Theoretical Chemistry,vol 2 (Specialist Periodical Reports) (v. 2)

A Specialist Periodical Report

Theoretical Chemistry Volume 2

A Review of the Recent Literature

Senior Reporters R. N. Dixon, Department of Theoretical Chemistry, Universify of Bristol C. Thornson, Department of Chemistry, University of St Andrews Reporters

R. F. W. Bader, McMaster University, Hamilton, Ontario, Canada B. J. Duke, University of fancaster R. A. Gangi, McMasfer University, Hamilton , Ontario, Canada J. G. Stamper, University of Sussex

0 Copyright 1975

The Chemical Society Burlington House, London, WIV

OBN

ISBN :0 85186 764 2

ISSN : 0305 9995

Library of Congress Catalog Card No. 73-92911

PRINTED IN GREAT BRITAIN BY ADLARD AND SON LIMTED, BARTHOLOMEW PRESS, DORKING

Foreword

This second volume of the Specialist Periodical Reports on Theoretical Chemistry is the first of the series covering the broader aspects of the subject, and contains four articles. Quantum chemical calculations on small molecules are reviewed by Thomson, and calculations on large molecules by Duke. It is hoped that the cIasses of calculation not dealt with in this report will be reviewed in Vol. 3. One of the growth areas in theoretical chemistry has been the attention paid to the potential energy surfaces for inolecules and interacting molecules, rather than just equilibrium properties. Bader and Gangi provide a very detailed and comprehensive review of ab initio calculations on potential energy surfaces, and Stamper deals with the subject of intermolecular forces. Other topics in theoretical chemistry will be covered in the next volume. May 1975

R. N. Dixon C. Thomson

Contents Chapter 1 Ab lnitio Calculation of Potential Energy Surfaces By R. F. W. Bader and R. A, Gangi

1

1 The Concept of a Potential Energy Surface

1

2 Scope of the Review

4

3 Hartree-Fock and Self-Consistent-Field Calculations Properties of Hartree-Fock Wavefunctions Roothaan’s Self-Consistent-FieldProcedure Comparison of Hartree-Fock and Near Hartree-Fock SCF Results with Experiment

6 10 10 12

4 Hartree-Fock Potential Energy Surface Calculations Asymptotic Behaviour of Single Determinantal Wavefunctions Early Examples of Surface Calculations for Closed-shell Systems Tests of Parallel Nature of HF and Correlated Surfaces for Closed-shell Systems HF Surfaces for A: + B. Interactions HF Surfaces for sN2 Reactions Other Closed-shell HF Results Corrections for Improper HF Asymptotic Behaviour HF Description of Addition to Double Bonds Electrocyclic Reactions Ion Solvation

18 18 20

5 BeyondHartme-Fock Minimal Configuration Interaction Truncated CI Wavefunctions Unrestricted Wavefunctions The IEPA Method

36 37 37

48

6 H + H2: A Benchmark Reaction Semi-Empirical Calculations Ab Initio Calculations

48 48 49

7 Potential Energy Surfaces from Correlated Wavefunctions

54 54 55 56

H3f

+ +

He Hi-, HeH+ HeH2 H2 D2-+2HD HeHf + Hz-, He He:+ He,+ + He

F

+H +

+ Ra4FH + H

21 24 25 28 29

34 35

35

41

57

Hi

59

60

60

vi

Contents

+

+

H Fz+HF F SurfacesforX + Mzand M CH2 + CH2+ C2H4 The Valence Bond Approach

+ X2

Chapter 2 Intermolecular Forces By J. G. Stamper

62 63 64 65

66

1 Introduction

66

2 General Theory

66

3 Short- and Intermediate-range Forces

70

4 Long-range Forces

74

5 Miscellaneous Topics Systems of more than Two Molecules Properties Hydrogen-bonded Systems Empirical Potentials and Related Topics

77 77 78 79 81

Chapter 3 Quantum Mechanical Calculations on Small Molecules By C. Thomson

83

1 Introduction

83

2 Diatomic Molecules containing up to Four Electrons A. H i a n d H z B. HeH, HeH+, and He: C. He2

84 84 90 91

3 Diatomic Hydrides of First-row Atoms, AH A. LiH and LiHf B. BeH, BeH+, and BH C. CH and CM+ D. NH,NHf, and OH E. H F a n d H F f

93 93 99 102 104 105

4 Hydrides of Second and Higher Rows, AH

107

5 Homonuclear Diatomic Molecules of First-row Elements A. Liz and Li; B. Be2, B2, CZ,Cl, and C, C. N2 andiNi D. 02, Oz, and 0, E. Fa, F;, and F,

108 109 110 110 112 114

Contents

vi i 6 Homonuclear Diatomic Molecules of Second and Higher Rows

115

7 Heteronuclear Diatomic Molecules A. Rare-gas Compounds B. Oxides of Elements of Groups I, 11, and I11 C . Metal Oxides of Other Groups D. Non-metal Oxides COYSiO, and CS NO, NO+, and PO C10 and FO E. Halides Metal Halides Halides of B, C, and Si Fluorides of N, P, S, Se, and As SF,SeF, and AsF F. Nitrides and Carbides G. Interhalogens H. Miscellaneous Calculations on Diatomic Molecules

116 116 118 119 120 120 121 122 122 122 123 124 124 125 125 126

8 Linear Triatomic Molecules A. He/H2 and H3 B. Hydrides AH2 C. Compounds of the Inert Gases D. The Bihalide Ions HX; E. HAB Molecules F. C02, CS2, Li20, and A h 0 G. CCO and TiCO H. Molecules containing B, C, and N

127 127 127 128 129 129 131 132 132

9 Non-linear Triatomic Molecules A. AH2 Molecules, including H3+ NH2, NH;, and NH; OH2 and OH$ H2Ff B. HAB Molecules C. ABz Molecules A02 AF2 D. ABC Molecules

134 134 137 138 141 141 143 143 147 149

10 Tetra-atomic Molecules A. A H 3 Molecules B. A B 3 Molecules C. A4 Molecules D. A2Bz and A2H2 Molecules E. Dimers of Diatomic Molecules, AB *AB F. Miscellaneous Tetra-atomic Molecules 9

150 150 153 154 155 156 156

viii

Contents

Chapter 4 Electronic Calculations on Large Molecules By B. J. Duke

159

1 Introduction

159

2 The Linear Combination of Atomic Orbitals-Molecular Orbital Approach Introduction Ab Initio LCAO-MO Methods Approximate LCAO Methods

160 160 162 184

3 The X, Exchange Approximation The SCF-X, Scattered-wave Method Other Related Methods

191 191 194

4 Beyond the Molecular Orbital Approach Introduction The PCLLO Method Other CI Methods

195 195 196 200

5 Concluding Remarks

20 1

Author Index

203

Abbreviations Average Natural Orbitals AN0 Best Atom (basis set) BA P Best Atom + Polarization functions BA Born-0 ppenheimer BO Contracted Gaussian Type Orbitals CGTO Configuration Interaction CI Complete Neglect of Differential Overlap CNDO DEM Distinguishable Electron Method DIM Diatomics in Molecules Double Zeta (basis set) DZ Polarization functions DZ P Double Zeta Extended Hartree-Fock EHF ESCA Electron Spectroscopy for Chemical Analysis FEGO Floating Ellipsoidal Gaussian Orbital FSGO Floating Spherical Gaussian Orbital Generalized Atomic Effective Potential GAEP GTO Gaussian Type Orbital GVB Generalized Valence Bond: particular forms are denoted G1, G1 & G F Hartree-Fock HF Hartree-Fock with Proper Dissociation HFPD Internally Consistent Self Consistent Orbitals ICSCF Independent Electron Pair Approximation IEPA Intermediate Neglect of Differential Overlap INDO Intermediate Retention of Differential Overlap IRDO Iterative Natural Orbital IN0 Ionization Potential IP Kolos-Roothaan (wave function for Hz) KR LED0 Limited Expansion of Diatomic Overlap Kolos-Wolniecwicz (wave function for H2) KW LCAO Linear Combination of Atomic Orbitals MBPT Many Body Perturbation Theory MCSCF Multi-configuration Self Consistent Field MIM Molecules in Molecules MIND0 Modified INDO procedure of Dewar, also MIND0/2, MIND0/3 Molecular Orbital MO Neglect of Diatomic Differential Overlap NDDO Natural Spin Orbital NSO Optimized Double Configuration ODC Optimized Valence Configuration ovc Potential Energy PE Perturbative Configuration Interaction over Localized Orbitals PCILO PNO Pseudo-natural Orbital or (PSNO)

+

+

+

Abbreviations

X

PDDO PRDDO RHF SAM0 SCF SOGI STQ STO-nG UA UHF VB VIP

Projectors of Diatomic Differential Overlap Partial Retention of Diatomic Differential Overlap Restricted Hartree-Fock Simulated ab initio Method Self Consistent Field Spin Optimized GVB method Slater Type Orbit a1 Slater Type Orbital expanded in terms of nGTO United Atom Unrestricted Hartree-Fock Valence Bond Vertical Ionization Potential

Units

A number of different sets of units are used throughout this volume. Conversions to SI units are as follows: Energy:

1 a.u. (Hartree) = 4.359828 aJ

1 eV

= 0.160210 aJ

= 2625.47 kJ mol-1

= 96.4868 kJ mol-l

1 cm-l = 1.98631 x 10-23J = 11.9626 J mol-l 1 K = 1.38054~10-23J= 8.3143 J mol-f

Length:

1 a.u. (Bohr) = 0.529177~10-lo m

Dipole moment: 1D (Debye) = 3.33564 x

Cm

I Ab initio Calculation of Potential Energy Surfaces ~

BY R. F. W. BADER AND R. A. GANG1

1 The Concept of a Potential Energy Surface This chapter is concerned with the calculation of potential energy surfaces by nonempirical methods, i.e. by obtaining solutions to Schrodinger's equation within the Born-Oppenheimer approximation. The concept of a potential energy surface is a consequence of the separation of the nuclear and electronic motions as proposed by Born and Oppenheimer.1 The nuclei may be considered to move under the infiuence of a potential determined by their mutual electrostatic repulsion and by the total energy of the electrons, an energy which is determined for every possible static configuration of the nuclei. The gain in conceptual simplicity afforded by the Born-Oppenheimer procedure is enormous and its use underlies many of our chemical concepts. For example, energies of activation or energy barriers in general, potential constants, the frequencies of vibrational and rotational motions, and bond lengths and bond angles as determined by an energy minimum, are all concepts defined in terms of the properties of a potential surface. To obtain information concerning a system of n electrons and N nuclei in the absence of external fields, one must solve the timeindependent Schrodinger equation

RTY~(x, R)

= EzPi(x, R)

(11

where the properties of the zTh stationary state of the system of energy Et are obtainable from the eigenfunctions Yi(x,R)which in turn depend upon the space and spin co-ordinates of all the electrons and nuclei (whose collective co-ordinates are denoted by x and R,respectively) in the system. The total Hamiltonian operator is

RT

=

R! + 3%

+

0

(2)

where Pe and PN are the kinetic energy operators of the electrons and nuclei, respectively, and where the potential energy operator in atomic units is,

One can distinguish three levels of approximation in obtaining solutions to equation (1):a (i) the Born-Oppenheimer approximation, often referred to as the a

M. Born and R. Oppenheimer, Ann. Phys., 1927,84,457. P. G. Wilkinson and H. 0. Pritchard, Cunad. J. Phys., 1969,47,2493.

1

2

Theoretical Chemistry

clamped-nucleus approximation; (ii) the adiabatic approximation; and (iii) the non-adiabatic approximation. The solutions to equation (1) provided by the first of these are entirely adequate for most problems of chemical interest. For example, use of this approximation in the calculation of the low-lying vibrational and rotational levels of Hl yields results which are correct within experimental In the Born-Oppenheimer approximation, one fist transforms to a centre-of-mass, molecule-fixedco-ordinate system, thereby yielding a kinetic energy operator for the electrons referred to nuclei that are fixed with respect to their centre of mass and a nuclear kinetic energy operator5 which refers only to their internal motions, i.e. their vibrations and rotations and the various couplings between them.* Because of the large disparity in the masses of the electrons and the nuclei, the average kinetic energy of the former will be many times that of the latter, the ratio being proportional to mK/me-2000 in the least favourable case when mKrefers to the mass of a proton. Thus, in the limiting classical case, one obtains a picture of the electrons undergoing very rapid motion relative to the nuclei and adjusting almost instantaneously to changes in the nuclear positions. This suggests that the electronic motions are determined to a good approximation by just the static field of the nuclei, i.e. the electronic motion is determined by where the nuclei are but not by how fast they are moving. The motions of the nuclei in this separation of co-ordinates are governed by a potential whose negative gradient is simply the electrostatic force of repulsion between the nuclei and the attractive force exerted on the nuclei by the electronic charge distribution, a distribution whose form is determined by the electronic wavefunction evaluated for each configuration of the nuclei. Mathematically, this separation of the kinetic motions of the electrons and nuclei amounts to approximating the total wavefunction !Pi(x, R) by a simple product of electronic and nuclear wavefunctions

The electronic function yi(x; R) is obtained by solving the electronic Schrodinger equation for a fixed nuclear configuration, where

The use of the semi-colon in the notation yi( x; R)is to denote the explicit dependence of yi on the electronic space-spin co-ordinates x and its implicit dependence, along with Ei on the nuclear co-ordinates R. Clearly, the solution of equation (5) for all spatial arrangements of the nuclei will generate an energy (hyper)-surface, the

*

For a recent discussion of the nuclear kinetic energy operator and of its application to nuclear motions in the water molecule the reader is referred to ref. 6. l

G. Hunter and H. 0. Pritchard, J. Chem. Phys., 1967,46,2153.

* C. L. Beckel, B. D. Hansen, and J. M. Peek, J. Chem. Phys., 1970, 53, 3681.

B. T. Darling and D. M. Dennison, Phys, Rev,, 1940,57, 128, C. W. Kern and M. Karplus, ‘Water-A Comprehensive Treatise’, ed. F. Franks, Plenum, New York, 1974, vol. 1, p. 21.

3

Ab initio Calculation of Potential Energy Surfaces

potential energy surface, which governs the motion of the nuclei as obtained by solving the nuclear eigenvalue equation

Equation (7) and its implied assumption of the separability of the electronic and nuclear motions is called the Born-Oppenheimer approximation. The exact Schrodinger equation of motion, equation (l), may be equivalently stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Ya(x, R) is expanded in terms of the complete orthonormal set of functions yi(x; R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions xt(R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the xi(R)6 [FN -k Ei(R)

+

< ~ a I ~ N I ~ t ) - ~ i ] X i (= R)

- ~ < W l ~ N ] ~ j ) x j ( R )(8) i #j

From this formal, but exact statement of Schrodinger's equation, which corresponds to the non-adiabatic approximation referred to above, one sees that the approximation made in obtaining equation (7)is to assume that

for all i and j. In another order of approximation, with the expansion of Yi(x, R) truncated to the single product term as given in equation (4), one obtains in place of equation (8),

[?"

+ E4R) + 2.8 bohr were found but the two curves had the expected hump between 2.6 and 2.8 bohr. Browne and Greena~ a l t ,using l ~ ~CI, however, did obtain a double minimum for the 1X+ state. Calculations on the N I T state at R = 2.316 bohr have been reported by Green.16 The SCF X-A transition energy of 2.48 eV compares with the experimental value of 2.86 eV. Molecular Rydberg states are currently of considerable interest to theoreticians. These states arise when the outermost electron occupies a diffuse orbital, such as, in the case of first-row atoms, 3s, 3p, . ..There have been a number of recent papers devoted to such states, including the BIZ+ and lowest 3 2 + states of BH. Near Re, such states are usually well described by a single configuration, but this is not the case at large R values. The BIZ+ and 3Z+states of BH arise from the orbital configuration la22a2304a,and the work of Blint and Goddard156 and of Mullikenl68 on these was described above. Schaefer et aZ.,lso using a contracted STO basis set, carried out extensive CI calculations on the X1Z+, Ill;)=+,and 3Z+ states. Included in the basis were a set of Rydberg functions. Electron correlation was introduced via the first-order wavefunction method. This is a particular form of CI which places special emphasis on valence orbitals not occupied in the SCF configuration.1, 161 The 102 core remained doubly occupied. The necessary proper dissociation behaviour for the BIX+ state required the inclusion of the three configurations (14). The iterative

.

1 a24a25a2 1023a4oSa2

(14)

natural orbital (INO) method was ~sed.10~ For the ground state, the ab initio dissociation energy obtained was 3.27 eV (expt. : 3.54 i-0.04 ev). Spectroscopic constants and AE for the X-B separation were also in good agreement with experiment. The double minimum predicted in ref. 159 was found. The lowest 3Z+ state was predicted to be Rydberg-like, with a minimum at R = 1.173 A, for short R but valencelike (repulsive) for large R. A maximum in the 3 Z + curve occurs at 1.45 A. 156

15' 168 159 160

R. J. Blint and W. A. Goddard, tert., Chem. Physics, 1974, 3, 297. E. L. Mehler, Internat. J. Quantum Chem., 1973, 7S, 437. R. S. Mulliken, Internat. J. Quantum Chem., 1971, 3, 83. J. C. Browne and E. M. Greenawalt, Chem. Phys. Letters, 1970,7, 363. P. K. Pearson, C. F. Bender, and H. F. Schaefer, tert., J. Cirem. Phys., 1971,55,5235. H. F. Schaefer, tert., J. Chem. Phys., 1971, 54, 2207,

Theoretical Chemistry

102

An interesting application of many-body perturbation theory using a discrete orbital basis has been reported by Robb.162In calculations on BH,comparison was made between the results and those of Houlden et ~ 1 . 1 5 2using CI. Most of the pairpair interaction energy can be recovered by this method. The ions BH+ and BH- have been relatively little studied in comparison with the neutral species. Cade and Huo 1°4 investigated BH+ several years ago, and recently Blustin and Li11nett1~~ reported FSGO calculationson the X2C+ and A2l-I states. The direction of the increase in bond length, R ( T I )> R(2Z+),was successfully accounted for. The dissociation energy for the process BH++ Bf H is positive, but not for BH++ H+ B. The ion BH- is not predicted to be stable in the 211 state, but the 4 Z - state might be bound. Gaussian lobe calculations on BH+ have also been carried out, and spin-orbit coupling constants successfully calculated.1~4 C. CH and CH+.-CH has been investigated in some detail recently, particularIy in view of its importance in astrophysics. Most of the earlier calculations have been SCF calculationson the ground state, X W ,and only a few calculationson the excited states appeared prior to a thorough study by Lie et al. on the A2A, BZC-, and a4Cstates, involving CI calcuIations.165 Computed PE curves were used to obtain vibrational-rotational levels. Results were in good agreement with experiment. The a4C- state, which has not been observed experimentally,was predicted to lie between 0.52 and 0.75 eV above the ground state. More accurate calculations by Lie et al.166 have also been reported, with bigger basis sets and more configurations. A variety of CI calculations were compared, and the X2rI and PI;+ states were also considered. 4147 Configurations were included in the most extensive calculation. Results in essentially quantitative agreement with experiment were obtained, but it should be noted that only the most extended CI predicts, correctly, the bound B2Z- state, whereas the less extensive CI calculations did not predict this. A further paper167 dealt with one-electron properties. Table 3

+

+

Table 3 C0rnputedl6~and experimental spectroscopic constants for the ground and various excited states of CHa State C2Z+

B2C-

A2A a 4 2 X2I-I

a

Zero-point energy 1403.1 (1381.7) 1015.1 ( N 1068) 1454.4 (1418.1) 1555.4 1424.9 (1415.5)

All quantities are given in cm-1.

162 163 164 165

b

weXe

W?

m0

32406.7 (31778.1) 25854.9 (25698.2) 23590.6 (23217.5) 5395.5 0.0

2887.5 (2840.2) 2147.7 ( 2250) 2970.3 (2930.7) 3145.7 2886.1 (2858.5) N

106.8 (125.96) 223.2 ( 229) 98.5 (96.65) 71.8 82.0 (63.0) N

Experimental quantities in parentheses.

M. A. Robb, Chem. Phys. Letters, 1973, 20, 274. P. H. Blustin and J. W. Linnett, J.C.S. Faraday II, 1974, 70, 826. P. W. Abegg and T.-K. Ha, Mol. Phys., 1974, 27, 763. G. C. Lie, J. Hinze, and B. Liu, J. Chem. Phys., 1972, 57, 625. This paper contains extensive

references to earlier work. 186 167

Be 14.763 (14.603) 13.51 (13.39)Q 14.976 (14.934) 5.364 14.498 (14.457)

G. C. Lie, J. Hinze, and B. Liu, J. Chem. Phys., 1973, 59, 1872. G. C. Lie, J. Hinze, and B. Liu, J. Chem. Phys., 1973, 59, 1887,

Quantum Mechanical Calculations on Small Molecules

103

compares some of these with experimental values. Most of the results are predictions, but the dipole moment of 1.41 D is in excellent agreement with the experimental value of 1.46 f.0.06 D. These studies represent highly accurate and useful calculations, and are a further landmark in computational chemistry. Again, Clementi's HFPD wavefunctions will give values comparable to the CI functions, at much lower cost.125 Analysis of the wavefunctionsof these calculations is not easy, and the GVB method seems to be somewhat more useful in this regard; the bonding in CH has been reviewed recently.13,168 It is of some significance first to consider the 3P state of C. The usual MO description (10)~(2s)~(2p)~ has doubly occupied orbitals. The GVB wavefunction has essentially the same 1s and 2p orbitals, but there is a splitting of the 2s orbitals into a pair which have the form of sp hydrids ( 17% p-character). These two orbitals and the two 2p orbitals are the four carbon orbitals involved in bonding. In CH, the H 1s orbital pairs with a 2p orbital to give a 2 I I state, which is the ground state. The GVB lone-pair orbitals are a split sx and SX pair and are bent back at 128" from the bond axis, whilst the 2pz orbital incorporates some s-character to maintain the orthogonality. The low-lying 4 2 state is also easily explained, since it arises from bonding of H to one of the sx lobes. This is calculated to be 0.36 eV above the X211 state. CI calculations can also be carried out using the GVB orbitals as basis. Such calculations of excitation energies for various states of CH are also reported by Hay et Many-body perturbation theory has proved very useful in atomic calculations, and a recent calculation on CH, using a single-centre expansion with a complete set of basis states appropriate to the neutral C atom, gives a total energy of - 38.482 10.02 hartree and Re = 2.10 b0l1r.l~~ This energy is lower than the previous best calculation and is very close to the experimental energy. The full PE curve, accurate to second order, was also calculated for the X211 state. The difference in correlation energy between C and CH was also evaluated in an earlier paper.170 Several excited states of CH that have Rydberg character have been studied by Mulliken.171 Rather surprisingly, an energy of -38.4255 hartree was claimed by Tandardini and Simonetta in a VB calculation using a minimal basis set, including 300 struo tures in the usual way.172 This low energy is surprising because of the use of an inflexible basis set, although many configurations were included. The authors also calculated proton and 13C hyperfine coupling constants. A Doubling constants148 and spin-orbit coupling constants have been reported for CH.164 The gaussian lobe calculations are inferior to the STO basis-set calculations of Richards and coworkers.148 The FSGO calculations on CH give a fairly accurate bond length.163 The bonding was also examined from an orbital standpoint :however, the conclusion that the C-H bond has almost entirelyp character is not in accord with the GVB results.l 6 8 The ion CH+, also important in astrophysics, has been the subject of a thorough study by Green et aZ.,173using both SCF and SCF/CI. The quoted errors in the results are 0.3 eV. There is no evidence for the quasi-bound 3Z+ Rydberg state, although N

N

N

168

P.J. Hay, W. J. Hunt, and W. A. Goddard, tert., J. Amer. Chem. SOC.,1974, 94, 8293.

T. E. H. Walker and H. P. Kelly, Phys. Rev., 1972, A5, 1986. T. E. H. Walker and H. P. Kelly, Internat. J. Quantum Chem,, 1972, 6, 19. R. S. Mulliken, Chem. Phys. Letters, 1972, 14, 141. 1 7 2 G. F. Tantardini and M. Simonetta, Chem. Phys. Letters, 1972, 14, 170. 173 S. Green, P. S. Bagus, B. Liu,A. D. McLean, and M. Yoshimine, Phys. Rev., 1972, AS, 1614. 169 170 171

104

Theoretical Chemistry

-

this was predicted for the isoelectronic BH.160 Oscillatorstrengths for the AlIT-XlC+ transition were computed by Yoshimine et al. to an estimated accuracy of 10%.I74 The agreement with calculations by the equations-of-motion method was very g00d.175 Earlier work on hyperfine splittings in CH has recently been reviewed.176 D. NH, NH+, and 0H.-The NH radical has been studied at various levels of accuracy in recent years. A survey of calculations prior to 1971 has been given by O’Neil and S ~ h a e f e rwho, ,~~~ using a contracted set of STO’s, computed PE curves for the three lowest PX-, alA, and blC+ states, using CI with the I N 0 rnethod.lO7 Various levels of CI were investigated. The ground-state energy is almost as good as an earlier calculation by Bender and Davids0n.1~~ Spectroscopic constants were calculated, and the dissociation energy, De. The latter quantity has been in dispute experimentally,and Steven~,l7~ using the Optimized Valence Configuration (OVC)179 version of the MCSCF method, has also computed De. O’Neil et al. obtained 3.06 eV, and Stevens 3.43 eV, both calculations supporting the lower of the experimental values. In a later paper, Das and co-workerslsOcomputed a variety of spectroscopic properties of NH. Using 10 configurationsin an OVC calculation, the energy obtained for X3C- of -55.0245407 hartree was slightly higher than O’Neil and Schaefer’s177 result (- 55.083968 hartree). However, of more significance are the spectroscopic constants. Solution of a one-dimensional Schrodinger equation for these, and the alternative (and usual) Dunham analysis yield different values, and the former are in better agreement with experiment. General rules for OVC calculations have been summarized by Karo et aZ.18l Spectroscopicconstants were also discussed by Lie and Clementi in their paper on correlation energies of the hydrides.125 SCF calculations on the l11 and 3rI states, 5C-,3C-, lX-, 3A, and 3C+Rydberg states have been reported for NH,17l more accurate thanearlier calculations by Liu and Verhaegen.182 Spin-orbit coupling constants ls3 and semi-empirical studies of the correlation energy have been reported.l51 A comparison of recent computations on the X211 state of NH is given in Table 4. Table 4 Comparison of some recent calculations on X211 NH CaIculation SCF CI HFPD

OVC(MCSFC) First-order CI Experiment

Elhartree -54.97806 -55.1620 -55.0010 -55.02454 -55.083968 -55.252

Rlbohr DeleV w,/cm-l 1.997 2.10 3556 1.976 2.73 3176 3.06 3287 1.967 1.967 3.06 3300 3125.6 1.9614 3.40

B,/crn-l 17.32 28.80 16.34 16.56 16.65

,u/D Ref: -1.627 104 -1.587 142 125 -1.537 178,180 177

-

M. Yoshimine, S. Green, and P. Thadeus, Astrophys. J., 1973, 183, 899. P. H. S. Martin, D. L. Yeager, and V. McKoy, Chem. Phys. Letters, 1974, 25, 182. 176 C. Thomson, in ‘Electron Spin Resonance’, ed. R. 0. C . Norman (Specialist Periodical Reports), The Chemical Society, London, 1974, Vol. 2, p. 1. S. V. O’Neil and H. F. Schaefet, tert., J. Chem. Phys., 1971, 55, 394. l i e W. J. Stevens, J . Chem. Phys., 1973, 58, 1264. G. Das and A. C. Wahl, J. Chem. Phys., 1972, 56, 1769. G. Das, A. C . Wahl, and W. J. Stevens, J . Chem. Phys., 1974, 61,433. lel A. M. Karo, M. Krauss, and A. C. Wahl, Internat. J. Quantum Chem., 1973, 7 S , 143. 182 H. P. D. Liu and G . Verhaegeen, Internat. J. Quantum Chem., 1971, 5S, 103. 183 R. K HinMey, T. E. H. Walker and W. G. Richards, MoI. Phys., 1972,24, 1095. 174 I75

Quantum Mechanical Calculations on Small Molecule,s

105

NH+ has not been investigated recently, apart from a FSGO calculation.1e3 The most recent large calculation is by Liu et uZ.lS4 The OH radical has been the subject of a large number of papers, partly prompted by recent detailed experimental information by both microwave spectroscopy and electron resonance.la5The theoretical calculation of these fine-structure parameters is a challenge to the theoretician; GVB wavefunctions for OH ( P U ) have been reported by Gukrman and Goddard1S6 and also OVC wavefunctions have been computed by Karo et aZ.lal The PE curves reported in the latter were obtained from a 14-configurationwavefunction and are N 0.1 eV off the experimental curve, indicating the high accuracy attainable by this method. Bondybey et have computed PE curves via ht-order wavefunctions for OH. The calculated De of 4.26 eV (expt. : 4.63 eV) is much improved over the SCF result. The bond length is rather accurately predicted. The detailed charge distribution has been studied for the SCF wavefunctions by Cade et aZ.la8The PE curves of the 2X- and 41=- Rydberg states have been computed by Lefeb~re-Brion~l8~ who found these states to be unstable. The current interest in the hyperfine structure and dipole moment of OH and OD has led to several attempts to compute these quantities accurately. Green190 has computed SCF wavefunctions with various basis sets, and also CI wavefunctions. The experimental and computed values were in good agreement for the CI wavefunction. The computed dipole moment has an uncertainty of kO.06 D and the high accuracy enables a choice to be made between alternative experimental values. The hyperhe splitting constants were in very good agreement with experiment.l’* Many-body perturbation theory (MBPT)lgl has been used to calculate the proton hypedine splitting constant in OH,lg2with results in better than 15 % agreement with experiment. The energy for this wavefunction was not reported. OH- has been studied by Lishka,lg3the only earlier calculation being that of Cade.194 Using the IEPA method, the bond lengths and harmonic force constants are corrected in the right direction, but Re tends to be too long. Protonation energieswere computed and were in reasonable agreement with experiment. The separated-pair approach has been used to calculate electronic wavefunctions for the lX+ state of OH-, yielding a geminal description of the bond.lg5 N

E. HF and HF+.-The HF molecule has probably been as much studied theoretically as any other hydride except for LiH. The ground state of HF was recently investigated by Bondybey et uZ.la7Using extended STO basis sets, their computed SCF wavefunction gave an energy which was within 0.0028 hartree of Cade and Huo’s va1ue.l O4 Correlated wavefunctions were obtained by the first-order pro1B4 185

H.P. D.Liu and G. Verhaegen, J. Chem. Phys., 1970, 53, 735. A.Carrington, ‘Microwave Spectroscopy of Free Radicals’, Academic Press, New York, 1974.

l a 6 S. L. Guberman and W. A. Goddard, tert., J. Chem. Phys., 1970, 53, 1803. 187 188

180 190

101 1g 2 193 19* 195

V. Bondybey, P. K. Pearson, and H. F. Schaefer, tert., J. Chem. Phys., 1972, 57, 1123. P. E. Cade, R. F. W. Bader, and J. Pelletier, J. Chem. Phys., 1971, 54, 3517. H. Lefebvre-Brion, J. Mol. Structure, 1973, 19, 103. S. Green, J . Chem. Phys., 1973, 58, 4327. 0. Sinan6glu and K. A. Bruekner, ‘Three approaches to Electron Correlation in Atoms’, Yale U.P., New Haven, Connecticut, 1970. J. E. Rodgers, T. Lee, T. P. Das, and D. Ikenberry, Phys. Rev., 1973, 7A, 51. H.Lishka, Theor. Chim. Acta, 1973, 31, 39. P. E. Cade, J. Chem. Phys., 1967,47, 2390. J. D. Allen and H. Shull, Chem. Phys. Letters, 1971, 9, 339.

106

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Chemistry

cedure, and a 43-configuration calculation via the IN0 method was used to compute the PE curves. The best dissociation energy obtainable by this approach (estimated to be between 5.95 and 6.00 ev) is in error by 0.14.15 eV. The spectroscopic constants are much improved over the SCF results. Lie and Clementi’s 125 semi-empirical calculation of the correlation energy gave reasonable agreement with the full ab initio results for HF. An extension of the HF calculations is reported in a later paper by Lie196 in which the theoretical dipolemoment function and i.r. transition matrix were computed for the X1X+ state. The value of p was calculated over a wide range of R values from two-coniiguration MCSCF wavefunctions, the second configuration ensuring proper dissociation as R-+ m. A large (13s8p2d/8s2p)GTO basis was contracted to a [7s4p2d/5s2p]CGTO set. The results show a qualitative and quantitative improvement in p and its derivatives. The calculated result for the vibrationally averaged dipole moment for the tl = 0 level is in excellent agreement with experiment. The Dunham method of analysis of PE curves is well known, but Simons and Parr 197 have presented an alternative method. Calculations on HF and CO show the method [which involves the expansion parameter (R- Re)/Rinstead of (R-&)/Re] to be superior. The IEPA calculations by Lishkalg3also included applicationsto HF. His total energy of - 100.4005 hartree was only 0.03 hartree from Bender and Davidson’s large CI r e ~ u 1 t . l ~ ~ A comparison of minimal-basis perfect-pairing and MO wavefunctions for HF and HFf has appeared,l98 in which complete exponent optimization was carried out. A series of OVC calculations on first-row hydrides has aimed at accurate values for dissociation energies and onselectron moments of the charge distribution, and we have already referred to the calculations on NH.1789 l80 Krauss and Neumann lS9 have recently examined HF at the experimental value of Re. Using eight configurations in an OVC calculation, the dominant correlation effects in HF are accounted for. Good agreement with experiment was found for the dissociation energy, dipole moment, and quadrupole moment. Table 5 compares the results with experimental values and other recent calculations. Table 5 Comparison of some recent calculations on HF Method Hartree-Fock HFPD First-order CI CI Perfect pairing min STO-MO MCSCF(0VC) IEPA

Energylhartree - 100.0703 - 100.0917 - 100.1274 - 100.3564 -99.5456 -99.5356 -100.1397 - 100.3258

DeJeV co,Jcm-l B,Jcm-l 4469 21.87 4.38 4128 21.06 4.95 4210 20.8 5.88 __ -_ 6.18 _ ~ _ I

pJD 1.934 --

1.816 1.72 1.41 1.805

-

Ref. 125,104 125 187 142 198 198 199 193

Localized MO’s have been computed by von Niessen200for HF and a variety of other small molecules (LiF, BF, BN, CO, etc.),using a previously described localizalgG I97 198 199 200

G. C . Lie, J . Cliem. Phys., 1974, 60, 2991. G . Simons, R. G . Parr, and J. M. Finlan, J. Chem. Phys., 1973, 59, 3229. R. E. Bruce, K . A. R. Mitchell, and M. L. Williams, Chem. Phys. Letters, 1973, 23, 504, M. Krauss and D. Neumann,Mol. Phys., 1974,27, 917. W. von Niessen, Theor. Chim. Acta, 1973, 29, 29.

Quantum Mechanical Calculationson Small Molecules

107

tion methody201* 202 in which the orbitals are density localized, i.e. the localization is based on the minimization of the sum of the charge-densityoverlap integrals of the orbitals. Agreement with the results of other localization methods was good. With the exception of LiF, BN, and COYthe localized orbitals are in agreement with the concepts of inner shells, lone pairs, and bonding pairs. One reason for interest in more accurate calculations on HF has been the measured dissociation energy Do(HF), which can be obtained from photoionization or photoelectron spectra. Since HF+ dissociates correctly within the Hartree-Fock approximation to H+ and F(2P) in both the X2II and 2C+ states, PE curves were calculated by Julienne et aLY2O3 and later by Bondybey et U Z . , ~ both ~ ~ of whom obtained values of D Oin good agreement with experiment for the S C F calculation. The bond length in HF+ of 1.OOO bohr is less than the experimentalresult, which the authors call into question. Julienne et al. give a detailed discussion of the adiabatic dissociation process. HF+ was also considered in the calculations of ref. 198. Richards and co-workers have reported several calculations of the PE curves of HFf, including the A2X+ state, which is correctly predicted to be b0und.~04 HF-has been little studied, but Bondybey et al. found that for their first-order calculations the PE curves were all rep~1sive.l~~ We may summarize the results on first-row hydrides by concluding that a variety of calculations with a large number of different methods have appeared, but, with rather few exceptions, most calculations have not attained the accuracy of Bender and Davidson's work of some five years ago. This latter fact emphasizes the fact that it is not easy to obtain a large fraction of the correlation energy for these. molecules, but fortunately many interesting molecular properties and potential curves can be computed with high accuracy despite this. 4 Hydrides of Second and Higher Rows, AH

A definitive paper dealing with near-HF calculations was presented some years in which the hydrides NaH, MgH, AlH, SiH, PH, and ClH were studied. Recent work has been rather fragmentary, but more experimental information on SHYSeH, and TeH has recently been obtained.185 Wirsam206has reported combined SCF and CI results on SiH, predicting the location of a variety of low-lying states and their properties. Gaussian lobe functions were used in this work. Bondybey et aZ.187 studied NeH+ and NeH. The ground-state curve of NeH is repulsive but those of the excited states are not. The Ar-H long-range interaction has been evaluated by the MCSCF method.207 These results are of use in the analysis of scattering experiments of H off Ar. Scott and RichardsaoScalculated approximate SCF wavefunctions for TiH, whose ground state is predicted to be 4@. The dissociation energy is predicted to be 1.6eV, similar to that observed for CaH. FeH has been 201 202

203 204

205 206 207 208

W. von Niessen, J. Chem. Phys.,-l972,56, 4290. W.von Niessen, Theor. Chim. Acta, 1972, 27, 9. P. S. Julienne, M. Krauss, and A. C. Wahl, Chem. Phys. Letters, 1971, 11, 16. J. Raftery and W. G. Richards, J . Phys. ( B ) , 1972, 5, 425. P. E. Cade and W. M. HUO,J. Chem. Phys., 1967,47, 649. B. Wirsam, Chem. Phys. Letters, 1971, 10, 180. A. F. Wagner, G. Das, and A. C. Wahl, J. Chem. Phys., 1974,60, 1885. P. R. Scott and W. G. Richards, J. Phys. ( B ) , 1974,7, 500.

Theoretical Chemistry

108

examined by Walker et aL210The high-spin

aZ.,209

and a detailed study of MnH reported by Bagus et

7X+and 'II states were computed at an accuracy near to the level

of HF calculations. A detailed discussion of the bonding was given. Of particular interest is the fact that the Mn 3d orbitals are essentially unchanged within the molecule. The magnetic properties of AIH have been investigated by Laws et ~ 2 . ~ 1 1 The (XeH)+molecule has not been previously studied, and SCF calculations using a STO basis and including spin-orbit interactions semi-empirically have been reported by Kubach and Sidk212 In a significant paper, Bauschlicher and Schaefer213 have examined the flexibility of atomic orbitals in a molecular environment, and they have shown in calculations on diatomics involving second-row atoms (among these the 3X- state of PH) that only the outermost orbitals are altered during molecular formation, and hence essentially fully contracted GTO can be used for the inner-shell orbitals. We will return to this point later. For PH the contraction procedure that was used recovered 89% of the energy obtained with an uncontracted basis set. The HCI molecule has been extensively studied by SCF methods in the past, and this work is referenced in a more recent paper by Petke and Whitten,214who have examined the effect of the size of the basis set on the geometry, bonding, and physical properties of HCI. The results were compared with previous calculations. A gaussian lobe basis set was used, including d-functions on C1 and p- and dfunctions on H, and finally a 206-configuration CI calculation yielded the most accurate wavefunction computed to date for HCI. Although the best SCF calculation gave 99.55 % of the experimental energy, the 206-configuration wavefunction only recovered "4% of the correlation energy. Inclusion of C1 d-functions improves the total energy and all molecular properties significantly.The d-functions result in well-defined charge shifts, charge being shifted from H to CI in MO's made up mainly of CI s-orbitals, while charge is shifted from CI to H and into the bonding region for orbitals containing Cl p-functions. The C1 d-functions are particularly important in the calculation of the dipole and quadrupole moments. The effect of CI was rather small, only the dipole moment being significantly altered. Finally, we mention briefly an interesting series of papers by Bader and cow0rkers,~16who have partitioned the charge distribution in a variety of hydrides in a particular way. Details and references to this series of papers are given in ref. 215. 5 Homonuclear Diatomic Molecules of First-row Elements We have referred above to Clementi and Lie's work on the correlation energy of the first-row hydrides.125 The authors 216 have also presented results for the homonuclear diatomics of the first row, namely the molecules H2(XlZi), Li2(X1C;), Be&VZ:), 209

210 211 212

213

214 215 a16

J. H.Walker, T. E. H. Walker, and H. P. Kelly, J. Chem. Phys., 1972, 57, 2094. P. S. Bagus and H. F. Schaefer, tert., J. Chem. Phys., 1973, 58, 1844. E.A. Laws, R. M. Stevens, and W. N. Lipscomb, J. Chem. Phys., 1971,54, 4269. C. Kubach and V. Sidis, J. Phys. (B), 1973, 6, 289. C. W.Bauschlicher, jun. and H. F. Schaefer, tert., Chem. Phys. Letters, 1974, 24, 412. J. D. Petke and J. L. Whitten, J. Chem. Phys., 1972, 56, 830. R. F. W. Bader and R. R. Messer, Canad.J. Chem., 1974,52, 2268. G. C. Lie and E. Clementi, J. Chem. Phys., 1974,60, 1288.

Quantum Mechanical Calculations on Small Molecules

109

Bz(X3Z;), C2(X1Zi), N2(X1Zi), Oz(X 3C,), and F2(X1Xi), and before considering calculations on individual molecules, it is convenient to describe this important work. Using the same functional of the electronic density as used in the calculations on atoms and on AH, PE curves were computed for these molecules over a wide range of R. The HFPD functions used initially were only those necessary to ensure that &+ 2A. The energies were then corrected by using equation (11). The computed values of De are again much improved over the Hartree-Fock values. However, it is found that for four cases, uiz. Liz, Bz, CZ,and 0 2 , agreement was not within 1 eV of experiment. The reason for this was that the chosen reference function, although correctly describing the dissociation behaviour, does not include configurations which have been shown to be important in CI calculations. For Liz for instance, the configurations 1oil ok30; and 1ail eln; must be added to the two original functions 10il0i20i (HF configuration) and 1 ~ ~ 1 4 2The 0 ~ mixing . coefficients of the added configurations add up to 0.214. When a more appropriate reference function is used for these four molecules, much improved agreement with experiment is obtained. The lC, state of Be2 is repulsive. Agreement with experiment is not, however, as good as for the hydrides AH,125but since it is a simple matter to use the method and the method is almost certainly capable of refinement, these two papers do demonstrate that it might be possible to compute quite reliable correlation energies for larger systems. We now review some of the more important calculations on specific homonuclear diatomic molecules. A. Liz and Liz . T h e PE curve of Liz at large values of R (between 5 and 30 bohr) has been investigated, using a multiconfiguration approach to valence-shell correlat i ~ n . The ~ l ~dispersion contribution to the binding energy was computed and turns out to be practically equal to the London energy, EL = - 66R-6for R 2 10 bohr. This work is interesting as being appropriate for the study of long-range intermolecular forces. The authorsZ1shave also examined by the same method the PE curve of the lowest (3C,')state of Liz, using a gaussian lobe basis. The van der Waals minimum is quite deep, N" 0.0012 hartree (380 K). The calculations confirm previous 8 6 values which are still in disagreement with experimental estimates. This paper also gives a refined calculation on the 1Ci ground state, the lowest binding energy obtained being - 0.0366 hartree (expt. : - 0.0385 hartree). Early calculations by the OVC method were presented by D a ~ , ~who l 9 obtained E = - 14.90260 hartree. Calculations by GoddardzZOof GVB wavefunctions for Liz and a discussion of the bonding have appeared. Using an STO basis set, the orbitals obtained are very similar to the Li atomic orbitals, except that there is small amount of sp mixing, and a subsidiary 2s peak comes from the other centre. Goddard's128t129later papers on the use of effective potentials (GAEP) in these calculations also discuss the XIZB+state of Liz. The orbital description implies, as is found, a rather weak bond. The close agreement between the orbitals in the GAEP and GI methods is gratifying. Electronic properties were also computed, agreement being good except for the field gradient at the Li nucleus. This method was also 217

als 219 220

W. Kutzelnigg and M. GClus, Chem. Phys. Letters, 1970, 7, 296. W. Kutzelnigg, V. Staemmler, and M.Gelus, Chem. Phys. Letters, 1972, 13, 496. G. Das, J. Chem. Phys., 1967,46, 1568. W. A. Goddard, tert., J. Chem. Phys., 1968, 48, 1008.

110

Theoretical Chemistry

applied to Lil, where it was shown that the valence orbital in this case has far morep character and larger amplitude in the internuclear region, leading to a stronger bond. 82% of the experimental binding energy was obtained even with an sp GTQ basis.221 Li: was also investigated by Hendersen et aZ.,222 who obtained a near-HF PE curve, and who give references to earlier work on Li:. Properties were calculated for the 2 X i and 211 states. An FSGQ 163 calculation gives 85 % of the SCF energy, and a reasonable value of De. Liz has not been previously studied, and Blustin and Linnett 16s have reported FSGO calculations on the 2X: and 211u states. The latter was calculated to be bound, with the unpaired electron in the lIIU bonding orbital. This result is surprising. SCF calculations for Liz using bond functions in the basis set have been reported by Chu.223 B. Be2, B2, C2, Ci, and Ci.-Apart from the calculations of Lie and CIementi,216 there has been no recent work on Be2 or B2. C2 has been re-investigated by B a r ~ u h n , ~who ~ 4 gives references to many earlier calculations. Using an extensive basis set of gaussian lobe functions, SCF orbitals were obtained and used in two CI calculations at R values near Re. A calculation involving only virtual a-orbitals agrees qualitatively with experimental data on the three known Rydberg states. The most extensive CI treatment over-estimates the excitation energies by up to 1.7 eV. Similar calculations by the same author were carried out on C;.225 These calculations were prompted by recent optical spectral observations. This ion is so far the only negative molecular ion known that has bound excited states. Using a contracted gaussian lobe basis set, the 2 X l , 211u, and 2X: states were investigated at eight internuclear distances. In the SCF approximation, the 211u state is predicted to be the ground state, which is not the case when CI is included. The excitation energy of the 2Ei state is over-estimated. Two hitherto unobserved states, 2nuand "i, are predicted. C. N2 and N;.-The nitrogen molecule is one of the most extensively investigated of homonuclear diatomics. Extended-basis-set SCF calculations by Cade et af. were reported some years ago (see ref. 1 for full references). Mulliken, who pioneered MO correlation diagrams, has carried out an SCF calculation at 17 internuclear distances from 0.1 bohr to Re and has optimized the 5 value of each exponent at Re, which results in a computed MO correlation diagram.226 A Mulliken population analysis shows how the composition of the orbitals changes with R . The correlation diagrams are as expected qualitatively in most respects, but the quantitative features are not all as expected. For instance, - d for the 2ag orbital at first increases with decreasing R, then decreases surprisingly to smaller values than at Re before again increasing steadily towards the 2s united-atom orbital values. The 3ag orbital is particularly interesting, especially the variation between p - and s-character as R decreases. Near 0.6 bohr, it is almost pure 3 4 i.e. it is of the form og3s, derivable from the 3s penetrating Rydberg atomic orbital, instead of being predominantly ag2p, as at Re. N

221 222 223 224

225 226

W. A. Goddard, tert., J. Chem. Phys., 1965, 48, 5337. G. A. Henderson, W. T. Zemke, and A. C. Wahl, J. C h m . Phys., 1973, 58, 2654. S. Y . Chu, Theor. Chim. Acta, 1972, 25, 200. J. Barsuhn, 2. Nuturfursch., 1972, 27a. 1031. J. Barsuhn, J. Phys. (B), 1974, 7, 155. R. S. Mulliken, Chem. Phys. Letters, 1972, 14, 137.

Quaraturn Mechanical Calculations on Small Molecules

111

Finally, the 20, orbital (which at Re resembles 2pa of the UA) shows steadily increasing s- and decreasing p-character as R decreases. The value of B increases until R = 0.9 bohr but then suddenly decreases to a surprising minimum at R-0.6 before increasing again to the 3pa form in the UA. This quantitative analysis is very interesting, and it is hoped that further results on other A2 will show if the subtle effectsare common to the other molecules. Other papers by Mulliken, dealing with N2, have also a~peared.~271 228 Another interesting class of molecular states are those are predicted.228 involved in V-N transitions. Two types, V, and V, (OD* and m*), Cederbaum has pointed out that the breakdown of Koopmans theorem for the ionization potentials of N2 and F2 can be rationalized using simple symmetry arguments.229 Kaijser et aZ.230have calculated momentum densities for the ground and singly ionized states of Nz.These are calculated from the Fourier transform of the NSO of the wavefunction, evaluated in a minimal STO calculation with valence-shell CI. Interpretation of these quantities is discussed in the paper. There has been little recent work on the excited states of N2, except for some calculations involving the . ~variety ~ ~ of excited states of N2,02, and CO use of a different SCF H a m i l t ~ n i a nA were investigated using a CGTO basis set. Results were mostly in good agreement with experiment, but the b' 1Xi state of N2 could not be reliably located by this method. Attempts to improve molecular wavefunctions so as to be able to calculate properties more accurately continue to be made, particularly via the constrained variational procedure. Two-particle hypervirial constraints were considered by Bjorna within the SCF formation,232and he presented a perturbational approach to their s0lution.~3~ Using Scherr's wavefunction, and constraining # to satisfy the molecular virial theorem, a calculation on Nz led to rapid ~onvergence.~s*~235 The constrained SCF orbitals are believed to be a closer approximation to the true t,b nearer the nucleus than further out. A later paper discussed the electron-density maps in comparison to the SCF derived maps, which confirm the conclusion that the wavefunction near the nucleus is impr0ved.~3~ Electron scattering from molecules is receiving increasing attention, and theoretically it can be treated by calculation of the static potential (the interaction potential of an electron with the unperturbed charge distribution). Ab initio calculationsfor NZ using wavefunctions varying between minimal-basis and near-HF quality have been reported by Truhlar et Q I . , ~and ~ ' compared with semi-empirical INDO calculations. The anisotropy of the potential is only correctly described if d-functions are included in the basis set. Probably the most extensive calculation yet carried out on N2 is that of Langhoff R. S. Mulliken, Chem. Phys. Letters, 1972, 14, 144. R. S. Mulliken, Chem. Phys. Letters, 1974, 25, 305. 229 L. S. Cederbaum, Chem. Phys. Letters, 1974, 25, 562. m o P. Kaijser, P. Linder, A. Andersen, and E. Thulstrup, Chem. Phys. Letters, 1973, 23, 409. 2s1 J. B. Rose and V. McKoy, J. Chem. Phys., 1971, 55, 5435. 2s2 N. Bjorna, J. Phys. ( B ) , 1971, 4, 424. 233 N. Bjorna, J. Phys. (B), 1972, 5, 721. 234 N.Bjorna, J. Phys. (B), 1972, 5, 732. 235 N. Bjorna, Mol. Phys., 1972, 24, 1. 236 N. Bjorna, Physica Norvegica, 1972, 6, 8 1. 237 D. G. Truhlar, F. A. Van-Catledge, and T. H. Dunning, J . Chem. Phys., 1972,57,4788. 227 228

112

Theoretical Chemistry

and D a v i d s ~ nwho , ~ ~obtained ~ 63 % of the correlation energy in a large CI calculation using a GTO basis set. In this set of calculations, a comparison was made between the sum of the pair-correlation energies and the total correlation energy. Calculations were carried out using both ICSCF 239 and canonical orbitals. The sum of pair energies was 17% bigger than the correlation energy in both cases. It was expected that N2 would be a particularly unfavourable case for the independent-pair method. The effect of quadruple excitations was also studied and found to be 8 % of the total. Second-order perturbation theory over-estimates Ec by 23-50 % depending on how 0 is chosen. In connection with calculations of pair-correlation energy, we should refer to an extensive set of OVC results, not published in full, carried out some years ag0.~40 We must finally mention two papers dealing with approximate methods in which NZhas been studied. Firstly, the X, multiple-scattering SCF method2419242 has been applied to N2,02, and F2 by Weinberger and Konowalow 243 and the PE curves have been calculated over a wide range of R values. Although qualitatively correct curves are obtained, the calculated values of De are much too small, and of Re are much too large. Note, however, that calculations by the same method on Liz244 give more realistic results. ‘Transition state’ calculations of the ionization excitation energies for N2 give ionization potentials virtually identical to those calculated directly, and are in fair agreement with experiment. Much further work is needed in order to establish and refine this method for calculations on small molecules. A comparison of properties calculated by the INDO method, such as dipole and quadrupole moments of the charge distribution, with ab initio results has been given for N2, H2, COYand HF.245Suggested improvements in the INDO procedure were given. N l has been the subject of two papers. The 2Xl, 2rIu, and 2 X i vertical ionization potentials of N2 were computed by Chen et aZ.246 The method 247 used permits direct calculation of ion-molecule energy differences, and contributions to the IP are analysed. Several low-lying quartet states of N2f have been studied using valence-shell CI with up to 270 configurations per symmetry.248 The 2s and 2 p exponents for the ground state of Nz were optimized for the molecule. Several states were found to be bound, and they have lower energies and larger Re than previously assumed, particularly the 411, state. D. 0 2 , Oi, and O;.-Although Cade, many years ago, obtained a near-HartreeFock wavefunction for the X3C; state of 0 2 , the wavefunction was not published, although a detailed discussion of the charge distribution and how it changes upon N

-

238

239 240

241 242 243

244 245

246 247

24*

S. R. Langhoff and E. R. Davidson, Internat. J. Quantum Chenz., 1974 8 , 61. E. R. Davidson, J. Chem. Phys., 1972, 57, 1999. P. Sutton, P. Bertoncini, G. Das, T. L. Gilbert, A. C. Wahl, and 0. Sinandglu, Internat. J . Quantum Chem., 1970, 3S, 479. J. C . Slater, A h . Quantum Chent., 1972, 6, 1 . K. H. Johnson, Adv. Quantum Chem., 1973, 7 , 143. P. Weinberger and D. D. Konowalow, Internat. J. Quantum Chem., 1973, 7 S , 353. See ref. 9, p. 161. F. A. Van-Catledge, J. Phys. Chem., 1974, 78, 763. T.-T. Chen, W. D. Smith, and J. Simons, Chem. Phys. Letters, 1974, 26, 296. J. Simons and W. D. Smith, J. Chem. Phys., 1973, 58, 4899. A. Andersen and E. W. Thulstrup, J. Phys. ( B ) , 1973, 6, 211.

Quantum Mechanical Calculations on Small Molecules

113

ionization or excitation has been given by Cade et a1.249 The computed HartreeFock De of 1.43 eV, is however, only 27 % of the experimental value. Schaefer 250 has gone beyond the Hartree-Fock approximation and computed the ground-state PE curve, using first-order wavef~nctions.~5l A contracted STO basis set has been used, with 128 configurationsincluded. The molecule now dissociatesto two oxygen atoms, and De was computed to be 4.72 eV (expt. : 5.21 ev). Spectroscopic constants were usually in better agreement with experiment than a previous minimal-basis full CI calculation. The value of Re obtained was close to the experimental value. Goddard and co-workers have also studied 0 2 by the GVB method.154 This molecule is particularly interesting for this method, since one of the difficulties of the earlier VB method was its failure to predict the triplet ground state. The GVB wavefunction is of the form (1 3,where ( o ~ ais~the ) 0-0 a-bonding pair. It differs from

by the presence of a split 0-0 cr pair, which involves the 3ag and 3au natural orbitals. The correct ordering of the 3X;, lZg, and lA; states was predicted. GVB-CI calculations give good results for De and the excitation energies, although the energy of Schaefer’s FO wavefunction is lower by -0.06 hartree. In a later paper Cartwright et aZ.252 have examined the n = 3,4, and 5 Rydberg series in 0 2 and correlated the results with experimental studies of electron energyloss spectra. The calculations were carried out using a method recently proposed by Hunt and Goddard.253 In this method, the form of the Hartree-Fock 2 is modified so that the virtual orbitals are good approximationsto the SCF excited-state orbitals, and are called improved virtual orbitals (IVO). A 4s3p CGTO basis set was used, with a set of diffuse functions added to obtain a good description of the Rydberg states. Results for the ordering of the states and the excitation energies were in generally good agreement with experiment. M ~ r u k u m has a ~ ~also ~ investigated the PZ;, clC& P A u , and A3Z: states of 0 2 with various CI wavefunctions. The oscillator strength of the B3&+ X3Z; transition was in good agreement with experiment. An interesting comparison of the use of 3d polarization functions and bond functions (GTO placed in the bond) has been presented by Vladimir0ff,~5~ using NZ and 0 2 as examples. It was found that three optimized bond functions perform about as well as six 3d-functions, and the computer time required is substantially less. A variant of the MsX, method has been proposed in which a is ~ a r i e d , ~and 5~ applicationsto N2 and 0 2 have been reported.257 For a = 0.70, results were in good agreement with experiment, mainly because of the removal of the muffin-tin approximations. #HF

249

250 251

252 25s 254

255 256

257

P. E. Cade, R. F. W. Bader, and J. Pelletier, J. Chem. Phys., 1971, 54, 3517. H. F. Schaefer, tert., J. Chem. Phys., 1971, 54, 2207. H. F. Schaefer, tert., and F. E. Harris, J. Chem. Phys., 1968, 48, 4946. D. C. Cartwright, W. J. Hunt, W.Williams, S. Trajmar, and W. A. Goddard, tert., Phys. Rev., 1973, 8A, 2436. W. J. Hunt and W. A. Goddard, tert., Chem. Phys. Letters, 1969, 6, 414. K. Morukuma and H. Kohnishi, J. Chem. Phys., 1971,55, 402. T. Vladimiroff, J. Phys. Chem., 1973, 77, 1983. E. J. Baerends, D. E. Ellis, and P. Ros, Chem. Physics, 1973, 2,41. E. J. Baerends and P. Ros, Chem. Physics, 1973, 2, 52.

114

Theoretical Chemistry

Calculations of the fine-structure parameters by V e ~ e t h , ~the ~ 8spin-orbit contribution to the zero-field ~plitting,~5~ and the Verdet constant for 0 2 have also been reported. ti O Photoelectron spectral measurements have prompted high-accuracy nearHartree-Fock calculations on the 1s hole states of 0,+.261Calculations were reported at Re for molecular 0 2 . The frozen-orbital approximation evaluated the energy of 0;from the RHF calculations of Schaefer250 reported above. Then the IP are the energy. The IP obtained difference between the 0 2 ground-state energy and the 0,' was 563.5 eV. Direct hole-state calculations for the relevant states of O,*,with the MO constrained to be of g or u symmetry, were also carried out. For the orbital occupancy (16), the computed IP was 554.4 eV. Finally, the restriction to g and u

symmetry was relaxed, giving an IP of 542.0 eV. The latter is in good agreement with the experimental value of 543.1 eV. The interpretation of this result shows that the singly occupied 1s orbital is essentially localized on one of the two oxygen atoms. The electron affinity of 0 2 is an important quantity, and its direct calculation by Zemke et ~1.262has been carried out, from a computation of De for 0; in conjunction with the Hess cycle (17). The OVC procedure gave De(Oz) = 4.14 eV and thus EA(02) = De(OC)-De(Oz)

+ EA(0)

(17)

EA(02) = 0.42 eV, in good agreement with the experimental value of 0.440k 0.008 eV. Krauss and co-workers263have also studied various excited states of 0; by both OVC and PNO methods. All excited states were found to have Re at least 1 bohr larger than the ground state. E. F2, Fi, and q.-"he status of calculations on F2 up to 1970 has been summarized by WahL4 who, together with Das, published a six-configuration OVC wavefunction which gives a De value in good agreement with experiment.264 Minimal-basis-set CI calculations on this molecule give rather poor results,265De being too high. In a recent paper, Das and Wahl266 discuss improved techniques for the computation of MCSCF wavefunctions, and discuss briefly the case of F2. A more recent267 minimal-basis-set full CI calculation of the PE curves has been and F; (see below). published, although this paper deals primarily with the ions F,* Kasseckart26shas carried out both SCF-CI and VB-CI calculations on the ground and lower excited states of F2.The results were no better energetically for the ground state than those from much earlier work, but the location of a variety of excited states was predicted. The experimentally observed orange bands are possibly due to the L. Veseth and A. Lofthus, MoZ. Phys., 1974, 27, 51 1 . J. A. Hall, J. Chem. Phys., 1973, 58, 410. 260 Y.J. I'Haya and F. Matsuka, walnternat. J. Quantum Chem., 1973, 7 S , 181. 261 P. S. Bagus and H. F. Schaefer, tert., J. Chem. Phys., 1972, 56, 224. 262 W. T. Zemke, G. Das, and A. C. Wahl, Chem. Phys. Letters, 1972, 14, 310. 2G3 M. Krauss, D. Neumann, A. C. Wahl, G . Das, and W Zemke, Phys. Rev., 1973, 7A, 69. 264 G. Das and A. C. Wahl, J. Chem. Phys., 1972,56, 3532. 265 F. E. Harris and H. H. Michels, Internat. J. Quantum Chem., 1970, 3S, 461. s0 G. Das and A. C. Wahl, J. Chem. Phys., 1972, 56, 1769. 267 D. J. Ellis, K. E. Banyard, A. D. Tait, and M. Dixon, J. Phys. ( B ) , 1973, 6, 233. 268 E. Kasseckert, 2. Naturforsch., 1973, 28a, 704. 268

259

Quantum Mechanical Calculations on Small Molecules

115

lrIut+fE; and 1ng4X: transitions. Goodisman269 has reported the results of Thomas-Fed-Dirac calculations of the one-electron energies of F2. Cederbaum et aL270have calculated the vertical ionization potentials (VIP’s) of F2 using a recently developed theory involving Green’s functions.271 Results were in good agreement with experiment, which is not the case if the VIP’s are derived via Koopman’s theorem. Calculations on FZ+have been carried out by Balint-K~rti2~2 and by Ellis and COw0rkers.26~References to previous work are given in ref. 267. Ellis et aZ.267carried out minimal-basis (STO) complete CI calculations of the PE curves. The ground state is 2r]Ig, and an energy of - 197.5625 hartree was obtained in a 4O-configuration calculation. Fa+has a shorter bond length than F2. Calculations on Fif predict it to be unbound. Balint-Kurti 272 has investigated 13 electronic states of FZ+using several different methods, with results in reasonable agreement with the two experimentally observed states. The anion Fi was also studied by Ellis et aZ.,267 but only using a small number of configurations. The results were inferior to those of Baht-Kurti and K a r ~ I u and s~~~ of Copsey et aZ.,274who used more extended basis sets. Gilbert and Wahl reported the results of near-HF calculations in 1971 for the ground and various low-lying excited states of Fa.275 6 Homonuclear Diatomic Molecules of Second and Higher Rows There have been only a few studies of these molecules, containing between 22 and 34 electrons, and the most extensive set of calculations has been on the interaction between two Ne atoms. For such species, the balance between the repulsive force due to the overlap of two neutral closed-shell atoms and the attractive dispersion force determines the position of the van der WaaIs minimum. Near-HF calculations of the repulsive curve were published several years The best a6 initio calculations, including electron correlation, produce good agreement for H e 0 2 Conway and Murrell 277 have calculated the interatomic energy for Ne2 directly from an antisymmetrized product of atomic Ne SCF wavefunctionsin the range R = 4-8 bohr. The best wavefunction was obtained using a 6s4p STO basis set. The binding energy is 20 % less than the experimental value. The dispersion energy was obtained from a multipolar expansion given by Starkschall and Gordon.278An ab initio calculation of the dispersion energy by K0chanski2~~ using a GTO basis set (including dfunctions) gave results close to the multipole-expansion results. Stevens et al.,280 using the MCSCF method, have also investigated this problem at R = 5.726, 6.0,

-

269 270

c71

273

273 274

276 277 278

279

J. Goodisman, Theor. Chim. Acta, 1972, 25, 205. L. S. Cederbaum, G. Hohlneicher, and W. von Niessen, MoZ. Phys., 1973, 26, 1405. L. S. Cederbaum, G. Hohlneicher, and W. von Niessen, Chem. Phys. Letters, 1973, 18, 503. G. G. Ralint-Kurti, Mol. Phys., 1971, 22, 681. G. G. Balint-Kurti and M. Karplus, J. Chem. Phys., 1969, 50, 478. D. N. Copsey, J. N. Murrell, and J. G. Stamper, MoZ. Phys., 1971, 21, 193. T. L. Gilbert and A. C. Wahl, J . Chem. Phys., 1971, 55, 5247. T. L. Gilbert and A. C. Wahl, J. Chem. Phys., 1967,47, 3425. A. Conway and J. N. Murrell, Mol. Phys., 1974, 27, 873. G. Starkschall and R. G. Gordon, J. Chem. Phys., 1971,56, 2801. E. Kochanski, Chem. Phys. Letters, 1974, 25, 380. W. J. Stevens, A. C. Wahl, M. A. Gardner, and A. M. Karo, J. Chem. Phys., 1974,60,2195.

116

Theoretical Chemistry

and 6.2 bohr. The results were in good qualitative agreement with the semi-empirical potential. An X, calculation of this potential has also appeared recently.281 Na2 has been investigated by the MCSCF procedure, with a computed De that is in good agreement with experiment.lS1Cl2 has been studied at near-HF leve1,275 and a complete VB-CI calculation has also been carried out by Heil et aZ.,282 using a basis of H F atomic orbitals. However, the calculated value of De was only 0.71 eV, which is only 29 % of the experimental value, and not very different from the nearHE; value of 0.87eV. Clearly, it is much more difficult to calculate this quantity accurately for second-row molecules. The ionic species CI;, Nei, and Arl were also investigated in ref. 275, and the results compared with experimental data. Finally, we mention an interesting paper dealing with P2 and PO. Mulliken and Liu283 have reported accurate SCF calculations on this molecule, with a view to examining the influence of 3d-orbitals on the bonding. At Re, the d population is -0.34 electron, and it increases markedly at smaller values of R. The increase in energy on deletion of the d-functions at Re is 2.53 eV. The calculations are compared with calculations on N2, C12, and F2. Participation of d- andf-functions in NZis smaller, but their effect on the energy is about the same (2.58 eV): in Cl2 the effect is somewhat smaller (1.61 eV). The bonding in N2 and P2 is discussed in some detail in this interesting paper. N

7 Heteronuclear Diatomic Molecules A. Rare-gas Compounds.-The recent preparation of a number of rare-gas compounds has led to a significant number of calculations on these molecules, particularly on diatomic species. The repulsive interaction and dispersion interaction between like rare-gas atoms has been dealt with above. Several years ago Matcha and Nesbet 284and Gilbertand Wahl 276 presented calculationson the species NeHe, ArHe, and NeAr. The interaction of two dissimilar rare-gas diatoms gives rise to a dipole moment, which leads to far4.r. collision-induced spectra of rare-gas mixtures. For this reason, the dipole moments of such species are of interest, but dispersion contributions are not described in the HF approximation and have to be computed by an empirical expression p = B exp (- R/p). Much more work is needed on this problem. Wahl and co-workers have reported the numerical results of OVC calculations on HeNeY240 but no detailed discussion was given. In cases like these, the H F model should yield reliable interaction potentials, since A and B are both closed systems. The interaction-energy curves for alkali metal-rare gas pairs are also of interest experimentally in scattering and radiation problems, and theoretically because of the expected reliability of the HF energy for this class of half-open-closed-shell systems. Calculations on LiHe and NaHe ( X 2 Z + , AzrI, B2C+)and their X1C+ ions have been reported by Krauss et al. 285 from R = 3 to 10 bohr. Both STO and GTO expansion bases were used, with comparable results except for theA2II state of NaHe. The variation of dipole and quadrupole moments with R was investigated. The X2Z+ curve is 281 282 283

284

285

13. D. Konowalow, P. Weinberger, J. L. Calais, and J. W. D. Connolly, Chem. Phys. Lettcm, 1972, 16, 81. T. G. Heil, S. V. O’Neil, and H. F. Schaefer, tert., Chem. Phvs. Letters, 1970, 5 , 253. R,S. Mullikeii and B. Liu, J. Amer. Chem. Soc., 1971, 93, 6738. R. L. Matcha and R. K. Nesbet, Phys. Rev., 1967, 160, 72. M. Krauss, P. Maldonado, and A. C. Wahl, J. Chem. Phys., 1971, 54,4944.

Quantum Mechanical Calculations on Small Molecules

117

purely repulsive, and the AZII and X1Z+curves are very similar owing to the penetration of the He within the sphere containing the 2p or 3p charge densities of Li or Na. The long-range repulsive behaviour of the B2C+ curve, compared with estimates of Ec, shows that the E(R)dependence in the region of 10 bohr is dominated by the H F repulsive curve. Chargedensity plots were presented. The long-range interaction has been studied by the MCSCF method in the case of HeH and LiHe (for R > 10 bohr).286 Results were in good agreement with Dalgarno’s estimate of a,but this is quite different from a semi-empirical result. The interactions between a rare gas and an alkali-metal ion and also between rare gas and alkaline earth dications are of interest in scattering calculations. Results have been reported for He-Lif287 and more recently for He-Be2+, using singleconfiguration SCF calculations, with a GTO bask288 There is a large discrepancy between theory and experiment for small internuclear distances, but the results are within 25 % of experiment between 0.4 and 0.8 A. The He-Li+ and He-Be2+ system has also been studied with the FSGO method.37 The computed binding energy of 1.5 x hartree was an order of magnitude larger than that found for HeH. A preliminary report of work on %He289 was followed by detailed calculations on the lowest lC+ states of BeNe and Be&, using a large GTO basi~.~90 Neither BeH, BeNe, nor BeAr were predicted to be stable. However, this paper showed that from the population analyses it should be possible to see which virtual MO’s correlate with which occupied orbitals. Also,contraction schemes for the second-row basis sets were investigated. We turn now to an interesting series of papers by Allen and co-workers on diatomic molecules containing a rare-gas atom and one of the atoms F, 0, B, or N. This work was initiated in 1966 by Allen et aZ.,291who showed by a6 initio valencebond calculations (including extensive CI) that the PE curves for the diatomics HeO, HeF, NeO, and NeF were repulsive, the least repulsion being found for HeO. However, the singly charged species should have at least one bound state, and in a subsequent paper most of the lower electronic states of Hex+ and NeX+ were investigated (X was F or N).292It was found that HeFf and NeF+ have bound 1C+ states of sufficient stability (140 and 130 kJ mol-1, respectively) that they might possibly participate in compounds such as HeF+PtF; or NeF+SbF;. However, HeN+ and NeN+ are not bound sufficientlystrongly to be likely cations. It is also expected that XeN+ will be bound. A separate paper on ArF+ and A r 0 2 9 3 showed ArO to be repulsive but ArF+ to be strongly bound ( 290 kJ mol-l), and hence ArF+PtF; might be isolable. Calculations on HeO+, NeN+, and ArO+294 show that for each of these ions De is small, but at least one stable state is predicted, which might be observed in an ionmolecule reaction. N

286

288 290

291 292

293 294

5

G. Das and A. C. Wahl, Phys. Rev., 1971,4A, 825. G. W. Catlow, M. C. R. McDowell, J. J. Kaufman, L. M. Sachs, and E. S. Chang, J. Phys. (B), 1970, 3, 833. S. W. Hamson, L. J. Massa, and P. Solomon, J. Chem. Phys., 1973, 59, 263. J. J. Kaufman and L. M. Sachs, J. Chem. Phys., 1970, 42, 3534. J. J. Kaufman, J. Chem. Phys., 1973, 58, 4880. L. C. Allen, A. M. Lesk, and R. M. Erdahl, J. Amer. Chem. SOC.,1966, 88, 615. J. F. Liebman and L. C. Allen, J. Amer. Chem. SOC.,1970, 92, 3539. J. F. Liebman and L. C. Allen, Chem. Comm., 1969, 1355. J. F. Liebman and L. C. Allen, Internut. J. Mass. Spectrometry lon Phys., 1971, 7 , 27.

118

Theoretical Chemistry

In the case of noble-gas compounds HeB+,NeB+, and ArB+, repulsive curves were c0mputed,~~5 but it was suggested that XeB+ might be bound. Finally, an extensive SCF calculation near the H F limit was reported by Liu and Schaefer for KrF and KrF+.296v297KrFf is predicted to be bound (Re = 1.68 A; De = 0.02 eV). Firstorder wavefunctions still predict KrF to be repulsive, but for KrF+, Re = 1.75 & De = 1.90 eV. The latter has been observed experimentally (D:xp> 1.58 eV) and KrSbFiz (KrF+SbFJ has been synthesized. Several molecular properties were computed in this work. More recently, potential curves for the lowest 2C+ and 211 states of XeF have been computed.29*Only a weak van der Waals attraction between Xe and F is predicted (- 0.15 kcal mol-I). This result is not consistent with the way in which a variety of experiments have been inter~reted.~g* Classification of the remaining heteroiiuclear diatomic molecules is somewhat arbitrary, and we have grouped together those molecules containing some particular electronegative element B, rather than consider isoelectronic species. B. Oxides of Elements of Groups I, 11, and 111.-LiO has been thoroughly studied recently. The PIE curves of the X z I l and A2C+states were computed using a large CI (valence shell only).299 The computed spectroscopic properties, including De,are in good agreement with the few experimental values which are known. In a later paper,sOo the problem of computing spectroscopic band intensities was reviewed in detail, and using the above wavefunctions and those computed in a similar manner for A10 (X2C+, A W , and B2C+),calculations were reported for these quantities for both molecules. One interesting observation is that there is no adequate single-configurationdescription of the 2Z,+ state of AlO. Predictions for the A2.Z-X2H system of LiO were also made by Wentink et aZ.,301using Yoshimine's wavefunction. Accurate SCF wavefunctionsfor the next alkali-metal monoxide, NaQ, have been reported for the 2 I l and 2C states, and also for the 3Cstate of NaO+ and the 3II, 3C, and l C states of NaO-.302 From the wavefunctions, several spectroscopicproperties and some thermodynamic data have been derived, particularly for several reactions involving NaQ and WaQ+. In addition to the paper cited above, dealing with AIQ, an extended DZ basis SCF calculation of the energy spectrum has been reported by S ~ h a m p sSix . ~ states ~ ~ of A10 and three of AIO+ were investigated. Correlation-energy dieerences were estimated semi-empirically. The alkaline-earth oxides are particularly interesting from a theoretical point of view, because of experimental uncertainty as to the nature of the ground state. Schaefer et al. have reported a series of calculations on Be0 which attempt to resolve J. F. Liebman and L. C. Allen, Innorg. Chem., 1972, 11, 1143. B. Liu and H. F. Schaefer, tert., J . Chem. Phys., 1971, 55, 2369. 297 P. S. Bagus, B. Liu, and H . F. Schaefer, tert., J. Amer. Chem. SOC., 1972, 94, 6435. 298 D. H. Liskow, H. F. Schaefer, tert., P. S. Bagus, and B. Liu, J . Anier. Chem. SOC., 1973, 95, 4056. 299 M. Yoshimine. J . Chern. Phys., 1972, 57, 1108. 300 M. Yoshirnine, A. D. McLean, and B. Liu, J. Chem. Phys., 1973, 58, 4412. 301 T. Wentink, jun., and R. J. Spindler, jun., J . Quant. Spectroscopy Radiative Transfer, 1973,13, 595. 302 P. A. G. O'Hare and A. C. Wahl, J. Chem. Phys., 1972, 56,4516. 303 J. Schamps, Chern. Physcis, 1973, 2, 352. a'J5

2y6

119

Quantum Mechanical Calculations on Small Molecules

earlier conflicting predictions. In the first paper,304 a first-order wavefunction for the lowest 1C+state was obtained using a contracted STO basis of DZ P quality. The calculated E = - 89.58455 hartree leads to De = 6.58 eV compared with the experimental value of 6.69 k 0.04 eV. All of the most important configurations involve the valence orbitals, and in particular the . . . 5021n2configuration is not as important as suggested in some earlier work. In a second paper,305 an investigation of the 3C- and low-lying 3C+ states with the same type of wavefunction was carried out. The 3X- state is repulsive and the (unobserved) 3C+state is predicted to lie 1.91 eV above the X1C+ state, and a variety of spectroscopic constants were calculated. In the third paper,306 the other possible contender for the ground state, the 3 l l state, was investigated. The authors show this state to be 0.73 eV above the lX+ state. This result is in contrast to that of a near-HF calculation,307where E(311)< E(lX+), and emphasizes the crucial role of electron correlation in determining the order of these states. A further calculation of the 3C- state confirmed it to be repulsive. It seems clear that the most obvious difference between Be0 and the isoelectronic C2 is that the 3X- state of the latter is attractive. It is also of interest that the 3C- state of the isoelectronic BN is bound but only 0.3 eV above the 311 state.308 MgO, the next oxide in this series, has been the subject of numerous experimental and theoretical studies. An extensive series of calculations at the SCF level for six differentconfigurations was reported by Schamps and Lefebvre-Bri~n,~O~ who give references to earlier work. These calculations used an extended DZ basis set and are consequently more accurate than previous work. The HF calculations predict a 311ground state, which is the same as found for CaO, but as noted above, CI might well reverse this ordering, and it is clear that very extensive calculations are needed before this question is definitely settled. A very recent paper reporting PE curves for BeO, Mgo, and CaO, using DZ P basis sets, has also appeared,3lO and the authors also discuss the dissociation behaviour of the ground state. Calculation of the spin-orbit matrix elements shows that 3II and lC+ states are not significantly mixed. Be0 has also been the subject of an OVC calculation by Wahl et aZ.,240 but full results have not appeared. C. Metal Oxides of Other Groups.-Despite the large number of electrons, some of these have been recently studied. The 90-electron molecule PbO has been studied with a minimal basis set in order to assess the importance of relativistic effects in such calculations.311Since for 2 = 82(Pb) relativistic energies are expected to be large (for the atom, = 1390 hartree), this molecule is an ideal test case, and comparison with CO, which has the same electronic structure in its vaIence shell, was carried out. The errors in the calculated De are comparable, and the predicted

+

+

Ez

304

306 307

308 309 310 311

H. F. Schaefer, tert., J. Chem. Phys., 1971, 55, 176. S. V. O’Neil, P. K. Pearson, and H. F. Schaefer, tert., Chem. Phys. Letters, 1971, 10, 404. P. K. Pearson, S. V. O’Neil, and H. F. Schaefer, tert., J. Chem. Phys., 1972, 56, 3938. W. M. Huo, K. F. Freed, and W. Klemperer, J. Chem. Phys., 1967, 46, 3556. M. P. Melrose and D. Russell, J. Chem. Phys., 1971, 55, 470. H. Schamps and H. Lefebvre-Brion, J. Chem. Phys., 1972, 56, 573. N. J. Stagg and W. G. Richards, Mol. Phys., 1974, 27, 787. G. M. Schwenzer, D. H. Liskow, H. F. Schaefer, tert., P. S. Bagus, B. Liu, A. D. McLean, and M. Yoshimine, J. Chern. Phys., 1973, 58, 3181.

120

Theoretical Chemistry

spectroscopic constants are in about as good agreement with experiment as in the case of CO. Thus the authors conclude that this type of wavefunction might well be useful for predictions of geometry. A near-HF calculation of the low-lying states of FeO has also appeared.312 Once again, experimental evidence is ambiguous on the ground-state configuration. Limited CI calculations were also carried out on various states, and extensive CI on the lowest 5X+state. It was concluded that this state is not the ground state, and at the moment the nature of this state is not completely clear. D. Non-metal Oxides.-CO, SiO, and CS. The literature on CO is very extensive, and much early work is discussed in ref. 1, together with an analysis of recent extensive calculations on the valence excited states. O’Neil and Schaefer313 carried out minimal-basis full CI calculations on 72 states of CO at nine values of R . Seventeen bound states were predicted, eight of which have been observed experimentally, and the ordering is in agreement with experiment except for the a3II and Aln states. Very detailed information is available in this investigation. Siu and Davidson,314 at about the same time, reported what is currently the most accurate ground-state wavefunction of CO, with a very large CI calculation. 70 of the correlation energy was obtained. The most important configurations were used to obtain the natural geminals. A pair-energy approach to the correlation energy gave significantly different results. Green16 has recently reviewed the accurate calculation of dipole moments, and, in a series of papers from 1970-1973, studied CO and CS. A large RHF calculation of the PE curve and the incorrect dissociation behaviour were discussed315before a series of papers dealing with CI calculations. A procedure for selecting

(ii)

yr = y f )

(iii)

YI

(c) Y I

= y(j,*)

and YJ are both di-excited configurations: yr = y(;$)

(ii)

yI = y($)

Each term leads to a simple contribution to the energy.151 The whole scheme is therefore fairly compact and, since there is no diagonalization, it is faster than the usual SCF approach using the CNDO approximations. It must be noted that even a simple extension from CNDO to INDO integrals increases the number of contributing configurations. The important question of the stability of the perturbation expansion to changes in hybridization and the bond polarity parameter is discussed by Jordan et The method has been extended to radicals with a well localized odd electronl53 and to some excited states.154 A similar ab initio treatment has been developed but can only be applied to small molecules.155 In a paper dealing with calculation of n.m.r. coupling constants, Dennis and Malrieu make some use of INDO terms rather than CND0.156 A complete derivation of the PCILO equations using diagramatic techniques has been given recently by Kvasnicka.157 The PCILO method has been extensively applied to conformational questions in biochemistry, mainly by Pullman and co-workers. This general field has been the subject of recent reviews.1589159 In only a few years a vast literature of such calculations has appeared. The following list of examples should, however, allow access to 152 153 l5* 155 151i

158

159

F. Jordan, M. Gilbert, J. P. Malrieu, and U. Pincelli, Theor. Chim. Actu, 1969, 15, 211. J. Langlet, M. Gilbert, and J. P. Malrieu, Theor. Chim. Actu, 1971, 22, 80. (a) J. Langlet, Theor. Chirn. Acta, 1972,27,223; (b)J. Langlet and J. P. Malrieu, ibid., 1973,30, 59. (a)A. Masson, B. Levy, and J. P. Malrieu, Theor. Chim.Acta, 1970,18,193; (b)J. P. Daudey and S. Diner, Internat. J. Quantum Chem., 1972, 6, 575. A. Dennis and J. P. Malrieu, Mol. Phys., 1972, 23, 581. V. Kvasnicka, Theor. Chim. Actu, 1974, 34, 61. A. Pullman, Fortschr. Chem. Forsch. (Topics Current Chem.), 1972, 31, 45-103. B. Pullman, Internat. J. Quantum Chem., Symp., 1971, 4, 319.

Electronic Calculations on Large Molecules

199

the majority of papers : nucleic acids,160,161 steroids,1S2 polypeptide~,l63-~6~ peptides, 167 acetylcholineand derivatives, nicot inamides, 69 bicyclo[2,2,1Iheptane alcohols and ketols,170 3-hydroxybornan-2-ones and 2-hydroxybornan-3-0nes,~~~ retinals,l72 disulphide bridges in proteins,l73 enniatin B,17* acetanilide,l75 diphenothiazines,l78 ribosaccharides,l76 2-(p-metho~ybenzyl)-1-acetylpyrroline,l7~ flavine,l?Qserotonin and histamine,l80 and monoamino oxidase.lS1 The method for radicals has been applied to malic acid free radicals.ls2Langlet and Malrieuls3have used the technique to discuss the Woodward-Hoffman rules for electrocyclic reactions. Arnaud et aZ.184 discuss the application of the method in comparison with CNDO and Extended Hiickel to donor-acceptor complexes. In all these studies the PCILO method gives surprisingly good results. It is sufficiently simple to allow complete conformational searches, and the resulting conformational maps appear to correspond well with experiment. Similar conformational maps have been produced using CNDO/2.185 For small molecules a few detailed tests have been carried out.la8 Langlet and van der conclude that a better calibration is required to give really accurate geometries. PCILO results do appear in most cases to be superior to those obtained by -0. For example, CNDO frequently predicts that the CHO or CO2H group is perpendicular to an aromatic ring rather than planar and it gives an incorrect geometry for butadiene. PCILO gives results more in accordance with experiment. Since PCILO uses the -0 integral approximations it corresponds to some configuration interaction expansion of CNDO configurations. Surprisingly,a full CI l60 181

162 163 164

165 166 167 168 169 170 171 172 173 174 175 1V6 177 178 179 180 181 182

(a) A. Saran, D. Perahia, and B. Pullman, Theor. Chim. Actu, 1973, 30, 31 ; (b) A. Saran, H.Berthod, and B. Pullman, Biochim. Biophys. Actu, 1973, 331, 154. (a) H. Berthod and B. Pullman, Biochim. Biophys. Actu, 1971, 232, 595; (b) H.Berthod and

B. Pullman, in ‘The Purines: Theory and Experiment’, Proceedings of the 4th Jerusalem Symposium, ed. E. D. Bergman and B. Pullman, Academic Press, New York, 1972. J. Caillet and B. Pullman, Theor. Chim. Acta, 1970, 17, 377. B. Maigret, B. Pullman, and M. Dreyfus, J. Theor. Biol., 1970, 26, 321. B. Pullman, J. L. CoGbeils, Ph. Courriere, and D. Perahia, Theor. Chim. Acta, 1971, 22, 11. D.Perahia, B. Pullman, and P. Claverie, Internut. J. Quantum Chem., Symp., 1972, 6, 337. B. Maigret and B. Pullman, Theor. Chim. Acta, 1974, 35, 113. P. R. Andrews, Biopolymers, 1971, 10,2253. (a) B. Pullman, Ph. Courritre, and J. L. Coubeils, Mol. Pharmucol., 1971, 7, 391; (b) B. Pullman and Ph. CourriBre, ibid., 1972, 9, 1972; Theor. Chim. Acra, 1973, 31, 19. J. L. Coubeils, B. Pullman, and Ph. Courri&re,Biochem. Biophys. Res. Comm., 1971,44, 1131. C. Coulombeau and A. Rassat, Tetrahedron, 1972, 28, 4559. C. Coulombeau and A. Rassat, Tetrahedron, 1972, 28, 751. J. Langlet, B. Pullman, and H. Berthod, J. Chim. phys., 1970, 67,480. D. Perahia and B. Pullman, Biochem. Biophys. Res. Comm., 1971, 43,65. B. Maigret and B. Pullman, Biochem. Biophys. Res. Comm., 1973, 50, 908. C. Decoret and B. Tinland, J. Mol. Structure, 1972, 12,485. M.Giacomini, B. Pullman, and B. Maigret, Theor. Chim. Acta. 1970, 19, 347. R. Cetina, M. Rubio, and 0 . A. Novaro, Theor. Chim. Acta, 1973, 32, 81. J. L. Coubeils and B. Pullman, Theor. Chim. Actu, 1972, 24, 35. Ph. Courriere and B. Puilman, Compt. rend., 1971, 273, D, 2674. Ph. Courri&re,J. L. Coubeils, and B. Pullman, Compt. rend., 1971, 272, D, 1697, 1813. J. L. Coubeils, Ph. Courritre, and B. Pullman, Compt. rend., 1971, 273, D, 1164. F. Letterrier, J. Capette, J. Langlet, C. Giessner-Prettre, and H. Berthod, J. Mol. Structure, 1971, 10, 75.

J. Langlet and J. P. Malrieu, J. Amer. Chem. SOC.,1972, 94,7254. lS4 R. Arnaud, D. Faramord-Baud, and M. Gelus, Theor. Chim. Actu, 1973, 31, 335. 1 8 5 F. A. Momany, R. F. McGuire, J. F. Yan, and H. A. Scheraga, J. Phys. Chem., 1971,75,2286. 1 8 6 J. Grignon and S. Fliszar, Cunad. J. Chem., 1974, 52, 2760. 187 J. Langlet and H. van der Meer, Theor. Chim. Acfa, 1971, 21, 410. 183

200

Theoretical Chemistry

over CNDO orbitals can give an incorrect result when PCILO successfully agrees with experiment.188The reasons for this are not clear and demand further study. In other cases, however, PCILO fails as dramatically as CNDO/2. Evleth and Feler l 8 9 have studied the endothermicities for breakdown of cyclobutane and oxetan to ethylene and 1,Zdioxetan respectively. The experimental values are 75.2 and 230 kJ mol-1, while CND0/2 gives 1504 and 1325 kJ mol-1 and PCILO 1388 and 1363 kJ mol-l. Like many other semi-empirical methods the PCILO method requires much theoretical analysis, apart from experimental experience, before it can be used with confidence. Other CI Methods.-Several w o r k e r ~ ~ ghave * - ~ used ~ ~ semi-empirical methods such as CNDO and INDO with limited configuration interaction. Such a technique is of course essential and widely used for excited states, but there are several applications to ground states. Such a technique is clearly valuable in studying potential energy surfaces involving bond-breaking processes where the simple MO approach is known to be inadequate. In general, however, one must have reservations about the use of CI for ground states with a parametrized model. Indeed Juglg3argues that since CNDO binding energies are too large it is meaningless to add a CI formalism directly. As discussed earlier in the section on PCILO, CNDO and CI do not always give satisfactory results. The need to have a semi-empiricalmodel which closely approximates ab initio techniques beyond the Hartree-Fock level remains an important problem. The use of large CI expansions based on an MO basis set of occupied and unoccupied orbitals involves the use of a large number of configurations. An alternative is the multi-configuration SCF technique where both the expansion coefficients of the configurations (Slater determinants) and the orbitals are optimized simultaneously. A much smaller number of configurations is required. In an ab initio framework this technique has been restricted to small molecules, but there have been two attempts to use the method with semi-empirical parameters.lg3.lg4No applications to large molecules have been reported, but this approach seems most promising. Juglg3intends to develop the technique using localized orbitals, thus allowing the ready selection of configurations to describe bond-breaking processes accurately. One interesting feature is that he develops an extended HartreeFock operator which has some resemblance to the usual Hartree-Fock operator. With CNDQ parameters the matrix elements of this operator can be cast into a form closely resembling the usual CNDO-SCF terms. The repulsion integrals yit and ytj are replaced by & and &, each depending on the original y’s and a variety of density-matrix-liketerms. A new term appears in the off-diagonalFtj element. This suggests that it may be feasible to parametrize semi-empirical schemes to include correlation. This idea has been 188 189

190

191

192 193 1 94

D. Perahia and A. Pullman, Chem. Phys. Letters, 1973, 19, 73. E. M. Evleth and G . Feler, Chem. Phys. Letters, 1973, 22, 499. (a)W. Jakubetz, H. Lischka, P. Rosmus, and P. Schuster, Chern. Phys. Letters, 1971, 11, 38; (6) D. R. Salahub, Theor. Chirn. Acta, 1972, 22, 325, 330; (c) C. Brabart and D. R. Salahub, ibid., 1972, 23, 285. (a) P. Cremaschi, A. Gamba, and M. Simonetta, Theor. Chim. Acta, 1973, 31, 1 5 5 ; (b) Z.-I. Yoshida and T. Kobayashi, J. Chem. Phys., 1973,58, 334; 59, 3444. (a) G . M. Maggiora and L. J. Weimann, Chem. Phys. Letters, 1973, 22, 297; (b) B. Tinland, R. Guglielmetti, and 0. Chalvert, Tetrahedron, 1973, 29, 665. K. Jug, Theor. Chim. A d a , 1973, 30, 231. C. W. Eaker and J. Hinze, J. Amer. Chem. Soc., 1974, 96, 4084.

Electronic Calculations on Large Molecules

201

widely believed, but Jug’s work raises a major difliculty. The ylj term in the diagonal Ff+term is not identical to the yij term in the off-diagonalFij term. This conclusion is similar to one reached by Brown and Roby195 in regard to the total energy expression. CI and MCSCF techniques with semi-empirical parameters appear to be a most promising field for much future development. For the conclusions to be clear and unambiguous it appears necessary to have a parametrization which fits ab initio results well at the singledeterminant level. In this way correlation corrections will be added in the correct place - by wavefunctions of superior quality. The alternative, such as MIND0 and CI,appears confused since it is claimed that even the singledeterminant energies include correlation corrections. 5 Concluding Remarks

The number of methods available for calculations on large molecules is now so large that the task of choosing the ‘best’ method for a particular study is full of difficulty. Inevitably, many workers stay with methods that are familiar or use computer programs that are most readily available. The problem is not improved by the fact that many advocates of particular methods base their advocacy largely on opinion. Furthermore, the choice of method may be strongly influenced by the individual’s skill in using one method (by parametrization, choice of basis functions, interpretations of results, etc.) rather than another. The Reporter cannot pretend to be free of these tendencies. The divergence of opinion is most strong in the decision between an ab initio MO approach or an approximate method which gives similar results and methods which are parametrized with a view to obtaining results superior in accuracy to Hartree-Fock MO results. The Reporter strongly holds the view that the most reliable progress will be made by the development of fully ab initio techniques. Where such methods fail to mirror the real world, theory rather than just experience will assist, firstly in uncovering the reasons for failure and secondly in developing new improved methods. This viewpoint leads to two important conclusions. Firstly, patience must be exercised in not attempting calculations in areas where the theory is known to be inadequate. Secondly, approximate methods should aim at mimicking or simulating ab initio methods, the objective here being merely a tradeoff between a cheaper calculation and a somewhat less reliable one. With this viewpoint the problem of choice of method becomes one of decisions of cost and accuracy. There are related questions of applicability since some methods are only feasible for certain classes of molecule. The cheapest general methods of wide applicability are semi-empirical methods such as CNDO and INDO. It is clear, however, that there is still much room for improvement of such methods so that they can give more precise agreement with ab initio results. In particular the ordering of orbitals, their orbital energies, and the total energy are often inadequate. Such methods are much less reliable than even the simplest ab initio technique for conformational problems. Methods, such as SAMO, which utilize transferability of matrix elements can be cheap if the problem of handling the large number of matrix elements is tractable for a particular case. They give reliable mimicking of ab initio orbital energies and charge distributions. How195

R. D. Brown and K. R. Roby, Theor. Chim. Ada, 1970, 16,291.

202

Theoretical Chemistry

ever, their ability to treat conformational problems does not look hopeful. Methods such as PRDDO are more elaborate and hence more expensive. PRDDO appears to mimic ab initio results well, although less accurately than SAM0 for orbital energies, and it has the clear advantage of fairly general applicability. This method has had only a few applications, being restricted almost entirely to Lipscomb’s group. Its wider use would be most welcome. The PCILO method suffersfrom the disadvantage that it does not aim to mimic a particular ab initio technique, although in principle it is closely related to an ab initio configuration interaction technique using localized orbitals. It is, however, extremely rapid and can be applied to very large organic molecules. Used with care it can assist in the elucidation of conformational problems. Its limitations, however, will only be clearly seen when the technique is more clearly understood in a theoretical sense to support the growing volume of practical experience with the method. In the Reporter’s opinion it would be valuable to study the method with a range of parametrizations, other than the original CND0/2. Would a version of PCILO at the N D D O level be feasible? In spite of the criticism of theorists it is clear that MINDO and related methods will continue to be used. Indeed if used with sufficient care and reservations, such methods can play a most valuable role in augmenting experimental evidence. Methods based on the X , exchange approximation have become very popular in recent years. Their status for inorganic molecules is perhaps similar to the status of PCILO and MINDO for organic systems. They can give excellent results but their limitations are not completely clear. It seems fair to say that their utility is not as high as some of their enthusiastic advocates maintain. The X , approximation, and in particular the muffin-tin approximation, appears to be best suited to molecules or clusters with high symmetry. They are in general more useful for inorganic complexes than LCAO semi-empirical methods such as CNDO and INDO. The Reporter, however, believes that their development should not deter the development of accurate ab initio studies, preferably going beyond the Hartree-Fock level for such systems, in spite of the high cost at present of ab initio techniques. Although not covered in this Report, very simple methods such as the Extended Huckel method and the Wolfberg-Helmholtz method remain popular, and for good reason. For very large molecules they can give a rough idea of the charge distribution and form of the molecular orbitals. They can be useful in suggesting systems for more detailed study. The Table was prepared in collaboration with Mr M. P.S. Collins, whose assistance is gratefully acknowledged,

Author Index Aarons, L. J., 153, 177, 182 Abdulnur, S. F.,77, 79 Abegg, P. W., 102 Abrahamson, A. A., 73 Abrahamson, E. W., 176, 190

Absar, I., 130, 131, 158, 184 Ackermann, F., 122 Adams, D. B., 177 Adams, W. H., 87 Ady, E., 151 Afzal, M., 139 Agrawal, V. P., 78, 150 Ahlrichs, R., 48, 55, 95, 99, 100,128,150,151,177,183

Akermark, B., 176 Alagona, G., 36, 80 Albright, R. G., 62 Alexander, M. A., 57 Allen, J. D., 105 Allen, L. C., 15,28, 117, 118, 136, 150, 151, 155, 156, 179, 188 Allevena, M., 148 Almemark, M., 176 Almlof, J., 16, 129, 152, 176, 177 Alvarez-Rizzatti, M., 75 Amdur, I., 57 Amos, A. T., 67 Amos, T.,42 Andersen, A., 111, 112, 124 Anderson, G. R., 129 Anderson, 0.K., 195 Anderson, P. W., 195 And& J. M., 177 AndrC, M. Cl., 177 Andrews P. R 199

Antoci, g., Archibald, R. M., 153 Arents, J., 183 Armour, E. A. G., 88 Armstrong, D. R.,125, 153, 157

b a u d , R., 199 Arrighini, G. P 76 Asbrink, L.,17; Ashe, A. J., 177 Ashmore, P. G 24 Aslangul, C., 93 Astier, M.,148 Aung, S., 13, 138 Averill, F. W., 193, 195 Aziz, P. A., 81 Backvall, J. E., 176 Bader, R. F. W., 11, 17, 27, 28,29,32,34,98, 113

Baer, S., 69

105, 108,

Baerends, E. J., 113, 195 Bagus, P. S., 99, 103, 108,

114,118,119,120,128,177 Baird, N. C., 157 Baker, D. A., 49 Balint-Kurti, G. G., 63, 115 Ballhausen, C. J., 188 Baloga, J. D., 82 Bandrauk, A. D., 122 Banyard, K. E., 91, 114 Barber. M.. 154. 177. 178. 179,‘180 ‘ Barber, M. S., 176 Barker, J. A., 81 Barker, R. S., 49 Barna. A. K.. 91 Barr, R. F., 157 Barsuhn, J., 110, 131 Bartlett, N., 129 Bartlett, R. J., 90, 91 Basch, H., 19, 64, 129, 155, 177, 178, 179, 180 Baskin, C. P., 33, 80, 130, 156 Bass, A. M., 33 Batana, A., 125 Batish, C.,177 Batra, I. P., 177 Bauer, S. H., 57 Bauschlicher, C. W., jun., 33, 55, 108 Bavbutt. P.. 179 BGhers; M:, 156 Beck, H., 71 Beckel, C. L., 2, 85, 88 Beebe. N. F.. 155 Bellum, J. C.’, 90 BeltrBn-L6pez, V., 28, 77 Bendazzoli, G. L., 139, 143, 179, 187 Bender, C. F., 24, 28, 33, 40, 41, 55, 57, 58, 60, 63, 80, 94, 99, 101, 130, 131, 134, 136, 137, 139, 141, 146, 151, 152, 154 156 Benedict, W. S.: 13 Ben-Shaul, A., 69 Benson, M. J., 21, 150 Benston, M. L., 96 Berard, E. V., 57 Bergman, E., 69 Berkowitz, J., 61, 123 Bernardi, F., 88 Bernstein, R. B., 90 Berrondo, M., 68 Berthier, G., 17, 148 Berthod, H., 199 Bertoncini, P., 71,72,91, 112 Besnainou, S., 148 Be&, G., 58 Beveridge, D. L.,84,180,184

203

Billingsley, F. P.,jun., 122, 145,185

Bkgel, W. A., 55 Biondi F.,76 B/rely,’J. H., 63 Birner. S.. 188 Birnstock; F., 188 Bishop, D. M., 85 Bjorna, N., 111 Blackman, G. L., 130 Blais, N. C., 5 Blint, R. J., 100, 101, 142 Bloor, J. E., 185 Blustin, P. H., 85, 102 Bock, H., 149 Body, R. G., 15, 186 Boer. F. P., 190 Boggs, J. E;, 156 Bohme, D. K., 17,26 Bologin, A. B., 91 Bonaccorsi. R.. 177 Bondybey, V., Borkman, R. F., 134 Born, M., 1 Borne, T. B., 5 Borrett, D. S., 98 Botcher, C., 91 Bowers, M. J. T., 79 Bowman, J. D., 67 Bowman, J. M., 54 Bovd. D. B.. 190 BOGS,’S. F.,’ 11, 49, 87, 88, 136

Brabart, C., 200 Brandas, E. J., 90, 91 Brailsford. D. F.., 176., 183 Braun, W:, 33 Breeze, A., 125, 128, 149 Brickmann, J., 151 Brillouin, L., 10 Brotchie. D. A.. 138. 142. I

_

146, 147

Broussard, J. T., 76 Brown. D. A.. 84. 159 Brown; P.J., h,127 Brown, R. D., 47, 138, 148, 187, 188,201

Brown, R. E., 90 Browne. J. C.. 84. 99. 101. I

,

_

~

183

Bruce, R. E., 106 Bruckner, K. A., 105 Brumer. P.. 73 Bruna, P. J., 157 Brundle, C. R., 155 Bruner, B. L., 50, 51 Buckingham, A. D., 79,92 Buenker, R. J., 35, 84, 132, 157, 176

Bugatur’yants, A. A., 188 Buhl, R. F., 131

,

Author Index

204 Bukta, J. F., 75 Bunker, D. L., 5, 28 Bunker. P. R.. 85 Bunton; C. A,’, 27 Bunyatan. B. Kh., 73 Burcat, A., 57 Burden, F. R., 148, 187, 188 Burgi, H. B., 28 Burnelle, L., 30,47, 143, 146, 153 Bums, G., 70, 78 Buss, V., 176 Byers Brown, W., 78 ‘

Cade, P. E., 11, 14, 93, 105, 107, 113, 182 Cadioli, B., 153, 179, 187 Caillet. J.. 199 Calais,‘J. ’L., 116 Capette, J., 199 Carlson, G. L., 176 Carney, G. D., 55 Carr, R. W., 33 Carrington, A., 105 Carrington, P. J., 92, 123 Carrington, T., 5 Carroll, T. X 126, 129 Cirsky, P., 126 Cartwright. D. C.. 113 Cashion, J.’ J., 58 Castex, M. C., 81 Catlow, G. W., 117 Caves. T.. 75 Cederbaum. L. S.. 111, 115, 141, 182 Certain, P. R., 79, 121 Cetina, R.,.199 Chalasinski. G.. 69 Chalvert, O:, 200 Chambers, W. J., 84 Chambers, W. S., 159 Chan, A. C. H., 86,99 Chan, S. I., 13 Chang, E. S., 117 Chang, H. M., 188 Chang, S.-Y., 68, 96 Chanmugan, J., 24 Chappell, G. A., 26 Chen, T.-T., 112 Cheney, B. V., 176 Child. M. S.. 5 Chiprkan, D: M., 67 Chong, D. P., 96, 143 Christoffersen. R. E.. 55. 163. 176. 177. 180 Chu A. H:-M ’ 8 8 Chu: S. Y.,99;’110, 136, 155 Chupka, W. A., 61 Cirule. Z.. 91 Ciiek,’J., ‘150 Clack, D. W., 188 Clark, D. A., 129 Clark, D. T., 157, 177 Claverie, P., 68, 92, 196, 199 Claydon, C. R., 32 Clementi, E., 9, 12, 15, 20, 35, 73, 80, 96, 108, 125, 151, 177, 183, 184, 193 Cobb, J. C., 148, 149 Cobley, U. T., 177 Cohen, M., 85, 162 Cohen, S. S., 82 Colbourne, E. A., 89 Collins, G. A. D., 128, 149 ‘

‘

I

_

Collins, M., 190 Colin, R., 99 Connollv. J. W.D.._ 116,_ 139._ 191, i92, 193 Connor, J. A., 154, 177, 178, 179, 180 Conroy, H., 50, 51, 57 Constanciel. R.. 98 Conway, A:, 73; 115 Cook, D. B., 83, 88,162, 187 Cook, T. J., 125 Copeland, D. A., 188 Copsey, D. N., 115 Coubeils, J. L., 199 Coulombeau, C., 199 Coulson, C. A., 10, 148 Courribre, Ph., 199 Coutibre. M.-M.. 178. 182 Cradock; S., 176. Cremaschi, P., 200 Cruickshank, D. W. J., 125, 128. 149 Csizmadia, I. G., 34, 35, 54, 55, 100, 141, 151, 155 Curtiss, L. A., 135 Dacre, P. D., 146, 178, 187 Dalgarno, A., 69 Damany, N., 81 Dandey, J. P., 92 Danese, J. B., 193 Darling, B. T., 2 Das, G., 41, 71, 86, 104, 107, 109, 112, 114, 117 Das, T. P., 89, 105 DasGupta, A., 189 Datta, R. K., 157 Daudel, R., 98 Daudey, J. P., 68, 198 David, C. W., 143 David, D. J., 17 Davidson, E. R., 29, 40, 41, 84, 89, 94, 99, 112, 120, 130, 137, 138, 157 Davidson, R. B., 188 Davidson. W. D.. 72 Davies, A. M., 88 Davies, P. B., 82 Deal, W. J., 75 Decoret. C.. 199 Dedieu,’A.,*17, 26 Degand, P., 190 De Greef, D., 99 de Haas, N., 54 Dehmer, J. L., 123, 153 Dejardin, P., 151, 152 Dekock, R. L., 149 DelacBte. G.. 176 Del Bene, J. ‘E., 79, 80, 135, 137, 156, 179, 188 Demuynck, J., 155, 178, 182 Denes. A. S.. 34 Dennis, A., 198 Dennison, D. M., 2 Deplus, A., 191 Derr, V. E., 13 Derrick, L. M. R., 178 Deutsch, P. W., 183 Devaquet, A., 144 Devaux, Ph., 176 Dewar, M. J. S., 28, 136, 184, 188, 189 Diamond, J. B., 194

Diercksen, G., 59, 141, 156 Diestler, D. J., 53 Diner, S., 196, 198 Dismuhe, K. I., 131 Ditchfield, R., 19 Dixon, D. A., 187 Dixon, M., 91, 114, 125, 139 Dixon, R. N., 34 Docken. K. K.. 93 Dodds, J. L., 187 Dodonov, A. F., 62 Doggett, G., 92, 125, 184 Domcke. W.. 141 Doolittle. J..’25 Doran, M.B., 76 Dreyfus, M., 36, 177, 199 Driessler, F., 48, 151 Duben, A. J., 155 Duff, J. W., 53 Dufty, A. N., 81 Duke, A. J., 17, 27, 151, 183 Duke. B. J.., 159., 176.- 184. 190, 191 Dunning, T. H., 12, 13, 15, 84, 111, 138, 140, 145 Durmaz, S., 33 Dutta. C. M.. 89 Dutta; N. C.,’ 89 Dyatkina, M. E., 188 Dycmons, V., 27, 152, 183 Dyke, T. R., 79, 82 Dyson, M. C., 74 DZidie, I., 73 Eakers, C. W., 138, 200 Easley, W. C., 122 Eastes, W., 57 Ebbing, D. D., 132 Eckarc‘ C., 54 Edmiston, C., 25, 40, 51 Ehrenson. S.. 34 Eilers. J. E.. ’176. 190. 191 Elbert, S. T:,138, 157 Elder, M., 146, 178, 187 Ell$ D.E., 113,178,194,195 Ellis. D. J.. 114 Ellis; R. L.’, 188 Ellison, F. O., 127 Empidocles, P. B., 63 Engelbrecht, A., 80 England W 84 162 Epstein,’I. R., l’k, 182 Epstein, S. T., 75, 88 Erdahl, R. M., 117 Ermler, W. C., 13, 138, 160 Eu, B. C., 53 Evans, S., 178 Evleth, E. M., 200 Ewig, C. S., 15 Ewing, G. E., 82 Eyring, H., 5, 49 Fano. U.. 153 Faramord-Baud, D., 199 Farnell, L., 163 Fateley, W. G., 176 Feler, G., 200 Fernandez-Alonso, J. I., 188 Fettis, G. C., 61 Feynman, R. P., 10 Field. R. W.. 121 Findlay, R. H., 176, 177 Fink, W. H., 15,32, 146, 155, 157

Author Index Finlan, J. M.,106

Fischer, C. R., 131 Fischer, J., 178 Fisher. C. J.. 82 Fitts, D. R.,*190 Fitzpatrick, N. J., 84, 159 Fleischhauer, J., 156 Flicker. M.. 73 Fliszar; S., 199 Flouquet, F., 32, 140 Fock, W., 81 Fortune, P. J., 79 Foster, J. M., 136 Franceschetti, D. R., 136 Freed, K. F., 119, 185 Freeman, A. J., 178 Freeman, D. E., 81 Fricker, H. S., 195 Frost, A. A., 85, 139 Fukui, K., 140 Funke, I., 59 Gailar, N., 13 Gallagher, J. J., 13 Gallup, G. A., 95, 128 Gamba, A., 200 Gangi, R. A., 29,32,34, 146 Gardner, M. A,, 71, 115 Garrison, B. J., 92 Gaskell, A. J., 176, 177 Gelius. U.. 179. 180. 183 Geller; M.; 128; 134' GClus, M., 100,109, 150,199 Genson, D. W., 176,177,180 George, T. F., 91 Gerloff, M., 151 Gerratt J., -14 Gershgorn, Z., 37 Giacomini, M., 199 Gianturco, F. A., 179 Giardina, M. P., 77 Giessner-Prettre, C., 199 Gilbert, M., 196, 198 G i f p p , T. L., 20, 73, 112, llJ

Gilman, R. R., 151 Ginsberg, A. P., 177 Gladney, H. M.,178 Goddard, W. A., tert., 57, 84, 89, 92, 94, 95, 97, 98, 100, 101, 103, 105, 1 09, 110 113 121, 131, 136, 138:*140,'!44, 145 Goeebiewslu, A., 188 Goethals, P., 99 Gole. J. L.. 141. 146. 148 Gombas, P., 96' . Gonzales-Tovany, L., 77 Goodisman, J., 14, 115 Goodman. L.. 155 Gordon, M.D., 23, 57 Gordon M. S., 142 Gordon: R. G., 58, 71, 74, 115 Gough, D. W., 81 Goutier, D., 153 Gouyet J. F., 68 GrahaG. W. R. M..131 Gray, J.; 91 Green, S., 74, 84, 103, 104, 105, 120, 121, 126 Greenawalt. E. M..101 Grein, F., 122 . Grice, R., 5, 63

205 Griffin, A. C.,189 Griffing, V., 49 Grignon, J., 199 Grimbert, D., 144 Guberman, S. L., 92, 105 Guest, M. F., 34, 125, 146, 153,155,177,178,179,182 Gugbelmetti, R., 200 Guidotti, G., 76 179 Gunning, H. E., 34 Gupta, S. K., 85 Gur'ev, K. I., 98 Guton, P. M., 61 Ha T.-K., 102, 143 Habdon, R. C., 136, 189 Hagstrom, S., 4, 128 Hahn, D., 177 Hains, W.J., 35 Halgren, T. A., 186, 187 Hall, G. G., 84,98, 139, 159, 180 Hall, J.A., 100,114,120,123 Hall, J. H., 178, 182 Hall, M. B., 125, 178, 179, 182 Hall, W. R., 137 Hameka, H. F., 137 Hammond, G. S., 27 Handy, N. C., 87,88, 134 Hankm, D., 129,179 Hansen, B. D., 2 Hardisson, A., 42 Harget, A. J., 84, 184 Hariharan, P. C., 137, 151 Harriman, J. E., 42 Harris, D: O., 15 Harris, F. E., 113, 114 Harris, H. H., 28 Harris, R. R., 121 Harris, S. J., 82 Harrison, J. F., 136,137, 138 Harrison, M.C., 32 Harrison, S. W., 70, 117, 128, 134 Hart, B. T., 148, 187 Hartree, D. R., 41 Hartree, W., 41 Hase. H. L.. 188 Haugen, J. A., 59 Hay P. J 84, 94, 100, 103, li6, ld, 145, 146, 196 Hayes, D. M., 157 Hayes, E. F., 23, 40, 48,18, 90, 127, 136, 141, 143, 145, 146. 148 H mi, 'A., 97 H eaton, M. H., 144 Hecht, K. T., 15 Hehre, W. J., 18, 19, 36, 135, 176,177, 185 H eil, T. G., 116, 121 Heilbronner, E., 177 Helfrich, K., 86 Hellman, H., 10 Hemsworth R. S., 26 Henderson,'G., 82, 110 H enriksson-Enflo, A., 176 He m , R. R., 63 Herman, F., 192 Herrjng, C., 73 H ernn F. G., 143 Herschtach, D. R., 63, 75 Herzberg, G., 3, 133

Higginson, B. R., 178, 182 Hillier I. H 125 144 146, 153,' 154,"155, ' 177,' 178, 179, 180, 182, 184 Hinchliffe A., 148, 149, 176 Hinds, A h.,41, 140 Hinkley, R., 83,100,104,159 Hinze. J.. 93. 102. 146, 163, 200'" . . Hirschfelder, J. O., 49, 67 Hobey W. D., 4 Hodason. B. A.. 5 Hoffkanh, R., 35 Hoheisel C.,22 Hohlnedher, G., 115, 141, 182 Ho Huck, S., 189 Holbrook, N. K., 141 Hollis, P. C., 187 Hollister, C., 129, 178, 179 Holloway, J. M.,61 Hopkinson, A. C., 141, 155 Hornback, C. J., 178 Hornung, V., 177 H$$ey, J. A., 32, 35, 140, 1J J

Hosteny, R. P., 41, 128, 140 Houlden, S. A., 100 Howell, J. M., 158 Hoyland, J. R., 176 Hsu, H., 157 Hsu, K., 35 Huestis, D. L., 89 Hunt, R. H., 15 Hunt W. J., 63, 84, 94, 100, lo;, 113,136,138,140,,154 Hunter, G., 2, 86 Huo, W. M., 14,93, 107,,119 Hurley, A. C., 10, 84 Hush, N. S., 188 Huzinaga, S., 26,189 Hyde, R. G., 149 Hylleraas, E. A., 37 Hylton, J., 183 Ibers, J. A., 15 I'Haya, Y. J., 114 Ikenberry, D., 105 Isacsson, P. U., 176 Ishimaru, S., 85 Iwatu, S., 80 Jackson, J. L., 49 Jaffk, H. H., 188 Jahn, H. A., 3 Jakubetz, W., 80, 200 Janoschek, R., 129, 176 Jansen, H. B., 178 Jean, Y., 35 Jefferts, K. B., 125 Jesaitis. R. G.. 188 Jeziorski, B., 69 Jiang, G. J., 129 Johansen, H., 155, 180 Johansson, A., 80, 156, 179 Johnson, B. R., 53 Johnson, K. H., 84,112,191, 193, 194 Johnson, R. E., 75 Johnston, H. S., 53 Jonkman, H. T., 176, 177 Jordan, F., 196,198 Jordan, P. C.H., 134 Jorgensen, W. L.,188

206 Jug, K.,89, 90, 200 Julienne, P. S., 107, 131 Jungen, Ch., 122 Jungen, M., 86, 95, 99, 134, 135 Kahn, L. R., 97 Kaijser, P., 111 Kaldor. U.. 15. 92 Kammer, W. E., 157 Kapral, R., 70, 78 Kari, R. E., 54, 86, 151 Karo. A. M.. 71. 104. 115 Karplus, M.,*2, 5 , 10,’24, 52, 53, 63, 73, 115, 138 Kasseckert, E., 114 Kato, H., 140 Kato, T., 11 Kaufman. J. J.. 70. 117. 128. 134, I&, 193 ‘ Kebarle, P., 73 Kelly, H. P., 103, 108, 140 Kelly, M. M., 180 Kemmey, P. J., 131 Kemp, J. D., 14 Kern, C. W., 2, 10, 13, 138, 160 Kestner, N. R., 69, 76 Kilcast, D., 177 Kim, H., 143, 154 Kim, Y.S., 74 Kimball, G. E., 49 Kindle. C.. 129 King, 6. W., 188 King, H. F., 21 Kinsey, J. L., 6, 82 Kirschner. S.. 189 Kirtman, B., 68, 95, 96 Ristenmacher, H., 35,73,80, 184, 193 Kitigawa, T., 63 Kleier, D. A., 187 Klemperer, W., 82, 119, 142, 191

K&& D. S., 143 Klint, D., 177 KloDman. G.. 184 Knight, L. B.; jun., 122 Knox, J. H., 61 Kobayashi, T., 200 Kochanski, E., 71, 73, 115, 151, 152, 154, 176 Koehler, H. J., 188 Koeppl, G. W., 54 Koetzle, T. F., 178, 182 Kohler. H. J.. 188 Kohn, ‘M. C.,‘ 28 Kohn, W., 191 Kohnishi. H.. 113. 151 Kollman,’P. A., 28, 80, 156, 179, 188 Koios, W., 3, 50, 72, 84, 86, 88 Komornicki, A., 14 Konowalow, D. D., 112,116, 193 Koopmans, T., 181 Korringa, J., 191 Kortzeborn, R. N., 129 Kosloff, R., 90 Koster, J. L., 90 Kottis, P., 98 Kouba, J. E., 75, 121, 125 Koutecky, V. B., 149

Author Index Kovner, M. A., 98 Kowalewski, J., 89, 176 Kraemer, W. P., 141, 156 Kramer, H. L., 75 Kramling, R. W., 176 Krauss, M., 20, 40, 51, 56, 104, 106, 107, 114, 116, 131, 140 Krohn, B. J., 138 Krumhansl, J. A., 71 Kubach, C., 78, 108 Kuchitsu, K., 13 Kuebler, N. A., 155 Kuehnlenz, G., 188 Kuntz, P. J., 5 , 56, 127 Kunz. A. B., 183 Kuppermann, A., 54 Kutzelnigg, W., 22, 27, 40, 48,55,71,84,95, 100, 109, 127. 128. 150. 151 Kuyatt, C.*E., 131 Kvasnicka, V., 198 Labarre J.-F., 153 Labib-dkander, I., 55 Lacey, A. J., 78 Ladner R. C., 94 Laidler: K. J., 48 Lamanna, U., 179 Land, R. W., 154 Langhoff, S. R., 112, 137, 138, 157 Langlet, J., 198, 199 Laplante, J. P., 122 Larsson, S., 192 Lathan, W. A., 18, 135, 151, 177, 185 Lavronskaya, G. K., 62 Laws, E. A., 108, 151, 178, 182 Leacock, R. A., 15 Lee, S. T., 138 Lee. T.. 105 Lefebvre-Brion,H., 105,119, 120, 121,122 Lehn, J. M., 28, 152, 176 Leibovici. C.. 141 Lempka, ‘H. J., 149 Lentz, B. R., 80 Leroi, G. E., 15 LeRoy, D. J., 52, 54 Leroy, G., 176, 190, 191 Lesk, A. M., 69, 92, 117 Lester, W. A., 23, 54 Letcher, J. H 184 Letterrier, F.,”IW Levine, R. D., 5, 53, 90 Levy, B., 198 Levy, D. H., 125, 146 Lewis, D., 57 Li, W. K., 189 Liberles, A., 65, 134 Lichtenstein, M. L. 13 Liebman, J. F., 117, 118 Liedtke, R. C., 137 Lie, G. C., 96, 102, 106, 108, 177 Lifshitz, A., 57 Light, J. C., 5 Lilley, D. M. J., 177 Limenko, N. M., 188 Linder. B.. 77 Linder; P.; 111 Lindgren, J., 177

Linnett, J. W., 85, 102, 139 Lipscomb, W. N., 14, 93, 108, 123, 151, 178, 182, 186. 187. 190 Lischka, H:, 28,71, 105, 143, 152, 156, 200 Liskow, D. H., 28, 1 18, 119, 141

Lister, D. G., 143 Liu, B., 51, 71, 91, 102, 103, 116,118,119,127,128,142 Liu, H. P. D., 104, 105 Liu, T. K., 132 Llaguno, C., 57, 85 Lloyd, D., 149, 177, 178, 182 Lloyd, J., 74 Lowdin, P.-O., 9,40,42, 185, 186 Loew, G., 130 Lofthus, A., 114 Longuet-Higgins, H. C., 134 Lory, E. R., 131 Losonczy, M., 70 Lowe, J. P., 89 Lykos, P. G., 89

M[cCain, D. C., 144 Mkclure, D. S., 15 M[cCullough, E. A., jun., 93 M [cDowell, M. C. R., 117 M [cEachran, R. P., 85 M[cEwen, K. L., 130 M[cGuire, R. F., 199 M!achZcek. M., 146 M‘cIver, J.. W.,. 14 M ackay, G. I., 17 M cKendrick, A., 125 M cKoy, V., 104, 111, 140, 1C7 121

Mackrodt, W. C., 77 McLachlan, A. D., 4 Maclagan, R. G. A. R., 128 McLaughlin, D. R., 21, 23, 91, 136, 150 McLean, A. D., 71, 89, 91, 103,118,119,121,122,126 McMahan. A. K.. 71 McRury, T. B., 77 McWeeny, R., 8, 187 McWilliams, D., 143 Maggiora, G. M., 176, 180, 188, 200 Magnasco, V., 77 Maigret, B., 36, 199 Maitland, G. C., 81 Maldonado. P.. 20. 116 Malinauskas, A. P:,57 Malli, G., 57 Malrieu, J. P., 68, 92, 196, 198. 199 Marchese, F. T., 80 Marsmann, H., 131 Martin, P. H., 79, 104 Marvnick. D. S.. 178. 182 Masmanidis, C. -A., 188 Mason, E. A., 75 Massa, L. J., 70, 117, 128, 134 Masson A., 198 Matcha: R. L., 20, 78, 116, 143 Matsen, F. A., 84 Matsukawa, F., 114 Matthews, G. P., 81

Author Index Meath, W.J., 75,77

Mehl, J., 177 Mehler, E. L., 101 Melius. C. F.. 98 Melrose, M. P., 119, 125 Mely, B., 177 Merlet, P., 35 Messer, R. R., 27, 108 MTxy, W.,14, 89, 140, 151,

207 Nieuwpoort, W.C.,154,176, 177,178 Noble, P. N., 129 Noor Mohammad, S., 126 Norbeck. J. M.. 95. 128 Norman; J. G.,*191 Norstrom, R., 155 Novaro, 0. A., 28, 77, 199 Novick, S. E., 82

1JJ

Michejda, C. J., 35 Michels, H. H., 114 Mielczanck. S. R.. 131 Mies. F. N.’. 56 ‘ Mihich, L., ’139 Miller, J. H., 140 Miller, W. B., 63 Miller. W. H.. 92 Milleur, M. BL, 78, 143 Millie, P., 148 Mills, I. M., 14 Minn, F. L., 24 Mishra, P. C., 188 Mitchell, D. N., 54 Mitchell, K. A. R., 106, 128 Mjoberg, P. J., 176 Moberg, C., 176 Moccia, R., 179 Moffat, J. B., 125, 133 Mok, M. H., 5 Moller, C., 10 Momany, F. A., 199 Morino, Y.,13 Morokuma K., 53, 80, 91, 113, 138,’151, 157 Morosov, I. I., 62 Mortola, A. P., 121, 180 Moser, C 35, 99, 125 Moshinsii M., 68 Moskowiti J. W., 32, 70, 121, 129,’178, 179, 180 Moulson, T., 89 Mowery, R. L., 68 Mrozek, J., 188 Muckerman, J. T., 61 Muenten, J. S., 79 Mukamel, S., 92 Mulder. J. J. C.. 26 Muller,-J., 177 Mullick, K., 91 Mulliken, R. S., 21, 90, 99, 101.103.110.111.116.185 Munsch, B., 152 ‘ Murphy, K., 25 Murrell, J. N.. 33, 34, 67,73, 84, 115, 156, 184 Musher, J. I., 67, 149, 153 MUSSO,G. F., 77 I

Nalewajski, R., 188 NAray-Szabb, G., 126 Nardelli, G. F., 139 Nee. T.-S.. 68 Nemeth, E. M., 5 Nesbet, R. K., 18, 20, 35, 42. 78. 116 Neufeld,. P. D., 81 Neumann, D., 106, 114, 131 Neumann, D. B., 121, 177 Newman. D. J.. 68 Newton, ’M. D.; 18, 34, 142, 177, 185, 190 Nicholson, B. J., 184

Ohm, Y.,121, 122, 124, 125, 131

O’Hare, P. A. G., 118, 122, 123, 124, 125 O’Leary, B., 176, 184, 190, 191

O’N& S. V., 6 24, 55, 60, 80, 104, 116, 119, 120, 130,

136, 146 O”ei1, T. G.. 125 Oppenheimer- R., 1 Orchard, A. I!.,178 Orltkowski, T., 86 Orloff. M. K.. 190 Ossa, E., 57 Ostlund, N. S., 185 Ozkan, I., 155

Pai. T. K. D.. 91 Painter, G. S’,194 Paldus, J., 150 Palke, W. E., 94, 96, 144 Palmer. M. H.. 176. 177 Palmieri, P., 139, 143, 179, 187 Pamuk, H. O., 100, 155 Parameswaran, T., 195 Pam, C. A., 5 Parr, R. G., 48,68, 106 Patch, R. W., 79 Patterson, P. L., 60 Pauncz, R., 162 Payzant, J. D., 17 Pearson, P. K., 24, 60, 63, 101, 105, 119, 130, 142 Pearson, W. B., 21 Peck, J. M., 85 Pedley, J. B., 33, 34 Peek, J. M., 2 Peel, J. B., 149 Peeters, D., 176, 190, 191 Pelletier, J., 34, 105, 113 Penzias, A. A., 125 Perahia D., 199, 200 Perkins: P. G., 153 Peslak, J., jun., 143 Peters, C. W., 15 Peterson, C., 139 Petke, J. D., 108, 152 Peyerimhoff, S. D., 35, 84, 132, 157 176 Pfeiffer, G: V., 139 Phariseau, P., 194 Philli~s.L. F.. 188 Pilling, .M., 33 Pincelli, U., 153, 179, 187, 198 Pipano, A., 144, 151 Pitzer, K. S., 14 Pimr, R. M., 13, 14,79,138 Piesset. M. S.. 10 Plyler,’E. K., ’13 Polanyi, J. C., 4, 5, 54, 55 Polanyi, M. ,5

P o k e r , P., 121 Popkie, H.,35, 73, 80, 99, 177 184 193 Pople: J. A., 17, 18, 19, 36, 42, 79, 84, 135, 137, 142, 151, 156, 159, 176, 177, 179, 184, 185, 187 Port, G. N. J., 29, 36, 177, 180 Port. M. J.. 36 Porter, R. F., 131 * Porter, R. N., 5 , 52, 54, 55 Poshusta, R. D., 55, 59, 65, 70.78. 134. 150 Present. ’R. D.. 81 Preston, H. J. T., 120 Preston, R. K., 4, 55 Preuss H., 36, 59, 176 Pritchird, H. O., 1,2, 86 Pugh, D., 74 Pulay P., 14, 150, 151, 157 Pullmk, A., 29, 36, 80, 177, 180, 198,200 Pullman, B., 177, 198, 199 Radna R. J., 180 Rado;, L., 17, 19, 159, 176, I81 Rae. A. I. M.. 74 Raffenetti, R.’C., 183 Raftery, J., 107 Rai, D. K., 188 Raimondiand. M.. 136 Ralowski, W.’M.,’ 176 Ramsden, C. A., 188 Ranck, J. P., 155 Ransil, B. J., 49 Rassat, A., 199 Rauk, A., 15, 151 Raynes, W. T., 88 Reid, R. V., jun., 88 Reira, A., 77 Renner. R.. 4 Rice S: A.; 97 Ricdards, W. G., 83, 100, 104, 107, 119, 123, 125, 159. 163 Ridley, B. A.. 54 Riehl, J. W., 82 Riley J. P., 8 Ritciie, C. D., 21, 26 Ritschl. F.. 73 Rittner E.-S., 74 Roach ’A. C., 5 54 55 Robb,’M. A., 54, 162 Robbe, J. M., 120 Robert, J.-B., 131 Roberts, C. S., 56 Robin, M. B., 155 Roby, K. R., 90, 126, 187, 188 201 Rochi, A. L., 122 Rode, B. M., 80 Rodgers, J. E., 105 Rosch, N., 193, 194 Roetti, C., 12 Rohmer, M.-M., 178 Rojas, O., 68, 92 ROOS,B., 71, 89, 130, 141, 152,154,176,179,180,183 Roothaan, C. C. J., 7, 88 Ros, P., 26, 113, 178, 195 Rose, J. B.,.lII Rosen, H., jun., 49

Author Index

208 Rosmus, P., 149, 157, 200 Rosner. S. D.. 5 Ross, J:, 5 ‘ Rostoker, N., 191 Rothenberg, S., 24, 80, 141, 144, 146, 147, 156, 179 Rothenstein. S. M.. 85 Rubinstein, ’M., 57’ Rubio, M., 199 Ruedenberg, K., 84,162, 185 Rundle. H. W.. 26 Russegger, P., 73 Russell, D., 119, 125 Ruttink, P. J. A,, 90 Ryan, J., 144, 177 Sabin, J. R., 131, 139, 143, 154, 155, 193 Sachs, L. M., 117, 128, 134 Safron. S. A.. 63 Sakai,-H., 140 Salahub, D. R., 200 Salem, L., 35 . Sales, K. D., 14, 182 Salmon. C.. 55 Salmon; L.; 84, 134, 162 Salotto, A. W., 30, 47, 143 Sannigrahi, A. B., 126 Santry, D. P., 187 Saran, A., 199 Sato, S., 5 Saunders, V. R., 144, 146, 153,154,155,178,179,184 Sawodny, W., 150 Saxon. R. P.. 76 Scanlan, I., 177 Schaefer, H. F., tert., 6, 11, 24, 28, 33, 55, 57, 58, 60, 63, 80,91,92,93, 101, 104, 105, 108, 113, 114, 116, 118, 119, 120, 121, 125, 128, 130, 134, 136, 137, 139 141 142 144, 146, 147: 148,’154, 362 Schamps, J., 118, 119, 120 Scharf, H. D., 156 Scheire, L., 194 Scheraga, H. A., 80, 199 SchiE, H. I., 26 Schleyer, P. von R., 176, 188 Schneider, B., 74 Schnuelle, G. W., 128 Schoenborn, M., 155 Schottler, J., 22 Schreiber, J. L., 4 Schug, J. C., 74 Schulman, R. G., 177 Schulte, K. W., 188 Schulz, W. R., 52, 54 Schuster, P., 36, 73 200 Schwartz, A. K., 83 Schwartz, M. E., 98, 150, 155. 177 Schweig, A., 188 Schwenzer, G. M., 119, 130 Scott, P. R., 107, 163 Scott, W. R., 96 Scrocco, E., 36, 177 Secrest, D., 23, 57 Segal, G. A., 32, 187 Seki, H., 177 Seligman, T. H., 68 Serafini. A., 153

Shafi, M., 85 Sharma, R. D., 5, 52 Shavitt, I., 15, 24, 37, 49, 52, 53, 57, 150, 151, 196 Shaw, G., 67 Shaw. R. W.., iun.. 129 Shih, ‘S., 35 Shillady, D. D., 153, 183 Shingkuo Shih, 35 Shipman, L. L., 176,177,180 Shull, H., 40,105 Sidis, V., 78, 91, 108 Siegbahn, P., 71, 89, 141, 152, 154, 179, 180, 183 Sieiro. C.. 188 Silbey, R.’,67Silver, D. M., 58, 154 Simonetta, M., 103, 136, 200 Simons, G., 85, 106 Simons, J., 112 Sinanoklu, O., 48, 95, 105, 112.155 Siu, A. K. Q., 90, 120, 136, 145, 148 Skancke, P. N., 156 Skillman. S.. 192 Sklar, A.‘L.; 186 Slater, J. C., 8, 10, 11, 112, 191, 194 Slepukhin, A. Yu., 98 Smith, E. B., 81 Smith, F., 81 Smith, F. C., jun., 191 Smith, V. H., jun., 81, 90 Smith, W. D., 112 Snow, R., 49 Snyder, L. C., 19,42, 177 Snyder, L. E., 131 Solarz, R., 146 Solomon, P., 70, 117, 128, 134 Solouki, B., 149 Spangler, D., 180 Spindler, R. J., jun., 118 Spohr, R., 61 Sprandel, L. L., 138 Srebrenik, S., 27 Stacey, M., 177 Staemmler, V., 22, 48, 86, 100, 109, 134, 136, 151 Stagg, N. J., 119 Stamper, J. G., 115 Stanton, R. E., 18 Starkschall, G., 71, 115 Stewart, R. F., 36,76, 85 Steiner, E., 69 Stenkamp, L. Z., 130 Stephens, M. E., 34,98 Stevens, R. M., 15, 24, 35, 58, 108, 151, 154, 178, 182 Stevens, W. J., 71, 104, 115 Stevenson, P. E., 123 Stillinger, F. H., 70 Stone, T. J., 82 Straub, P. A., 126 Strausz, 0.P., 34, 155 Streitweiser, A., jun., 123, 188 Strich, A., 179 Struve, W. S., 63 Student, P. J., 189 Sundbom, M., 176 Sustman, R., 188 Sutcliffe, B. T., 8 I

Sutton, P., 112 Swalen, J. D., 15 Swanstrflm, P., 139 Swenson, J. R., 157 Swirles, B., 41 Switalski, J. D., 98 Switkes, E., 178, 182 Tait, A. D., 91, 98, 114, 139 Tal’rose. V. L.. 62 Tanaka,. K., 85 Tanaka, Y.,81 Tang, K. C., 25 Tang, K. T., 5, 53 Tantardini. G. F., 103, 136 Tapia, D.,‘58 Taylor, H. S., 32 Taylor, W. L., 81 Tegenfeldt, J., 177 Teixeira-Dias. J. J. C.. 76 Teller, E., 3, 32 Terauds, K., 149 Thadeus, P., 104 Thakkar, A. J., 81 Thirunamachandran, T., 128 Thomas, T. D., 126, 129 Thompson, D. L., 23 Thomsen, K., 139 Thomson, C., 104, 129, 130, 131,132,138,142,147,149 Throne, C. J., 49 Thornton, E. R., 27 Thulstrup, E., 111 , 112, 122 Tiberghier, A., 176 Tinland, B., 199, 200 Toennies, J. P., 22, 81 Tomasi, J., 36, 80, 177 Tomlinson, R. H., 91 Tosatti, E., 153 Tossell, J. A., 178, 182, 190 Trajmar, S., 1f 3 Trindle, C., 145, 153 Trotman-Dickenson, A. F., 61 Truhlar. D. G.., 53., 54., 89. 111, 140 Trulio, J. G., 49 Tsapline, B., 71, 127 Tseng, T. J., 122 Tully, J. C., 4, 127 I

.

Van-Catledge, F. A., 111, 112 van der Avoird, A., 67 Van der Lugt, W. Th. A. M., 26 van der Meer, H., 199 van der Velde, G. A,, 176,177 Van Putters, A. A. G., 188 Van Wazer, J. R., 131, 158, 184 Varandas, A. J. C., 76 Vauge, C., 60 Veillard, A., 14, 17, 26, 151, 152, 153, 155, 178, 179, 181, 182 Venanzi, T. J., 68, 95 Verhaegen, G., 99, 104, 105, 125 Veseth. L.. 114 Vestin; R.; 89 Vinot, G., 153, 178 Vladimiroff, T., 113, 120,156 Voigt, B., 188

209

Author Index von Niessen W

106, 107, 115, 141, '156; 176, 177, 182, 191 VuEoliC, M., 131

Waber, J. T., 193 Wachters, A. J. H., 154, 178 Waddington, T. C., 129 Wadt, W. R., 145 Wahlgren, U., 130, 193 Wagner A. F., 71, 107 Wagner: E. L., 132, 176, 179 Wagnitre, G., 155 Wahl A. C., 14, 20, 41, 71,

72,'73, 86,89,91, 104,107, 110, 112, 114, 115, 116, 117, 118, 122, 123, 124, 125, 140, 182 Wahlgren, U., 142, 152, 176, 178 Walker, J. H., 108 Walker, T. E. H., 83, 100, 103.104.108.153.159.163 Walker. W., 152 ' ' Wall, F. T.,' 54 Wallach, D., 57 Walton, P. G., 123 Wang, P. S. C., 96 Wasserman. E.. 135 Watts, R. O., 35 Watts, R. S., 79, 92 Webster, B. C., 76 Weeks, J. D., 97 Weimann, L. J., 188, 200 Weinberger, P., 112,116, 193 Wejner,.R. K., 136,189 Weinstem, H., 162 Weiss, C., 188

Weiss, R., 178 Weiss, S., 15 Weissman, S., 81 Weltner, W., tert., 122, 131 Wentink, T. jun., 118 Westenberg A. A., 54 Weston, R. E., jun., 52 Wetmore, R. W., 85 Whisnant, D. M., 78 White, R. A., 23 Whitehead, J. C., 5 Whitman, D. R., 190 Whitten, J. L., 12, 60, 84, 108, 140, 152, 157, 177

Wilhite, D. L., 140, 152 Wilkinson, P. G., 1 Williams, G. R., 47, 133, 138, 148, 188

Williams, J. E., 123, 156, 188 Williams, M. L., 106 Williams, W., 113 Wilson, C. W., jun., 57, 70, 78,95

Wilson, K. R., 63 Wilson, M., 122 Wilson, R. W., 125 Wilson. W.. 25 Winter,' N. 'W., 15, 131 Wipff, G., 28, 176 Wirsam, B., 107, 141, 149, 153. 155

Wishart,- B. J., 132 Wishart, B. W., Wishart. W., 131 Wolniewicz, L.; L., 3, 50, 72, 86 Wong, D. P., 155 Wong, W. H., 5, 55 Wood, M. H., 71, 154, 178, 180

Woods R. C., 121 WoodGard. R. B.. 35 Woolley, R. G., 195 Woolsey I. S., 177 Wright, i. S., 26, 35, 144 Wright, W. M., 89 Wu. F. M.. 88 wu; s., 53 ' WU, S.-F., 53 Yadav, J. S., 188 Yamabe. S.. 140 Yamabk T.; 85, 140 Yan, J. F., 199 Yandle, J. R., 188 Yarkony, D. R., 6, 80, 148 Yates. A. C.. 54 Yates; K., 141 Yeager, D. L., 104, 157 Ye,K. K.,82 Yoshida, Z.-I., 200 Yoshimine, M., 103, 104, 118, 119, 121, 122

Yoshimo, K., 81 Young, C. E., 5 Young, R. H., 75,141 Yue, C. P., 96 Yurtsever, E., 183 Zahradnik, R., 146 Zare, R. N., 43 Zemke, W. T., 89, 110, 114 Zerner, M. C., 184 Zeroka. D.. 88 Zetik, D. F., 59, 65, 70 Zhogolev, D. A., 73 Ziegler, T., 188