The requirement of the definition of a gradient norm to be extremal

(minimal) along a contour hypersurface E=const has to include that the

search direction is constrained to this subspace, which is connected

with the original gradient of E(x). The projection of any vector a

onto the direction of grad E is written by

po a = ( ~ P~j a j , •.• , ~ P~j a j , •.. , ~ P~j a j )T,

where i=I, ... ,n numbers the vector components which are written in

the round brackets, and where the projection matrix elements are

109

and where we assume that 2U=L(gk)2*0. As expected, we obtain po grad E = grad E.

Consequently, the matrix with the elements

(12)

generates the projection into the linear space which is orthogonal to the gradient, in such a way that it is tangential to the contour hypersurface. We get

p grad E = 1 grad E - grad E = 0 •

By constructing PV we obtain the derivative in the tangential plane of the hypersurface E=const. In the direction of this tangential plane we have to satisfy the extremal condition

( PV ) CT (x) = 0 = ~ L P.. CT ~ j 1.) Xj

for an extremal point of CT(X) along a contour of E(x). We have n

CTX • = LEE , ) k=l ~ xkxj

(13)

which is an inner product of rows of the Hessian H of E(x) multiplied by the gradient g of E(x)

with j=l, ••. ,n.

If CT*O the Eq.(13) leads to

o = ~ L Pij kL gkHkj ~ , j

i=l, ••• ,n

~ l ~ ( ,sij - gigj/pgl) 2 ) ( gkHkj) ~

(14)

We define

A(X) := l ~ gkgj Hkj I 2CT

and obtain from Eqs.(13,14) the eigenvalue equation

H g A g • (15)

It is the equation for a gradient extremal depending from the point x. Hand g are terms given with the PES. This means, a point x lies on a gradient extremal (GE), if the gradient of E is also an eigenvector of

110

the Hessian matrix of E at this point. This is the original criterion of panc1~4 for a valley ground. Remark: In the 2-dimensional case we actually obtain from Eqs.(14) the one single GE equation7- 11

(16)

which results from both components i=1 or 2 of the Eqs.(14). We proof

it for the first one with xi=x, Xj=y:

o = (Hxxgx+Hyx9y)-gx(Hxxgxgx+2Hxygx9y+Hyy9y9y)/(gx2+gy2)

= gy ~ Hxy (gy 2 - gx 2) + gxgy (Hxx -Hyy) ~.

Equations (14-16) are implicit functional relations of the coordinates x (possibly) giving a one-dimensional line. Equation (16) does not be a differential equation. We have to differentiate the given energy E=E(x) and then to probe the test point x either to give the value zero (then it lies on the GE) or to give nonzero. In the 20 case we have a simple example against the expectation to get always a line as GE. This is the model potential of a paraboloid of rotation.

EXAMPLE 1: E(x,y) x2 + y2

Since E =0, xy Exx=2, E =2 yy we have

GE(x,y) 0 (4X2 2 - 4y ) + 0 (4xy) O.

Every point (x,y) fulfills the GE equation. Thus, the whole plane itself is a 20 gradient extremal. From the equations of definition we can conclude a further property:

All stationary points are on a GE because there gi=O' i=1, ••• ,n. The direction of a GE curve shows a somewhat surprising behavior: It is not necessarily parallel to the gradient of the potential in the corresponding point, but the direction of the gradient and of the tangent to a GE can form a non-zero angle. This can be explained in terms of examples, but also by a formal derivation. We. look, again in the 20 case, at GE(x,y)=O as an implicit definition of the curve y=ge(x) and assume a displacement (dx,dy) along the GE. From GE(x,y)=O we get

GExdx + GEydY 0 or

dy dx = ge'(x) = - GE /GE , if GEy"O. x Y

(17)

The gradient components (in the kind of dy/dx ) are Ey/Ex' They are

111

in general different from the value -GEx/GEy which contains third order derivatives of E. It is to remark, that the GE, Eqs.(14) or (15), give more solutions than expected. Solutions are not only valley floors, but also crests of a ridge, and other interesting curves in a potential landscape. To discuss further details we look for some interesting model surfaces.

3.2.2 Gradient Extremals and Bifurcation Points

EXAMPLE 2 E(x,y) = 1/2 (xy+2) (y-x) (Fig.6) (18)

We find three valleys and three ridges meeting in a central region. We denote valleys by v1 ,v2 ,v3 and ridges by r 1,r2 and r 3 , clockwise. From Eq.(16) we find the formula

GE(x,y) = (x+y) (2+XY-(Y-X)~ (Y_X)2_2~1/2)

*(2+XY+(Y-X)~ (Y_X)2_2~1/2) = 0,

where the three factors give three different branches of the solution of the GE condition. The valleys v1 and v3 are connected by a GE over saddle point 2 (SP2) numbered GEl' the ridges r3 and r 1 are connected by GE2 over SP1 • Note that SP2 is lower than SP1 • The third straight GE3 , y=-x, coming uphill out of v2 meets the SP1 , and then going downhill through point (0,0) to SP2 and at last it goes again uphill along the ridge r 2 • The GE3 in v2 traces a cirque and on r 2 it traces a cliff. 10 The point (0,0) is a so called valley-ridge inflection point,14 but it is better to term it cirque-cliff inflection (eel) point11 to make it consistent with Ref.10. Here the potential function fulfils the condition PV(PVE)=O. The three GEs of Fig.6 exactly meet at the two SPs and no further bifurcation emerges. The eel point does not be a bifurcation point, see Fig 8. In Fig. 6, steepest descent trajectories orthogonal to the contour lines are possible in the six

sections of the plane divided by the GEs. Thus, except for GE3 itself, there exists no steepest descent line from any ridge to its opposite valley.

EXAMPLE 3 (PES with bifurcation points):

E(x,y) = 1/2 (xy-2) (y-x) (Fig.9) (19)

We find the SPs to be on the same contour line, y=cl(x)=x of height zero, as the eel point. Fig.9 may be discussed in terms of a double "monkey" saddle where the monkey SP is flatten to two SPs and a eel

112

y

2

o

-2

-2 o 2 x

Fig.6. Model surface (18) where vi is a valley and r i a ridge. GE are gradient extremals (fat curves). contour lines are solid except for those at -0.5,0, and 0.5 (dot-dashed)

y=-x

G~

Fig.7. Energy profile over GE3 in Fig.6

113

Fig. 8. Surface with CCl point. Contours are solid lines, GEs are dashed for a valley, dot-dashed for a ridge, and dotted along a cirque or cliff from bowl B to summit S

point in between, cf.Fig.l0. One readily ascertains from Eq.(16) the

condition

GE(X,y) (x+y) (2-XY-(Y-X)~ (Y_X)2+2~1/2)

*(2-XY+(Y-X)~ (Y_X)2+2~1/2) = ° which again gives three GE branches. GE3 from northwest to southeast goes steadily uphill through the CCl point (0,0). SP1 connects v1 and

v 3 ' or r 1 and r 3 . Two branches of GEl coming from v1 and v3 go over SP1 and SP2 , as expected, and flow into valley v 2 • There they meet at the point

BP = (-(2/5) 1/2, (2/5) 1/2) • 1

Analogously, two branches of GE2 , going down along the ridges r 1 and r 3 , cross the SPs and meet at

BP = ( (2/5)1/2,_(2/5)1/2) 2

on the ridge r 2 crossing GE3 •

114

y

2

o

-2

-2 o 2 x

Fig.9. Model surface (Eq . 19), where BP means a bifurcation point

Fig.10. View uphill on the surface of Fig.9 from valley v2 along GEl

115

If we look at the valleys, we expect a bifurcation of v2 into v1 v3 • The bifurcation point (BP) on the ascendinq GE3 indeed exists the two side valleys emerqe only very sliqhtly. We qet two BPs of by the condition

and but

GE3

(20)

where u is the qradient norm (11) and t the tanqential direction of a contour line in Eq.(12). (Note du/dt=O is the condition (13) for aGE, but Eq. (20) contains already third derivatives of the PES!). Ansatz (20) qives the formula system15

Ett (Ett - Eqq) + Eq Eqtt

Etq

o

0, (21)

where q,t denotes direction derivatives in the qradient and the tanqential direction of the contour line. In the symmetric model of Eq.(19), if we are interested in the points of the diaqonal" GE3 with y=-x, the direction derivatives can be easily obtained by a -n/4 rotation of the (x,y) coordinates in (q,t) "coordinates":

x = (t+q) Ivz y = (t-q)/vz We qet the representation of the surface, Eq.(19)

2 2 E(t,q) = -q (t -q -4)/~ ,

and the derivatives

Eq 2 2 Et tq Ivz -(t -3q -4)/~ =

Eqq = 3q Ivz Ett= -q Ivz Eqtt= -1/ vz .

With condition Eq.(21) we qet t=O, 5q2+t2_4=0, thus,

q = ± (4/5)1/ 2

and, by a back-transformation, the BPs result in (x,y) coordinates. They are the triple points ~ and m in the Basilevski classification.7

The trident (or pitchfork) form of the three branchinq GEs is a standard fiqure of catastrophe theory (see Sect. 2.6.2). It is a normal case in branchinq events of reaction paths that the BP does not be a SP.7 But the cusp of the bifurcatinq path normally assumed is an erroneous -imaqination (as it is found in Ref.8).

CT

10

8

6

4

2

°

116

........ 9.. . ...................... -.................................... .

°

.... ' . .....

Fig.ll. CT profile over contour lines (cl) of Fig.9,. cl through point (-1,1) or (1,-1); b cl through BP1 or BP2 ; c cl through point (0,0)

What happens in a bifurcation point which is no SP? On GE3 ' the gradient takes an extremal value if we test it over any contour line (see Fig.11). Thus, v2 is a valley uphill to BP1 , between BP1 and (0,0) v2 ends in a cirque. In the eel point (0,0), the potential is flat in tangential direction, i.e., Ett(O,O)=o. Between (0,0) and BP2 there is a cliff, and r 2 is a ridge from BP 2 uphill. In BPs the CT-curve over the contour line flattens to a nearly horizontal bottom, Fig.12. The shape of the CT-curves has the form of a Ginzburg-Landau potentia19 of forth order CT(t)=at2+bt4 where the coefficient a changes its sign in a BP, but the coefficient b is positive, see Fig.12. In a local neighborhood of the BP, the points of the contour line show the same CT value. Locally we have a situation analogous to Example 1, since we get the GE property for a piece of a contour line. Between BP1 and the eel point (0,0), in addition to the maximal CT on the contour line indicating GE3, we get two minimal CT points indicating the branches of GEl (see Fig.11c). On GE3, the straight y=-x, we have the values

CT = (1+ 3/2 X2)2,

and on the contour line zero, y=x, we have

CT = (1- X2/2)2.

117

Leaving SP1 along a steepest descent path, we observe the same

direction as for GEl. Thus, GEl indicates a real valley over SP1 , and

later GEl turns to the right with its valley ending by flattening in

BP1 on the slope of cirque v 2 • The valley of GEl near BPI is

imperceptible to such a degree that the gradient of the surface nearly

suppresses the influence of the valley curvature. Orthogonal

trajectories, flowing over the axis y=x along the piece between the

SPs, cross the GEl in cirque v 2 and the GE2 on cliff r 2' almost

without distortion. The reason is the possible divergence of the

direction of the gradient and of the vector tangential to a GE as

mentioned above. Again in the CCI point we do not find any branching.

In (0,0) we have a non-zero gradient, but the Hessian

H [ -y y-X) y-x x

has two zero eigenvalues, as in Example 2.

Fig.12. Ginzburg-Landau potential

function9 f(a,t)=at2+bt4 ,b>0, the

sections of which are shown in

Fig .11. The point t=o, a=O is a

parabolic umbilic (cf.Sect.2.6.2)

118

EXAMPLE 4: (22)

has an additional parameter ~. The case ~=1 is the HNR surface. 10 It

is easy to execute derivations of the GE condition, Eq. (16), using

model surface (22). It giyes a polynomial of fifth degree in x and y

which implicitly defines the GE curves. If we use the advantage of a

2D model, we can solve it by a point-by-point calculation over a ciose

grid and draw the contour curves GE.(x,y)=const=zero. This is done in

Figs.13-1S. Model surface 4 (Eq.22) is an asymmetric distortion of

models 2 and 3. This results in a dramatic change of the GE behaviour

in the central region of the surface. With the experience of Examples

2 and 3 we can give an explanation of the strange central pieces of

the GEs in Figs.13-1S.

In Fig.14 we observe a straight GEl from south to north, viz from v 1 to r 1 • The two other GEs cross GEl at true BPs of the type in Example

3, but with a nonorthogonal crossing angle showing a "skew trident".

If we vary the parameter ~ we shift mainly the SP2 • For ~>1.08 it is

raised, for ~<1.08 it is lowered. In both cases we "lacerate" the BPs

and only retain some turning points of the corresponding GEs. The

y

2

o

-2

-2 o 2 x

Fig.13. Surface of Eq . (22) with ~=l.lS

119

y

2

°

-2

-2 ° 2 x

Fig.14. Surface of Eq.(22) with ~=1.08

returning form of the central GE in the former bifurcation region can

be better observed in case ~=1.15 (Fig.13). Here, the branches meet at

an acute angle. In the HNR case (Fig.15) we get obtuse-angled

branches. with the valley floor path GEl and the crest ridge path GE2 ,

the representation in Fig.15 gives a pattern like that in Example 2.

Only the GE3 seems very strange, showing a strong deviation from the

straight path (Example 2) in the central region. On contour lines in

valley v1 , which are lower than -1.5, we have at the crossing with GEl

by definition a minimal value of u. On the contour line through (0,0),

the u profile suffers a change: a shoulder emerges. We have the

gradient

Ex(O,O) = 0, Ey = 3/2,

and the Hessian elements

The gradient actually pointing in the y-direction is parallel to an

eigenvector (which in addition is a zero eigenvector). Hence, the

120

point (0, 0) is a GE point. It is a double point in the Basilevski

classification. 7 If we go uphill to the left from (0,0) we can yet

trace a new valley floor of GE3 up to SPl which must be characterized

by a second minimum of u. The u shoulder splits into a minimum and a

maximum on higher contour lines. In contrast to Figs.9 and 14, the

origin of the two new u extremals is outside the old GEl. A maximal

value of u belongs to the central arc of GE3 between (0,0) and (2,2).

It begins at (0,0) in tracing the end cirque of v l . Going uphill it

does not find a simple eel point but a broad intermediate region which

is characterized by a contact line of a flank of the central ridge r l and of a flank of valley v l leading to SP2 • The GE3 passing this eel belt, and the contact line dividing the flanks of ridge r l and valley

v l ' do not coincide, see Fig.15. They do, however, cross at the point

(0.685, 0.920), where the Hessian H has a zero eigenvector parallel to

a tangent t in a steepest descent direction. At this point the GE Eq.

(16) becomes

o

-2 o +2

Fiq.15. surfacelO of Eq. (22) with 1l=1.0, including GEs and a piece of the contact line (dotted)

121

(t H) 9 = 0 9 = O.

Thus, the point is on the GE3 • On the remainder of the contact line we

have still a zero eigenvalue of H, but the corresponding eigenvector

is not parallel to t. From the crossing point downhill to (0,0) the

GE3 traces the maximal cirque region of the flank of vI' and uphill to

(2,2) the GE3 traces the maximal cliff region of the flank of r l . So,

over three contour lines, the CCl belt sUbstitutes for the CCl point

as in Examples 2 and 3.

Note that a GE which is a flank can complicate the understanding of numeric results. The reader can find an example (Ref.16) in case of BeH2-dissociation, where such a flank is misinterpreted as a valley.

The model surface Fig.15 consists of three valleys and three ridges

meeting in an asymmetric central region. An intuitive concept of a

valley vI bifurcating uphill to SPI and SP2 , and of a ridge r l also

bifurcating downhill to SPI and SP2 , is not supported by the GE result.

The valley floor from SPI downhill ends in (0,0), analogously the

crest to the right from SP2 uphill on r 1 in northwest direction ends

in (2,2) (see also Ref.17). The end of a valley floor downhill is

characterized by Basilevski's double point7 which in mathematical

terms is 'a turning point. Of course, from the end point of the floor

we can draw a steepest descent line and thus get a connection, or a

bridge, over the old lacerated bifurcation region downhill.

As a conclusion we note, that for ceasing or beginning valleys the GEs

are perhaps of importance if electronic reorganization of bonds in

molecules acts somewhere in the geometry change, and not exactly in

the equilibrium geometry or at the saddle point.

3.2.3 Gradient Extremals for Higher Dimensional Cases

In the last point we have treated 2D examples, which are only of

interest for visualization, not yet for real chemistry. The minimum of

degrees of freedom of a triatomic molecule is three. We return to

Eqs.(14)

0

thus,

0

o

in Sect. 3.2.1- We have

~ 1: gkHki - g, 1: 1: g,gk k l.jk 1.

~ 1: 2 1: gkHki -1

(gl) k

gi

1: g, ~ gJ' (H g) i j J

Hkj I 1: 2 }, i=l, ••• ,n, (gl)

I

1: 1: g,gkHk' j k J J

}, yet, 1 rename in j

(23)

122

In case n=3, we can introduce the vector product, and we find in the 3

equations the three components of gxv, where v=gx (H g). All the

components of gx(gx(H g» have to be zero.

REMARK: There are some preliminary concepts for a numerical tracing of a GE (Ref. 12 , 18,19). It is the feeling of the authors that these approaches are not fully satisfactory. This is because of the extent of computational effort, or by the oversimplification of the PES studied in Ref.18.

3.3 Steepest Descent Paths (SDP)

In Sect.3.2 some examples are given as to whether the valley succeeds

in developing or whether it becomes flattened, finally disappearing at

the ends. The latter in general occurs in regions of the PES far away

from the stationary points, but bifurcations can also emerge in an

SP.14 To detect such bifurcational behavior, we need the mathematical

tool of gradient extrema Is and their bifurcation points, because

gradient extremals give a local characterization of the corresponding

path. In contrast, a steepest descent path cannot bifurcate on a

continuously differentiable PES. In mathematical terms a gradient

system is an autonome, differential equation (Le., the gi do not

explicitly contain the "time" t), which cannot allow a bifurcation in

points x other than those values of x with grad E(x)=O. But in a

global view on a PES, a gradient path is perhaps the only practicable

way for constructing a potential hypersurface section for reactions of

polyatomic systems with N>3. 20

DEFINITION: A gradient system is a system of ordinary differential equations

8E(x)

8Xi i=l, ..• ,n (25)

for n functions of the coordinates xi=xi(t) which describe the path of

steepest descent in the configuration space, corresponding to the

descent of the potential function E=E(x). If mass-weighted cartesian

coordinates are used (see Sect. 3.4), and if this path is defined as

starting at the transition structure (SP) from which it goes down to

the reactants and down to the products, we have the intrinsic reaction coordinate (IRC).21

Note, that the value of the energy E(x) may only decrease in t in all

solutions of Eq.(25). This is because

123

_ ~ (dXi) 2 = t aE (x) dXi = dE th ddEt :5 0, L at L aXi dt at' us

and as t -+ flO, th$ solutions of Eq. (25) must conv$rg$ to any stabl$ minimiz$r.

computation 2i SDP

Equation (25) S$$ms w$ll adapt$d to a num$ric application by changing th$ diff$r$ntial $quation into a diff$r$nc$ $quation, thus giving th$ oPP'ortunity to traC$ point by point a st$$P$st d$sc$nt path with an Eul$r int$gration. How$v$r, a corr$sponding proc$dur$ giv$s ris$ to som$ probl~s; in flat r$gions of th$ PES it shows a prop$nsity to zig-zagging about th$ tru$ path (cf. S$ct. 2.4.4). Th$r$for$, as int$nsiv$ t$sts hav$ shown, th$ st$p-siz$ must b$ V$ry small,22 th$ computation of too many $n$rgy valu$s b$com$s too $xp$nsi V$ for ab initio calculations, in spit$ of th$ simplicity of th$ approach.

W$ r$port h$r$ an improv$d algorithm for th$ proc$dur$. It follows th$ d$sc$nt path with comparativ$ly larg$ st$P siz$s, which dO$S not r$quir$ th$ full calculation of th$ H$ssian at points along th$ path. 23 ,24 This algorithm is W$ll suit$d for mol$cular orbital calculations, wh$r$ th$ en$rgy and th$ gradient calculation are costly and must be k$pt to a minimum. First, W$ change th$ param$t$rization of Eq.(25) by a normalization of th$ right hand side of Eq.(25) into a unit l$ngth V$ctor

(26)

Th$ solution curv$ x=x(s) is param$t$riz$d by th$ curV$ l$ngth s (S$$ also S$ct. 3.4 below, r$garding this and other in variants). An Euler step would be given by

(27)

where s is th$st$P siz$ and k th$ numb$r of a corr$sponding point on th$ path. In changing this strategy, W$ choos$ th$ n$xt point xk+l so that th$ d$sc$nt path b$tw$$n xk and xk+l is an arc of a circl$, and so that the gradi$nts gk and gk+l ar$ tangent to this arc path. (W$ assum$ h$r$ that both gradi$nts ar$ in the normal plan$ of th$ descent path in xk. This is th$ plan$ spann$d by th$ tang$nt and th$ normal V$ctor of xes) at th$ point xk. Hence, we ignore a possibl$ torsion of xes) in going from xk to xk+l. This is an $rror of third order. ) Th$n, th$two tang$nts to th$ circl$ int$rs$ct at a point ;k+l on th$

124

normal plane. Because gk and gk+1 are the two tangents, the points xk, ~k+1, and xk+1 form an isosceles triangle. The pivot point ~k+1 is found by first taking an Euler step (27) of length s/2 along the gradient gk

(28)

At the point ~k+1 no calculation of E is· performed. A constrained minimization is carried out to determine the desired point xk+1 on the surface of a sphere of radius s/2 and centered at ~k+1. The result of this minimization is xk+1. Because of the constraint, the residual gradient gk+1 is parallel to (xk+1 _ ~+1). Hence, xk and xk+1 actually lie on an arc of a circle with tangents ~ and gk+1. The algorithm requires a constrained (n-1)-dimensional optimization. However, very larger step sizes can be used to achieve the same accuracy in following the descent path.

",

'I'~k+l_ xk~ = s/2

a 1 2

Fig. 16. Descent path following algorithm (see text). The vectors point downhill, indicating the orientation of -g

In order to perform a constrained optimization, it is convenient to make use of the Taylor expansion of energy (in second order). We have

(29)

where E* is the energy at a point x*, (we omit the superscript k+1). Let Ax be the displacement vector from x* toward the optimization

k+1 * k+1 , point x , where both x and x are on the sphere of radl.us s/2

125

"k+1 centered at x • We define (see Fig.17)

* * "k+1 P x - x

xk+1 * Ilx - x ,

P xk+1 "k+1 - x *

P + Ilx.

"k+l X

Fig.17. Definition of vectors used in the constrained optimization

(30)

since the radius of the sphere must be fixed at s/2, the energy E has to be minimized with the constraint

T 2 P P = (s/2) • (31)

Therefore, with Eqs.(29) and (31), we have a Lagrange function.

(32)

where i\ is the undetermined multiplier. Since 8L (i\) 181lx and 8L/8i\

must be zero at the minimizer xk+1, the following expression for Ilx is

obtained:

Ilx = -(H-i\I)-l (g - i\p*) . (33 )

The value of i\ can be obtained from Eq. (33), when substituting Eqs. (31) and (30) into (33)

(s/2)2 , (34)

126

which can be solved iteratively for A with a suitable initial

estimate. A must be less than the lowest eigenvalue of H in order to

ensure that the path is followed in the descent direction (cf.Ref 25).

A simple Newton-Raphson root search is sufficient for the present

algorithm. To start the optimization, we use the true Hessian at the

transition structure. At each subsequent step, however, the Hessian H

can be updated, for example, by the BFGS formula (see Sect. 2.4.2). An

advantage of this method is that the change in the Hessian may be

managed to be positive definite.

A new technique for the calculation of the SOP between two given

extremizers (minima or saddle points) has recently been given. 24a The

algorithm is based on discretizing the SOP by a grid of M points xk,

k=l, ••• ,M where each of the points is a replica of the complete

system. We choose

as a unit vector. If the xk. are on the SOP, and if the sk point

towards the slope of the path at xk then

The potential is minimized in all directions except in the direction

of the path. If the xk are sufficiently close to each other, but not

on the SOP, we can calculate gproj*O' The essence of the method is to iteratively refine the initial guess xk (which can be quite poor) to

the complete SOP. It uses the coupled "quenched" trajectories for all

the M grid points

8/ Xk{t) = • 8t gpro] , k=l, ••• ,M. (37)

The limit of large t provides the SOP; but note that the estimate in

Eq. (35) for the slope of the path is changing at each step of

integration in Eqs.(37).

3.4 The Independence of Steepest Descent Paths from

Parameterization and Coordinate System

3.4.1 Parameterization

We assume a solution curve x=x{t) of the gradient system Eq.(25) in Sect. 3.3

127

i=l, ••• ,n, (38)

starting in any non-stationary point x°=x(O) , and usually leading to a minimizer xmin. The solution x=x(t) is a one-dimensional curve in Rn

and t is the curve parameter describing the functional connection of an interval [O,tend] with the steepest descent curve from XO to xmin

in the confiquration space corresponding to the steepest descent on the surface E=E(x). The curve has a curve lenqth s (which we measure beginning in XO along the path) with

2 n 2 ds - E (dxi ) •

i

By a corresponding parameter transformation dt=f(s)ds, i.e. s

t = J f(s)ds o

we obtain

dxi f(s)ds

8E 8xi

or

(39)

(40)

nota bene, with a still unknown transformation function f(s). Yet, we change Eq.(39) in

dxi ds = L ( ds )dxi

by Eq.(40), and further

1 = -L f(s) 8E dxi • 8Xi ds

If we assume E=E(x(s» along the path then we can construct

dE = L 8E dXi , ds 8Xi ds

thus we get

(41)

as normalization factor in a generalized Eq.(40) where the parameter s is now the curve length in Eq.(39). On the other hand, we can further change Eq.(41) again using Eq.(40)

or f(s) 1/ Dqrad En (42)

128

Hence, the differential equation system for a steepest descent path in

the curve length parameterization of the cartesian coordinates is

aE / ~grad Ell , axi

and the tangent vector dxi/ds has unit length.

(43)

Remark that we treat one and the same curve line of steepest descent

from x O , namely x=x(t) or x=x(s) in the configuration space Rn of the

cartesian coordinates x. The difference is the parameter interval

[O,tend ) or [O,Send) to describe this piece of the curve.

If we. have a PES of purely harmonic character (in the neighborhood of

a minimizer as starting point)

E(x)

we obtain an analytic solution of Eqs. (38), dxi/dt=-wixi' by

xi(t) = x i o exp( -Wit) , i=l, ••. ,n. (44 )

Here, the steepest descent interval starting in t=o with point x O , is

finished at the minimum E=Eo at the minimizer xi=o at the "end of the

interval" t end=", • There is grad E(x) =0, and f(s) is singular. The

simple ansatz (38) leads to a difficult parameterization of the

descent curve (cf. Ref.27) but the approach (43) gives the descent in

a numeric available way outside the near neighborhood of stationary . t 22

po~n s.

3.4.2 Invariance From Coordinate System

Foremost, we have to remark ~hat a steepest descent path is in general

not identical with a valley floor at all, even if it is coming from a

saddle point (see Sect. 3.3). This is due to the fact that a valley

floor has to be described by a gradient extremal curve which can have

a direction other than the gradient itself (see Sect. 3.2, Example 3,

and Examples 4-6 with valleys which flatten out somewhere at the

slope). In Fig.18 we give a view of such a drastic case. 23

We discuss the influence of a coordinate transformation on the path

definition using the case of steepest descent, because the definition

of a gradient path is a mathematically very simple, and we may here

better understand the problems of coordinate dependence than in a more

complicated path formulation.

129

E

Fig.ll. A valley not leading to a saddle point (SP)23(the dotted curve is the steepest descent path from the saddle point)

What does the independence from coordinate system mean? 28 , 29 The potential E(x) is defined by the mutual positions of the atoms in a chemical system. A configuration xa of 'atoms corresponds uniquely to an energy Ea=E(Xa ), any other to Eb and so on. If we have Ea>Eb, so this relation holds in every system of coordinates chosen for the description of the positions of xa and xb • Furthermore, we must ensure that all positions of the molecule giving an equipotential hypersurface, have to be independent from a coordinate transformation. From this it follows that the description of stationary points is independent from the choice of the coordinate system. The definition of descent or other paths, however, includes the use of angle relations. For example, the steepest descent path goes orthogonal to equipotential hypersurfaces. Here, orthogonality is a measure coupled to a given coordinate system and the corresponding definition of the scalar product of vectors. In general, a coordinate transformation changes the angle relations of the coordinate axes. Hence, any definition of a path is changed which uses angle relations. The simplest model potential to understand these circumstances is the 2D Example 1 (Sect. 3.2) and a linear transformation of one axis only:

E(x,y) = x2 + y2 , p=mx , q=y, with m > 1 • (45)

130

q

Fig. 19. The change of gradient paths by a

transformation. 29 ,30 In the (x,y) system, gradient lines, where in (p,q) system they are curvilinear. the transform of the path on the left hand side

p

linear coordinate

paths are straight The dashed line is

We obtain E(p,q)= p2'm2 + q2, where circles, as the original equi­

potential lines, become ellipses. In Fig.19, we show the relations of

corresponding gradient paths.

The mathematical conception of an independent definition of geometric

subjects (as reaction paths) in the configuration space starts with

the idea of an analogous transformation of the coordinates as well as

the angle relations in the new system. The distortion of equipotential

lines in the new system should be compensated by an inverse distortion

of the scalar product defining the angles.

If we treat two coordinate systems, x and q, we should have

the transformation equations

and we

B-matrix

can determine the

elements31 which

tangent vectors of a path

k=l, ••• ,n

8qk,8Xi expressions giving the

act as a transformation matrix

( 46)

Wilson

of the

( 47)

If we write E(x)=E(q(x»=E(q) we obtain a transformation of grad E by

131

In the gradient system (38), every equation

r k

j=l, ••• ,n

has to be multiplied by Bqi/BXj • Summing up all n equations, we

obtain with Eq.(47)

( 48)

( 49)

We define the contravariant metric coefficient of transformation (46)

We arrive the gradient system

dq gki BE i=-r _, i=1, ••• ,n • (50) dt k Bqk

The solution curve q=q(t) of system (50) is developed from Eq.(38) by

pure mathematical steps, hence, per definition, q(t) is the same path

as the original x(t). We obtain an independent definition of this

path! The prize is a more complex system of Eqs.(50) against (38). In

the new coordinate system q, the solutions of Eqs. (50) do not cross

the equipotential lines of E(q) under an orthogonal direction.

The solution curves with orthogonal crossing should be given from the

simpler system of equations

dqi = _ BE i=1, ... ,n (50' ) dt Bqi

analogous to definition (38) •

In case of transformation (45) we have

gll m2 g22 = 1 g12 = g21 0 , and the Equations (38) become

dx = -2x, dy -2y, but with (50) we get dt dt

dp

dt

11 8E -g 12 8E - g

132

= -2p and dq

dt -2q

The new equations are equivalent, as expected. The distortion by the

coordinate transformation (45) in E is compensated by the action of

the metric coefficients.

The way to define a path independent from any coordinate system used

must start with a genuine PES, which is given in the reference

coordinates. This is a question of physical foundation of the genuine

coordinate system. If we deal with pure cartesian coordinates we have

the genuine potential energy of the nuclei in the field of electrons,

in the original 3-dimensional space of our universe, (where we ignore

the Einstein 40 space-time, for simplicity).

The unfortunate complication of the possible independent definition of

a steepest descent path is that this path may not agree with the

location of the valley floor.

3.4.3 Mass-weighted cartesian Coordinates

A convenient choice32 ,33 of a coordinate system to describe

approximately a line from a saddle point to a minimum as a transformed

steepest descent path, is a mass-weighted set of coordinates, as used

implicitly in the normal mode analysis. Mass weighting originates

with the spectroscopy of small and stable vibrations of a molecule. 31

It takes a diagonal mass tensor which is then invariant to any

subsequent unitary coordinate transformation. 27 - 29 Truhlar34 describes

these as "isoinertial" coordinates. The mass-weighting transformation

does not necessarily introduce units of mass into the coordinates.

Each coordinate can be re-scaled, if desired, to some unit mass

choice. This amounts to a relative mass-weighting. 35

DEFINITION

If xi are the cartesian coordinates of the N atoms of a molecule, then

with '_ [i+2] ]- , -3-

i=l, ... ,3N (51)

are the mass weighted Cartesian coordinates, where mj is the mass of

the j-th atom, (and the angular bracket (z] means the entire part of

the real number z).

REMARK: Eqs. (51) are linear transformations in the way outlined in Sect. 3.4.2 (see example of the Eq.45).

133

Finally, we treat the action of a mass-weighting transformation for

the PES in a diatomic and a triatomic molecule.

EXAMPLE 1. Diatomic potential

We treat the potential of CH in Murrell's HCN-PES. 36 There is an

equilibrium distance of r CH=1. 0823 A , and atomic weights of 12 au

and 1 au, respectively. The zero of the laboratory frame should be the

equilibrium structure, and a 2D coordinate system is chosen with the

x-axis for displacements of the C-atom, and the y-axis for

displacements of the H-atom, along a fixed line without any rotation.

Any shortening or lengthening of the CH-distance produces a reactive

force. But, a change of the position of the molecule as a whole,

against the laboratory frame, does not change the potential of CH.

Thus, the equipotential lines of constant CH potential are the

straight lines y-x=constant, in our coordinate system. If the

CH-distance is distorted then the force of the potential energy acts

as a push on both nuclei of exactly the same amount29

II Fx II = II Fy II·

Fig.20. The potential

energy "surface" for

CH. Coordinate x is the

distortion of C, and y

is the distortion of H

against a laboratory

frame. The equipoten­

tial lines 0 to 3 in -1 1000 cm steps are the

lines y=x+constant, the

gradient is the ortho­

gonal direction y=-x, but CH vibrates along

the v=-12x line

y[1q

x[iq

134

In Fig.20 we obtain the resulting force vector along the direction

y=-x orthogonal to the equipotential lines, which is the direction of

steepest descent as well. Contrary, the Newton law of "force=mass

times acceleration" enforces that the two accelerations of the real

atoms C and H are very different. Because a relaxation takes place

under preserving of the center of mass, the resulting direction y=-12x

is the geometric location of a CH vibration. In the 20 configuration

space (x,y), the gradient of the potential points in a very other

direction than the real vibration goes on, because the potential force

and the kinetic energy are coupled by very different masses.

The key idea of mass weighting, for instance in Fukui's term the

intrinsic reaction coordinate (IRC) as a mass weighted steepest

descent,32 is the transformation of the (x,y) coordinate system such

that the transformed gradient direction coincides with a real

distortion under conservation of the center of mass.

EXAMPLE 2. A linear triatomic molecule

We treat the molecule HCN. There is a variation between 1 au and 14 au

of the masses included. Thus, the mass weighting should have an

enormous influence on the shape of the PES. This indeed is the case.

We choose a 20 section of the 30 configuration of internal modes, the

two linear stretching modes vI and v 3 • We have the equilibrium

structure r CH=1. 065 1, r CN=1.153 1 , and in ~-axis configuration

(1}i=<i=O) 000 ~ 000 ~ 000 .Q

~1 =~H =x1 =1.622 A, ~2 =~C =x4 =0.557 A, ~3 =~N =x7 =-0.594 A.

This is a point in the 90 space of all 3N cartesian coordinates xi' or

a point in a 3D subspace (X1 ,X4 ,X7 ), treated below in Fig.21. The zero

of the coordinate system is the center of mass, and we allow only

collinear movements without ~ change of the center of mass. Hence, the

3D problem is constrained to two degrees of freedom in a 20 plane. The

center of mass equation is

(52)

and the 20 plane of collinear movements is depicted in Fig.21 by the

dashes

and

For changes of r CH and rCN there are the retrograde force constants

F33 = 6.25 and

Every change of the linear configuration, which leads to the same

135

change of potential energy, gives a point on an equipotential line in

the 2D collinear configuration plane of HCN. This line is, at the

harmonic level of approximation of the force constants F11 and F33 , an

ellipse with half-axes of a ratio of nearly 1: 3 . Our equipotential

ellipse is depicted in the left part of Fig.21. The thick arrows are

the pure CH or CN stretches. Because of Eq. (52), in a pure diatomic

stretch the remaining third coordinate is changed as well, and the

system point remains on the plane of the dashes. Mass weighting (51)

with ~ = 1.008, mC = 12.0, ~ = 14.003

gives Q 0= 1.929, c

o QN =-2.223 •

Fig.2l. Change of an equipotential ellipse of linear HCN stretching by mass weighting (see text)

We nearly obtain the QH-value by a parallel transport of ~, but the

ratio between the Xc and ~ values is slightly turned in direction of

-xN for Qc and QN. In the right-hand side of Fig.21, we transform the

ellipse from xi to Qi coordinates. We find a large stretching of the

two directions in the (xN,xC)-plane. The result comes out with a quite

different equipotential ellipse in the Qi-coordinates, where the half

axes have changed their order! The v3 vibration of CH is now stronger

(3311 cm-l ) than the v 1 vibration. of CN (2097 cm- l ) in HCN. The

136

"minimum energy path" in xi-system would be the CH distortion

direction, but in the Qi-system it is now a CN stretch. Thus, the mass

weighting "changes" the MEP! (Remark, we have ignored the third degree

of freedom of HCN, the bending mode, which really is the lowest mode

and indicates the true MEP in the xi as well as in the Qi system).

References (Chapter 3)

1 Babic Z, Light JC (1987) J Chem Phys 86:3065

2 Peric M, Mladenovic M, Peyerimhoff SD, Buenker RJ (1983) Chem Phys 82:317

3 Urban M, Bartlett RJ, (1988) J Am Chem Soc 110:4926

4 panci~ J, (1975) ColI Czech Chem Corom 40:1112, (1977) 42:16

5 Scharfenberg P (1982) J Comput Chem 3:277

5a Anet FAL (1990) J Am Chem Soc 112:7172, Kazansky VB (1982) in: Kolotypkina JM (ed) Physical Chemistry -Modern Problems, Chimiya, Moscow, p 7, (russian)

6 Mezey PG, Pederson MR, Csizmadia IG (1977) Can J Chem 5:2941

7 Basilevski MV, (1982) Chem Phys 67:337

8 Baker J, Gill PMW (1988) J Comput Chem 9:465

9 Venkataraman G,Balakrishnan V (1988) in: Caglioti G (ed) synergetics and Dynamic Instabilities, North Holland, p 175

10 Hoffman DK, Nord RS, Ruedenberg K (1986) Theor Chim Acta 69:265

11 Quapp W (1989) Theor Chim Acta 75:447

12 Basilevski MV, Shamov SG (1981) Chem Phys 60:347

13 McIver JW, Komornicki A (1972) J Am Chem Soc 94:2625

14 Valtrazano P, Ruedenberg K (1986) Theor Chim Acta 69:281

15 Basilevski MV (1987) Theor Chim Acta 72:63

16 O'Neal D, Taylor H, Simons J (1984) J Chem Phys 88:1510

17 Rowe DJ, Ryman A (1982) J Math Phys 23:732

18 Jorgensen P, Jensen HJA, Helgaker T (1988) Theor Chim Acta 73:55

19 Shida N, Barbara PF, Almlof JE (1989) J Chem Phys 91:4061

20 Ischtwan J, Collins MA (1988) J Chem Phys 89:2881

21 Fukui K (1980) J Phys Chem 74:4161

22 Garrett BL, Redmon MJ, Steckler R, Truhlar DG, Baldridge KK, Bartol D, Schmidt MV, Gordon MS (1988) J Phys Chem 92:1476

23 Schlegel BH (1987) in: Lawley KP (ed) Ab Initio Methods in Quantum Chemistry I, Wiley, New York, p 249

24 Gonzales C, Schlegel BH (1989) J Chem Phys 90:2154

24a Ulitsky A,Elber R (1990) J Chem Phys 92:1510

25 Banerjee A, Adams N, Simons J, Shepard R (1985) J Phys Chem 89:52

26 Simons J, Jorgenson P, Taylor H, Ozmut J (1983) J Phys Chem 87:2745

27 Tachibana A, Fukui K (1979) Theor Chim Acta 51:189

131

28 Quapp W, Heidrich D (1984) Theor Chim Acta 66:245 29 Quapp W, Dachsel H, Heidrich D (1990) J Mol struct (Theochem)

205:245 30 Mezey PG (1987) Potential Energy Hypersurfaces (studies in

Physical and Theoretical Chemistry 53) Elsevier, Amsterdam 31 Wilson EB, Decius JC, Cross PC (1955) Molecular Vibrations,

MCGraw-Hill, New York 32 Fukui K (1974) in: Daudel R,Pullman B (eds) The World of Quantum

Chemistry, Reidel, Dordrecht, p 113 33 Truhlar DG, Kupperman A (1971) J Am Chem Soc 93:1840 34 Truhlar DG, Brown FB, Steckler R, Isaacson AD (1986) in: clary DC

(ed) Theory of Chemical Reaction Dynamics, Reidel, p 285 35 Dykstra CE (1988) Acc Chem Res 21:355 36 Murrell IN, Carter S, Halonen LO (1982) J Mol Spectrosc 93:307 37 Quapp W (1987) J Mol Spectrosc 125:122

4 QUANTUM CHEMICAL PES CALCULATIONS: THE PROTON TRANSFER REACTIONS

In this chapter we discuss

~ the characterization and visualization of selected reaction PES

(proton transfer reactions, Sect. 4.1),

~ how approximations in the quantum chemical methods may determine

the quality of reaction PES (Sect. 4.2),

both along with ~ recent results for bimolecular single proton transfer reactions in

comparison to experimental data.

Many proton transfer reactions are of great importance due to their

occurrence in a wide variety of processes in chemistry and biology,

e.g. in acid-base catalysis, electrophilic sUbstitutions and

additions, reactions with kinetic deuterium isotope effects for the

elucidation of reaction mechanisms, etc. The interest in these

processes intensified as gas phase studies of the structural and

energetic properties of hydrogen-bonded complexes and their reactions

were increasingly reported in the last decade. It can be noted that a

considerable number of these reactions may be described in a good

approximation based on the PES properties of the isolated reacting

system (in vacuo). This is intelligible when one refers to actual gas

phase processes (occurring with low kinetic energy) as well as to

other related processes, such as reactions at enzyme active sites, if

water is largely excluded, or reactions on heterogeneous interfaces,

etc.

In other cases, the theoretical gas phase characteristics (instead of

results of complicated or unrealizable gas phase experiments) can be

considered as a description of the intrinsic properties of the system

allowing a clear separation from temperature and medium effects.

corresponding to the importance of hydrogen bonding, a large number of

monographs or textbooks concerning this field of research are

available which frequently involve the problem of proton transfer

reactions. Here, we mention in particular those outputs which consider

theoretical PES analyses and interpretations with respect to proton

transfer processes, thereby creating a new basis and quality of

interrelations between experimentalists and theoreticians when dealing

139

with H-transfer processes. 1

Here, we focus our attention to the analysis of theoretical PES studies of a number of single proton transfer (PT) reactions, in particular those between electrically neutral molecules

A-H ••• B ~ A- ••• H-B+ •

4.1 The Problem in Visualization of PES Properties

High-dimensional PES cannot be represented in a visual manner. However, it is possible to o~ercome this difficulty to a considerable degree by determining their one- and two-dimensional cross-sections (curves and surfaces, respectively). Thus, a two-dimensional function as a PES section of the total PES may. be visualized as a three­dimensional relief or mapped by a two-dimensional contour diagram. In this manner, one may obtain an intuitive view from the shape of a PES which is an important factor in its interpretation, especially with respect to the course of chemical reactions. Economic considerations require general restriction of the PES 'cal­culations to these one or two-dimensional cross-sections. The special kind of calculation and visualization depends upon the size of the system (PES dimension) and the number of degrees of freedom which directly contribute to the process investigated, and the computing possibilities. Beyond quantitative representations, there are a number of more qualitative illustrations which -are of importance for a presentation of theoretically or experimentally derived potential energy data for the purposes of everyday chemistry. We shall characterize the advantages and disadvantages of some of these representations, selecting single proton transfer as example.

4.1.1 RP Energy Profiles and Surfaces Derived from Usual PES Sections

By selecting one or two internal .. leading" coordinates, which should sufficiently determine the course of a chemical process, one may generate sui table cross-sections of the total PES. For each process the possible leading coordinates have to be selected carefully. Energy curves or surfaces may be produced by freezing all other coordinates. The result is an energy curve over one internal leading coordinate, e.g., R1, or a surface over the leading coordinate plane (R1 ,R2). These rigid cross-sections (cf.Ref.2) may represent the important features of the PES in a sufficient approximation, if the

140

reaction is objectively governed by processes along the selected coordinates. If all r i , i>l, are reoptimized for each R1 (curve) or (R1 ,R2)-pair (surface) the energy curves or two-dimensional re­presentations are generated as relaxed cross-sections2 • the problems conn~cted with a visualization of PES properties shall be

elucidated in terms of single proton transfer reactions, as given in Eq. (1)

:NI + H-X --+ (:NI ••• H-X, :NI •• H •• X, ::-H ••• X-) --+ ~N-H + X- (1)

with R(N-H)= R1 , R(X-H)= R2, R(N-X)= R1+R2, which we additionally illustrate by

R1 R2 H N==H==X

3 '-..::..../

1800

possible stationary points on the reaction PES are featured in the parentheses of Eq. (1). One of them forms the basis for a single minimum (SM) potential; three occur in the case of a double minimum (DM) potential 0

(1) Rigid two-dimensional cross-sections: The ammonia (amine) molecule may be frozen at monomer geometry. Then R1 and R2 are the leading coordinates (e3V symmetry and linear H­bonding, respectively). Such rigid cross-section may be used to obtain an idea of the critical region of the PES. The restrictions may be lifted after locating the region. 3

In the given example, it is reasonable to assume that the rigid section should sufficiently characterize those features of the PES which are important for the proton transfer process. This approximation was already used in Clementi's famous calculations of the H3NoHCl system. 4

A SM potential may be featured by surfaces as given in Figs. l(a,b), which are based on calculations of X-Hoamine systems. 3 ,5 One of the corresponding representations for a DM potential is featured in Fig.4. It is also based on ab initio calculations, in this case found for the H3NoHCl system (Hartree-Fock level)5 •.

(ii) Relaxed two-dimensional cross-sections: These surfaces may be obtained if the geometry of the ammonium (amine) fragment is reoptimized for all R1,R2 combinations. This gives certain improvements to the visualized surface, at least in the marginal parts of the surface region. In the H3N •• oHCl system, the relaxation of the

141

a b

Fig.1. Two-parameter representations of a SM potential of reaction (1) and the result of one-dimensional cross-sections through the surfaces: (R1+R2=const.) lines in R2 vs (R1+R2) (a) and R2 vs R1 (b) plots

Fig.2. Artificial OM energy profiles along the one-dimensional cross­sections through the SM potential surface as indicated in Figs. 1a,b

142

internal degrees of freedom in NH3 seems to be negligible (Refs.3,5

and literature therein).

(iii) Rigid one-dimensional cross-sections:

Apart from dissociation processes, a simple reduction of the leading

coordinates from two to one generally fails to yield an adequate PES

section. The rigid one-dimensional cross-sections correspond to one­

dimensional slices through the surfaces as shown in Figs. land 4.

In Figs. la,b one-dimensional cross-sections straight through the

SM-PES of reaction (3) produce an artificial OM potential as shown in

Fig.2. Thus, proton relaxation at a constant (Rl+R2) distance of the

heavy atoms cannot be used to explore the energy profile for instance

for reactions as represented by single proton transfer, 3 unless a

larger distance (mean value of Rl+R2) of the main atoms is maintained

by steric hindrance or solvent effects.

The character of rigid one-dimensional cross-sections through a OM

potential of reaction (3) is illustrated in Fig. 4 • The section may

afford qualitatively correct results in favourable cases.

(iv) Relaxed one-dimensional cross-sections:

Relaxation of the one-dimensional sections through a reaction PES

along the second significant coordinate (or the second and the

following coordinates r i ) may significantly improve the results.

Relaxation of one-dimensional cross-sections means that one drives the

reaction toward products along a leading coordinate (here for instance

the R2 distance in Fig.3) by varying this coordinate in small

increments from its reactant value to its product value and minimizing

the energy with respect to the second important coordinate (R1 , Fig.3)

or all remaining degrees of freedom, r i (i*2), at each incremental

value. The result is also known as "intuitive" RP6. The underlying co­

ordinate driving procedure has been constantly used due to its

simplicity. One gets a false sense of security; the hidden problems in

it are often overlooked.

The differences of the relaxed one-dimensional cross-sections and the

corresponding rigid cuts are evident when comparing Figs.lb and 3 for

the SM potential, and considering Fig.4 for the OM. The erroneous

one-dimensional section of the SM potential is greatly improved by an

optimization with respect to the second coordinate. For the important

OM potential, relaxed one-dimensional cross-sections cannot uniquely

reproduce the RP (Fig.4). Here we find the so-called chemical

hysteresis,7 i.e. the RP generated in this way and beginning at the

reactants, may be different from the one obtained when starting at the

product(s); for a fine illustration cf. also Ref.8. However, in fa-

143

Fig.3. Relaxed one-dimensional cross-section (solid curve) produced by minimization along Rl (or along all r i , i~2) for fixed R2

Fig.4. A two-parameter representation of a OM potential for Eq.(l) and the result of a one-dimensional section at a constant heavy atom distance, Rl+R2. The relaxed one-dimensional RP (solid curves) is produced by minimization along the straight lines (dashed) for fixed R2

144

vourable cases the relaxed one-dimensional cross-sections path may be

qualitatively correct, and for reaction OM potentials like that in

Fig.4, the path may pass through or near the transition structure.

Problems arise particularly for paths which are sharply curved, i.e.,

dominated by one of the two coordinates, which are anticipated to

determine the course of the reaction.

4.1.2 Graphical Presentation of Three-center Problems

The geometry of a three-center problem is specified by the three

internal degrees of freedom, e.g.,R1 ,R2 and a for the system discussed

above. The PES E(R1 ,R2 ,a) is a hypersurface in a four-dimensional

space for which a complete graphical visualization is not possible.

However, the total energy can be shown in contour diagrams using

combinations of a number of suitably arranged two-dimensional

cross-sections. From the equipotential lines on these surfaces, one

may interpolate the behaviour in between. Examp.les are represen­

tations for the H3-system9a , or visualizations with three selected

coordinates in larger systems, as used now in molecular modeling, 9b

too. The perimetric coordinates10 RA=(1/2)· (rAB+rAC-rBC)

RB=(1/2) (rAB-rAC+rBC)

RC=(1/2) (-rAB+rAC+rBC)

are especially suitable when using different planes for drawing

contour diagrams in this coordinate system.

4.1. 3 Interaction Surface of an Attacking Species with a Fixed

Valence System

Surfaces derived from the interaction of a chemically active unit with

a stable basic molecular framework can be used for illustrating the

qualitative features of the reaction PES. The topology of the chemical

network of the system is only changed with respect to the bond(s)

addi tionally formed by the attacking species. The PES illustration

indicates the favoured positions for the attack of an agent as well as

the course of isomerization reactions in the attacked systems. This

kind of visualization is of particular interest for conformationally

relatively stable (especially plane) systems with the peculiarity of

having a direct connection to the (projected) three-dimensional

structure of the system. In the displays one may additionally indicate

the topologically equivalent atomic positions of the attacked

reactant. The pictures are similar to presentations of electrostatic

potentials11 which are based upon a fixed molecular topology or geometry.

145

The reactions of aromatics are suitable examples: If a proton or any

other electrophilic agent passes over a benzene molecule in minimum

Fig.5. Benzene-H+ surface12- 15

energy positions, one can draw a potential energy surface over the

benzene framework similar to that given in Fig.5. Here, the moving

species is simply a proton. The illustration shows the central energy

maximum (proper saddle point of index 2, P-SP 2,16 with respect to the

whole dimension), and the 6 ~-complexes and 6 saddle points along a

peripheral migration of the proton. The proton shifts in the system

may be formulated in the following manner:

(2)

SP ~-complex ~-complex

At present, the upper limit of the gas-phase potential barrier for the

process described in Eq.(4) was estimated to be between 20 and

30 kJ/mo1. 13- 15 This gives qualitatively the same picture as it was

found for instance in super-acid solution, where a barrier of about 45

kJ/mol has been derived. 17

A three-dimensional representation of such a surface looks formally

like that given in Fig.6. However, since the calculations have not yet

reached the required high level of sophistication with respect to the

basis set used, the inclusion of correlation energy and the

146

Fig.6. Three-dimensional representation for the benzene-H+ surface type of Fig.5

Fig.7. Computer graphics of a model function that qualitatively + represents a benzene-X surface showing an isoenergetic "peripheral"

channel for X+ "motions"

calculation of zero-point vibrational energy, it cannot be excluded

that the potential barriers (or the free energy barriers) are near

zero. Such a surface would resemble the computer graphics in Fig.7.

The display in Fig.6 is typical for the PES appearance when benzene is

147

attacked by (cationic) electrophilic species E+, such as H+, CH;, F+

(see also Ref. 16), in contrast to Li+, Na+ as representatives of

the first groups of the periodic table where the face structures

Fig.6 is qualitatively

(cf. SP structures in

become minima on the benzene-cation PES. 18

similar to that when the bridged structure

Fig.5) form the minima along the peripheric

species. In this case the underlying atomic

motion of a

arrangement

cationic

for the

surface has to be changed in such a way that the minima are localized

in the middle of the C-C bonds. This has been semi-empirically

determined for the migration of the NO+ cation over the benzene ring

thereby forming 6 stable cationic n-complexes along the six-fold

degenerate topomerization reaction. Figure 6 utilizes a presentation

describing this process. 19

In order to draw a picture of the essential PES features, it is

sufficient to know all stationary points in the regions of interest,

as well as a certain number of route points connecting the stationary

points (for instance along the descent paths from the saddles). Thus,

for any aromatic compound including heterocycles, one can produce PES

sections well-suited for textbook illustrations, and as a basis for a

systematic comparison of the chemical and physical properties of the

systems. Such figures focus the striking features of the shape of the

PES and do not aim at unimportant details. These representations

should in particular stimulate chemist's intuition for the reactive

processes in a given system.

4.1.4 Empirically Derived Diagrams for more Complex Reaction PES

A special type of visualization for more complex systems has been

developed by More O'Ferral120 and Jencks21 ,22 (free energy or

potential energy representations), mostly generated on the basis of

empirical data.

These schematic two-dimensional diagrams are based upon two guiding

(bond-breaking) internal coordinates in a rectangular form suchwise

that the lower left corner generally represents reactants, whereas the

148

upper right is identified with the products. Points near the other

corners may indicate the existence of intermediates. A concerted

process will be featured by a direct passage diagonally from reactants

to products with a possible inclination towards one or other corner

due to asymmetry of forming and breaking bonds.

The displays are very useful in interpreting concerted vs step-wise RP

or in discussing imbalanced transition states23 , etc. They are

particularly suited for representing potential energy surfaces for

x­I

-c­I N+ I H

N I H

B

x­I

-c-I N

Fig.S. More O'Ferrall-Jencks plot for the general base catalyzed addition of a nucleophilic to an unsaturated center with a preferred concerted pathway

( .. ep .. lnted with p .... mlsslon f .. o .. Ch .....

Rev. (1972)72:702.Copy .. lght 1990

Ame .. lcan Chemical Society)

proton transfers which depend upon two variables, the distances A ••• H

and B ••• H (see Eq. 1). A more complex example from real chemistry is

given in Fig.S.

It is possible and desirable to use such diagrams in connection with

quantum chemical PES calculations giving them a stronger quantitative

character.

4.1.5 Energy Profiles from Mathematically Defined RP Calculations

At present, the RP energy profiles may be obtained by procedures using

no coordinate constraint by local assumptions in its basic concept.

Of course, this approach may be extended to large systems by ignoring variations in those distant parts of the molecular system which can only negligibly influence the reaction site.

149

Here, the underlying RP is defined as a curve in the multi­dimensional coordinate space. Until now, its calculation is possible at least for small- and medium-sized systems (cf. the reaction ergodo­graphy) 25. The methods. have been outlined in Sect.l.3 and discussed in detail in Chap. 3. This more sophisticated approach is of great importance for a precise description and visualization of reaction PES, especially in those cases, where the reaction cannot be modeled sufficiently well on calculating a surface considering two guiding coordinates only. Of course, in this way the calculation and visualization is restricted to one-dimensional presentations (energy vs RP, cf. also the term "reaction coordinate", Sect. 1.3). If the RP can mathematically and physically be defined in a suitable manner, the resulting energy profile (Fig.9) would be the first true quantitative counterpart to the frequently used potential energy profiles in chemistry where mostly either an intuitive RP or a theoretically approximated one have been used. Then it may form the basis for the determination and interpretation of details of the progress along the energy profile,26 among them the barrier-width which is of importance when calculating tunneling. 27

RC(in Bohr or in Vamu-Bohr)

Fig.9. Energy profile along a RP determined without geometry con­straints: A pure geometric (1 Bohr=o.529177-10-10m) or a mass­weighted RP (amu: atomic mass units, Cf.IRC24 )

Additionally, by extending these diagrams, the "walls" perpendicular to the RP (mass-weighted or not) as well as the RPcurvature may be visualized, illuminating the character and stability of the path, and possibly allowing an interpretation of certain specific dynamic properties of the reaction in question. 28 Based on the empirical

150

valence-bond (EVB) model,29 Chang and Miller30 recently proposed a

fine procedure for constructing and visualizing reactive PES in order

to analyse reactions of "highly vibrationally excited molecules, where the dynamics tends not to be localized about anyone reaction path," and reactions of systems "with a number of low-frequency modes orthogonal to the reaction path, which allows for large-amplitude motion far away from any reference path."

4.1.6 Summary

In order to reduce the efforts for calculations of reaction PES

aiming at a RP, different coordinate driving procedures have been used

until now. They allow direct visualization by one- and two-dimensional

cross sections. However, particularly for rigid one-dimensional

slices, but also for the relaxed ones, one has to be cautious in

interpreting and visualizing PES properties. Rigid, especially relaxed

two-parameter representations are frequently adequate descriptions for

certain types of chemical reactions (e. g. single proton transfer

reactions, Eq.1). Relaxation may significantly improve one-dimensional

cross-sections, but does not remove unsuited coordinate constraints

and the errors which may arise therefrom. Thus, we must learn that the

accurate selection of the representative coordinates is of really

great importance for each visualization.

At present, we find a considerable increase in calculations Which try

to trace the RP by mathematical procedures defined in the whole or in

an "active" coordinate space. It is of high chemical interest to

profit from these calculations by finding further possibilities for

visualizing properties such as RP curvature, RP bifurcation31 and

RP stability etc. in order to explore or to interpret relations to

experimental peculiarities of the chemical reactions. The develop­

ments in this field are still affected by a number of problems in

defining a suitable static RP.

References (Section 4.1)

1 Schuster P (1976) in: Schuster P, Zundel G, Sandorfy C (eds) The Hydrogen Bond, Vol 1, North-Holland, Amsterdam;

Zeegers-Huyskens Th, Huyskens P (1980) in: Ratajczak H, Orville­Thomas WJ (eds) Molecular Interactions, Vol 2, Wiley, New York;

Scheiner S (1983) in: Wyn-Jones E, Gormally JA (eds) Aggregation Processes in Solution, Elsevier, Amsterdam, p 462;

Hibbert F (1986) in: Gold V, Bethell D (eds) Advances in Physical Organic Chemistry, Vol 22, Academic Press, p 113

2 Mezey PG (1977) in: Csizmadia IG (ed) Applications of MO Theory in Organic Chemistry, Vol 2, Elsevier, Amsterdam, p 137

151

3 Raffenetti RC, Phillips DH (1979) J Chem Phys 71:4534 4 Clementi E (1967) J Chem Phys 46:3851;

Clementi E, Gayles IN (1965) J Chem Phys 42:2323 and (1967) 47:3837

5 Brciz A, Karpfen A, Lischka H, Schuster P (1984) Chem Phys 89:337 6 Havlas Z, Zahradnik R (1984) Int J Quantum Chem 26:607 7 Dewar MJS (1971) J Am Chem Soc 93:4294 8 McIver JW Jr, Komornicki AJ (1972) J Am Chem Soc 94:2625 9a Howeler U, Klessinger M (1983) Theor Chim Acta 63:401; 9b Howeler U (1990) MOBY, version 1.4: Molecular Modelling on the

PC, Springer Verlag, Heidelberg 10 Davidson ER (1977) J Am Chem Soc 99:397 11 Tomasi J (1981) in: Politzer P, Truhlar DG (eds) Chemical Applica­

tions of Atomic and Molecular Electrostatic Potentials, Plenum Publishing Corp, New York

12 Hehre WJ, Pople JA (1972) J Am Chem Soc .94:6901 13 Heidrich D, Grimmer M (1975) Int J Quantum Chem 9:923 14 Heidrich D, Hobza P, ~arsky P, Zahradnik R (1978) Collect Czech

Chem Commun 43:3020 15 Kohler H-J, Lischka H (1979) J Am Chem Soc 101:3479 16 Heidrich D, Quapp W (1986) Theor Chim Acta 70:89 17 Olah GA, Schlosberg RH, Porter RO, Mo YK, Kelly DP, Mateescu GD

(1972) J Am Chem Soc 94:2034 18 Jemmis ED, Schleyer PvR (1982) J Am Chem Soc 104:4781;

Heidrich D, Deininger D (1977) Tetrahedron Lett :3751 19 Minkin VI, Minyaev RM, Zhdanov YuA (1987) Nonclassical Structures

of organic Compounds, Mir Publishers, MOscow, p 175 20 More O'Ferrall RA (1970) J Chem Soc B:274 21 Jencks WP (1972) Chem Rev 72:705 22 Jencks WP (1977) J Am Chem Soc 99:451, 7948 23 for further references see: Bernasconi CF (1987) Acc Chem Res

20:301 24 Fukui K (1970) J Phys Chem 74:4161, see also Chap. 1 25 Kato S, Fukui K (1976) J Am Chem Soc 98:6395;

Fukui K (1981) Acc Chem Res 14:363 26 cf. for instance: Yamabe T, Koizumi M, Yamashita K, Tachibana A

(1984) J Am Chem Soc 106:2255. 27 Kato S, Fukui K (1976) J Am Chem Soc 98:6395

28a see for instance: Tachibana A, Okazaki I, Koizumi M, Hori K, Yamabe T (1985) J Am Chem Soc 107:1190;

b Tachibana A, Fueno H, Yamabe T (1986) J Am Chem Soc 108:4346 29 Warshel A (1981) Acc Chem Res 14:284 30 Chang Y-T, Miller WH (1990) J Phys Chem 94:5884 31 see also: Quapp W (1989) Theor Chim Acta 75:447

152

4.2 PES Properties Along the Bimolecular Single Proton Transfer

Proton transfer reactions were used as examples in Sect. 4.1 to

discuss possibilities of visualization and calculation of reaction PES

by taking suitable cross-sections, or by the calculation of

mathematical well-defined curves in the coordinate space of the PES.

In this section, we choose the single proton transfer reaction between

neutral molecules to show the problems and success of quantum

chemistry in calculating energy profiles of chemical r~actions on the

one hand, and their interplay with the experimental data and research,

on the other hand.

4.2.1 Formulation of the Reaction Mechanisms

In general, a single H-transfer may occur by interaction of a proton

donator and a proton acceptor following several possible mechanisms

(A-H: proton donator, B: proton acceptor, Me: molecular complex,

IP: contact ion pair)

A-H + B A-H ••• B Me

A-H ••• B E' ~ Me

- + A ••• H-B IP

A ••• H ••• B MC-IP

- + A ••• H-B IP

(1)

(2)

(3)

(4)

For a general formulation of such reaction PES we prefer the

short-hand notation

A-H + B ~ [A-H ••• B, A: •• H-B+, A ••. H ••• B) ~ A- + H-B+ •

In the parentheses one can find the structures which should be probed

as stationary points along the proton transfer reaction PES. They have

to be identified by quantum chemical calculations for each system in

153

order to find the reaction mechanism. It is noteworthy that bi­

molecular association reactions are important examples of chemical

systems which are relevant to unimolecular rate theories.

a c d i--l 1

§ ~ 2A ;-i

3 .... , 0 Q, --

LLI 2B

4

RC AH+B MC IP

Fig.lO. Potential energy curves for the single proton transfer mechanisms, Eqs.(l) to (4). Th~ shaded areas schematically show, be­ginning from the left, the regions of a) the isolated systems b) a possible molecular complex c) a possible IP and d) the separated ion pairs . Curves (2) design double minimum (OM) potentials with larger stability of the MC (2A) or larger stability of the IP (2B). The curves are placed in arbitrary distances on the energy axis

Equations (1)-(4) can be used for a general discussion of transfer

mechanisms. Thus, the potential curves in Fig.l0 may form a suitable

basis for studying the equilibria in the vapor phase (or hypothetical

systems without environments) as well as for the analogous competition

between the molecular complex and the ion pair in solution when our

154

attention is restricted to the central part "MC-IP" of the curves.

The equilibrium of Eq.(2) between a hydrogen bonded molecular complex

(MC) and a hydrogen bonded ion pair (IP) may be regarded as one kind

of molecular complex - ion pair tautomerism. The formation of contact

ion pairs is favoured under the influence of a medium, but even the

question of their occurrence already without medium influence is of

fundamental chemical interest and has become a subj ect of intense

theoretical and experimental investigation.

Experimental gas phase investigationsl - 3 as well as theoretical

studies4- 10 (using more sophisticated quantum chemical methods) on

proton transfer in the favourable hydrogen-halide amine systems have

been carried out only since the late seventies. They represent a good

example for a fruitful interaction between theory and experiment.

Based on theoretical research and gas phase experiments, it has been

derived that acid base pairs without environment show transfer of the

proton from the acid to the base molecules only between strong acids

and bases. Schuster et al. 11a suggested by using gas phase acidities

that combinations of ammonia and HBr or HI should form candidates for

vapour-phase ion pairs. We note that already in 1973 Ault and

pimentel12 postulated the proton affinity of the amine bases to be

>940 kJ/mole for producing ion pairs in the HCI-amine system.

In the next section we briefly repeat the use of gas phase acidities,13,14

the further qualitative discussion

transfer systems.

4.2.2 The Proton Transfer Energy AEpT

some definitions in context with

which play an important role in

of PES properties of proton

The proton affinity (PA) of a species B is defined as the negative of

the enthalpy for the gas phase reaction

B + H+ ~ B-H+ (5)

The PA of a deprotonated species A- is given by the negative of the

enthalpy for the reaction

A-H (6)

which corresponds to the deprotonation energy, Eop' of A-H.

The values defined in this manner are frequently corrected to zero

Kelvin (0 K) and for zero-point vibrational energy, thus yielding

powerful measures of the i n t r ins i c Broensted acidity (the

potential energy difference AE t (OP) and the standard heat of de-po protonation energy at 0 K: AH~p[O] = AEpot(OP) + AEzpv ' cf. also Eq.5

and Fig.4 in Chap.l) and basicity.

155

It should be added that the absolute gas phase basicity of B and the absolute gas phase acidity of A-H are analogously defined as the negative standard f r e e energies A.Go of the processes given by Eqs. (5) and (6). NOW, the difference of the experimental (or theoretical) proton affinities of Band A- in a reaction

A-H + B ~ A- + H-B+

defines a proton transfer energy EPT in the case of infinite se­paration of the ions:

PA(A-) - PA(B)

EDP(AH) -PA(B).

(7)

Equation (7) is applied to two examples. The values are given below the reaction equation (kJ/mol, corrected for A.Ezp and 0 K, non­corrected values in parentheses) 14

HCl + NH3 -+ Cl-+ NH4+

519 = 1411 892 (Eq.7) (536) = (1394) - (858)

HF + H20 -+ F- + H30+

825 = 1575 750 (Eq.7) (833) (1558) - (724)

EpT is always positive due to the relation PA(X-) > PA(Y). The smallest EpT has been successfully used as a guide in the search of prime candidates of IP formation in the vapour phase. The Coulomb-type stabilization EC forms the main component of opposite sign when using the real distance of the ions within the contact ion pair. The occurrence of a second minimum at the geometry of the contact ion pair or the position of the proton between the main atoms in a SM potential critically depends on the balance between EPT and EC (cf.also Refs.11a and 15a).

4.2.3 Discussion of most Recent PES Data of Bimolecular Single Proton Transfer

Theoretical methods play an outstanding role to analyse detailed properties of the potential curves for Eqs.(1)-(4). The ab initio studies on this subject were pioneered by Clementi's SCF investigation16 of

156

The subject has been revisited by a number of other authors. 4- 7 The

best hitherto existing large-scale calculations on the SCF level

predict a SM potential

in formal agreement with Clementi's result. The inclusion of electron

correlation6 ,7 confirms the SM potential in the sense of a stable

hydrogen bonded molecular complex (MC).

The interaction energy amounts to AEpot'" -45 kJ Imol including a con­

tribution of probably 40% correlation energy;8 if the estimate of 13

kJ/mol for the EZpv correction4 is adopted, sufficient agreement is

achieved with a value (33.5±l2 kJ/mol)17 derived experimentally for

the theoretical dissociation energy of the complex.

In full agreement with the theoretical results, a molecular complex

was experimentally derived for ClH-NH3 from an analysis of the

rotational spectrum. 3a We note, that there are recent experimental

conclusions3b, e that all H3N-HX dimers are of the hydrogen bonded

type.

Two problems concerning the PES calculations for systems of the given type shall be discussed in more detail:

(i) The influence of basis sets, correlation energy, zero point vib­rational energy and superposition err-or on the zero point energy Eo: ~ It is not always possible to use large basis sets in MO theory as it is required. In such cases, the sensitivity and the kind of dependence of the SCF result from the quality of the basis set has to be studied. It is of interest to know that the frequently used 3-21G split­valence set erroneously postulates an ion pair, whereas the related 4-31G basis predicts a OM potential (Ref.18). Finally, the above mentioned calculations with larger basis sets give a molecular complex. Thus, when going from small to the required larger basis sets in SCF computations, we observe a reduction of the ion pair character.

~ Recent studies6- 8 show that the inclusion of correlation energy does considerably influence the potential energy data especially in search of OM potentials for proton transfer. Note that it has been well-known for a long time that correlation energy should favour structures with the proton in a central position of the main atoms. utilizing this experience· (cf.Ref.19), a potential wall between MC and IP derived from SCF calculations (cf.Refs.5-7) for very strong proton donor-acceptor systems is expected to be flattened by the inclusion of electron correlation, or changed to a SM potential. Having an SCF SM potential, which may represent a molecular complex or an ion pair, the correlation will broaden the corresponding minimum towards the middle of the bond.

~ The role of the zero point vibrational energy E (ZPV) for systems considered here, has not been systematically studied until now. However, in order to get a final result, it is necessary to carry out the more extensive calculations connected with a vibrational analysis, because the energy differences between the two minima and the transition structure in a possible OM potential (cf. Refs. 5-7,9) are expected to be small or very small. A first important contribution to this problem wa's given by Jasien and Stevens (Ref.7) who discussed the rather dramatic effect of the

157

(harmonic) E(ZPV) correction for HI-ammonia. E(ZPV) greatly handicaps the stability of the IP in comparison to the MC. The example is discussed below.

~ It can be expected that superposition errors (Sect.1.6) playa role in the study of such systems. However, the changing definition of the sub-units as the proton is shifting from one unit to the other does not allow an inambiguous determination of the BSSE (the problem is discussed in more detail in Ref.9). Thus, the most promising way to circumvent BSSE effects in MO theory is the employment of suitable extended basis sets (for hydrogen bonded systems see e.g. Refs. 20,21). comparing the stationary points along the DM potentials, the geometric differences seem to be too small to assume a critical influence of the BSSE.

(ii) Geometrical restrictions vhen examining the reaction PES

When exam1n1ng the DM potentials, the given systems have been generally reduced to a C3V geometry:

! X - H ••• IN'--="H

\ For a deeper understanding we have to mention that the rocking potential (along the angles given above) is very soft. Thus, the system has to be considered as a non-rigid one undergoing large­amplitude motions. It is worth noting that Dannenberg (Ref. 22a) for the water dimer concluded that "on so flat a surface the free energy surface may be largely determined by entropic factors." Here, trifurcated and similar structures only slightly differ in the energy from the linear hydrogen bonded complex. The experimental results show a linear structure to be most stable (Ref.22b). However, the experimental determinations were performed at 350-400 K.

O-H---

j linear

If one estimates a cyclic structure

H\ o

H/ \~ , " " "

trifurcated' 'F "to be 10 eu lover than the linear structure, at 400 K the linear structure vould be favoured by 4 kcal/mol from the entropic con­tribution to the free energy surface. This vould be more than enough to render the linear structure lover in energy at the experimental temperatures."

Next we look for the influence of an increased acceptor strength by

taking alkyl-substituted ammonia derivations, so modelling the

following systems

158

On the SCF level, double minimum potential surfaces with small

barriers may be obtained for HCl-amine systems which by inclusion of

electron correlation change into SM potentials of considerable

broadness. 6 The secondary amine complex still seems essentially to be

described as hydrogen bonded form. In the case of the tertiary amine

complex, the proton is transferred to the nitrogen atom suggesting an

IP structure for the gas phase. These conclusions about the nature of

the dimer are consistent with deductions from recent studies of ground

state rotational spectra by Legon et al. 3b, c Both theoretical and

experimental data favour a variation of the potential energy as given

in Fig.II (see Ref.3c).

1·50 1·00 R (N"'H)fA

Fig.1!. A qualitative representation (see Ref.3c) of Epot vs R (N-H

distance) with an ionic SM potential at 1.00 A (the dashed line refers

to the zero-point energy of the D species)

The zero-point energy level was assumed to span the hydrogen bonded

form as well as the ionic structural range. In (CH3)3N'DCl the lower

zero-point energy would confer more ionic character since the deuteron

would spend more time in the locality of the minimum. We mention that

Fig.II characterizes the actual appearance of the 2B-, 3- and 4-type

curves of Fig.I0 for a number of ionic systems studied under gas phase

conditions.

Using similar high level ab initio calculations for

159

with the stronger proton donor HBr, Latajka, Scheiner and Ratajczak9

(1987) also derived broad SM potentials with characteristics similar

to that described by Eq.(3). After the inclusion of electron

correlation, the proton position is shifted towards the middle of the

main atom distance. Thus, the resulting structure for both BrH-amine

systems cannot be simply characterized as either a neutral complex or

an ion pair (cf.Fig.10, curve 3).

Analysis of their calculations led Latajka et al. 9 to a description of

the situation for the BrH-NH3 and BrH-NH2 (CH3 ) as follows:

" Thus, the proton would oscillate rapidly over a fairly wide range between the Br and N nuclei. Each complex could hence be described more approximately as one in which the central position is shared more

or less equally between the halide and the amine rather than as a

strict neutral or ion piHr. As the basicity of the amine increases relative to the halide, ... , the time-averaged location' of the proton

would shift smoothly away from Br and towards N, corresponding to a

gradual transition in the character of the complex between the

extremes of neutral and ion pair."

Although these' new calculations9 will not be the last ones, we can

certainly conclude that the favoured candidates for gas phase ion

pairs essentially result from combinations of tertiary amines with

HCI, HBr or HI7.

similarly to the (CH3)3oHCI system, a very recent interpretation of

the rotational spectrum of trimethylammonium bromid vapor actually

shows ,this heterodimer lying close to the limiting model associated

wi th the ion pair. 3d The above-presented

suggested by the observation of a NMR 2 BrH-N(CH3)3.

conclusions were already

gas phase ion pair for

Jasien and Steven (1986) found a double-well proton transfer potential

with a small barrier at the heavy halide system7

1HoNH3

using high-level calculations including a certain extent of electron

correlation. However, the change of the low-lying degenerated bending

vibration of the hydrogen bonded complex (~ 400 cm-1 ) into modes of

the IP (NH: 1-) which are now essentially represented by an HNH

bending (~1640 cm-I ) leads to a considerable difference in the

harmonic EZPV for the two minima of the proton transfer PES. The

second minimum (the IP) of the double-well proton transfer potential

with its small barrier is not maintained with respect to the zero

point energy Eo (IP). Based on these results, we have to note that

160

E(ZPV) may finally prevent the formation of a double energy minimum

(in E ) when the potential barrier in between is only slightly marked o as in the case of the HI-ammonia complex in the gas phase.

We particularly analysed the hydrogen halide - amine systems because

they have the advantage to be accessible by high-level theoretical

methods as well as by gas phase experiments.

Theoretical results with weaker bases and hydrogen halides such as

benzene-HF, show weakly hydrogen bonded complexes in a SM potential23

as already expected from the proton affinities of the bases (PA

benzene: 186, ethylene: 168 kcal/mol) 14. This should also be true when

using HCOOH24a , HCI (for a PES study of the weak bonded complexes see

Ref. 24b) or acids of similar strength. The systems including

unsaturated organic species

level of sophistication. We

suggested that the second

were not yet calculated on a suitable

note that already early CNOO/2 studies + -minimum (benzene-H· .. F) of the OM

potential "disappears in the steep slope of the potential curve for

the HF stretching vibration".25

After all, it is not surprising that gas phase experiments of HF with

weak bases such as benzene only show weak hydrogen bonded complexes26

in a SM proton transfer potential as it has been found by calculation.

Gas phase experiments on others as hydrogen halide/amine systems are

reported only for some exceptional cases. For example the interaction

between CF3COOH and N(CH3) 3 was surprisingly found to reach only a

molecular complex. 1 The IR study was carried out at high temperatures

because the vapour pressure of acid-amine compounds at room

temperature is very low. Conclusions concerning the form of the proton

transfer potential cannot be drawn alone from these data. We have to

consider that the tertiary ammonium salt may be disfavoured by entropy

(steric hindrance) at higher temperatures. On the other hand, for

primary and secondary amines, the situation may be changed because

additional binding energy may be gained through a second H-bond in a

cyclic contact ionic pair (cf.Sect.4.2.4).

with respect to a free energy surface, increasing temperatures will

generally hamper the detection of the possible two minima when the OM

potential is characterized by a small barrier (and a high probability

of tunneling, respectively) quite apart from the effect that complex

formation is a priori disfavoured by the reaction entropy terms.15~

161

summary: For gas phase conditions the possible occurrence of DM potentials at

equilibrium distances should be extremely rare because of two main

reasons:

~ Ion pairs can be hoped to occur only for combinations of acid base

pairs with extremely strong donor-acceptor properties.

One interesting consequence seems to be that only amines with a PA~ 900 kJ/mol should exhibit a sufficient PA to stabilize ion pairs with strong acids in contrast to the weaker oxygen acceptor systems (PA ~ 750 kJ/mol). We mention that experimental studies of proton transfer reactions in neutral gas phase clusters (reaction of an acid with solvent clusters, (HOR)n and (NHR2)n ; n=1,2 ••• ) give an interesting

insight into the influence of increasing proton affinities and the effect of electrostatic stabilization on the formation of micro­solvated ion pairs formed by an acid in combination with Nand 0 acceptor systems, respectively (Ref.27).

~ The PES properties have the tendency to allow only one stable

complex which may cover the range between a Me and an IP (cf. Fig.1).

This is theoretically manifested by the generally small barrier

between them in the possible SCF-DM potentials together with the

strong effect of stabilizing the transition structure by the final

inclusion of electron correlation which as a rule leads to an extended

single minimum. Thus, in the gas phase, proton transfer between

neutral species is in general no actual chemical reaction, but rather

a barrier less formation of an ion pair corresponding to Eqs. (2) and

(4). Additionally, the binding energy remains far from chemical

bonding. Hence, in the gas phase the proton transfer does not

represent a suitable process for testing conventional TST.

In this context theoretical suggestions should be mentioned which are related to particular hydrogen transfer reactions between heavy atoms, e.g. I + HI ~ IH + I. Here, the stretching vibration perpendicular to the RP may give cause for a new type of bond, the vibrational bonding (cf. Ref.28). The small potential energy barrier should be over­compensated by the zero-point vibrational energy difference between the broad saddle and the narrow valleys I+HI.

Furthermore, we note that in certain systems a competition between a one-step proton transfer and an electron transfer followed by hydrogen atom transfer must be taken into consideration (cf.Ref.29).

162

4.2.4 Gas-Phase Results and Medium Influenced Experimental Data

The gas phase characteristics may be influenced by medium effects of

different kind and strength:

(i) reaction on catalytic interfaces, especially in zeolites, (ii) influence of matrices at lov temperatures, (iii) influence of inert (less polar or less nucleophilic)

solvents, (iv) influence of strongly polar and nucleophilic solvents.

We are able to show qualitatively the consistency of a number of

experimental results under medium influence on the one hand, and

gas-phase results including a qualitative estimation of entropic or

medium effects when necessary, on the other hand.

(i): The study of Bronsted acidic sites of solid catalysts in their

interaction with proton acceptor molecules (heterogeneous catalysis)

represents a subject of continuously high interest in chemistry.30 The

heterogeneous reaction of surface sites with single molecules may well

be described by the interaction of a suitable surface cluster (cluster

model) with the reactant molecule. The process may be considered as a

generalized gas phase problem. In a first approximation, this may be

also true for the interaction of zeolite molecular sieves with gases

or liquids because the reactions are starting inside the cavities

after the reagent is infiltrated and adsorbed. In the idealized case

the reacting species is isolated from other adsorbates and

non-adsorbed molecules.

Actually, the reaction may be influenced by environmental effects

which are comparable with gas phase solvation by further infiltrated

molecules. FUrthermore, a direct (field effects) and/or indirect

(mediated by the associates and other adsorbates) interaction of , the

adsorbat with other surface centres will occur. This can theoretically

be treated and interpreted on the basis of an extended supermolecule

approach.

The intrinsic acidity of such alumosilicates and related systems can

best be estimated and compared with others by ab initio calculations

of cluster models of the acidic sites (Fig.12). The bridged hydroxyls

are found to be the origin of the strong acidity in zeolitic

catalysts. Estimates concerning the intrinsic acidity of the terminal

and the bridged acidic sites have been given by using theoretically

derived energy differences for minimal cluster sizes. 31

163

~ ~

H H I I O'-S~ .... 0,-- .0, /0 ........

....... ' "AI" si

~o ~o ~o 0' 0' 0' , , ,

Fig.12. Particular acidic sites ('" terminal, "bridged) in zeolite frameworks

The bridged sites have been treated by a boron-modified zeolite model

which is suggested to be of similar intrinsic acidity in comparison to

the corresponding AI-cluster. 32 Further calculations33 on Al­

zeolite, using larger basis sets for treating the bridged acidic

sites, modified the energy differences but did still suffer due to

computational restrictions such as basis set limitations and

geometrical constraints. In spite of some uncertainties in the MO

calculation of the clusters, the most acidic sites of the zeolites can

now be valued by the comparison with other in first line experimental

deprotonation energies: 14

H20» CH30H > HF » (HCN ~ phenole ~ SiH30H) > CH3COOH > HCl > HBr ~

\. /H CF3COOH > HI -Si-O , '.

'BH3 (or AIH3 )

The intrinsic acidity of the most acidic sites on zeolites has been

estimated to be in the order of magnitude of the superacids:

~Epot(DP) ~ ~H~p[O] ~ 1250±80 kJ/mol)31 when relating the ab initio

results to the experimentally derived values of H20 and CH30H.

Small basis set SCF calculati~ns performed for larger cluster models

of surface sites, show similar results when correcting them by a scale

factor found by the above mentioned relation to the experimental

results on water and methanol. 30 Hence, amines (for instance pyridine)

may form ion pairs in a SM potential. Early IR experiments showed that

the pyridinium ion is the stable high temperature form in zeolites. 34

IR studies on the interactions of pyridine with the surface of y-type

decationized zeolite over a wide temperature range indicate the

occurrence of a hydrogen bonded MC and hydrogen bonded IP, so possibly

suggesting a DM potential as formulated in Eq.(2).35 Furthermore, on

H-ZSM-5, methanol was found to form the coverages. 36

+ CH30H2 methoxonium ion at low

164

(ii): Proton transfer equilibria in inert matrices occur under conditions which are also in between the solution and the gas phase. They demonstrate a sensitive dependence of the kind of complex from the nature of the matrix37 , so showing the properties of the complex to be strongly dependent on relatively small energy influences from the environment in agreement with the results of theoretical gas phase calculations.

(iii) and (iv): In the condensed phase, the equilibria of Eqs. (1)-(4) are shifted to the right, mainly by the solvation of the ions, and may finally give solvent-separated ion pairs (SS-IP) and/or the completely dissociated ion pairs (CD-IP) in the case of strongly polar solvents

AH + B + solv -+ ( AH ••• B, - + ) A ••• BB , •••

MC IP solv

and/or ( A-n HB+, (A-Isolv + HB+lsOlV) ) (8)

SS-IP CD-IP

Thus, the ion pairs may occur in complexes between much weaker acids and bases which is confirmed by comprehensive experimental material as well as by theoretical estimations of solvent influence. It is well-known that the charges of the contact ion pairs can be stabilized by solvent molecules to a significant extent, even when using low dielectric (aprotic) solvents. In solvents, the ion pairs occur more frequently in equilibrium with the neutral complexes (MC). This fact has been explained as a consequence of the interaction with the environment which should increase the intermolecular distance (the mean value at a temperature T, so finally being an effect of the free enthalpy surface) between the main atoms of the hydrogen bond in comparison to the gas phase data (cf.for instance Ref.1l~). In this manner, a second minimum can be produced at all or is separated by a larger barrier from the MC.

The change of a hydrogen bond from a "SM molecular complex", Eq.1, to a DM potential energy profile of proton transfer (with MC and IP, Eq.2) can be obtained arbitrarily in theoretical gas phase calcula­tions by using a main atom distance fixed at 1 a r g e r distances in comparison to the equilibrium one. This fact also demonstrates the sensitive dependence of the potential form (and the barrier height) from the distance between the main atoms of the hydrogen bond. Results of this kind have been frequently obtained involuntarily by many older ab initio calculations where the main atom distances were not included in the geometry optimization (for instance constant at the larger experimental values) due to the lack of effective optimization procedures and computational restrictions (cf. also Sect; 4.1.1).

A comparison of the main atom distances in the MC and the contact IP (Eq.2) both in solution seems to yield individually different results

165

due to the competition between ion pair contraction and lengtheninq by interaction with the solvent shells.

It has to be mentioned that the second minimum of the DM potential in solution in certain cases is not easily accessible for a direct experimental observation because the formation of the ionic pair is accompanied by a favoured arranqement of solvent molecules around it so producinq a stronq decrease in entropy (20-30 e.u.). This may lead to an extremely small equilibrium constant. Independent of that it is well known, that even a second hiqh-lyinq minimum (IP) may 'con­siderably affect the kinetics of proton exchanqe (Ref.38a).

On the other hand, symmetric or nearly symmetric DM potentials may experimentally be detected more easily. For broad DM potentials with barriers which are not too larqe, protons are extremely mobile and qive rise to the so-called infrared continua. The extremely larqe "proton polarizabilities" are due to the influence of the continuous­ly chanqinq electrical fields of the environment on the structures forminq DM or broad SM potential enerqy profiles of hydroqen bonds (Ref.39).

We illustrate the occurrence of hydroqen-bonded molecular complexes and ion pairs as well as of equilibria between them by results first obtained by Barrow and Bell (1956/59)40,41 on pyridine and haloqenated acetic acids. 'The study was carried out in solvents of low polarity usinq infrared spectroscopy. The reactions are hiqhly sensitive to the qradual increase of the intrinsic acid strenqth. The results were later confirmed by Gusakova and Denisov et a1. 42 and a number of subsequent papers (for leadinq references cf.Refs. 38~,43a):

Acid Base Reaction products

CH3COOH + .. MC (cf.Eq.1): CH3COOH" 'N(Py)

CH2C1COOH + )=\ MC-IP tautomerism (cf.Eq.2): CHC12COOH IN ~pyridine .. - +

\}~ R-COOH"'N(Py)~ R-COO "'H-N (Py)

CC13COOH + ion pair IP (cf.Eq.4):

CF3COOH .. - + RCOO "'H-N (Py)

A similar picture is obtained when usinq one acid (trifluoracetic acid) and varyinq the basicity of pyridine by sUbstituents (in benzene).44

The intrinsic acidity of HCl seems not yet to be sufficiently hiqh to qet ion pairs with trimethylamine in liquid oxyqen but with (HC1)2 or hiqher associates (1:2, 1:3 ••• compositions) which are stronqer acids. 1

In this connection we note that with primary and secondary amines, six- and hiqher-membered doubly hydroqen bonded cyclic ion pairs may be formed. For instance, a six-membered contact ion pair has been

166

found experimentally in combination with CF3COOH (Ref.4S) in inert media:

Based on PES calculations on the isolated complexes, the existence of further cyclic ion pairs has been suggested, among them for ammonium salts of zeolite-clusters or ammonium tetrafluoroborate (Ref.46, 3-21G and 6-31G** ab initio basis sets). In these cases, an asymmetric hydrogen bonded form did not occur as a second minimum on the PES. surprisingly, the proton transfer in the eight-membered formamidine dimer is supposed to pass a transition structure of lower symmetry to form an ion-pair like intermediate

of higher symmetry and high energy even in the absence of solvent (6-31G* ab initio method, Ref.47)! The two bridging hydrogens are much closer to one of the two amidine SUbunits. The intermediate will easily be stabilized in solvents.

With increasing dielectric constant (polarity of the environment) and

increasing influence of specific interactions

(iv) the most polar form of the system as well as the dissociation of

ion pairs into solvent-separated ions is favoured: In the latter case,

A- + H-B+ in Eqs.(1)-(4) has to be interpreted as solvent-separated

ion pair (SS-IP) A-II H-B+ (see Eq. 8) •

If B is identical with the solvent, the process represents the

dissociation reaction in solutions. There are further interesting and

important possibilities for interactions of the hydrogen bonding

cations with environmental molecules39a which we do not discuss here.

The reader interested in a complete story on "Mechanisms of proton

transfer between oxygen and nitrogen acids and bases in aqueous solution" together with the most important experimental techniques

used, should see the ~xcellent review of Hibbert48 .

superacid solutions (HF:BF3 , HF:SbFs etc.)49 are able to proton ate

weak bases, i.e. the hydronium ion was found to be stable, but even

methane is pro.tonated (PA: 127 kcal/mole). Such cations formed by

protonation reactions may be the actual reactive species in PT

reactions or undergo intramolecular proton transfer. This behaviour

167

allows theoretical reaction modelling by molecule-H+ PES (inter­action PES of an attacking ion and a molecule, see Sect. 4.1.3).

4.2.5 Theoretical Approach to Medium Influence and the PES Concept

In the last years the theoretical organic chemistry has been increasingly extended beyond the gas phase realm of quantum mechanics to the study of the course of chemical reactions in solution. The success of these methods will indicate the begin of a new period for modeling chemistry in solution. Here, we mainly restrict our attention to a static solvent treatment. The discussion of the limitation of this approach was recently continued. 500:,51 Such studies assume the solvation to be in equilibrium with the chemical system at each point along a RP. This basic hypothesis may first be questioned from possibly different time scales of solvent relaxation and the chemical process and, secondly, from the motion of a (limited number)' of solvent molecules which may form an important part of the motion of the whole system along the RP. But apart from dynamical, "non­equilibrium" solvation effects and other limitations in the applica­tion of TST to reaction in solvents (see Chap. 1.4), static approaches will give much information on the intermolecular interactions and may represent a suitable ansatz for the estimation and interpretation of solvent effects in many cases. The extension of the PES concept to medium effects is realized using the supermolecule approach (for the inclusion of charged systems we generalize to the term supersystem when necessary). In this approach, molecules or fragments which represent the medium, are explicitly involved in the calculation of the potential energy. The main drawback is the enormous computational expense arising from the growing number of degrees of freedom by each additional solvent molecule included in the supersystem. FUrthermore, the incorporation of a large number of solvent molecules into the supermolecule to simulate a condensed phase reaction, requires the selection of one geometric arrangement among a large number of minima on the PES which are close in energy to t~e absolute minimum (which normally cannot be found). Finally, we have to consider that in the presence of a very limited number of solvent molecules (gas phase clusters) the direct medium-substrate inter­actions are overestimated with respect to the interaction between the molecules in the environment, i.e., the effect of bulk solvent re­organization is not included. Unfortunately, even this effect may contribute significantly to the barrier in'solution. Apart from all these difficulties, the extension of the computational possibilities by using molecular mechanics, approximated ab initio

168

theory, or point charge and dipole approximations to the sol vent

molecules, respectively, as well as results from Monte-Carlo

calculations in suitable methodic combinations opens a wide field of

promising research. For many questions of chemical reacti vi ty , the

large quantity of minima (structures) with similar statistical weight

in the supersystems indicates the value and the necessity of

statistical or dynamical studies in this field.

Now, we outline a number of possibilities for extending the

calculations of gas phase reactions or processes occurring in small

clusters of molecules into the liquid phase. Once more, the first

problem is the determination of a suitable RP. For certain cases, it

may be solved by using suitable coordinate driving procedures (see

Sect. 4.1.1) in the complete dimension of the supersystem, or, within

the dimension of the system described by the corresponding gas phase

reaction (for instance, by using two guiding coordinates in case of

the proton transfer reactions, Sect. 4.1.1). In the latter case, we

have the potential surface (two guiding coordinates) for the gas phase

reaction and, over the same coordinate space, a corresponding (free

enthalpy) surface for the solvation energies and the possibility and

the problem to combine both surfaces. The guiding coordinates are

assumed to describe the actual chemical process in the gas phase as

well as in solution. On the other hand, it is possible, although more

critical, to restrict the attention to a RP which has been found for

the gas phase reaction directly by mathematically well-defined

procedures (see Sect. 1.3 and Chap. 3) with a subsequent addition of

the solvation energies for points only along this path.

An important variant of the methods mentioned above may be the ab

initio calculation of points along a gas phase RP and modeling the

medium by using statistical mechanical simulation techniques to

determine solvent structural data along the gas phase RP geometries.

The selection of one of the methods has to be made for each case

separately.

At present, methods are frequently

chemical calculations of the reacting

preferred, where the

system are completed

quantum

by

(i) Simplified statistical treatments such as continuum models for considering the solvent influence.

In these methods the medium is represented by a structure less di­

electric continuum. This is reasonable because the largest effect

frequently arises from electrostatic fields emanating from

neighbouring ions or dipoles. The method may also be applicable for a

series of related reactions when specific interactions are of the same

169

type and size. The models are developed in different versions (for

references see for instance50 ,52-55), e.g. in Tomasi's mode152a ,b;53

the solute is placed inside a cavity accurately defined by the solute

geometry and surrounded by a continuous polarizable dielectric with

the permittivity, c.

special attention should be paid to the fact that the supersystem

should include especially those solvent molecules which are expected

to play an active role in the chemical process studied. The inclusion

of statistical methods in simulating medium influences opens the

possibility of changing from an internal potential energy profile to a

free energy (enthalpy) one.

We note that the reacting system may already be a supersystem in case of bi- or termolecular reactions. Then, this reacting (super) system may further be extended for instance over the first solvation shell by a corresponding supermolecule approach, now additionally involving the solvent molecules.

Here, we give place to a review concerning the effect of solvation on

the - in comparison to H3N"HCl - more covalent proton transfer complex

H3N"HFlsolv considering the methods mentioned till now:

The system belongs to the normal type of hydrogen bonded . molecular

complexes forming a SM potential in the gas phase. 8,10 In water,

NH3 "HF is most stable as ionic complex NH:+F:- However, most of the

calculations using different levels of quantum theory as well as

different kinds of solvation models find a SM potential representing a

molecular complex. 56-58 An exception forms the application of the

solvaton-model (based on the CNOO/2 method),59 where the stabilization

of the ionic subunits is apparently overestimated.

It is important to note that in the calculations cited above, 56-58

e.g. those of Schuster et ale 56 on a supermolecule formed by NH3"HF

and six water molecules,

H 0 2

H \ " " ANI---H---F

HJ " H

an evolution towards the appearance o£ a contact ion pair within a OM

potential could be derived. The same trend was found by using

fractional point charges at similarly arranged atomic centers to

represent solvent molecules. 57 In this case, a two-dimensional search

170

of the PES has also been undertaken varying both the N-F separation

and the position of the proton in between. The configuration of

solvent molecules has been derived by extrapolation of gas phase ab

initio results.

More recently, Burshtein60 suggested that the assumed structures of

the supermolecule prevent the formation of the alternative

sol vent-separated ion pairs. His continuum type model (point dipole

model) shows the solvent-separated ionic complex as most stable

minimum along with the molecular complex in another type of OM energy

profile. Based on the MNOO/H version, the intimate ion pair does not

occur as minimum in this model. Thus, the theoretical studies on

H3N'HF indicate that the occurrence of gasphase type OM potentials

(Eq.2) may be questioned in presence of a solvent for hydrogen bonded

systems where these OM profiles are mostly accepted.

For· completion we mention the systematic comparison of amine-HX

systems in solution in combination with the continuum approach of

Tapia61 using fixed internal amine geometries and MINI-l ab initio

optimizations. 62 This basis set gives SM potentials for all 'of the

hydrogen halide systems in the gas phase. It is confirmed that the

shift of the bridging protons toward the nitrogen is enhanced by

increasing solute-solvent interaction, for NH3 interaction with HCI

and especially with HBr, easily leading to SM complexes of the contact

ion pair type. -

In more sophisticated approaches,64 the treatment of the solvent in­

fluence may be extended by using quantum mechanical methods and

condensed-phase simulation procedures such as

(ii) Monte Carlo (MC) statistical mechanics63 and molecular dynamics (MD) methods:

" Monte Carlo or molecular dynamics simulations are typically carried out for one or two s:Jlutes in a cube ... with 200-400 solvent

molecules ... Equilibrium properties are obtained in Monte Carlo calculations by averaging over millions of instantaneous geometrical

configurations of the system that are selected by the Metropolis

algorithm, and in molecular dynamics calculations by solving the

Newtonian equations of motion and performing time averages ... Another

element at the heart of the simulations is the selection of inter­molecular potential functions that describe the interactions between the components of the system. ,,65

Successful MC or MD calculations presuppose good potential energies

for the intermolecular interactions. These PES data can increasingly

be gained from high-level ab initio computations using selected models

171

of intermolecular interaction PES.

MC studies based on such an extensive supermolecule are not yet

published for proton transfer reactions. However, we may consider MC . + -. 66 stud1es such as on (CH3)4N 'CI 1n water. Here, the occurrence of a

OM free enthalpy profile for both contact and solvent separated ion

pairs has been calculated by a series of simulations at points along

the RP determined for the gas phase PES (see also the example below).

By reviewing the literature related to MC and MO calculations, the occurrence of such free enthalpy minima has been derived to be the general rule for oppositely charged ions in water (for references cf.Ref.66). The free enthalpy barrier between contact and solvent separated forms has been estimated to be no more than 15 kJ/mol.

Until now, the application of the MC approach has been used as follows

(3 steps):

- determination of the RP for the gas phase PES

- development of the potential functions to describe solvent-solute interactions, mostly on the basis of PES ab initio studies

- MC simulations along the RP.

This Monte Carlo approach has been pioneered by Jorgensen63a ,b to

investigate the degenerate SN2 reaction:

\1 _1* Cl. .C .. CII I

)

ClCH3 •.. CII -+

In this case, .the gas phase reaction was examined at the 6-31G(d) ab

initio level. Only the RP for the gas phase featuring collinear,

E

,~ W , , • 1 ........ _- .... : . ---, ~ "

, "9 \ , ...

RC

Fig.13. Potential energy (gas phase) and free energy (in water)

profiles for the SN2 reaction63a ,b (see text)

172

backside substitution, has been solvated. The RP in the gas phase,

which may simply be defined here as the difference between the two

c-halogen distances thereby enforcing symmetry about the transition

state, is not supposed to be seriously perturbed in solution.

Monte-Carlo simulations were then carried out for the solute cluster

solvated by 250 water molecules at 298 K/1 atm. The calculated free

enthalpy of activation is significantly increased and is in agreement

with the experimentally available value. The influence of water leads

to a unimodal energy surface instead of a double well (double minimum)

profile found for the gas phase reaction (Fig.13).

The double well potential in the gas phase is characterized by two equivalent ion-dipole minima separated by a symmetric transition state. However, the ion-dipole interactions are nullified in water by partial desol vation processes. Subsequent calculations lead to the result that in nonaqueous solution the reaction may also proceed with the presence of intermediates (Ref.63b).

It is of interest

~ that results with Tomasi's continuum model in this case can be

compared to those obtained through MC calculations (Ref.50~)

~ that the Tomasi-Miertus model (along with good gas phase reference

energies) also seems to allow the reproduction of the order of amine

basicities in solution (Ref.67).

The relative order of basicities of ammonia and methylamines in aqueous solution is well-known to be

• NH3 < NH2Me ~ NHMe2 > NMe3 in contrast to the regular sequence with the methyl substitution found by experimental studies in the gas phase (Ref.68). The relative order of basicities of methylamines in water is reproduced by the theoretical model when the internal characteristics of the systems is given by a sufficient basis set in the quantum chemical ab initio method.

Both continuum model and Monte Carlo approach suggest the solvent to

be in equilibrium with the reacting system at each point along the RP.

The effects of non-equilibrium solvation were tested by variational

transition state (VTST: see Chap. 1) calculations of the microsolvated

SN2 reaction (supermolecule approach)

(n=1,2) •

Here, "the extent of nonequilibrium solvation is tested by comparing calculations in which the water molecule degrees of freedom par­

ticipate in the RC to those in which they do not". 69a

In spite of the evident success, Bertran50 summarized 1989:

"In conclusion, although spectacular advances have been made on how the ~olvent influences the SN2 reaction, we are far from being close

173

to fully understand the solvent effect on it." (We refer to a last

paper analysing the subset of trajectories that are reactive events in

molecular dynamics calculations on a model SN2 barrier-climbing . t) 69b process 1n wa er •

4.2.6 Proton Transfer, Transition state Theory, and Quantum Chemistry

It is a weighty and unpleasant experience that just single proton

transfer reactions between hydrogen bonded molecules do not represent

a suitable probe for the efficiency of transition state theory in

chemical PES calculations presupposing gas phase species. The

following reasons may be responsible:

(1) The peculiarities of PES describing the single PT in hydrogen

bonded complexes

(li) The particular problems of semi-empirical as veIl as Hartree-Fock

(i. e., noncorrelated level) quantum chemical methods in reproducing

the properties of hydrogen bonded systems.

(i): The quantum chemical calculations first concern the gas phase reaction, i.e., the individual system in vacuo. As discussed above, a DM potential energy profile (involving the occurrence of a transition structure) along the proton transfer reaction can surprisingly be expected only for exceptional cases. Furthermore, for these cases the barrier between the minima should be very small. Thus, the objective supposition for a check of con­ventional TST is usually not fulfilled. Only by considering en­vironmental effects, transition structures may frequently be found at the price of a time-consuming use of computers.

(ii): Only a few versions of the early semi-empirical procedures of quantum chemistry succeeded for hydrogen bonding and other aspects of proton transfer reactions. For a long time, the original CNDO/2 (Ref.70) method has been considered as most suitable for calculating hydrogen bonding. However, for describing the structures of organic cat ion s and the proton transfer reactions in it, CNDO/2-FK (Ref.71), MINDO/2 and MINDO/3 methods (Refs.72,73) as well as the MNDO method (Ref.74) have been used with considerable success. In a later phase of establishing semi-empirical theory, one attempted to improve Dewar's semi-empirical versions in order to avoid the erroneous and artificial representations of hydrogen bonding and proton transfer. Methods such as MINDO/3-H (Ref.75), MINDO/3-HB (Ref.76), different versions of MNDO-H (Refs.77,78), MNDO/M (Ref.79) and AM/1 (Austin model, Ref.80) have been presented in which empirically modified terms for calculating the nuclear repulsion have mainly been proposed (for instance in MINDO/3-H, one has additionally modified the bonding parameters). It is easy to see that empirical changes in the representation of .the core-core repulsion cannot compensate all approximations in the calculation of the electronic terms, therefore new or related problems have to be expected in describing hydrogen bonding (cf. also recent surveys and studies comparing the different methods, such as Refs.81-88). However, apart from certain shortcomings, these methods are still of considerable interest. This is due to the fact that only highly sophisticated ab initio theory including electron correlation and geometry optimization is able to describe correctly the proton transfer potential in a neutral or cationic system. The use of such

174

methods is only common since the beginning of the eighties of this century and still limited to smaller systems.

In the preceding chapter, current PES ab initio results on single

proton transfer reactions along with experimental data has been

presented studying neutral complexes. The PES studies on that examples

clarify why such single proton transfer reactions do not provide those

suitable examples for theoretical testing of conventional TST for

which has been hoped. -

In contrast to the bimolecular single proton transfer reactions, large

barriers may be calculated when studying double proton transfer reactions, for instance in 4- or 6-membered rings. Here, the

analysis of the experimental data in gaseous or inert media

demonstrates the need for utilizing quantum chemistry in order to

explore the mechanism of the exchange processes. 38

A prototype reaction is the proton exchange on a protonic center B

mediated by the bifunctional unit, A-H. The exchange reactions

A-H + B-H* ~ A-H* + B-H (9)

are most easily investigated for systems which offer symmetric cyclic

structures with sets of equivalent bonds

(10)

Structure type (10) is always a stationary point of the PES due to its

symmetry properties. For isolated 4-membered rings studied hitherto,

(10) represents a transition structure of proton exchange. 46 ,89,92 The

potential energy barrier is defined by the energy difference between

(10) and the most stable molcular complex, MC:

~E* = E(10) - E(MC). pot

The highest barriers are found for four-membered homoassociates,

e.g. (HF)2' with a symmetric D2h transition structure89 ,90,46

and a barrier height of about 42 kcal/mol ( .. 50 kcal/ mol for the

corresponding barrier in the HCI dimer) using the MP2/6-31+G** ab

initio level. 46 Lower but still significant potential barriers are

expected for mixed dimers formed by combinations of protonic species

175

such as NH3 or amines, H20 or alcohols, H-Hal etc. For instance, the energy difference between the c3V form of NH3 ·HF and the transition structure

is calculated to be = 34 kcal/mol91 (for the NH30HCl cf.Ref. 92 ). In the important 6-membered rings,93 (10) forms either a transition structure89 ,46 or an ion pair46 (gas phase). The reaction involves a D3h transition structure for (H-Hal)3 systems89 ,46 (Hal=F,Cl) with A-H formed by (H-Hal)2. For instance, the reaction with (HF)3 (multiple cyclic proton exchange, model for proton exchange on [!])

---gives a barrier of about 20 kcal/mol compared with 42 kcal/mol in the four center reaction of (HF) 2 (all calculations are related to the MP2/6-31+G** PES). Ab initio models of four-center and six-center

46 proton exchange in NH3 (HF) 2' NH3 (HCl) 2' NH3oHCloBX3 (X=Hal) represent simple examples of electrophilic substitution. The reaction mechanism may be analysed in terms of an intrasupermolecular PES. The results allow to discuss to what extent the exchange of the hydrogens is concerted and simultaneous, and what suppositions have to be

fulfilled in order to produce (intermediate) cyclicly hydrogen bonded ion pairs. The computations finally aim at the treatment of the electrophilic SUbstitution on aromatic compounds (here represented by the c2V structure of a BF3 catalyzed proton exchange)

Using very strong proton donor-acceptor systems, we observe a tendency to change the 6-membered transition structures into ion pairs already

176

in the gas phase (Refs.93,46). The interest in such double PT transfer

reactions may also be related to the 1,3-proton shifts in the

bifunctionals. Here, we only mention the double PT in formamidine­

water. The system was subject to RP calculations (IRC) 94,95 with a

subsequent use of the chemical reaction molecular dynamics method (CRMD) 96.

Following these lines, it is hoped to interpret important chemical

reactions first by cyclic proton transfer models in terms of

RP calculations.

References (Section 4.2)

1 Kulbida AI, Schreiber VM (1978) J Mol Struct 47:323

2 Golubev NS, Denisov GS (1982) Khim Fiz 5:563

3a Goodwin EJ, Howard NW, Legon AC (1986) Chem Phys Letters 131:319;

b Legon AC, Rego, CA (1989) Chem Phys Letters 162:369;

c Legon AC, Rego, CA (1989) J Chem Phys 90:6867;

d Legon AC, Wallwork AL, Rego CA (1990) J Chem Phys 92:6397;

e Howard NW, Legon AC (1988) J Chem Phys 88:4694

4 Rafenetti RC, Phillips DH (1979) J Chem Phys 71:4534

5 Brciz A, Karpfen A, Lischka H, Schuster P (1984) Chem Phys 89:337

6 Latajka Z, Sakai S, Morokuma K, Ratajczak (1984) Chem Phys Letters

7

8

9

10

110:464

Jasien PG, Stevens WJ (1986) Chem Phys Letters 130:127

Latajka Z, Scheiner S (1984) J.Chem Phys 81:4014

Latajka Z, Scheiner S, Ratajczak (1987) Chem Phys Letters

Latajka Z, Scheiner S (1987) Chem Phys Letters 140:338;

Latajka Z, Scheiner S (1987) J Comput Chem 8:674;

Del Bene JE (1989) J Comput Chem 10:603

135:367

11 Schuster P, Wolschann P, Tortschanoff K (1977) in: Pecht I, Rig­ler R (eds) Molecular Biology, Biochemistry and Biophysics, Vol 24, springer-Verlag, Berlin-Heidelberg-New York, alp 113, ~) P 119

12 Ault BS, Pimentel GC (1973) J Phys Chem 77:1649

13 See for instance: Trombini C, Bonafede S (1976) Annali di Chimica 66:19

14a Deprotonation energies: Bartmess JE, McIver RT,Jr (1979) in: Bowers MT (ed) Gas Phase Ion Chemistry, Vol 2, Academic Press, New York, p 1

b Proton affinities of neutral systems: Aue DH, Bowers MT (1979) in: ibid., p 87

c Hehre WJ, Radom L, Schleyer PvR, Pople J (1986) Ab initio Molecular Orbital Theory, Wiley, New York

15 Beyer A, Karpfen A, Schuster P (1984) Topics in Current Chemistry, Vol 120, Springer, Berlin-Heidelberg, a) p 5ff, ~) P 12

177

16 Clementi E (1967) J Chem Phys 46:3851 Clementi E, Gayles IN (1967) 47:3837

17 Goldfinger P, Verhaegen G (1969) J Chem Phys 50:1467 18 Latajka Z, Scheiner S (1985) J Chem Phys 82:4131 19 Schuster P (1978) in: Pullman B (ed) Intermolecular Interactions:

From Diatomics to Biopolymers, wiley, New York, p 387 20 Latajka Z, Scheiner S (1987) J Comput Chem 8:663,674 21 Tomasi J, (1987) ~nt J Quantum Chem 32:207 22a Dannenberg JJ (1988) J Phys Chem 92:6869;

b Dyke TR, Mack KM, Muenter JS (1977) J Chem Phys 66:498 23 For leading references see: Bredas JL, Street GB (1988) J Am Chem

Soc 110:700 24a Bredas JL, Street GB (1989) J Chem Phys 90:7291;

b Cheney BV, Schulz MW, Cheney J, Richards WG (1988) J Am Chem Soc 110:4195

25 Jakubetz W, Schuster P (1971) Tetrahedron 27:101 26 Baiocchi FA, Williams JR, Klemperer W (1983) J Phys Chem 87:2079 27 Knochenmuss R, Cheshnovsky 0, Leutwyler (1988) Chem Phys Letters

144:317 28 cf.the surveys: Manz J, R6melt J (1985) Nachr Chem Tech Lab.33:210;

Lefebvre R (1990) Ann Phys Fr 15:1 29 Edidin RT, Sullivan JM, Norton JR (1987) J Amer Chem Soc 109:3945;

Han Ch-Ch, Brauman JI (1988) J Amer Chem Soc 110:4048 30 Sauer J (1989) Chem Rev 89:199; 31 Heidrich D, Volkmann D, Zurawski B (1981) Chem Phys Letters 80:60 32 for a summary see for instance: Sauer J (1989) in: Klinowski J,

Barrie J (eds) Recent Advances in Zeolite Science (Studies in Surface Science and catalysis, Vol 52) Elsevier, Amsterdam, p 73

33 Sauer J (1987) J Phys Chem 91:2315 34 see for instance: Kiselev AV, Lygin VI (1972) IR Spectra of Inter-

face Compounds (Russ), Nauka, Moscow ' 35 Paukshtis EA, Karakchiev LG, Kotsarenko NS (1977) React Kinetics

Lett 6:147 36 cf. Mirth G, Lercher JA, Anderson MW, Klinowski J (1990)

J Chem Soc Faraday Trans 86:3039 and references therein 37 For literature cf. Ref.9 38 Denisov GS, Bureiko SF, Golubev NS, Tokhadze KG (1980) in:

Ratajczak H, Orville-Thomas WJ (eds) Molecular Interactions, Vol 2, Wiley, New York, p 107, a) p 125 ~) P 122

39a Zundel G, Fritsch J (1984) J Phys Chem 88:6295; b Zundel G, Eckert M (1989) J Molec Struct (Theochem) 200:73

40 Barrow GM (1956) J Am Chem Soc 78:5802 41 Bell CL, Barrow GM (1959) J Chem Phys 31:300 and 31:1158 42 Gusakova GV, Denisov, GS, Smolyansky AL, Schreiber VM (1970) Dokl

Akad Nauk SSSR 193:1056

178

43 Zeeqers-Huyskens Th, Huyskens P (1980) in: Ratajczak H, orville­Thomas WJ (eds) Molecular Interactions, Vol 2, Wiley, New York, p 1, ex) p 50

44 Deqa-Szafran D, Szafran M (1982) J Chem Soc (Perkin Trans 2) 195 45 Denisov GS, Golubev NS (1981) J Molec struct 75:311 46 Heidrich D, van Eikema Hommes N, Schleyer PvR (1991), in prepara­

ti"on; 47 Svensson P, Berqman N-A, Ahlberq P (1990) J Chem Soc, Chem

Commun :82 48 Hibbert F (1986) in: Gold V, Bethell D (eds) Advances in Physical

Orqanic Chemistry, Vol 22, Academic Press, p 113-212 49 Olah GA, Prakash SGK, Sommer J (1985) Superacids, Wiley-Inter­

science, New York; and references therein 50 Bertran J (1989) in: Bertran J, Csizmadia IG (eds) Concepts for

Understandinq Orqanic Reactions, Vol 267, NATO ASI Series (Series C), Kluwer Acad Publ, Dordrecht, ex)p 231 ~)p 234

51 Tucker SC, Truhlar DG (1989) in: Bertran J, Csizmadia IG (eds) 'Concepts for Understandinq Orqanic Reactions, Vol 267, NATO ASI Series (Series C), Kluwer Acad publ, Dordrecht, Vol 267, p 291

52a Bertran J (1983) J Mol struct (THEOCHEM) 93:12; b Persico M, Tomasi J (1984) Croat Chiu Acta 57:1395; c Constanciel R, Contreras R (1984) Theor Chim Acta 65:1

53 MiertuB S, Srocco E, Tomasi J (1981) J Chem Phys 55:177 54 Bonaccorsi R, Cimiraqlia R, Tomasi J (1983) 4:567 55 Karlstr6m G (1988) J Phys Chem 92:1315 56 Schuster P, Jakubetz W, Beier G, Meyer W, Rode 8M (1974)

Jerusalem symposia on Quantum Chemistry and Biochemistry, VI:257 57 Noell JO, Morokuma K (1976) J Phys Chem 80:2675 58 Anqyan J, Naray-Szabo G (1983) Theoret Chim Acta 64:27 59 Miertus S, Bartos J (1980) Collect Czech Chem Commun 45:2308 60 Burshtein KYa (1987) J Mol struct (Theochem) 153:203 61 See for instance: Tapia 0, BrKnden C-I, Armbruster A-M (1982) in:

Daudel R, Pullman A, Salem L, Veillard (eds) Quantum Theory of Chemical Reactions, Reidel, Dordrecht, Vol 3: 97-123 and Tapia 0 (1981), Vol 2: 25-72; Tapia 0, Stamato FMLG, Smeyers YG (1985) J Mol Struct (THEOCHEM) 123:67

62 Kurniq IJ, Scheiner S (1987) IntI J QuantUm Chem, Quantum BioI Symp 14:47 .

63a Chandrasekhar J, smith SF, Jorqensen WL (1984) J Am Chem Soc 106:3049, 107:154

b Chandrashekar J, Jorqensen WL (1985) J Am Chem Soc 107:2974 c Weiner SJ, Sinqh UCh, Kollman PA (1985) J Am Chem Soc 107:2219

64 cf. the recent survey: Van Gunsteren WF, Berendsen HJC (1990) Anqew Chem 102:1020

65 Jorqensen WL (1989) Acc Chem Res 22:184 66 Bruckner JK, Jorqensen WL (1989) J Am Chem Soc 111:2507 67 Pascual-Ahuir JL, Andres J,Silla E (1990) Chem Phys Lett 169:297

179

68 Aue DH, Webb HM, Bowers MT (1976) J Am Chem Soc 98:311

69a Tucker SC, Truhlar DG (1990) J Am Chem Soc 112:3347

b Gertner BJ, Whitnell RM, Wilson KR, Hynes JT (1991) J Am Chem Soc 113:74

70 Pople JA, Beveridge DL (1970) Approximate Molecular Orbital Theory, McGraw-Hill, New York

71 Fischer H, Kollmar H (1969) Theor Chim Acta 13:213

72 Bodor N, Dewar MJS, Harget A, Haselbach E (1970) J Am Chem Soc 92:3854

73 Bingham RC, Dewar MJS, Lo DH (1975) J Am Chem Soc 97:1285

74 Dewar MJS, Thiel W (1977) J Am Chem Soc 99:4899

75 Mohammad SN, Hopfinger AJ (1982) Int J Quantum Chem 22:1189

76 Zhanpeisov NU, Pelmenshchikov AG, Zhidomirov GM (1987) Zh Struct Khim 28:3

77 Burshtein KYa, Isaev AN (1984) Theoret Chim Acta 64:397 and (1986) Zh Struct Khim 27:3

78 Goldblum A (1987) J Comput Chem 8:835 and J Molec Struct (Theochem) 179:153

79 Voityuk AA, Bliznyuk AA (1987) Theoret Chim Acta 71:327 and 72: 223

80 Dewar MJS, Zoebisch EG, Healy EF, Stewart JP (1985) J Amer Chem Soc 107:3902

81 Bliznyuk AA, Voityuk AA (1988) THEOCHEM 43:343

82 Zhanpeisov NU, Zhidomirov GM (1989) React Kinet Catal Lett 38:395

83 Ventura ON, coitino EL, Lledos Au, Bertrand J (1989) THEOCHEM 56:55

84 Buemi G, Gandolfo C (1989) J Chem Soc, Faraday Trans 2 85:215

85 Voityuk AA, Bliznyuk AA (1988) Zh Fiz Khim 62:991

86 Williams (1987) J Am Chem Soc 109:6299

87 Kass SR (1990) J Comput Chem 11:94

88 Rzepa HS, Yi MY (1990) J Chem Soc, Perkin Trans 2, :943

89 Gaw JF, Yamaguchi Y, Vincent MA, SchaeferIII HF (1984) J Am Chem Soc 106:3133

90 Heidrich D, Ruckert M, Volkmann D, Kohler H-J (1984) Z Chem 24:419

91 BUbl M, Heidrich D, Schleyer PvR, unpublished results

92 Haase F, Heidrich D (1991) submitted for publication

93 Heidrich D, Ruckert M, Kohler H-J (1987) Chem Phys Lett 136:13

94 Yamashita K, Kaminoyama M, Yamabe T, Fukui K (1981) Theor Chim Acta 64:303

95 Yamabe T, Yamashita K, Kaminoyama M, Koizumi M, Tachibana A, Fukui A (1984) J Phys Chem 88:1459

96 Nagaoka M, Okuno Y, Yamabe T (1991) J Am Chem Soc 113:769.

Index

19-21,156ff ab initio MO SCF absolute reaction acidity

rates 17-19

intrinsic -anharmonicity ascent path

B-matrix basicity

- of amines basis

- orthogonal - set+

benzene-H+ -x

BFGS-method modified -

BFGS-update bifurcation Born-Oppenheimer Broyden-method

154,162 86

8

97,130 155 172

85 20,21,156

145-147,160 147

63,126 65

54f,59f 94,111ff,114,122

approximation 4 62

Broyden's class of updates Broyden's second update Broyden update

55 51

49,50 21,157 BSSE

catastrophe center of mass chemical hysteresis concave (function) condition number continuum model convex (function) coordinates

Cartesian -curvilinear -guiding/leading -isoinertial -- driving procedure perimetric -- transformation

corner cutting correlation energy critical points cross-section

rigid -relaxed -

curvature main direction of -

CHnClmCOOH

defect functional definite

86f,94f 84

142 35 56

168ff,172 35

97,128,134 95,97,130

9,139ff,147 132

9,142 144

30,132

10 20,156

9,10

139-143 140-143

35,78,85 35

165

39,108

negative - 35 positive - 35

degree of freedom 78,83,86,95f,136 deprotonation energy descent method descent vector OFP-method

45,66-71,124 7,66f,125

63

OFP-update dimensionality problem directional derivative directional vector direction of steepest

51,52,54,58 3

34,35 34

- ascent - descent

dyadic product

Eckart condition eigenvalue eigenvector energy profile

34 7,34,122ff

49

85 9,78,82f,86,96

9,86,101

4,82,86,106,112,141,171 entropic factors 23,157,165 equipotential line 104,108,133 equipotential section 107 Euler's formula 81 extremizer 128

Fletcher-Powell update 52 force field calculations 20 force constant 6,83,97,137 formamidine dimer 166 formamidine-water 176 free enthalpy surface 13,157,169

geometry optimization 83 gradient 6,19,22,33,79,88,91,102,

108ff,117,122f,131 - extremal 7,8,108ff,121f - field 33,130 - norm 108,115 - revolution 19-22

graph 32

Hessian

hill

matrix 6ff,35,36,78ff,86, 97,104f,109,117,120f,126

91ff HCN HX'amines (HF) 2 (HF)3 H+

3

index of inertia inflection point inner product invariance

rotational -translational -

83,101,134ff 140,155ff,175

174f 175

93

63f 41,105,111

32

- from coordinates

80ff 78ff

11,128ff ion pairs

solvent separated - - 164ff inner/contact - - 152-166

kernel of a matrix 40,80,83

Lagrange function least-change update level surface line search

181

locally convergent procedure LiCN

128 49 33

66,68 62 95

malonaldehyde mass-weighting maximizer medium effect metric minimizer minimum energy path molecule~ linear

quasilinear -molecular dynamics molecular complex

- structure

15 9f,132ff

36 16ff,162ff

131,134 36,37,43,127f

4-10,136 78,137

96 170

152ff,156ff

More O'Ferrall-Jencks plot Monte-Carlo method mountaineer's algorithm Murtagh-Sargent (MS-)

2 147f

170ff 108

method update

62 50,53,56-58,64

neighborhood of a point Newton-like method

global --Newton process Newton vector norm normal number

modes of efficiency

36

71-76 41-48,126

42,43,62 32,108,115

6,21,24 77 96 NH3

NH3"HX 140,155ff,169f

optimal conditioning orthogonality

56 79,84

paths relief -

perturbation points

86,95,101,124ff,130 4f

88,94

catastrophe - 86 stationary - 13f,35-41,83,86f,

124 potential

- energy barrier 18,19,145 double minimum - 95,140ff,152ff

- function 86f,96 harmonic - 128 Henon-Heiles - 93 single minimum - 140ff,156ff

Powell's symmetric Broyden update

principal axes projector

proton affinity proton transfer

energy

51 103

83,108

154

154

single proton transfer 139ff,152ff double -­

PSB-method PSB-update quasi-Newton equation quasi-Newton method quasi-Newton vector

166,174ff 63 51 48

11,45,60-66 47,62,67

rate of convergence 43,44 reaction coordinate 5,124,134

intrinsic -- (IRC) 7,14,122 reaction mechanism 11,23,140,152 reaction path (RP) 4,6,83,96

bifurcation/branching

curvature Hamiltonian in ascent in descent

9,111ff,114,122 14,83 12-15

7ff,101£f 7ff,122ff

142 intuitive -­onset orientation

reaction theory ridge rotation

problem 7,10 11-19

111 78,80f,85

saddle point 5,36-38,43,101,105f, 129,132

monkey -- 88,92f,111 -- of index 2 37,145 -- of index v 38 proper -- 37,145 virtual -- 37

secant equation 48 semi-empirical quantum mechanics

25,173 Sherman-Morrison-Woodbury-formula

50 sigma-complex 145 SM2 reaction 171ff

solution reaction 16ff,162,167 stationary point 13f,35-41,83 steepest descent 7,10ff,124,129ff

-- method 48,70 -- vector 34,43,66

stream bed path 7,9 structurally stable 95 super linear convergence 44f supermolecule approach 16,162,167f

tangential plane total energy transition structure

5,18,23, transition state

-- spectroscopy -- theory (TST)

translation tripel point

102,109 3

129,161,174ff 5,17,18

16 5,11,12,17,18

78,80

truncated Newton method trust region method

115 46 46

182

umbilic update

88ff,92,94,117

BFGS -Broyden -

54f,59f 49,50

55 51

Broyden's class of -Broyden's second -DFP - 51,52,54,58 dual -Fletcher-Powell -general single-rank -

54 52 49 49 least-change -

MS - 50,53,56-58,64 optimally conditioned - 58 PSB - 51

valence bond theory 21 valley floor line

6-9,91,94,101,105f,111,119,132 variational TST 12ff vibration 85f,132,134

zeolites 162ff zero point vibration/energy

18,20f,155-161

closing remark

Another solution of the problems of PES would be part of a more

general perspective indicated by the scriptural verse:

Every valley shall be exalted,

and every mountain and hill made low,

the crooked straight and

the rough places plain.

Isaiah XL,4

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