Theoretical Chemistry Volume 1 (Specialist Periodical Reports)

A Specialist Periodical Report

Theoretical Chemistry Volume I - Quantum Chemistry

A Review of the Recent Literature

Senior Reporter

R. N. Dixon, Department of Theoretical Chemistry, University of Bristol

Reporters

D. Garton, University of Yorh

J. Gerratt, University of Bristol

R. K. Hinkley, University of Oxford

1. M. Mills, University of Reading

J. Raftery, University of Oxford

W. G. Richards, University of Oxford

B. T. Sutcliffe, University of York

0 Copyright 1974

The Chemical Society Burlington House, London, W1 V OBN

ISBN : 0 85186 754 5

Library of Congress Catalog Card No. 73-92911

Printed in Great Britain by Adlard & Son Ltd.

Bartholomew Press, Dorking

Foreword

This is the first volume in the biennial series of Specialist Periodical Reports devoted to Theoretical Chemistry. Theoretical Chemistry is an extremely wide subject, since it provides the background for the interpretation of so many chemical phenomena, and it is therefore necessary to define the scope of these volumes. Quantum theory plays an important role in theoretical chemistry, both through the application of valency theory to the interpretation of molecular structure, and also in the development of spectroscopic models based on quantum mechanics, which are used in the determination of structural information from experimental spectroscopy. Indeed, to many chemists theoretical chemistry is synonymous with quantum chemistry. Quantum Chemistry will thus constitute a major part of this series. There is, in addition, a second important aspect of theoretical chemistry, particularly concerning chemical reactions, where the dynamics of molecular motion and their statistical behaviour is more important than specific quantum effects. This aspect will also be included in the general coverage of the series. The intended coverage of the series may thus be summarized as: the quantum theory of valence, with application to the calculation of the structure and properties of molecules, and to the calculation of potential energy surfaces for chemical reactions ; theoretical aspects of spectroscopy; the dynamics of chemical reactions; intermolecular forces ; and developments in fundamental theory and in computational methods. However, since these topics have differing rates of progress it is not planned that each topic will be included in every volume.

The present volume was conceived under the narrower coverage of ‘quantum chemistry’, and deals with four aspects of the application of quantum mechanics to chemistry and spectroscopy.

R. N. Dixon

Contents Chapter 1 The Calculation of Spectroscopic Constants

By W. G. Richards, J. Raftery, and R. K. Hinkley

1 Introduction

2 Spectroscopic Constants

3 Molecular Wavefunctions

4 Electronic Term Values

5 Rotational and Vibrational Constants The Born-Oppenheimer Approximation The Calculation of Vibrational and Rotational Energy

Levels from Potential-energy Curves Rigorously By Use of Model Potentials

Spectroscopic Constants The Accuracy of Computed Rotational and Vibrational

6 Spin-Orbit Coupling Constants

7 A-Doubling and Spin-splitting Constants

8 Magnetic Constants

9 Calculation of Hyperfine Interaction Constants

10 Transition Probabilities

11 Conclusions

Chapter 2 Direct Minimization Methods in Quantum Chemistry By D. Garton and B. T. Sufcliffe

1 Introduction

1

1

2

4

5

7 7

9 9

12

15

16

21

26

26

32

33

34

34

2 A Sample Problem 35

Con tents vi

3 Optimization Non-derivative Methods

Multivariate Grid Search Univariate Search Pattern Search Conjugate Directions

Derivative Methods Steepest Descent Conjugate Gradient Variable Metric Newt on- type

Implementing Optimization Schemes

4 The Realization of Direct Methods in Quantum Chemistry

5 Experience with the Use of Optimization Schemes

Chapter 3 Valence Bond Theory By 4. Gerratt

1 Introduction

2 Construction of Antisymmetric Wavefunctions The Exact Wavefunction Approximate Wavefunctions Construction of Spin Functions

Relationship between Valence Bond and Spin Valence

Spatial Symmetry in VB Theory Improvements in the VB Description : Ionic Structures ;

Single Configuration of Non-orthogonal Orbitals Computation of Matrix Elements

Theory of Pair Functions Atoms in Molecules

3 Valence Bond Theory

Theories

Hybridization

4 Extensions of VB Theory

5 Appendix: Energy of the Separated Pair Function

Chapter 4 Harmonic and Anharmonic Force Field Calculations By L M, Mills

1 Introduction

2 Definition of Force Constants

38 39 39 39 40 41 43 43 44 46 46 47

50

54

60

60

61 61 64 65

68

68 71

75 87 91

98 98

104

107

110

110

112

Contents vi i

3 Diatomic Molecules 115

4 Polyatomic Molecules : Method of Calculation Transformation from Internal to Normal Co-ordinates Symmetry, and the Number of Independent Force

Contact Transformation for the Effective Hamiltonian Relation to the Observed Spectrum; Resonances

Constants

Fenni Resonance Coriolis Resonance

Anharmonic Force Constant Refinements

5 Results and Discussion Linear Symmetric Triatomic Molecules Linear Unsymmetric Triatomic Molecules Linear Tetra-atomic Molecules Bent Triatomic Molecules More Complicated Molecules Discussion

121 123

131 132 137 138 139 140

143 144 146 151 152 157 1 59

Author Index 1 60

1 The Calculation of Spectroscopic Constants

~~ ~ ~

BY W. G. RICHARDS, J. RAFTERY, AND R. K. HINKLEY

1 Introduction

The answer to the question ‘why calculate spectroscopic constants?’ is not merely ‘to find the value’. There are in fact two possible reasons why one should use theoretical methods to compute spectroscopic constants. Firstly, there are cases where an experimental value has been measured for some constant, but the observed magnitude or even sign is not comprehensible in terms of the idea of electronic structure which the spectroscopist has in mind. In such cases a relatively crude calculation which only reproduces the observation to an order of magnitude may offer explanations in terms of perturbations by unobserved states or the atomic constitution of molecular orbitals.

These crude calculations which assist the spectroscopist in his interpretation have been possible for some time and are increasingly used almost as an experimental tool. Recently, however, the power of electronic computers has advanced to the stage where, for small molecules, wavefunctions of a very accurate nature can be calculated. For very good wavefunctions, energies and, more especially, other expectation values can be computed very accurately. Indeed in some instances spectroscopic constants can now be computed to greater precision than they can be measured.

Typically such an improvement is counterbalanced by some corresponding loss. In this case the loss is a loss of understanding in terms of simple pictorial concepts. It is possible to calculate a number very accurately but it may be no longer possible to explain its magnitude in terms of major components.

With these more recent calculations the work is done with the intention of finding a numerical value. As a result there is a change of emphasis in the type of problem attempted. Simple calculations which go hand in glove with experiment remain in the area of problems where both experiment and computation are possible. The accurate work is different. If there is only a number resulting, then it is perhaps pointless doing the work if that number is experimentally accessible, save for testing the method. The accurate work attempts to extend spectroscopy by computing properties which cannot be observed, or have eluded the experimentalist. Frequently this will mean calculations on excited electronic states and on the electronic levels of molecular ions. From the

1

2 Quan t imi Chemistry

standpoint of theoretical chemists who compute wavefunctions and molecular properties this is a welcome impetus away from the relatively boring computation of properties of closed-shell ground states of molecules.

Although some accurate work has been attempted for polyatomic molecules, the majority of published papers1 deals with diatomic molecules. This is not only for the reason that they are simpler but also because the level of understanding of diatomic molecule spectra is so much higher than that for polyatomic species. In the latter case relatively few spectra have been analysed to the level where all the rotational and fine structure is assigned.

This Report deals with the calculation of spectroscopic constants both for diatomic and for polyatomic molecules, while concentrating on the former. The spectroscopic constants included are restricted to those measured in high-resolution gas-phase work. We cover the whole range of complexity in computation since this is determined not by the method used in computing the expectation value, but by the quality of the wavefunction used. Generally the wavefunctions used are of the ab iizitio type, but their quality will depend on the size and type of basis set employed as well as the method.

A spectroscopic term value can be considered to be made up of energy terms which are separable as electronic (T), vibrational (G) , rotational (F), and other hyperfine effects, i.e.

E= T+ G + I;+ hyperfine terms.

In this Report we will consider constants under all these headings but concentrate on the smaller energy terms, Electronic energy levels will only be given a cursory treatment as they are really a special case and the calculation of excited electronic energy levels really demands a review article to itself.

2 Spectroscopic Constants

Although it may seem more like a problem for a linguistic philosopher it is necessary briefly to make quite clear what is meant by a spectroscopic ‘constant’. Unfortunately spectroscopic constants are not immutable quantities measured by spectroscopists to an accuracy solely dependent on the quality of his measuring instruments and the precision of wavelength standards. The measured quantities which are in this position are the spectroscopic lines and hence the term values. These may be measured and quoted in terms of wave numbers with statistical limits on their accuracy.

The term values, or in some cases the actual lines, are then fitted to energy expressions containing the ‘constants’ and functions of quantum numbers. The constants may not be correct or meaningful for one of two reasons. The theoretical formula to which the energy levels are fitted may not be appropriate for every part of the potential-energy curve of a diatomic molecule.

W. G. Richards, T. E. H. Walker, and R. K. Hinkley, ‘Bibliography of Ab Initio Molecular Wave Functions’, Oxford University Press, 1971.

a W. Aslund, Arkiv. Fysik, 1965, 30, 377.

Tlze Calculation of Spectroscopic Constants 3

Just how much of the data are fitted is largely arbitrary and variations can cause dramatic changes in the so-called constants. A striking example in a seemingly simple situation is the case of the ground-state vibrational constants of H2. Here fourteen vibrational energy levels can be observed. They may then be fitted to a formula of the type

G(o) = U ~ ( U + +) - X ~ U ~ ( U + +) + ewe(^ + +)3 - . . . It is entirely arbitrary just how many vibrational levels or how many 'coa- stants' one includes. Table l shows just what variations are possible when fitting all fourteen levels to a variety of polynomials. In a situation like this the theoretician must be careful to realize just what worth the constants have. They are merely expansion coefficients and not necessarily physically significant. He must also realize that the same problem will amct his own work If he calculates a spectroscopic constant from a computed potential curve without considering which part of that curve it is appropriate to use.

A similar problem in giving physical meaning to spectroscopic constants arises when the situation is theoretically complex and it is not possible to devise a model which will reproduce the data. A striking case of this has been found3 in a re-analysis of the A2X+- X2rI1r system of OH in an attempt to provide experimental constants to test calculations of A-doubling constants. None of the theoretical formulae4 fit the data, probably because the coupling case is not near an extreme. The final formula used to fit the fi andf2 components of the ground state was

E t ~ = B x - D [x2 + (x+ l)] +Hx3 + -$[o + & P + ~ ] X + I)] + k 1+(+P + 4) (J+ 3)

(2B- ~ D x + 6Hx2) 2 -- - ( + P + d V+3) +[+-I

+ I* 1 (3P+4) (J+3)I2 9 1 l J2

where x = J ( J + 1). E. A. Moore and W. G. Richards, Phys. Scripta, 1971, 3, 223.

' R. S. Mulliken and A. Christy, Phys. Rev., 1931, 38, 87.

4 Quantum Chemistry

Table 1 The vibrational constants of H2 X 1 X$ for various energy expressions

No. of terms in power series

3 4 5 6 7 8 9

10

W e

4380.4 4405.3 4399.7 4400.5 4399.8 4400.1 4401.1 4403.6

X e W e

107.1 126.8 119.6 121.1 119.5 120.2 123.5 132.2

Yeme -0.85

2.42 0.31 0.95

0.56 3.73

13.69

-0.04 -

ZeWe - 0.14 0.07 0.04

-0.22 0.01 1.41 6.80

Even an expression of this complexity does not reproduce the experimental data and exclusion of small constants or addition of further ones alters many of the others quite significantly.

Spectroscopists do not always have enough data to test whether their constants are meaningful or not. Consequently it must be borne in mind that calculated constants may not always agree with experiment either because the published experimental work has defects or because the theoretician is making the same type of error: using a model which is unsatisfactory or applying a formula beyond the limits of its validity.

3 Molecular Wavefunctions

The starting point for any calculation of spectroscopic properties will be an appropriate molecular wavefunction Y. The property may then be computed as an expectation value

(Yloperatorl u/>

Alternatively, perturbation theory may be used, or energies and hence potential curves are produced with the constants then being derived from the po tential-energy curve.

The starting point for the vast majority of molecular wavefunctions is the Hartree-Fock framework as developed by Roothaan. The molecular wavefunction Y is expressed as an antisymmetrized product of molecular orbitals $t

each multiplied by its appropriate spin function,

-

<!PI V } -- *

Y=d$1$2$3. . . &. The MO q5i are in turn expanded in terms of a basis set of atomic orbitals

(AO), 4 Ic = 2 Cikx k .

k

The basis set may be constructed from radial functions of either Slater (exponential) or gaussian form, multiplied by appropriate angular (0 and 4) functions. This scheme is frequently referred to by the acronym LCAO-MO- SCF.

The Calculation of Spectroscopic Constants 5

When the basis set contains many terms (effectively infinite), one obtains the best possible result from solution of the Hartree-Fock equation: the Hartree-Fock limit. Improvements beyond this limit are most usually achieved by allowing the molecular wavefunction Yf to be a linear combination of antisymmetrized products of orbitals @I,

Y= Co@o+ Cl@l+ . . . The @I correspond to different electron configurations. In ‘configuration interaction’ @ O is the Hartree-Fock function (or an approximation to it in a truncated basis set) and the other @I are constructed from virtual orbitals which are the by-product of the Hartree-Fock calculation. The coefficients Ci are found by the linear-variation method. Unfortunately, the @i so constructed are usually an inadequate basis for the part of the wavefunction not represented by @o, and the process is very slowly convergent as more configurations are added.

In ‘optimized multi-configuration’ or ‘multi-configuration SCF’ (MCSCF) methods the coefficients CZ and the orbitals from which the @t are constructed are varied simultaneously. This is in principle a much more satisfactory process. It achieves energies comparable with those from configuration interaction calculations and yet provides much simpler wavefunctions (i.e. fewer configurations); however, the variation problem is non-linear, and the resulting technical difficulties have inhibited wide use of the method.

4 Electronic Term Values

The calculation of potential curves or surfaces for several electronic states of a molecule should yield by difference the spectroscopic constants Te and voo.

There is considerable interest in such properties as the vast majority of excited electronic states are not observed experimentally and yet may be of keen interest in a number of experimental areas, notably photochemistry and photoelectron spectroscopy.

In the case of photoelectron spectroscopy the experimental data provide electronic term values for the excited states of the ion. By the use of Koop- mans’ theorem,5 estimates of these quantities can be obtained from orbital energies found in calculations on the closed-shell ground state of the molecule. This procedurecan give a useful hint at the analysis of a spectrum but is on rather shaky ground theoretically,G involving a number of assumptions of doubtful validity and giving spurious answers in several well-documented instances.

The calculation of term values directly by treating each state as a separate variational problem is also fraught with difficulties, or rather with two difficulties. These are the problems of relativistic and correlation energy. As Figure 1 shows, the potential-energy curve computed by solution of the

T. Koopmans, Physica, 1933, 1, 104. * W. G. Richards, Internat. J. Mass Spectrometry Zon Phys., 1969, 2, 419.

6 Quantum Chemistry

t x

Q) c P w

Limited basis set calculation \ // limit

Hartree -Fock

curve

\

- /

Internuclear distance - Figure 1 Schematic calculated and experimental potential curves for a diatomic

moIecule

Hartree-Fock equations for a series of distances is neither coincident with nor parallel to the true potential curve. Any curve computed in the LCAO- MO-SCF scheme with a small basis set of atomic orbitals will be even further from the truth. This discrepancy has two origins. Firstly the Hartree-Fock equations are non-relativistic and consequently there is a relativistic energy error. The virial theorem tells us that inner-shell electrons have the greatest kinetic energy, so that relativistic effects become important for these electrons. For example, in the case of the Mg atom the relativistic energy of a Iselectron is 0.2 a.u. and that for a 2s electron is 0.03 a.u. (1 a.u. = 27.21 eV). Relativistic energies are thus massive contributors to total energies, but since they are contributed largely by the core electrons they are reasonably constant between valence electronic energy states of a molecule.

The second source of error is correlation energy. This is defined7 as

&xa l experimental = EHartree-Fock + Erelativistic -!- Ecorrelation*

The physical origin of correlation energy is in the nature of the Hartree-Fock equations. The inter-electronic interaction is represented by coulombic and exchange terms; each electron has a direct interaction with the average charge of all the others obtained by squaring the one-electron functions (the molecular orbitals), but an exchange interaction only with electrons of the same.

' P. 0. kowdin, Adv. Chem. Phys., 1959, 2. 207.


spin. In reality an electron in a molecule will have instantaneous interactions with all the other electrons, which will not be the same as the average interaction included in the SCF procedure.

Although the origin of the error is easy to visualize, its circumvention is less simple. From a systematic examination of the magnitudes of Ecomelation in a series of atoms and ions it has been shown* that the predominant effects are between pairs of electrons in the same orbital. Tables of estimates of corre- gation effects have been published9 and occasionally used to predict electronic term values.1°

Although some theoretical work aims at direct calculation of the correlation energy,ll the most widely used approach is to go beyond the Hartree- Fock limit by the use of configuration interaction. This process is now fully automated and it is not unusual for several thousand excited configurations to be incorporated into the calculation. By these brute-force methods, good estimates of electronic term values can be calculated; similar results using fewer configurations can be obtained by optimized multiconfiguration methods. Table 2 shows a few random examples which are of general interest and typical of the sort of accuracy currently expected.

Table 2 Calculated electronic excitation energies

Term Valuelcm-l Molecule State Observed Calculated

BeH A2II 20 039 20 215 CH+ AlII 24 146 24 970

Apart from spectroscopic constants, other observable spectroscopic features may be studied by consideration of calculated potential-energy surfaces. Notable in this respect is the understanding of observed Jahn-Teller splittings, a good example being the methane ion, CH4+.12

5 Rotational and Vibrational Constants

The Born-Oppenheimer Approximation.-The total wavefunction for a molecule may be approximated by separation into two factors,

where YIelectronic is the solution of the ‘clamped nucleus’ electronic Schrodinger equation,

E. Clementi, J. Chem. Phys., 1963,38,2248. A. Veillard and E. Clementi, J. Chem. Phys., 1968, 49, 2415.

lo G. Verhaegen, W. G. Richards, and C. M. Moser, J. Chem. Phys., 1967, 47, 2595. l1 R. K. Nesbet, Phys. Rev., 1967, 155, 51. a s R. N. Dixon, Mol. Phys., 1971, 20, 113.

8 Quantum Chemistry

Here Y(R,r ) symbolizes the potential energy of electrons in the field of nuclei whose co-ordinates are given by R. This equation may be solved for different values of R, since the hamiltonian contains no differential operators with respect to R. The eigenvalues plotted against R give the potential-energy curve or surface referred to above. Ynuclear is then a solution of the nuclear Schrodinger equation,

( ( - f i2/2)xvk2/Mkf U ( R ) ynuctear=EYnuctear- 1 Born and Oppenheimer13 showed that this approximation is equivalent to the neglect of terms in the hamiltonian of the order of 1/Mk, i.e. about 10-3

even for the lightest nuclei. In the absence of external fields, we may take axes that move laterally with

the molecule, thus eliminating translational motion. In these co-ordinates a diatomic molecule becomes equivalent to a single particle with mass p = MAMB/(MA + MB), moving in a spherically symmetrical potential U(R), where R is the internuclear separation. The Schrodinger equation is therefore

This equation may be separated analogously to that for the hydrogen atom, the angular functions being spherical harmonics Yzm( 8, $),

{(ti 2 / 2 ~ ) V 2 + u(R )I ynuclear = Eynuc~ear-

Yzm(8, $) is readily identifiable with a ‘rotational wavefunction’, and since p2(R) is the probability of finding an internuclear distance r , p(R) may be identified with a vibrational wavefunction. p(R) is a solution of the equation

{ ( - h 2 / 2 p ) V 2 + U(R)+ J (J+ 1)h2/2pR2}p(R)=Ep(R). (1)

Given U(R), this equation may be solved to obtain energy levels dependent upon u, the vibrational quantum number, and J.

Tf the rotational quantum number J is zero, the molecule possesses no angular momentum arising from the motion of the nuclei ; nuclear motion is purely vibrational. The vibrational energy levels depend on the shape of the potential function U(R), most often of the well-known diatomic form. Near the minimum the potential function approximates to a parabola. The eigenvalues and functions are thus approximately those appropriate for a harmonic oscillator,

or, in wavenumbers,

This approximation is the basis for expressing vibrational energy levels as a power series in ( v + &), as discussed above. l 3 M. Born and J . R. Oppenheimer, Ann. Physik, 1926, 79, 361.

E(4 = VO& + 31,

G(D)= O J ~ ( Z , 4- f).


The term J (J+ 1)ti2/2pr2 in equation (1) is typically very small compared with the other terms, for small values of J, and may be regarded as a perturbation on the vibrational levels, owing to rotational motion. In spectroscopic notation, the correction due to this term is often labelled F(J).

Including the perturbation to first order (the vibrational wavefunction for level v king written as Iv>)

The quantity ti2/2p <1/R2) is called Bv, the rotational constant for level v. Thus the expression for F(J ) becomes

F(J )=BJ(J+ 1).

Comparison with the theory of the rigid rotor identifies Bv with the expectation value of the inverse of the moment of inertia.

If the perturbation is included to second order, the expression becomes

This may be written P(J)=BvJ(J+ 1)- D,J2(J+ 1)2. If rotational levels are required to a greater degree of accuracy, higher terms may be included, but this is rarely justified by the experimental data. Thus, in principle, the energy levels for nuclear motion may be calculated exactly from a potential-energy curve, either by numerical solution of equation (1) for different values of J, or by numerical solution of the simplified rotationless equation,

and including rotation as a perturbation.

The Calculation of Vibrational and Rotational Energy Levels from Potential- energy Curves.-Rigorously. If a potential-energy curve has been calculated then, in principle, all the vibrational and rotational levels may be extracted by solution of the clamped-nucleus Schrodinger equation [equation (l)]. (The terms neglected in the separation of the hamiltonian into nuclear and electronic parts give rise to a small diagonal correction term U’(R) which may be added to the potential energy.) This equation may be solved by numerical integration, following the technique of Cooley14 and Cashion.15

l 4 J. W. Cooley, Math. Comp., 1961, 15, 363. l 6 J. K. Cashion, J. Chem. Phys., 1963, 39, 1872.

10 Qiruntum Chemistry

In practice, the number of eigenvalues of this equation is so large that the procedure has until recently only been employed for the hydrogen molecule. 14-16

In the most accurate work, Wolniewiczlc used the 85 points on the potential- energy curve of ground-state (lC+) H2 calculated by Kolos and Wolniewicz17 to obtain the energy levels. He made a careful study of the interpolation methods, and deduced that the error introduced by these was very small: different techniques gave rise to differences of less than 0.5 cm-1 in the levels. In two computations, the correction U’ was neglected and included, respectively; the former gave errors in the energy levels of less than 2 2 cm-l, the latter errors of less than 2 1 cm-1. These were ascribed to slight inadequacies in the curve, and possibly to the failure of the Born-Oppenheirner approximation. This work clearly shows that, given extremely accurate potential energy curves, it is possible to reproduce observed energy levels very well.

This method appears to have been applied to few other systems. In extremely thorough work Sahni et a2.18919 calculated between forty and fifty points on the potential-energy curves of ground-state CO and various states of CO+, and used the Cooley-Cashion method to obtain the energy levels G ( v , J ) . Unfortunately, the quality of their wavefunctions (minimal basis, with exponents re-optimized at each distance) was insufficient to give results in agreement with experiment. It is regrettable that the thoroughness they showed has not generally been applied by calculators of more accurate wavefunctions. Their results are of great interest however, for several reasons. Comparison of the rigorous solution of equation (1) with the perturbation treatment outlined above gave good agreement for the values of J considered, thus providing a justification for the work based on that treatment. An alternative method of obtaining the G(v,O) was also used, based on the WKB approximation,2* and gave results in ‘fair’ agreement. This work was followed up by work on LiH which used21 a minimal basis and polarization functions, together with extensive CI and optimization of exponents at each of 25 distances. This gave quite good agreement with experiment, as shown in Table 3.

The major drawback with such methods is the requirement of the Cooley- Cashion method of a very large number of points on the curve, to obtain the necessary accuracy. Thus some form of interpolation is inevitable. The calculation of Kolos and Wolniewicz gave values of the derivative as well; Wolniewicz was therefore able to use an interpolation method utilizing these. However, comparison with a simple polynomial interpolation showed that the errors introduced by the latter were very small.

A simpler method of calculation which is still of high accuracy is to treat the term J (J+ 1)h2/2R2 as a small perturbation? and to solve the vibrational

la L. Wolniewicz, J . Chem. Phys., 1966, 45, 515. l 7 W. Kolos and L. Wolniewicz, J. Chern. Phys., 1966, 45, 509.

R. C. Sahni, C. D. LaBudda, and B. C. Sawhney, Trans. Faraday SOC., 1966,62, 1993. l S R. C. Sahni and B. C. Sawhney, Trans. Faraday SOC., 1967, 63, 1. a o D. C. Jain and R. Sah, Proc. Phys. SOC., 1962, A80, 525. l 1 R. C. Sahni, B. C. Sawhney, and M. J. Hanley, Trans. Faraday SOC., 1969,65,3121.

The Calculation of Spectroscopic Constants

Table 3 Observed and calculated vibrational levels for LiH

G (u)/cm-l P Observed Calculated 0 698 .O 710.2 1 2058.0 2041.3 2 3376.0 3291.1 3 4643 .O 4550.0 4 - 5796.1

11

equation obtained by putting J=O,

This equation may be solved either by the Cooley-Cashion technique or by a variational approach, where p(R) is expanded in terms of the eigenfunctions of some approximate hamiltonian. It has been shown that the form of the basis functions is critical ; use of eigenfunctions of a harmonic oscillator22 gave unsatisfactory results for higher vibrational levels, but if the eigenfunctions of a Morse oscillator were used, results comparable with numerical techniques were obtained.23 Recently, potential-energy curves for various states of CH from very accurate configuration interaction wavefunctions have been analysed in this manner,2* with results in close agreement with experiment.

Even the solution of this simpler equation has not been used frequently. Browne and Greenawalt25 have solved it for their multiconfiguration potential- energy curves for BH, including the difficult case of the lC+ state, which has q curve with a double minimum. The agreement with experiment is good, and th‘kqcan be little doubt that the use of this method gives a better indication of the accuracy of their potential-energy curves than more approximate methods, even if the latter were to obtain nominally better agreement with observed constants. The ‘experimentally observed’ constants depend on the particular model used, so direct comparison of calculated energy levels with those observed is on a much sounder footing.

Use of this method enables a first-order prediction of the effect of rotation. The first-order rotational energy correction is F(J ) = B,,J(J+ l), where Bv= <vl 1/R21v)ti2/2p. The expectation values of 1/R2 are readily calculated from the eigenfunctions obtained by the Cooley-Cashion method. Again, those calculated by Browne and Greenawalt are in good agreement with experiment. Fitting the observed B,, values to a linear expression in (v++) yields values of Be and Re from the equation B,=Be-ae(v++). These ‘constants’, while useful for comparison with other approximate methods, are not observable and have no theoretical significance outside a particular model.

B a E. Zetik and F. A. Matsen, J. Mol. Spectroscopy, 1967, 24, 122. 23 E. Greenawalt and A. S. Dickinson, J. Mol. Spectroscopy, 1969, 30, 427. p 4 G. C. Lie, J. Hinze, and B. Liu, J. Chem. Phys., 1972, 57, 625. 2 6 J. C. Browne and E. Greenawalt, Chem. Phys. Letters, 1970, 7 , 363.

12 Quantum Chemistry

The calculation of the second-order correction, - D,J2(J+ 1)2, is in principle possible, since it is a sum of off-diagonal matrix elements of 1/R2, but will be extremely lengthy, and has not been performed. If divergence from a linear dependence on J (J+ 1) in the rotational levels is observed, it is probably no more difficult to solve the equation with rotation explicitly included.

Browne and Greenawalt calculated 23 points on their potential curve for the ground state of BH, a much larger number than is generally done for diatomic molecules. Obviously, the more points which are known, the less reliance placed on interpolation. Similar calculations have been performed on the 2C+ state of HCl+ using fifteen points,26 but in spite of the comparative simplicity and accuracy of the method, few other calculations of this type have been reported. The results for HCl+, despite the fact that the potential- energy curve, being derived from a single configuration, was probably much less accurate than that of Browne and Greenawalt, gave good agreement with experiment, giving vibrational term values accurate to better than 1 % (a systematic error) and B-values of comparable accuracy.

By Use of Model Potentials. Most of the calculations performed on diatomic molecules have preferred the approach of fitting the calculated points to some model curve, and extracting the spectroscopic constants from the known relationships between them and the parameters of the model potentials. Most frequently used has been the polynomial in R. Generally an expression in positive powers of R is used,

U(R)= Uo+AR+BR2+ . . . ,

but M ~ L e a n ~ ~ has shown that a much better approximation is given by an expression with a term in R-I. The curve of Kolos and Roothaan28 was used to compute spectroscopic constants in a number of ways. Tn Table 4 some of

Table 4 Comparison of results for the hydrogen molecule (potential curve of Kolos and Roothaan)

(0 e Culculation 4393.8 4402.3 4412.67 Numerical solution (Cooley) 4400.39 Experiment

Polynomial fit, positive powers of R Polynomial fit, including R-1 term

his results are compared with those of Cooley14 and with the experimental answer. As can be seen, inclusion of a R-1 term gave substantial improvement. The closeness of this answer to experiment is of course fortuitous; the error in the curve-fitting approximately cancels that in Kolos and Roothaan’s curve. Using Cooley’s procedure on the better curve of Kolos and Wolniewicz gives better agreement with experiment. This work may be regarded as

J. Raftery and W. G. Richards, J. Phys. (B) , 1973, 6, 1301. A. D. McLean, J. Chem. Phys., 1964,40,2774.

2 8 W. Kolos and C. C. Roothaan, Rev. Mod. Pliys., 1960, 32, 205.


optimum in fitting a polynomial; the fact that it still differs from the value of Cooley by an appreciable amount indicates that this approach is less satisfactory. The methods outlined by McLean have been applied to some calculations on light diatomics.29~ 30

Basic to much of the work in which calculated points are fitted to a polynomial is the Dunham analysis. Dunham31132 showed that if a potential energy curve is expressible as a power series,

U= Umin+aoS2(1 +a15+a2t2+ . . . ), where

R-Re Re

(=-,

then the energy levels are F(u,J) =C Yz~(u + +)-)IJi(J+ l)i the Y25 being related

to the coefficients ao, a1, . . . etc. Clearly, the lower Y z ~ are identifiable with the spectroscopic constants we, X e w e , Y e w e , Be, me, etc. The Dunham analysis enables the spectroscopic constants to be calculated directly from the polynomial coefficients. This analysis has been widely 2 9 y 3O9 33-52 however, the degree of the polynomial used is often not given. This is a point of some importance, since the computed constants may differ considerably, depending on the degree used. It has been pointed out53 that the presence of random errors in polynomial curve-fitting requires that the number of points be considerably larger than the degree of the polynomial used, a requirement not satisfied in much of the work published. Another difficulty is that if different combinations of points on the curve are fitted, different answers are obtained. In unpublished work on OZ+, Raftery and Todd found that fitting

ki

Is R. K. Nesbet, J. Chem. Phys., 1964, 40, 3619. s o R. K. Nesbet, J. Chem. Phys., 1965,43,4403. 8 1 L. Dunham, Phys. Rev., 1932, 41, 713. a s L. Dunham, Phys. Rev., 1932,41, 721.

A. C. Wahl, J. Chem. Phys., 1964,41, 2600. W. M. Huo, J. Chem. Phys.. 1965,43, 624. G. Das and A. C. Wahl, J. Chem. Phys., 1966,44, 87. R. K. Nesbet, J. Chem. Phys., 1966, 44, 285.

I' P. E. Cade, K. D. Sales, and A. C. Wahl, J. Chem. Phys., 1966,44, 1973. 3 8 W. M. Huo, K. F. Freed, and W. Klemperer, J . Chem. Phys., 1967, 46, 3556. a 0 P. E. Cade and W. M. Huo, J. Chem. Phys., 1967, 47, 614.

O P. E. Cade and W. M. Huo, J. Chem. Phys., 1967, 47, 649. d l P. E. Cade, J. Chem. Phys., 1967, 47, 2390.

G. Das and A. C. Wahl, J. Chem. Phys., 1967,47,2934. R. L. Matcha, J. Chem. Phys., 1967, 47, 4595.

4 4 R. L. Matcha, J. Chem. Phys., 1968, 49, 1264. 4 6 K. D. Carlson, K. Kaiser, C. M. Moser, and A. C. Wahl, J. Chem. Phys., 1970, 52,

4 e R. L. Matcha, J. Chem. Phys., 1970, 53, 485.

' * P. Julienne, M. Krauss, and A. C. Wahl, Chem. Phys. Letters, 1971, 11, 16. 4 9 T. L. Gilbert and A. C. Wahl, J . Chem. Phys., 1972,55, 5247. e o G. Das and A. C. Wahl, J. Chem. Phys., 1972,56, 3532. 51 P. A. G. O'Hare and A. C. Wahl, J . Chem. Phys., 1972,56,4516. 5 2 M. Yoshimine, J. Chem. Phys., 1972, 57, 1108. .W E. L. Mehler, K. Rudenberg, and D. M. Silver, J. Chem. Phys., 1970, 52, 1181.

4678.

P. A. G, O'Hare and A. C. Wahl, J. Chem. Phys., 1970, 53, 2469.


a quartic to different sets of five, six, and seven points selected from seven calculated could produce values of we ranging from 2140 to 2470 cm-1. A more reliable method (see below) gave an answer of 2493 cm-1. The experimental answer is 1903 cm-l, but the calculated curve was very bad. In further work,54 Todd took arbitrary points on an experimental RKR curve for the curve-fitting, and again obtained wildly varying results.

Possible solutions to this were given by N e ~ b e t , ~ ~ ? 303 36 who performed calculations on distances selected according to the roots of a Chebyshev polynomial, thus facilitating interpolation, and by Peyerimhoff 55 and Mehler et c11.,~3 who have fitted several different sets of points to different degrees of polynomial and averaged the results.

and, most usually, the quartic.29~ 309 369 59-76 The quartic has the advantage that it gives analytical expressions for the most commonly recorded constants, we, X e w e , Be, me.

Probably a high proportion of the work quoting a Dunham analysis uses this function too.

Surprisingly, little work has been done by fitting the calculated points to morc realistic model potentials. McLean’s R-1 potential, in spite of its proven advantages, and the fact that the fitting procedures are still linear, has been little used. Other model potentials, which, unlike these, do not ‘blow up’ outside the fitted range, have been little used because of the complexity of fitting a function which is non-linear in the coefficients. Todd54 has devised a method of fitting points to a Morse function, which has been used with some success.77-80 It has been found not to have the defect of the quartic

Commonly used polynomials are the

6 4 J. A. C. Todd, B.A. Thesis, Oxford, 1967. 6 6 S. Peyerimhoff, J. Chem. Phys., 1965, 43, 998. 6 6 S. Fraga and B. J. Ransit, J . Chem. Phys., 1961, 35, 669. b7 S. Fraga and B. J. Ransi!, J. Chem. Phys., 1962, 36, 1112. b g H. P. D. Liu and G. Verhaegen, J. Chem. Phys., 1970, 53, 735. h S H. H. Michels and F. E. Harris, J. Chem. Phys., 1963, 39, 1464.

S. B. Schneideman and H. H. Michels, J. Chem. Phys., 1965, 42, 3707. €3. F. Schaefer and F. E. Harris, J. Chem. Phys., 1968, 48, 4946.

6 2 F. GuCrin, Theor. Chim. Acta, 1970, 17, 97. 6 s H. F. Schaefer, J. Chem. Phys., 1970, 52, 6241. I4 H. H. Michels, J. Chem. Phys., 1970, 53, 841. O 6 S. V. O’Neil and H. F. Schaefer, J. Chem. Phys., 1970, 53, 3994. I s M. F. Schaefer, J. Chem. Phys., 1971, 54, 2207.

H. F. Schaefer and T. G. Hei!, J. Chem. Phys., 1971, 54, 2573. O e €3. F. Schaefer, J. Chem. Phys., 1971, 55, 176.

S. V. O’Neil and H. F. Schaefer, J. Chem. Phys., 1971, 55, 394. 7 0 B. Liu and H. F. Schaefer, J. Chcrn. Phys., 1971, 55, 2369.

H. F. Schaefer and W. H. Millcr, J. Chem. Phys., 1971, 55, 4107. 7 f P. K. Pearson, C. F. Bender, and H. F. Schaefer, J. Chem. Phyzys., 1971,55,5235. 7 3 J. Kouba and Y . Ohrn, J. Chem. Phys., 1970, 52, 5387. 7 4 J. Kouba and Y. Ohrn, J. Chem. Phys., 1970,53, 3923. 7 5 V. Bondybey, P. K. Pearson, and H. F. Schaefer, J. Chem. Phys., 1972, 57, 1123. 7 8 S. Green, P. S. Bagus, B. Liu, A. D. McLean, and M. Yoshimine, Phys. Rev. ( A ) ,

7 7 J. Raftery and W. G. Richards, Internnt. J. Mass Spxtrometry Ion Phys., 1971, 6, 269. 7 8 J. Raftery and W. G. Richards, J. Phys. (B), 1972, 5, 425.

J. A. Hall and W. G. Richards, Mol. Phys., 1972, 23, 331, P. R. Scott, B.A. Thesis, Oxford, 1972.

1972, 5, 1614.

The Calculation oJSpectroscopic Constants 15

fit, in that different sets of points give similar answers, provided the points do not lie too far up the left arm of the curve, where the Morse potential is known to be inaccurate. The method gives results in good agreement with the numerical method, e.g. for the calculation on the HClf A 2 C f state mentioned above, a value of we of 1610 cm-l was obtained, compared with 1574 cm-l by the numerical method. Only six points were used. A comparison also exists between the Morse curve-fit and the Dunham analysis. Raftery and Richards78 and Julienne et aZ.48 have published potential energy curves for the X 2TI state of HF+ which are virtually identical. Morse curve-fitting by Todd’s method gave we= 3453 cm-1, and Dunham analysis 3492 cm-1. The observed we is 3340 cm-l.

The simplest method of obtaining values of we and Re is fitting a parabola.81-93 This suffers from the defects of other polynomials but has the advantage of requiring only three points ; this can be important if calculations are exceedingIy lengthy. A qualitative guess at we and Re may be obtained, provided the points are close to the minimum and approximately equally spaced about it. In calculations on ScF, Scott80 has shown that fitting a parabola to three well-chosen points gives answers close to those obtained by fitting a Morse curve. In this case calculations were rather lengthy, so calculating enough points for a rigorous analysis was economically impracticable.

The Accuracy of Computed Rotational and Vibrational Spectroscopic Con- stants.-The work of Wolniewiczl6 shows that given an accurate potential energy curve, the energy levels may be obtained to very high accuracy, the accuracy being limited only by the Born-Oppenheimer approximation. Expressions for the correction terms, diagonal and off-diagonal, have been given by Sidis and Lefebvre-Brion.94~ 95

In general, however, potential-energy curves cannot be calculated accurately, and approximate methods must be used. Hartree-Fock calculations are often employed; these may give potential curves greatly in error, since they often dissociate to the wrong limits. When this is not the case, Hartree-Fock curves may be quite accurate.78 When it is, the effect is generally to cause these curves to give high we and low Re values.

R. K. Nesbet, J. Chem. Phys., 1962, 36, 1518. R. K. Nesbet, J. Cliem. Phys., 1964, 41, 100. K. D. Carlson and R. K. Nesbet, J. Chem. Phys., 1964, 41, 1061.

8 4 H. Lefebvre-Brion and C. M. Moser, J. Chem. Phys., 1965,43, 1394. 8 6 K. D. Carlson, E. Ludena, and C. M. Moser, J. Chem. Phys., 1965, 43, 2408.

K. D. Carlson and C. M. Moser, J. Chern. Phys., 1966, 44, 3159. 8 7 G. Verhaegen and W. G. Richards, J. Chem. Phys., 1966,45, 1829.

W. G. Richards, G. Verhaegen, and C. M. Moser, J. Chem. Phys., 1966, 45, 3226. K. D. Carlson and C. M. Moser, J. Chem. Phys., 1967, 46, 35. G. Verhaegen, W. G. Richards, and C. M. Moser, J. Chem. Phys., 1967, 46, 160.

s 1 K. D. Carlson, C. R. Claydon, and C. M. Moser, J. Chem. Phys., 1967,46, 4963. O a W. G. Richards and R. C. Wilson, Trans. Faraday SOC., 1968, 67, 1729.

* * V. Sidis and H. Lefebvre-Brion, J. Phys. (B), 1971, 4, 1040. G. Verhaegen, J. Chem. Phys., 1968,49,4696.

V. Sidis, J. Chem. Phys., 1971, 55, 5838.


The dissociation to the wrong limits may be eliminated by incorporation of selected configurations in the wavefunction; if this is done, the we value generally becomes too low by a small amount, since one is effectively only allowing correlation of the ‘near-degenerate’ type. A curve which reflects better the true curve may be obtained, in principle, by including in the wavefunction configurations which take account of that correlation which is not present in the dissociated atoms. Results from beyond Hartree-Fock calculations have in general been rather disappointing. Mehler et aZ.53 performed calculations using separated pair theory on LiH and BH, giving results which they regarded as good but cannot be objectively so viewed. For LiH they obtained we= 1483 cm-I (cf. expt. 1405 cm-l) and for BH 2928 cm-I (cf. expt. 2367.5 cm-I). The last answer especially is rather disappointing. The method used was Dunham analysis using an average of results from polynomials of 4th, 5th, 6th, and 7th degree. The CI calculation of Pearson, Bender, and S~haefer7~ on BH gave an we value of 2138 cm-1, almost as disappointing, using a quartic fit. To some extent the fitting methods may account for some of this discrepancy, since the results of Browne and Green- awalt, when their levels are fitted to spectroscopic constants, give results equally far from the experimental answers, although the energy levels are in fairly close agreement. Similarly, calculations on Be0 yield68 disappointing we

values. However, Schaefer’s calculation on 0 2 , 6 6 using the same method, gave fairly high accuracy in (1614 cm-1 compared with 1580 cm-l), and a recent multiconfiguration SCF calculation on F2 gave50 very good results (942cm-l compared with 932cm-l). This work showed that the type of replacements chosen in the multiconfigurational calculation was crucial in obtaining an accurate potential-energy curve.

More recent work has shown that given massive configuration interaction, typically involving thousands of configurations even for first-row hydrides, will give extremely accurate results for vibrational constants. Work on CH,24 BeH96 and CH+,76 using very nearly complete CI (the latter work coming within 1 eV of the true Born-Oppenheimer energy), gives consistently good results for equilibrium distances, vibrational frequencies, and even for the smaller correction terms.

An interesting corollary of Schaefer’s work66 was the discovery that whereas Hartree-Fock Re values were generally small, and extended basis CI values larger, minimal basis CT results were generally too large. From this he deduced that minimal basis SCF results were probably near experimental, and suggested that this might explain the otherwise extraordinary success of minimal calculations in predicting bond lengths in small polyatomic molecules.

6 Spin-Orbit Coupling Constants

The spin of an electron gives rise to a magnetic moment which can interact with effective magnetic field set up by its orbital motion about the nucleus.

O b P. S. Bagus, C. M. Moser, P. Goethals, and G.Verhaegen, J. Chem. Phys., 1973,58, 1886.


Although this spin-orbit interaction is essentially a relativistic effect it may be approached classically. For hydrogen-like atoms the spin-orbit hamiltonian is

Hso = (a2/2) ( 1 / r a U p r ) s. Z,

with cc the fine-structure constant and U the potential in which the electron moves.

When this expression is extended to many-electron systems, two related problems arise. Firstly, what is the effective spin-orbit hamiltonian for the electron in open shells? Secondly, what is the potential in which they move? For a hydrogen-like atom the field would be written

- grad U= (Z/r3) r;

for a many-electron atom the nuclear charge Z will be shielded by the inner electrons. This shielding is produced by the spin-other-orbit interaction involving the spin of one electron with the orbit of another. Now,

yielding the hamiltonian

Blume and Watson,9719* using spherical tensor methods, were able to reduce the second part of this expression to a form suitable for calculation. They derived expectation values of this operator from non-relativistic atomic Hartree-Fock wavefunctions and hence the spin-orbit coupling constant, 5, for many atoms and ions. Consistent results for atoms were also obtained by Hinkleyg9 using wavefunctions both from exponential basis sets100 and from gaussian basis sets.101 Agreement with experiment is good.

Lo et a1.102 have calculated spin-orbit coupling constants for first- and second-row atoms and for the first transition series, results agreeing with the work of Blume and Watson. Karayanislo3 has extended the calculation to triply ionized rare earths. However, with very heavy atoms relativistic effects on the part of the wavefunction near the nucleus become severe, leading to a breakdown of the conditions under which simple perturbation theory ought to be applied. Lewis and co-workers104 have used relativistic self-consistent Dirac-Slater and Dirac-Fock wavefunctions to evaluate spin-orbit coupling

M. Blume and R. E. Watson, Proc. Roy. SOC., 1692, A270, 127. M. Blume and R. E. Watson, Proc. Roy. SOC., 1963, A271, 565. R. K. Hinkley, D. Phil. Thesis, Oxford, 1971.

l o o P. S. Bagus and T. L. Gilbert, unpublished work. l o l S. Huzinaga, J. Chem. Phys., 1965, 42, 1293. la* B. W. N. Lo, K. M. Saxena, and S. Fraga, Theor. Chim. Acta, 1972, 25, 97.

O a B. Karayanis, J. Chem. Phys., 1970, 53, 2460. O 4 W. B. Lewis, J. B. Mana, D. A. Liberman, and D. T. Connor, J. Chem. Phys., 1970,53,

809.

18 Q uan tum Chemistry

constants of actinides. They found that for such cases, e.g. Th2-+, there are large discrepancies between values computed with relativistic and non- relativistic wavefunctions.

For diatomic molecules the expressions are complicated by the non- central nature of the field. Walker and Richardslo5 extended Blume and Watson's technique to diatomic molecules by expanding the molecular integrals in terms of integrals over atomic Slater-type basis functions, When these matrix elements involve atomic orbitals centred on different atoms, the integrations must be performed numerically. This can be a lengthy process for integrals involving two electrons. Table 5 gives the results of their calcu-

Table 5 Spin-Orbit coupling constants/cm-l

Molecule State Observed Walker core polarization Kobori FValJcer + Ishiguro and

BeH BH+ CH NH- OH FH+ MgH AlH+ SiH PH- SH

A 2 H ~ 2.14 A 211r 14.0 x " I T , 28.0 x 2 r I i - X 'ITi -139.7 x 21Ti -

A 'IIr 35 A 2I I r 10s X 2 1 1 T 142 x 2 I I i - X 211t -378.6

2 .3 - 15.9 - 30.4 --

-57.8 - - 141.4 _-

305.8 - 27.2 36.4 96.4 110.9

139.2 155.1 -178.5 -211.7 -362.0 -375.7 -

2.02 15.2 29

151 -

-

40.51 124.9 - -_

,382.4

lations for molecules containing first- and second-row elements. All integrals are included except for the two-electron, two-centre integrals. This method gives good results for hydrides of first-row elements but the values obtained for those with second-row atoms are all too small. Blume and Watson showed a similar divergence in atoms. The error has been attributed to the effects of 'spin polarization'. In the usual restricted Hartree-Fock approximation, p-orbitals have the same spatial form whatever the values of the quantum numbers rrrz and ms. If an atom has an open-shell electron this degeneracy should be removed. In unrestricted Hartree-Fock calculations, the p-orbitals of different mt and ms are allowed to have different spatial parts of their wavefunctions and this results in closed p-shells contributing to the one-centre matrix elements <Z/r3). Walker and Richards*OG used atomic values obtained from unrestricted Hartree-Fock calculations to correct the molecular constants, justifying this approximation on the grounds that there are only small changes in the orbitals involved on forming a diatomic molecule. The resulting improvement is shown in Table 5.

A detailed inspection of these results shows that all the two-centre integrals tend to be self-cancelling. Accordingly, the neglect of such integrals in the

(a) T. E. H. Walker and W. G. Richards, Symp. Faraday SOC., 1968, no. 2, p. 64; (6) T. E. H. Walker and W. G. Richards, Phys. Rev., 1969,177,100.

l o 6 T. E. M. Walker and W. G. Richards, J . Chem. Phys., 1970, 52, 1311.


calculation of coupling constants may be justified. With this method, Walker and Richardslo7 investigated the spin-orbit coupling in BeF and MgF. Previous work had indicated that the A 211 states were derived from inverted configurations (open-shell more than half-full), i.e. mx3na2 with a negative spin-orbit coupling constant. However, the calculation did not support this view, suggesting that the states were regular (m7r4[m + lln). This was confirmed by a subsequent investigation of the A-doubling.

More recently, Hall and Richards108 investigated the states of CF. Using different wavefunctions, they calculated the constants in the X211 state and obtained good agreement with experiment. However, for B 2A, they were unable to reproduce the experimental result of 4.5 cm-1. Instead they gave a value of - 2.4 cm:l; including two-centre integrals produced a change of less than 0.1 cm-l, while configuration interaction with singly excited states gave a contribution of 1 cm-l. It would appear that with small effects like these either a more rigorous treatment of this spin-orbit coupling is necessary or better wavefunctions.

Finally, one may note recent work on the zero-field splitting of the ground state of the oxygen molecule. The major contribution to this quantity is a spin-orbit coupling matrix element diagonal in A. Zamani-Khamiri et aZ.1099110 obtained values far smaller than the experimental result. They suggest that the reasons for the discrepancy were (i) the inadequacy of the single configuration wavefunction used, and (ii) the application of second-order perturbation theory truncated after one term. Halllll has suggested that the reason lies in the size of the basis set.

Zamani-Khamiri used a minimal Slater-type basis whereas Hall included ten Q and four x functions. He was able to produce a result of 1.25 cm-l, very close to that obtained from experiment.

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et ~21.~~2-1~5 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules.

Vesethll7 has also derived a formula convenient for ab initio calculations

1 0 7 T. E. H. Walker and W. G. Richards, J. Phys. (B) , 1970,3, 271. 1 0 8 J. A. Hall and W. G. Richards, Mol. Phys., 1972,23, 331. 1 0 9 0. Zamani-Khamiri and H. F. Hameka, J. Chem. Phys., 1971,55,2191. 110 R. H. Pritchard, C. W. Kern, 0. Zamani-Khamiri, and H. F. Hameka, J. Chem. Phys.,

1972,56, 5744. ll1 J. A. Hall, J. Chem. Phys. 1973, 58, 410. 11% R. L. Matcha, C. W. Kern, and D. M. Schraeder, J. Chem. Phys., 1969, 51, 2152. 118 R. L. Matcha and C. W. Kern, J. Chem. Phys., 1969,51, 3434. 114 R. L. Matcha and C. W. Kern, Phys. Rev. Letters, 1970, 25, 981. l16 R. L. Matcha, G. Malli, and M. B. Miller, J. Chem. Phys., 1972, 56, 5982; R. L.

Matcha and C. W. Kern, J. Chem. Phys., 1971,55, 469. 110 J. A. Hall, T. E. H. Walker, and W. G. Richards, Mol. Phys., 1971, 20, 753. I17 L. Veseth, Theor. Chim. Acta, 1970, 18, 368.

20 Q uan turn Chemistry

of the spin-orbit coupling constant and includes a method for theoretical determination of the sign.

When the number of atoms in the molecule increase above two, three- and four-centre integrals enter the expressions. Matcha and Kernli5 have given expressions for these. However, there is also the problem of the number of integrals to be calculated; with large molecules this will be extremely time- consuming.

A few triatomic molecules have been investigated using ab initio techniques. Horsley and Hall118 have used the methods of Walker et aZ.lo57 116 to calculate the spin-orbit coupling in some triatomic molecules, neglecting multicentre integrals. Calculations on states of COz+, COS+, N3, CCN, NzO+, and NCO gave answers close to experiment, excepting the B state of NCO, where it was suggested that a Fermi resonance was giving an unrealistic value for the observed constant. Another example is the relativistic calculation by Varga et a1.119 of the spin-orbit coupling constant in AmOzf, which they used to produce a fit of the spectroscopic data.

For polyatomic molecules, semi-empirical methods offer possibilities. Tshiguro and Koboril 2O developed an extremely simple method of obtaining spin-orbit coupling constants in diatomics from the component atom values and appropriate Slater-type overlap integrals from coefficients of atomic orbitals. They neglected multicentre integrals but, as has been seen, this is a justifiable assumption for these molecules. Representative results are shown in Table 5 . They explained discrepancies qualitatively in terms of the electronegativity differences between the atoms.

Leach121 has proposed a method suitable for use in linear molecules. He uses the free atomic ion values modified by a method based on Mulliken’s population analysis. Some account is taken of differences in electronegativity of the constituent atoms. Thus for COzf he obtains constants of - 157 cm-1 and - 101 cm-l for the X 21T, and A 2rIlL states. The appropriate experimental values are - 159.5 cm-i and - 95.5 cm-l. Although such calculations do not explicitly include the full hamiltonian, their success should allow extension to larger or less symmetrical molecules.

Both of these methods essentially use ab initio wavefunctions to deduce the composition of the MOs. An alternative approach is to use semi-empirical functions, obtained from the MIND0 or CNDO methods, to calculate the constants directly from the hamiltonian. A major problem in such work is that in many cases only valence shell electrons are included and consequently spin-other orbit effects cannot be calculated directly. Hinkley, Walker, and Richards122 have shown from ab initio calculations that this shielding for a given atom often bears a constant ratio to theZ/r3 terms. Such a ratio could be

1 1 8 J. A. Horsley and J. A. Hall, Mol. Phys., 1973, 25, 483. 1 l S L. P. Varga, J. B. Mann, L. B. Asprey, and M. J. Reisfield, J. Chem. Phys., 1971, 52,

l Z o E. Ishiguro and M. Kobori, J. Phys. SOC. Japan, 1967, 22, 263. l a 1 S. Leach, Actu Phys. Polon., 1968, 34, 705. 1 2 2 R. K . Hinkley, T. E. H. Walker, and W. G. Richards, Mol. Phys., 1972, 24, 1095.

4230.


used to determine the shielding if the MIND0 function gave good values for the nuclear terms. Unfortunately this did not prove to be the case. The wavefunctions are optimized for heats of formation, whereas to derive good values for spin-orbit coupling constants the electron distribution near the nucleus has to be accurately represented. It must be remembered that these semi-empirical methods are approximations to minimal Slater-type basis calculations; given the known inadequacies of such basis sets, as shown by the work on o ~ y g e n , ~ ~ ~ - l l l it is unlikely that the semi-empirical methods presently available will predict spin-orbit coupling effects reliably.

To summarize, therefore, it is reasonable to say that ab initio calculations of spin-orbit coupling constants may be successfully performed on atoms (although relativistic wavefunctions will be necessary for the heavier ones) and diatomic molecules (especially hydrides). For larger molecules, such methods may be too time-consuming and resort to semi-empirical techniques will be necessary. The ‘atoms-in-molecules’ approach has proved extremely successfuI, but it should be possible to use semi-empirical wavefunctions with the full harniltonian before long. This will be probably more useful with very large molecules.

7 A-Doubling and Spin-splitting Constants

Although the theory of these related phenomena is fairly well established, very few calculations of the constants involved have been reported. Presumably this is because they involve the calculation of off-diagonal matrix elements. A detailed account of the theory of A-doubling in 2 I I states of diatomics has been given by Van V l e ~ k , l ~ ~ Mulliken and Christy,12* and Hinkley, Hall, Walker, and Richards.lZ5 A brief synopsis is presented here.

As the component of electronic angular momentum along the internuclear axis (A) can have two orientations, each component of a 2 I I state will be doubly degenerate. This degeneracy is removed because the orbital angular momentum (L) interacts with the rotational angular momentum; the splitting (Avcd) is called A-type doubling. It may be calculated by considering the perturbations of the 211 by 2Z* states. Two of the operators giving matrix elements between such states are those describing spin-orbit coupling (see above) and the coriolis interaction,

To second order, Van Vleck123 and Kovacsl26 have shown

A Vcd= ( J+ *)( ( f 1 - Y/X+ 2 / X ) (4P + 4) + 2X-YJ-f $) ( J - %)q},

where Y= A/B, and X2= { Y( Y- 4) + 4(J+ +)2}. A is the spin-orbit coupling lL8 J. H. Van Vleck, Phys. Rev., 1929, 33, 467. 1 9 4 R. S. Mulliken and A. Christy, Phys. Rev., 1931, 38, 87. 1*6 R. K. Hinkley, J. A. Hall, T. E. H. Walker, and W. G. Richards, J. Phys. (B), 1972,5,

120 I. Kovacs, ‘Rotational Structure in the Spectra of Diatomic Molecules’, Hilger, London, 204.

1969.


constant in the TI state and B, is the B value in the uth vibrational level of the 11 state. The constantsp and q are given by

< 2 I I 1 HSOl2C) ( 2 2 1 BL+I 2II> 4 A Ccs P=2C 9

and

where u(l3, C) = E( 211) - E(zC) and the sum is over all vibrational levels of all interacting zE states. 2C+ States give contributions of opposite sign to those of 2c-.

These expressions appear extremely difficult to evaluate, involving as they do summations over states whose spectroscopic constants are unknown. The situation is much simpler in the case of an atom because the selection rules for matrix elements of the Lf operator reduce all but one (i.e. that in which the ml value of an appropriate electron is raised by one) to zero. In certain molecules, notably diatomic hydrides, the molecular orbitals are closely related to these in atoms and similar orthogonality conditions will pertain. Thus, only one C state (i.e. that related to the II state by a change of 1 in the A-value of a given electron) need be included in the summation. In this situation, Van Vleck’s case of ‘pure precession’ is said to apply. As an example, one may cite the A211 state of BeH with configuration la22021n, the related 2X state is the ground-state 1~1~20~30, and both states correlate with ls22s22p of the united atom B.

In certain cases, the formulae for p and q may be simplified further. If the spectroscopic constants for the interacting states are similar, many of the vibrational matrix elements will reduce to zero. Consider the u=O level of a given state. The principal matrix element will be with the u = O level of 2C and the value <2rIIB I 2C> between vibrational functions will be approximately the 63 value of By the orthogonality rules, the remaining matrix elements should be zero. This has led Van Vleck to suggest

and

Here Z is an integral quantum number and ~(11, 2) is the energy difference between the v’= d’= u levels of the interacting states.

A similar mechanism gives rise to spin doubling in z X states. Consideration of the perturbations between II and C states yields an expression for the splitting between the spin doublet of

Au12 = y(N+ +).

y is made up of two contributions, one derived from the interaction of L


with the rotational angular momentum N, and the other derived from the interaction of electronic spin with N. The former has been shown by Van Vleck to be the more significant and he derived an expression for this part, i.e.

V W ,

<2C I H S O l 2 I I ) (2II IBL'I 2 2 > Y = 2 X Y

_ _ _ ~

where the summation is now over all vibrational levels of all 211 states interacting with the 2C state. If the case of pure precession holds, only one II state need be considered and then the further approximation shown above yields

y = 2AB&+ 1)/ v(n, C).

The expressions for Aycd and A m are more complicated for states with higher orbital or spin angular momenta. However, the principles underlying their derivation are similar.

Very few ab initio calculations of these constants have been published. Mulliken and Christy gave a series of values derived from the simplified expressions but, of course, these involved the use of diagonal constants rather than off-diagonal matrix elements. Similarly, Knight and Weltnerl27 have derived y from Ag, values determined in e.s.r. experiments. The first attempt to calculate the constants from molecular wavefunctions was made by Walker and Ri~hards.10~ These authors considered interaction of the A2II state in BeF and MgF with X2C+ and B 2C+ to resolve the anomaly in the spin-orbit coupling mentioned above. To calculate the A-doubling constants they assumed that electronic motion could be treated separately from vibrational- rotational motions. The off-diagonal matrix elements of the spin-orbit and L+ operators between electronic wavefunctions were required. As it is more convenient to have all wavefunctions constructed from the same orthogonal MOs, the 2CS- states were derived from virtual orbitals of the 211 state. For <Hso>, methods similar to those used for diagonal constants were employed; the matrix elements of L+ were evaluated numerically after expressing the operator in suitable spherical tensor form. The matrix elements of B and the overlaps were calculated with vibrational wavefunctions from simple harmonic oscillator potential curves. The results showed that p and q differed in sign for BeF, implying that the A 2II state is inverted, although the wavefunction was derived from a regular configuration. It was found that the positive contribution to <ITso> from the fluorine atom outweighed the negative contribution from the metal. This was not the case for MgF, where both constants had the same sign. As the agreement with experiment for both molecules was good, these authors concluded that the regular configuration was the most abundant contributor to the wavefunction.

Hinkley et al.125 improved these calculations by using Morse and RKR curves to produce vibrational wavefunctions. Using the concept of 'pure precession' they calculated A-doubling constants for 211 states of OH, BeH,

l a7 L. B. Knight and W. Weltner, J. Chem. Phys., 1970, 53, 4111.


CH, and NO (Table 6). Agreement for the first two was excellent because the values of A in the interacting states were determined essentially by one electron. The situation is very similar to that in an atom. However, in CH and NO orbital motions of the outer electrons are more complex and interaction with one state alone is not sufficient to explain the observed splitting.

Table 6 A- Doubling constants

P 4 Molecule Calculated Observed Calculated Observed

BeH 0.0041 a 0.0134 0.0142

CH 0.0386 0.0374 0.0254 0.038 NO 0.0039 0.0117 0.0002 O.ooOo8

OH 0.242 0.246 -0.0391 -0.0384

* Too small to measure.

Ha11128 has investigated the spin splitting in the ground-state X 2Z+ of A10. He summed over a large number of 211 states but performed a detailed vibrational analysis of the first two only. These gave the main contributions. He found y to be 0.0029 compared with an experimental result of 0.008 k 0.006. Pure precession gave a value 0.005. Although the former theoretical prediction is probably low, both lie within experimental error.

This work has shown that ab initio calculations can assist in the explanation of many diatomic spectra. Of course, there are limitations to the methods described here. The treatment of electronic and vibrational motions indepen- dently may not always be valid. Furthermore, the use of virtual orbitals and crude potential functions may provide a very poor representation of the interacting states. Refinement of the model may be necessary in cases such as NO where the lack of information on excited states may be responsible for the poor answers.

One important feature of these phenomena is their variation with rotational quantum number. Plots of these changes are often used in the assignment of MO configurations. Simple formulae for Hund’s cases (a) and (b) are readily derivable, but they often break down at high J. Mulliken and Christy suggested that this might be due to rotational stretching of the molecule. Accord- ingly they corrected p and q to produce better results in some cases. Veseth, in a series of paper~,129-~3~ has improved on these formulae by including the rotational stretching in the perturbation calculation considering high-order terms. Thus, in order to get an adequate fit of data for the spin splitting in B 2Z states of Ca, Sr, and Ba hydrides, fourth-order terms are included.130 Sixth-order terms are necessary for an adequate treatment of the A-doubling in the A2lX states.131 In a more recent paper132 he has investigated the fine

I p s L. Veseth, J. Phys. (B) , 1970, 3, 1677. * * O L. Veseth, Mol. Phys., 1971, 20, 1057. l a l L. Veseth, Mol. Phys., 1971, 20, 287. I * * L. Veseth, J. Phys. (B ) , 1972, 5, 229.

3. A. Hall, D. Phil. Thesis, Oxford, 1971.

The Calculation of Spectroscopic Constants

- /+ ‘ + - /+

/+ - i

/+ - Theoretical curve - 4 Experimental points

- /+

25

c

‘E 0 \

3 a

10

0

6

4

2

0 0 10 20 30 40 50

N Figure 2 Theoretical A-doubling in BeH (A2n, v=O)

structure of the 311 and 3Z- states in ND, NH, and OHf, and resolved certain anomalies in the A-doubling.

It is not always necessary to go to high order to explain these features in the spectra. Horne and Colin133 have re-investigated the spectrum of BeH and confirmed the existence of a maximum in the plot of A-doubling against N(J= N f 3). However, if p is small, as is the case here, Mulliken and Christy showed

Avcd=qN(N+ 1).

Even when they included rotational stretching, q was not lowered sufficiently at high N to produce the maximum. Should higher-order terms be included? Hinkley, Walker, and RichardP4 modified their procedures for calculating vibrational matrix elements to include the rotational energy in the potential curve. They then determined q over a range of N. The resulting A-doubling is compared with experiment in Figure 2. A maximum is seen although at slightly too high a value of N. However, the matrix element of L+ was assumed constant when in fact it should decrease. Consequently it appears that for BeH the A-doubling is adequately explained by a second-order treatment.

It will be gathered from this summary that, although the formulae for determining spin-splitting and A-doubling are available, very little quantitative work has been done. Results obtained to date are encouraging. If more refined treatments are available it should be possible to resolve many of the out- standing problems. In particular, molecules such as NO, where simple concepts of ‘pure precession’ do not apply, are worthy of study. Likewise, little work has been done on A,@, etc. states where effects are smaller and quantitative accuracy more difficult to obtain.

1 3 3 R. Horne and R. Colin, Bull. SOC. chim. belges, 1972, 81, 93. l a 4 R. K. Hinkley, T. E. H. Walker, and W. G. Richards, J. Phys. (B) , 1972, 5, 2016.

B


8 Magnetic Constants

Although it is not the purpose of this Report to consider magnetic properties in detail, the effect of magnetic fields on the A-doubling will be briefly considered because further insight is given into the phenomenon of pure precession. Detailed formulae have been given by R a d f ~ r d . l ~ s r ~ ~ ~ It is found that gas-phase e.p.r. spectra will yield information about differences in ‘g’ values, quantities which depend essentially upon p and q, as well as on two similar constants,

<2IIIBL+I 2c> <2CIL+l2rI)

dJJ, a Bz = 2c-

Recently, Clough, Curran, and Thrush137 have looked at such spectra of OH. They determined the constants for the first five vibrational levels of the ground state and hence found the matrix elements of L+. These decreased with increasing u much too slowly to be accounted for by pure precession arguments (i.e. interaction of A2X+ alone). However, Hinkley et al.,l38 using the methods described above, were able to obtain constants in accord with the observed data. They considered interaction solely between A 2C and X2JJ and concluded that other interactions were smaller than suggested by Clough et al.

Ha11139 has investigated similar matrix elements for the X zC+ ground state of AID. The difference between the perpendicular component of g and the free spin value is given by

<X~C+~Hso~2rI) <2rI[L+ I X2C+) v ( K v -_____~~_-_ 47, = 4 2

an He considered the interaction with A2IIi and C2rII, at five internuclear distances. The negative contribution from C2111, remained fairly constant but that from A21b (positive) rose rapidly with increasing r, causing a change of sign in AgL at large distances.

The work of Haydon and McCainl40 must also be mentioned. They calculated similar quantities for the x-radicals 03-, NO2*-, and NF2. Although they used ab initio wavefunctions, the spin-orbit coupling constants were derived from atomic data. Providing that they used accurate experimental values for the energy differences, agreement with experiment was good.

9 Calculation of Hyperfine Interaction Constants

The isotropic hyperfine coupling observed in e.s.r. spectra of radicals arises from the Fermi-contact interaction between the electronic spin angular l a 6 H. E. Radford, Phys. Rev., 1961, 122, 114. l a 6 H. E. Radford, Phys. Rev., 1962, 126, 1035. 1 3 7 P. N. Clough, A. H. Curran, and B. A. Thrush, Proc. Roy. Soc., 1971, A323, 541. 1 3 s R. K. Hinkley, T. E. H. Walker, and W. G. Richards, Proc. Roy. SOC., 1972, A331,553. 1 3 9 J. A. Hall, D.Phi1. Thesis, Oxford, 1971. 1 4 0 D. W. Hayden and D. C. McCain, J . Chem. Phys., 1972,57, 171.


momentum and the nuclear spin. (The direct dipole-dipole interaction gives an anisotropic component which averages to zero in fluid phases.) The form of the interaction operator is, in atomic units,

where ~ ( R N ) = <electronic] ~ M ~ l S z d ( r k - RN)lelectronic),

and rk: is the position of electron k, RN the position of nucleus N, 8 a Dirac &function, and IN the nuclear spin operator for nucleus N. This is generally written Hcontact=aN IN. S ; aN is the hyperfine coupling constant. Strictly, CZN is an energy, and should be quoted in energy or frequency units (commonly MHz); however, it is often quoted in units of magnetic field (gauss, or G) and the conversion factor is 2.8026 MHz per gauss. We have given values in the units used by the various sources.

Values of this constant (aN) have been calculated from wavefunctions in various ways. In early work on atoms, Bessis, Lefebvre-Brion, and Moser1419142 calculated aN for 14N, llB, and 170 . The restricted Hartree- Fock value for N in the 4S ground state is zero, since the interaction arises from spin polarization. In order to take account of this, they used restricted Hartree-Fock functions augmented by determinants arising from single replacements of s electrons. The answer was extremely basis-set dependent. Not only was a very large basis required (seven s-functions, cf. two for a minimal calculation, four for a double- 5 calculation), but it was necessary to optimize the seven s-exponents. An indication of the importance of the basis was that six s-functions gave an answer of 13 MHz, and seven functions of 11 MHz (experimental 10.45 MHz). It has been ~ h o ~ n ~ ~ ~ - ~ ~ ~ that, to first order, only single replacements contribute to the hyperfine coupling. Bessis et al. found, in contrast to later workers,l46 that use of first-order perturbation theory in the CI was most unsatisfactory. Ideally they suggested, ‘extended Hartree-Fock‘ functions (now more commonly termed ‘multi-configuration SCF’ functions) should be used. In further work on 19F they find that, for this atom, projected unrestricted Hartree-Fock functions give better results than restricted Hartree-Fock with CI;146 however, they admit this may be fortuitous. In the work142 on 1lB the s-electron contribution actually changed sign between two different s-bases which gave ‘essentially identical energies’.

The comparative failure of this work on atoms to give conclusive indications of a reliable method for calculating hyperfine constants did not deter people

k

1 4 1 N. Bessis, H. Lefebvre-Brion, and C. M. Moser, Phys. Rev., 1961,124, 1124. 1 4 1 N. Bessis, H. Lefebvre-Brion, and C. M. Moser, Phys. Rev., 1962, 128, 213. 1 4 9 R. Lefebvre, J. Chim. phys., 1957, 54, 168. 14’ R. Lefebvre, in ‘Modem Quantum Chemistry’ Istanbul Lecture, ed. 0. Sinanoglu,

Academic Press, New York, 1965. 1 4 8 H. Longuet-Higgins and J. A. Pople, Proc. Phys. SOC., 1955, A@, 591. 1 4 1 N. Bessis, Cahiers Phys., 1962, 16, 345.


Table 7 Hyperfine coupling constants: comparison of calculations

Atom Method 1lB ( 2 P ) 14N (”) l’0 (3P) IgF ( z P )

RHF 0 0 0 0 RHFf CI - 3 . 8 11 -11.4 I

UHF 5.1 24 - 34 221 PUHF 1.7 14 - 17 74 Experiment 0.11 10.45 - 18 101

from extending the methods to molecules. Both the RHF+CI and the projected UHF methods have been applied to small molecules. Chung147 used RHF + CI and a one-centre basis on 13CH3, 14NH2, and 170H. Obviously a one-centre basis cannot give results for the proton coupling. The basis sets contained 27, 22, and 23 functions, respectively. Results were good, but only after she had arbitrarily removed configurations which gave a ‘bad’ contribution. This is hard to justify. An extensive study of the RHF + CI method has been made by Chang, Davidson, and on CH and CH3. They came to the familiar conclusion that the results were extremely basis- sensitive. Minimal STO basis gave uniformly bad results; a double-5 basis gave tolerable results for 13C but not for IH. Results for l3C were considerably improved by taking into account zero-point vibration ;l5O the temperature dependence was also obtainable. This correction has been shown to be important for many molecular pr0perties.15~9 152 It has been suggested153 that the disappointing results obtained by the RHF+CI method could be over- come by incorporating basis functions with a large density at the nucleus. For H, a 2s Slater-type function with high exponent was suggested. Poling, Davidson, and V i n ~ o w l ~ ~ examined the effect of basis set on CH. They also suggested that the reason for the bad answers might be that the wavefunctions did not satisfy the electron-nuclear cusp constraint. This constraint155 imposes additional boundary conditions on the wavefunction to the effect that

for all positions of the other electrons. Here ria is the distance between electron i and nucleus a, and Z, is the charge on nucleus a. Poling et al. have proposed a method of imposing this constraint on RHF + CI wavefunctions. Unfortunately, no experimental values were available, but the familiar wild variation with basis was obtained. A wide variety of bases were used,

14 ’ A. L. H. Chung, J. Chem. Phys., 1967,46, 3144. 1 4 0 S. Y. Chang, E. R. Davidson, and G. Vincow, J. Chem. Phys., 1968,49, 529. 1 4 9 S. Y. Chang, E. R. Davidson, and G. Vincow, J. Chem. Phys., 1970, 52, 1740. 1 5 0 S. Y. Chang, E. R. Davidson, and G. Vincow, J , Chem. Phys., 1970, 52, 5596. l S 1 D. M. Schraeder and M. Karplus, J. Chem. Phys., 1964,40, 1593. l S a D. M. Schraeder, J. Chem. Phys., 1967,46, 3895. l U a H. Konshi and K. Morokuma, Chem. Phys. Letters, 1971, 12, 408. 164 S. M. Poling, E. R. Davidson, and G. Vincow, J. Chem. Phys., 1971, 54, 3005. 1 6 6 T. Kato, Comm. Pure Appl. Math., 1957, 10, 151.


including three minimal sets (four functions), a double- 5 set (eight functions), a 14-function set, and the 24-function set of Cade and Huo, with and without the cusp constraint added; this set gave a value of ac = - 20.4 G and aH = - 1.2 G. Results for other bases varied from - 34.6 to - 18.3 (LEH) and from 43.66 to -3.82 (ac). Application of the cusp constraint to the CadeHuo set reduced the value of ac from 24.48 G to - 1.2 G, a sure indication that even this very large basis set is insufficient to account for hyperfine constants correctly. For if this basis set were complete, the orbitals would be the Hartree-Fock orbitals, which automatically satisfy the constraint. Fortui- tously good results were obtained for a H using minimal basis with Slater exponents (aa: = - 22.06 G). Optimizing the exponents gave an unreasonable value (- 34.4), but if the carbon exponents were optimized, and the hydrogen exponent left at 1.0, a better value was obtained (-21.2). Roughly similar results, but with a better energy, were obtained by optimized carbon exponents and two functions (those of Cade and Huo, 5= 1.34, 2.89) on the hydrogen ( a H = -23.5 G), compared with the much worse results of the conventional double-4 basis ( a ~ = 29.8 G). Similar results were recorded for CH3 (Cade- Huo exponents, aH= - 25.3 G ; conventional double-5, U H = - 38.2 G ; experiment - 23 G). The inescapable conclusion from this work is that either even larger basis sets, or numerical wavefunctions, are required to calculate hyperfine coupling constants accurately, the great problem being the representation of the wavefunction at the nucleus.

After consideration of these difficulties it is perhaps extremely surprising that calculations have been performed using basis sets of gaussian-type functions, since it is well known that such functions are particularly inadequate at representing the wavefunction near the nucleus. Rothenberg and Schaeferl56 have performed a single-determinant SCF calculation on NO2 using a near- Hartree-Fock gaussian basis, giving results remarkably close to experiment for 14N, and results about half the experimental for 170. This agreement must be considered fortuitous. They also calculate expectation values for the quantitiesinvolvedin thedipole-dipole interaction, (3ra2 - l)/r3 and(r0 - irc)/r3, in reasonable agreement with experiment.

The unrestricted Hartree-Fock method has been used extensively, mostly with gaussian basis sets. The results for atoms140s 1*29 l b 7 clearly showed the necessity of projecting out the required spin component. [An alternative procedure, which is approximately equivalent, is to annihilate the next highest spin component (usually the quartet), on the assumption that higher components are negligible contributors.] However, Salotto and Burnelle,l5* using a small gaussian basis (roughly comparable in energy terms with minimal Slater basis), obtained results closer to experiment with unprojected functions. They were, however, ready to admit that their inadequate basis, or neglect of vibrational corrections, might lead to this misleading conclusion.

H. F. Schaefer and S. Rothenberg, J. Chem. Phys., 1971, 54, 1423. Is' N. Bessis, H. Lefebvre-Brion, and C. M. Moser, Phys. Rev., 1963, 130, 1041. lSB A. W. Salotto and L. Burnelle, J. Chem. Phys., 1970, 53, 333.


Claxton and co-~orkers16~-~~7 have performed an extensive series of UHF calculations in which the quartet state was annihilated. They use a minimal basis of ‘contracted gaussian functions’ in which each basis function is a linear combination of gaussian functions. They suggest159 that such a basis is preferable to the use of Slater-type functions. This seems to be difficult to justify, in that to even reproduce closely the forms of atomic SCF orbitals a large number (nine or ten) of gaussian functions is required, whereas three or four Slater functions will suffice. It will be recalled that to obtain acceptable values for hyperfine constants, six or seven Slaters had to be used for atorns,143 corresponding to perhaps twenty gaussians. Furthermore, restriction of the basis to minimal will prevent the representation of much of the change on molecule formation. An example from Claxton’s own work160 illustrates this. For a hydrogen atom, a basis of nine gaussians gives a value for a~ of 496.4 G ; ten gaussians give 502.1 G, still short of the experimental value, 508.4 G. It is surely superfluous to point out that one Slater-type function with exponent 1.0 is an exact solution for the hydrogen atom and will reproduce this result exactly. Claxton’s results for proton hyperfine coupling constants are generally fairly close to experiment, and where available, to RHF+ CI results.161 They were not as close, for CH3, as the best RHF+CI calculations using an extended basis.14g Results for heavier nuclei were erratic. Although discrepancies for 14NH3+ were largely removed by inclusion of zero-point vibration,162 and results for l*NHz were fairly close to experiment, results for 11BH3- were not.166 The shape of BH3- was wrongly predicted to be bent, so obviously the vibrational corrections were unreliable. Altering the basis set to a minimal set appropriate for B- gave more consistent, but still unsatisfactory, results. It is interesting to note here the failure of Bessis et al. to reproduce the l1B coupling constant with the projected UHF method. Results for CN and for vinyP7 showed errors due to contamination by sextets and higher states, showing the inadequacy of simply annihilating the quartet component. This is a particularly distressing defect of the method, since the contamination not only varies substantially with internuclear distance, but increases markedly when larger basis sets are used. It would seem much more satisfactory to use the fully projected wavefunctions and/or larger and more flexible basis sets, especially since the previous work has shown such a strong basis-set dependence. Undoubtedly the contracted gaussians Claxton uses are better minimal functions than single Slater-type functions in most cases, but it would be preferable to use more flexible sets, constructed from gaussians

1 6 9 T. W. Claxton, D. McWilliams, and N. A. Smith, Chem. Phys. Letters, 1970,4, 504. T. W. Claxton and D. McWilliams, Trans. Faraday Sue., 1970, 66, 513.

1 6 1 T. W. Claxton, Trans. Faraday Suc., 1970, 66, 1537. T. W. Claxton, Trans. Faraday SOC., 1970, 66, 1540.

163 T. W. Claxton and N. A. Smith, Trans. Faraday SOC., 1970, 66, 1821. $ 8 4 T. W. Claxton, Trans. Faraday SOC., 1971, 67, 897. 1 6 6 T. W. Claxton and N. A. Smith, Trans. Faraday SOC., 1971, 67, 1859. lE1 T. W. Claxton, M. J. Godfrey, and B. L. J. Weiner, Trans. Faraday Sue., 1972,68, 181. 1 6 7 T. W. Cla.xton, M. J. Godfrey, and B. L. J. Weiner, Trans. Faraday SOC., 1972,68, 366.


or Slater functions. Cook et a1.168 and Bendazzoli et have used contracted gaussians and have used gaussians to simulate minimal and double5 Slater- type sets, using both projected UHF and RHF+CI. They conclude that although the minimal basis is generally adequate for proton coupling, it fails for heavier nuclei, a conclusion which fits objective judgement of Clax- ton’s work. This is true for both projected UHF and RHF + CI wavefunctions. Double-5 sets give quantitative agreement using projected UHF on NH2 and OH for the 14N and lH coupling, good agreement from RHF+ CI for 1H, but worse agreement for 14N. Both methods failed to give any agreement with 1 7 0 coupling constants (UHF spin-density= 0.015, RHF+ CI= 0.107, cf. experiment 0.142) was to a large extent due to the contribution of configurations in which a l o electron is replaced. Their gaussian calculations gave rather high results uniformly; however, it is significant that the contractions used are much less accurate than Claxton’s. For these (minimal) gaussian calculations, the RHF + CI wavefunctions gave the better results. Thus again the conclusion was reached that very large basis sets are required to predict coupling constants at nuclei other than hydrogen. Given this point, no calculations accurate enough have been performed to allow distinction between the projected UHF and RHF+CI methods, save those on at0ms.1~~. 1429 157 The growing feasibility of MCSCFcalculationsshould provide a more acceptable alternative method; however, it seems that at present and in the future there will be no short cut to calculation of these sensitive quantities, and larger basis sets than hitherto will have to be used. Two further methods of calculation are worthy of mention. A minimal basis of Slater-type functions was used in a valence-bond calculation on CH.170 A result of - 21.54 G for CZH

was obtained, close to other r e s ~ l t s , ~ ~ ~ , ~ ~ ~ but the value of 42.7 G for QC

(cf. the values quoted earlier from ref. 154) was obviously wildly wrong. This is only to be expected from the smallness of the basis. Finally, the INDO semi-empirical method has been usedl71 widely to calculate proton coupling coefficients for hundreds of radicals. The approximate ‘McConnell relationship’ is

where Pkk is the unpaired spin density of the H 1s orbital in the singly- occupied orbital. This quantity p k k is only meaningful in a minimal Slater-type basis calculation; Q is estimated empirically or the free-atom value is used. A similar correlation has been used for 13C coupling constants. Although agreement for protons is fairly good, it must be recalled that INDO is an approximation to a UHF calculation in a minimal Slater basis; given in addition the approximate nature of the McConnell relationship, and the defects of minimal basis calculations, especially for heavier nuclei, the agreement obtained must be regarded as a coincidence. l a @ D. B. Cook, A. Hinchliffe, and P. Palmieri, Chern. Phys. Letters, 1969, 3, 223. 1 1 9 G. L. Bendazzoli, F. Bernardi and P. Palmeiri, MoZ. Phys., 1972, 23, 193. 1 7 0 G. F. Tantardini and M. Simonetta, Chem. Phys. Letters, 1972,14, 170. 1 7 1 D. L. Beveridge, P. A. Dobosh, and J. A. Pople, J. Chem. Phys., 1967,47, 3026. l?l H. M. McConnell, J . Chem. Phys., 1956,24, 764.

a$$= QPkk,

32

10 Transition Probabilities

Quantum Chemistry

In the calculation of transition probabilities, theoretical chemists could provide a valuable service to the wider scientific community. Interest in the intensities of spectral lines lies not only in the relative scarcity of experimental data but more because of the importance of such measurements or calculations. This importance is in the realm of astrophysics.

Theories about the origin and nature of the universe have as test data the composition of stars and even of our own sun. The abundance of, say, boron in the atmosphere of the sun can be estimated from measurements of the BH electronic spectrum in the sun’s emission coupled with known values of the oscillator strength of the transition. If the spectrum cannot be produced in the laboratory then only calculated values of the transition probability can be used.

Unfortunately the calculation of transition probabilities has not been attempted in many instances and the results are not encouraging. Apart from a valiant early attempt by D a v i e ~ l ~ ~ for Naz in a calculation where not all electrons were specifically considered, the first serious attempt to compute transition probabilities from ab initio wavefunctions was due to La Paglia.174 In this important paper La Paglia considered the dipole strengths of various 1X-JX transitions for some first-row diatomic molecules.

Oscillator strengths can be defined either as dipole velocity quantities or dipole lengths,

or

Clearly,f(V)=f(R), but only if Y k and Yn are the exact wavefunctions of the initial and final states. This last relationship is sometimes used as a criterion for the quality of wavefunctions, but it is only a necessary condition for good wavefunctions not a sufficient condition.

La Paglia demonstrated convincingly how the computed electronic transition probability is very sensitive to configuration interaction which, as we have stated earlier, is only very slowly convergent. Further, the results are also sensitive to the basis set employed, and agreement between f( V) and f ( R ) is not satisfactory.

If really good wavefunctions can be employed, then the results are convincing. Wolniewicz,l75 with very accurate wavefunctions for H2, has calculated transition probabilities for the B-X,C-X and E,F-B systems. He has even considered individual vibrational and rotational lines and has shown that owing to significant variation of the electronic moments with internuclear distance, the use of Franck-Condon factors is not permissible.

Wavefunctions which are not as accurate as those available for the unique case of H2, but nevertheless very good, are to be found for some small mole-

f ( V ) = 3 2AE-11<~k lv l~n>12

f(R) = -yq< R I y n > I 2.

1 7 3 D. W. Davies, Trans. Faraday Soc., 1958, 54, 1429. S. R. LaPaglia, Theor. Chim. Acta, 1967, 8, 185. L. Wolniewicz, J. Chem. Phys., 1969, 51, 5002.


cules such as diatomic hydrides. Following the work of Bender and David- for HF, and, later, BeH and MgH,177 ' 178 very good wavefunctions

involving extensive configuration interaction are available. Using these wavefunctions for HF, oscillator strengths for transitions between the lowest thirteen states have been computed, as well as dipole velocity and length transition moments for BeH and MgH.178

For polyatomic molecules even less work has been done on intensities. A single exception179 is a calculation of a generalized oscillator strength for the Rydberg transition lB1-lA1 in H2O.

Clearly much remains to be done on this problem and it is to be hoped that the calculations will be performed for cases where there is real astrophysical need for the results rather than randomly computing numbers of no interest beyond that of computation.

11 Conclusions

In many instances spectroscopic constants can now be computed to an accuracy comparable with that which they can be measured. The choice between calcula- lation and measurement then rests solely on grounds of convenience and cost. It is in the realm where the constants are needed but cannot be measured that the potential of this work lies. For full exploitation there needs to be some change of emphasis in the work of theoreticians; it is ceasing to be a matter of wonder if the calculations can reproduce the experiments. It is now time to seek out the problems wherever the application of computation is going to be really beneficial, providing answers to questions for which the answer is really sought, perhaps in photochemistry or astrophysics.

The recent history of this work has been satisfyingly successful but for the future, unless the problems are chosen with care, it could become routine and tedious.

C. F. Bender and E. R. Davidson, J . Chem. Phys., 1968,49,4989. 17' A. C. H. Chen and E. R. Davidson, J . Chem. Phys., 1970,52,4108. 1 7 0 H. E. Popkie, J. Chem. Phys., 1971, 54, 4597. 1 7 * K. J. Miller, S. R. Miekzarek, and M. Krauss, J. Chem. Phys., 1969, 51, 26.

3 Direct Minimization Methods in Quantum Chemistry

BY D. GARTON AND B. T. SUTCLIFFE

1 Introduction

Historically, one of the central problems of quantum chemistry has been to minimize the expression

E(x1, x2.. . x , ) ~ E ( x ) = < d , ( S I@} subject to the constraint (d, I d,) = 1, where S is the ordinary fixed nucleus electronic hamiltonian for the system being considered and @ is a trial wavefunction. This trial function is an explicit function of the electronic variables and is also a function of certain parameters denoted above by the collective variables x, with respect to which the minimization is to occur. To take a simple example, we could regard the d, as an expanded function,

m

where the expansion functions @t are fixed and the coefficients are thus the minimization (or variation) parameters. In this particular example, as is well known, the constrained minimization of E with respect to the cz may be effected by solving the problem

HC = EMc, (3)

where Hij=(@fl&f]@j) and Mij=(@i[ @j), with c a column matrix Qf coefficients and E the required energy.

There are, however, many situations in which it is inappropriate to regard the @sb as fixed functions. For example, if we imagine that the @Z are Slater determinants of spin orbitals yi,

the problem naturally arises as to how the yi are to be chosen, and clearly one solution is to regard them as minimization variables, whose ‘best’ forms are to be determined by minimizing E with respect to both the ct and yi

34

Direct Minimization Methods in Quantum Chemistry 35

for fixed m in equation (2). If we take this view, however, there is no simple expression like equation (3) from which we can actually determine the a,ht because of the complicated implicit dependence of E on lyt.

Besides the kind of minimization parameters that we have so far discussed, there are also the internuclear distances which occur in 2 and implicitly in 0, with respect to which it is often desired to minimize E, and here again no simple minimization form is readily available because of the implicit dependence.

It should be noticed also that minimizing the energy is not the only minimizing problem that is of interest in quantum chemistry, and that problems of minimizing electron repulsion integrals, differences between functions, best least-squares fitting, and so on, frequently arise at intermediate stages in established computational schemes.

It is therefore appropriate perhaps at this time to attempt to survey what is known about minimization in relation to quantum chemistry and to try to assess in particular what role some of the more modern minimization methods are playing, and might in future play, in quantum chemistry. How- ever, before attempting to talk more generally about optimization, it would perhaps be appropriate, at the risk of making our exposition appear less general than in fact it is, to consider a fairly simple and well known optimization problem in quantum chemistry, as an example to be held in mind throughout the subsequent discussion.

2 A Sample Problem

Suppose that we are interested in minimizing the energy function of a molecule, which we are describing by a close-shell one-determinant function, in the usual LCAO-MO approach, with respect to the linear coefficients and the atomic orbital exponents. The energy function is simply

E(T, a) = 2 tr RRS tr RG(R), (9

R= TTt, (6)

where the notation is that of McWeeny and Sutcliffel with

where T is the m by n matrix of coefficients relating the MOs br to the AOs qt and h is the matrix of the one-electron operator in the A 0 basis.

The matrix G(R) = 2J(R) - K(R) and

and

where < z j I g I k l ) is a two-electron repulsion integral in the Dirac notation.

R. McWeeny and B. T. Sutcliffe, ‘Methods of Molecular Quantum Mechanics’, Academic Press, London and New York, 1969.


The above expression for the energy is valid, however, if and only if

and if

where S is the overlap matrix in A 0 basis and cq the exponent of the ith orb; tal.

If we are interested simply in minimizing equation ( 5 ) with respect to T, subject to the constraint of equation (9), then from the classical theory of optimization we know that we can incorporate this constraint by constructing the lagrangian

LW, a, x)=E(T, a)+ c Ars(T!STs- a,,), (1 1) r> 8

where Tr is the rth column of T, and by finding the extremal of this function, that is, solving the simultaneous equations :

and aL/aTir=O; i= 1 , 2 . . . m ; r = 1 , 2 . . . n,

aL/ail,,=o; s ~ r = i ,2 , . . . n.

(1 2)

(1 3)

If the resulting simultaneous equations are linear, their solution is a straightforward matter, but it is easy to see that in our case the equations will not be linear, and thus the minimization of equation (5) subject to equation (9) is an example of a non-linear programming problem.

In conventional practice, as is well known, in this case we linearize the resulting simultaneous equations, to produce the iterative self-consistent field (SCF) scheme of calculation, according to the eigenvalue equation,

f T= STE,

where f= h + G and the linearization consists of the assumption that G can be considered as a constant matrix at any step. The diagonal matrix E is effectively a matrix of the lagrangian multipliers A,,, of which only n are strictly necessary (see, e.g. chap. 5 of ref. 1). The minimization scheme represented by equation (14) in general works very well, but it does not invariably converge, and McWeeny2$3 was the first to attempt to analyse the reasons for this. He came to the conclusion that the difficulty lay essentially in the linearization assumption, and he suggested that a better way of treating the problem would be to attempt to minimize equation (5) directly instead of via equation (14). The difficulty in doing this lies in incorporating the constraint of equation (9).

It might be thought at first sight, perhaps, that one could incorporate the constraints by minimizing equation (11) with respect to T, a, and A instead of minimizing equation (3, but unfortunately this is not the case. In fact

R. McWeeny, Proc. Roy. SOC., 1956, A235,496. R. McWeeny, Rev. Mod. Phys., 1960, 32, 335.


one may have to look for a saddle point in equation (11) (minimizing L with respect to the T and a and maximizing it with respect to A) and it is not easy to find this point directly. Indeed, even were it possible to use the lagrangian directly, it would not be very appealing because of the extra variables which it contains, which increase the need for computing store and lengthen a calculation. However, M ~ W e e n y ~ 9 ~ showed that if one adopted an iterative procedure for minimizing equation (5) one could incorporate the constraint in forming 6R, the variation in the density matrix R. We discuss this method in Section 4. An alternative approach (first considered by Fletcher5) would be to incorporate the constraints by means of a transformation of the linear variables directly. Thus one could introduce a new set of variables Y via the relation

T = YU, (1 5 )

and require that U be chosen so that

T ~ S T = U ~ ( Y ~ S Y ) U = r. (1 6 )

Any choice of U for which UUt=(YtSY)-l will satisfy equation (16), so that

R= Y(YWY)-lY+, (1 7)

and using this in equation (5) we can regard the resulting function as a function of Y and a and minimize it directly.

Many other ways suggest themselves of incorporating the constraints of equation (9), some of which we shall consider in more detail later, but perhaps enough has been said at the moment to indicate it is by no means impossible to incorporate constraints like equation (9) into the quantum chemical minimization problem.

Let us now turn to the problem of minimizing equation ( 5 ) with respect to T and a, subject to both equations (9) and (10). It is usual in practice simply to perform a sequence of calculations optimizing E against T for discrete sets of exponents cc1, cc2, cc3. . . and so on, and to determine the set am that minimizes E by some kind of interpolation procedure. This means that the constraint of equation (10) causes no trouble at all, since we can simply choose all our discrete sets to obey it, and forbid interpolation into the excIuded regions. However, it is also easy to see that a simple transformation

cci-+yi=ln mi, (1 8)

effectively removes equation (10) as a constraint since yi now lies in the range (- co, + co).

It is therefore perhaps not unreasonable to suggest that it will be a relatively easy matter to incorporate or to avoid constraints like equation (10) in most quantum mechanical problems.

R. McWeeny, Phys. Rev., 1959, 114, 1528. R. Fletcher, Mol. Phys., 1970, 19, 55.


From the foregoing discussion we hope that it appears at least plausible that it will in general be sensible to regard the quantum mechanical minimization problem as a special case of the problem of finding the minimum off(x) = f(x1, x2 . . . xn) with all the xi in the interval (- co, + co), recognizing that because of constraints it may be necessary to solve a sequence of such problems. We return later to the problem of incorporating the constraints in specific problems. With this in mind, in the next section we present a short introduction to the literature and a brief survey of some methods in this field.

3 Optimization

The theory of finding minima of functions is just one aspect of a branch of mathematics: optimization theory, which has had a long history com- mencing with the discovery of the calculus and extending through the development of the calculus of variations up to the present day in the theory of non-linear programming. We are interested only in the most recent developments in the theory, which yield results that are in forms appropriate for computer realization. An elementary introduction to modern optimization theory can be found in a book by Cooper and Steinberg,6 which contains references back to earlier work and to other more advanced contemporary books. As the field is a rapidly developing one, much of the current work can be found in collected papers from symposia, published as books (e.g. Fletcher7 and Murray8) and research monographs (e.g. Kowalik and Os- borneg).

Most modern minimization methods are designed to find local minima in the function by search techniques: characteristically they assume very little knowledge of the detailed analytic properties of the function to be minimized, other than the fact that a minimum exists and therefore that, close enough to the minimum, the matrix of the second derivatives of the function with respect to the minimizing variables (the hessian matrix) is positive definite.

In books on optimization methods, sections are devoted to discussions both of unconstrained and constrained optimization methods but, as we have suggested in the previous section, there are often convenient ways of introducing constraints into the quantum chemical problem, so that we can, without any serious loss, concentrate our attention entirely on unconstrained methods and ignore the more general discussions involved in the

L. Cooper and D. Steinberg, ‘Introduction to Methods of Optimization’, W. B. Saunders Company, Philadelphia, London and Toronto, 1970. ‘Optimization (Symposium of the Institute of Mathematics and its Applications)’, ed. R. Fletcher, Academic Press, London and New York, 1969. W. Murray, ‘Numerical Methods for Unconstrained Optimization’, Academic Press, London and New York, 1972. J. Kowalik and M. R. Osborne, ‘Methods for Unconstrained Optimization Problems’, Elsevier, New York, 1968.


theory of constrained optimization. We shall also assume that all the variables of interest may be chosen to be real.

Given this, it is convenient to divide minimization methods into two categories : non-derivative methods and derivative methods. In the former category, no explicit use is made of the derivative of the function, whereas in the latter category derivatives are used, and we now proceed to examine the method under these headings. Our examination is far from general in that we examine in detail only methods which (to our knowledge) have so far found a use in quantum chemistry. We have tried, however, to discuss the methods in as broad a context as possible and to give some indication, by means of selected algorithms, as to what computation is required by a met hod.

Nonderivative Methods.-Multivariate Grid Search. The oldest of the direct search methods is the multivariate grid search. This has a long history in quantum chemistry as it has been the preferred method in optimizing the energy with respect to nuclear positions and with respect to orbital exponents. The algorithm for the method is very simple. In this and subsequent algorithms we use x to indicate the variables and a to indicate a chosen point.

(i) Select a grid Aa. (ii) Select a point x = a. (iii) Evaluatef(x) at the 3%- 1 points surrounding x= a determined by the

(iv) Select the point ci for whichf(x) is smallest. (v) Repeat steps (iii) and (iv) until ri yields the least value f(6) of f ( x ) .

The method needs no explicit justification. It is apparent that if a minimum exists it is, in the long run, possible to find it by this strategy.

grid Aa.

Uniuariate Search. A variant on the multivariate grid search is the univariate search, sometimes called cyclic search, which again has had a long history in the context of nuclear position and orbital exponent variation. This method is based on the idea that the individual variables refer to co-ordinate axes el = [l, 0 , O . . . O]T etc., in the n space, and we can thus perform succes- sive one-dimensional searches along each of the axes. The algorithm is:

(i) Select a point a, set i= 1. (ii) Minimize f ( a + lei) with respect to l to obtain a, the value of il that

(iii) Replace a by a + aiei. (iv) If i#n, set i: = i+ 1 and repeat from (ii). If i= n, set i: = 1 and return

It is perhaps fairly obvious that if the variables of the problem are strongly dependent, then a minimum need not emerge from this process, and that

minimizes the function along ei.

to (ii) unless the CIZ are less than some pre-set tolerance.


it is only in the case of strictly independent variables that one can be sure of eventually reaching the required minimum.

The crucial step in the univariate search procedure is undoubtedly niini- mizing along the line ec. In situations where the gradient of the function along the line is not readily available, direct search procedures along the given line fix. one-dimensional direct search procedures) must be employed to find the minimum. Many such procedures are available (see, e.g. Cooper and Steinbergs pp. 136-151), but one of the more efficient procedures seems to be quadratic interpolation. This may briefly be described as follows.

If we wish to find the minimum of f ( x ) in the direction r, we construct the function about the point a,

F( A) = f ( a + Ar),

and find three points along the line, A17 Az7 and A3 such that

F(Ai) > F(A2) < F(A3), A1 < A2 < A3,

and the minimum of the quadratic fitting these three points is then at

This point may be accepted if F(a)<F(Az), or cc can be used together with A2 as initial points for a re-estimate of the minimum. Some care must be exercised in the use of this method to avoid rounding error in the cases where A1 z A2 z A3 and/or I;(&) z F(A2) z F(&) and the difficulties encountered in such situations are discussed in section 7.3 of ref. 8.

Pattern Search. A more sophisticated non-derivative method is the pattern search method of Hooke and Jeeves.10 The general form of this method can be regarded as the paradigm of direct search methods by rational strategy, in that it involves two classes of moves, exploratory and pattern moves. We offer first a broad description of the general kind of approach and then a more detailed account of what is actually involved in the pattern search method, though within the broad general approach we could also discuss the method of Rosenbrockll and the Simplex method (see e.g. section 2.5 of ref. 9). Since, however, (as far as we know) there has been no use made in quantum chemistry of either of these last two methods, we shall not discuss them further here.

The essence of the method is very simple. We start from a base point a and by a series of exploratory moves build up a pattern from which it is possible to infer a direction in which it is likely that a successful move may be made. On the basis of our pattern we make such a pattern move, to establish a new base point for further exploration. We continue the cycle

l o R. Hooke and T. A. Jeeves, J . Assoc. Comput. Machin., 1961, 8, 212. l 1 H. H. Rosenbrock, Comnpirt. J., 1960, 3, 175.


of pattern moves followed by exploratory moves until we are satisfied with the outcome.

In the actual pattern search method the exploratory moves are made in a way very similar to the univariate search method; however, instead of minimizing along the line, we proceed as follows:

(i) Calculate f (a+ Aet) with preset A, if this is less thanf(a), a+a+ Aei;

(ii) If not, try f ( a - Aei) if this is less than f(a), a+a- Aet; i: = i + 1 and

(iii) If not a-w, i: = 1 and repeat from (i).

i: = if 1 and repeat.

repeat from (i).

If we assume that we began the exploratory cycle with a point ZB and ended it with a point aB, f(aB) .c f ( Z B ) , then our pattern move is to a point

a= (ZB + ( a B - -B a 1.

The point a is used [without further testing of f(x)] as the base point for further exploration. If a is not a satisfactory point, then the exploratory moves will take us back to aB, and if this happens, A is reduced and further exploration about aB is attempted. The process is terminated when the exploratory moves yield, for small enough A, no further improvement in the function.

It is perhaps useful to think of the pattern search method as an attempt to combine the certainty of the multivariate grid method with the ease of the univariate search method, in the sense that it seeks to avoid the enormous numbers of function evaluations inherent in the grid method, without getting involved in the (possibly fruitless and misleading) process of optimizing the variables separately.

Conjugate Direcblions. A potentially more powerful method than any of the ones described above is the method of conjugate directions and before giving an algorithm using this method it is appropriate to discuss its background in a little more detail. Suppose for the moment we assume our function to be quadratic, i.e.

f ( x ) = c + X T b + f X T A X , (1 9)

where c is a scalar, b a column vector, and A a positive definite real symmetric matrix. Let us denote the gradient matrix of this function by the column vector g(x) where

then g(x )=b+Ax ,

and at a minimum point, n=ao, g(ao)=O. After a little manipulation it is easy to see that the direction vector p = a* - a, which extends from an arbitrary


point a to the minimum, is given by

p= - A-lg(a).

Now suppose that we start our minimization procedure from a point ao, with gradient go [= g(ao)] and move in a direction PO [=p(ao)] with PO arbitrary, then it can be shown (see, e.g. Cooper and Steinberg6 p. 163) that providing we choose the subsequent directions of descent so that

and

then g,=O, so that the function has been minimized in just iz steps without recourse to inverting matrix A explicitly. Directions chosen according to the prescriptions of equations (23) and (24) are called conjugate directions, and any algorithm which results in convergence in just rz steps is called a quadratically convergent algorithm.

In fact it is generally not even necessary to know A explicitly in order to construct a set of conjugate directions, nor is it generally necessary to know the gradiants explicitly either. That the gradients are not necessary is easy to see, because the condition of equation (23) is simply the condition that the function is minimized along the linepi from the point ai and that minimum may easily be found by an ordinary one-dimensional search. That equation (23) is equivalent to a linear minimum follows because the linear minimum condition is just that I shall be chosen so that

and therefore that

where cc is the value of 1 that makes the derivative vanish. It should be noticed also that any method employing a linear search of this

kind will be a stable method, that is, the function will decrease at each iteration, provided that a minimum exists alongp. Now it is reasonable to hope that sufficiently near the minimum the objective function f ( x ) may be expanded in a Taylor series to second order, so that the quadratic approximation is good, with A the matrix of second derivates offtaken at the minimum point (the hessian matrix), Thus a rational strategy in minimizing an arbitrary function would be to choose those directions which would be conjugate if the function were quadratic and, because the condition of equation (23) ensures stability, we will eventually enter a quadratic region of the function and in this region the minimization will terminate quadratically.

There are in fact a number of algorithms known for choosing conjugate directions when neither A nor g are known and as an example of these methods we give an algorithm due to Powell.12

gi+l= g (ai + .pi>, (26) *

M. J. D. Powell, Comput. J., 1964, 7, 155.

Direct Minimization Methods in Quantum Chemistry

1-4, i = 1, . . . n (e.g. the unit vectors ei).

(where as = as-l + Asrs).

43

(i) Select an initial point a0 and a set of it linearly independent vectors

(ii) For s= 1, . . . n, determine the A8 such that f(as-l + A&) is minimized

(iii) For s= 1, . . . n- 1 set rs=rs+l. (iv) Set rn=an-ao. (v) Minimize f(an+ilnrn) to yield an+l and set ao=an+i and return to

A proof that the directions generated by this algorithm are conjugate may be found in ref. 6 pp. 164-165, where there is also a discussion of more complicated forms of the algorithm less subject to rounding error.

(ii) unless a satisfactory minimum has been found.

Derivative Methods.-The most well developed of the derivative methods are univariate in nature, that is, they approach the minimum of the multivariate function along a sequence of lines (directions) in the many-dimensional space, and the problem is then to determine an algorithm for the choice of these directions. Usually (but not always) it is required that the current direction be followed until a minimum of the function in that direction is found. One may say that these methods are based on a sequence of one- dimensional searches.

There are some modern methods (such as the memory13 and super- memory14 gradient methods) which are not univariate in nature but which approach the minimum in a sequence of many-dimensional searches. So far, however, such methods have found no use in quantum chemistry and we shall not discuss them further.

In the one-dimensional search methods there are two principle variations : some methods employ only first derivatives of the given function (the gradient methods), whereas others (Newton’s method and its variants) require explicit knowledge of the second derivatives. The methods in this last category have so far found very limited use in quantum chemistry, so that we shall refer to them only briefly at the end of this section, and concentrate on the gradient methods. The oldest of these is the method of steepest descent.

Steepest Descent. The algorithm for the steepest descent is really very simple :

(i) Select an initial point a. (ii) Calculate g(a). (iii) Minimize f(a- Ag) with respect to 1 to yield a minimum 1 = a, exit

(iv) Select a new point a= a- ag and return to (ii). if this minimum is satisfactory.

The reason for the name ‘steepest descent’ is obvious from this algorithm

A. Miele and J. W. Cantrell, J. Opt. Th. Appl., 1969, 3, 459. l 4 E. E. Cragg and A. V. Levy, J. Opt. Th. Appl., 1969, 4, 191.

44 Quaitturn Chemistry

since the direction defined by - g is clearly the direction along which the greatest local decrease about a in the function can be obtained. [It is perhaps appropriate to note here that it is not necessary in this, or indeed in any of the derivative methods to be discussed, to be able to obtain an analytic expression of g ; in most methods it is possible to show that a particular numerical approximation will do (see, e.g. Stewart 15).] Furthermore, the method has clearly attractive aspects: for instance, it is obvious that if the contours of f ( x ) are hyperspheres in n dimensions, one iteration of this process will suffice to determine the global minimum. We notice also that a characteristic of the steepest descent method is that it is stable in the sense that if there is a minimum in the region near x = a, steps (ii) and (iii) guarantee a lowering of the function value each time that they are executed (though not necessarily a lowering in magnitude of g). However, the method does not generally have finite termination properties and, indeed, experience suggests that the convergence of the method is generally rather poor. This seems to be because in the limit the steepest descent procedure ensures that the minimum is approached in a two-dimensional subspace of the full space (see p. 30 of ref. 9).

Conjugate Gradient. As we have seen in the previous section, conjugate direction methods do have finite termination properties, and thus are attractive methods. We now consider an example of a conjugate direction method where the gradient is used but calculations neither of A nor of A-1

are required. The method we shall consider is due to Fletcher and Reeves16 and is based on an earlier method of Hestenes and Stiefell7 (see also Beck- man,18 in which a proof can be found that the directions generated in the Fletcher-Reeves method are conjugate). The algorithm is as follows :

(i) Choose an initial point a, find g and let p = - g . (ii) Minimize f(a + Ap) to yield a minimum at A = a, exit if this minimum

(iii) Construct ci = a + ap, find 2 = g@). (iv) Construct f i = - 8+ pp where p= 6TQ/gTg and return to (ii). (v) Set a = ri, p =@ and return to (ii).

is satisfactory, otherwise go to (iii).

Methods of this type are often known as conjugate gradient methods. Other examples are the ‘Partan’ methodlg and the method of Polak and Ribikre.20

Since the gradients are calculated in this method, they may with advantage be used in step (ii), to estimate the linear minimum. A widely used procedure

l o R. Fletcher and C. M. Reeves, Comput. J., 1964, 7 , 149. M. R. Hestenes and E. L. Stiefel, J. Res. Nat. Bur. Stand., 1952, 44, 409.

lB F. S. Beckman, in ‘Mathematical Methods for Digital Computers, Vol. I’, ed. A. Ralston and H. S. Wilf, Wiley, New York, 1960.

l S B. V. Shah, R. J. Buehler, and 0. Kempthorne, J. SOC. Ind. Appl. Math., 1964, 12, 74.

2 o E. Polak and G. Ribikre, Rev. Fr. Inform. Rech. Operation, 1969, 16-R1, 35.

ir G. W. Stewart, tert., J. Assoc. Comput. Machin., 1967, 14, 72.


seems to be cubic interpolation (see e.g. Fletcher and Reevesl6). As in quadratic interpolation, one constructs the function F(Ac) = f(a + Air) but here one also constructs their derivatives FA(&) according to equation (25). One then finds two points A1 and A2 (A2 > Al) on either side of the presumed minimum, that is, points such that FA(A1)<O and FA(A2)>0. The estimate of the minimum is then

where

and

If F(Am) is less than both F(A1) and F(Az), then 1, may be accepted as the minimum, or used as a base point for reinterpolation between either A 1 or A2, according to whether FA(Am) is positive or negative.

There are also a large number of conjugate direction methods which, although using only the function and gradient values, make estimates of A-1 at each stage and choose the directions of descent according to

where HZ is the estimate of A-1 at the ith iteration. These methods are perhaps best considered in the light of some results of Huang,21 who was able to show that the necessary and sufficient conditions for an algorithm, which determines pi by updating H and utilizes linear searches to produce conjugate directions, may be written as

pi= - Higi,

HiApj=ppj i- 1 2 j Z 0 ,

where p is an arbitrary scalar, so that in the limit

Hn =PA- '. Huang showed that it is possible to find an infinite class of algorithms for constructing H matrices from the pi and gi alone which have this property. Fortunately he was also able to show that for quadratic functions all the algorithms in this class have precisely the same convergence and descent properties from a given point in a given direction, in the absence of rounding error and assuming accurate linear searches. This result was later generalized by Dixon,22 who was able to show that providing the linear searches were so conducted as to yield the absolute minimum along the line searched, then all algorithms with the same p value had the same convergence and descent properties from a given point in a given direction, on an arbitrary function. It can be shown that most of the currently used algorithms based

I 1 H. Y. Huang, J. Opt. Th. Appl., 1970,5,405. I P L. C. W. Dixon, Math. Programming, 1972,2, 383.


on updating an H matrix are members of Huang’s class (see D i ~ o n ~ ~ and also Broyden in chapter 6 of ref. 8).

Variable Metric. Perhaps the most widely used of the H updating (the so called variable-metric or quasi-Newton) algorithms is that due to Fletcher and following an earlier suggestion of Davidon. This algorithm is a member of Huang’s class with p= 1 and goes as follows:

(i) Choose an initial point a, and an initial matrix HO which must be

(ii) Minimize f (a+ Ap) to yield a minimum at A = a, exit if this minimum

(iii) Construct Li = a + ap, find (iv) Construct A= H+ A + B,

find = --fig (v) Set a=&., p =f i , H = A and return to (ii).

positive definite. Find g and let p = --Hog.

is satisfactory, otherwise go to (iii). = g(ci).

In step (iv) A = .PPT/PTY,

B= - HyyTH/yTHy,

y = g - g .

and

A special proof of the conjugacy of the directions generated here can be found in the paper of Fletcher and Powell.

In most variable metric methods in the limit, H approaches PA-1 so that the optimum step length, a, approaches a constant as the procedure converges, and it may be argued that there will come a time in the calculation when linear searches are no longer required. In the discussions of the necessary and sufficient conditions for quadratic termination, we have, however, required that an exact linear search should be carried out, so that quadratic termination cannot generally be guaranteed without accurate linear searches. However, there are some members of the class of variable metric methods which will terminate quadratically even if the function is merely decreased in the direction of search. An example is the method of Murtagh and Sar- gent25 (see also FletcheP).

Newton-type. Finally, we come to those algorithms which depend on a knowledge of A and A-l (the Newton-type algorithms). If we are dealing with quadratic functions, then once we know A-l it follows immediately from equation (22) that we can reach the minimum in just one step, so that we need not trouble about directions of descent. However, if the function is not quadratic, then the problem of optimal directions again becomes

2 3 L. C. W. Dixon, unpublished results. 2 4 R. Fletcher and M. J. D. Powell, Comput. J., 1963, 6, 163. 2 5 B. A. Murtagh and R. W. H. $argent, Comput. J., 1970, 13, 185. 2 g R. Fletcher, Cornput. J., 1970, 13, 317.


important. The most widely used approaches to the problem in this case seem to be methods based on the Marquardt-Levenberg method (see, e.g. Murray in chapter 4 of ref. 8), and it is perhaps appropriate to exhibit the algorithm for this approach before attempting further discussion.

(i) At the point a, find g and test the hessian A at this point to see if it is positive definite; if so put A = A and go to (iii).

(ii) Construct A= A + PQ, where @ is a positive scalar and Q a positive definite matrix, such that A is positive definite.

(iii) Construct p = -2-1s. (iv) Minimize f(a+ Ap)to yield a minimum at A= a; exit if this minimum

(v) Construct a=a+ap and go to (i). is satisfactory otherwise go to (v).

It is clear that if the method does descend into a quadratic region, then may be chosen as zero in step (ii) and an exact minimum will be found at step (iv). The incorporation of step (iv) ensures that the algorithm is stable (if a minimum exists alongp) so that, barring accidents, we can be sure of eventually entering a quadratic region of the function.

There are many variants of this kind of algorithm and examples of some of them may be found in chapter 4 of ref. 8. It should also be pointed out that such methods may be combined with those variable metric methods which estimate A-1, so that instead of calculating A-1 at every stage, an estimate of it may be obtained merely by updating the previously calculated matrix. Some examples of studies undertaken by such a combined method may be found in the review by Yde.27

Implementing Optimization Schemes.-There are a number of problems that are common to most of the schemes that we have outlined above which should be discussed in a little more detail before leaving this general account of optimization theory. Perhaps the foremost of these problems is that of knowing when to stop a method. In theory we should not stop a method until we reach a point where the elements of the gradient matrix are all identically zero and the hessian matrix is positive definite. In practice, of course, these requirements are much too strong and at best we must content ourselves with satisfying these requirements subject to rounding and truncation errors introduced by finite computation in a machine of a given word length. What such errors will in fact be depends, of course, on the algorithm chosen and the machine used, and acceptable error levels must be determined by the user in every particular case.

However, in non-derivative methods, neither the gradient nor the hessian matrices are readily available, and to compute these matrices wherever a minimum was thought to have been reached would remove, to some extent, precisely those advantages that the methods possess. In consequence it is customary, in the use of such methods, to content oneself with a value for

P. B. Yde, J. SOC. Ind. Appl. Math. Rev., to be published.


the minimum that seems reasonable in the context of the problem. Even in derivative methods it is often thought much too time-consuming to examine every element of the gradient matrix to test its size and it is usual to test on an ‘average value’ of the gradient (say gTg/n), stopping the procedure when this falls below a preset value. It is not usual, even in gradient methods where the inverse hessian is estimated, to conduct further tests at this point on the hessian, for again testing for positive definiteness is a time-consuming business.

Deciding where to stop is therefore almost completely a matter for the judgement of the user of a method. He must himself determine convergence conditions which are sufficient for his needs. These conditions must not be so sIack that he stops too far from a minimum nor so strong that he wastes time computing quantities made up mostly of rounding and truncation errors.

In methods which employ linear searches, how close one can in fact get to the minimum depends entirely on how accurately one can locate a linear minimum. Locating a linear minimum, as we have indicated above, is most often done by bracketing the minimum and interpolating. Bracketing the minimum starting from the point x = a along the line r is generally done by extrapolation. A step length h is chosen and the function values f ( a + nhr), n=0, 1, 2, 4, 8, etc., are calculated until a set of bracketing points is found. The difficulty here is in choosing the step length h. If h is chosen too small, then many time-consuming function evaluations must be made, and if h is chosen too large the interpolation will not be accurate and subsequent re-interpolations may well be necessary using the interpolate points as members of the set of bracketing points. In either case the process is time consuming. If one has available an estimate of the minimum value of the function fest and the gradient of the function, then an estimate of h may be made. On the assumption that the function is quadratic along r the minimum value lies a distance t from a, given by

Fletcher and Reeves,lG for example, recommend that this value, t, should be taken for h, unless it is greater than ( rT r ) - I / 2 or less than 0, and in either of these cases (rTr)-l/2 should be chosen (in which case hr is a unit vector). It may be, of course, that in particular problems other schemes to estimate step length suggest themselves, but this scheme is known to work in most cases without too much trouble if a reasonablef&t is available. Ifyest can be updated at each cycle to become a closer lower bound to the true minimum as the process converges, then t becomes a very effective choice of step length. However, care must be taken to make sure that, near convergence, t is not computed solely from rounding and truncation error, and to avoid this h is often kept constant after a certain point in the calculation has been reached.

Near convergence one must also take great care with the interpolation


formulae, for although neither of those formulae given above (for quadratic and cubic interpolation) are inherently unstable near a minimum, they again can yield estimates of the minimum from information that is essentially rounding error, so that they must not be pressed too far. If one is close to a minimum and the interpolation formulae apparently cease to give consistent results, then this is almost certainly due to rounding error and the lowest- value point so far found will be the closest that one can get to a minimum. Thus for example if, in cubic interpolation, one interpolates to a point Am such that

where the function values are close together and the points close together, but one finds that FA(Am) and FA(&) are both small positive numbers, it is very likely rounding error has set in, as it is unlikely that the function is really cubic along the line so close to a minimum.

It should be stressed again here (as mentioned earlier) that a great deal of work is being done (especially in variable-metric methods) to find ways of dispensing with linear searches and yet still retaining good convergence properties. In such work it is usually proposed that a step length be chosen in advance, instead of interpolating at each cycle, with the metric appropriately modified to ensure stability and convergence. Such work as has been done so far is inconclusive and many of the methods developed are no more efficient than ones employing linear searches. However, the work is developing fast and it is not inconceivable that in the very near future efficient implementations will be available.

It should also be noted that the performance even of a stable, quadratically convergent method employing linear searches may well be adversely affected by rounding error, even before rounding makes further interpolation fruitless. Thus for example in Powell’s method it is quite possible, in large problems, that owing to rounding error, the directions chosen become linearly dependent. They cannot then be properly conjugate directions even in the case of a quadratic function, and in these circumstances the quadratic convergence of the algorithm will be lost. In situations where this happens there is little that can be done short of abandoning the calculation at the point where the failure is discovered, and trying a complete restart from that point or, perhaps better, switching to a new method.

In this context it should be noted that, in the Fletcher-Reeves method, there is something to be said for periodic restarting anyway. If a minimum has not been reached with such a method in n or so iterations, then the function cannot have been quadratic in the region of investigation, and therefore the chosen directions will not have been conjugate with respect to the required hessian, and will thus have no special properties. It is arguable, therefore, that the information so far obtained from them is at best irrelevant or at worst misleading in a quadratically convergent method and so should be discarded at this stage and the point reached simply regarded as a completely fresh starting point for the method. Experience seems to indicate

F(A1) > F(Am) > F(&),


that restarting speeds convergence frequently in the Fletcher-Reeves method, but clearly it is a matter of experience in a particular problem as to precisely where and on what criteria such restarts should be made, for their effectiveness may well be very problem and technique dependent. A discussion of these points has been made by Huang and Levy.28

4 The Realization of Direct Methods in Quantum Chemistry

In Section 2 we outlined briefly how the ordinary closed-shell SCF problem could be recast as a problem in direct minimization. In this section we consider the problem in a little more detail and also consider its generalization, particularly in respect of incorporating constraints and finding the relevant gradient expressions.

In McWeeny’s3s4 realization of the steepest descent to the method for coefficients, constraints were included in rather an oblique manner. McWeeny concentrated on the fact that in the ordinary closed-shell SCF problem the ‘physical’ variables were the elements of the R matrix [see equation (6)], and he regarded these as the variables of his problem. To first order the change in energy induced by a charge R 4 R + 6R is simply:

so that in the absence of constraints:

-- aE -2f i i . - aRij

It is clear that in the absence of constraints the energy function may not possess any minima, so that constraints cannot be completely neglected. McWeeny observed, however, that orthogonality could be preserved to any order by requiring that

with

and

where A is an arbitrary rn by rn matrix at the disposal of the user. It follows that

6E = 2 tr [pS-1/2fS-1/2(l - p) + (1 - ~)S-1/~fS-l/2p]A.

McWeeny then regarded the elements of d as determining the minimization problem and clearly the steepest descent is along a negative multiple (- A, say) of the quantity in square brackets in equation (32). Given that there is a convergent power-series expansion for the inverse in equation (29), it then

3 8 H. Y. Huang and A. V. Levy, J. Opt. Th. Appl., 1970, 6, 269.

(32)

Direct Minimization Methods in Quantum Chemistry

follows that, to second order, the change in R is

51

SR= - A(s + S T ) + A'(sSS~ - s~SS), (33)

S= (S- l - R) fR, (34)

with

and it is now possible to show that the optimum value of A (a, say) for a steepest descent is given by

where

where

a = - 1/(2m - m'), (35)

1 = tr Lf, m = tr LSMf, m' = tr LG(L), (36)

L = s + s ~ , M = s - s ~ , (37)

and G(L) is made up from L as G(R) is made from R [see equations (7)

It is easy to see that in general the new R matrix created from equation (33) will not be accurately idempotent (i.e. orthogonality will be lost), and that therefore one will not be able to use this matrix at the next iteration without correcting it for this defect. However, as the minimum is approached the new R matrix will become more and more accurately idempotent.

The method in this form was investigated by other authors, e.g. Sleeman,29 and by and large it was found to be very much inferior in convergence characteristics to the conventional procedures when the conventional procedure worked. Suggestions were made for modifying the method to speed up convergence (see also, for example, Hillier and Saunders30).

In 1970 Fletcher5 observed that slow convergence was a characteristic of the steepest-descent method and that a more modern method would probably work better. However, he noted that the scheme for incorporating constraints used by McWeeny was unsuitable for most modern methods, since modern methods often needed information from the previous cycle and this information would be misleading if it had been necessary, after the previous cycle, to restore idempotency. Fletcher therefore suggested the method we have already referred to in equation (15) leading to equation (17). If we denote the matrix (YtSY) by A, it can easily be seen from equation (17) that, to first order,

and @)I.

6R= YA-16Yt(l-SR)+(Z- RS) GYA-lYt, (38)

and that hence (assuming real elements of Y) , from equation (27),

6E=4 tr(Z-SR) fYA-16YT, and

(39)

(40) -_ aE - [4(1- SR) fYA-l]zj. a Yij

** D. H. Sleeman, Theor. Chirn. Acfa, 1968, 11, 135. 1 0 I. H. Hillier and V. R. Saunders, Proc. Roy. Soc., 1970-71, Mu), 161.


Equation (40) then gives the required expressions for the derivatives with respect to variables that satisfy the constraints.

It is assumed in this method, of course, that the initial estimate of Y yields a non-singular, positive-definite, A matrix, but given these starting conditions, the matrix A remains positive definite (non-singular) throughout the course of the minimization (given accurate arithmetic) precisely because the constraints are built in.

It should be noted here that the variables Yij are not the ‘physical’ variables of the problem; in fact the minimizing point of E in Y is not, as Fletcher observed, even unique. Furthermore, there are p = nm variables in the matrix Y and in fact only p - +n(n+ 1) of these are independent. The first point is of no consequence in the closed-shell case because, as is well known, the energy is invariant under orthogonalization of the orbitals by whatever scheme, so that all Y matrices that minimize the energy will lead to the same energy and density matrices. The second point does not matter as long as it does not affect the chosen minimization scheme, and there is no reason to suppose that it will as long as the energy, sufficiently close to the minimum, can be expanded in terms of the dependent variables.

It is possible to extend Fletcher’s method for incorporating orbital orthogonality constraints to the more general LCAO-MO-MC-SCF wavefunction and a discussion of how this may be done has been published by Kari and S~ tc l i f f e .~~ Of course, in the multi-configuration case one no longer has the invariance of the wavefunction under transformation of the orbitals, so that different expressions for the energy and for the gradient matrix are obtained with different orthogonalization schemes. However, given that a unique minimum exists, in any MC-SCF scheme that minimum should be found irrespective of the orthogonalization scheme used. Kari and Sutcliffe32 have investigated minimization in the usual Roothaan open- shell scheme (which can be regarded as a special case of an MC-SCF scheme) and found that no difficulties or ambiguities arose in practice.

Tt is not necessary, however, to follow Fletcher and include dependent coefficients in the minimization scheme, and it is possible to use the orthogonality relations to remove them. A scheme for doing this has been proposed by Raffenetti and Ruedenberg,33 who refer to it as a generalized Euler-angle scheme, and this method has been used by KouteckJi and BonaEiE34-36 in some semi-empirical calculations, in which Powell’s method was the chosen minimization method. There is, however, no doubt that there would be considerable difficulties in the way of using Ruedenberg and Raffenetti’s scheme in derivative methods, because of the complexity of the

R. Kari and B. T. Sutcliffe, Internat. J. Quantum. Cliem., 1973, 7, 459. 5 2 R. Kari and B. T. Sutcliffe, Chem. Phys. Letters, 1970, 7 , 149. y 3 R. C. Raffenetti and K. Ruedenberg, Internnt. J. Qrrantirm Chem., Symp., 1970, 3,

625. J. Koutecky and V. Bonaci;, Chem. Flzys. Letters, 1971, 10, 401.

3 5 V. BonaEic‘ and J. Routecky, Internat. J. Quantum Chem., Syrnp., 1972, 6, 171. 3 8 V. BonaEit and J. Kouteck?, J. Chem. Phys., 1972, 56, 4563.


energy derivative expressions required. In this connection it should be noted that with constraints incorporated even in the Fletcher manner, one obtains extremely cumbersome formulae for the second derivatives of the energy with respect to the coefficients. This poses considerable problems in the use of a Newton-like method for the coefficients.

So far we have considered the more usual orthogonal orbital type wavefunctions in which the constraints are those of orbital orthogonality. How- ever, for wavefunctions in which orbital orthogonality is not required (or for more general wavefunctions) the above discussion need not apply since in these cases it is possible to use an unconstrained minimization method directly on the functional

However, the complexity of this functional even in simple cases makes it seem unlikely that much use could be made of such an approach.

We turn now to the problem of optimizing the non-linear parameters in a wavefunction. As mentioned in the introduction, for non-linear parameters (such as orbital exponents or nuclear positions) traditionally, non-derivative methods of optimization are used. However, if we wish to use a gradient method, for example, we must be able to obtain the required derivatives, subject to the constraints on the non-linear parameters and also subject to the condition that the constraints on the linear parameters continue to be bound during the variant of the non-linear parameters. In the usual closed- shell case, Fletcher5 showed how the linear constraint restriction could be incorporated, providing that one started from a minimum in the linear parameters. Assuming for the moment no particular constraints on the non-linear variables, then starting from a linear-minimum it is easy to see that

6E=2 tr RBR+tr R6G-2 tr RfRSS,

where 6h, 6G, and 6s are first-order changes in h, G, and S, respectively, induced by the changes in the non-linear parameters.

The precise gradient expressions will, of course, depend on exactly which non-linear parameters are varied, but if we take the example of orbital exponents a&, assuming one exponent to each orbital and assuming that they are independent parameters, then it follows that

(42)

where the derivatives of the matrices are taken to mean the matrix of the derivatives. This expression may be considerably simplified by a little manipulation to give


where E and are made up from the integrals - ht3= (aipccg 1 h Ij>, &= (ailaai I j } , (45)

Gijkl=(ai/accijlgI kl>. (46)

and is made up as in the ordinary G matrix, but from the integrals

The notation aipcci is used to denote the orbital arising from ith A 0 on differentiation with respect to cci.

The only constraint on the orbital exponent is that it should be greater than zero and, as explained in the introduction, it is easy to incorporate this constraint by the transformation

ori-+yi = In xi, so that

The generalization of these results, to the LCAO-MO-MC-SCF case, can be found in the paper of Kari and Sutcliffe.3l

For other non-linear parameters, such as nuclear positions, more complicated gradient expressions are needed (see, for example, Gerratt and Mills37 for nuclear position expressions) and also sometimes more complicated constraints (for instance, constraints to prevent effective translation and rotation of the molecule as a whole in the nuclear position case), but there are no essential differences in principle.

5 Experience with the Use of Optimization Schemes

In this, the final section of our Report, we attempt to summarize the practical experience gained in the use of the optimization schemes we have talked about in Section 3. We also attempt to provide pointers to the use of the schemes in new contexts. In describing practical experience of optimization schemes we shall, wherever possible, take our examples from quantum chemistry. Unfortunately for our purposes, however, in many instances of papers reporting the use of a direct minimization scheme in quantum chemistry the authors regard the scheme used (quite properly) as incidental to the main purpose of the paper, and do not therefore give details of its performance, the chosen starting points, the convergence criteria used, and so on. It is thus often very difficult for the Reporters to make sensible statements about performance of the methods in the chosen context. As a com- promise, therefore, when we do compare performance we do so in the context of very carefully performed computations, done generally by workers in the field of optimization theory on what are undoubtedly, from a quantum chemical point of view, highly artificial functions containing a trivial number

1 7 J. Gerratt and I. M. Mills, J. Clzem. Phys., 1968, 49, 1719, 1730.


of variables. These comparisons must therefore be treated with great caution for the purposes of generalization to quantum chemical problems, bearing in mind our earlier remarks about the possible problem dependence of linear search techniques and similar difficulties.

Experience with non-derivative methods of optimization has been long and very mixed and it has not been until recently, with the advent of more sophisticated techniques such as Powell's method, that much effort has been made to compare their performance characteristics, both among themselves and with derivative methods. The attractions of a non-derivative method, particularly in a quantum chemical context, are fairly obvious. They are, on the whole, conceptually simple, the algorithms for them are easily programmed, and they make only small storage demands on the computer. Some comparisons among various non-derivative methods have been made by B o p and he concludes (in the case of only a small number of variables) that Powell's method is the most efficient and he further estimates that in this kind of case that it may be as good as some derivative methods. The conclusions of Box are largely in accord with those of Fletcher,39 who also found Powell's method to be the best.

In the context of quantum chemistry the most widely used of the non- derivative methods has, however, undoubtedly been sequential univariate search (perhaps sometimes preceded by a preliminary grid search). This method has been most used in optimizing orbital exponents. In this kind of case the constraints are easily satisfied simply by rejecting any points that violate them, and this makes the process even more attractive from the point of view of ease of programming then some other non-derivative methods. However, as mentioned before, the method is not a good one in the case of strongly dependent variables, and orbital exponents even in quite small basis sets are generally agreed to be strongly dependent. It is interesting in this context to compare the results obtained by Kari and S~tcliffe,3~ optimizing the exponents in a 'double zeta' basis for the first-row atoms using the Davidon-Fletcher-Powell method, with those obtained by Clementi,** using a univariate search method. The exponent values found by Kari and Sutcliffe to minimize the energy at a point where the gradient of the energy with respect to the exponents was small were quite different in some cases from those of Clementi. It is perhaps fair to summarize the situation by saying that it would seem that a sequential univariate search procedure is quite adequate for locating a fairly approximate minimum in a function of dependent variables, but it must be used with great care if it is desired to locate the minimum at all exactly, and in general other methods are to be preferred for exact location of a minimum.

Pattern search has also found some use in quantum chemistry. An early use was by Miller and Ruedenberg4I in their geminal calculations on the ' 8 M. J. Box, Comput. J., 1966, 9, 67. a * R. Fletcher, Comput. J., 1965, 8, 33. 4 0 E. Clementi, J. Chem. Phys., 1964, 40, 1944. 4 1 K. J. Miller and K. Ruedenberg. J. Chem. Phys., 1965, 43, S88.


beryllium atom and more recently Bishop and Leclerc42 have used it in optimizing non-linear parameters in basis set construction, but it is difficult to ascertain from these papers the performance characteristics of the method.

Powell’s method has been used in optimizing exponents by Solomon43 and for optimizing the non-linear parameters associated with defining the orthogonal matrix in Raffenetti and Ruedenberg’s33 method of incorporating orthogonality constraints. Mehler, Ruedenberg, and Silver44 used Powell’s method in a practical application of the Raffenetti-Ruedenberg parametriza- tion. It has also been employed by Pan and King45 in the optimization of non-linear parameters in geminal calculations. Kouteckp and BonaCiE3*-36 have also used Powell’s method for linear coefficients in some semi-empirical SCF calculations. Once more, little comment is made by the authors who have used Powell’s method on its efficiency, beyond a comment in the paper by RafKnetti and Ruedenberg33 that they found the method better than the ‘Partan’ method.19 In the absence of specific comment one assumes that the authors were satisfied with the method and this in itself is interesting because the problems that these authors considered often involved many (>20) variables, and it is sometimes stated (see, for example, Fletcher on p. 78 of ref. 8) that Powell’s method is quite ineffective for more than 10 variables. The Reporters also know of cases where workers in the field of quantum chemistry have attempted to use Powell’s method in large problems and had to abandon its use, because of convergence difficulties probably associated with rounding errors and the like.

The situation with respect to Powell’s method is therefore far from clear and, in the absence of any systematic work to determine its effectiveness in a quantum chemical context, it seems to the Reporters that the method is always worth a preliminary try in an optimization problem particularly as it is so easy to program and so compact.

However this may be, it is thought generally that derivative-based methods are faster than non-derivative methods. Indeed, Fletcher5 estimates that a good method based on first derivatives should take of the order of l/n of the time taken by a good non-derivative method. This is, in a sense, perfectly reasonable since derivative methods make use of more information about the function than do non-derivative methods. However, derivative methods do require the calculation of that extra information (in addition, that is, to the function values) and the cost of calculating this extra information in a quantum chemical situation is hard to assess. Thus, for example, if one is using a gradient method to optimize exponents then one must calculate extra electron-repulsion integrals. This may mean extending an existing program, for if one has say, p-type functions in one’s basis, then one must have integrals involving d-type functions to evaluate the gradient elements.

D. M. Bishop and J.-C. Leclerc, hlol. Phys., 1972, 24, 979.

E. L. Mehler, K. Ruedenberg, and D. M. Silver, J. Chem. Phys., 1970,52, 1174, 1181, 1206.

a 3 C . E. Solomon, Internat. J . Quantum. Chem., 1971, 5, 319.

4 s K.-C. Pan and H. F. King, J . Chem. Phys., 1972, 56, 4667.


The cost of extending integral programs is usually far from small, if one wishes them to perform efficiently, and if the program does not perform efficiently then the time taken in the optimization program to evaluate the gradient elements can become prohibitive.

If we confine ourselves for the moment to comparing methods employing only first derivatives, it is clear from the theoretical work of Dixon22 that there should be nothing to choose between most variable-metric methods, and indeed this is confirmed by numerical work by Dixon23 and by Huang and Levy.28 The situation of the Fletcher-Reeves method in its context is, however, a little difficult. Huang21 was able to show that for a quadratic function the Fletcher-Reeves method is a special case of Huang’s class with p = 0, and so the Fletcher-Reeves method is just as good as a variable-metric method here. However, when the function is not quadratic the Fletcher- Reeves method is not a member of Huang’s class and nothing can be said theoretically about its performance in comparison with the Huang’s-class methods. Such numerical work as has been done (see, e.g. Huang and Levy28)

does not help to resolve the problem either; sometimes the method is found to be faster and sometimes slower than a variable-metric method; it seems to depend entirely on the problem. It can be said, however, (see McCormick and Pearson in chapter 21 of ref. 7) that the Fletcher-Reeves method with periodic restarts has, in the general case, super-linear convergence.

It is widely believed that, generally speaking, methods such as the Davidon-Fletcher-Powell method are superior to the Fletcher-Reeves method and, indeed, Fletcher suggests (see p. 82 of ref. 8) that typically the Fletcher-Reeves method will take about twice as many iterations as the Davidon-Fletcher-Powell method.

Comparing gradient methods with Newton-like methods (that is, second- derivative methods), the general opinion is that Newton-like methods are superior. Indeed, if we follow Murray (see section 4.12 of ref. 8) we should be persuaded that such methods are definitely the ‘best buy’, providing that it is not too difficult to construct a positive definite hessian for the problem. Again this seems intuitively a reasonable assertion because of the extra information used in such methods, but once again there is the problem (as between gradient and non-derivative methods) of the cost of constructing this extra information and it is very difficult to decide this in general terms.

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable


one. Given these difficulties it is not surprising that there has been little or no experience in quantum chemistry of optimizing linear coefficients by variable-metric or Netwon-like methods. One of the very few examples of such a use of these methods is the MCSCF calculation by Hinze and R ~ o t h a a n , ~ ~ who used a Newton-like method developed by Wesse11,47 in which an iterative Newton-Raphson scheme is used to update the inverse estimate. The authors make no comment on the characteristics of the method.

The use of Newton-like methods in optimizing non-linear parameters is rather more widespread. used a Newton-like method in some of the first work on orbital exponent optimization, and more recently both Boys and Handy49 and Bishop and L e c l e r ~ ~ ~ have used Newton-like methods with numerical estimates of the derivatives to optimize exponents, and again no comments are given on the efficiency of the methods. Kari and Sutcliffe32 and Fletcher5 have both used the Davidon-Fletcher-Powell method for optimizing exponents, and McIver and Komornicki50 have used Murtagh and Sargent’s method for optimizing nuclear geometry in some semi-empirical calculations. McIver and Komornicki comment that they found Murtagh and Sargent’s method to be faster than the Davidon-Fletcher- Powell method, and both methods much faster than Powell’s method. Both Fletcher, and Kari and Sutcliffe observed quadratic convergence from their chosen starting points in exponent optimization with the Davidon-Fletcher- Powell method, and it is difficult to see how this could be improved on by any other method. However, it must be admitted that in both cases work began with rather good guesses at the exponents.

In summary, therefore, there is too little work with Newton-like methods to make any assertion about their utility in quantum chemistry, but there is enough work with variable-metric methods to make it possible to assert with some confidence that they are worth very serious consideration by any worker wishing to optimize orbital exponents or nuclear positions in a wavefunction.

As far as the Fletcher-Reeves method is concerned, it must clearly be the method of choice in linear coefficient optimization as it involves only the storage of gradient and direction vectors between iterations. It has been used by a number of authors (Sleeman,29 F let~her ,~ Kari and Sutcliffe,32 Claxton and Smith,S1 and Weinstein and P a u n c ~ ~ ~ ) . It [is unfortunately possible, however, to sum up the experience so far gained of the method in quantum chemistry as disappointing, in the sense that in SCF caclulations the authors have found that the calculations proceed significantly more slowly than the conventional iterative procedure, when the conventional procedure converges at all.

d a J. Hinze and C. C. J. Roothan, Siippl. Progr. Tlteor. Phys. (Kyoto), 1967, 40, 37. 4 7 W. R. Wessel, J . Chem. Phys., 1967, 47, 3253. 4 8 B. J. Ransil, Rev. Mod. Phys., 1960, 32, 239. 4 0 S. F. Boys and W. C. Handy, Proc. Roy. Soc., 1969, A310, 63; 1969, A311, 309. 10 J. W. McIver, jun., and A. Komornicki, Chern. Phys. Letters, 1971, 10, 303. 5 1 T. A. Claxton and N. A. Smith, Theor. Chim. Acta, 1971, 22, 399. 6 * H. Weinstein and R. Pauncz, Chem. Phys. Letters, 1972, 14, 161.

Direct Minimization in Methods in Quantum Chemistry 59

In most cases the authors have used as a starting set of coefficients a set of vectors chosen by solving the zeroth order eigenvalue problem [that is, for example, in the closed-shell case equation (14) with G(R) set zero]. From this starting point it is usually found that the descent into a quadratic region is very rapid, and from there on to very near the minimum the performance of the method is precisely that predicted by theory. However, to move from the point obtained after about n iterations to the true minimum, or at least to a point obtained in the conventional procedure, is a very slow process indeed. It would seem as if one is approaching a minimum here in a subspace of the full n-dimensional space, much as one would do if one were using a steepest- descents technique. The Reporters would agree with Claxton and Smith51 that the easiest way to avoid these difficulties in an SCF calculation is to change to the conventional procedure once one is in a stable quadratic region, only returning to the Fletcher-Reeves method in the event of divergence. If one chooses to use such a ‘mixed’ method in the solution of an SCF problem, then it is clear from reported performance characteristics that a user would be wise to consider as a possible mixed method the method proposed by Hillier and Saunders30.53 as implemented by Saunders in the program suite ATMOL.

However, in MC-SCF problems there is not generally the possibility of using ‘mixed’ methods and the Fletcher-Reeves method will in practice be one of the few available methods in this case. If open- and closed-shell SCF experience is anything to go by, then one would certainly not expect spec- tacular convergence from its application here, but it would be a stable and quite certain method of reaching the minimum. There is, however, no practical experience yet of using the Fletcher-Reeves (or any conjugate-gradient) method in MC-SCF calculations.

Non-linear programming is a fast growing subject and much research is being done and many new algorithms appear every year. It seems to the Reporters that the current area of major interest in the field is the area of variable-metric methods, particularly those not needing accurate linear searches. Unfortunately, from a quantum chemical point of view, such methods are liable to be of use only in exponent and nuclear position optimization and in this context, as we have seen, Newton-like methods are also worth serious consideration.

Undoubtedly more effective conjugate-gradient techniques would be of most use in quantum chemistry, perhaps some methods tailored to suit particular functional forms common in the field, but this area of research seems unlikely to be developed further by workers principally interested in optimization and is perhaps a suitable field of endeavour for quantum chemists.

Note added in proof. The Simplex plethod (see p. 40) has now in fact been used in some semi-empirical calculations optimizing nuclear positions. 64 The authors, however, do not comment in the paper on the observed performance characteristics. li I. H. Hillier and V. R. Saunders, Internat. J . Quantum Chem., 1970, 4, 503.

M. J. S. Dewar and M. C. Kohn, J . Amer. Chem. SOC., 1972, 94, 2704.


BY J. GERRATT

1 Introduction

Within a few weeks of their arrival at a university, most chemistry students know - or believe that they know - why the ground state of the 0 2 molecule is a triplet, whereas by contrast the ground states of N2 and F2 are singlets and form closed shells. Some may even be able to illustrate their argument with the aid of a cunningly drawn molecular orbital (MO) correlation diagram.

Yet, in 1937, Whelandl had shown that the valence bond (VB) theory was quite capable of accounting for the observed spectroscopic states of 0 2 ,

and at about the same time Nordheim-Poschl2 had come to the same conclusion using the spin valence theory." However, the MO description of the lowest states of 0 2 is so direct and elegant that one is hardly surprised at the almost total disregard paid to VB theory in this matter. For although VB theory does in fact predict the ground state of 0 2 to be a triplet, one reaches this conclusion only as the result of a detailed calculation. To obtain from a very large calculation a result which agrees with experiment is only a part of science (as one likes to expound to students when in a suitably expan- sive mood); we require, even more, a convincing model for the phenomena under study.

By the same token, it is not often that one can say anything useful on the basis of MO theory about the shapes of potential surfaces for molecules. Consider the following example: the A lli state of the BH molecule is observed to possess a maximum.3 As shown by Hurley,* the abnormal shape of this curve can be very simply interpreted on the basis of spin-valence theory by the crossing of two zeroth-order curves. The first curve arises from a wavefunction constructed from the ground (2s22p; ZP,) state of the B atom and the (1s; 2S) state of the H atom and is strongly repulsive. However, the second curve, which arises from the interaction of the excited ( 2 ~ 2 ~ 2 ; 2 0 , )

state of B with (1s; 2s) of H, is bonding but only begins to contribute significantly to the total wavefunction when the internuclear distance is quite

* The distinction between the VB and spin valence theories is fully described in Section 3, p. 68.

1 G. W. Wheland, Trans. Faraday SOC., 1937, 33, 1499.

s G. Herzberg and L. G. Mundie, J. Chern. Phys., 1940, 8, 263. G. Nordheim-Poschl, Ann. Physik, 1936, 26, 258.

A. C. Hurley, Proc. Roy. Soc., 1961, A261, 237.

60

Valence Bond Theory 61

small (- 3 a.u.). One would no doubt obtain a correctly shaped curve from MO theory if a large enough configuration-interaction calculation were performed,? but the simplicity of the model is then lost.

It seems, therefore, with the current renewal of theoretical interest in atomic and molecular collision problems, reactive scattering, and predissociation phenomena, that it is worthwhile to examine the VB theory as a useful model that is capable of yielding accurate potential energy surfaces.

For this purpose it would seem profitable to review briefly in the next section some of the, perhaps, less well-known properties of the exact non- relativistic wavefunction, but which are, nevertheless, important when discussing VB theory. Also in this section a short description is given of the construction and manipulation of antisymmetric wavefunctions of more general form than a simple Slater determinant. This is then followed by a brief survey of some of the more commonly used spin functions.

In Section 3, the Reporter has attempted to cast VB theory into as compact and unified a form as possible by making considerable use of group theoretical techniques. This is followed by a discussion of the various improvements and extensions that have been made over the past few years. The basic difficulty in VB theory is the calculation of the matrix elements of the hamiltonian when there is no orthogonality between the orbitals involved. This problem is also discussed at some length in this section, together with a survey of the various approaches that have been tried or proposed for its solution.

Several important developments in the straightforward VB theory have occurred in the past 10 or 15 years. These include the atoms-in-molecules method of Moffitt as modified by Hurley and others, the pair function model of Hurley, Lennard-Jones, and Pople, and the general group function model of McWeeny. These theories can all be usefully discussed within the framework developed in Section 2, and this is done in Section 4.

2 Construction of Antisymmetric Wavefunctions

The Exact Wavefunction.-We consider a molecule consisting of N electrons and A nuclei. We assume the Born-Oppenheimer adiabatic approximation, 6

and within its framework choose a suitable molecule-fixed co-ordinate system with which to characterize the positions of the electrons and nuclei and the momenta of the electrons. The electronic hamiltonian can now be written (in atomic units) as:

N N A

p = l p>v=1 J > E = 1 H= x ( - + V ~ + u , > + rG1+ x Z,Z,R;J, (1 1

t A single configuration LCAO-MO-SCF calculation for this state shows no maximum in the potential energy curve.6

5 J. L. Harrison and L. C. Allen, J . Mol. Spectroscopy, 1969, 29,432. * M. Born and K. Huang, ‘Dynamical Theory of Crystal Lattices’, Clarendon Press,

Oxford, 1954, Appendix VIII.

62 Quaiztcim Chemistry

where up is the potential experienced by electron ,u owing to the nuclei,

and rtL J , ribV, and RJIf are respectively the electron-nucleus, electron-electron, and internuclear distances. The hamiltonian (1) does not contain any electron spin interaction terms, and these will not be considered in this article.

The eigenfunctions of Hare written as Y, and theeigenvalues as E. Although H contains no electron spin interactions, the Y do depend upon both the spatial and spin co-ordinates of the electrons:

Y= Y'(r1, r2, . . . , riv; GI, ~ 2 , . . . , G N ) ,

in which the otL denote spin co-ordinates. A consequence of the absence of spin operators in His that the hamiltonian commutes with the operator for the square of the total spin, s2, and with the operator for the projection of the total spin upon some external, laboratory-fixed, axis, &:

(3)

[ H, S2]=0; [ H, &]=O,

in which the external axis is conventionally taken to be the z-axis. As a result the eigenfunctions Y can always be chosen to be simultaneously eigenfunctions of the operators s2 and sz with eigenvalues S and M, respectively:

where !P is now written with the two eigenvalues which characterize the function as subscripts.

In addition, one requires that the YSM satisfy the Pauli principle which, for electrons, states that the wavefunctions must be antisymmetric under any simultaneous permutation of space and spin co-ordinates. Thus it is required that

P Y S M 5 P r P g Y S M

= E P P S M , (6)

in which Pr, P are operators which respectively permute spatial and spin co-ordinates, and EP is the parity of the permutation (EP= + l for p even, - 1 for P odd).

In order to satisfy equations (5) and (6) simultaneously, Wigner7 showed that Y S A ~ must have the following form:

7 E. P. Wigner, 'Group Theory', translated by J. J. Griffin, Academic Press, New York, 1959, chap. 22.

Valence Bond %ory 63

in which the @Sk are a set of purely spatial N-electron functions,

@ S k = @ S l c ( r l , r2, . . . , r N ) , (8)

(9)

and the @$

The number of terms in the sum of equation (7), fy, is given by

a set of spin functions : N a,$, M ; E = @g M ; k(01, 02, - ON)*

(2S+ 1) N ! f R = ( * N + s + l ) ! (+N-S)!.

The spatial functions @m are orthonormal and are each individually eigenfunctions of H with eigenvalue E:

<@SkI@SI> = a k l , (1 1)

H @ s k = E@sk, (12)

k , l = l , 2 , . . . , f#.

The significance of this set of spatial functions will become clear shortly.

of $2 and sz with the eigenvalues S and M, respectively, The spin functions @gM; k are also orthonormal, and are eigenfunctions

<@: M ; kl @f M ; k> = BkZ, (1 3)

(14) s2@,f M ; k=s(s+ 1) @,f M ; k,

s;?@#, $1; k=M@gif;k, (1 5 )

k, 1=1,2,. . . , f y , where the integration in equation (1 3) is now, of course, over the spin co-ordinates.

M ; k fUnCtiOnS is that they possess permutational symmetry, meaning that under permutations of the space or spin co-ordinates, these functions generate representations of the group of N! permutations, YN. Thus,

An important property of the @,$k and

It can be shown that the set of matrices Us(P) generated in this way constitute an irreducible representation of the group 9”. The set of functions disk: hence form a basis for this representation, and the degeneracy in the level E implied by equation (12) is termed the permutational degeneracy; it has no physical significance.

The representation of the group 9” generated by the spin functions is


the set of matrices E P U ~ ( P ) being said to form the dual representation. If equations (16) and (17) are substituted into the left-hand side of equation (6), it is easily seen that a function of the form of equation (7) does indeed satisfy the Pauli principle.

Approximate Wavefunctions.-We now turn to the problem of constructing an acceptable wavefunction from an arbitrary spatial function @(r l , r2, . . ., r N )

which we might select according to some model. This is achieved by forming the following functions:

yi&f; k = dmd(@@l&f; k) (1 8) for k = 1,2, . . .f{,

where d is the antisymmetrizing projection operator,

That the functions Y ~ M ; k are in fact of the form of equation (7) can be easily demonstrated by substituting equation (19) into equation (18) and making use of equation (17). One obtains the following alternative form for the approximate wavefunction [equation (1 S)] :

in which the operators WE are given by

Except for a trivial normalization factor, these are just the usual group theoretical projection operators. Equation (20) may now seem to be of the same form as equation (7).

The most general wavefunction that can be formed from a given approximate spatial function @ is a linear combination of the functions Y&;k:

fg %,= c Ck%,,; x-7 (22)

k= 1

the coefficients ck being determined by solving the secular equation

in which Eo is the normalized expectation value of H given by the function (22). For this purpose it is necessary to calculate the matrix elements

and

Vdence Bond Theory 65

Substituting expression (18) into these equations, noting that H and d commute and that d2=d, and using equations (17) and (13), one arrives at the expressions

and

These expressions illustrate at once the basic difficulty of any general N-electron theory such as VB theory, for unless some assumptions are made about the form of the function @, each expression consists of N! terms. This is a matter to which considerable attention will be devoted in this article. However, there is to date no satisfactory general solution to this problem.

Construction of Spin Functions.-It can be seen from expressions (26) and (27) that, besides the various integrals required, the basic group theoretical quantities which one needs are the matrices US(P). The form of these in turn is determined by the way the set of spin functions of equation (9) are constructed. We note that the form of the wavefunction (7) is unchanged by any simultaneous unitary transformation of the functions and @g k. There is therefore an infinite number of possible bases of spin functions, a specific choice often being dictated by the particular problem under investigation.

The simplest method of constructing the functions @gM;k is by coupling together successively the spins according to the usual rules for coupling angular momenta in quantum mechanics. The index k on the spin functions in this basis may then be thought of as a set of partial resultant spins,

k= (Sl s2 . . . sp . . . S N - 1 )

in which S, is the resultant spin of the function after coupling together the spins of the first p electrons. Thus SI must always be 3, and it is unnecessary to specify SN as this is just the total resultant spin S. This basis is very common, and we shall refer to it simply as the ‘standard basis’. The totality of spin functions constructed in this way is most conveniently visualized with the aid of the ‘branching diagrarn’,s-lO and is shown in Figure 1. In this, the resultant spin S is plotted against the number of electrons, N. The integer f,# is seen to be the total number of ways of starting from N= 1,S= 3 on the diagram and arriving at a given resultant N,S. Each circle in the figure contains the value off$ for that position. We note that

For a further description of this basis and how the V ( P ) matrices in it are constructed, the reader is referred to the article by Kotani et aZ.8

a M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, ‘Tables of Molecular Integrals’, Maruzen, Tokyo, 1963. R. M. Corson, ‘Perturbation Methods in the Quantum Mechanics of n-Electron Systems’, Blackie, London, 195 1.

lo J. H. van Vleck, ‘The Theory of Electric and Magnetic Susceptibilities’, Clarendon Press, Oxford, 1932.

66 Quuntum Chemistry

t s

N - Figure 1 The branching diagram

(Reproduced by permission from ‘Advances in Atomic and Molecular Physics’, Academic Press, London and New York, 1971, Vol. 7)

Another basis of some importance is one in which two standard functions of N1 and N2 electrons, respectively, are coupled together:

N @S, M ;SISlklk,= c <s1s2M1M21sM>o~M1; k , @ Z 31,; k,, (28)

311, M2

(A41 + A42 = M; ivl + N2 = N ) ,

in which the <SiS2MiM21SM> are just the usual Clebsch-Gordan or vector- coupling coefficients.11 The spin functions in this basis, which now require the set of four indices (SlSzklk2) to specify them, are useful in describing an atomic or molecular system which consists of two well-defined subsystems with resultant spins SI and S2. This basis will be referred to as the YN, x 9 ’ ~ ~ basis, since the two spin functions on the right-hand side of equation (28) form bases for irreducible representations of the permutation groups 9~~ and Y N ~ , respectively.

The standard basis and equation (28) are connected by an orthogonal transformation of the form

(29) iv @s, M ; h?= c c (SlS2klk2ISk) @: M;S1Sek lL , ,

SI,S, k ,k ,

I * D. M. Brink and G. R. Satchler, ‘Angular Momentum’, Clarendon Press, Oxford, 1962.


the transformation coefficients (SlSzklk2 I Sk) being purely group-theoretical in nature. In this way it is possible to transform wavefunction (22) from one basis of spin functions to another (Section 3).

Similar but more complicated spin functions may be constructed by coupling together several standard functions to form an 9”, x 9 ’ ~ ~ x 9 ’ ~ ~ x . . . basis (NI + N2+ N3+. . . = N ) , depending upon how many groups of electrons it is desirable or physically reasonable to distinguish (see Section 4).

An example of a basis of this kind is the 9’2 x 9’2 x . . . 9’2 ( x 9’1) basis in which pairs of electron spins are first coupled to form singlets or triplets, the pairs then being coupled to form the desired resultant S. This is, of course, the natural basis to use when constructing pair wavefunctions (Section 4), and will be referred to as the ‘Serber basis’ since it was first used by him in VB theory.12s13 It should be noted that in this basis the matrices Us@) representing simple pair interchanges Pr-lr (p even) are all diagonal,

the positive sign referring to a singlet pair and the negative sign to a triplet pair. One should mention briefly the classical VB basis in which, for N even,

all possible pairs of electrons are first coupled to form singlets, and then +N of these pairs are taken at a time to form a total of N!/(23N(+N)!) complete singlet spin functions. Of course, only f{ of these functions are linearly independent, but there is a well-known graphical method, originally due to Rumer,l* for selecting out an independent set. The main advantage of this basis is that the elements of the Us(P) matrices in it can be determined very easily by hand. However, we shall not consider this basis any further, since the spin functions are not orthogonal and expressions (26) and (27), which are already the source of major difficulties, would have to be replaced by ones still more complicated.*

Finally it shouId be noted that if the molecular system has any spatial symmetry, i.e. if there is a point group 93’ whose operations W all commute with H, then each function @sk: in equation (7) must be replaced by a set which forms a basis for an irreducible representation A of g:

in which the letter tc denotes a specific member of the basis for A. Thus for any spatial symmetry operation 9’ of the point group 9,

for all k= 1,2, . . ,f$. * See, however, the article by Shull.16

R. Serber, Phys. Rev., 1934,45461. la R. Serber, J. Chem. Phys., 1934, 2, 697. l‘ G. Rumer, Goettingen Nachr., 1932, 377. l6 H. Shull, Pnternat. J. Quantum Chem., 1967, 3, 523.

68 Quian turn Chemistry

The dimension of the irreducible representation @I(&') is fA. Note that since H contains no spin interaction terms, the operations L%' affect only the spatial functions @gia) in equation (31) and not the spin functions.


Relationship between Valence Bond and Spin Valence Theories.-We consider first for simplicity a diatomic molecule AB. The basic physical idea behind all the variants of VB theory is that the wavefunction for the molecule, YAB, should in some way be written as a product of the wavefunctions FA, YB for particular states of the participating atoms. Thus

YAB ~ ~ ( Y A Y B ) , (33)

in which the operator at need consist only of permutations between the functions YA and YB if these latter are already antisymmetric with respect to their own electrons. If the !PA,!€% possess non-zero resultant spins, SA and SB respectively, then these must be coupled to a definite overall resultant for the molecule. Similarly, it may be necessary to form a linear combination of functions (33) for PAB to possess the correct molecular spatial symmetry. These procedures are described in more detail below and in succeeding sections.

The approximation (33) corresponds to taking as the spatial function 0 in equation (18), the product form

@(ri, r2, . . . , ~N)=@A(YI , . . . , ~ N J @ B ( Y N A + ~ , . . . , ~ N A + N & (34) where

NA+ NB = N,

in which @A,@B are approximate spatial wavefunctions for the atoms A and B, respectively. Most often, the @A,@B are taken as products of atomic orbitals, thus representing specific configurations of the participating atoms :

and similarly for atom B. If we now choose as the set of spin functions the standard basis, the most

general VB wavefunction that can be constructed from the two configurations @A,@B is, according to equation (22),

It is important to note that as long as we take the most general linear combination of spin functions as above, the ordering of the orbitals 4, is immaterial, and we are free to put all the doubly occupied atomic orbitals from both @A and @B together, followed by all the singly occupied orbitals, as has been done in equation (35). This circumstance is very convenient both for nota- tional and computational reasons. A single function from the sum (35)


does not have this property, for then the index kZ((SiS2 . . . SN-1) implies a specific mode of coupling the spins, and a different ordering of the orbitals will lead to a different function.

However, from a physical point of view, it is clear that one particular ordering of the orbitals in equation (35) is very important, for then the single coupling k=(jO+O . . . 3) (i.e. in which pairs of orbitals are coupled to form singlets) makes the overwhelming contribution to the total function YSM. Consider as an example the NZ molecule. The function (35) for this case assumes the form

YO0 = C C, 'd~ !yd ( lS~2s~ ~S~~S~~P,.~PZB~PX.~P,B~P,.~P,.'B~,; k). (3Q k

With this particular ordering of the orbitals, the coupling k= (w . . . 4) describes the (2pzA, 2pzB), (2pxA, 2pXB), and (2pyA, 2puB) orbital pairs as each coupled to form singlets, thus constituting the triple bond between the two atoms. At the equilibrium internuclear distance, this coupling is expected to form the major contribution to the total function (36) and, indeed, a VB calculation by Kopineckl6 with this coupling alone gives a binding energy of 9.261 eV* compared with the experimental value of 9.756 eV.17

Since only one spin function is possible for the eight electrons in the four doubly filled orbitals, we have, as far as the spin functions are concerned, effectively a system for which N=6, S=O, and from equation (10) we see that there are a total of five possible spin functions. From Figure 1 these may be enumerated as (again ignoring the spins corresponding to the doubly filled orbitals): 0 1 = ($1+1+), 0 2 = (+l+l+), 0 3 = (+O-$l*), 0 4 = (+ lm) , and the perfect pairing function 05=(+0$03- ) . However, owing to the Z$ symmetry of the N2 molecule, there are in fact only four independent functions since it is necessary to form the linear combination 43(03+ 0 4 ) in order to preserve the correct spatial symmetry (see Section 3, p. 71).

The role of the three spin couplings 01, 0 2 , and 43(03 + 04) in the total wavefunction (36), though certainly less than that of the perfect pairing function 0 5 , is still significant, and becomes more important as the internuclear distance increases. This may be seen more clearly by adopting, instead of the standard basis, the 9'7 x Y7 basis of spin functions. In physical terms, this means that the orbitals constituting the two configurations @A,@B are now coupled to a specific resultant spin, SA or SB, the spins of these two subsystems being then coupled to the overall resultant S. If, further, we choose @A,% to form also eigenfunctions respectively of the operators zi, eZ, and &:, &,,, then the VB function (35) is now formed from atoms

* It should be noted that this is an approximate VB calculation in which the Is electrons on the two atoms are ignored, and all permutations higher than simple interchanges are also neglected. The neglect of the 1s core electrons is probably not important, but the effect of the higher order permutations is not predictable. These may give contributions to the energy of either sign, so that inclusion of these permutations could well worsen the result somewhat. H.-J. Kopineck, 2. Naturforsch., 1952, 7a, 314.

l7 G. Herzberg, 'Spectra of Diatomic Molecules', van Nostrand, New Jersey, 1950.


in definite L-S coupled states. We denote a general function formed in this way as

In the case of our N2 example, the most general wavefunction that can be so constructed is

Ul(nl2, ~ A ~ A + ~ L A ; nl;fH,2sB+1Lp,I 2s+1dd).

Y ~ ~ = C ; ' Y ( ~ P ~ , 4s; 2p;, 4spx;) i- cLY(2pi, 2D; 2p$, 2D11Cl-)

-I- cLY(2pi, 2P; 2p;, 2PpC.g)

+ Cl { Y(2p i , 2P; 2p;, W [ l C + )

+ !P(2&, 2 0 ; 2p;, 2PllZ+)}. (3 7) There is thus a total of four linearly independent functions, the linear combination with the coefficient c; being necessary to produce the correct g symmetry.

Wavefunctions (36) and (37) are completely equivalent, and indeed according to equation (29) there exists a linear transformation between them, the coefficients in this transformation, moreover, being determined purely by group theory.* For problems in which the internuclear distance R is close to its equilibrium value, it is physically more meaningful to solve the 4 x 4 secular equation in the basis (36), since one of the coefficients (that corresponding to 0 5 ) will be far larger than the others. But as R increases, it is better to transform to the basis (37), for in the limit as R 4 CO, the hamiltonian matrix will become diagonal in this basis, the function Y(2p1, *S; 2p& 4S11C:) corresponding to the lowest energy.

In essence, then, the general linear combination of couplings in the function (36) allows one to describe the dissociation process as a smooth recouphg of the orbitals from the perfectly paired state to the atomic coupling (2p3, 4S) on each atom.

The choice of a single function from either set (36) or (37) does not permit such a useful physical interpretation, and may indeed lead to difficulties as the internuclear distance is varied. Thus if one chooses just the perfectly paired function from the set (36), as R-+ co one finds each N atom is described by a curious non-stationary state - the so-called 'valence state' of the atom, about which there has been so much discussion in the 1iterature.ls The choice of the set of functions (36) in which orbitals participating in a bond are directly coupled to each other is just the VB theory as proposed by Slater and Pauling,lg whereas the set (37) formed from atoms in specific L-S coupled states corresponds to the spin-valence theory employed by HeitIer.20 * In the general case, the actual transformation coefficients are fairly complicated quanti-

ties, since one has to take into account also the coupling of the orbital angular momenta. However, this still remains a group theoretical problem which can be solved algebraically once and for all and the results embodied in a set of tables.

l8 J. H. van Vleck and A. Sherman, Rev. Mod. Phys., 1935,7, 167; W. E. Moffitt, Reports Progr. Phys., 1954, 17, 173; G. Doggett, Theor. Chim. Acta, 1969,15, 344. J. C. Slater, Phys. Rev., 1931,37,481; 1931,38, 1109; L. Pauling, J. Amer. Chem. SOC., 1931, 53, 1367; Phys. Rev., 1931, 37, 1185.

ao W. Heitler, Marx Handb. d. Radiologie, 1934, 11, 485.


Note that, within the framework of the spin-valence theory, one could choose more complicated atomic functions (PA,% in equation (34) so that

and (38)

are either exact eigenfunctions of their respective atomic hamiltonians, or are at least good approximations to the exact atomic solutions. One hopes in this way to prevent errors inherent in the description of the atoms from propagating into the molecular calculation. This is just the motivation behind the atoms-in-molecules methods which is discussed in Section 4.

v s A , M A ; k A = 1 / N A ! d ( @ A @ F t , MA; kA)

p S B , M B ; k g = 1/NR!d(@B@fz, M g ; kB)

Spatial Symmetry in VB Theory.-We take as our VB wavefunctions the set given in equation (39 , in which the orbitals constituting the spatial part are all atomic in nature and are centred on one or other of the nuclei forming the molecule. In order to keep the discussion as simple as possible we confine our attention at first either to molecules of the form ABn in which several equivalent atoms are bonded to a central atom, or to the electrons in planar conjugated molecules which one usually thinks of as forming the conjugated system. These are general enough examples for many actual situations, and in any case the arguments given here can easily be generalized to more complicated circumstances.

The point symmetry group of the molecule is denoted by %' (Dnh or Cnv in the present case), and it is necessary to produce from the functions (35) wavefunctions which form bases for irreducible representations A of '3. We note first of all that since all the orbitals are localized on one or other of the atoms forming the molecule, the application of a spatial symmetry operation W of 3 is equivalent to a permutation of the orbitals on the equivalent atoms amongst themselves, possibly multiplied by a rotation of the orbitals on the central atom. Hence with every operation 92 we may associate a certain permutation of the orbitals, PR, in which the bar emphasizes that one permutes the orbitals themselves and not the electron co-ordinates. Thus,

FMrl) $2(r2) . . . q j ~ ( r ~ ) = $pl(r1) $pZ(r2) . . . 4pN(riv), whereas

P'Mr1) 42(r2) . . . $iv(rN)=$l(rP1) $2(rPz) . . * $ N ( r P N ) ,

(39)

(40)

so that Pr and P commute, and P = P - 1 . This connection between spatial symmetry operations and permutations of orbitals was first used in VB theory by Serber,I3 and has since then been developed extensively by Kaplan in a series of papers21922 and also in a recent book.23

l1 I. G. Kaplan, Liet. Fiz. Rinkinys, 1963, 3, 227; Teor. i eksp. Khim., 1965, 1, 608, 619; 1966, 2, 441; Zhur. eksp. i teor. Fiz., 1966, 51, 169 (Sov. Phys. JETP, 1967,24, 114).

la I. G. Kaplan and 0. B. Rodimova, Zhur. eksp. i teor. Fiz., 1968,55,1881 (Sov. Phys. JETP, 1969, 28, 995); Teor. i eksp. Khim., 1970, 6, 435.

'* I. G. Kaplan, 'Symmetry of Many-Electron Systems', translated by J. Gerratt, Academic Press, New York, 1973.

72 Quan turn Chemistry

We may disregard the closed-shell cores of the atoms since these play no role in the construction of symmetry-adapted wavefunctions, and concentrate attention upon the valence electrons. In the simplest case, with one valence electron per atom, we have a configuration 4142 . . . 4~ of N singly- occupied, non-degenerate valence orbitals which is then said to form a covalent structure for the molecule. Then under any spatial symmetry operation 9, a VB function ?Ps M ; k transforms as

B P S M ; k = d x d ( p ~ 4 1 4 2 . . . # ~ @ g 111; k)

= dmd(P&- ‘4142 . = dmd(4142 . 4 N E p R p g @ $ M ; k)

. $,@: ;&lk)

= c U W n ) Y s Y l f ; l , (41) 1

the first equality resulting from the commutation of PR with any permutation QrQ‘ and hence with&, and the last equality from equation (17). In order to obtain a set of spatially symmetry adapted VB functions, we apply to PSM; the projection

In this expression, g is (41), we obtain

operator

the order of the group 3, and making use of equation

The coefficients a(SZ I A&) are given 23

from which it can be seen that they are purely group-theoretical in nature and so can be determined algebraically.

The set of matrices US(PR) in equation (41) form a reducible representation of the group 3, which is reduced into its irreducible components A by the coefficients a above. If we denote the irreducible representations of 9~ and 3 by Ufs9 Nl andD@), respectively, then this reduction can bewritten symbolically as

The coefficients csn determine the number of times a particular molecular multiplet 2s+lA occurs in this decomposition. This is just the dimension of the secular equation (23) which has to be solved in the symmetry-adapted basis. The csA can be determined in the usual manner from character tables for the groups YN and 9’*. * Character tables for the permutation groups have been given by Lyubarskiiz4 and by

2 4 G. Y. Lyubarskii, ‘The Application of Group Theory in Physics’, translated by S. Kaplan.z3

Dedijer, Macmillan (Pergamon), New York, 1960.


As an example, we consider the six electrons in the benzene molecule which form the conjugated system. These are accommodated in six 2pz orbitals 61, $2, . . . , $6. The symmetry point group is &h, but since the 2pz orbitals all change sign under the reflection bh, we regard the point group as being simply D6. We adopt the standard basis of spin functions, and on carrying out the reduction (45) for S= 0, 1,2, and 3, we obtain the following multiplet states :

21Alg, lBzu, 1E2g; 3A2g, z3B1u, 23E1u, 3E2g; 5Alg, 5 E ~ u , 5E2u; 7 B ~ u .

By using equations (43) and (44), we derive the following symmetry-adapted VB functions for the singlet states :

y(a1o) 0 , 0 ; 1 - - 1 / & p % i - d3p3-k \ / 3p4+53V5)

ph$Pd= 1/-&(5pi-21/2p2+ 1/6p3+ 1/6Y4)

'Y$,B$'i= 1 / 2 - ( - 3 y z - 2/3y3- 2 / 3 y 4 + 3 y 5 ) ,

Y $ , ~ ~ ~ ~ ) = 2/2s(2/6Y1+ 1 / 3 Y 2 - 3 3 3 ) ,

p ~ ~ $ ' ~ ) = 1 / ~ - ( - 1/2p1- p2+21/3p3- 1 / 3 y 4 ) ,

and

in which p k stands for Yoo;~, and the spin functions are numbered as described in Section 3, p. 69. The most general covalent wavefunction for the ground state is then a linear combination of the two 1Als functions above.

It is perhaps worth remarking that had we chosen, instead of the standard basis, the Rumer basis of spin functions, then the five VB singlet covalent functions are just the two well-known KekulC structures and the three Dewar structures.

The extension of this method to the case of a molecule of the form ABn, taking into account several valence electrons on the central atom, is straightforward. We consider q such valence electrons stemming from a configuration (n2)q on atom A (1 <q< 22+ 1). The set of covalent wavefunctions are now written as

Y(SMkImlm2 . . . mq)= 1/Rd[#g\#g2 . . . #)&,+, . . . $@g Lv; E ] , (46)

in which the quantum members mi ( i= 1,2, . . . , q) may vary from - 2 to 1. On applying the projection operator (42) to functions of this kind, we obtain the symmetry-adapted functions

Y(nqSMkImlm2. . . mq)

= C C a(Sjm;mL . . . mJAakmlm2 . . . m,) Y(SMj(m;mi. . . mi). mi,mg,. . . rnir j

(47) The coefficients on the right-hand side of this equation are given by

a(Sjm;mi. . . m;lAukmlm2. . . m,)

(48) w


in which the Dam(W) are elements of the matrices for the irreducible representation D(1) of the three-dimensional rotation-reflection group 0 3 . In order to carry out the reduction analogous to equation (43, it is necessary to derive the characters of the representation of 99 which is generated by the set of functions (46). This has been done in very general form by Kaplan and Rodimova,Z2 and adapting their method to the present case we obtain for an operation 9 of g,

%(8) = %IS* q P R ) [ X ( y W ) ] q , (49)

in which the symbols are self-explanatory. Had we chosen, instead of the VB functions (46), the spin-valence form in which the orbitals of atom A are coupled in a definite L-S state:

P(SMk[(nl)qLML)= 1/N!d{@[(n l )qLM~]$~,1 . . . $N@$ (50)

then, in place of equation (49) we obtain the somewhat simpler equation

We recall that the spin functions in equation (50) are from the Sqx Y N - ~ basis, the index k standing for (SlSzklkz).

As an example, we consider the H2O molecule (point group C2,) constructed from O(2p4) and a 1s orbital on each H atom. From equations (51) and (45) we obtain the states: 4lA1, 2lA2, lB1, 2lB2; z3A1, 33A2, 43B1, 33B2; %, 5A2, and 5B2. There are thus four 'A1 covalent VB structures in equation (35).

In general, the spin-valence theory provides a very elegant method of constructing symmetry-adapted wavefunctions as shown long ago by Kotani2 but has, unaccountably, been hardly used. Consider as an example the CH4 molecule (point group Td). We ignore the ls2 electrons on the C atom, and choose the Y 4 x 9'4 basis of spin functions. Thus the molecule is regarded as being built from two groups of electrons: a group of 4 1s electrons from the H atoms placed at the corners of a tetrahedron, and 4 valence electrons from the C atom. Denoting the 4 lsH orbitals as hl, h2, h3, and h4, we form the functions

However, we now note that the two groups Td and 9 4 are isomorphic. Hence the representations (41) of Ta generated by the functions (52) are already irreducible and correspond to the states lE, 3 T ~ , and 5A2.

Since the symmetry of the ground state of methane is 1A1, we may only couple to the functions (52) those from the C atom which possess the same symmetry. We thus obtain the following seven spin-valence functions derived

2 5 M. Kotani, Prac. Phys.-Math. Sac. Japatr, 1937, 19, 460.

Valence Bond Theory

from the 2s22p2, 2s2p3, and 2p4 configurations of the carbon a t o m 9

R = Y[s2p2, 3T1(3Pg) ; h4, 3T~ I %I, Y2= Y[s2p2, lE(1Dg); h4, 1EI1A11,

Y4 = [sP~, 3~1(3&) ; h4, 3T1 I 1 4 ,

y3= [sp3, 5A2(5Su); h4, 5A211Ai],

y 5 = P[sp3, 'E(lDu); h4, 'EIIA1],

y 6 = [p4, 32"1(3pg) ; h4, 3T1 I 'Ail,

Y7= b4, lE(lDg); h4, 1El1Ai],

and

75

in which the principle quantum number has been dropped from the C atom configuration in order to simplify the notation somewhat, and the state of the isolated C atom is indicated in brackets. A linear combination of these seven functions may well provide an excellent description of the ground state of the molecule for almost any nuclear configuration, the functions P3, Y 4 , and Y5 no doubt playing the major role close to the equilibrium configuration. No such calculation has so far ever been carried out, owing of course to the N! problem in the calculation of the necessary matrix elements of equations (26) and (27). However, there is now no reason why this should not be done (p. 91). It might be pointed out here that a general linear combination of the seven functions above is invariant under any linear transformation of the 2s and 2p orbitals amongst themselves.* These may therefore be taken to be orthogonal without any loss in generality, thus reducing substantially the N! problem. Moreover, this invariance, as pointed out by Kotani and Siga,26 means that the total wavefunction remains unaltered if one transforms the carbon 2s and 2p orbitals to the tetrahedrally hybridized representation (p. 80). One can then examine how good or otherwise is the popular representation of CH4 as a single configuration of four tetrahedral hybrid orbitals each coupled to a hydrogen 1s function. A similar calculation was in fact carried out in 1953 on the X 2A1 state of the CH3 radical by Itoh, Ohno, and KotanL27 However, this is still not a complete ab initio work since all overlaps between atomic orbitals on different centres are neglected and all multicentre integrals similarly dispensed with. The results predict that the molecule is slightly non-planar (which is at variance with the experimental evidence presently available), and that the hybridized electron-pair representation is only a fair approximation.

Improvements in the VB Description: Ionic Structures; Hybridization.-We have so far considered the problem of coupling together the atomic orbitals. from various configurations of the participating atoms so as to form a

* The presence of the p 4 configuration, although very high in energy, ensures this invariance. M. Kotani and M. Siga, Proc. Phys.-Math. SOC. Japan, 1937, 19, 471.

I' T. Itoh, K. Ohno, and M . Kotani, J. Phys. SOC. Japan, 1953, 8,41.


molecular wavefunction which possesses the correct symmetry. We now turn to the question of how to allow for the fact that, under the influence of chemical bonding, the atomic wavefunctions themselves distort.

This object is traditionally achieved in VB theory by taking into consideration ionic VB structures in which the electrons are redistributed amongst the valence orbitals, some of these now being doubly occupied. The simplest example is, of course, the H2 molecule, where one adds to the Heitler-London covalent function

yl= 2 / 2 d ( $ a $ b @ i , 0 ; I)= ‘d/f(&$b -k $b$a) @:, 0 ; 1

y 2 = d?d$arba -t #b$b) @8,0; 1

(53)

the ionic configuration

(54)

suitably symmetrized so as to possess the required E: symmetry. The linear combination clYl+ c2YJ2 was first considered by Weinbaum28 in 1933. However, in 1949 Coulson and Fischer29 pointed out that this combination is in fact equivalent to a single configuration of the kind (53) in which the functions #a, #b are replaced by two non-orthogonal orbitals #:,#; of the form :

$;=$a+ A$b; &= A$a+#b, (55)

in which A is a variable parameter. The &,$; functions are essentially distorted atomic orbitals, and possess the same symmetry properties as the undistorted orbitals: they are each invariant under the group C,,, a subgroup of the full molecular point group Dmh, and are reflected into each other by the operation m.

It is perhaps worth emphasizing how good a wavefunction the Weinbaum- Coulson-Fischer function (WCF function) is. The total energy given by this function, which contains only one linear variable parameter, is - 1.14796 a.u. and yields 85% of the observed binding energy. By contrast, the molecular Hartree-Fock function gives a total energy of - 1.133629 a.u. and yields 77% of the observed binding energy. If one adds to the WCF function a third configuration constructed from 2px orbitals (but with exponents adjusted so as to equal roughly the 1s exponent values) the resulting wavefunction is now essentially the Hirschfelder-Linnett function,30 which yields an energy of - 1.1561 a.u.* (90% of the observed binding energy). This wavefunction, moreover, yields a potential energy curve which remains closely parallel to that derived from the observed vibrational levels over a very large range of the internuclear distance R.30

The Weinbaum and Coulson-Fischer functions, though formally identical, have provided the prototypes for two somewhat divergent developments in

* This energy is a corrected value given by Eliason and Hirschfelder.”““

as S . Weinbaum, J. Chem. Phys., 1933, 1, 593. a D C. A. Coulson and I. Fischer, Phil. Mag., 1949, 40, 386. 30 J. Hirschfelder and J. Linnett, J . Chem. Phys., 1950, 18, 130. 30a M. Eliason and J. 0. Hirschfelder, J. Chem. Phys., 1959, 30, 1397.


VB theory. Thus the Weinbaum function has led to the realization of the importance of taking into account many structures, both ionic and covalent, in a VB calculation, similar to configuration-interaction (CI) calculations in MO theory. On the other hand, the Coulson-Fischer form has led to the development of the model in which each valence electron is accommodated in a distinct orbital, the valence orbitals all being non-orthogonal to one another and having the form essentially of heavily distorted atomic orbitals. The multistructure approach is surveyed in this section and the configuration of non-orthogonal orbitals theory is discussed in the following section.

We continue to ignore the core electrons and consider N valence electrons distributed amongst the valence orbitals in configurations of the form

in which m of the orbitals are now doubly occupied. There are a total of

such configurations K @ ) , and for a given spin S they give rise to the following number of VB structures:

The total number of structures with this resultant spin is

It is necessary to form from this large number of functions

YSM; k [K(m) l = 2 / T ! s $ ( @ [ ~ ( m ) ] @,f a,; k) (58)

linear combinations which belong to definite multiplet states z S + l A of the molecule. This problem was first considered by Craig.32 However, for our purpose, we make use once more of the projection operator SL.l,> [equation (42)], and obtain, from equation (58),

In order to make use of this expression, one now divides all the configurations K ( m ) into sets such that the configurations within any one set are all transformed into one another by the operations 3' of the point group 9f. The number of configurations in a given set depends upon the symmetry possessed by the K ( m ) [it is clear that all the K ( m ) within a set must have a common value of m].

G. J. Mulder, Mol. Phys., 1966, 10,479. a s D. P. Craig, Proc. Roy. SOC., 1950, A200, 390, 401.


If a particular configuration K(m) does not possess any symmetry under 9, the operations W generate g other, similar, configurations so that the set consists of g members which form a regular representation of the group. The application of equation (59) to this case is then straightforward, the desired symmetry-adapted functions consisting of linear combinations of all g functions, the coefficients being simply (fA/g)1/2D&t)(9?).

Other configurations K ( m ) may possess some symmetry under 9, i.e. there is a subgroup2 of 9 under the operations of which some of the functions (58) are invariant. In such cases a given K ( m ) is converted into other configurations in the set by operations ga of 9 which are not contained inyf. The number of configurations in such a set is therefore equal to (g/h), where h is the order o f z . In this instance, equation (59) may be written in the form

If we assume the simplest possible case in which the orbitals comprising K(m) are all non-degenerate, the coefficients on the right-hand side of equation (60) are given by

The characters of the representation of 9 which is generated by the functions (58) are just2lY23

where ~ ( 9 ~ ) is the number of configurations K(m) left unchanged by the operation Wa.

As an illustration of this method we consider once more the six 2pz orbitals which are used to describe the conjugated electrons of benzene. We ignore the operation crh and continue to regard the molecular point group as D6. There are four types of configuration: the covalent configuration K(O) considered in the last section, and the ionic configurations K(Q, K(3) which we denote by p4p+p-, ~ 2 ( p + ) ~ ( P - ) ~ , and ( P + ) ~ (p-)3, respectively. We examine in detail just the p4p+p- types of configuration. According to equation (56) there are a total of 30 such configurations and, as shown in Figure 2, these can be divided into 3 sets which do not mix under the operations of Ds. Configurations of types 1 and 2 possess no elements of symmetry and form regular sets with 12 members in each. Configurations of type 3 are invariant under the group (E, UZ), where 0 2 is a rotation through x about the axis shown in the figure. There are hence only (12/2)= 6 configurations in this set. Application of equation (62) shows that these configurations give rise to 120 multiplets as follows:

x ( 9 ) 3 X(W,Q) = P' "(Pd~(9?d, (62)

6lA19, 41A2g7 6lB1.u~ 4lBzu, 1O1Eiu, 1o1Ez~,

73Aig, S3Azg, 73Biu, S3BzU, 153E i~ , 153E2g,

35Aig, Z 5 A 2 g , 3 5 B i ~ , Z5B2u, 55Ei~, 55Ezg.


Figure 2 The three types of the p4p+p- ionic configuration of benzene

The symmetry-adapted functions themselves may then be constructed directly from equations (59), (60), and (61). The same procedure can be applied without any difficulty to the other types of ionic configuration. The results show that there are altogether a total of 268 multiplets of which 22 correspond to lAlg states. A similar result, of course, would be obtained from MO theory with full configuration interaction.33

This treatment can be easily extended to more complicated configurations K ( m ) consisting of several valence orbitals &!$) situated on the various atoms, and a general formula can be derived for the characters of the representation generated by the corresponding VB functions. For this purpose it is simplest to adopt the spin-valence scheme so that each atom comprising the molecule is characterized by a specific resultant spin S, and resultant orbital angular momentum La. The permutation of the atoms PB corresponding to the symmetry operation W is broken down into a product of cycles of length 1,2, . . . , k :

the notation indicating that pse consists of n1 cycles of length 1, n2 cycles of length 2, . . . , etc. All atoms within a cycle ni must possess a common spin Si and resultant orbital angular momentum Li as otherwise the character of the operation, x(W), is zero. The atoms within each cycle are coupled to a resultant spin for the cycle, S t ) , and these are then finally coupled to the overall spin for the molecule, S. With this coupling scheme, the formula for the character assumes the form

. P 9 ~ ( 1 ~ 1 2 ~ 2 . . . k n k ) ,

x(@= { c c WnJS1, S(1)] x(L1) (B.1) . . . SU), ... .,S(Q S a t

. . . wnk[Sk, Sck)] x(Lk) ( 2 . k ) z(@), (63) i in which the summations are over all the S(i) and also over all the allowable intermediate spins, Sint, which occur as the S(C) are successively coupled.

L. F. Mattheiss, Phys. Rev., 1961, 123, 1209.


The symbol z ( 9 ) denotes, as before, the number of VB functions which remain unchanged by the operation 9. The details of the derivation of this formula are described in reference 22, in which the authors also give a table of the permutational factors wlZi [Si , SCi)]. As an example of the application of equation (63), we consider all the VB structures that can be constructed for the HzO molecule from three 2 p orbitals on the oxygen atom and a Is orbital on each hydrogen atom. There are six types of configuration: the covalent configuration O(2p4) H( 1 s) H( 1 s) (which gives rise to the multiplets described earlier), and the ionic configurations O(2p5) H( ls) , O(2p4) H( ls2), O(2p3) H( ls2) H( ls) , 0(2p2) H(ls2) H(1s2), and 0(2p6) . The characters of the representations of the point group C2, which is generated by these functions are obtained from equation (63), and on reducing these into their irreducible components, we derive the following 100 multiplets: l8lA1,

The equations (59)-(63) solve the formal problem of which and how many multiplets 2s+1A arise from a given set of covalent and ionic configurations, and of how to construct the corresponding symmetry-adapted VB functions. However, one seeks to express at least some part of an expansion in many VB functions in a more compact and, hopefully, physically suggestive form. This is essentially the motivation behind the introduction of hybrid orbitals.

In order to illustrate this argument, let us consider a linear molecule of the form AX2 in which the ground-state configuration of the valence electrons on the central atom A is of the type ns2npq (q = 0, 1 , . . . , 6), or s2pq for short. We assume for simplicity that the ligand atoms X each possess only one non-degenerate valence orbital, Z1 and 12, respectively. We represent the electronic state of the molecule by a linear combination of VB wavefunctions (58) constructed from the configurations (s2pq1112), ( s ~ ~ + ~ Z I Z ~ ) , and (pq+2Z112) in the first instance, the functions being suitably adapted to the Dmh symmetry of the molecule. However, as is well-known, we may form from the s and one of the p orbitals the orthogonal transformed functions

10U2, 12%, 10%; g3A1, 12%, 123&, 123B2; 5A1, 2 5 A ~ , z5Bz.

and

which are reflected into each other under the operation oh, and then construct VB functions from the configuration (pqd1d~ZlZ~). This single configuration represents to a fair approximation the linear combination of the three original configurations, as can be seen by substituting in it the explicit forms (64) of the two hybrid orbitals d1,dz. Moreover, the function

where the coupling k corresponds to the state in which the two pairs of orbitals (d1, Zl), (d2, Z2) are each coupled to form a singlet, has the simple visual interpretation of an electron-pair bond between each ligand X and the atom A. This function by itself is often taken to represent the electronic


state of the molecule, the other possible couplings between d1, h, d2, and 12

being said to be negligible. This may well be so, but by taking only the single function (65) corre-

sponding to the perfect pairing approximation, as the ligands 11 and 12 are removed the atom A is left in an ill-defined non-coupled state which is said to constitute the ‘valence state’ of atom A. If, however, we form the most general linear combination of spin couplings from equation (65), then, as 21 and 22 are removed, it is fairly easy to convince oneself that the atom A is left in a definite stationary state corresponding to the coupling

which is, in general, an excited state of A. This state may then be regarded as the ‘valence state’ of A. A similar conclusion, though employing more complicated arguments, has been reached by Craig and Thirunamachand- ran.34

These considerations are well illustrated by the BeH2 molecule. Denoting the 1s orbitals on each of the hydrogen atoms by hl and hz, we represent the VB function involving a linear combination of the configurations (s2h1h2), (sphlhz), and (p2hlh2) by the functions constructed from the single configuration (dldzhlha). The most general linear combination of spin couplings for this case (S=O, N = 4 ) involves only two couplings, k=(+l+) and k=(+O+) in the standard basis, and as the H atoms are removed, the function

leaves the Be atom in the (sp, 3Pu) state.* (Recall that in such a general linear combination, the ordering of the orbitals is immaterial). An interesting recent calculation by Mitchell and Thirunamachandran35 (see also McLagan and Schnuelle36) shows that the single perfectly paired structure gives a minimum energy of - 15.7613 a.u. when the BeH distance is 2.55 a x . This is very close to the minimum energy of - 15.7617 given by the linear combination of the three configurations (s2h1h2), (sphlhz), and (p2h1h2) at virtually the identical Be-H distance of 2.56a.u. Thus the perfect pairing approximation, for this molecule at least, seems well justified. Distortion of the s andp orbitals on the Be atom owing to the bonding is taken into account by including in the calculation ionic configurations of the form (sh:h2),

* Note that we may also represent the hybrid orbitals as d;=(l +A*)-”’(s+Ap), d;=(l +A*)-112(s--Ap),

which have the same directional properties as equation (64) but are non-orthogonal. VB functions constructed from the configuration (d;d;h,h,) will now predict the correct ground-state dissociation products, since as the H atoms are removed, the parameter A+O. See p. 88.

‘4 D. P. Craig and T. Thirunamachandran, Proc. Roy. Soc., 1968, A303,233; 1970, A317, 341.

a * K. A. R. Mitchell and T. Thiranamachandran, Mol. Phys., 1972, 23, 947. R. G. A. R. McLagan and G. W. Schnuelle, J. Chem. Phys., 1971,55, 5431.


(ph fh~ ) , the coefficients of which are larger than those of the covalent configurations (s2hlh2), (p2hlh2). The complete VB calculation includes the six configurations (s2hlh2), (sphlh2), (p2hlh2), (sh;h2); (ph:h2), and (h;h;) (out of a possible total of 20 functions for these four orbitals), and yields a minimum energy of - 15.7710 a.u. at a Be-H distance of 2.57 a.u. It is interesting to note that this entire wavefunction is very well represented by a function based on the single configuration (4142hihz), where $1, 4 2 are two non-orthogonal semi- localized orbitals of the form $1 z (di + phi), 4 2 z (d2 + ph2). These are reflected into each other by the operation ~ h , the parameter p giving a measure of the distortion of the Be orbitals owing to bonding. Wavefunctions of this kind form the subject of the next section.

These considerations apply equally well to more complex molecules. Thus in the case of tetrahedral molecules, such as CX4, one constructs the linear combination

C k 1 / ~ ~ ( t l t 2 t 3 t 4 1 1 1 2 1 3 1 4 ~ ~ , ~ ; k) b

from a configuration based on the tetrahedral hybrid orbitals ti (i= 1,2, 3,4) centred on the carbon atom and four non-degenerate ligand orbitals. Applica- tion of equations (45) and (44) shows that there are only three linearly independent spin-couplings consistent with lA i symmetry. At the equilibrium nuclear configuration, the perfect pairing function no doubt plays the dominant role, but as the ligands are removed, the orbitals are smoothly recoupled to leave the C atom in the state (tlt2t3t4; 5Su)=(sp3; 5Su). This state, which lies N 4.2 eV above the ground (s2p2; VS) state, therefore constitutes the valence state, One expects that ionic configurations of the form (tzt3t&121&) will play an increasingly important role as the ligand X becomes increasingly electronegative.

Similarly, in the case of six-co-ordinated molecules of the form sx6, a general VB function based on six octahedral hybrids Oi (i= 1,2, . . . , 6),

Ckd/12!rc4(0102. . - O6lll2. . . z6@6%;k), k

leads to a description according to which the S atom is left in the (sp3d2; 7FU) state when the six ligands are removed. This state of the sulphur atom has not been observed, but a Hartree-Fock ca l~ulat ion~~ predicts that it lies at 24.48 eV above the ground (s2p4; 3.Pg) state.* This seems an improbably high value for a ‘valence’ state, and indicates that a description of the bonding in such molecules based on the corresponding (sp3d2) configuration of the central atom is not likely to be useful. The whole question has been discussed in a recent book,38 in which it is pointed out that whereas SH6 is unknown,

* A similar calculation state predicts that it lies at 7.21 eV above

8 7 D. W. J. Cruickshank, B. C. Webster, and D. F. Mayers, J . Chem. Phys., 1964, 40,

81 G. Doggett, ‘The Electronic Structure of Molecules: Theory and Application to

on the S(sapsd; the ground state, compared with the observed separation of 8.38 eV.

3733.

Inorganic Molecules’, Pergamon, Oxford, 1972.


SF6 is quite stable. This suggests that ionic configurations, which cannot safely be neglected even in BeH2, now play a dominant role. The bonding in such molecules as SF6 can then be understood on the basis of the following description, which does not require any participation of d orbitals. The dominant configurations are assumed to be of the form

in which two electrons have been transferred from S to ligand orbitals. The two X- ligands are assumed to occupy cis positions around the central atom, as shown in Figure 3. Such a structure is then invariant under the operations

X

X

X Figure 3 Ionic configuration of an sx6 molecule

E, oh, oV, and CZ (i.e. under the operations of the group Gv), and since the order of the molecular point group, Oh, is 48, one can construct from each of the three configurations (66) a set of (48/4)= 12 equivalent structures. These are then combined according to equations (60) and (61) into the required function with lAls symmetry. Now, a linear combination of the three configurations (sp3), (s2p2), (p4) which occur in equation (66) can be represented by the single hybridized configuration (dldzp2), particularly if all allowed spin-couplings are taken into consideration. This configuration, moreover, provides for the correct disposition of pair bonds to the remaining 4 ligand orbitals, 13, 14, 25, and IS, without any participation of d orbitals being necessary.

In Table 1 are collected details of the major applications of ab initio VB theory over the past decade or so. A number of general comments may usefully be made. For a given basis set, a properly optimized VB calculation with non-orthogonal orbitals converges on to the result given by full interaction of all possible structures considerably more rapidly than a calculation

84 Qlcantum Chemistry

by MO theory with CI. This appears to be true both for ground and excited states. For example, in the case of the CH molecule, minimum basis set LCAO-MO-SCF calculations actually predict the wrong ground state ( 4 X instead of 2n) ,42 this only being corrected in the Hartree-Fock limit.

Table 1 Recent major applications of ab initio VB theory

Molecule Remarks

BH X 1Z+, 311, 1II, 3 C+, C-, A, C-, 3A, C, 511 states calculated; minimum basis STOs, best atom C values; full interaction of all structures obtainable to leave 1s; ; Hurley i.c.c.a applied to all states; E= -25.110 (-25.150)yh*c Dez2.22 (3.29) eVb*c for X 1Cf state.

X I Z f state; minimum basis STOs, Slater P values; 5 structures; i.c.c.a applied; De=2.10 (2.72) eV.bgC

311. 1II. 3 2 - states: basis set as in ref. 39: 7, 6. and 5 structures respectively for the 3 states; i.c.c.a applied. .

6 structures used; i.c.ca applied; DO= 3.39k0.04 eV; P.E. curve calculated, showing a maximum.

X 2II, 4C-, 2A, 2C-, 2C+, 211 states calculated; minimum basis as in ref. 39, but AOs taken to approximate Hartree AOs for C(s2p2); <lsc I h H > assumed zero; structures arising from (s2p2), (sp9, and (p4) configurations on C, no ionic configurations; De=2.60 eV for X 211 state;" excitation energies: 4 2 , 0.12; 2A, 2.99 (2.87); 2C-, 3.41 (3.22); 2 9 , 4.15 (3.95);e 211, 7.93 eV.

X 2II state; basis set as in ref. 39 but with Slater C values; 6 structures; i.c.cSa applied; De= 1.60 (2.94) eV.b*"

4C-, 2A, 2Z+, 2C- states; basis set as above; 7, 6, 6, and 7 structures used, respectively, for the 4 states; i.c.c. applied ;a

excitation energies: 4C-, 0.60;a 2A, 2.83;a 2C-, 3.36;a 2X+, 3.97 eV.a

basis set on C, 1s on H; exponents optimized; 10 configurations used: 24 structures for X 211 state (including all spin-couplings), 1 1, 1 1, 12, 18 structures respectively for the other states; E= -38.2269 a.u.; De=2.362 eVf excitation energies: 4Z-, 0.17; 2A, 3.59 (2.87); 2C-, 3.39 (3.22); 2E+, 4.86 (3.95) eV.e

X 2TI state; minimum basis of STOs, Slater C values; 102 structures; E= - 38.4255 a.u. ; spin densities at nuclei calculated as CIH= -21.54, ac=42.73 G.

state, E= -37.917858 a.u., De=3.30 (4.11) eV.e

A 1II state; STO basis as in ref. 39, with best atom C values;

CH 42

40

41

X ZII, 4X-, 2A, 2C-, 2C+ states calculated; double4 STO 43

CH+ X l X + , lII, 3II , 3X;f states; 37 structures included; for ground

Ref.

39

40

41

4

44

45

3@ K. Ohno, J. Phys. SOC. Japan, 1957, 12, 938. 40 A. C. Hurley, Proc. Roy. SOC., 1958, A248, 119.

A. C. Hurley, Proc. Roy. Soc., 1959, A249, 402. J. Higuchi, J. Chem. Phys., 1954, 22, 1339.

4s N. Grun, 2. Naturforsch., 1967, 22a, 1228. G. F. Tantardini and M. Simonetta, Chem. Phys. Letters, 1972, 14, 170.

4 K P. L. Moore, J. C. Browne, and F. A. Matsen, J. Chem. Phys., 1965, 43, 903.

Valence Bond Theory

Molecule Remarks Table 1 Continued

LiH

HF

OH

NH

Liz

He2

0 2

X1X+ state; basis: Is, 2s, 2 p o 1 2 p n ~ , 3da STOs on Li, Is, 2s, 2pu, 2pzh on H; partially optimized C values; 20 structures; E= -8.04379 a.u.; De=1.793 (2.52) eV, ,u= - 5.57 (- 5.882) Dee

X 1Z+ state; basis: Is, 2s, 2pz, 2py, 2p2 STOs on each centre; Li best atom C values; H 5 values varied; 13 structures; E= -8.009 a.u., De=2.26 (2.52) eV, p= -5.949 (-5.882) D.e

X1X+ state; minimum STO basis, Slater C values; 6 structures; i.c.c.0 applied; De=5.84 (6.12) eV,a,e, ,u=l.91 (1.82) D . e

X 1Z+ state; large gaussian basis; spin valence representation with 100 structures deriving from F(s2p5, zPu; sp6, 2Ss),

p2, 3Pg, IDg, lSs); OMj and i.c.c.a applied; Dez4.63, 5.44,j 6.19" (6.12) eV.6

XZII state; minimum STO basis, Slater C values; 5 structures; i.c.c.0 applied; De=1.14 (4.00) eV,bJ p=0.82 (2.16) D.6

2X+ state; minimum STO basis, Slater C values; 6 structures; i.c.c.0 applied; E= - 74.854 (- 75.609) a.u.;b excitation energy=4.01 (4.08) eV.e

X 3Z- state; minimum STO basis, Slater C values; 5 structures; i.c.c.0 applied; De=l.Ol (3.21) eV,b*g p=0.83 (1.94) D.6

1A, 1Z+, 3II, 1II states; minimum STO basis, Slater 5 values; 5, 5, 7, 5 structures respectively for the 4 states; i.c.c. applied;~ excitation energies: 'A, 1.76 (1.85); 1C+, 3.04 (2.90); 3II, 3.68 (3.69); 1II, 5.76 (5.80) eV.e

X 1Z$, 3 C t , 1x2, lII, states; minimum basis of STOs, fixed C values; many calculations, both VB and MO; for ground state, VB with 8 structures gives E= - 14.8614 a.u., De=0.770 (1.05) eV; for 3Z2, E= - 14.8145 a.u., De= -0.507 eV, excitation energy= 1.28 eV; for lCz, E= - 14.7912 a.u., De=0.880 (1.04) eV, excitation energy= 1.99 (1.86) eV; for l I I u , E= - 14.7409 a.u., De= -0.487 (0.31) eV, excitation energy=3.35 (2.59) eV.e

3 C z state; large double4 STO basis: Is, 2s, 2p0, 2pnh, 3d0, 3dn& on each centre; exponents varied; 12 structures; E= - 5.1 1346 a.u., De = 1.24 (2.6) eV;e P.E. curve possesses a maximum of 0.081 eV (rel. to R= a) at R=4.5 a.u.

x 1 C; state; configurations: (Iszlsi), ( 1 ~ ~ 2 p ~ ~ l ~ t , 2 p u ~ ) , (1su2pn$ lst,26n$), Hirschfelder-Linnett type function ;30

P.E. curve shows van der Waals minimum at -3.2 A. X 3C;g, lAg, 1Z;, lX;, 3 Q , 3C; states; minimum basis of

STOs, fixed C values; VB and MO calculations.

F+(sap4, 3 P g , 'Dg, IS,; sp5, 3Pu, 'Pu), F-(s2p6, 'Sg), H(ls, 2 ~ , 2S; 2p, 'Pu), H-(s2, 'S; SS, IS, 3s; SP, 'Pu, 3Pu;

85

Ref. 46

47

48

49

40

41

40

41

50

51

52

53

F. A. Matsen and J. C. Browne, J. Phys. Chem., 1962, 66,2332. 4 7 J. Thorhallsson, 2. Naturforsch., 1967, 22a, 1222.

A. C. Hurley, Proc. Phys. Soc., 1956, A69, 301. '0 G. G. Balint-Kurti, Ph.D. Thesis, Columbia University, New York, 1969. '0 E. Ishiguro, K. Kayama, M. Kotani, and Y. Mizuno, J. Phys. SOC. Japan, 1957, 12,

61 R. D. Poshusta and F. A. Matsen, Phys. Rev., 1963, 132, 307. L t N. Moore, J. Chem. Phys., 1960,33,471. La M. Kotani, Y. Mizuno, K. Kayama, and E. Ishiguro, J. Phys. SOC. Japan, 1957, 12,

1355.

707.


Molecule

N2

F2

CO

LiF

BeH2

CH2

NH3

H2O

Table 1 Continued

X 1 X ; state; minimum basis STOs, fixed C values; 6 structures,

X 1 ZJ state, large gaussian basis; spin-valence representation

Remarks

i.c.c.a applied; De= 3.29 (9.18 k0.2) eV.a* h

used with 18 structures deriving from F(sZp5; 2Pu),

OM inethodj and i.c.c.a applied; De= 1.388 (2.033;j 1.837a) eV.k

X I D - state; minimum basis of STOs, best atom C values; 8 structures with spin couplings restricted to 'valence-type coupling'; i.c.c.a applied; E= - 113.368 a.u., De= 11.00 (11.24) eV."

X 1X+ state; large gaussian basis ; spin-valence representation used with 13 structures deriving from 5 F atom configurations as for Fa above, and Li(ls22s; 2S), Li(ls22p; 2Pu), Li+(ls2; 1%; OM methodj and i.c.c.a applied; De=2.990 (6.120,j 5.33Ia) eV.I

X 1 ZC,. state (linear) ; minimum basis of STOs, 5 values optimized; 18 structures, but Be(sp) missing; E= - 15.7202 a.u.

X 12; state (linear); minimum basis of STOs, C values optimized; 6 structures; E= - 15.7710 a.u. at RBeH=2.57 a.u.; see also ref. 36.

X3B1, 1A1, 1B1 states; very large basis of gaussian lobe functions; 48 structures; energies: X 3B1, -38.9151 a.u., 8= 138 O ;

1A1, -38.864 a.u., 8=108 "; lB1, -38.833 a.u., 8 ~ 1 4 8 O ;

1A1+1B1 excitation energy 1.52 (-0.88?) eV." Large no. of properties calculated : ,u(O), O,p, diamagnetic susceptibility. Extensive review of previous calculations on this molecule.

X 1A1, 1E, 3A1, 3A2, lA2 states. Minimum basis of STOs, t values optimized; 12 structures; for ground state, E= -56.0565 a.u., p=1.919 (1.48) D;e excitation energies: lE, 24.4; 3 4 27.4; 3B, 29.0; 3E, 29.9; lB, 30.4 eV; none of these are bound.

X 1A1 state; minimum basis of STOs, best atom 5 values; 16 structures; De=4.4 (10.1) eV; p = 1.46 (1.85) D,e E= -75.703 a.u.

F(~p6; 2Sg) , F-(P'; ' S g ) , F+(s2p4; 'Dg; ' S g ) , F+(s~'; 'PZJ;

Ref. 54

5s

56

55

57

35

58

59

60

a Intra-atomic correlation correction. See ref. 60n. b I.c.c. value in brackets. C Experimental dissociation energy of BH(X IX+), De 3.58 k0.08 eV. See ref. 60b. d Experimental dissociation energy of CH(X TI), D,= 3.65 eV.l7 6 Experimental value in brackets. f Experimental dissociation energy of OH(XaII), D, =4.58 eV." g Experimental dissociation energy of NH(X "3, De = 3.9 0.2 eV.1 h Experimental dissociation energy of NB(X lE$), 0,=9.756 eV.17

Orthogonalized Moffitt (atoms-in-molecules) method. See ref. 55; and also p. 104. E Experimental dissociation energy of F ,(X lX;), D , = 1.649 eV. 1 Experimental dissociation energy of LiF(X lZ+), De = 5.883 eV.

64 A. C. Hurley, Proc. Phys. SOC., 1956, A69, 767. 6 0 G. G. Balint-Kurti and M. Karplus, J. Chem. Phys., 1969,50, 478.

A. C. Hurley, Rev. Mod. Phys., 1960, 32, 400. ~7 H. H. Michels and F. E. Harris, Internat. J. Quantum Chem., Suppl., 1967, 1, 329. 6 8 J. F. Harrison and L. C. Allen, J . Amer. Chem. SOC., 1969, 91, 807. 60 H. Moraw, Z. Naturforsch., 1966, 21a, 2062. 60 B. Kockel, D. Hamel, and K. Ruckelshausen, 2. Naturforsch., 1965, 20a, 26. eoaA. C. Hurley, Proc. Phys. SOC., 1956, A69, 49. rap. G. Wilkinson, Astrophys. J., 1963, 138, 778.

Vdence Bond Theory 87

By contrast, even a single VB structure shows the correct ordering of the levels. The semi-empirical intra-atomic correlation correction method of Hurleygl seems to be outstandingly successful, making possible on two celebrated occasions (that of N2 and CO) actually to predict the binding energies of molecules. However, some care in the choice of ionic configurations is necessary, as can be seen by comparing the 20-structure calculation on LiH by Matsen et aZ.46 with the smaller 13-structure calculation by Thorhallsson.47 Analysis of the binding energy of the F2 molecule in the calculation by Balint-Kurti and K a r p l ~ s ~ ~ is of some interest since, as is now well-known, the molecular Hartree-Fock function predicts no binding at all. In this case also, one finds that the single covalent configuration (paupab + p b p u ) predicts only 0.071 eV binding energy (compared with the experimental value of 1.649 ev), and that it is necessary to take into account both ionic configurations (pai+pa%) and configurations of the form (sapsb + S b p u ) , corresponding to sp hybridization. These three configurations can be condensed into a single configuration of the type (#u#b+#b#a) in which $a and #b are distorted sp-hybridized orbitals of the form dU+p&, and pdu +db, respectively, the hybrid orbital du being given by (1 + i22)-1/2 (pau+ Asa), and similarly for db.

Single Configuration of Non-orthogonal Orbitals.-We have shown that when hybridized orbitals are used it is often possible to condense a linear combination of several configurations into a single configuration in a physically meaningful way. The orbitals of the new single configuration are non- orthogonal and represent distorted atomic orbitals.

This was first demonstrated for the H2 molecule by Coulson and Fischer (see previous section). Their calculation was repeated by Mueller and Eyring62 who chose for the two basis orbitals #b in equation (55) the functions

#u=exP E-(w++41, #b=exP [-(v-+~)], in which p = (ru + rb)/R, v = (ru - T b ) / R , and a,+ are variational parameters. These two functions are semi-localized and represent distorted 1s orbitals ($a and C$Z, become pure Isu and 1sb functions respectively when at=B). The total energy which is now obtained amounts to - 1.1541 a.u., or N 89 % of the binding energy. This probably represents the basis set limit obtainable by a function of this form.

A similar calculation was carried out by Mueller on the HF molecule63 in which the two orbitals &, qb which participate in the bond are expressed as

so that the single configuration (#&b) represents a linear combination of the three configurations (2pa&b), (2~03, and (hg). The function #u represents a

'1 A. C. Hurley, Proc. Phys. SOC., 1956, A69, 49. 4' C. R. Mueller and H. Eyring, J. Chem. Phys., 1951,19, 1495. I* C. R. Mueller, J. Chern. Phys., 1951, 19, 1498.

#a 2pOu 4- h h b , #b !Z M p a u + hb,


2pa orbital of F which is distorted by the bonding and $b a similarly distorted H 1s function. In contrast to the Hz case, the two functions &,$b now possess the same symmetry, i.e. they are both invariant under the molecular point group Cmv. This simple function, in which the core orbitals are taken as the Hartree-Fock orbitals for atomic fluorine and all double interchanges between core and valence are neglected, yields a binding energy of 4.48 eV, compared with the experimental value of 6.12 eV,64 and the HartreeFock value of 4.38 eV.64

One is thus led to propose that the N valence electrons in a molecule be described by a single configuration of N distinct non-orthogonal orbitals (61, $ 2 , . . 9 $ N :

The orbitals $p ( p = 1,2, . . , N) are taken essentially to represent distorted atomic orbitals and hence possess the same symmetry properties as the undistorted orbitals. A general investigation into functions of this form65 shows that provided the function (67) belongs to a spatially non-degenerate state of the point symmetry group 99 of the molecule (2s+1A or 2s+lB for polyatomic molecules, 2s+1X for linear molecules), the orbitals $ p generate a permutation representation of 99 :

The factor C V p ( 9 ) is just a phase factor and is equal to -t 1. This implies the existence of a subgroup=% of 3 for which one may write a finite left coset expansion

3=*+91*+. . .+9tn*, (69)

where n=(g/h), the ratio of the orders of the two groups. The orbitals $ p

may be divided up into sets, each containing IZ members, orbital $1, say, belonging to an irreducible representation of&', $2 to a similar representation of WI&'L%i', . . . The members $1, $2,. . . , #R are interconverted according to equation (68) by the operations 91, a 2 , . . . , 9 n which are not contained in&@. This means that there is in fact a choice of symmetries for the #p corresponding to each subgroupz for which the expansion (69) is finite. However, since the physical interpretation of the orbitals is that of distorted atomic orbitals, this usually decides the choice of subgroup.

The Coulson-Fischer function for HZ serves as a simple example of these arguments, the orbitals $a,$b being taken to be invariant under the subgroup C,, of the molecular point group Dmh. By contrast, in the case of HF there is no subgroup of C m v for which the expansion (69) is finite, so that the valence orbitals are both invariant under the full group.

In the molecule CH4 (symmetry Td), the orbitals are taken to be invariant under the subgroup C3v, this being the group under which an orbital mainly

84 P. E. Cade and W. M. Huo, J . Chem. Phys., 1967,47, 614. '6 J. Gerratt, Adv. At. Mol. Phys., 1971, 7, 141.


associated with one of the H nuclei transforms as that nucleus is smoothly removed from the molecule.

In general, the orbitals in this method are expanded as linear combinations of a basis set of Slater functions in the same spirit as in the LCAO-MO-SCF method. However, in the present case the orbitals are essentially localized, and a description such as equation (67) is clearly equivalent to a linear combination of a great many VB structures, both covalent and ionic. Thus, in the case of methane this should provide a very good description of the ground state, particularly of the potential surfaces for such processes as

CH4(lAi)+ CH3(2Al) + H(2S).

A number of recent computations on molecules by this method have been collected together in Table 2. The calculations on H3 and H4 form

Table 2 Recent molecular calculations using the single configuration of non- orthogonal orbitals method

Molecule Remarks

H2

He2

LiH

X 1ZJ state; basis set: Is, 2s, 2pa STOs on each centre, partially optimized 5 values; $1 = q&; E= - 1.151 526 a.u.;a basis set nearly linearly dependent, det (S)z 10-3.

X 1Z; state; basis set : 2 elliptical orbitals, = exp (- L Y , ~ - PV), 4a=exp(-cxp+ B v ) ; E= - 1.1541 a.u.

X1C; state; basis set: Is, 2s, 2s', 2po STOs on each centre, fixed 5 values; linear combination of the 2 spin couplings. Both possible symmetries for the $4 examined. C,, symmetry found to be the most favourable, with coupling almost exactly perfect pairing; N 40 % of correlation energy obtained (rel. to Hartree-Fock).

A 1E$, CIX; states; basis of 9 contracted gaussians on each nucleus; for AIZ$, De=1.88 eV (2.5 eVC or 2.34 eV) ; potential curve possesses a hump, calc. height 0.061 eV, exptl. value 0.03-0.06 eV; for C "C,. state, De=0.643 eV, height of hump 0.22 eV.

X 1Cf state; basis set of 10 STOs, all t values optimized; perfect pairing coupling (+&> gives E= - 8.017326 a.u. (76 % of obs. De) ;O linear combination of 2 spin couplings decreases E by 3 x a.u. (ref. 72); ,u=5.645 (5.882) D;e field gradients at Li and H nuclei calculated.

5 1Z+, 4 3 X + , 3 lII, and 311 states; basis set of 16 STOs, fixed C values; calcns. all carried out at ground state Re=3.015 a.u. spin couplings optimized, but energies virtually indistinguishable from pergect pairing; excitation energy of A lZ+=3.2 eV; obs. value (at Re=4.9 a.u.)=3.3 eV; no other stable states of LiH known.

Ref. 66

62

67

68

69

70

6 6 W. A. Goddard, jun., Phys. Rev., 1967, 157, 81.

6 8 S. L. Guberman and W. A. Goddard, jun., Chem. Phys. Letters, 1972, 14, 460. as W. E. Palke and W. A. Goddard, jun., J. Chem. Phys., 1969, 50, 4524.

D. Kunik and U. Kaldor, J. Chem. Phys., 1972, 56, 1741.

C. F. Melius, and W. A. Goddard, jun., J. Chem. Phys., 1972,56, 3348. D


BH X lZ+ state; large gaussian basis contracted to 11 functions; E= -25.1801 a.u.f for linear combination of all spin couplings; indistinguishable from energy given by (w) couplings; De=3.24 eV (obs. value=3.58 eV; see footnote c of Table 1).

5 values; when RAB= 1.470 a.u., R~c=2.984 am, min. energy for 2 spin-couplings is - 1.642236 a.u. ; for single coupling (w), E= - 1.641 824 ax.; ca1c.d height of energy barrier to reaction is - 17 kcal mol-1 (exptl. value=9.8 kcal mol-l).ff

X lBlg, 3A1g states; minimum STO basis, t= 1.05; symmetry DU assumed, with R=2.54 a.u.; for general spin-coupled function, E(lBlg)= -2.057685 a.u., E(3Alg)= -2.036985 a.u. ; for single coupling (m), E(lBl,)= - 1.96926 a.u., E(3Alg) = - 1.983 395 a.u.

71

Hs X 2X state; basis: 9 STOs, 1s and 2p on each centre, fixed 72

H4 72

a Hartree-Fock energy of H is - 1.133 629 a x . See ref. 72a. b Hartree-Fock energy of LiH is - 7.9873 a.u.04

Ref. 72b. d Ref. 72c. c Experimental value in brackets. f Hartree-Fock energy of BH is -25.131 37 a . ~ . ~ ~ 0 Ref. 72d.

interesting examples of the symmetry properties of the orbitals as expressed by equations (68) and (69). In the case of H3, which was calculated in the linear configuration, when the two internuclear distances RAB, RBC are unequal, the molecular point group is C,,, and there is no choice for the symmetries of the orbitals. But when RAB=RBC, the point group is Dcah and there are now two choices, corresponding to2=Ccov or%=D,h itself. In the first case, two of the orbitals are reflected into each other by the operation oh, while the third is unique with symmetry G,. In the second case, all three orbitals are symmetry orbitals of D,h and form the configuration (o,oio,). The calculations show that at the saddle point RAB = RBC = 1.765 a.u., the Cmv choice (odou) yields an energy of - 1.59978 a.u., whereas the Dmh alternative (o,o;~,) gives - 1.62382 ax. This second choice therefore affords a better description, and moreover the orbitals in it correlate smoothly with those for unsymmetric nuclear configurations, as can be recognized by symmetry arguments. However, this calculation does not provide a particularly good description of the reaction Hz+H-+H+Hz, since it yields a value for the barrier height of N 17 kcal mol-1, compared with the experimental estimate of 9.8 kcal mol-1 (Table 2, footnote 8).

71 R. J. Blint and W. A. Goddard, jun., J. Chem. Phys., 1972, 57, 5296; see also R. 3. Blint, W. A. Goddard, jun., and R. C. Ladner, Chem. Phys. Letters, 1970, 5, 302.

7 % R. C. Ladner and W. A. Goddard, jun., J. Chem. Phys., 1969, 51, 1073. ?=aW. Ko€os and C. C. J. Roothaan, Rev. Mod. Phys., 1960, 32,205. ?nbK. M. Sando, Mol. Phys., 1971,21,439. 7*c A. L. Smith and K. W. Chow, J. Chem. Phys., 1970,52,1010. "dI. Shavitt, J. Chem. Phys., 1968, 49, 4048.


The calculation on H4 was carried out assuming the nuclei at the vertices of a square of side 2.54 a.u. (symmetry D4h). The four orbitals fall into two sets of two members, each set forming bases for non-degenerate representations of the subgroup D2h. The sets are therefore interconverted by a rotation C4. The configuration for the lowest lBls state is of the form (asbzu) (a;&), the symmetry designations of the orbitals referring of course to the D2h

subgroup . Although only a few of these calculations have been carried out for mole-

cules so far, the results in Table 2 do seem to show that they are capable of yielding useful estimates of binding energies and of other molecular properties. In addition, the model itself provides a sensible interpretation of chemical bonding in terms of a distortion of the participating atomic orbitals combined with a recoupling of the spins. The main difficulty is in the calculation of the matrix elements of the hamiltonian, since there is no orthogonality between the orbitals. Thus the BH molecule with six electrons leads to 720 terms in expressions (26) and (27), so that this is the largest system (with no orthogonality occurring owing to symmetry) that has so far been attempted. However, there now exists some hope of extending the method to larger systems, as described in the following section.

Computation of Matrix Elements.-We consider for simplicity a single function of the form

[cf. equation (58)]. Since the main difficulty in calculating matrix elements with functions of this type is due to the non-orthogonality of the orbitals, it is necessary to state first one or two simple theorems whose application will at least alleviate some of the complexity.

Firstly, the function (70) is invariant under a linear transformation of the rn doubly occupied orbitals amongst themselves. A proof of this statement seems hardly necessary as, in the case m = +N, equation (70) is equivalent to a Slater determinant, and this property of a determinant is well-known. The rn orbitals +1,+2, . . . , +m may therefore be orthogonalized amongst themselves by a linear transformation, without altering the total wavefunction. This, of course, may be done in several ways, by transforming to MOs for example, but perhaps the most convenient method is to employ the Lowdin symmetric orthogonalization method :73

in which d is the matrix of the overlap integrals <#,ld,). The advantage of the set is that it possesses the same symmetry properties as the original set 4,. Thus, without any loss in generality, the +, (p= 1,2, . . . , m) may be regarded as orthonormal, and this will henceforth be assumed to be the case. 1’ P.-0. Lowdin, Adv. Quantum Chem., 1970,5, 185.


Secondly, each of the singly occupied orbitals $ m + ~ , . . . , $ N - ~ may be orthogonalized to the core, without changing the total wavefunction (70). This fact does not seem to be widely realized, but the proof of it is very simple: let us replace each valence orbital # p (,u=m+ 1, . . . , N-rn) by its counterpart 4; which is Schmidt orthogonalized to all the core orbitals,

where Np=<$Ll#L>. Then whenever $p is replaced by 4; in equation (70), we see that only the first term in equation (72) survives, and hence the total wavefunction is changed only by a constant, N;1/2. We may therefore assume in general that all the core orbitals are orthogonal to one another, and that all the valence orbitals are orthogonal to the core. The only remaining non- orthogonality is between the valence electrons themselves.

We may therefore focus attention upon the valence electrons alone. This greatly reduces the problem in some cases but does not by any means eliminate it. Most of the methods employed to calculate the matrix elements (26) and (27) are based on writing out a function of the form (70) explicitly in terms of Slater determinants. Thus assuming there are N valence electrons, and considering only these, we have

2/N!d(45142 . . . ~ N @ ~ J L ; 1;)

= C c ( i l i z . , . iiv(SMk) 1/N%d(41$2.. . $Arci1ofZ. . . of,\-) i l , i 2 , . . iN

= C c(i l i2. . . iivlSMk) D(iii2 . . . irv). (73) i l , i 2 , . . . i N

In this equation Ui lL is the spin of electron p and may be either a: or /?, and D(i1i2. . . iy) represents a Slater determinant constructed from the set of spin orbitals y ip =

D ( i h . . . iN)= d N ! d ( y i l y i 2 . . . y ~ ~ ) . (74)

The coefficients c(i1i2 . . . iivlSMk) depend of course on the coupling scheme, but may be assumed known. Thus, for example, for a system of four electrons with S=O we have, from the formulae given by Kotani et aZ.,8

2/4!d(4142d344@$,0 ; + I t )

= d&D(aa:BP) + D(BBaa)>

- d&ID(.SP4 + W B a : P ) 3- W a P 4 + D(,sol.B)>, and

1/41d(414243$4@8,0 ; f**)

= + { W B a : P ) - W P P 4 - m.4 + mw.)). The problem then reduces to the calculation of matrix elements between


Slater determinants constructed from non-orthogonal spin orbitals. If we now denote the matrix of overlap integrals <y, ]yv) between spin orbitals by r, then we obtain

<ole)= EP<Pyl- - . ? f )N lY l . . - Y N ) P C Y N

= det (r) (75)

[cf. equation (27)], and

(Dl3f lD>= Ep<Pyi . . . Y i v I x I Y 1 . - . Y N > P C Y N

N

p,w=l = c w+)<Plhlv>

N N

+ c c r(PI at)(<Plgl at> - <vi.lgl at>). , u > v = l o > t = l

(76)

In these equations, r(pIv) represents the first cofactor of (r), obtained by striking out the pth row and vth column from I-', and r(pvlat) the second cofactor analogously defined. The terms (plhl v>, <pvlgl or) denote one- and two-electron integrals over the spin orbitals yp, yv, yo, and yt. Formulae (75) and (76) were first obtained by L O ~ d i n . ~ ~ However, it is important to note that these cofactors are all related to one another,

N

t = l (t. f v )

N

v = l

r ( p l v)= C J'(pol v t )<a lz> , (a= 1,2, - . . N ; # p ) (77)

det(r)= C PI v)<pl v>, (p=1,2, . . . , A 9 (78)

in which <pl v> = r,, represents the overlap between the relevant spin orbitals. Thus it is only necessary to determine the second cofactors of I-', the other quantities then being obtained from these by using equations (78) and (77). This is essentially the method employed by McLagan and Schnuelle36 and Tantardini and Si rn~net ta .~~

There are up till now two main methods available for the evaluation of the necessary cofactors of r. The first, due to Prosser and Hagstrom,75 depends upon finding two non-singular matrices R and L such that

LTR = d, (79)

where L is a lower triangular matrix with unit diagonal, R an upper triangle with unit diagonal, and d a diagonal matrix. It can then be shown that the matrix of T(p1 v), adj(l)(r), is given by

adj(l)(r) = R adj(l)(d)L,

74 p.-O. Lowdin, Phys. Rev., 1955, 97, 1474. 7 6 F. Prosser and S. Hagstrom, Internat. J. Quantuin Chem., 1968, 2, 89.


the cofactors of a diagonal matrix being, of course, trivial to find. The matrix of second cofactors, adj@)(r), is similarly obtained :

adj@)(r) = R adj@)(d)L.

The second method, due to King et aZ.,76 is well illustrated by considering the overlap matrix TAB between two Slater determinants DA, DB. These two determinants remain unaltered by unitary transformations U and V of the sets of spin orbitals which constitute DA and DB, respectively, while in the new representation TAB becomes UrABV. One may now choose these transformations so as to diagonalize FAB, giving

Matrix elements of the form { D A ~ NIDB) assume a particularly simple form in this special representation, the necessary cofactors being given by products of the elements of dAB. The relative merits of the two methods have been compared in the l i t e r a t ~ r e . ~ ~ , ~ * However, too few calculations of any size have been performed by either method so far for one to make any useful statement about their relative computation efficiencies.

An alternative procedure first proposed by Moffitt79 is to expand a determinant D(ili2 , . . i ~ ) of non-orthogonal spin orbitals directly in ternis of determinants composed of orthogonal orbitals. Let us denote the non- orthogonal spin orbitals by y i and the orthogonalized counterparts by E,L. Each ELL may then be written as a linear combination

in which the orthogonalization coefficients y ip remain to be determined, and we assume a set of A7 spin orbitals ( M a N ) . Hence

the determinant on the left-hand side consisting, of course, of the E,’s. Now on the right-hand side of this equation, each index in the set il, i2, . . . iN

must be distinct (as otherwise the determinant vanishes), and the set represents a particular selection of N different spin orbitals y i from the total of M. Hence, for a given set (id2 . . . i ~ ) , the summations over these indices is equivalent to a sum over all permutations of (i1i2. . . z“). However, the determinants D(ili2. . . i ~ ) which correspond to differing orders of the i’s are all the same except for a multiplication factor, EP, where this is just the parity of the permutation. Each distinct determinant D(ili2. . . i ~ ) in (82)

H. F. King, R. E. Stanton, H. Kim, R. E. Wyatt, and R. G. Parr, J . Chetn. Phys., 1967,47, 1936.

7 7 F. Prosser and S. Hagstrom, J . Client. Phys., 1968, 48, 4807. 1 8 H. F. King and R. E. Stanton, J. Chem. Phys., 1968,48, 4808. 7 9 W. Moffitt, Proc. Roy. SOC., 1953, A218,486.

Valence Bond neory 95

therefore becomes multiplied by I‘(i1i2. . . i iv lpip2. . . p ~ ) , where this is a cofactor of order (a- N ) of the matrix of the yzp’s. The transformation thus becomes

D(p lp2 . . . p ~ ) = T(ili2 . . . iivlpulpa. . . piv) D(hi2. . . zk), (83) i l . i 2 , . . . iN

in which the sum is now over just the i @ ! / [ N ! ( n - N ) ! ] cofactors r. This transformation can of course be inverted, and the various matrix elements between determinants of yt’s are computed in terms of matrix elements between determinants of tp ’s . The efficiency of this method depends to a large extent upon the ease with which the cofactors r can be evaluated. For this purpose, Schmidt orthogonalization has much to recommend it, for then the matrix of ytp’s is in triangular form and the cofactors are determined very easily. This procedure has been used extensively by Hurley,40* 4 1 p 5 4 ~ 5 6 and on a large scale by Balint-Kurti and Karplu~.~9*5~.

In all the methods based upon the expansion (73) of the wavefunction in Slater determinants, the N! problem is transmuted into the calculation of very large numbers of cofactors. This enables one to apply well-known techniques of linear algebra. However, recently some progress has been made in the direct calculation of matrix elements of the hamiltonian with functions of the form of the left-hand side of equation (73). It can be shown65 that

N

c”,v=l ( Y S M ; kl HI ~ S M ; z>= C @&(PI v)<plhl v>

N

+ 5 c C~%z(P4..><P4gl 0.) p > v = l a>t=l

+ ~ ~ ~ z ( ~ V 1 ~ ~ ~ < ~ ~ l g l ~ ~ > ~ , (84)

in which D&(plv) is an element of the one-electron density matrix, and D$,Yz(pvl UT), D!&\(pcl.] TO) are the two components (Coulomb and exchange) of the two-electron density matrix. The one- and two-electron integrals are denoted as in equation (76). These density matrices are connected with one another by relations similar to those between cofactors [equations (77) and (78)]. Thus

N

v = l < ~ S M ; ~ ~ Y S M ; I > = A $ ~ = C D,$$’i(~lv)<~lv>, (p=1,2,. . . , N ) (85)

N

t.=l (fv)

D&l(PI v) = c D$,?lWl ..)<.I.>. (0=1,2,. . . , N ; # p )

(86)

D&!&pipsl vivz)= c D$&plp2p31 V i v 2 v 3 ) < ~ 3 1 VQ> (87)

However, we may now continue this sequence of relations, and obtain

N

v a = l ( + VlrV2)

( ~ 3 = 1 , 2 , . . . , N; # p i , p2)


D,&1)(pi/r2 . . . px-11 V ~ Y Z . . . vN-i j=

D , & ~ ) ( P I P ~ . . . .P~v~ V ~ W . . . viv)<pU,vj V A T ) .

(88) It should be noted that as we progress down the sequence, each density matrix is formed from fewer and fewer electrons. Thus consists of N- 1 electrons [and hence ( N - 1)! terms], D2) of N - 2 electrons, . . . , and D(N) of no electrons. The N-electron density matrix depends only upon N, S, and consists of just the representation matrices Us(P) defined in equation (16) in symmetrized form:

D&g(plpZ . . . p d y1v2 . I . Y N j = Ufk(P)+ U&(P), (89)

where the permutation P is given by

The D(A+) elements may thus be computed once and for all and kept on a file. The elements of D(N- l ) are formed from this by multiplying each entry in the list of DN) elements by the appropriate overlap integral <p i l v t> , the D(lv-2), . . . D(3), elements being formed in succession in a similar way. Once the elements of DN) have been formed, this is an extremely fast process and, moreover, is independent of the size of the basis set in which the orbitals are expanded. A particularly convenient feature of this method is that the 3- and 4-electron density matrices, IN3) and 0 ( 4 ) , are formed simultaneously, and these are necessary in constructing the equations from which the 4y are determined,G5 or in minimizing the energy directly. The N ! problem is of course still present, but this mainly occurs in the calculation of DA') which may well take a considerable time. However, this has to be carried out once only, and in this way the purely group theoretical aspects of the problem are separated from those quantities which depend upon the physical details of the situation.

In a useful recent development, Dacre and McWeenyso have shown how to calculate directly the interaction energies between atomic or molecular systems at short to medium range. The method is well illustrated by taking as an example two atoms for which we write the wavefunctions

and

The two configurations are of the form

6 o P. D. Dacre and R. McWeeny, Proc. Roy. SOC., 1970, A317,435.


and @B=v:Y$ - - - v : n , w m B + 1 - . . (92)

in which we assume for simplicity that all the orbitals centred on any one atom are orthogonal to each other. The normalization integrals for these functions are given by

A8 1%- - c Ufk(P)<P@A@BI@A@B), (93) P ~ ~ N A + N B

in which we adopt the spin-valence representation (28) so that the index k is equivalent to ( S A S B ~ A ~ B ) , where SA, SB are the spins of the two atoms A and B, respectively. We now divide the sum over P into three summations:

C P 3 c c C P A P B h (94) P ~ ~ N A + N B P A E ~ N A P B ~ ~ ” B Qq

where Q, is a set of (NA+ NB)!/(NA!NB!) permutations which interchange electrons between the two atoms. These can be taken in the formal

eq=PilklPiZkZ . . . PZqkq, (95)

min(NA,NB). Substituting (94) and (95) into (93) we obtain, for the A s , in which 1 < il < i2. . . < iq< NA, NA+ 1 < k l < k2 . . . 9 NA+ NB, 4 = O , 1 , 2 , . . .,

d $ = 2 m ~ + m ~ &k+ U&Pilkl)<iilki)2 ( i l ,kl

+ c c U~(PilklPiZkZ)<il lk1)2 < i 2 1 k ~ ) ~ i Z > & k , > k l

+. . .). (96)

This is a finite sum in squared powers of overlap integrals between the orbitals of the two atoms. A similar development can be carried through for the 1- and 2-electron contributions to the matrix elements of the hamiltonian, from which one obtains immediately an expression for the matrix elements of the interaction in the form

- AS lk - Zk+ ,z AMillkl) <illk1)2

t l , k ,

+ C C ~fk(i3iilk2ki) < i i l k ~ ) ~ <i2lk2)2 (97) i2>il k,>k,

+. . . Even for overlap integrals <illkl) as large as -0.3, the second term in equation (97) is of the order -0.1A& and the third term 0.01AE, so that except for very close interactions, this series may be safely cut off after the third term. This may be useful for evaluating the interaction energy between the closed-shell atomic cores within a molecule, for which just the first term may well be sufficient.

H. Horei, J . Phys. SOC. Japan, 1964, 19, 1783.


A similar development had already been carried through in the VB theory of solids by Arai.82~~3 An expansion of the form (97) is still likely to be useful for solids even though the series is now formally infinite since the overlap integrals decrease as exp(-R), where R is the distance from the atom which is at the origin. Arai has also shown how various types of term may be exactly summed with the aid of diagram techniques. In this way he has demonstrated that in general no ‘orthogonality catastrophe’ occurs as N+oo and that the series (97) converges as long as a certain (fairly stringent) inequality is satisfied by the overlap integrals [equation (98)].83 As a result he has been able to derive rigorously the well-known Heisenberg expression for the energy of a ferromagnetic subs tan~e ,~~

E= -2 c J,,S,S,, P’>’

from first principles so that the ‘exchange integrals’ JPv may now also assume negative values and can be given a precise definition.

4 Extensions of W3 Theory

In this section, we discuss briefly two of the important developments of VB theory that have occurred over the past 10 or 15 years, namely the theory of separated pair functions, and the atoms-in-molecules theory.

Theory of Pair Functions.-We now turn to the model in which the electronic state of a molecule is represented by wavefunctions of the form

the spatial part of which is written as a product of two-electron functions of the type ypp(2p- 1,2p), the spatial co-ordinates now being written simply as numerals. The number of electrons is assumed to be even for simplicity; an extra electron would be described by an orbital &+I. With this form of the spatial function, it is particularly convenient to use the Serber basis for the spin functions (see p. 67) SO that the spins of each pair function y,, are coupled to a definite resultant, S,L, which may be 0 or 1 . The pair functions may be assumed to be normalized,

but the central feature of the theory is that one imposes upon the P,~,‘ the property of strong orthogonality,

1 ypp( 1, 2) y,,( 1 , 3) d VI = 0 unless p = v ,

T. Arai, Pliys. Rev., 1962, 126, 471 ; 1964, 134, A824.

J. H. van Vleck, ‘Theory of Electric and Magnetic Susceptibilities’, Clarendon Press, Oxford, 1932.

st. T. Arai, Prog. Theor. Phys. (Japan), 1966, 36, 473.

Valence Bond Theory

in addition to the more conventional orthogonality between pairs

99

The strong orthogonality condition was first introduced by Hurley, Lennard- Jones, and Pople, and the notation employed by them is largely followed in this section. The importance of this condition is that it eliminates at a stroke* almost all of the difficulties due to the N! problem, while allowing for a certain non-orthogonality within a pair which may be important for the description of a chemical bond. Nevertheless, the strong orthogonality property is undoubtedly a severe condition, and one cannot expect that all molecules will be well described by a wavefunction of the form (98). Some further light may be thrown on to this problem by considering the possible symmetries of the pair functions ypp. An analysis similar to that briefly described on pp. 88-89 shows that, for spatially non-degenerate molecular states, the pair functions must also either be transformed one into the other by the operations of the point group or must remain invariant. Consequently pair functions may always be taken to be real, and are likely to be highly localized, so that the model may be expected to produce the best results when applied to molecules with well-separated pair bonds such as in ethane, SFs, or BCk. However, calculations upon such systems as the Be atom,*s LiH, and BHS7 have yielded surprisingly good results, in which moreover all the pairs are assumed coupled to form singlets.

The

in which (pull H f lip) is the one-electron contribution to the total energy given by the pair function yPP and (pllGf Ilp) is the sum of the Coulomb and exchange interaction of this pair function with itself. The last two terms, (pvllG$Ilpv) and (pvllG:fII vp), represent respectively the Coulomb and exchange interactions between the functions yPP and y,,,,. The total energy is thus of the form of a sum of energies due to the pairs plus an interaction energy between pairs. Explicit expressions for these terms are given in Appendix 1. The use of a star to represent an exchange operator should not cause confusion since the pair functions may always be taken to be real.

The use of the Serber basis for spin functions combined with the strong orthogonality condition (100) ensures that separated pair functions (98)

* E. R. G. Heath, June 1970.

u A. C. Hurley, J. Lennard-Jones, and J. A. Pople, Proc. Roy. SOC., 1953, A220, 446. 8a R. McWeeny and B. T. Sutcliffe, Proc. Roy. SOC., 1963, A273, 103.

E. L. Mehler, K. Ruedenberg, and D. M. Silver, J. Chem. Phys., 1970, 1181. 88 J. Gerratt and W. N. Lipscomb, Proc. Nut. Acad. Sci. U.S.A., 1968, 59, 332.


with different values of the spin-coupling index k are orthogonal. For the same reason, off-diagonal matrix elements of the hamiltonian between such functions are given simply by (see Appendix 1)

n

, U > Y = = = l

= C U$Lp2p,2J (1~~riG;‘Ilv~) (103)

The standard procedure for determining the pair functions would be to vary the energy (102) together with Lagrange multipliers in order to maintain the normalization of the yPu, and the strong orthogonality requirement between them. The resulting equations are of the forms9

in which the E~ and E are the necessary multipliers. However, since these are two-electron equations their solution may be as arduous as that for the helium atom or the H2 molecule (but not quite, since the y,, may not contain interparticle co-ordinates : see the following paragraph). The process is further complicated by the existence of the off-diagonal multipliers ; these are a sufficient nuisance even in orbital equations.

The most common approach is to note that since the pair functions ypp may not contain any interparticle co-ordinates (as then the strong orthogonality requirement cannot be sustained), they may be expressed as bilinear expansions in one-electron orbitals:

1””

The orbitals themselves are further expanded in a basis set of atomic orbitals XP

and sets of equations can then be derived straightforwardly for the coefficients u$ and c:~. It has been shown by Araigo that the effect of the strong orthogonality condition upon the set of functions xp (which we assume for the moment to be orthonormal) is togartition the basis into n subsets, such that each set of functions y ~ r , yf for a given ,u is expande,d only in terms of basis functions x p from a single subset. When the basis set is small, it often seems obvious on simple physical grounds as to how the basis should be partitioned. This is no longer the case for larger basis sets, and it then becomes very important indeed to allow the basis to find its own optimum partitioning during the minimization process.

It has so far been usual to consider almost exclusively the perfectly paired function (98), in which the spins of all the pairs t ~ , ~ , ~ are coupled to form

J . M. Parks and R. G. Parr, J . Chem. Plzys., 1958, 28, 335. O 0 T. Arai, J. Chem. Phys., 1960, 33, 95.


singlets, Si=O. In this case the matrix a$ in equation (105) can always be diagonalized without any loss in generality, so that the expansion now assumes the form

The orbitals y r in this representation are known as ‘natural orbitals’, (NOS) and the first terms in the expansion (106), yf(1) yr(2), are thought to resemble closed-shell SCF MOs which have been transformed to a localized representation (by the method of, for example, Edmiston and Ruedenberggl). Indeed, in a whole series of calculations on the molecules NH3, NHZ, NH,, NH, N, and CO,92 Robb and Csizmadia have approximated the yPP by expanding them in a bilinear series of such localized MOs, the coefficients in the expansion being determined by the solution of a certain integral eigenvalue equation. Ahlrichs and K~tze ln igg,~~ on the other hand, have attempted to determine the NOS directly from a given basis set (2,) in calculations on the Be atom, and on the LiH and BeH2 molecules. However, they dropped a number of small terms from the (coupled) equations for the af and c$ in order to facilitate their solution; the energy obtained for the X 1 Z i state of BeHz with six NOS of 0 symmetry is - 15.761 16 a.u. A similar calculation was carried out on the Liz0 molecule by Lin and Ebbing,g4 who found that the most stable nuclear configuration is in fact linear. This is in accordance with the observation that a beam of Liz0 molecules is not deflected by an electric field, indicating that the molecules possess a zero dipole moment. A similar approach to the calcuIation of the yPP by expansion in NOS has also been developed in a series of papers by Silvers95 (see also ref. 87).

However, it can be argued that to use an NO expansion for the pair function y p p destroys the purpose of the model. Instead it is more physical to use the direct expansion (105) in which, moreover, two functions constituting a pair y r , yjP) are non-orthogonal. Thus when this expansion is now cut off after a single term, it describes each electron pair by a function of the form y:(l) yf(2) + yf(2) yf(l), which is just the Coulson-Fischer form, the individual yi’s representing distorted atomic orbitals. Higher terms in the expansion would then be constructed from functions of diferent symmetry, thus main- taining automatic orthogonality between different ‘configurations’ ylf(l)y$(2), and yf(1) y?(2), while at the same time introducing a new kind of correlation effect not provided for by a single configuration, This is tantamount to describing a single bond by a Hirschfelder-Linnett type of wavefunction. Thus, for example, each C-H bond in methane would be described by a linear combination of two configurations of the form al(yfyr t ) + a2(y:y$),

M. A. Robb and I. G. Csizmadia, Internat. J. Quantum Chem., 1970, 4, 365; 1971, 5, 605; 1972, 6, 367.

OS R. Ahlrichs and W. Kutzelnigg, J. Chem. Phys., 1968, 48, 1819; Theor. Chim. Acta, 1968, 10, 377.

g4 T. K. Lin and D. D. Ebbing, Internat. J. Quantum. Chem., 1972, 6 , 297. *3 D. M. Silver, J. Chem. Phys., 1969, 50, 5108; 1970, 52, 299; 1971, 55, 1461.

O 1 C. Edmiston and K. Ruedenberg, Rev. Mod. Phys., 1963, 35, 457.


in which the functions yfyft belong to the A1 representation of the subgroup C3v (and hence being symmetric about the bond axis), and y$&, belong to the E representation. The energy expression given by representing each y,+, as a single product (yy,”y,”.) has of course been derived by Hurley et aE.85 for the perfectly paired case, and was later generalized to cover any spin coupling.88 Energy expressions and orbital equations for the more general case of several configurations have also been derived, and applications to various molecular systems are in hand.96

A calculation with a wavefunction of this kind has been carried out recently for the three lowest states of the CHz radical ( X 3 B ~ , 1A1, and lB1 states).97 The total energies obtained using a fairly large basis set of contracted gaussian functions are X zB1, - 38.9483 a.u.; 1A1, - 38.9362 ax.; lB1, - 38.8818 a.u. The occurrence of off-diagonal Lagrange multipliers was avoided during the minimization process by the device of using as the basis set for the current iteration the eigenvectors of the previous iteration. This appears to be equivalent to a generalized Newton-Raphson method due to Hinze and R0othaan.~8

It should be pointed out that in general the dissociation products predicted by a separated pair function of the form (98) will be excited states of the constituent atoms. These states are in fact just the coupled valence states discussed on pp. 81-83 (provided of course that one uses the general linear combination of spin couplings). Thus the BeH2 molecule, as a simple example, would be described by an inner core pair corresponding to Be(ls2) electrons, and two two-electron pair bonds, the three pairs of course being strongly orthogonal to one another. As the two H atoms are symmetrically removed, each bond pair function goes over asymptotically to the form - d ~ ( l ) h1(2)+ dl(2) hl( l ) , where d1 is one lobe of the orthogonal sp hybrid functions, d1= d $ ( s +pz). This implies that the Be atom will be described as RB~-H+ 03

by the state Be(&&; S= 1)= Be(sp; 3P). Similar considerations apply to other molecules of the form ABn, the central atom being left in an excited state of multiplicity n + 1 .

This kind of approach in which one distinguishes electron pairs as physically or chemically distinct subsystems has been extended by McWeeny in his theory of generalized product functi0ns.~9 In this model, one now distinguishes between several groups of electrons and writes the spatial wavefunction in the form of a product of group functions:

where YSM; k(a, b, C, . . .)= 1/N!d{@a@b@pc . . . @ g ~ ; k], (108)

@a=@a(l, 2 , . . . na),

@b=@b(na+l, . . - na+nb),

@c=@c(na+nb+ 1, . . . na+na+nc),

Qs S . Wilson and J. Gerratt, unpublished work. Q7 P. J. Hay, W. J. Hunt, and W. A. Goddard, jun., Chem. Phys. Letters, 1972, 13, 30. s8 J. Hinze and C. C. J. Roothaan, Prog. Theor. Phys. (Japan), Suppl., 1967, 40, 37.

R. McWeeny, Proc. Roy. SOC., 1959, A253, 242.


are the spatial wavefunctions for groups which consist of nu, n b , nc, . . . , electrons, respectively. It would be sensible to employ an Yna x y t z b x 9% x . . . basis for the spin functions so that each group is characterized by a definite resultant spin S,, s b , Sc . . . etc. The most general type of function of this kind is of the form

F S M = 2 Cube . . . C s k Y S M ; k(a, b, C, . . .), (109) a,b,e k

in which not only are all possible spin couplings present, but excited ‘configurations’ Qu, @b, . . . etc., are also allowed. As in pair theory, it is essential to impose strong orthogonality between the group functions :

JQa(l,i1,i2,. . . ,ina)Qb(l,j1,j2,. . . jnb)dVi= Ounlessa=b (110)

in order to preserve a tractable theory. With this provision, an expression for the total energy given by the function (108) may be derived which is of the same form as expression (102) for the separated pair function. Thus

E S k = C [(ail ~~IJa)+(aIIGfIIa)I+ C [(aBIIG&IIaB)+ Cu~(P,~s>(aBIIG~~IIBa)I, U a>D 1

(1 11)

in which the indices a,p run over the groups a, b, c, . . . , which constitute the function (108). The terms (all H$ /la) + (allG$ IJa) represent the contribution to the total energy ESk given by the group Qu by itself, and the remaining terms (aBllG&IlaB), (a@IIG$sIIB~) represent respectively the Coulomb and exchange interactions between the groups Qa, Qp The matrix Us(P,tp) refers to the exchange of electron co-ordinates a’,p’ between the groups a

and p. If all the groups are coupled to form a zero resultant spin, Sa=Sb= Sc = . . . = 0, then this matrix simplifies to the form - 41, where I is the unit matrix. It should be noted that because of the strong orthogonality condition, the two-electron density matrices which occur in the terms (apllG&llap), (apllGZsllBa) are just products of the one-electron density matrices for the separate groups a and @.

For spatially non-degenerate molecular states, the symmetry properties of the group functions Qu, %, Qc, . . . , etc., are similar to those of the pair functions : equivalent group functions belong to non-degenerate representations of a subgroup S of the molecular point group and are transformed into each other by the operations of 3 not contained i n s . Unique groups belong to non-degenerate representations of the whole group The functions cDU, Qb, die are therefore likely to be highly localized. This property is consistent with the generalized strong orthogonality requirement (1 lo), since it would be unreasonable to impose this requirement between strongly over- lapping groups.

Actual applications of this model so far have, in fact, been almost exclusively pair-function calculations. This is the case in a calculation by Klessinger

104 Quan t urn Chemistrj?

and McWeeny on the CH4 molecule,100 using a minimum basis set of Slater orbitals with Slater exponents. A total energy of - 40.0980 ax. was obtained. A series of similar calculations on HF, H20, N2, C0,101 ethane, methyl- amine, methyl alcohol,l02 ethylene, formaldehyde, acetylene, and HCN103 has been carried out by Klessinger. In all these calculations, the basis set was first srthogonalized so as to fulfil the strong orthogonality requirements, i.c. the basis set was first partitioned into orthogonal subsets 011 physical grounds. It is therefore not certain that the wavefunctions have been truly optimized, since a slightly different partitioning may give a slightly lower energy and have a substantial effect on other calculated molecular properties. This is most probably the cause of the result that, according to these calculations on ethane, the eclipsed conformation is more stable than the staggered.Io2 The indications are that if the basis set is allowed to partition itself during the minimization process,97 the staggered conformation is found to be the more stable, with an energy barrier of -3.1 kcal mol-1, in good agreement with experiment. Expectation values of various spin-dependent operators have also been derived for a group function of the form (108).Io4

Atoms in Molecules.-In this approach, which was first proposed by Moffitt,lo5 a wavefunction for a particular electronic state of a molecule is constructed from products of atomic wavefunctions, these, moreover, being taken to be exact eigenfunctions of their respective atomic hamiltonians. We confine our attention to the case of diatomic molecules AB so that, according to this procedure, the wavefunction is written as

y/,b(2s+1A)= Cabpab(2SA+1L~; 2sB+1Ln12Sf1A). (1 12) a,b

The functions on the right-hand side of this equation are termed ‘composite functions’ and consist of products of atomic eigenfunctions properly coupled and antisymmetrized as follows:

(N= NA+ NB),

in which d is a partial antisymmetrizing operator that only exchanges electrons between the functions !Pa, Y b .The orbital angular momentum of the

l o o M. Klessinger and R. McWeeny, J. Chern. Phys., 19 42,65, 3343. M. Klessinger, Cliem. Phys. Letters, 1968, 2, 562; 1969, 4, 144; Frrraday SOC. Sytup., 1968, no. 2, 73.

l o 2 M. Klessinger, J . Chem. Pliys., 1970, 53, 225. lU3 M. Klessinger, Intemnt. J. Quantum Chem., 1970, 4, 191. lU4 W. J. van der Hart, hfol. Phys., 1971, 20, 385, 399, 407. IUi W. Moffitt, Proc. Roy. Soc., 1951, A210, 245.


molecule, A, is just the sum M L ~ + M L ~ , and a,b denote all the other indices (such as electron configurations, internal spin-couplings kA, k ~ ) which are necessary to specify a molecular state completely. The motivation for using exact, or at least very accurate, atomic eigenfunctions as the basis set is to prevent the propagation into molecular calculations of errors which are inherent in the description of the atoms. Several discussions of the foundation of the method may be found in the literature, and for this reason will not be pursued here.lO6, lo7

The coefficients Call in (112) are determined by the solution of the secular equation

where the matrix elements are defined over the composite functions in (112), and indices such as LA, M L ~ , SA, MA, etc., are temporarily dropped for clarity. The energies are found from the associated equation

det (Nab, cd - Ed ab,cd) = Q. (115)

According to the original method of Moffitt, the necessary matrix elements are evaluated according to the following prescription :106,107

1)1

d a b , c d = &b,cd. (1 16) - - The matrix elements Hab,cd, d a b , c d are calculated using spin valence functions constructed from atomic orbitals. The energies E t , EF, E t , E z are the exact values for the particular states of the participating atoms and may be taken from spectroscopic tables. The corresponding quantities E:, etc., with tildes are the values for these same spectroscopic states obtained from calculations using the orbital wavefunctions. In essence, equation (1 16) prescribes how

the matrix elements Hall,&, which are obtained from an ab initio spin valence calculation, are to be corrected in order to eliminate known atomic errors.

According to Hurley,log the matrix elements fiub,cd, ~ I ~ b , ~ d are to be calculated with all orbital exponents optimized so as to achieve the best possible molecular energy.* This takes into account to some extent the defor- mations of the atoms when the molecule is formed, so that the expansion

(112) might converge rapidly. In the same way, the quantities g$ etc. are

* This was not, however, always done in those cases shown in Table 1 , since it was almost certainly technically impossible to optimize exponents in molecular wavefunctions at the time when the method was proposed.

lo6 M. Kotani, K. Ohno, and K. Kayama, in ‘Quantum Mechanics of Electronic Structure of Simple Molecules’, Handbuch der Physik, vol. XXXVII/2, Springer, Heidelberg, 1961; A. C. Hurley, Rev. Mod. Phys., 1963, 35, 448. G. G. Baht-Kurti and M. Karplus, ‘Atoms in Molecules’, in ‘Orbital Theories of Molecules and Solids’, Clarendon Press, Oxford, 1973.

- -

loB A. C. Hurley, Proc. Phys. SOC., 1956, A69, 49.


to be calculated from similarly optimized atomic functions which result when the internuclear distance increases to infinity in the linear combination (1 12). This is just the ‘intra-atomic correlation correction’ (i.c.c.) method, and a glance at Table 1 shows that it has been very successful indeed in predicting binding energies.

However, in spite of these accomplishments, there is some evidence that the Hurley i.c.c. method is not always reliable, particularly in the calculation of potential energy surfaces. The corrections to the elements Hab,cd become unreasonably large at very small internuclear separations, and erroneous dissociation products are frequently predicted for charged species.lo7 In order to remedy this drawback, the ‘orthogonalized Moffitt (OM) method‘ has been lo7 which starts from the observation that the composite functions form a non-orthogonal basis except at infinite internuclear separations. This circumstance makes the identification of a particular composite function with given atomic states ambiguous. The composite function basis is therefore orthogonalized by the Schmidt procedure, for example, and all calculations are carried out in this new basis. This transformation, of course, has no effect when the atoms are separated to infinity. However, at finite

internuclear separations it can be seen in equation (116) that, since i u b , c a

is now just dab,cd , there are no longer any atoms-in-molecules corrections to the off-diagonal elements of the hamiltonian matrix. It is the very large and negative corrections to these elements that is apparently responsible for the breakdown of the i.c.c. method in certain circumstances. The potential curves obtained by the OM method for the LiH, Liz, Lil, HF,49 LiF, F2,

and F2 molecules55 give fairly good agreement with experiment for the binding energies (see also Table 1). The major application of this method, however, has been to the calculation of the potential energy surface for the reaction Li + Fz-, LiF + F.109

It seems that atoms-in-molecules methods may develop in two separate directions. The basis of composite functions (112) is not the most suitable in which to expand the molecular wavefunction as many terms are needed in order to express adequately the distortion of the atoms upon the formation of molecules. Thus the OM calculations above require very large basis sets (18 composite functions for Liz, 100 for HF, and 204 for LiFz) in order to obtain reliable results. The i.c.c. method, on the other hand, which does allow for some distortion, achieves in several instances as good or even better results with much smaller basis sets. An attempt has been made by AraillO to develop a ‘method of deformed atoms in molecules’, in which the composite functions are multiplied through by a certain spatial function, A(r1, r2, . . . r ~ ) , to be determined, which expresses the desired distortion. Thus, in hydride molecules, the 1s wavefunction of H, which is of the form exp( - r), is multiplied by a function A(r) = exp( - 6r), so that the combined function, exp [ - (1 + Qr], with 6 z 0.2, is now an adequate representation of lo9 G. G. Baht-Kurti, Mol. Phys., 1973, 25, 393.

T. Arai, J. Chem. Phys., 1957, 26, 435; Rev. Mod. Phys., 1960, 32, 370.


a distorted (contracted) 1s orbital. However, the final equations of the method (see e.g. Kotani et al., ref. 106) unfortunately appear to be hardly less tractable than the execution of an ab initio VB calculation with optimization of exponents. Applications of this method have been made to the Hz and Liz molecules.111I1l2 In the case of Liz, a binding energy of 0.96 eV was obtained, compared with the experimental value of 1.04 eV.

An interesting recent attempt to allow for the distortion of atoms within the atoms-in-molecuies framework has been made by Ellison.113,114 This approach makes use of the fact that an atomic navefunction may be scaled by multiplying all electron co-ordinates rp by a factor 5. It is easily shown that the energy Eso calculated with the scaled waverunction is related to the unscaled energy EO by Esc=5(2-E)Eo. The scaling factors may then be chosen so as to concentrate the different atomic wavefunctions which constitute the composite functions in a common region of space, so that they express more adequately the necessary deformation. The few results obtained so far are fairly encouraging. In the case of H2, a calculation with five scaled composite functions (one of which was a quite sophisticated function for H-) gave a binding energy of 4.46 eV, compared with the experimental value of 4.73 eV. The method, however, is still in the process of development; in order to apply it to larger systems, it will be necessary to find a way of scaling just the part of an atomic wavefunction which describes the valence electrons, leaving the core electrons unchanged.

5 Appendix: Energy of the Separated Pair Function

We rewrite the electronic hamiltonian (1) in the form

where h, = (- +V; + up), g,, = r;>, and the nuclear repulsion term has been dropped for simplicity. The expectation value of H with a function of the form (98) is then given by

The only permutations Pr which give non-zero contributions to (118) are those which permute the co-ordinates of an electron pair function, and transpositions PLY which correspond to a term g,, from (1 17). The normalization integral is given simply by

where (119) A S - d S k S k . . .

kk- 1 d 2 d:k,

d;k= <w,,(l, 2)l@Iw,p(L 2)) (120) l L 1 T. Arai, J. Chem. Phys., 1957, 26, 451. 118 T. Arai and M. Sakamoto, J. Chem. Phys., 1958, 28, 32.

F. 0. Ellison, J. Chem. Phys., 1965,43, 3654; F. 0. Ellison and Ay-Ju A. Wu, J . Chem. Phys., 1967, 41, 4408; 1968, 48, 1103.

ll' F. 0. Ellison and J. A. Slezak, J. Chem. Phys., 1969, 50, 3942.

the pair-pair Coulomb interaction by

Matrix elements of H between separated pair functions which differ in the spin coupling index k are given by

mi= < y S M ; 11 YS M ; k)

= c ug(Pz,,, 2v) ~P~~l lGt " , " l !~d , (131) / f > z . = 1

the only surviving contribution being that due to the pair-pair exchange interactions. This is defined by

(invIIG$' 1 1 ' ~ ) = (d~~dS,ld~'~:z)-1'2<~~,~,,(~, 2) ~ ~ , , ( 3 , 4 l ~ ~ ~ l ~ , , , ( ~ , 2) ~,,(3, 4)>, (1 32)

Valence Bond Theory

in which

109

4 Harmonic and Anharmonic Force Field Calculations

BY I . M. MILLS

1 Introduction

Within the Born-Oppenheimer approximation for the separation of nuclear from electronic degrees of freedom there exists a potential energy function V(Y) for every molecule, giving the energy as a function of the internal nuclear configuration defined by the nuclear co-ordinates r. It is an important property of this function that it does not involve the nuclear masses, so that the same function holds for all isotopic species (provided of course that the co- ordinates Y are geometrically defined). The function V has (in general) a deep minimum at the equilibrium configuration Y = r e , which defines the equilibrium geometry of the molecule. All the details of the vibration-rotation energy levels and wavefunctions, and hence of the vibration-rotation and pure rotation spectra, may be calculated (in principle) from a knowledge of the masses of the atoms and the shape of the potential energy function around this minimum. Conversely, the analysis of vibration-rotation spectra should ultimately lead to experimental information on the shape of the function. In the simplest analysis the equilibrium geometry is related to the moments of inertia and hence to the rotational spectrum; the second derivatives - or curvature - in V at the minimum define the harmonic force field which is related to the vibrational spectrum; the detailed shape of the surface for increasingly large displacements from equilibrium defines the anharmonic force field and is related to the various manifestations of vibration-rotation interaction.

This Report is concerned with progress in the attempt to determine the equilibrium configuration and the harmonic and anharmonic force field of a molecule, i.e. the shape of the potential surface V(r ) around the equilibrium configuration, from observed spectroscopic data. This subject has developed considerably in the past ten years, through a wider understanding of the necessary theory, through advances in the experimental techniques of high- resolution spectroscopy, and through the application of modern computers to the calculations. However, there is still a great need for more experimental measurements of vibration-rotation spectra at the highest possible resolution, and even in the most favourable cases the data are generally insufficient to

110

Harmonic and Anharmonic Force Field Calculations 111

determine all the symmetry-allowed force constants characterizing the force field without some simplifying assumptions about the nature of the potential surface.

The earliest anharmonic force field calculations on polyatomic molecules were made by Pliva and co-worker~.~-~ More recently, Kuchitsu and Morino and co-worker~,~-~ Overend and co-workers,s-10 Cihla and Chedin,ll and Hoy, Mills, and Strey12 have developed the techniques of calculation and applied them to a wide variety of molecules; many other workers have reported calculations on particular molecules, as discussed in Section 5. A recent review by Pliva l3 contains reference to most published calculations on particular molecules up to 1973.

The primary motive for attempting calculations of this kind is simply our desire to determine the potential function V(r) more accurately and over a wider range of co-ordinate space. Even if our immediate ambition is only to determine the equilibrium configuration and the harmonic force field, our ability to withdraw this information from spectroscopic data is limited by the need to make corrections arising from the cubic and quartic anharmonic force field.

A secondary motive is our general desire to verify and extend our understanding of vibration-rotation interactions in molecular spectra, and particularly to interpret data on different isotopic species in a consistent manner. Consider, for example, a: constants (which measure the dependence of the rotational constant B on the vibrational quantum numbers vr) determined experimentally for several isotopic species of the same molecule. It is clear that these constants are not all independent, since they are related to the potential function which is common to all isotopic species. However, the consistency of the data and of our theoretical formulae can only be tested through a complete anharmonic force field calculation (there are at this time no known relationships between the a values analogous to the Teller-Redlich product rule). Similar comments apply to many other vibration-rotation interaction constants.

This Report will be mainly concerned with the analysis of data on the force field obtained from high-resolution spectroscopy. There are other useful

J. Pliva, Coll. Czech. Chem. Comm., 1958, 23, 777. * J. Pliva, Coll. Czech. Chem. Comm., 1958, 23, 1839, 1846.

Z. Chila and J. Pliva, Coll. Czech. Chem. Comm., 1963, 28, 1232. ' K. Kuchitsu and L. S. Bartell, J . Chem. Phys., 1962, 36, 2460, 2470.

K. Kuchitsu and Y. Morino, Bull. Chem. SOC. Japan, 1965, 38, 805, 814. ' K. Kuchitsu and Y . Morino, Spectrochim. Acta, 1966, 22A, 33. ' Y. Morino, K. Kuchitsu, and S. Yamamoto, Spectrochim. Acta, 1968, U A , 335.

M. A. Pariseau, I. Suzuki, and J. Overend, J. Chem. Phys., 1965, 42, 2335. S. G. W. Ginn, S. Reichman, and J. Overend, Spectrochim. Acta, 1970, 26A, 291.

l o D. F. Smith and J. Overend, J. Chem. Phys., 1971, 54, 3632. l1 A. Chedin and 2. Cihla, J. Mol. Spectroscopy, 1971, 40, 337; 1973, 45, 475. l a A. R. Hoy, I. M. Mills, and G. Strey, Mol. Phys., 1972, 24, 1265. la J. Pliva, in the 'Proceedings of the Conference on Critical Evaluation of Chemical and

Physical Structural Data', ed. D. R. Lide, National Academy of Sciences, Washington, D.C., 1974.

112 Quantum Chemistrj)

sources of data on the force field. Gas-phase electron-diffraction studies give information on the molecular structure and the force field, which is to some extent complementary to spectroscopic data; this subject has been reviewed recently by Kuchitsu and Cyvin,14 Kuchitsu,15 and Robiette.lG The study of spectra associated with large amplitude internal motions in non-rigid molecules (such as inversions and internal rotations), the study of collision dynamics from molecular beam scattering experiments, and even the study of chemical kinetics and the mechanisms of chemical reactions all give information on the shape of the potential surface along particular displacement co-ordinates. A priori calculations of potential energy surfaces from electronic wavefunctions l7--I9 also give information which may be compared with that determined spectroscopically.

This Report is arranged as follows. Section 2 is concerned with the representation of force fields: the definition of force constants, choice of units, etc. Section 3 is a brief discussion of the theory and interpretation of diatomic vibration-rotation spectra, and is intended to act as an introduction to the greater complications of polyatomic molecules. Section 4 is concerned with the theoretical and mathematical problems involved in relating the spectra of a polyatornic molecule to its force field, and in trying to calculate the force field from observed data. Finally, in Section 5 we discuss some of the calculations carried out at this time, with examples, and we consider some of the problems involved in finding useful model force fields.

2 Definition of Force Constants

It is usual to write the potential function V as a power series expansion in displacement co-ordinates from the equilibrium configuration, so that the coefficients - which are the force constants - give a mathematical representation of the potential surface in terms of the co-ordinates used. Tt is convenient to write the expansion in the form of a Taylor series:

V -- ( 1 / 2 ! ) f i j y i y j + (1 /3! ) fQkr i~ j~1C + (1/4!)fijh“ui~jrkri + . . . ( I )

where an independent summation is implied over each repeated index in each term, and the indices on the force constants are written as superscripts to conform to the tensor notation developed more fully in Section 4. (It is more

‘ I K. Kuchibu and S. J. Cyvin, in ‘Molecular Vibration and Structure Studies’, ed. S. J. Cyvin, Elsevier, Amsterdam, 1973, Chapter 12.

I K. Kuchitsu, In ‘MTP International Review of Science, Physical Chemistry Series One’, Vol. 2 : ‘Molecular Structure and Properties’, ed. G. Allen, Butterworths, London, 1973.

* * A. G. Robiette, in ‘Molecular Structure by Diffraction Methods’, ed. G. A. Sim and L. E. Sutton (Specialist Periodical Reports), The Chemical Society, London, 1973, Vol. 1, p. 160. H. F. Schaefer, tert., ‘The Electronic Structure of Atoms and Molecules’, Addison Wesley, New York, 1972. R. Ditchfield, in the ‘Proceedings of the Conference on Critical Evaluation of Chemical and Physical Structural Data’, ed. D. R. Lide, National Academy of Science, Washing- ton, D.C., 1974. W. J. Helare, R. Illitchfield, and J. A. Pople, J . Cliew. Phj’s., 1972, 56, 2257.

Harmonic and Anhurmonic Force Field Calculations 113

usual to write all the indices as subscripts; however, the present notation is helpful in considering some of the general equations.) The factor (l/n!) on the nth order term ensures that the force constants are equal to the true derivatives of Y in equilibrium:

where quadratic, cubic, and quartic force constants will have 2, 3, or 4 superscripts, etc. This method of defining force constants will be used throughout this Report, but other authors do not always include the factors (1/3!) = (1/6) and (1/4!) = (1/24) in the cubic and quartic terms, nor do they always treat the multiple summations as unrestricted: they are often taken in the form

2 C C . . . etc., i < J < k

so that each term occurs only once regardless of the order of indices. Overend and co-workers *--lo even omit the factor (1/2!) = 3 from the quadratic term and treat this also as a restricted summation, although this is unusual. The consequence is that care is needed in comparing numerical values of force constants reported in different papers, since the results may differ through numerical factors arising from the different definitions used.

For small displacements, of the order of vibrational amplitudes at room temperature, the terms in the power series expansion (1) converge fairly rapidly, and higher-order terms are related to successively smaller-order effects in the spectrum, so that they become more and more difficult to determine. Almost all calculations to this date have been restricted to determining quadratic, cubic, and quartic force constants only [the first three terms in equation (111, and in this Report we shall not consider higher-than-quartic terms in the force field. The paper by Cihla and Chedinll is one of the few exceptions in which force constants involving up to the sixth power have been determined for a polyatomic molecule, namely COB.

For large amplitude co-ordinates involving inversion or internal rotation a power series expansion in the displacements may not be appropriate (for example, a Fourier expansion is clearly preferable for internal rotation), and the force constants will then be defined in a different way. However, such examples are not considered here.

Finally, there is the question of units and dimensions for co-ordinates and force constants. The customary usage at this time is to write bond-stretching co-ordinates in A, and angle-bending co-ordinates either in radians, so that they are dimensionless, or in A, by multiplying by an appropriate scaling factor (usually the equilibrium bond length of one of the two bonds on either side of the angle concerned). The units of quadratic force constants in common use are mdyn A-l, mdyn, or mdyn 13, for stretch-stretch, stretch-bend, or bend-bend quadratic constants, respectively; this corresponds to measuring


the energy in equation (1) in mdyn A = aJ units.? The units of cubic and quartic force constants follow in a logical way in this scheme, and Table 1 shows the quadratic, cubic and quartic force constants of the hydrogen halide molecules in these units.

Table 1 Force constants for hydrogen halide molecules

f i r1 f r r r l frrrrl r",rl rgfrrrl refrrrrl

HF 0.9171 (2) 9.640(2) -71.1 477.7 8.116(2) -54.8 337.7 HCI 1.2746 (2) 5.162 (1) -28.7 139.7 8,388 (2) -59.5 368.7

HI 1.6090(2) 3.140(1) -15.0 59.3 8.130(2) -62.4 397.7

re /A mdyn A-1 mdyn A-2 mdyn A-3 aJ aJ aJ

HBr 1.4143 (2) 4.116 (1) -21.2 94.0 8.234 (2) -60.0 376.0

This procedure is untidy in several respects. Firstly, there is often ambiguity about whether an angular displacement has been scaled or not, and about what length has been used to scale the angles when ;hey have been scaled. Secondly, although the quadratic constants come out to be conveniently in the range 1-10 mdyn A-l, cubic and quartic constants are usually larger, typically 10-100 mdyn A-2 and 50-400 mdyn respectively, for bond stretches. This apparently odd result is due to the fact that it is the terms, not the coeflcients, that converge in equation (l), and the vibrational displacements 6rt are typically around 0.05 A; hence the terms do indeed decrease by a factor of 10 or so from quadratic to cubic and cubic to quartic for typical displacements. In a sense there is nothing wrong with this situation, although the relative magnitudes of the force constants appear to be misleading. Finally, it is clearly desirable to have a system of units more closely related to SI, although a straight conversion to N m-l, etc., yields many awkward powers of 10 in the conversion factors for angle-bending co-ordinates and for anharmonic force const ants .

A possible solution to some of these problems, which is not at present in general use, is to define all co-ordinates to be dimensionless by taking the angle- bending co-ordinates in radians and writing the bond-stretching co-ordinates as &/re where re is the equilibrium bond length. This might be described as 'descaling the bond stretches'. All force constants then have the dimension of energy (bond stretches or angle bends, harmonic or anharmonic); they may be conveniently quoted in aJ, or sometimes cm-l. For unscaled angle-bending co-ordinates the force constants remain unchanged in this new system when quoted in aJ; for bond-stretching co-ordinates they change by scaling factors close to unity since most bond lengths are close to 1 A. Table 1 also shows the force constants of the hydrogen halides written in aJ in this way, for comparison with the more familiar units; Tables 13 and 15 in Section 5 (pp. 155 and 158) show a similar comparison for some bent triatomic molecules. The comparison shows that trends which should be associated with changes in

t A = 10-lOm; mdyn = 10-adyn = 10-8N; aJ (attojoule) = J. Note that 1 aJ m6.2415 eV = 0.2294 hartree w hc (50 340 cm-*).


equilibrium bond length are removed in the new system, perhaps allowing us to observe more fundamental similarities and differences in electronic structure.

Although the Reporter favours this way of writing force constants, it has not been adopted in the later tables in this Report because it is felt that it would be premature at this time, and would be so unfamiliar to most readers that it might make the Report more difficult to follow. All numerical results in this Report are given in the customary mdyne and Angstrom units, using unscaled (dimensionless) angle-bending co-ordinates measured in radians, bond-stretching co-ordinates measured in A, and energies measured in aJ. However, units such as mdyn A-l have been written in the equivalent form aJ A-2, etc., in order to emphasize the close relation to the aJ as a unit of energy, and the possibility of making all co-ordinates dimensionless and quoting all force constants in aJ units.

3 Diatomic Molecules Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration- rotation spectrum. The essential theory was worked out by Dunham 2o as long ago as 1932; however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term values being given by the formulae

E v , ~ / h c = TV,J = G(v) + Fv(J)

G(u) = W(U + ~ ) - x ( u + &)2 + Y(U + 4)3 + . (3)

(4)

Fv(J) = BvJ(J + 1)-DvJ2(J + 1 ) 2 + HvJ3(J + 1)' + . . . ( 5 )

Bv = Be-gB(v + 3) + ~ B ( v + +)2 + . . . (6)

Dv = De-P(t) + +) + . . . . (7)

The coefficients w,? x,f Be, aB, De, . . . are the quantities determined from a spectroscopic analysis of a diatomic molecule, and our problem is to relate these to the force constants in the potential energy function.

The potential energy is written as a power series expansion in the displacement of the bond length from equilibrium, as follows:

(8) V = +f2p2 + (1/6)f@3 + (1/24)f&4 + . . . 7 These two coefficients are more commonly written me and

** J. L. Dunham, Phys. Rev., 1932,41, 721.

but the notation used here is chosen to parallel the notation for polyatomic molecules.

I16 Quantum Chemistry

where p = r - Y e , (9)

Y is the bond length and re its equilibrium value, and f2,f3,f4 . . . are the quadratic, cubic, and quartic. . . force constants. It is convenient to write the vibration-rotation hamiltonian in terms of a normal co-ordinate Q defined by mass-adjusting the co-ordinate p as follows :

Q = m1J2p, m = mlm2/(ml + m21, (1 0)

where m, and m2 are the atomic masses. The vibration-rotation hamiltonian is then

ii = + V(Q) + ny2r. (1 1)

Tn this expression, PQ = -iha/aQ, V(Q) is the potential energy of equation (8) re-expressed in terms of Q by using equation (lo), I? is the total angular momentum operator, and I is the instantaneous moment of inertia given by

r = my2 = mr;(l + p/r,)z, (12)

Thus I is itself a function of Q, obtained by substituting equation (10) in equation (12). In practice it is also convenient to re-write I? in terms of dimensionless co-ordinates and momenta defined by

J = D/Ii

q = y1/2Q = ~ 1 1 2 ~

where

y = (jJm)lJ2/l i and N = L1f2 m1/2/l i,

p = -ia/ag = PQ,Wi.

and

With these substitutions the hamiltonian (1 1) takes the form

H/hc = j w p 2 + V(q)/hc + (h2/2hcl)J2

where w = ( f 2 / m ) 1 / 2 / 2 ~ ~ is the harmonic vibration wavenumber. Finally, we expand V(q) and (111) in powers of the co-ordinate q:

Y(q)/hc = 392,' + (1/6)43y3 + (1/24)$,y4 + . . . (1 8)

$2 = f 2 / h c ~ = W , (1 9)

& = jJhcci3/', (20)

$4 = fJhcCC2, etc., (21)

where

The symbol aB is used to denote the vibrational dependence of the rotational constant, equation (6) , and a is used to denote the scaling factor defined in equation (15); these quantities should not be confused.

and Anharmonic Force Field Calculations 117 Harmonic

and

where

(h2/2hcI) = Be [ 1 - 2 F)l” __ + 3 C) - q 2 - . . .] (22)

Be = ti2/2hcIe = (h/8x2cmr:).

Substituting equations (18) and (22) into equation (17) gives the hamiltonian in terms of j 2 , P A 2 , and powers of q.

The leading terms in this hamiltonian are those for a one-dimensional harmonic oscillator ; this is treated as a zeroth-order hamiltonian, whose eigenvalues and eigenfunctions may be written down by inspection, and the remaining terms are then treated by perturbation theory. Alternatively, we may say that the hamiltonian is subject to a contact transformation to give a new effective hamiltonian having the same eigenvalues which operates indepen- dently within the wavefunctions of each vibrational state. In either case the result of the perturbation treatment is to give a series of formulae which relate the coefficients in the term formulae ( 3 ) - ( 7 ) to the force constants 5b2, 43, 44, . . ., orfi, f3, fa . . ., and the reduced mass m. The most important of these formulae are as follows:

and De =

Cfi/m)lJ2/2xc = fZ/hca,

4 4 1 1 6 - 5&/48m

f4/16hca2- 5f ;/48h2c 2 a 3 ~ ,

h/8x2cmr:,

The higher-order coefficients involve f5 and other higher-or der force constants . Equation (26) has been written in terms of d b b ) = (aI/aQ)e = I t / 2 , and equation (27) in terms of Tbbbb = -+(a(bb))2/AI,4, where A = 4 x 2 ~ 2 ~ 2 , in order to emphasize the analogy to the corresponding formulae for polyatomic molecules.

The two terms in the formula (24) for x have a simple physical interpretation: the first is a correction to the energy levels owing to quartic anharmonicity in the first order of perturbation, and the second is a correction due to cubic anharmonicity in the second order of perturbation. It is well known that these

Table 2 Observed spectroscopic constants and calculated potentiaE constants for a seIection a f diatomic molecuIesa

Molecule HZ

D2

LiH llBH CH NH OH FH

NaH SiH SiD PH SH ClH

BrH IH

“Bz BN BO BF BCl BBr

cz CN co CF CP cs CCl

co/cm-l 4402.74 3118.46

1405.65 2366.90 2859.1 3125.5 3735.21 4137.25

1172.2 2041.80 1469.32

2689.6 2990.95

2649.97 2308.60

1051.3 1514.6 1885.44 1402.13 839.12 684.31

1854.71 2068.70 2169.82 1308.1 1239.67 1285.08

-

875

x1crn-l 121.55 64.10

23.20 49.39 63.3

82.81 88.73

19.72 35.51 18.23

45.5 52.82

45.22 39.36

9.4 12.3 11 -76, 11.84 5.11 3.52

-

-

13.340 13.144 13.29, 11.10 6.86 8.46 -

Belcrn-l 60.8550 30.429

7.5131 12.021 14.448 16.344, 18.871 20.9456

4.9012 7.4996 3.8840 8.412 9.601

10.5934

8.4665 6.5111

1.212 1.666 1.7803 1.5174 0.6838 0.490

1.8198 1.8986 1.9312, 1.4172 0. 79868 0.8200, 0.693

@/cm-’ 3.0581 1.0492

0.2132 0.412 0.530 0.646 0.714 0.7888

0.1353 0.2190 0.0781

0.285 0.3072

0.2313 0.1689

1.4 2.5 1.648 1.98 0.646 0.35

1.765 1.735 1 .7505 1.84 0.597 0.5922

-

-

re/A

0.741,

1 .595, 1.232, 1 .no9 1 .O4Y4 0.970, 0.9170

1.887 1. 5201 1. 519, 1.433 1.341 1.274,

1.406, 1 .609,

1.589 1.281

1.262, 1.715, 1.89

0.7414

1 .2O50

1.242, 1.171, 1 . 128, 1. 2718 1. 5623 1.534, 1.650

filaJ A-2 5.755 5.770

1.026 3.048 4.478 5.41 7.793 9.651

0.782 2.389 2.390

4.164 5.163

4.166 3.140

3.585 8.331

13.658 8.074 3.474 2.66,

12.160 16.294 19.019 7.415 7.831 8.490 4.03

-

f&J A-3 -37.4 -37.1

-3.64 -15.8 -26.5 -35.0 -54.1 - 70.7

-2.61 -11 .o -10.7

-22.2 -28.7

-

-21.5 -14.8

-18.1 -63.8 -89.6 -57.7 -17.8 -11.3

-17.7 - 110 - 136 -52.4 -44.1 -47.8 -

h/aJ A-4 237 220

11.5 70

136

337 475

-

7.5 44.7 41 .O - 110 140

95.8 58 .O

63.7 516 481 3 72 81.6 36.6

3 66 600 807 230 194 153 -

a,/A-l 2.17 2.14

1.18 1.72 1.97 2.161 2.32 2.44

1.11 1.53 1.49

1.77 1.86

1.72 1.57

1.68 2.55 2.18 2.38 1.71 1.41

2.13 2.27 2.39 2.36 1.88 1.88

-

-

a4/A-l E 00

2.43 2.34

1.27 1.81 2.08

2.48 2.65

1.17 1.63 1.56

1.94 1.97

1.81 1.63

1.59 2.97 2.24 2.57 1.83 1.40 !$

-

-

2.07 5 2.29 2.46 $ 2.52

1.60 1.88 (t‘

-

SIN SiO SiF Sip SiS Sic1

N8 NO NF

NP NS NC1

PO PF Pa PS PCI

Oa 0s oc1 s2

F* Cla Bra 11

1151 -68 1241.44 857.20 510.98 749.69 535.89

2358.07 1904.03 1141.31

1337.24 1219.1 827.0

1233.38 846.75 780.89 739.1 577

1580.36 1148.19

725.67

891.85 559.71 323.33 214.52

(780)

6.560 5.92 4.74 2.02 2.58 2.29

14.188 13.97 8.99

6.98, 7.5 5.1

6.56 4.48, 2. 820 2.96 3.5

12. 073 6.12 - 2.844

- 2.70 1 .081

-0.607

0.7310 0.7267, 0.5813, 0.2390 0.3035, 0.2562

1.9987 1.7049 1.2057

0.7864,

0. 64686

0.7337 0.5665

0.77364

0.30356 - -

1.4457 0.7208 0.622 0.29541

0.8828 0.240, 0.081 1 0.0373,

0.567 0.504 0.490 0.13 0.147 0.163

1 .TO9 1.755 1.492

0.5536 0.612 -

0.55 0.456 0.143 - -

1.579 0.5736

0.158 -

- 0.153 0.0321 0.0121,

1 . 5720 1.509, 1 .6008 2.246 1.929, 2.058

1.097, 1.150, 1. 317,

1.4900 1.495* 1 .6144

1.4750 1 .589, 1 ,893, - -

1.2075 1.4811 1. 571 1. 8894

1.417

2.281 2.665

1 .9876

7.292 9.241 4.898 2.152 4.941 2.630

22.940 15.948 6.187

10.160 8.527 4.029

9.454 4.974 5.564 5.06 3.22

11.768 8.281

(3.93) 4.960

- 3.227 2.461 1.720

-42.3 -54.6 -28.2 -8.5

-23.0 -12.3

- 168 - 121 -41.6

-61.2 -52.6 -

-59.6 -28.2 -26.6 - -

-87.4 -52.2

-25.1 -

- -16.5 -11.8 -7.9

196 274 146 26 88 52

972 746 254

290 246 -

315 142 97.0 - -

543 292

105 -

- 69.0 43 .O 92.3

1.93 1.97 1.92 1.30 1.55 1.56

2.44 2.53 2.24

2.01 2.06 -

2.10 1.89 1.59 - -

2.48 2.10

1.69 -

- 1.71 1.59 1.54

1.96 2.06 2.06 1.32 1.59 1.68

2.46 2.59 2.42

2.02 2.03

2.18 2.02 1.58 - -

2.57 2.25

1.74 -

- 1.75 1.58 2.77

a The data are mostly taken from ref. 21. When a value for x or aB is not quoted, the values for w or Be should be read as Y (1-0) or Be, respectively, with a correspondingly increased uncertainty in the derived values off, or the bond length.

1 20 Quantum Chemistry

two terms in an anharmonic vibration generally give corrections to the energy levels of comparable magnitude, and hence that the effects of cubic and quartic anharmonicity cannot be distinguished from experimental observations of the vibrational energy levels alone.

The two terms in ctB, equation (26), also have a physical interpretation. The first arises from the second derivative of the effective rotational constant in the hamiltonian with respect to the internuclear distance, by first-order perturbation theory; i.e. it arises from the third term in square brackets in the expansion (22) and from the fact that there is a mean square displacement in q in the zeroth-order model given by

< q 2 > = (v + 3).

The second term in ctB arises from second-order perturbation theory; it may be described as due to the fact that cubic anharmonicity results in a mean first- power displacement in q given by

and this contributes to aR through the first derivative of the effective rotational constant in the hamiltonian with respect to the internuclear distance, i.e. the second term in square brackets in equation (22). The two terms in equation (26) are of comparable magnitude, but for almost all known diatomics the second term is dominant, and sincef, is always negative, ctB is positive.

The conclusion is that if the spectrum can be analysed in terms of equations (3)-(7), then the force constants can be determined. The bond length re can be determined from the equilibrium rotational constant Be; then the quadratic force constant fi can be determined either from the harmonic wavenumber cu

or from the equilibrium centrifugal distortion constant De ; then the cubic force constant f3 can be determined from aB; and finally the quartic force constant f4 can be determined from x. It is necessary to determine the force constants in this order since in each case we depend upon already knowing the preceding constants of lower order. The values of r e , f 2 , h, andf, calculated in this way for a number of diatomic molecules are shown in Table 2.

It is common practice to present the cubic and quartic force constantsf, and f4 in a slightly different way, in terms of so-called Morse p a r a r n e t e r ~ . ~ ~ ~ The Morse function is an empirical diatomic potential function of the form

V = De(1 -e-ap)2 (30) t Expanding this function about p = 0 gives Vin the form of equation (8) with:

J2 = 2a2De,

.h = -6a3D, = -3af2, (32)

-f The symbol d b b ) is used to denote the derivative of the moment of inertia about the 6 axis in equation (26), and a is used to denote the Morse parameter in equation (30); these quantities should not be confused.


and

f4 = 4- 14a4De = + 7a2fi. (33)

Real molecules do not fit these formulae accurately, but it is convenient to express& andf, in terms off2 in a manner that draws attention to the magnitude of the discrepancy. This is done by defining two Morse parameters a3 and a4 such that

and & = -3a3fip (34)

f p = + 7a4f2. (35)

Both a, and a, have dimension (length)-l, like a, and typical values are close to 2 A-l. Values of a3 and a4 are also shown in Table 2.

In practice, the analysis of experimentally observed spectroscopic data for diatomic molecules in the manner described above is straightforward, and is internally self-consistent in the sense that independent experimental values of o, Be, and De fit equation (27) within experimental uncertainty, and data on different isotopic species give values of re, fi, &, and f p which are indeed found to be independent of the atomic masses. This self-consistency is also reflected in the isotope relations described by Herzberg in his book on diatomic molecules.22 A more complete treatment of the perturbation theory, to high order, shows that the theoretical formulae (23)--(27) are incomplete in that certain small terms have been neglected. These missing terms are given by Dunham, and are discussed by Herzberg on p. 109 of ref. 22; however, they are of magnitude (&/OI)~ x the terms included, and are almost always small compared with experimental uncertainties. They have been neglected throughout the above discussion. When sufficient experimental data are available to make the inclusion of such terms practical, it is usually more profitable to analyse the data by a Rydberg-Klein-Rees calculation 23 to obtain the potential function V ( r ) explicitly in tabulated form. However, this kind of analysis is not possible for polyatomic molecules and is not discussed here.

Table 2 is an extended and re-calculated version of table VI in ref. 5. The values of the anharmonic constants of diatomic molecules as tabulated here are often used to estimate the expected bond-stretching anharmonicity associated with bonds between corresponding pairs of atoms in polyatomic molecules, as discussed in Section 5.

4 Polyatomic Molecules: Method of Calculation

The relation of the anharmonic force field to the spectroscopic observables for a polyatomic molecule is similar to the calculation described above for a

B. Rosen, ‘Spectroscopic Data Relative to Diatomic Molecules’, International Tables of Selected Constants, Pergamon Press, Oxford, 1971, Vol. 17.

Ia G. Herzberg, ‘Spectra of Diatomic Molecules’, Van Nostrand, New York, 1950, Chapter 3.

Is A. L. G. Rees, Proc. Phys. SOC., 1947, 59, 998. E

Potential f u nction

Hami ltonian Effective ham i I tonian

0 bserved spect ru m

V(R): R2 I Fij,

Fijk,. . .

L matrix L tensor

Degenerate pe r tu r bat ion theory

Spectra I analysis

A,BVC,

(A tt) " D , etc ... perturbations

(3

1 Figure 1 Diagrammatic representatioii of the relationship between the potential energy frmctionirig of'a polyatoniic molecule and the spectroscopical observables

4


diatomic molecule, but as one might expect it is more complicated in a number of respects, making the calculation correspondingly more difficult to

The calculation is summarized diagrammatically in Figure 1. One starts with the potential function V( r ) expressed in geometrically defined internal displacement co-ordinates. The first stage is to transform into normal co-ordinates, in terms of which the hamiltonian is formulated, and thus to obtain the potential V(q). For an N-atomic molecule this co-ordinate transformation is a non-linear transformation in 3N - 6 co-ordinates, p rather than a linear transformation in 1 co-ordinate as for a diatomic molecule in equation (14). The second stage is to transform the hamiltonian by a contact transformation or by perturbation theory to obtain an efective rotational hamiltonian for each vibrational state, expressed as a power series in the appropriate vibrational and rotational quantum numbers. The coefficients in the effective hamiltonian are essentially the spectroscopic observables obtained from analysing the spectrum : the w and x values, B and a values, etc., for the molecule. The contact transformation gives formulae relating these constants to the anharmonic force constants in normal co-ordinates; these formulae are similar to, but more complicated than, thecorrespondingformulae(23)--(27) for diatomicmolecules.

The final stage is to relate the coefficients in the effective hamiltonian to the observed spectrum; this is essentially the problem of assignment and analysis of an observed spectrum, and is already discussed in many places in the spectroscopic literature.

The vibration-rotation hamiltonian of a polyatomic niolecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for tc and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance: situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum.

These various aspects of the calculation will now be discussed in more detail, and the whole calculation is reviewed on pp. 140-142. Transformation from Internal to Normal Co-ordinates.-It is an important property of the internal co-ordinates, in terms of which we wish to represent the force field, that they should be geometrically defined, i.e. their definition should be made only in terms of the internal distances between the atoms, and should in no way involve the atomic masses. This is necessary to ensure that the force constants are unchanged for different isotopic species. It is clear that

carry through.

t 3 N - 5 for a linear polyatomic molecule. E*


for a diatomic molecule the bond length Y, and the displacement p = r - Ye,

satisfy this requirement. For a polyatomic molecule the situation is more complicated.

For harmonic force field calculations it is usual to express the force field in terms of internal co-ordinates Ri defined through a linear transformation from Cartesian displacement co-ordinates GxnL (z = x , y , or z ) in the molecule- fixed axis The molecule-fixed axes are located using the Eckart

The equations are written in the form

Ri = C Bgrnhn, arm

0 = c p p x r n , urn

or in matrix notation

(3 6)

(37)

In these equations, i is an index numbering the 3N- 6 internal co-ordinates in equation (36), j is an index numbering the 6 Eckart conditions in equation (37), and rn is an index numbering the N atoms. The (3N- 6) x 3N elements of the B matrix are obtained by standard methods,24 and the 6 x 3N elements of the /? matrix are the coefficients in the Eckart conditions. The co-ordinates Ri defined in this way are rectilinear in the sense that varying the displacement in Ri, holding all other Rj undisplaced, results in varying the displacement of each atom in a straight line from its equilibrium position. It is customary to describe the Ri as ‘bond-stretching’ or ‘angle-bending’ co-ordinates and to choose the B matrix to give them these properties for infinitesimal displacements, but it is clear that each angle-bending co-ordinate will involve some bond stretching in higher order, as shown in diagram (1) illustrating a displacement in the rectilinear bending co-ordinate of a linear triatomic molecule.

I ‘. I

The internal co-ordinates Ri are also related by a linear transformation to the normal co-ordinates, through the L matrix:

Rr = CL:Q,, or R = LQ. (39) T

E. B. Wilson, J. C. Decius, and P. C. Cross, ‘Molzcular Vibrations’, McGraw-Hill, New York, 1955.


In terms of the co-ordinates Rc, 6am, or Qr, the kinetic energy is represented by an effective mass matrix whose elements are constants. In a momentum representation this effective mass matrix is the G matrix, a diagonal matrix of the reciprocal atomic masses m;', and a unit matrix 1, for the co-ordinates Rg, 6am, and Qr, respectively.

At first glance it appears that the internal co-ordinates Rr are geometrically defined in the equations (36)--(38), since the B matrix elements do not involve the atomic masses. However, careful consideration shows that this is not quite true, owing to the mass dependence of the Eckart conditions which appears in the /?matrix elements. The consequence is that the same geometrical configuration of the atoms is generally represented by different values of the displacement co-ordinates Rc for different isotopic species, owing to the fact that the Eckart conditions give rise to a different orientation of the molecule-fixed axes for the different isotopic species. It can also be shown that the transformation between the co-ordinates Rf for different isotopic species is non-linear and that the mass dependence enters only in the higher-order (i.e. the non-linear) terms.

The detailed justification of the preceding paragraph is not easy to give in a few sentences. The problem has been discussed in a recent paper by Hoy, Mills, and Strey;12 the Reporter is unable to add usefully to that discussion at the present time. However, the consequence of the statements in the preceding paragraph is clear. It is that although the co-ordinates Rt provide an acceptable representation of the harmonic (or quadratic) force field, they do not provide an acceptable representation of the anharmonic force field: in terms of the Ri the cubic and quartic force constants will vary from one isotopic species to another. Moreover, it is clear that any truly geometrically defined co- ordinates must be related to all of the co-ordinates Rg, 8am, and Qr by a non- linear transformation. (At least, the transformation cannot be linear for all isotopic species.)

In practice the co-ordinates which are customarily used in anharmonic force field calculations are described as true curvilinear bond stretches and angle bends. For infinitesimal displacements they are identical to the Rt discussed above, but for finite displacements they are defined such that the bond lengths remain unaltered however large the displacement in an angle-bending co- ordinate; hence the adjective true. It is also clear that varying the displacement in one of these co-ordinates will generally cause the atoms to move along a curved path, hence the adjective curuilinear. Diagram (2) illustrates a displacement in the curvilinear bending co-ordinate of a linear triatomic molecule. There may clearly be some choice about how these co-ordinates are defined in more complicated cases ; Hoy et al. l7 have discussed their definition

t


for the five common cases of (i) bond stretching between two atoms; (ii) angle bending between three atoms; (iii) linear angle bending between three atoms; (iv) out-of-plane bending of four atoms; and (v) torsion between four atoms. Further complications arise when redundant co-ordinates are used, since it is clear that redundancy relations among curvilinear co-ordinates will themselves necessarily be non-linear (unlike redundancy relations among the Ri co- ordinates, which are necessarily linear); these problems have also been discussed by Hoy et a1.12 and by Mills.25

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Ri co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction; they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear co- ordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et a1.12 and write the curvilinear co-ordinates with the symbol J?i.

There is a further difference between the rectilinear co-ordinates Rz and the curvilinear co-ordinates JTi which should be emphasized. Figure 2 shows the potential surface of CO, as a function of displacements of one oxygen atom from equilibrium, holding the other two atoms fixed, as determined by Pariseau et a1.* from an anharmonic force field calculation. The contours show the familiar fact that bending is a much softer distortion than stretching; they also show a result which we intuitively expect : that the potential surface tends to make the oxygen atoms bend in the arc of a circle around the central carbon atom. The consequence is that a quadratic force field in curvilinear co- ordinates, without any cubic or quartic terms, gives quite a good representation of the anharmonicity; in rectilinear co-ordinates it is clear that a large cubic term will be required to represent the curvature of the valley. Thus we may expect a simpler representation of the anharmonic force field in curvilinear co-ordinates than in rectilinear co-ordinates. Similar results hold for more complicated molecules.

'The discussion so far may be summarized as follows. There are two reasons for using curvilinear co-ordinates to represent the anharmonic force field of a polyatomic molecule, despite their apparent complexity. The first is that it is only in this way that we obtain cubic and quartic force constants which are independent of isotopic substitution. The second is that in terms of curvilinear bond-stretching and angle-bending co-ordinates we obtain the simplest expression for the force field, in the sense that cubic and quartic interaction terns are minimized. The first reason is compulsive; the second reason is not compulsive, but it does make the curvilinear co-ordinates very desirable.

Lz I . M. Mills, Chem. Phys. Letters, 1969, 3, 267.


0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

2 - 0 B

-0.1

-0.2

-0.2

-0.L

-0.5

-0.t

-0:

-0.1

2

'.

0- 1 0 ' 1 1 I l l 1 I

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Distance from carbon atom/Angstrom

Figure 2 Coiitour map ofpotential energy in COz as a function of the displacement of one oxygen atom, holding the other two atoms undisplaced (Reproduced with permission from J. Chem. Phys., 1965,42, 2335)

The remainder of this section is devoted to formulating the algebra of the transformation from the curvilinear co-ordinates to the normal co- ordinates Qr, and to making the corresponding transformation in the representation of the potential energy.

The normal co-ordinates Qr are first determined from a harmonic force field calculation, in the usual way. They are chosen to diagonalize the kinetic


energy and the quadratic terms in the potential energy; they are defined by equation (39) where the L matrix is chosen to ensure that the vibrational hamiltonian takes the form

where Pr = --ih@/aQT), and the Ar are related to the harmonic vibration wavenumbers by the equation

Ar = ~ X ~ C ~ U ) : . (41)

It is also convenient to define dimensionless normal co-ordinates q r and conjugate momenta pr by

q r = yr1I2Qr, and Pr = -i(a/aqr) = Prly!J2hy (42)

where

in terms of which the vibrational hamiltonian may be written

H = C Shcwr(pf q:) + Vanh(qr.. .). (43) T

The co-ordinate transformation from the f l i to the Qr (or q r ) is a non-linear transformation which we may write in the form

where the * is used to draw attention to the non-linearity of the transformation involved.12

and Lrst, etc., will be called the second and third derivative elements of the L tensor, following the notation introduced by Hoy et ~ 1 . l ~ In equation (44) an unrestricted summation is to be understood over all indices repeated as a superscript and a subscript in the terms on the right-hand side; this convention will be followed in all the later equations of this section. The use of subscript/ superscript notation for the indices on the L matrix and L tensor elements, which is used throughout the equations of this section, simplifies the rather complex algebra involved in the non-linear co-ordinate transformations. Equation (44) may be compared with equation (39) for the co-ordinates Ri, which contains only linear terms.

The first derivative L tensor elements, L:, are determined as the elements of the L matrix from the preliminary harmonic calculation. The second and third derivative L tensor elements have been determined in one of two ways in the literature of anharmonic calculations. The first method involves setting up an

The Li are the familiar L matrix elements; the coefficients


intermediate transformation through the Cartesian displacement co-ordinates Sari. This may be written in terms of the B tensor:

or

H = B*Sa

[compare with equation (3 611. The transformation from Cartesian co-ordinates to normal co-ordinates is

determined in the preliminary harmonic force field calculation; in the customary notation it is written

Sa = M-lf21Q,

where the I matrix is determined from the relation

or

in terms of the elements of the B matrix and the L-l matrix. Substitution of (46) into (45) gives the transformation from Xi to Q r , and in effect determines the L tensor. Most authors have followed this method, using varying degrees of numerical and analytical programming to determine the transformation. Formulae for the B tensor elements may be obtained by differentiating Hg with respect to the Cartesian displacements 6a,, in much the same way that B matrix elements are obtained - except that it is necessary to differentiate to a higher order. Pariseau et aL7 have listed formulae for the B tensor elements for bond stretches and angle bends, although unfortunately their expressions contain many errors and misprints.

The second method of determining the L tensor is that proposed by Hoy et aZ.,12 in which closed analytical formulae are obtained for the second and third derivative L tensor elements in terms of the first derivative elements, by direct differentiation of 19i with respect to Qr. The resulting formulae are given in equations (22)-(36) of ref. 12, and they will not be repeated here. These formulae, which may be directly programmed, greatly simplify the determination of the L tensor. The existence of these formulae emphasize that the L tensor may be entirely determined from a knowledge of the L matrix.

The relationship between the co-ordinates l 7 t , Ri, 6an, and Qr is summarized diagrammatically in Figure 3.

Finally, it is necessary to transform the potential energy function V ( m into the corresponding function in normal co-ordinate space, V(Q) or V(q). The


Cartesian displacements in mol-fixed axes

T B tensor, R=B+ba

True curvilinear internal co-ordinates

B matrix

R.=.Bdn

L tensor.

Rectilinear internal co-ordi na tes

f matrix,

R = L Q I Qr

Normal co-ordinates r H = L * Q

scaling factor T,% 1

Qr

D i mensi on less normal co-ordinates

Figure 3 Relationship bet ween digevent co-ordinates in a polyatomic molecrrle

original expansion is written as in equation (1):

V = $f ijXiHj + (1/6)f'jkXiXjX1c + (1J24)f i j k 'X iX jX~X~ + . . ., (48)

and the corresponding expansion in dimensionless normal co-ordinates is written

V/hc = +mrq; + ( 1 / 6 ) P t q r q a q t + ( 1 / 2 4 ) P U q r q s q t q u + . . (49)

(The quadratic force constants in the normal co-ordinates are, of course, diagonal, i.e. r j s t = co,S:SF.) The expansion (49) is then obtained by substituting (42) and (44) into (48). In fact it is possible to write closed formulae for the force constants 4 in terms of the force constants f, as is done in equations (1 1) of ref. 12. It is an important property of this transformation that a purely quadratic force field in the curvilinear co-ordinates gives rise to quadratic,


cubic, and quartic, etc., force constants in the normal co-ordinates, as antici- pated in our earlier discussion. t

The whole of this section has been concerned with the problem of transforming the potential energy Y from a representation in geometrically defined internal co-ordinates H to dimensionless normal co-ordinates 4, a transformation achieved in the single equation (14) for a diatomic molecule. It will be clear to the reader that programming this transformation is a considerable part of the task of performing an anharmonic calculation on any polyatomic molecule.

Symmetry, and the Number of Independent Force Constants.-As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a1.12 Thus we may choose the curvilinear co-ordinates JI in linear combinations that span the irreducible representations of the point group; we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear co- ordinates :

The symmetrized L tensor, which we denote 2, defines the transformation from S to Q in a manner analogous to equation (44):

Substituting (44) into (50) shows that

The advantage of the symmetrized9 tensor is that, since the two sides of equation (51) must transform in the same way under all point group operations, the element 2 y t will be non-zero only if the proauct of co-ordinates QrQsQt . . . transforms in the same way as the co-ordinate SZ.

If the molecule has no symmetry, it is easy to see that the total number of independent quadratic, cubic, and quartic force constants is given in terms of

t Note that, since the presence of cubic and quartic force constants depends on the co- ordinate representation in use, the description ‘harmonic force field’ and ‘quadratic force field‘ are not synonymous and should be used with care. The adjective ‘harmonic’ should be used only for a force field which has no higher-order force constants in the normal co-ordinate representation, so that the vibrations are truly separable and harmonic in each normal co-ordinate; such a force field will generally have non-zero cubic and quartic force constants in curvilinear internal co-ordinates. The adjective ‘quadratic’ should only be used with a qualification making clear the co-ordinate system to which it refers.


the number of independent co-ordinates n = 3N-6 as follows:

n(n + 1)/2 quadratic, f i j

n(n + l ) (n + 2)(n + 3)/24 quartic, f i j k l .

n(n + l)(n + 2)/6 cubic, f i j k (53)

These are simply the total numbers of combinations of indices i, j , k . . . taken without regard to order. If the molecule has symmetry, the problem is more complicated; the recipe for the number of independent force constants in this case has been given by Watson.26 If we denote the force constants in symmetry co-ordinates Si by the symbols Fij, Fijk, etc., then it is clear that since the potential energy V must be invariant to all point-group operations, F t j k - . - can be non-zero only if the product sisjsk . . . contains a totally symmetric term. Since the order of factors sisjsk . . . is immaterial, the number of independent force constants F of the 2nd, 3rd, 4th . . . power of the co-ordinates is simply the number of totally symmetric terms in the symmetric square, symmetric cube, or symmetric fourth power of the representation based on the n vibrational co-ordinates. Watson 26 has given formulae for determining the number of totally symmetric terms in each case.

Although symmetry reduces the number of independent force constants, they still grow rapidly with the order of anharmonicity. For example, for a CH,X molecule of C,, symmetry there are 12 independent quadratic constants, 38 cubic constants, and 102 quartic constants.

The normal co-ordinates Qr have, of course, the same symmetry properties as the co-ordinates Si, and the force constants qVst - * * are subject to the same restrictions as the Fijfi. . -.

Finally, we should note that in cases where a redundant set of curvilinear co-ordinates 112 are defined, the transformation to curvilinear symmetry co- ordinates becomes more complicated. This difficulty is discussed briefly in ref. 12, but it will not be developed here.

Contact Transformation for the Effective Harni1tonian.-The vibration- rotation hamiltonian of a polyatomic molecule, expressed in terms of normal co-ordinates, has been discussed in particular by Wilson, Decius, and Cross, 24

and by Watson.27* 28 It is given by the following expression for a non-linear? polyatomic molec&e, to be compared with equation (17) for a diatomic molecule :

t The corresponding hamiltonian for a linear molecule is given by Watson in ref. 28.

2 5 J . K. G. Watson, J . MoI. Spectroscopy, 1972, 41, 229. 2 7 J . K. G. Watson, Mol. Phys., 1968, 15, 479. 28 J. K. G. Watson, Mol. Phys., 1970, 19, 465.


The first term is the vibrational kinetic energy and the harmonic part of the vibrational potential energy, and has the form of n = 3N-6 independent harmonic oscillators in n independent co-ordinates. The second term is the anharmonic part of the potential energy. The third term is the rotational kinetic energy and terms arising from vibration-rotation interaction, and the final term U is a small mass-dependent correction to the effective potential energy function, arising out of the derivation of equation (54) as discovered and discussed by The hamiltonian in (54) is expressed in wavenumber units, in terms of dimensionless normal co-ordinates qr and conjugate momenta Pr, and dimensionless angular momentum operators Ja and j,. The mr are the harmonic wavenumbers of vibration, and the terms

3 C cIJ,qf Vanh/hC T

constitute the vibration potential energy in normal co-ordinates as determined in equation (49). The pub (a, 16 = x , y , or z) are the elements of the so-called modified inverse inertia matrix; they are functions of the normal co-ordinates and may be expanded in powers of qr, as discussed by Oka29 and If the molecule-fixed axes are chosen as principal axes in equilibrium, as is usual, this gives

In these equations, I , = I:) is the equilibrium moment of inertia about the a principal axis, and CZFB) = (i31aa/aqr). The scaling factor Yr is given by yr = 2xcwr/Ft. Finally, returning to equation (54), J , and j , are the components of the total angular momentum and the vibrational angular momentum about the a axis, respectively, j , being given by

where the Ct\ are the Coriolis interaction zeta constants.30 The correction term U has been shown by Watson to be given by the expression

U = -(h2/8)(~zz + , ~ y y + pzz) , (60)

which can be expanded in powers of the co-ordinates using equation (55);

2 9 T. Oka, J . Chem. Phys., 1967, 47, 5410. 3 0 J. H. Meal and S . R. Polo, J. Chem. Phys., 1956, 24, 1119.

F


however, order-of-magnitude considerations show that the term U can in practice generally be neglected.

The hamiltonian (54) is expressed as a power series in the co-ordinates q r , the conjugate momenta pr, and the components of the total angular momentum J,. In principle it could be represented as a matrix in (harmonic oscillator) x (rigid rotor) basis functions, in terms of which the matrix elements of the co-ordinates and momenta are known, and the energy levels and wavefunctions could be determined by diagonalizing this matrix. In practice the matrix is infinite and has to be truncated, and although this procedure has been followed by some authors (see, for example, Foord and Whiffen31) it is not generally the most convenient. It is more usual to treat interaction matrix elements between different vibrational basis functions by perturbation theory, and thus to obtain a separate effective rotational hamiltonian within each vibrational state, whose coefficients are slowly varying functions of the vibrational quantum numbers. This effective hamiltonian is closely related to the empirical formulae used to interpret observed vibration-rotation spectra. The coefficients in the effective hamiltonian are given in terms of the coefficients in (54) by formulae which are determined in the perturbation calculation.

The perturbation calculation may also be described as a contact transformation. The original hamiltonian is transformed to a new effective hamiltonian which has the same eigenvalues but different eigenfunctions, to some carefully chosen order of magnitude. This contact transformation of the vibration- rotation hamiltonian was originally studied by Nielsen and co-workers. 3 2 9 33

In more physical language, the effective rotational hamiltonian in each vibrational state is obtained by averaging the original hamiltonian over the vibrational co-ordinates using the true vibrational wavefunctions, obtained by an appropriate perturbation of the harmonic oscillator basis functions. It is an extension of the Born-Oppenheimer separation of the electronic from the nuclear motion, to achieve a separation of the vibrational from the rotational motion.

s1 A. Foord and D. H. Whiffen, Mol. Phys., 1973, 26, 959; A. Foord, J. G. Smith, and

33 G. Amat, H. H. Nielsen, and G. Tarrago, ‘Rotation ’.:?zit% of Polyatomic Molecules’,

34 I. M. Mills, in ‘Molecular Spectroscopy - Modern Research’, ed. K. N. Rao and

8 6 €3. T. Darling and D. M. Dennison, Phys. Rev., 1940, 57, 128. a6 H. H. Nielsen and G. Amat, J. Mol. Spectroscopy, 1958, 2, 152. 37 M. L. Grenier-Besson, J. Phys. Radium, 1960, 21, 555; 1964, 25, 757.

D. H. Whiffen, to be published. H. H. Nielsen, Rev. Mod. Phys., 1951, 23, 90.

Dekker, New York, 197!.

C. W. Mathews, Academic Press, New York, 1972.

W. Gordy and R. L. Cook, ‘Microwave Molecular Spectra’, Interscience, New York, 1970.

3 9 T. Oka and Y. Morino, J. Mol. Spectroscopy, 1961, 6, 472. 4 0 D. Kivelson and E. B. Wilson, J. Chem. Phys., 1952, 20, 1575. 4 1 E. B. Wilson, J. Chem. Phys., 1957, 27, 986. 4 1 J. K. G. Watson, J. Chem. Phys., 1967,46, 1935; 1968, 48, 181,4517. 48 E. B. Wilson and J. B. Howard, J. Chem. Phys., 1936, 4, 260. 44 S . Maes, Cahiers Phys., 1960, 14, 125, 164. 4 6 K. T. Chung and P. M. Parker, J. Chem. Phys., 1965,43, 3865.

Table 3 Spectroscopic constants used in anharmonic force field calculations, and their relation to the force field

Spectroscopic constant tor; harmonic wavenumber xrs; anharmonic constant gttf ; anharmonic constant 'yrs ; Darling-Dennison constant

rtt'; vibrational 1-doubling constant

Ae, Be, Ce; asymmetric top

Ae, Be; symmetric top rotational

(ACt)e; symmetric top z axis (B(t)e; asymmetric top or symmetric

a+, a?, a?; asymmetric top a$, aB; symmetric top a P ; symmetric top qt ; I-doubling constant

rotational constants

constants

top xy axis Coriolis constants

rt; 1-doubling constant

DJ, DJK, D K ; centrifugal distortion

dJ, ~ J K , A K , 65, 6 ~ ; centrifugal

r=srs HJ etc. ; sextic distortion constants

constants, symmetric top

distortion constants, asymmetric top

"?tJ, 'OK

Order of magnitude

1 x2

x2

x2

x2

x2

x2

x2

x2

x4

x4

x4

x4

x6

x6

xe x6

x10

Quantum numbers in efective hamiltonian

(or + $4) (or + +dr)(vs + 3ds) Itltf Off-diagonal: hug = +2,

Off-diagonal: Alt = + 2, AuS = -2

Art* = -2 J:, J,a,J,a k2, J(J + 1)-k2

- 2klt Off-diagonal: Av7 = + 1 ,

Jl(vr + &dr), etc. k2(of + idf ) , etc.

Off-diagonal: Art = +2,

Off-diagonal: Art = +2,

J2(J + 1)2, J(J + l )k2, k4

A v ~ = -1, also Ak = 1 . . .

-2klt(vr + 3dr)

Ak = +2

Ak = - 1

See ref. 42

Ja Jb J y Jd J(J + l)klt, k31t J6

Ref. 24 34 34 35

29, 36, 37

38

38

30, 34 30, 34

39,34

34 29, 34, 37

37

40-42

42

40,43 44 42,45

I36 Quaiiturn Chemistry

In practice, the result of the perturbation treatment may be expressed as a series of formulae for the spectroscopic constants, i.e. the coefficients in the transformed or effective hamiltonian, in terms of the parameters appearing in the original hamiltonian, i.e. the wavenumbers mr, the anharmonic force constants 4, the moments of inertia I,, their derivatives app), and the zeta constants 52;. These formulae are analogous to equations (23)-(27) for a diatomic molecule. They are too numerous and too complicated to quote all of them here, but the various spectroscopic constants are listed in Table 3, with their approximate relative orders of magnitude, an indication of which parameters occur in the formula for each spectroscopic constant, and a reference to an appropriate source for the perturbation theory formula for that constant.

It is instructive to consider two of the formulae for the spectroscopic constants in more detail, and for this we choose -a: and xrs for an asymmetric top, these being respectively the coefficients of (ur + +)J:, the vibrational dependence of the rotational constant, and (vr + +)(us + +), the vibrational anharmonic constant quadratic in the vibrational quantum numbers. As for diatomic molecules these two types of spectroscopic constant provide the most important source of information on cubic and quartic anharmonicity, respectively. The formulae obtained from the perturbation treatment for these two coefficients in the effective hamiltonian are as follows:

Equation (61) may be compared with equation (26) for a diatomic molecule. Apart from complications arising from the presence of 3N- 6 rather than just 1 normal co-ordinates, the first and third terms in square brackets in equation (61) correspond exactly to the two terms for a diatomic molecule, and their physical origin is similar (see Section 3). The second term in square brackets

-i This formula is correct only for Y is; see ref. 34 for more complete formulae.


arises from Coriolis interactions, which do not occur in a diatomic molecule. They arise from cross terms in equation (54) of the type

as discussed further under the heading of Coriolis resonance (p. 139). It is clear from equation (61) that if the harmonic force field of a molecule is

known, each observed a: constant gives a linear relation between the cubic anharmonic constants p r s , where s ranges over all normal modes in the molecule (subject to symmetry restrictions).

Equation (62) may be compared with (24) for a diatomic molecule. Note first that there is a change of sign in the conventional definition of x for a polyatomic molecule: the term in the vibrational energy is written +xrs(ur + +)(us + 3) for a polyatomic molecule, but it is written -x(u + +)2 for a diatomic. Again the two expressions are comparable, and the physical origins of the terms are similar (see Section 3), except for the extra terms involving C2 in (62) arising from Coriolis effects due to the terms in j in (54). In particular we note that Xrs is linear in one quartic 4 only, and quadratic in a number of cubic + values. Thus given the quadratic and cubic force field, each Xrs determines the single quartic constant q W S (and xrr determines 4rrw). Unfortunately none of spectroscopic constants normally observed to the accuracy of Xrs involves the constants 4rsss with Y # s, and in fact there is no experimental method of determining such constants at this time.

There are many other spectroscopic constants, as listed in Table 3, which provide information on the anharmonic force field in particular cases. Some examples are discussed in Section 5.

Relation to the Observed Spectrum; Resonances.-The contact transformation described briefly in the last section yields an effective hamiltonian which, ideally, gives directly an expression for the vibration-rotation energy levels of the molecule in the form of a convergent power-series expansion in the vibrational and rotational quantum numbers. The coefficients in the effective hamiltonian are the spectroscopic constants determined from the analysis of vibration-rotation spectra (the final step in the diagram of Figure 1, p. 122). The analysis of the spectra is, of course, a considerable step in the chain: our present understanding of this step is the culmination of many years of research by many spectroscopists on the spectra of polyatomic molecules. Much of the subject is covered in Herzberg’s book on infrared and Raman spectra of polyatomic molecules, 4 6 although the subject has also advanced considerably since that book was written.47

Although it is not our purpose to review this subject here, one aspect of this step in the calculation must be mentioned because of its importance in determining the anharmonic force field and its relationship to the contact

4 6 G. Herzberg, ‘Infrared and Raman Spectra of Polyatomic Molecules’, Van Nostrand,

47 A. G. Maki, Ann. Rev. Phys. Chem., 1969, 20, 273. New York, 1945.


transformation. This is the presence of various kinds of resonance between the different vibrational levels of a molecule when they happen to lie close together in energy. This phenomenon becomes more and more common as the number of atoms and hence the number of vibrational levels in the molecule increases, and it complicates both the analysis of the spectrum and the theory of the contact transformation discussed in the last section.

Resonances occur when there are interaction terms in the original hamiltonian connecting two different vibrational levels of the harmonic basis functions, whose magnitude is not sufficiently small compared with the separation of the unperturbed energy levels to make a perturbation treatment of the terms appropriate. It is clear that some element of judgment is called for in deciding the point at which perturbation theory is no longer appropriate. If one persists in using a perturbation treatment even when the resonance interaction is comparable with the unperturbed energy level separation, it becomes necessary to go to a high order of perturbation theory to achieve an acceptable level of accuracy in the theory; there comes a point when exact solution of the hamiltonian matrix for the pair of interacting vibrational levels is preferable - even though it may be necessary to solve the problem numerically.

The complication in analysing the spectrum is that the energy levels, and hence the spectral lines, can no longer be fitted by a power series in the quantum numbers that converges stcficiently rapidZy to form a simple pattern of energy levels and spectral lines, so that simple analytical formulae cannot be used, and assignment becomes a problem. Usually the spectrum can only be analysed in conjunction with an appropriate theoretical treatment of the hamiltonian for the interacting levels.

Despite the complication which resonances introduce into the analysis of a spectrum and the theoretical treatment of the hamiltonian, when they can be analysed they often give valuable information on the force field which cannot be obtained directly in the absence of a resonance. We consider briefly the two commonest types of resonance interaction, Fermi (or anharmonic) resonance and Coriolis resonance, to illustrate this point.

Fermi Resonance. Suppose the anharmonic potential V(q) in equation (54) contains a cubic term 4rTSqFq8; the effect of such a term is to produce an interaction between vibrational states differing by Aur = + 2 and Aus = T- 1, since

<ur + 2, usl#rrsq;qslur, us + 1> = $ r r s [ ( ~ r + l ) ( ~ r + 2)(us + 1)/811/’.

(63)

Here Iur, us> denotes an unperturbed harmonic oscillator function, and we assume neither vibration is degenerate. For example, typical values of qW might be of the order 30cm-l; if the separation between the unperturbed vibrational levels (vr = 2, us = 0) and (ur = 0, us = 1) were also about 30 cm-1 the interaction would result in a ‘pushing apart’ of the energy levels of about 7 cm-l each way, giving an observed separation of about 44 cm-l.


A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule Avr = f 2, Avs = T 1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory; it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants

Xrr(Dr + %)2 and XTS(Ur 4- $)(a8 4- (64)

in the formula for the vibrational energy levels, and also to the terms

in the quadratic vibrational dependence of the rotational constants. Terms such as (65) are regarded as too small to detect in the analysis of most vibration-rotation spectra. Terms like (64) are of course important as a source of information on the force field, and up to a certain point Fermi resonance may be thought of an an exceptionally large contribution to particular terms in the expression for certain of the x values.

It is clear that a strong Fermi resonance of the type described above may in principle be analysed directly to give an ‘observed’ value of the single anharmonic constant $rrs; in Section 5 (p. 143) we give examples of such analyses, and their use in anharmonic calculations. In the absence of Fermi resonance, information on the cubic constants like qWS comes mainly from the vibrational dependence of the rotational constants a:, which determine linear combinations of the cubic constants $rrs as described earlier.

Coriolis Resonance. The hamiltonian (54) contains cross terms in the angular momenta of the type:

- 2(h2/2hcIa) J, j , =

where the a axis has been identified with the b axis of inertia to put the term into a more familiar notation. Such terms produce an interaction between vibrational states differing by AuT = k 1 and Avs = T 1, from the matrix elements of the operators qrp6 and q6pr. The order of magnitude of the coefficient is that of a rotational constant, and the magnitude of the cross term is roughly given by multiplying the rotational constant BC!;: by an appropriate rotational quantum number representing the magnitude of A. When the cross term is large compared with the unperturbed separation of the interacting levels, the situation is described as a Coriolis resonance, and the perturbation treatment of these terms adopted in the usual contact transformation is inappropriate. It is then necessary to diagonalize the hamiltonian matrix for the


interacting levels without the use of perturbation theory. If, on the other hand, the interaction (66) were small or the separation of the unperturbed levels were large compared with the interaction, it can be shown that, in second-order perturbation theory, equation (66) would contribute terms to the vibration- rotation energy of the type represented by the C2 term in equation (61).

The observed a values are of course generally an important source of information on the cubic anharmonic force field. However, in the presence of a Coriolis resonance the particular oc values involved are dominated by the harmonic Coriolis contribution arising from equation (66), and analysis of a Coriolis resonance essentially gives information on the constant B&::, or rather the effective value of this constant in the vibrational states involved. Up to a certain point a Coriolis resonance may be thought of as an exceptionally large contribution to two particular a values, and in the presence of a Coriolis resonance it is not generally possible to obtain reliable information on the anharmonic contributions to these a values. However, the information on the zeta constant may be valuable in determining the harmonic force field.

In symmetric top and spherical top molecules there are exact degeneracies in the unperturbed vibrational levels arising from symmetry, and Coriolis perturbations between such levels produce ‘first order’ effects in the rotational structure. For symmetric top molecules the constants A[::), t e , relating the components of a degenerate mode (qtl, qt2), may be quite accurately determined for appropriate vibrational states of each degenerate ~ i b r a t i o n . ~ ~ A similar situation holds for spherical top molecules. 46 Symmetric and spherical top inolecules also exhibit a number of other resonances which essentially arise from vibrational degeneracies due to symmetry. Examples are I-doubling effects, vibrational 2-doubling effects, and a number of similar terms listed in Table 3 (p. 135). In many cases these have been analysed theoretically and observed experimentally to give useful information on the force field, as shown by some of the examples in Section 5.

Anharmonic Force Constant Refinements.-The preceding parts of this Section 4 constitute an outline of how the vibration-rotation spectrum of a molecule may be calculated from a knowledge of the force field in some set of geometrically defined internal co-ordinates, denoted V( r ) in general in this Report [but denoted V ( R ) in the special discussion on pp. 126-1321. In practice we wish to solve the reverse problem: we observe the vibration- rotation spectra, and we wish to deduce the force field.

The problem is similar to that involved in harmonic force field calculations, but more difficult in almost all respects. In simple cases one may attempt to solve directly, or graphically, for some of the anharmonic Q values using the observed values of the spectroscopic constants in equations like (61) arid (62). These may then be related to fvalues through the L tensor as described on pp. 124-132. However, such methods are of only limited value. The more general method of calculation is to attempt an anharmonic force field refinement, in which a trial force field is refined, usually in a large non-linear least- squares calculation, to give the best agreement between the observed and


calculated spectra. The necessary jacobian matrix elements (derivatives of the spectroscopic constants with respect to the anharmonic force constants) may be calculated by finite difference methods, or analytically.12 As in harmonic force field refinements the calculation is plagued by problems of indetermi- nancy in all but the simplest cases: it is hardly ever true that the data are sufficient to determine all possible symmetry-allowed force constants without imposing some constraints in the form of a model force field, or without transferring force constants from simpler molecules in which they have been previously determined.

The refinement calculation may be carried out in a variety of ways, and a few general remarks should be made before we consider particular examples. We wish to determine re, fi, f3, and f4, where these denote symbolically the equilibrium structure (which may be thought of as the linear force field), the quadratic, cubic, and quartic force field. (Terms higher than quartic are not considered here.) Each set of data depends on all constants up to a certain order, as shown in Table 3 ; for example, Ae, Be, and Ce depend only on re, the w values depend on re and&, the a values on re, fi, and f3, and the x values on re, fi, f3, and f4. Ideally one should refine all data simultaneously to all force constants (including the equilibrium structure), but in practice the calculation has to be broken down into steps. Thus usually the equilibrium structure re, or some approximation to re, is determined first from the rotational constants; then the quadratic force field fi is determined from the w, C, and z values holding re constrained; then the cubic force fieldf, is determined from the ct values holding re and fi constrained; and finally the quartic force field f, is determined from the x values holding re, f i , and& constrained. (This should be compared with the discussion for diatomic molecules at the end of Section 3.)

Often the original structure determination will have involved some un- corrected vibrational averaging effects: it may be an ro or an rs However, once fs, or some approximation to fs, has been determined it is possible to correct ro to re and obtain an improved equilibrium structure (in most cases this correction can be made directly from the ct values without going through a cubic anharmonic calculation, but in some cases the calculation will allow unobserved a values to be determined, perhaps for other isotopic species, etc.). Similarly, it is often true that the harmonic fieldf, is calculated from the observed fundamentals (the v values) rather than the harmonic vibration wavenumbers (the w values), for want of information on the corrections. However, oncef,, or some approximation tof,, has been determined, it may be used to calculate a complete set of x values and hence to calculate all the corrections to obtain the w values. Thus the calculation of re and fi may be improved from a knowledge of f3 and f4.

Some calculations have been reported in which f3 and f4, and occasionally f 2 , f 3 , andf,, have all been refined simultaneously, although almost all calculations have been made with re constrained. It is not clear that simultaneous refinement of fi, f,, and f4 has any advantage. Hoy et al. l2 have observed that there is a technical advantage in constraining re and fi while refiningf, and f4,


since the L tensor depends only on re and fi (see p. 129) and thus need only be calculated once. Refining the cubic force field alone is particularly simple, since the observed data on& (notably the a values) are generally linear functions of the cubic force constants, so that the refinement converges in a single step.

Another choice appears in the alternative of refining either to the observed wavenumbers or frequencies of spectral lines, or to spectroscopic constants (w, x, B, a . . .) determined from a preliminary analysis of the spectrum. We may think of the first method as a singZe-step refinement, and the second as a two-step refinement in which the many observed lines in the spectrum are first reduced to a smaller number of spectroscopic constants, and in the second step these are further reduced to the force constants. If all refinements are made by least squares the two-step method should give the same results as the single-step method, but only if the weight matrix used in the second step is taken as the inverse of the variance-covariance matrix of the spectroscopic constants determined from the first step; this is the statistically correct procedure for weighting the input to the second step. In practice the second stage of a two-step refinement is often carried out using a diagonal weight matrix, which is not correct ; single-step and two-step refinements may then give somewhat different answers. t

However, in practice there may be good reasons for doing the refinement in two (or more) steps. Often, computer limitations make this essential. Also the different steps of the calculation may properly be carried out by different workers in different laboratories at different times, each using their own particular expertise. For example, the three rotational and five centrifugal distortion constants of an asymmetric top molecule may be determined from a preliminary analysis of as many as 100 observed microwave lines: it is simply not feasible to treat this as a small part of a huge single-step refinement in which all observed spectroscopic data on all isotopes are considered simultaneously. Moreover, small higher-order constants (e.g. sextic centrifugal constants) may be included in the first step of analysis in order to obtain more reliable values of the lower-order constants, but the higher-order constants may not be used in the second step because they relate to higher-order force constants which we are not able to determine.

Finally, as in all least-squares calculations, differences may arise through different methods of weighting the input data, and through different methods of analysing errors.

Most of these problems are considered further in relation to calculations on particular molecules in the followiiig sections.

t Albritton et al.*% have recently discussed these effects inla slightly different context.

48 D. L. Albritton, W. J. Harrop, A. L. Schmeltekopf, R. N. Zare, and E. L. Crow, J. Mof. Spectroscopy, 1973, 46, 25, and following papers.


5 Results and Discussion Table 4 gives a list of molecules on which anharmonic force constant calculations have been reported. Perhaps the first notable fact is that the list is not long; although we have attempted to make the list complete as of 1973, there

Table 4 Molecules for which anharmonic force field calculations have been reported

Molecule

co2 cs2 ocs "0 FCN ClCN BrCN ICN HCN HCP

CsOH RbOH }

Ref. 1, 2, 8, 11,49; a * d

2, 5, 10; a * d

30, 52; b * d

54,55; 57; c * d

57, 59; c , d

57; G d

57; cyd

63-65; a

63; c

68; b * e

1, 2, 5, 60-70; a

60; a

60; a

73; 5,75-77; a

69,76; C

78,79; c

Ref. 50; 51; e 53; 56; e

58 ; e 60, 61; b

62;

66; 67; c

7; = 71 ; cpe

72; 74; c

0 Well determined general quadratic + cubic + quartic force field. b Moderately well determined force field with some assumptions about the model. C Limited anharmonic force field with assumptions. d All these linear triatomics show a strong Fermi-resonance between vl and 2vp, see text. C Cubic force field only.

I. Suzuki, J. Mol. Spectroscopy, 1968, 25, 479. T. Tanaka and Y . Morino, J. Mol. Spectroscopy, 1970, 33, 538.

a1 Y. Morino and S. Saito, J. Mol. Spectroscopy, 1966, 19, 435. Y. Morino and T. Nakagawa, J. Mol. Spectroscopy, 1968, 26, 496.

ba H. Takeo, E. Hirota, andY. Morino, J. Mol. Spectroscopy, 1970,34,370; 1972,41,420. I. Suzuki, J. Mol. Spectroscopy, 1969, 32, 54. D. F. Smith, J. Overend, R. C. Spiker, and L. Andrews, Spectrochim Acta, 1972, 28A, 87. H. Shoje, T. Tanaka, and E. Hirota, J. Mol. Spectroscopy, 1973, 47, 268.

67 C. B. Muchison and J. Overend, Spectrochim. Acta, 1971, 27A, 1509, 1801, 2407; V. K. Wang and J. Overend, ibid., 1973, 29A, 1623.

68 H. Takeo, R. F. Carl, and P. W. Wilson, J. Mol. Spectroscopy, 1971, 31, 464. A. G. Robiette and A. D. Haner, to be published.

O 0 G. Strey and I. M. Mills, to be published. I. Suzuki and J. Overend, Spectrochim. Acta, 1969, 25A, 977.

6* M. Bertram, Ph.D. thesis, University of Reading, 1973. G. Strey and I. M. Mills, Mol. Phys., 1973, 26, 129.

64 T. Nakagawa and Y . Morino, Bull. Chem. SOC. Japan, 1969,42, 2212. O 6 I. Suzuki, M. A. Pariseau, and J. Overend, J. Chem. Phys., 1966, 44, 3561.


are doubtless a number of ommissions, but nonetheless the list reflects the difficulties of these calculations, and the fact that the number of laboratories in the world attempting such calculations can be counted in single figures. Linear Symmetric Triatomic Molecules.-The simplest polyatomic molecules for such calculations are linear symmetric triatomics ; specifically, CO, and CS, are the only molecules on which results have been reported. The general quartic force field contains Ire + 3fi + 3f3 + 6f4 parameters; Table 5 shows the relationship of the cc and x values (the primary spectroscopic observables) to (i) the anharmonic force constants 4 in normal-co-ordinate space, and (ii) the anharmonic force constants f in curvilinear internal co-ordinates.

Table 5 Dependence of spectroscopic constants on anharmonic force constants for CO,-type moleculesa

a:

CI 2

4 2 2 2 2 , f : * l ,

.YI I

a Note that the quartic force constants do not contribute to the combination xa% + 3x11.

6 e S. G. W. Ginn, S. Reichman, and J. Overend, Spectrochim. Acta, 1970, 26A, 291. 6 7 A. J. Dorney, A. R. Hoy, and I. M. Mills, J. Mol. Spectroscopy, 1973, 45, 253. 6 8 D. R. Lide and C. Matsumura, J. Chem. Phys., 1969,50, 3080. 6 D D. F. Smith and J. Overend, J. Chern. Phys., 1971, 55, 1157; Spectrochim. Actn, 1972,

28A, 2387. D. F. Smith and J. Overend, Spectrochim. Acta, 1972, 28A, 471. M. Otake, C. Matsumura, and Y . Morino, J. Mol. Spectroscopy, 1968, 28, 316; M. Otake, E. Hirota, and Y. Morino, ibid., 1968, 28, 325.

7 2 S. Reichman and J. Overend, J. Chem. Phys., 1968, 48, 3095. i 3 R. N. Dixon, J . Mol. Spectroscopy, 1969, 30, 248. 7 1 A. R. Hoy, M. Bertram and I. M. Mills, J. MoI. Spectroscopy, 1973, 46, 429. 7 6 A. Barbe. C. Secroun, and P. Jouve, J. Phys. (Paris), 1972, 33, 209. 7 8 Y. Morino, Pure Appl. Chem., 1969, 18, 323. 7 7 79. Morino, Y . Kikuchi, S . Saito, and E. Hirota, J. Mol. Spectroscopy, 1964, 13, 95.

J. C. D. Brand, R. W. Redding, and A. W. Richardson, J. MoI. Spectroscopy, 1970, 34, 399. D. Papousek and J. Pliva, Cod. Czech. Chem. Comm., 1964, 29, 1973.

Harmonic and An harmonic Force Field Calculations 145

The notation is essentially self-explanatory; e.g. &$lll is the coefficient of 413, +4122 is the coefficient of q1(qZa + q;J, etc., where the normal co-ordinates are numbered in the conventional order; +&-* is the coefficient of (6r16r$ + 8r:6r2), etc. It should be realised that only the dependence of a? or Xrs on the force constants of highest order is given (e.g. each Xr8 depends on the cubic and quadratic constants and the structure, and each a? on the quadratic constants and the structure, in addition to the relations given in the table). In addition, both CO, and CS2 show a strong Fermi resonance between w1 and 2v2, and corresponding higher polyads, which must be taken into account directly in analysing and fitting the data; the resulting Fermi resonance parameter relates directly to &, and hence &.

Table 6 gives numerical results for CO, and CS, from the best recent papers. (It is notable that the first calculations on any polyatomic molecule were those reported by Pliva for C02 in 1958; he obtained closely similar results.)

Table 6 Anharmonic force fields in curvilinear internal co-ordinates for CO, and for CS2a

CO, cs2

1.160 (1) 1.553 (1) Ref. 1 1 Ref. 49 Ref. 10

j JaJ 16.032 16.022 (4) 1.553 (1) jLt/aJ 1.25 1.261 (4) 0.636 (4) faalaJ 0.7826 0.7850 (4) 0.2849 (1)

j L r / a J A-3 -113.16 -113.94 (16) -44.64 (8) frrr '/aJ A-3 -1.64 - 3.91 (20) - 1.58 (8) f r aJaJ A-1 -1.195 - 1.218 (6) - 0.734 (8)

frrrr/aJ A-4 606.6 630.0 (4.0) 212 (3.6) frrrrt laJ A-4 -6.48 22.1 (1.3) 8 . 3 (3.1) frrrfr /aJ -8.96 12.1 (1.6) 6.7 (3.2) frraa/aJ A-2 0.734 2.01 (25) 1.72 (16) frrpaa/aJ 2.838 3.74 (24) 2.28 (20) f a a J i f . J 4.363 4.264 (50) 1.908 (72)

a Standard errors are given in brackets in the last digits quoted, except for Chedin and Cihla," who do not give uncertainties. Note that fa,,, is based on a definition of the bending co-ordinate as sin 0, where 0 is the bending angle; see discussion in refs. 12 and 63.

Table 5 suggests that one might hope to determine all the constants in the most general anharmonic force field without too much difficulty. The comparison of Suzuki's with Chedin and Cihla's results in Table 6 gives some feel for the reliability of the results obtained. These two calculations were made in different ways (see the original references); although both refined the force field to fit all observed vibrational levels and rotational constants, Suzuki used an up-to-quartic force field, where Chedin and Cihla used an up-to-sextic force


field and a quite different method of calculation. Theirs is the only reported work on any polyatomic molecule that goes above quartic terms in the potential.

It would appear that the uncertainty in these constants might realistically be thought of as about 10 x the standard errors obtained from the calculation. This is possibly due to correlation effects, which are too complicated to present in these results. It is not clear which of the calculations is more reliable where they differ, and probablyfrrrf and most of the four quartic interaction constants are uncertain even as to sign.

One conclusion is clear: the dominant cubic and quartic interaction force constants are those associated with bond stretching, and these are not dis- similar to those of the corresponding diatomics. The same conclusion follows from a study of all other published data, and comparisons between bond- stretching anharmonicity in related molecules are discussed further below (see Table 15).

Cubic and quartic interaction force constants may be visualized in the following way, e.g. for CO,:

Thus if we plot the force constant f r r in one CO bond against the stretching co-ordinate in the other, frrr l gives the gradient of the plot at equilibrium. Since& only changes from 16 to 19 aJ A-2 in going from OCO in equilibrium to 0 - - * CO, i.e. the CO diatomic, see Tables 2 and 6, we expect the gradient f r r r l to be of the order of about 2 aJ A-3. This and similar arguments suggest that the magnitudes of the interaction cubic and quartic force constants obtained are exactly what should be expected, although the numerical values are often uncertain to something of the order of their magnitude.

Linear Unsymmetric Triatomic Molecules.-Reducing the symmetry from Dtoh to CmV, as in N20, OCS, and HCN, increases the number of parameters in the general quartic force field to 2re + 4f2 + 6f3 + 9f; Table 7 shows their relationship to the primary spectroscopic observables. It is clear that problems of insufficient data to determine the general force field are already on the horizon: for example, data from at least two different isotopic species must be combined in order to determine&-, f r r R , f r ~ ~ , and f R R R from the observed values of a: and a:. In practice, of course, substitutions like 14N for 15N tend to change the spectroscopic constants by only a small fraction, and conversely the observed data on the constants of such isotopic species tend to give nearly parallel information on the force field to that obtained from the parent species. For these reasons the anharmonic force field of molecules like N,O is much less well determined than that of COz. These effects are apparent in the uncertainties obtained on the force constants in the refinement calculations referred to in Table 4.


Table 7 Dependence of spectroscopic constants on anharmonic force constants for OCS-type molecules

Xl2 41122,

For HCN the situation is somewhat better, because the data on DCN are much more effectively independent of the HCN data. This molecule has also been the subject of much high-resolution spectroscopic study, so that the vibration-rotation energy levels are particularly well known and its vibrational spectrum is free of accidental resonances. Table 8 compares the results of three quite different calculations. The calculation by Strey and Mills is the most recent, and was based on the latest spectroscopic data; the refinement was made to tc and x values rather than to the vibrational levels and rotational constants as used by both the earlier workers. Strey and Mills also constrained 3 of the quartic interaction constants to zero, and refined to cubic and quartic force constants in a separate calculation to the quadratic refinement. The level of agreement between the calculations leads to conclusions rather similar to those made above for COz; in particular, standard errors should be multiplied by at


Table 8 Force constants in curvilinear internal co-ordinates for HCN a

Ref: 63 1.0655 I . 1532

6.251 (1)

18.703 (3) -0.200 (1)

0.2596 (1)

- 35.37 (48) +0.04 (19) +0.41 (67)

-0.19 (12) - 0.65 (9)

-125.95 (I .35)

+ 181.4 (9.8) +1.4 (3.5)

f580.2 (29.9) zero* zero”

+0.28 (72) zero*

+ 1.07 (3)

+0.11 (74)

Ref. 64 - -

6.244 (3)

18.707 (16) 0.2598 (4)

-0.211 (6)

- 33.76 (42) - 0.99 (22)

- 125.09 (96) + 0.09 (44)

-0.11 (6) -0.66 (4)

+153 (7)

+650 (28) +4.3 (1.6)

f 8 . 1 (2.4) +4.4 (7.2) -0.44 (28) - 0.04 (40) +0.28 (12) + 1.07 (2)

Ref. 65

6.230 (30)

18.776 (17) -0.216 (80)

0.2592 (30)

-36.51 (90) - 0.64 (54) - 1.33 (80)

-115.66 (2.16) - 0.26 (14) -0.54 (10)

+211 (17) +4.7 (12)

+420 (165) +6.3 (12.0)

+28.3 (46.2) + 0.80 (92)

+ 0.39 (86) 4-1.11 (10)

-0.76 (1.84)

a Standard errors in brackets in the last digits quoted; * indicates a constrained force constant.

V(rcHI for HCN

Morse parameters: kJ moi-1 103cin-1 a = 1.961 A-j

De = 0.8130 aJ = 5.074.eV = 0.1865 hartree = (490 kJ mol-1) NA-1

= (40927 cm-l) hc

600 -

400 -

ZOO -

Morse parameters: kJ moi-1 1 O3 cm-1 a = 1.961 A-j

De = 0.8130 aJ = 5.074.eV = 0.1865 hartree 800 - = (490 kJ mol-l) NA*I

I = (40927 cm-l) hc 60 -

600- 50- v ( 2 + 3 + 4 )

40 - 400 -

30 -

20 -

10-

ZOO -

I ” ” I ”

0.0 0.5 1 .o 1.5 2.0 A Figure 4 Potential energy as a function of C-H stretching in HCN


least 5 to obtain a realistic assessment of the uncertainty in the force constants. The fit achieved to the observed spectroscopic data, for which the reader is referred to the original papers, is impressive by any standards.

Three different ‘sections’ of the potential function of HCN are shown in Figures 4,5, and 6. In Figures 4 and 5, which show V(YCH) for fixed RCN and vice versa, V(2t3+4) denotes the potential calculated from Strey and Mills’ quadratic + cubic + quartic force field. V ( m ) denotes a Morse curve calculated by taking the Morse constant to be $(a3 + a*), calculated from the cubic and quartic force constants using equations (34) and (35). The deviation of Vm) from V2+3+4) indicates the stage at which higher than quartic terms in the force field become important, and also presumably the point at which higher-order spectroscopic constants such as yrst and y: become important in their effect on the observed spectrum. It is clear that these effects may already be important for z1 = 3 of the CH stretching vibration, although they are probably unimportant for any of the observed excited states of CN stretching. The dissociation constants given are those obtained by extrapolating the Morse curve, and they should certainly not be regarded as significant to better than +lo%. The results can, however, be compared with thermochemical

V(rcN) for HCN

kJ mol-1 1 O3 cm-1

1200

1000

800

600

400

200

1 00

90

80

70

60

50

40

30

20

l o 1

Morse parameters: a = 2.175 A-7 D, = ( 1 187 kJ mol-1) NA-I

= (99523 cm-I) hc

I , - ‘ I ~ * . * I . . . * , * . , S , . \

0.0 0.5 1 .o 1.5 2.0 A

Figure 5 Potential energy as a function of C-N stretching in HCN

\

\ \ \

- 0.5

- A

- 0.0

- - 0.5

-0.8 \

by extrapolation

r I C N ) = 1.1735 A at 60" i.e. + 0.0203 A at 60"

Figure 6 Potential energy contours as a function of the position of the N nitrogen atom in HCN, holding C and H atoms fixed

data from the JANAF tables,*O from which we find

HCN + H(2S) + CN(X2Z), D,"/hc = 42 800k900 cm-1

HCN + N(4S) + CH(X"), D,"/hc = 77 6004900 cm-1.

In the first case the agreement is certainly as good as might be expected. The discrepancy in the second case is probably due to the fact that a linear dissociation of the CN bond could not lead to a ground-state X ZIT CH diatomic: if the N atom is obtained in the *S ground state, the CH would have to be in a 4C- state. This is not known experimentally, but it must of the order of 20 000 cm-1 above the ground state, which would explain the large discrepancy from Figure 5.

JANAF tables, 2nd edn, NSRDS-NBS 37, June 1971.


a; PI22

NP

NP

””] $239 -

Linear Tetra-atomic Molecules.-Acetylene and cyanogen appear to be the only linear tetra-atomic molecules on which calculations have been made. In this case the number of parameters in the general force field becomes 2re + 6f2 + l l f , + 23h. The problem of determining all of the quartic force constants in the general force field looks unlikely to be well determined, although there are apparently good data on a variety of isotopic species for

r;frrr

frrrt

j ; ra

f r r ’a

fra,

--fama

x11

X I 2

x22

XlS

x 2 3

frrrr

x 3 3 43333 F 3 3 3 3 * frrrrt f7rrfrr

a E.g. HIO, SO,. Fzjr . . . denote force constants in curvilinear symmetry co-ordinates; &, S,, S, denote symmetric stretch, angle-bend, and asymmetric stretch. The force constants marked with an asterisk are those constrained to zero in the force fields of Table 10.

152 Q uan ttim Chemistry

acetylene. Strey and Mills’ work6* was based on a general cubic force field, but only 8 quartic force constants, the remaining 15 being considered to zero; Suzuki and Overend 61, however, have reported an unconstrained calculation for acetylene,

Bent Triatomic Molecules.-Calculations have been reported for many bent triatomic molecules (see Table 4). The general force field contains 2re + 4fi + 6f3 + 9f4 parameters, the relation to the primary spectroscopic constants being shown in Table 9. The fact that these are asymmetric top molecules, for which a$, a:, and a: can all be determined (generally from the microwave spectrum for the heavier molecules), means that 9 a values are available from each isotopic species to determine the 6 cubic force constants, so that the cubic force field is generally well determined. For the quartic force field the situation is much less satisfactory; the experimental data on the anharmonic constants X r s are generally incomplete, and are in any case insufficient to fix all the quartic constants without good isotopic data.

Table 10 shows the force fields of HzO, H2S, and H,Se as determined by Strey and Mills,6o with 3 quartic interaction constants constrained to zero. This calculation was refined to co, a, and x values, the quadratic force field

Table 10 Force constants60 in cirruilinear internal co-ordinates a for H20, H2S, and H,Se

Force constant H2O H,S H,Se relA 0.957 1.336 1.460 4 d e g 104.55 92.12 90.53

frr/aJ A-2 8.454 (1) 4.284 (2) 3.507 (10)

fralaJ A-1 +0.219 (2 ) + 0.054 (30) +O. 130 (50) facYlaJ 0.697 ( 1 ) 0.758 ( 5 ) 0.710 (10)

.frrf/aJ -0.101 (1) -0.015 ( 5 ) -0.024 (10)

-58.2 (2.3) -23.4 (0.3) -16.7 (0.3) frrr/aJ frrrs/aJ A-“ -0.8 (0.3) -0.1 (0.1) 0 . 0 (0.01) frralaJ A-‘ +0.4 (0.2) -0.4 (0.1) - 0 . 3 (0.1) frrpcYlaJ A-‘ -0 .6 (0 .2 ) -0.2 (0.1) 0.1 (0.1) jAa/aJ A-1 -0 .2 (0.1) -0.2 (0.1) -0.2 (0.1) f aora/aJ -0 .9 (0.1) -0.1 (0.1) -0.7 (0.04)

&rrr/aJ +367 (50) + 120 (6) +63 ( 5 ) frrrr’/aJ A- f 7 (3) +0.7 (0.1) + 0 . 3 (0.1)

+6 (2) 4-0.1 (0.1) +0.3 (0.1) frraa/aJ A-2 -2 (1) - 1 .G (0.7) - 1.8 (0.6)

fwraalaJ -0.1 (0.2) -0 .9 (0.1) -0.09 (0.03) frrfacuIaJ k2 +0.9 (0.3) -0.2 (0.2) 0.0 (0.1)

frrrfa constrained to zero I- frrrm

fraorv

a Standard errors are indicated in brackets after each force constant. Note that these quartic force constants were constrained to zero.


Table 11 Fit to data used in refining to the harmonic and anharmonic force field of Table loa

H2O H2S Observed- Observed-

Observed calculated Observed calculated 3832.2 +0.1 2721.9 +1.1 1648.5 -0.4 1214.5 0.0 3942.5 0.0 2733.4 -1.1

+0.75 -2 +0.125 -4 -2.94 - 5 -0.335 -2 +1.25 +3 +0.180 +4

+0.238 +17 +0.163 -5 -0.160 -66 -0.216 -2 +0.078 +14 +0.129 +5

+0.202 +14 +0.073 -5 +0.139 -5 +0.065 +8 +0.144 +9 +0.060 +4

-42.6 -0.2 -25.1 0.0 -15.9 -0.1 -19.7 0.0

-165.8 -0.2 -94.7 0.0 -16.8 0.0 -5.7 0.0 -20.3 -0.3 -21.1 0.0 -47.6 +0.4 -24.0 0.0

-77.5 -0.1 -45.7 0.0

H2Se Observed-

Observed calculated

1057.9 +0.3 2438.7 -0.3

2453.8 -0.4

+0.125 +2 -0.230 +3 +0.145 -3

f0.092 +1 -0.190 -5 +0.062 -3

+0.055 +13 +0.040 -7 +0.045 -2

-21.4 0.0 -17.7 0.0 -84.9 0.0 -2.4 0.0

-20.2 0.0 -21 -7 0.0

-41.6 0.0

a For the water molecule, data on D1O and HDO were also used; for H B and H S e no isotopic data were used. y is the Darling-Dennison resonance parameter.86 All data are in cm-l; errors in a values are given in the last digits quoted.

being refined in a separate calculation to the cubic and quartic force field. The fit to the data is shown in Table 11. The quartic force field was refined to fit the Darling-Dennison constant y (which is related to &3 and various cubic constants) as well as the x values. The apparently perfect fit to the x values for H,S and H2Se is presumably due to the absence of isotopic data; for water the x values of H20, D20, and HDO are probably not quite consistent owing to experimental error (the fit to D,O and HDO is not shown in Table 11).

Table 12 shows the normal co-ordinate force constants 9 for H20, calculated from the force field of Table 10, and it shows the major contributions of the internal co-ordinate force constants f to each 4. This table illustrates one important and general point : that large contributions to the cubic and quartic q5 values come from the quadratic f values. For example, the bending co- ordinate Q,, in which the atoms move in straight-line displacements, involves a positive displacement in the bond stretch 6r which increases as the square of


Table 12 Potential constants &-t. . . in normal co-ordinate space for the H,O molecule, calculated from the force field of Table I0 a

Total frr faa f r r f .fk J;rr f a m f r r r r O h r I - - + 11 3832 +3879 0 -46 0 -

22 1648 +27 +1676 0 -54 - - - -- 33 3942 +3896 0 -1-46 0 - - - - .

4 111 -1880 0 0 0 0 -1803 0 - - 77 112 +96 -2 +l 0 -46 +152 0 - -9

222 -384 -166 +60 + 2 +164 + l -463 - + 18 133 -1785 0 0 0 -2 -1810 0 - + 27 233 +292 +48 -8 + I 0 +152 0 - + 99

122 3-338 +657 -237' -7 + 7 -13 + 1 - - 69

4 1111 i 8 6 4 0 0 0 0 0 0 +768 +96 1122 -350 -93 +50 +1 -2 -303 0 + 5 -8 2222 f 5 5 +330 -158 -4 +10 -13 -33 0 -77 1133 +758 0 0 0 0 0 0 +772 -14 2233 -403 -91 +33 - 1 - 1 -303 + 2 +5 -47 3333 -1-752 0 0 0 0 -2 0 +775 -21

a Also given are the principal contributions to each 9 from the force constants in curvilinear internal co-ordinates through the non-linear L tensor transformation. All force constants are in cm-I.

the displacement in Q,:

In this case the second derivative L tensor element L;2 is particularly important. Thus on substituting for 6r in the stretching potential V = -$hr8r2, we obtain large terms in Q!, Q,Q& and Q;, showing that &, #122, and 42222 may all be expected to show large contributions from&. Table 12 confirms this.

Table 13 shows the force field obtained for a number of non-hydride bent triatomics. In most of these cases the cubic force field should be regarded as much more reliable than the quartic force field; indeed, in many cases the latter remains completely undetermined, owing to lack of experimental data on the x values. The table gives force constants both in the customary units, based on stretching co-ordinates in A and dimensionless angle bends, and in energy units of aJ based on the dimensionless stretching and bending co- ordinates discussed in Section 2. A possible advantage of the latter is that they remove the scaling effect of different bond lengths when comparing force constants for different molecules. Thus, for example, comparing the force constants&,. for 03, SO2, and SeO,, we see that the values in aJ A-2 give the misleading impression that SO, has a much stronger bond than either O3 or SeO,; the values in aJ give what is in fact the correct impression that SO2 and SeO, both have strong bonds compared with 0,.

The pure bending potential function in these molecules is also of some


Table 13 Anharmonic force fields of some bent triatomic molecules in curvilinear internal co-ordinates, up to cubic terms onlya

so2 6o

1.4308 (3) 119.33 (4)

10.42 (5) 0.124 (20) 0.525 (8) 1.680 (8)

-71.2 (2.1)

-3.3 (0.4) -1.3 (0.4) -2.8 (0.6) -2.4 (0.2)

-2.1 (1.0)

21.33 (10) 0.254 (40) 0.75 (12) 1.680 (8)

-208 (6.1) -6.1 (3.0) -6.8 (0.8) -2.7 (0.8) -4.0 (0.9) -2.4 (0.2)

SeO, 7 6

1.6076 113.83

6.92 0.04 0.04 1.38

-41.2 -1.9

1.4 -1.0 -2.9 -0.5

17.88 0.10 0.06 1.38

- 171 -7.9

3.6 -2.6 -4.7 -0.5

0, OF,76 N0299

1.2717 1.4053 1.1934 116.78

5.74 1.57 0.50 2.05

-50.6 -2.6 -2.7 -1.1 -3.6 -3.8

9.28 2.54 0.64 2.05

- 104 -5.3 -4.4 -1.8 -4.6 -3.8

103.07

3.98 0.83 0.21 1.42

-27.1 -1.6 -1.3 -0.4 -2.6 -3.3

7.86 1.64 0.30 1.42

-75.4 -4.4 -2.6 -0.8 -3.6 -3.3

134.25

11.05 2.13 0.65 1.62

-94.8 -6.4 -1.2 -1.2 -2.2 -2.4

15.74 3.03 0.77 1.62

- 161 - 10.9 -1.7 -1.7 -2.6 -2.4

a The force constants are given both in the usual units and in energy units in terms of dimensionless co-ordinates (see text).

interest; we would like to be able to determine V(a) throughout the range from the equilibrium configuration to a linear configuration, and thus to determine the height of the hump in the potential function for a linear configuration. However, the experimental data from vibration-rotation spectroscopy only relates to quite small displacements from equilibrium, over the range spanned by the amplitude of vibration in the highest energy levels studied. Several workers have, however, attempted to fit the bending potential over a wide range using analytical functions, and it is possible to compare such functions with the experimentally determined 2nd, 3rd, and 4th derivatives at equilibrium. For example, Hougen, Bunker, and Johns used a lorentzian function in the angle,

KB (c2 + P2)

V(p) = +kp2 + where p = n - 01 = x - ae - 6a, and k, KB, and c are adjustable constants.

81 J. T. Hougen, P. R. Bunker, and J. W. C. Johns, J. Mol. Spectroscopy, 1970, 34, 136.

Quan trim Chemistry

10 -

8 -

156

kJ mol-1

120

1 oc

80

60

40

20

0 -

103 cm-1

F-

0

\

\ \ \ \ \

180 170 160 150 140 130 120 110 100 90 80 70

degrees

Figure 7 Potential energy as a jhct ion of the bending co-ordinate iit H20. SHM = Strey, Hoy, and Mills (Mol. Phys., 1972, 24, 1265); HBJ = Hougen, Bunker, and Johns (1. Mol. Spectroscopy, 1970, 34, 136)

The potential calculated from the constants which they quote is compared in Figure 7 with that calculated from the constants of Table 10. There is a significant discrepancy for negative ~ C W , although the curves are coincident for positive displacements in 6cc (up to the maximum displacement to which it is reasonable to extrapolate V(2+3+4)). The discrepancy from a Lorentzian function is shown in another way in Table 14. The three parameters in the Lorentzian can be calculated from the experimental values of Pe, fa,, and &,,, and then the function can be used to calculatef,,,, for comparison with the value determined directly from the x values. It is seen that, even allowing


Table 14 The pure bending potential function of H 2 0 and SO,

ore

f au/aJ

faaalaJ

folaualaJ

H a 0 so2 104.52' 119.33'

0.697 & 0.001 1.680+0.010 - 0.88 k 0.04 -2.44k0.14 -0.06+0.15 -15.4k2.8

for the experimental uncertainties, the Lorentzian function is not a good fit to either H20 or SO,.

More Complicated Molecules.-Calculations have been reported on a number of more complicated molecules, as indicated in Table 4. The work on BF3 and SO3, and on NH3 and NF,, is of particular interest since these are the simplest symmetric top structures for which the calculation is practical, and for which there exist sufficient spectroscopic data to make it worthwhile; for symmetric top molecules there are extra observable vibration-rotation interaction constants associated with the vibrational degeneracy that provide further information on the force field (see Table 3). Formaldehyde and ethylene, and their simple halogen derivatives, and also methane, are obvious candidates for further work.

However, it will be clear from the selected results that have been quoted in this section that calculations on more complicated molecules face a serious problem of insufficient data to allow a determination of the general anharmonic force field; indeed, experience of attempting to determine harmonic force fields uniquely makes one approach this problem with caution. Further progress depends on the assumption of some model to reduce the number of parameters, and most workers in the field have been searching for a simple and appropriate model.

Kuchitsu and co-workers5~ were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes SO% of all that we have learnt so far about anharmonic force fields.

Efforts have also been made to devise more sophisticated models, with a few more parameters, because some of the spectroscopic constants generally show a sensitivity to features of the force field which are not present in the valence force model. They range from extended Urey-Bradley models, through semi- empirical valency models,1° to an ad hoc introduction of extra anharmonic


constants that seem likely to be imp~r tant .~ '~ 74 However, although all of these more sophisticated models achieve an improved fit to the data, they all remain somewhat unconvincing - mainly because there are so few cases in which the data are adequate to provide a convincing test. The uncertainties in the force constants obtained in Tables 6 and 8, for example, and the comparison between the results from different laboratories, almost force one to the conclusion that we cannot really claim to know much about any of the interaction anharmonic force constants. . . except that they are generally small, even if they are not quite zero as they are assumed to be in the valence force model. There seems to me to be little prospect of a breakthrough in more sophisticated models at this time.

Table 15 compares some of the results obtained on bond-stretching anharmonicity for corresponding bonds in different molecules. The force constantsf2,f3, and& are expressed in energy units, in terms of a dimensionless stretching co-ordinate, as discussed in Section 2, so that the scaling effect of changes in bond length on the force constants is taken out.

Table 15 Comparison of the stretching potential of CH, CN, and CC bonds irz various molecules with the corresponding diatomics a

CH diatomic CH in HCN CH in HCP CH in C2H2

CN diatomic CN in HCN CN in C,N,

CC diatomic CC in C,N, CC in C2H2

OH diatomic OH in H20

SO diatomic SO in SO,

CO diatomic CO in CO, CO in OCS5,

CS diatomic CS in OCS52 CS in CS,

reiA 1.120 1.066 1.069 1.062

1.172 1.153 1.57

1.243 1.384 1.203

0.971 0.957

1.481 1.43

1.128 1.160 1.155

1.535 1.563 1.553

h/aJ 5.62 7.10 7.14 7.19

22.38 24.87 23.00

18.78 14.0 23.64

7.34 7.74

18.16 21.34

24.20 21.57 21.51

20.00 18.18 18.97

AlaJ - 37 - 43 - 49 - 4-2

- 177 - 193 - 165

- 149 - 87 - 170

-49.5 -51 .O

- 169 - 208

- 195 - I76

(- 241)

- 173 (- 463) - 167

h/aJ 214 23 3

(249) 205

1132 1026 (928)

873 (420) 91 3

299 307

1404 1589

1306 1098

(3155)

849 (1 173) 1233

+(a3 + d l A-1 2.03 1.96 2.13 1.86

2.28 2.17 2.07

2.10 1.49 1.97

2.40 2.39

2.17 2.27

2.42 2.34 (3 .60)

1.74 (1.87) 1.93

a Note that all force constants are in terms of dimensionless stretching co-ordinates; see text. Brackets indicate particularly uncertain constants.


Discussion.-The technique of relating cubic and quartic anharmonic force constants to the spectroscopic observables for polyatomic molecules has been successfully developed, and has convincingly demonstrated the success of vibration-rotation theory in its present stage of development. However, the results obtained up to this time have been largely restricted to triatomic molecules, owing partly to the complexity of the calculations and partly to the lack of sufficient high-resolution spectroscopic data. The information gained on anharmonic force fields may be summarized in the statement that, in a true curvilinear co-ordinate representation, the predominant anharmonicity is associated with bond stretching. In a number of cases the study of the anharmonic force field has enabled more accurate vibrational corrections to be made to the structural parameters and the harmonic force field, thus giving more accurate results for the lower-order force constants re and fi. Future progress will probably develop in the wider application of simple model anharmonic force fields, particularly the valence force model, to calculations on more complicated molecules.

The Reporter is grateful to many colleagues for help in gathering data for this Report, and particularly to Dr. P. D. Mallinson for recalculating the results in Table 2.

Author Index

Ahlrichs, R., 101 Albritton, D. L., 142 Allen, L. C., 61, 86 Amat, G., 134 Amemiya, A., 65 Andrews, L., 143 Arai, T., 98, 100, 106, 107 Aslund, N., 2 Asprey, L. B., 20

Bagus, P. S., 14, 16, 17 Balint-Kurti, G. G., 85, 86,

105. 106 Barbe, A., 144 Bartell, L. S., 11 1 Beckman, F. S., 44 Bendazzoli, G. L., 31 Bender, C. F., 14, 33 Bernardi, F., 31 Bertram, M., 143, 144 Bessis, N., 27, 29 Beveridge, D. L., 31 Bishop, D. M., 56 Blint, R. J., 90 Blume, M., 17 BonaEiE, V., 52 Bondybey, V., 14 Born, M., 8, 61 Box, M. J., 55 Boys, S. F., 58 Brand, J. C. D., 144 Brink, D. M., 66 Browne, J. C., 11, 84, 85 Buehler, R. J., 44 Bunker, P. R., 155 Burnelle, L., 29

Cade, P. E., 13, 88 Cantrell, J. W., 43 Carl, R. F., 143 Carlson, K. D., 13, 15 Cashion, J. K., 9 Chang, S. Y., 38 Chedin, A., 1 1 1 Chen, A. C. H., 33 Chila. Z.. 111 Chow, K: W., 90 Christy, A., 3, 21 Chung, A. L. H., 28 Chuiie. K. T.. 134 Claxton, T. A,. 30, 58 Claydon, C. R., 15 Clement], E., 7, 55 Clough, P. N., 26 Colin, R., 25 Connor, D. T., 17 Cook, D. B., 31 Cook, R. L., 134

Cooley, J. W., 9 Cooper, L., 38 Corson, R. M., 65 Coulson, C. A., 76 Cragg, E. E., 43 Craig, D. P., 77, 81 Cross, P. C., 124 Crow, E. L., 142 Cruickshank, D. W. J., 82 Csizmadia, I. G., 101 Curran, A. H., 26 Cyvin, S. J., 112

Dacre, P. D., 96 Darling, B. T., 134 Das, G., 13 Davidson, E. R., 28, 33 Davies, D. W., 32 Decius, J. C., 124 Dennison, D. M., 134 Dewar, M. J. S., 59 Dickinson, A. S., 11 Djtchfield, R., 112 Dixon, L. C. W., 45, 46 Dixon, R. N., 7, 144 Dobosh, P. A., 31 Doggett, G., 70, 82 Dorney, A. J., 144 Dunham, J. L., 13, 115

Ebbing, D. D., 101 Edmiston, C., 101 Eliason, M., 76 Ellison, F. O., 107 Eyring, H., 87

Fischer, I., 76 Fletcher, R., 37, 44, 46, 55 Foord, A,, 134 Fraga, S., 14, 17 Freed, K. F., 13

Gerratt, J., 54, 88, 99, 102 Gilbert, T. L., 13, 17 Ginn, S. G. W., 11 1, 144 Goddard, W. A., Jun., 89,

Godfrey, M. J., 30 Goethals, P., 16 Gordv. W.. 134

90,102

Gree&’S., 14 Greenawalt, E., 11 Grenier-Besson, M. L., 134 Griin, N., 84 Guberman, S. L., 89 GuCrin, F., 14

Hagstroni, S., 93, 94

Hall, J. A., 14, 19, 20, 21,

Hameka, H. F., 19 Hamel, D., 86 Handy, W. C., 58 Haner, A. D., 143 Hanley, M. J., 10 Harris, F. E., 14, 86 Harrison, J. F., 61, 86 Harrop, W. J., 142 Hay, P. J., 102 Hayden, D. W., 26 Heil, T. G., 14 Heitler, W., 70 Helare, W. J., 112 Herzberg, G., 60, 69, 121,

24, 26

137 Hestenes, M., 44 Higuchi, J., 84 Hillier, I. H., 5 1, 59 Hinchliffe, A., 31 Hinkley, R. K., 2, 17, 20,

Hinze, J., 11, 58, 102 Hirota, E., 143, 144 Hirschfelder. J.. 76

21, 25, 26

Hooke, R., 40 Horie, H., 97 Horne, R., 25 Horsley, J. A., 20 Hougen, J. T., 155 Howard, J. B., 134 Hoy, A. R., 11 1, 144 Huang, H. Y., 45, 50 Huang, K., 61 Hunt, W. J., 102 Huo, W. M., 13, 88 Hurley, A. C., 60, 84, 85,

86, 87, 99, 105 Huzinaga, S., 17

Ishiguro, E., 20, 65, 85 Itoh, T., 75

Jain, D. C., 10 Jeeves, F. A., 40 Johns, J. W. C., 155 Julienne, P., 13

Kaiser, K., 13 Kaldor, U., 89 Kaplan, 1. G., 71 Karayanis, B., 17 Kari, R., 52 Karplus, M., 28, 86, 105 Kato, T., 28 Kayama, K., 85, 105 Kempthorne, O., 44

1 60

Author Index

Kern. C. W.. 19

161

Kikubhi, Y.,’144 Kim, H., 94 Kimura, T., 65 Kjng, H. F., 56, 94 Kivelson. D.. 134 Klemperer, W., 13 Klessinger, M., 104 Knight, L. B., 23 Kobori, M., 20 Kockel, B., 86 Kohn, M. C., 59 Kolos, W., 10, 12, 90 Komornicki, A., 58 Konshi. H.. 28 Koopmans,‘ T., 5 Kopineck, H.-J., 69 Kotani, M., 65, 74, 75, 85,

105 Kouba, J., 14 Kouteckq, J., 52 Kovacs, I., 21 Kowalik, J., 38 Krauss, M., 13, 33 Kuchitsu, K., 111, 112 Kunik, D., 89 Kutzelnigg, W., 101

La Budda, C. D., 10 Ladner, R. C., 90 LaPaglia, S. R., 32 Leach, S., 20 Leclerc. J.-C., 56 Lefebse-Brion, H., 1

Lefebvre, R., 27 Lennard-Jones, J., 99 Levy, A. V., 43, 50 Lewis, W. B., 17 Liberman, D. A., 17 Lide. D. R.. 144

29

Lie, b. C., 1 1 Lin, T. K., 101 Linnett, J., 76 Lipscomb, W. N.. 99 Liu, B., 11, 14 Liu, H. P. D., 14 Lo B. W. N., 17 Lohdin. P. 0.. 6. Longuet-Higgihs; Ludeiia, E., 15 Lyubarskii, G. Y

91, H.,

., 72

5, 27,

93 27

McCain, D. C., 26 MCCOMell, H. M., 31 McIver, J. W., Jun., 58 McLagan R. G. A. R., 81 McLean, D., 12, 14 McWeeny, R., 35, 36, 37,

96, 99, 102, 104 McWilliams, D., 30 Maes, S., 134 . Maki, A. G., 137 Malli, G., 19 Mann, J. B., 17, 20 Matcha, R. L., 13, 19 Matsen, F. A., 11, 84, 85 Matsumura, C., 144 Mattheiss, L. F., 79

Mayers, D. F., 82 Meal. J. H.. 133 M M M M M M M M M

M M M M M M M M

M M

M M M M M M

ehler, E. L 13, 56, 99 elius, C. F.: 89 ichels, H. H., 14, 86 jekzarek, S. R., 33 iele, A., 43 iller, K. J., 33, 55 iller, M. B., 19 iller, W. H., 14 ills. 1. M.. 54. 111. , 126, 134, 143, 144 itchell, K. A. R., 81 izuno, Y., 85 offitt, W. E., 70, 94, 104 oore. E. A.. 3 oore; N., 85 oore, P. L., 84 oraw, H., 86 orino, Y., 1 1 1 , 134, 143, 144 orokuma, K., 28 oser, C. M., 7, 13, 15, 16, 27 ueller, C. R., 87 ulder, G. J., 77 ulliken, R. S., 3, 21 undie, L. G., 60 urray, W., 38 urtagh, B. A., 46

Nakagawa, T., 143 Nesbet, R. K., 7, 13, 15 Nielsen, H. H., 134 Nordheim-Poschl, G., 60

Ohm, Y., 14 O’Hare, P. A. G., 13 Ohno, K., 75, 84, 105 Oka, T., 133, 134 @Neil, S. V., 14 Oppenheimer, J. R., 8 Osborne. M. R.. 38 Otake, M., 144 Overend, J., 111, 143, 144

Palke, W. E., 89 Palmeiri, P., 31 Pan, K.-C., 56 Papousek, D., 144 Pariseau, M. A., 111, 143 Parker, P. M., 134 Parks, J. M., 100 Parr, R.-G., 94, 100 Pauling, L., 70 Pauncz, R., 58 Pearson, P. K.. 14 Peyerimhoff, S., 14 Pliva, J., 111, 144 Polak, E., 44 Poling, S. M., 28 Polo, S. R., 133 PoDkie. H. E.. 33 Pople, J. A., 27, 31, 99, 112 Poshusta, R. D., 85 Powell, M. J. D. 42, 46 Pritchard. R. H.. 19 Prosser, F., 93, 84

Radford, H. E., 26 Raffenetti, R. C., 52

Raftery, J., 12, 14 Ransil, B. J. 14 58 Redding, R.’W.: 144 Rees, A. L. G., 121 Reeves, C. M., 44 Reichman, S., 1 1 1, 144 Reisfield, M. J., 20 Robiette, A. G., 112 Ribithe, G., 44 Richards, W. G., 2, 3, 5, 7,

12, 14, 15, 18, 19, 20, 21, 25,26

Richardson, A. W., 144 Robb, M. A., 101 Robiette, A. G., 143 Rodimova, 0. B., 71 Roothaan, C. C. J., 12, 58,

Rosen, B., 121 Rosenbrock, H. H., 40 Rothenberg, S., 29 Ruckelshausen. K.. 86

90, 102

Ruedenberg, K:, 13,52,55, 56, 99, 101

Rumer, G., 67

Sah, R., 10 Sahni, R. C 10 Saito, S., 145, 144 Sakamoto, M., 107 Sales, K. D., 13 Salotto, A. W., 29 Sando, K. M., 90 Sargent, R. W. H., 46 Satchler, G. R., 66 Saunders, V. R., 51, 59 Sawhney, B. C., 10 Saxena, K. M., 17 Schaefer, H. F., 14,29, 112 Scbneideman S. B 14 Schnuelle, G.’ W., $1 Schraeder, D. M., 19, 28 Scott, P. R., 14 Serber, R., 67 Shah, B. V., 44 Shavitt, I., 90 Sherman, A., 70 Shoje, H., 143 Shull, H., 67 Sidis, V., 15 Siga, M., 75 Silver, D. M., 13, 56, 99,

Simonetta, M., 31. 84 Slater, J. C., 70 Sleeman, D. H.. 51

101

Slezak, J. A 107 Smith, A. L:: 90 Smith, D F 1 1 1 143,144 Smith, J. G:: 134 Smrth, N. A., 30 Smith, W. A., 58 Solomon. C. E.. 56 Spiker, R. C., 143 Stanton, R. E., 94 Steinberg, D., 38 Stewart, G. W. tert., 44 Stiefel, E. L., 44 Strey, G., 1 1 1 , 143

162

Sutcliffe, B. T., 35, 52, 99 Suzuki, I., 111, 143

Takeo. H.. 143

Author Index

Tanaka, T., 143 Tantardini, G. F., 31, 84 Tarrago, G., 134 Thirunamachandran, T., 8 1 Thorhallsson. J.. 85 Thrush, B. A:, 26 Todd, J. A. C., 14

van der Hart, W. J., 104 Van Vleck, J. H., 21,65,70,

Varga, L. P., 20 Veillard, A., 7

98

Verhaegen, G., 7,14,15,16 Veseth, L., 19, 24 Vincow, G., 28

Wahl, A. C., 13 Walker, T. E. H., 2, 18, 19,

20, 21, 25, 26 Watson. J. K. G.. 132. 134 Watson; R. E., 17 Webster, B. C., 82 Weinbaum, S., 76 Weiner, B. L. J., 30 Weinstein, H., 58 Weltner, W., 23 Wessel, W. R., 58 Wheland, G. W., 60 Whiffen, D. H , 134

Wigner, E. P., 62 Wilkinson, P. G., 86 Wilson. E. B.. 124. 134 Wilson; P. W:, 143 Wilson, R. C., 15 Wilson, S., 102 Wolniewicz, L., 10, 32

Wyatt, R. E., 94

Yamamoto, S., 11 1 Yde, P. B., 47 Yoshimine, M., 13, 14

Zamani-Khamiri, O., 19 Zare, R. N., 142 Zetik, E., 11

WU, Ay-Ju A., 107

Theoretical Chemistry Volume 1 (Specialist Periodical Reports)

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