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. ...

Author: R.N. Dixon | C. Thomson

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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) +
(10)

for the nuclear eigenfmction xiv(R). The term
4

Tireoretical Chemistry

or Rennerll effects. In these cases, the non-adiabatic terms, < y i l p ~ l y j >can , become very large as the energies approach one another. The large mixing of the electronic levels by the nuclear kinetic energy operator implies a strong dependence of the electronic wavefunction on the nuclear motion and results in a complete breakdown of the Bom-Oppenheirner approximation.l2 2 Scope of the Review

The theoretical study of any chemical problem which falls within the domain of the Born-Oppenheimer approximation involves two basic steps : (a) the solution of Schrodinger’s electronic equation, equation (3, for one or more electronic states over the range of nuclear configurations demanded by the problem; and (b) solving the nuclear equations of motion on the potential energy surface, or surfaces, thus obtained.* The calculations in part (b) may be of two types; the determination of the nuclear energy levels for bound states of the system, i.e. the quantized vibrational and rotational levels of the system, or the study of the dynamics of the chemical changes described by the surface in terms of quantum reactive scattering or classical trajectory calculations. It is the purpose of this review to discuss and illustrate the methods presently employed to obtain potential energy surfaces by approximate, but non-empirical solutions to Schrodinger’s electronic equation. In addition to discussing the different levels of approximation employed in these ab initio calculations, we emphasize the type of chemical system (in terms of its electronic structure) to which each level of calculation may be expected to yield usable results, i.e. results with acceptable errors or with predictable bounds on the error. Our interest will be primarily in surfaces which have been determined for the prediction and understanding of chemical reactions. This will include a survey of those calculations which have concentrated on determining the reaction path, and the geometry and properties of the system at points on this path, as well as those in which an essentially complete surface has been determined. The latter type of calculation coupled with either classical or quanta1treatments of the nuclear motion on such a surface provides a total theoretical prediction of a chemical reaction. This ultimate objective has been achieved in the case of the H + H2 exchange reaction. A recent review of the dynamics of bimolecular reactions by Polanyi and Schreiberl4 provides a detailed account of the cZassicaE trajectory method for treating the motion of a reactive system on a potential energy surface. Such classical motion studies on three-dimensional surfaces are now commonplace and have yielded a great deal of information regarding the microscopic dynamical behaviour

*

When the nuclear motion may be described as occurring on a single surface, it is referred to as being adiabatic. In some systems, more than one potential surface must be employed in the description of the nuclear motion and these cases are referred to as 110n-adiabatic.l~

l1

12 13 14

R. ReMer, Z . Physik, 1934, 92, 172. W. D. Hobey and A. D. McLachlan, J. Chem. Phys., 1960,33, 1695. J. C. Tully and R. K. Preston, J. Chern. Phyx., 1971, 54, 4297. J. C. Polanyi and J. L. Schreiber, ‘Physical Chemistry - An Advanced Treatise,’ ed. H. Eyring, W.Jost, and D. Henderson, Academic Press, New York, 1974, vol. 6.

Ab initio Calculation of Potential Energy Surfaces

5

of reactive systems. The majority of these studies, those by Polanyi et a1.,15 Bunker et al.,le Grice et aZ.,17 and Karplus et a1.,18 for example, have employed potential energy surfaces determined by semi-empirical methods. Most notable of these is the 20 Other semi‘extended LEPS (London, Eyring, Polanyi, and Sato) function’.fQ* empirical or purely empirical surfaces have been introduced and used in trajectory studies by Blais, Bunker, and and by Porter and Karplw22 Roach and Child 23 have developed a pseudopotential method for semi-empirical variational calculations of a potential surface. Quantum calculations of reactive scattering on a potential surface are much more diflicult than the corresponding classical trajectory studies and are not as far advanced. Progress in this field has been reviewed recently by L e ~ i n e , ~ ~ Karplus,26 and Light.27 Fortunately, it appears from the limited comparisons of the quanta1 and classical studies and from a comparison of the classical results with experimental findings, that the classical approach to the study of reaction dynamics is sufficiently accurate to yield informative and useful res~lts.1~ The classical trajectory studies in particular have resulted in a cataloguing of the properties of potential surfaces with respect to the possible observed types of reaction dynamics. Thus, one desires a knowledge of the principal features of a surface which determine the distribution of energy in the products, whether it appears primarily as vibrational excitation of the newly formed bond, as rotational energy, or as relative translational motion of the product molecules. Information regarding the product energy distribution over vibrational and rotational states is obtained from the i.r. chemiluminescence method developed by Polanyi (for a recent review, see ref. 28). In addition, crossed molecular beam studies, by measuring the angular distribution as well as the translational energy distribution of the scattered molecules, require that predictions be made regarding the relative probabilities of ‘forward‘, ‘back’ and P. J. Kuntz, E. M.Nemeth, J. C. Polanyi, S. D. Rosner, and C. E. Young, J. Chem. Phys., 1966,44, 1168;J. C. Polanyi and W. H. Wong, ibid., 1969,51, 1439; M. H.Mok and J. C. Polanyi, ibid., 1969,51, 1451; 1970, 53, 4588;B. A. Hodgson and 3. C. Polanyi, ibid., 1971, 55,4745. 1 6 D. L . Bunker, ‘Methods in Computational Physics,’ ed. B. Adler, S. Fernbach, and M. Rotenberg, Academic Press, New York, 1971,vol. 10, p. 287;T.B. Borne and D. L. Bunker, J. Chem. Phys., 1971,55,4861. 17 J. C . Whitehead and R. Grice, Mol. Phys., 1973,26, 267. 18 M. Karplus, R. N. Porter, and R. D. Sharma, J. Chem. Phys., 1965, 43, 3259; M. Karplus and K. T. Tang, Discuss. Faraday Soc., 1967,44, 56. 1s H.Eyring and M. Polanyi, 2.phys. Chem., 1931,B12, 279; for a review and summary of earlier references, see J. C. Polanyi, J. Quant. Spectroscopy and Radiative Transfer, 1963,3, 471. 20 S. Sato, J. Chem. Phys., 1955,23,592,2465. 21 D.L. Bunker and N. C. Blais, J. Chem. Phys., 1964,41,2377;D.L. Bunker and C. A. Parr, ibid., 1970,52, 5700. 22 R.N.Porter and M. Karplus, J. Chem. Phys., 1964,40,1105. 23 A. C. Roach and M. S. Child, Mol. Phys., 1968, 14, 1. 24 R. D.Levine, ‘Quantum Mechanics of Molecular Rate Processes’, Clarendon Press, Oxford, 1969. 26 J. Ross, in ‘Molecular Beams and Reaction Kinetics’, ed. Ch. Schlier, Academic Press, New York, 1970,p. 392. 26 M. Karplus, ref. 25, p. 407. 27 J. C. Light, in ref. 16, p. 11 1. 28 T. Carrington and J. C. Polanyi, in ‘Chemical Kinetics’, ed. J. C. Polanyi, MTP lntemational Review of Science, Physical Chemistry Series One, Medical and Technical Pub. Co., Oxford, 1972, p. 135. 15

Theoretical Chemistry

6

angularly uniform scattering of the product molecules. For a recent review of the crossed molecular beam method, the reader is referred to an article by K i n ~ e y . ~ ~ This type of categorization of the properties of potential energy surfaces is well documented in the recent review by Polanyi and Schreiberl4 and will be referred to in discussing the potential energy surfaces obtained for particular reactive systems. Through the use of the Rayleigh-Ritz variational principle, approximate solutions to Schrodinger’s electronic equation, equation (5), may be obtained to any desired degree of accuracy for a many-electron system. Because of the formidable computational problems involved in obtaining these solutions, chemical accuracy (generally taken to imply errors of a few kcal mol-1 or less) has been achieved in only a few two- and three-electron systems. Considerable progress is being made, however, by methods reviewed here, and complete potential surfaceswith an accuracy acceptable for trajectory studies for systems with up to 30 electrons and four nuclei now lie within the present bounds of accomplishment.30 In addition, of course, many questions concerning the electronic mechanisms of particular reactions may be answered by calculations performed at lower levels of approximation. Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called ‘Hartree-Fock limit’ represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger’s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation ( 5 ) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces, 3 Hartree-Fock and Self-Consistent-FieldCalculations The orbital approach to the problem of electronic structure reduces a many-electron problem to a corresponding number of oneelectron problems. The Hartree-Fock solution represents the best attainable description of the electronic structure of a many-electron system in terms of the one-electron orbital approach. In the orbital description of electronic structure, the motion of each electron is determined by a spin-orbital, ut, a simple product of a space function q5i and a spin function crt

where ut may denote either an ct or /Ispin function. By analogy with a system in which each electron moves independently of the others, the total wavefunction is taken to be a product of the oneelectron spin orbitals, one for each electron in the system. To satisfy the additional restrictions of the Pauli principle, and simultaneously to allow for the indistinguishability of the electrons, an antisymmetrized sum of such product functions must be used to L. Kinsey, in ref. 28, p. 173. D, R. Yarkony, S. V. O’Neil, and H. F. Schaefer, J. Chem. Phys., 1974, 60, 855.

as J. 30

7

Ab initio Calculation of Potential Energy Surfaces

approximate the many-electron wavefunction for the system. This is most conveniently done by expressing the wavefunction in the form of a determinant, O(x; R) = (n!)-1I2det [ul(l)u2(2).. .un(n)]

(1 1)

since on expansion, a determinant equals a sum of such product terms [asrepresented in equation (11) by its diagonal element] representing all possible permutations of the n electrons among the n spin-orbitals(n! in number), each term in the sum being multiplied by (+ 1) or (- 1) depending on whether the number of permutations is even or odd. The factor (n!)- 1 is ~ required to normalize @(x; R) to unity when the spin-orbitals are orthonormal, i.e. Jui(k)uj(k)drk =

&j

(12)

where the orthonomality obtains for both space and spin functions separately. The energy of such a determinantal wavefunction, @POI; a),is obtained by the quantum mechanical averaging of the electronic Hamiltonian operator given in equation (6) E(R) = <@(x; R)lfieI@(x;R)>

(13)

or N

R

where hi = Jaj

=

and Kaj

= [z&k)uj(k)(

I rk - rl 1) -l4(l)ui(Z) dtr dtt

(16)

The kinetic energy of the electron described by ug together with its energy of interaction with all of the nuclei in the system is represented by hi. The s u m over Jij and Kij in equation (14) represents the energy of repulsion between the electrons in the system, the Coulomb integral Jij representing the electrostatic repulsion between the two charge distributions g(k)Ui(k) and uj*(Z)uj(Z)and the exchange integral Ksj correcting this classical estimate of the repulsion in a manner dictated by the antisymmetry principle. The forms of the orbitals us are determined by demandingthat the energy obtained in equation (13) be a minimum with respect to arbitrary variations in the orbitals. This procedures1leads to a set of n coupled equations, the Hartree-Fock equations, of the form F(k)ui(k) = eiut(k) (1 7) The HartreeFock operator p(k) is generally expressed as a sum of the two terms, 81

C. C . J. Roothaan, Rev. Mod. Phys., 1951, 23, 69.

8

Theoretical Chemistry

+ G(k) where H(k) is the Hamiltonian for motion of the electron in orbital u&) in the field of the bare nuclei and where the operation of G(k) on ut(k) is given by

p(k) = k ( k )

(1 8) The first summation in equation (18) determines the repulsive Coulomb field exerted on electron k at r k by the total charge distribution including the distribution z$(Z)ut(Z). This self-repulsion of the electron described by ut is removed throughout space in a particular manner as described by an exchange distribution Cj ui(Z)ui(Z) which vanishes on integration over the spins unless ut and uj possess identical spin functions. Thus, the effect of antisymmetrization on the simple product wavefunction is to replace the intuitively obvious coulomb repulsion of the form Z j +t &Z)&(Z) with a repulsion involving the charge distribution of all the orbitals, from which is removed the equivalent of the repulsion from the one electron in 4i.32This selective removal of charge density throughout space of electrons with the same spin is called the 'Fermi hole'? Thus, each orbital in the system is determined by a Schrodinger-like eigenvalue equation, equation (17), with a Hamiltonian whose potential field is determined by the attractive field of the nuclei and the average repulsive field of the remaining electrons in the system. The equations for all n orbitals are coupled since the potential field for any one orbital is determined by the distribution of charge described by the remaining orbitals. Thus, to solve the Hartree-Fock equations, one must fkst assume some form for all the orbitals in the system, construct the resulting potentials G(k),solve the equations to generate a new set of orbitals which are then employed over again in the same manner. This procedure is repeated until the potentials &(k)used in the determination are indistinguishable from or converge to the potentials obtained in the calculation. That is, the process is continued until self-consistency in the assumed and calculated potentials is achieved. When the orbitals are determined in this manner, with the only restriction being the orthonormality constraint, equation (12), they yield the best possible antisymmetrized product wavefunction (i.e., a single determinant wavefunction) for the system in question since the resulting @(x; R) yields the lowest possible value for E(B). The heart of the Hartree-Fock model is to replace the detailed and accurate description of the repulsions between every pair of electrons in the system by the average field that each electron exerts on every other. This is a consequence of the oneelectron or orbital basis of the model which leads to a product wavefunction. The probability, P2(rl7ra)dr1dr2,that electron 1 be in some small volume element about r l when electron 2 is simultaneously in some small volume element about r2 is, in a simple product-type wavefunction, given by the product of the two singleparticle probabilities,

sa

J. C. Slater, 'Quantum Theory of Molecules and Solids', McGraw-Hill, New York, 1963, Vol. 1.

33

R. McWeeny and B. T. Sutcliffe, 'Methods of Molecular Quantum Mechanics', Academic Press, New York, 1969, p. 76f.

Ab initio Calculation of Potential Energy Surfaces

9

where, for a closed-shell system nl2

mrl)

=

(2/n)

c $i+(l)+r(l)

i=l

This result for Pz(r1,rz) can only be true in the limit that the motion of electron 1 and hence its probability of being at a certain point in space is totally independent of the motions of electron 2, i.e. there are no forces acting between the two electrons and their motions are totally uncorrelated. The best that can be done in a model which employs an expression of the form of equation (19) for the pair probability function is to determine the ui (which in turn determine the one-particle probability function) in such a way that the electron it describes experiences the average field of the other electrons, a situation which is attained in the Hartres-Fock limit of the orbital model. The Hartree-Fock wavefunction is not a simple product function, but an antisymmetrized sum of such products. As pointed out above, the effect of the antisymmetrization is to selectively remove, from the region of space about a given electron, charge density of electrons with the same spin. Thus, the pair probability function for electrons with the same spin is not of the form given in equation (19) but instead is

where

and where na denotes the number of orbitals occupied with a spins in a closed-shell system. Notice that ps"(rl, r2) [or Pfb(rl, rz)], equals zero when rl = 1-2, and hence the antisymmetrizationof a simple product wavefunction results in a 100 % negative correlation between the motions of electrons with identical spins as required by the Pauli principle.ss The pair probability of electrons with opposite spins predicted by a Hartree-Fock wavefunction is, however, given by the simple product expression, equation (19). This is the source of the inherent error in a Hartree-Fock wavefunction and because of its fundamental importance, it is given a special name, the correlation error. Correspondingly, the correlation energy is defined34*35 as the difference between the Hartree-Fock energy and the exact energy of the system.* The methods for obtaining approximate solutions for Schrodinger's electronic equation which go beyond the Hartree-Fock limit are specifically designed to reduce the correlationerror, i.e. they are constructed so as to obtain a more detailed and more accurate description of the pair probability function for electrons of opposite spin.

(a,

The electronic Hamiltonian, equation does not include relativistic effects and hence cannot yield the true total energy of a system. Thus, its contribution to the energy is accounted for in the definition of the correlation error.85 84

P.-0. Lowdin, Adu. Chem. Phys., 1959,2, 207. E. Clementi, J. Chem. Phys., 1963, 38,2248.

10

Theoretical Chemistry

Properties of Hartree-Fock Wavefmctions.-A Hartree-Fock wavefunction satisfies36,37 the virial theorem38 and the Hellmann-Feynman theorem,39, theorems which have been referred to by Slaterg1as being two of the most powerful theorems applicable to molecules and solids. Hartree-Fock wavefunctions also satisfy a theorem derived by Brillouin,42 which is discussed in Section 5. As a result of this very important theorem, one can prove that a molecular charge distribution obtained from a Hartree-Fock wavefunction and the properties determined by it are predicted with surprisingly high accuracy. For example, the charge distribution is probably accurate to within 2--5z.43 As explained below, all properties predicted by a Hartree-Fock wavefunction are, because of Brillouin’s theorem, correct up to second-order. A comparison of Hartree-Fock and experimental results for molecular properties must be prefaced with the methods employed in obtaining wavefunctions at or close to the Hartree-Fock limit in molecular systems. Roothaan’s Self-Consistent-Field Procedure.-While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by R ~ o t h a a nIn . ~Roothaan’s ~ approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. N

Two kinds of basis functions (or basis orbitals) are commonly used in the expansions of the orbitals in both atomic and molecular calculations, being either of the Slatertype (STO’s) or of the Gaussian-type (GTO’s). Substituting the expansion, equation (22), for each molecular orbital in the Hartree-Fock expression for the energy, and again demanding that E m ) be a minimum with respect to variations in the uz, results in a set of matrix equations (of dimensions N x N ) for the coefficients cij. These equations are solved by an iterative method starting with some initial guess of the cip, until self-consistency has been achieved. This method is termed (Roothaan’s) LCAO-MO SCF approximation to the Hartree-Fock limit.*

* The term Restricted Hartree-Fock (RHF) is applied to those cases in which all the possible spin pairing in a system is allowed for by having electrons of both a and B spin occupy the same space orbital. If this restriction is relaxed in writing out the determinantal wavefunction, the method of calculation is referred to as the Unrestricted Hartree-Fock (UHF) method. Unless otherwise stipulated, the calculations referred to in this chapter are of the RHF variety.

36

37

C. W. Kern and M. Karplus, J. Chem. Phys., 1964,40, 1374. A. C. Hurley, ‘Molecular Orbitals in Chemistry, Physics, and Biology’, ed. P.-0. Lowdin and B. Pullman, Academic Press, New York, 1964, pp. 161-191; C. A. Coulson, Mol. Phys., 1971,20, 687.

J. C. Slater, J. Chem. Phys., 1933, 1, 687. se H. Hellmann, ‘Einfuhrung in die Quanten-chemie’, Franz Deuticke, Leipzig, Germany, 1937, 38

41 42

p. 285. R. P. Feynman, Phys. Reu., 1939,56, 340. J. C. Slater, J. Chem. Phys., 1972, 57, 2389. (a) L. Brillouin, Actualities Sci. I d . , 1933-34, 71, Secs. 159, 160; (b) C. Moller and M. S. Plesset, P h p . Reu.. 1934, 46, 618.

Ab initio Calculation of Potential Energy Surfaces

11

Use of a complete (necessarily infinite) set of basis orbitals in the expansion for the molecular orbitals would insure absolute convergence to the Hartree-Fock limit. In practice, this is both impossible and unnecessary, and only a finite number of basis functions is employed in the expansion. The selection of the finite basis set is, therefore, of crucial importance in determining how closely one approximates the true Hartree-Fock solution. A complete review of the characteristics of various types of basis sets has been given recently by Schaefer.44The radial form of STO’s is similar to the (nodeless) hydrogenic atomic orbitals, rn -le -Cr where n is the principal quantum number and C is a variable exponent.45Their angular dependence is described by multiplication by a spherical harmonic Y z , ~ ( ~ , The $ ) . use of GTO’s in molecular calculations was first suggested by Boys.46 These functions are usually employed in their Cartesian form, xpyrrZ8e-cr2 where different angular dependences are obtained by assigning values to the integersp, 4,and s. STO’s with their exponential dependence on r are better able to approximate the atomic solutions in the region of a nucleus than are GTO’s with the r2 dependence. In particular, STO’s can properly describe the cusp behaviour which the wavefunction should exhibit at the positions of the nuclei,47 while GTO’s cannot, Thus, an expansion of the molecular orbitals in terms of an STO set yields a better approximation to the Hartree-Fock orbitals than one expressed in terms of the same number of GTO’s. However, as pointed out by B o ~ s , ~ ~ the two-electron multicentre integrals which are encountered in molecular calculations are much more rapidly evaluated when the orbitals are expressed in terms of GTO’s rather than STO’s. This saving in the time required for integral evaluation is obtained even when the size of the GTO set is increased to the point where it yields results of accuracy comparable with those obtained using an STO set.44 One may use any size basis set above some minimal size in the SCF procedure to obtain orbitals, a wavefunction and an energy of an atomic or molecular system. However, only very large basis sets possess the flexibility required to obtain good approximations to the true Hartree-Fock orbitals for a system. Thus, one must distinguish between those SCF calculations which use basis sets large enough so that the results closely approximate the Hartree-Fock limit, and those which do not. One finds that the SCF energy, since it always represents an upper bound to the true energy, converges smoothly to the limiting Hartree-Fock value as the size of the basis set is increased. The values of properties other than the energy, such as the Hellmann-Feynman force or the electric field gradient, however, may fluctuate erratically around the Hartree-Fock value until this limit is very nearly attained.48 The minimum size basis set which can be used in an SCF calculation includes one ST0 for each occupied atomic orbital with distinct n and I values as determined by the electronic configuration of the atom in question. While of conceptual and 43

44

45 46 47

48

R. F. W. Bader, in ‘Theoretical Chemistry’, ed. C. A. Coulson and A. D. Buckingham, MTP International Review of Science, Physical Chemistry Series Two, Butterworths, London, 1975. H. F. Schaeler, ‘The Electronic Structure of Atoms and Molecules’, Addison Wesley, Reading, Mass., 1972. J. C. Slater, Phys. Rev., 1930, 36, 57. S. F. Boys, Proc. Roy. SOC.,1950, A200, 542. T.Kato, C ~ m mPure . Appl. Math., 1957,10, 151. P. E. Cade, Trans. Amer. Cryst. Assoc., 1972, 8, 1.

12

Theoretical Chemistry

historic importance, such a minimum basis set is usually inadequate for molecular calculations. A great improvement in calculated energy and properties is obtained by doubling the size of the minimum basis set, i.e. representing each occupied atomic orbital with distinct n and I values by two STO basis functions. This is termed a double-zeta basis set. The use of such a basis set in atomic SCF calculations yields results close to the Hartree-Fock limit. Roetti and Clementi49 have determined double-zeta basis sets with optimized exponents for the ground-state functions of all atoms from He(lfl through to Xe(W). In all cases, the calculated energies are close to the Hartree-Fock limiting values. For example, the double-zeta SCF energies for C(3P), F(2P)and Br(2P) lie above the Hartree-Fock values by 5, 21, and 6 kJ mol -l, respectively. As’ discussed below, the double-zeta set when complemented with suitable polarizing functions is a widely used and rather accurate set for use in molecular SCF calculations. Basis sets composed of GTO’s which are equivalent to the doublezeta Slater sets have been devised by Whittens* and Dunning.51 A considerable saving in computation time can be achieved with only a small sacrifice in flexibility by employing what are called contracted sets of GTO’s. That is, the original set of GTO’s, referred to as primitives, are grouped into a smaller number of linear combinations with fixed coefficients. Thus, the double-zeta set of Dunning51 which consists of nine s- and five p-type Gaussian functions is, in a typical calculation, usually contracted to four s- and twop-type functions. This information is summarized in the notation (9sSp)+ [4s, 2pl. Dunning52has also provided (10s6p)-+[5s,4p] or [5s, 3p] sets which exceed double-zeta quality. While basis sets composed of a sufficient number of s- and p-type functions will yield the Hartree-Fock results for atoms up to scandium,53such s-p saturated sets are not suitable for molecular SCF calculations. Instead, the basis sets must be augmented with ‘polarizing’ functions, i.e. those functions, p for hydrogen and d and f f o r first- and second-row atoms, which are absent in the description of the isolated atoms, but whose presence is essential for a proper description of the distortions in the charge distribution caused by chemical bonding. As the examples in the following section illustrate, the inclusion of polarizing functions in the basis set is essential in the calculation of potential surfaces. Comparison of H-Fock and Near Hartree-Fock SCF Results with Experiment.The properties of the water molecule described by an SCF single determinant wavefunction have been determined with a number of GTO and STO basis sets, ranging in size from minimal to near Hartree-Fock. A detailed tabular summary of these results, which are typical of such SCF calculations on molecular systems, has been given by Kern and Karplus.6 The Hartree-Fock estimate of the total energy for water is estimated to be - 76.066 & 0.002 a.u.,54 compared to the experimental value of - 76.431 a u . An SCF calculation using a Gaussian (1ls7p2d/5slp)primitive set 49

50

51

j2 j3

54

C. Roetti and E. Clementi, J. Chem. Phys., 1974, 60, 4725. J. L. Whitten, J. Chem. Phys., 1966, 44, 359. T. H. Dunning, J. Chem. Phys., 1970, 53, 2823. T. H. Dunning, J. Chem. Phys., 1971, 55, 716. E. Clementi, ‘Tables of Atomic Functions’, International Business Machines Corp., San Jose, California, 1965. T. H. Dunning, quoted by C. W. Kern and M. Karplus in ref. 6.

Ab initio Calculation of Potential Energy Surfaces

13

contracted to a [6s5p2d/3slp]* yields an energy of -76.0621 I L . u . A ~ ~smaller set of STO's (5s4pld/3slp) yields an energy of -76.0631 a.u., in very close agreement with the estimated Hartree-Fock limit.55 A minimal STO set yields an energy more than 11 eV above that obtained from the best function.56 The Gaussian basis set which is equivalent to the doublezeta Slater set and augmented with two d functions, the set (9s5p2d/4slp) contracted to [4s3p2d/2slp], yields an energy of - 76.0504 a.u., or 0.44eV above the HartreeFock limit.55 In addition to the energies, the values predicted by these S C F calculations for a number of the one-electron properties of the water molecule are also compared by Kern and Karplus.6 As anticipated, the values predicted for these properties by the near Hartree-Fock functions are in good agreement with the experimental values. The molecular dipole moment, for example is calculated to be 2.054 D by the best STO set compared with an experimental value of 1.884D.57(HartreeFock estimates of molecular dipole moments are generally too large by ca. 0.2 D.) Ermler and Kern58 have computed the potential energy surface for the water molecule in the neighbourhood of its energy minimum using the [4s3p2d/2slp] contracted set of GTO's (equivalent to a double-zeta set plus the polarizing functions). The theoretical minimum in the energy surface was found to occur at a geometry with bond lengths of 1.779 a.u. and a bond angle of 106.1", compared with the experimental values60 of 1.8089 a.u. and 104.52".The energy in the neighbourhood of the equilibrium geometry was expanded in a Taylor series so as to obtain all the harmonic and cubic potential constants. A comparison of the calculated and experimental results59 is given in Table 1. The harmonic constants are overestimated by amounts ranging from ca. 10% to 20 %. This agreement is typical for Table 1 Comparison of (SCF)calculated and observedforce constantsfor H20 Force constanta fr

fa f i p

ha firr faaa

fm* fira

h a fraa 4

b

*

Computed valueb 9.876 0.881 -0.079 0.258 - 10.505 -0.145

- O.Oo04 -0.033

-0.526 -0.149

Observed value" 8.454 0.761 -0.101 0.228 - 9.55 -0.14 -0.32 0.16 - 0.66 0.15

% Error 16.82 15.82 21.39 12.98 - 10.00 -3.64 99.88 - 120.63 20.26 - 199.20

All values quoted in units of 105 dyn cm -1 (see ref. 59 for the definition of the force field). Ref. 58. C Ref. 59.

The solidus separates the basis functions centred on oxygen from those centred on each of the hydrogens.

55 56 58

59 6o

T. H. Dunning, R. M. Pitzer, and S. Aung, J. Chem. Phys., 1972,57, 5044. S. Aung, R. M. Pitzer, and S. I. Chan, J. Chem. Phys., 1968,49, 2071. M. L. Lichtenstein, V. E. Derr, and J. J. Gallagher, J. Mol. Spectroscopy, 1966, 20, 391. W. C. Ermler and C. W. Kern, J. Chem. Phys., 1971,55,4851. K. Kuchitsu and Y . Morino, Bull. Chem. SOC.Japan, 1965, 38, 814. W. S. Benedict, N. Gailar, and E. K. Plyler, J. Chem. Phys., 1956,24, 1139.

14

Theowticnl Chemistry

results obtained from wavefunctions at or near the Hartree-Fock limit, for polyatomic or diatomic619 62 systems.* Pulay65has developed a method for calculating the potential force constants for polyatomic systems, by analytically evaluating the first derivatives of the SCF determined energy surface, a problem also considered by Gerratt and Mills.66 Forces for only a few different nuclear configurations are required to determine the complete force field by this method, as opposed to the many such configurations required for obtaining second derivatives of the energy surface by numerical methods. This method has been applied to NH3,67aC2H6, C2H4, and and CH4.68 Using a large basis set of GTO’s, Meyer and Pulay68 obtained theoretical estimates of the symmetry force constants for CH4 which differed from the experimental values by amounts ranging from 0 to 12%.They concluded that the values obtained at the true Hartree-Fock limit would deviate by no more than 5 % from the experimental ones. A similar approach, for use with extended Huckel and MIND0 wavefunctions has been developed by McIver and Komornicki.69 While non-negligible errors are found in the curvatures of a HartreeFock potential energy surface in the region of the equilibrium geometry, the heights of small energy barriers separating geometrical conformers are accurately predicted. It was to explain the low-temperatureheat capacity data of ethane that Kemp and Pitzer70 first postulated the existence of barriers to internal rotation. The same system was studied in the first ab initiu calculation of an internal rotation barrier by Pitzer and Lipscomb.71 They used a minimum Slater basis set to determine the relative energies of staggered and eclipsed ethane and obtained a barrier of 13.8 kJ mol -l, compared with the then best experimental value of 12.7k 1.3 kJ mol -1. A number of other calculations have since been carried out which confirm this prediction and indicate that the barrier height in this case is insensitive to basis set composition. Thus, Veillard,72 using a large set of GTO’s for carbon and hydrogen, (1 ls7pld/6slp) contracted to [5s3pld/3slp], in a calculation which included careful optimization of the geometries of both eclipsed and staggered forms of ethane,

*

A relatively recent review of the theoretical calculation of potential curves and their derived properties for diatomic molecules has been given by Wahl.G3A massive review of the theoretical determination of the ground state properties of diatomic molecules by Goodisman 6* has recently been published. The reader is referred to these two sources, particularly the latter, for information regarding the calculation and properties of potential curves for diatomic species.

62

63 64

6s 66

13’

69

70

71 72

P. E. Cade, K. D. Sales, and A. C . Wahl, J. Chem. Phys., 1966, 44, 1973. P. E. Cade and W. M. Huo, J. Chem. Phys., 1967,47, 614, 649. A. C. Wahl, in ‘Theoretical Chemistry’, ed. W. Byers-Brown, MTP International Review of Science, Physical Chemistry Series One, Medical and Technical Pub. Co., Oxford, 1972, p. 41. J. Goodisman, ‘Diatomic Interaction Potential Theory’, Academic Press, New York, 1973, VOl. 1. P. Pulay, Moi. Phys., 1969, 17, 197. J. Gerratt and I. M. Mills, J. Chem. Phys., 1968, 49, 1719, 1730. (a) P. Pulay and W. Meyer, J. Chem. Phys., 1972, 57, 3337; (b) P. Pulay and W. Meyer, Mol. Phys., 1974, 27, 473. W. Meyer and P. Pulay, J. Chem. Phys., 1972,56,2109. J. W. McIver and A. Kormornicki, Chem. Phys. Letters, 1971, 10, 303. J. D. Kemp and K. S. Pitzer, J. Chem. Phys., 1936,4, 749. R. M. Pitzer and W. N. Lipscomb, J. Chem. Phys., 1963, 39, 1995. A. Veillard, Chem. Phys. Letters, 1969, 3, 128.

Ab initio Calculation of Potential Energy Surfaces

15

obtained a barrier height of 12.8 kJ mol-I. This value is in excellent agreement with the most recent experimental estimate of 12.3 kJrnol-1,73 and is nearly unchanged in value from the minimal basis set result. Even the more complicated double barrier present in HzO2 is well predicted at the near Hartree-Fock limit. The experimental geometry of Ha02 lies between the cis- and trans-forms, the two 0-H bonds forming a dihedral angle $ 0 of 111.5". The cis- and trans-conformers lie 31.8 kJ mol-1 and 4.6 kJ mol -l, respectively, above the energy minimum of the equilibrium ge0metry.7~~ 75 Dunning and Winter,76 using a large and carefully contracted set of GTO's for 0 and H, (9s5pld/4slp) contracted to [4s3pld/2slp], and performing a detailed geometry variation, obtained the results listed in Table 2. The barriers are very well predicted as are the parameters defining the equilibrium geometry. Table 2 Comparison of Calculated (SCF)a and observedb barriers and geometry for H202

Barriers (in cm-1) V(cis) V(trans) Geometry 90

L HOO

ROH Roo

CaIculated 2921 384 113.7' 102.5' 1,788 a.u. 2.632 a.u.

Observed 2649 3 86

111.5" 94.8" 1.795 a.u. 2.787 a.u.

Ref. 76. b R. H. Hunt, R. A. Leacock, C. W. Peters, and K. T. Hecht, J. Chem. Phys., 1965,42, 1931; C . S. Ewig and D. 0. Harris, ibid., 1970, 52, 6268.

a

The theoretical determination of the barrier height to inversion in NH3 by an SCF single-determinant calculation illustrates the importance which the inclusion of polarizing functions may have in such calculations. The experimental value for the inversion barrier in NH3 is 24.3 kJ mol-1.77 Three early attempts to account for this barrier using the SCF approach with either STO or GTO basis sets without polarizing functions centred on nitrogen yielded discouraging results.78~ 80 Using a GTO basis set not too different from one employed previously79 but with a d function centred on nitrogen, Body, McClure, and Clementi81obtained a theoretical value for the barrier which was ca. 20 % too large. With this encouragingresult, two near Hartree-Fock determinationsof the barrier were attempted, one using GTO'S,~~ the other S T O ' S , ~and ~ both yielded barrier values in very close agreement with 799

75 74

75 76 77 78 79

Bo sl 82 88

S. Weiss and G. E. Leroi, J. Chem. Phys., 1968, 48, 962. C. S. EWig and D. 0. Harris, J. Chem. Phys., 1970,52, 6268. R. H. Hunt, R. A. Leacock, C. W. Peters, and K. T. Hecht, J. Chem. Phys., 1965,42, 1931. T. H. Dunning and N. W. Winter, Chem. Phys. Letters, 1971, 11, 194. J. D. Swalen and J. A. Ibers, J. Chem. Phys., 1962, 36, 1914. U. Kaldor and I. Shavitt, J. Chem. Phys., 1966, 45, 888. E. Clementi, J. Chem. Phys., 1967,46, 3851. W. H. Fink and L. C. Allen, J. Chem. Phys., 1967,46, 2276. R. G. Body, D. S. McClure, and E. Clementi, J. Chem. Phys., 1968,49, 4916. A. Rauk, L. C. Allen, and E. Clementi, J. Chem. Phys., 1970,52, 4133. R. M. Stevens, J. Chem. Phys., 1971,55, 1725.

16

Theoretical Chemistry

experiment. The calculation employing GTO's was performed by Rauk, Allen, and Clementis2 using a (13s8p2d/8s2p) basis set contracted to [8s5p2d/4s2p]. The predicted equilibrium geometry, R = 1.90 a.u. and 8 = 107.2" agrees well with experiment, R = 1.91 a.u. and 8 = 106.7'. The barrier obtained is 21.3 kJ mol-l, also in good agreement with experiment. These authors repeated the calculation of the barrier with the d functions omitted and obtained a value of only 5.0 kJ mol --l. The calculation of the NH3 inversion barrier using a large set of STO's, (5s4pld/2slp), was performed by Stevens.83 The energies obtained by both Stevens and Rauk et al. for the pyramidal equilibrium geometry, - 56.2212 a.u. and - 56.2219 a.u., respectively, are believed to lie within 8 kJ mol-1 of the Hartree-Fock limit. Stevens' calculated barrier height of 24.7 kJ mol-1 is in almost perfect agreement with the experimental value. Stevens also computed a potential curve for the inversion motion in NH3 by determining the minimum energy path. Schrodinger's equation for nuclear motion, equation (7), was solved for this system, treating the inversion motion as a one-dimensional problem. The vibrational energy levels thus obtained are listed in Table 3 for both NH3 and ND3, together with the experimental values.

Table 3 Computed (SCF)a and observed inversion energy levels in NH3 and N D 3 Level

Computed 0 1.133 987.96 1037.97 1684.9 2030 2579.50 3 154.02

0s OA

1s fA

2s 2A 3s 3A a

Ref. 83.

b

NHs Experimental 0 0.7935 932.51 968.32 1597.6 1910 2383.46 2895.48

ND3

Computed 0 0.068 804.7 809.5 1451 1543 1960 2286.0

Experimentalb 0 0.053 745.7 749.4 1359 1429 1830 2106.6

W. S. Benedict and E. K. Plyler, Canad. J. Phys., 1957, 35, 1235.

There is fair agreement between the theoretical and experimental values. Stevens points out that the inversion frequency is very sensitive to the barrier width, and it is because the calculated barrier is too narrow, that the predicted inversion frequencies are too large. Almlof 84 has computed the SCF potential surface for the linear degrees of freedom in the neighbourhood of the equilibrium geometry for HF; using an (9,5,1/4,1)set of GTO's contracted to [4,2,1/2,1].A potential function was generated by fitting the SCF energies with a nine-term polynomial in the symmetry co-ordinates Ql and Q z . The vibrational Schrodinger equation was then solved by expanding the nuclear wavefunction in products of linear harmonic-oscillator eigenfunctions, that is, products of the form &(Ql)$j(Q2). From a thorough investigation of the convergence properties of this expansion of the two-dimensional nuclear wavefunction, it was concluded that an expansion employing 160 such product terms was sufficient to insure an accuracy of k 5 cm -l for frequencies up to 5000 cm -l from what would 84

J. Almlof, Chern. Phys. Letters, 1972, 17,49.

Ab initio Calculation of Potential Energy Surfaces

17

be obtained using the full, infinite set. While the calculated values of the vibrational frequencies tend to be high [e.g. vl(ca1c.) = 660 cm-1, vl(exp.) = 631 cm-11, the general agreement between the theoretical and calculated frequencies appears to be about as good as the agreement between the different sets of experimental values.84 Radom and Pople 85 have provided a comprehensivesummary of SCF calculations of rotation and inversion barriers and a comparison of these results with known experimental values. It appears that Hartree-Fock theory yields acceptable values for the heights (but not necessarily the curvature and shape) of such barriers. In fact, barriers which result from the passage of the system through a series of geometries not involving a change in hybridization about some central atom, such as the barrier to internal rotation in C2H6, for example, are well-predicted by SCF calculations above the Hartree-Fock limit. The inversion barrier in NH3, which in a simple model involves geometries with very different hybridizations on the nitrogen, is well described only when the SCF calculation approaches the HartreeFock limit. In particular, polarization functions must be centred on the atom which undergoesthe change in hybridizationif the height of the barrier, or even its presence, is to be properly predicted. As a further example of the importance of d functions in determining barrier heights we refer to calculations by Veillard et a1.86-87 and by Duke and Bader88 of the classid energy of activation for s N 2 reactions. While these calculations are discussed in more detail below, we mention here that both sets of authors found that basis sets which did not include polarizing functions gave results for the activation barriers which were radically different from those obtained near the Hartree-Fock limit. For example, both sets of authors found that in SCF calculations using large basis sets but with d functions absent, the F CH3F displacement reaction was predicted to proceed via a symmetrical (FCH3F) - intermediate stable by ca. 13 kJ mol -1 relative to reactants. With d functions included, the same reaction is predicted to occur via a transition state species with a symmetrical structure lying ca. 29 kJ mol-1 above the energy of the reactants. The existence of activation barriers for s N 2 displacements of F- from CH3F in the gas phase has been verified experimentallyby Bohme et ~1.89One associates a considerable change in the hybridization of carbon in the formation of the transition state for an s N 2 reaction, and the inclusion of d functions in the basis set is again found to be essential if a single determinant SCF calculation is to provide a proper description of the accompanying change in energy. The examples discussed so far serve to illustrate the ability of a Hartree-Fock or near Hartree-Fock SCF wavefunction to describe a polyatomic system in the neighbourhood of its equilibrium geometry (or geometries). Equilibrium geometries are predicted rather well by the Hartree-Fock SCF procedure. Predicted bond lengths are, in general, too short by 0.01--0.03a.u. and bond angles too large by ca. one degree. This is found for systems larger than those reported above. For example, Duke and Bader 88 performed an extensive geometry

+

85

86 88

L. Radom and J. A. Pople, in ref. 63, p. 71. G. Berthier, D. J. David and A. Veillard, Theor. Chim. Acta., 1969, 14, 329. A. Dedieu and A. Veillard, Chem. Phys. Letters, 1970, 5, 328. A. J. Duke and R. F. W.Bader, Chem. Phys. Letters, 1971, 10, 631. D. K. Bohme, G. I. Mackay, and J. D. Payzant, J. Amer. Chem. SOC.,1974,96,4027.

18

Theoretical Chemistry

optimization for CH3F and CH3CN using an s and p GTO basis set of above double-zeta quality. The calculated and experimental (in brackets) values of the geometry parameters are for CHsCN; R(C-N) = 2.1539 a.u. (2.1864 a.u.), R(C-C) = 2.7661 a.u. (2.7553 a.u.), R(C-H) = 2.0471 a.u. (2.0853 a.u.), = 110.09" (109.31"); for CH3F; R(C-F) = 2.6498 a.u. (2.6147 6(H-C-CN) a.u.), R(C--H) = 2.0387 a.u. (2.0731 a.u.), 6(H-C-F) = 109.19"(108.45'). Since Hartree-Fock theory underestimates bond lengths, slightly better estimates of the molecular geometries are usually obtained using a basis set which falls short of that required for the HartreeFock limit. The set used in CH3F and CH3CN, for example, did not contain polarizing functions. A minimal basis set frequently yields better results, with errors of ca. 1 or 2 z.90As a further example, Pullman and Port (see ref. 137) predict that the preferred conformation of the acetylcholine molecule is gauche, on the basis of minimal basis set SCF calculations. This result is in agreement with the observed conformations of this molecule in the crystalline phase and in solution. The absolute energies obtained in a Hartree-Fock calculation are, of course, in considerabIe error (the correlation error), but for most problems of chemical interest one is concerned with energy differences and errors in the absolute energies are unimportant. It is now well established that the correlation error in the HartreeFock energy remains relatively constant in the neighbourhood of the minimum or minima of the potential energy surface. Consequently, the HartreeFock energy surface approximately parallels the true surface in these regions, and the ability of HartreeFock theory to provide acceptable predictions of molecular geometries and estimates of rotational and inversional barrier heights is understandable. Nesbet pointed out some time ago that the correlation energy should be insensitive to changes in molecular conformation. Stanton 92 has argued, on the basis of the fact that a HartreeFock wavefunction satisfies the Hellmann-Feynman theorem, that a Hartree-Fock potential energy surface and the exact surface should be parallel over short distances. There are certain types of system for which this approximate parallelism between the Hartree-Fock and exact potential energy surfaces holds not only in the region of the equilibrium geometry, but over the total surface. In such cases, one can obtain a Hartree-Fock description of the complete chemical change, a description at a level of approximation, which, from the above examples, is clearly adequate for answering many if not most of the questions concerning the static behaviour of the system along the so-called reaction pathway. It is to a discussion of such systems that we now turn. 4 Hartree-Fock Potential Energy Surface Calculations Asymptotic Behaviour of Single Determinantal WavefunCtions.-In general, a single determinantal wavefunction, whether or not it is at the Hartree-Fock limit, does not provide an adequate description of a molecular system over the complete range of internuclear separations, because of the failure of such a function to describe go 91 g2

M. D. Newton, W. A. Lathan, W. J. Hehre, and J. A. Pople, J. Chem. Phys., 1970,32,4064; J. A. Pople, Accounts Chem. Res., 1970,3, 217. R. K. Nesbet, J. Chem. Phys., 1965,43, 530; Ado. Chem. Phys., 1965,9, 321. R. E. Stanton, J. Chem. Phys., 1962, 36, 1298.

Ab initio Calciilation of Potential Energy Surfaces

19

properly the possible dissociation products of the system. The classic example of this behaviour is provided by the H2 molecule in which case the energy of the single determinantal wavefunction corresponds to the mean of the energies of H+ H-(lS) and 2H(2S)rather than that of 2H(2S).32A corresponding situation always arises when one attempts to describe the formation of a single bond in a singlet molecular state from two doublet states, e.g. H2, lX; from 2H(2S) or F2(lZ;) from 2F(2P). The energy at the dissociated limit for such a wavefunction is always greater than that of the true dissociation products calculated properly at the same level of approximation. Thus, in general, a Hartree-Fock potential energy surface increases too steeply along a co-ordinate which leads to dissociation. This over-steep behaviour on the dissociativearm or portion of a Hartree-Fock potential surface is usually cited as the cause for the too-short bond lengths and too-large force constants usually obtained in the Hartree-Fock description of a system. There are, however, certain types of system which constitute exceptions to this incorrect asymptotic behaviour of Hartree-Fock wavefunctions. An important example is one in which the electronic structures of the dissociation products correspond to closed-shell systems, i.e. each spatial orbital (including spatially degenerate sets) is doubly occupied in their molecular orbital descriptions. In this case, the single determinantal wavefunction reduces properly to a product of wavefunctions which describe the dissociation products. The elementary example of this situation is provided by the dissociation of He2(1C,') into 2He(1S),32but the correct behaviour is also obtained in more complicated closed-shell systems. For example, in the reaction of CN-(lZ+) with CH3F(lA) to form CH3CN(lA) F-(lS), the reae tants, products, and the nature of the reactive complex along the reaction coordinate joining the two ends of the reaction are all properly described by a single determinantal wavefunction. One way of accounting for the correct asymptotic behaviour of the Hartree-Fock (or single determinantal) wavefunction in closed-shell systems is via the observation that in such systems the number of electron pairs remains constant. In those cases where dissociation leads to a decrease in the number of electron pairs, a single determinantal wavefunction is unable properly to describe the products. Thus, by the definition of correlation energy, as the difference between the true (non-relativistic) energy and the Hartree-Fock energy, one anticipates that major changes in

+

+

the correlation energy will occur only when the number of electron pairs predicted by the orbital model of the system changes during the course of the reaction. Therefore, reactions involving closed-shell reactants and products should be accompanied by only small changes in the correlation energy, and consequently, as first pointed out by Snyder,g3 HartreeFock energies should yield reasonable estimates when used to calculate the heats of reaction for such systems. Snyder and Baschg4have confirmed this prediction in an extensive series of calculations using an extended basis set. Pople and co-workers95 have since employed this concept of near conservation in the correlation energy to determine energies of reactions in which not only the number of electron pairs, but also the number of bonds of each particular type, is gS 84

95

L. C. Snyder, J. Chem. Phys., 1967,46, 3602. L. C. Snyder and H. Basch, J. Amer. Chem. SOC.,1969,91, 2189. R. Ditchfield, W. J. Hehre, J. A. Pople, and L. Radom, Chern. Phys. Letters, 1970, 5, 13; W. J. Hehre, R. Ditchfield, L. Radom, and J. A. Pople, J. Amer. Chem. SOC.,1970,92,4796.

Theoretical Chemistry

20

conserved. Mean absolute errors of ca. 13 kJ mol -l are found in these calculated energies of reaction using a rather small GTO basis set. The nuclear motions involved in internal rotation and inversion do not, in general, correspond to dissociative co-ordinates. Thus, one anticipates that the accompanyingchanges in correlation energy should be small. The ability of HartreeFock theory to provide reasonable estimates of the barriers to these motions as referred to previously, is thus understandable. In the case of diatomic systems, numerous calculations of the interaction potentials for two closed-shell atoms, e.g. He2, Ne2, A r 2 , Q 6 HeNe, ArNe, ArHe,97 or an atom and an ion, e.g. Li +Ne,Na +He,gs have demonstrated that the Hartree-Fock SCF method yields results which are accurate to within 0 . 1 4 . 5 eV over the total range of internuclear separations. Wahl 63 classifies the Hartree-Fock potential energy surfaces obtained for such systems as being ‘semi-quantitative’. The dispersion forces which operate between systems at large distances are a result of a small but significantcorrelativeinteraction of the motions of the electrons in the two systems. Thus, the polarization of the two systems resulting from this long-range correlative effect and the accompanying small change in energy are not accountedfor in the Hartree-Fock descriptionof the combined system. For example, the Hartree-Fock potential curves for He2, Ne2, and A r 2 , yield very good agreement with experimental data except for phenomena related directly to the van der Waals region of the curves.96The inability of the Hartree-Fock method to describe the long-range dispersive interactions is a relatively unimportant failing when one or more of the interacting systems is a polar or ionic species. In these instances, the shallow van der Wads minimum is eclipsed in importance by the much larger charge-dipole, or dipole-induced dipole interactions. In summary, the results in this and the preceding section illustrate that the LCAO-MO SCF approximation to the Hartree-Fock limit is a method which is capable of yielding semi-quantitative potential energy surfaces for closed-shell systems, with geometrical parameters characterizingminima in the surface given to within 1 or 2 %, with energy barriers given to ca. 4 kJ mol -l, and energies of reaction given to f.12 kJ mol-1 in favourable cases, and usually to within k 30 kJ mol -1. Early Examples of Surface Calculations for Closed-shell Systems.-The first LCAOMO SCF calculation of a reaction pathway on a potential surface for a reaction for the system between two closed-shell species was performed by ClementiQQ NHs HCl to form NH4CI. The calculation employed contracted sets of GTO’s; (I ls,7p)-+[5s,3p]for N, (6s,p)-+[3s,p] for H and ( 1 1 ~ ~ 7 ~[5s,4p] )for C1. Since the basis set does not contain any polarizing functions on the N and C1 centres, it will not yield results at the Hartree-Fock limit. For example, this basis set predicts a barrier height for the inversion motion in NHs which is almost twice as large as the experimental value, while as pointed out before, this same barrier is quantitatively accounted for at the Hartree-Fock limit. The HCl molecule was constrained to

+

98

y7 98

g9

T. L. Gilbert and A. C. Wahl, J. Chem. Phys., 1967, 47, 3425. 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. E. Clementi, J. Chem. Phys., 1967,46, 3851.

Ab initio Calculation of Potential Energy Surfaces

21

approach, H end first, the N of NH3 along the CsVsymmetry axis. Any accompanying relaxation of the N H 3 geometry during the approach of HC1 was not allowed for. NO potential barrier was found along this reaction pathway, the energy decreasing monotonically to yield the N H K l molecule, with a predicted stability relative to separated NH3 and HCl of ca. 80 kJ mol-1. It had been anticipated that the reaction pathway for a proton transfer reaction should be characterized by a double minima in the potential surface.100 The failure of Clementi’s calculation to predict a barrier along the reaction pathway for the transfer of H + from HCl to NH3 to form NHaCl -, must be accepted with some caution in view of the lack of polarizing functions on N and C1 in the basis set. This work, however, broke new ground and Clementi99 includes in his paper a detailed discussion of the merits and faults to be expected for the Hartree-Fock method in the different regions of the potential surface. In the year following Clementi’s work, Ritchie and King reported LCAO-MO SCF calculations of the reaction pathways for the abstraction of a proton by Hfrom H2 and HF,101 from H20,102 and from NH3 and CH4.103 Only in the case of H- reacting with H2, did they predict a barrier to reaction (of 63 kJ mol-l in the linear approach). Intermediates were predicted in the reaction of H - with HF, a linear HHF- stable by 33 kJ mol-1 relative to reactants and with HzO, an HOH‘Hspecies, with an HOH bond angle of 105” and a linear arrangement for OHH predicted to be stabilized by 42 kJ mol -l relative to the products OH- and H2. The basis sets used in these calculations were reasonably large sets of GTO’s but consisted only of s- and p-type functions on all centres, While no d-type functions were included on the C, N, and 0 centres, no large changes in the bond angles about these centres were involved along the reaction pathways. Thus, their prediction of the absence of an energy barrier in the abstraction of a proton from CH4, N H 3 , or H2O may be correct, but the absence of any polarizing functions on the heavy-atom centres leaves the question in some doubt. The (5s,3p) set of GTO’s used in the HH2 calculation should yield results reasonably close to the Hartree-Fock limit (p-type functions account for most of the polarization in the region of a proton) and, a barrier to reaction was predicted just for this case. Tests of Parallel Nature of HF and Correlated Surfaces for Closed-shell Systems.To illustrate the sufficiency of an s- and p-type basis set for the description of a potential surface for a reaction involving only hydrogen and helium centres and to further illustrate the ability of a near Hartree-Fock surface for closed-shell systems to parallel a surface with much of the correlation included, we consider the reaction of H2 with HeH+ to form He H:. The relevant portions of this surface have been calculated by Benson and McLaughlinlo4in the near Hartree-Fock approximation using a GTO basis set consisting of (9s,lp) on He and (4s,lp) on H. The minimum energy pathway was found to exhibit a continuous CzVsymmetry with HeH + approaching along the bisector of the H2 internuclear axis. The reaction was

+

+

R. S. Mulliken and W. B. Pearson, ‘Molecular Complexes; A Lecture and Reprint Volume’,

loo

John Wiley and Sons, New York, 1969. C. D. Ritchie and H. F. King, J. Amer. Chem. SOC., 1968, 90, 825. C. D. Ritchie and € F. I. King, J . Amer. Chem. SOC.,1968, 90, 833. lo3C. D. Ritchie and H. F. King, J. Amer. Chem. SOC.,1968, 90, 838. 104 M. J. Benson and D. R. McLaughlin, J. Chem. Phys., 1972,56, 1322. lol

2

22

Theoretical Chemistry

predicted to be exothermic by 2.63 eV and no energy barriers or local minima were found along the reaction pathway. Benson and McLaughlin104 also calculated the energy of the system beyond the Hartree-Fock limit for five nuclear configurations; for the separated reactants and products and for three intermediate geometries along the reaction pathway.* These extended calculations yielded ca. 80% of the total correlation energy (i.e. 80% of the error in the Hartree-Fock approximation) for both sets of separated twoelectron species. The correlation energy calculated at each of these five points on the surface differed by no more than 10 kJ mol-1 from point to point. Thus, the Hartree-Fock SCF and nearly exact potential surfaces in the regions of the minimum energy pathway probably parallel one another to within 10 kJ mol-l. This result would not obtain over the complete surface for HeH;, a portion of which would correspond to the extension and breaking of the bond in H2. Fortunately, motion along the minimum energy pathway involves only a slight extension in the internuclear separation of H2, where Re(calc.) = 1.4 a.u., to form the product Hi, where Re(calc.) = 1.6 a.u. The calculations performed beyond the Hartree Fock limit sample geometries with both extremes of the H-H separation on the surface and in the product. The gross extension of one bond in a closed-shell system without the concomitant shortening of another, or a radical change in the manner in which the charge density of one bonded pair is distributed, will result in a loss of the parallel nature of the Hartree-Fock surface and the correct one. This situation has been thoroughly investigated and nicely illustrated for the four-electron system LiHi by Kutzelnigg, Staemmler, and Hoheisel.105 The scattering of Li+ ions by H2 molecules has been extensively studied by molecular beam techniques.106 The energies used in the experimental scattering studies are sufficient to allow the H2 molecule to be vibrationally excited into its sixth quantum level. Since the classical turning points of this vibrational level in H2 are 0.9 and 3.0 ax., it was necessary to calculate the potential hyper-surface over a region which included this range of H-H internuclear separations. Kutzelnigg et aZ.105 performed calculations for the LiHi system by a method which accounts for most of the correlation energy, the so-calledIEPA PNO method.? The implementation of this method of calculating the correlation energy, however, requires as its starting point a conventional Hartree-Fock SCF calculation. Consequently, these authors provide a description of the LiH; potential surface at the Hartree-Fock SCF level and at the (nearly) fully correlated level. A basis set of Gaussian lobe functions was employed, whose size was sufEcient to yield energies very close to the Hartree-Fock limit for H2 and Li+. The potential surface calculated with correlation included is claimed to parallel the true surface to within a maximum overall error of less than 0.002 a.u. (ca. 5 kJ mol -l).

* t

The calculations beyond the Hartree-Fock approximation were configuration interaction calculations (as discussed in Section 5) which employed the configurations arising from all the single and double excitations from the single, ground-state dete~ninant.10~ The details of the IEPA PNO method are given in Section 5 which deals with calculations beyond the Hartree-Fock limit.

lo5 108

W. Kutzelnigg, V. Staemmler, and C. Hoheisel, Chern. Phys., 1973, 1, 27. J. Schottler and J. P. Toennier, 2.Physik, 1968,214, 472; Chern. Phys. Letters, 1972, 12, 615.

Ab initio Calculation of Potential Energy Surfaces

23

The LiHl potential energy surface was found to be characterized by the presence of an absolute energy minimum of 0.22 eV with respect to Li + H2 corresponding to an intermediate of CzVgeometry with an H-H separation r = 1.4 a x . and a Li to H-H bond midpoint separation R = 3.75 a.u. There is also a saddle point in the linear H-H-Li geometry for Y = 1.4 a.u. and R = 4.7 a.u. with a binding energy of 0.06 eV. The same LiH: surface has been investigated by Lester107,10* at close to the Hartree-Fock SCF limit. Lester’s calculations, however, were performed for only three values of the H-H separation r, 1.0. 1.4, and 2.0 a.u. Lester argued that the H2 correlation energy should be independent of R for a given r value, i.e. independent of the distance of the Li + ion from the midpoint of H2 for a given H-H separation. Kutzelnigg et ~71.105were able to demonstrate that this assumption, while nearly correct for small r, introduced errors for values of r greater than 2.0 a.u. Thus, in the linear approach of Li+ to H2 they found the correlation energy of the H2 electron pair for r = 3.0 a.u. to decrease from 0.0680 a.u. for R = 00 to 0.0437 a.u. for R = 2.0 a.u. This is explained by noting that in linear (LiHH)+, the bonded pair of Ha for close approach of Li+ is strongly polarized towards Li+ and the correlation energy approaches that of H-. For the CzV geometry, the correlation energy of the H2 electron pair changed by only 1.5 and 4.6 kJ mol-1 for R = a0 to R = 2.0 a.u. when r = 1.0 and 1.4 a.u., respectively. This same change in R values for r = 2.1 a.u. causes a change of 12.0 kJ mol-l in the H2 correlation energy, a value which increases to 30.2 kJ mol-1 for r = 3.0 a.u. Kutzelnigg et aZ.lo5also provided an analytical fit for their surface calculations suitable for trajectory calculations. White and Hayes lo9 have carried out quantum mechanical scattering studies on the Li+ + H2 system using a parametrized form of the Hartree-Fock SCF surface calculated by Lester.108 Gordon and Secrestl10 in calculating the potential surface for the He-H2 system also find that the near-Hartree-Fock SCF results parallel those obtained using correlated wavefunctions for those portions of the surface for which the H-H separation is held fixed at its equilibrium value. The portion of the surface for the HeHf-Hz system calculated by Benson and McLaughlin,lO* and referred to above, lies within the range of H-H separations found by Kutzelnigg et ~1.10~ to be adequately described at the Hartree-Fock limit. Their Hartree-Fock SCF results should, therefore, be accurate enough to warrant their use in classical trajectory calculations. Such calculations have indeed been performed by McLaughlin and Thompson.111 Spline interpolation functions were used to express the tabular near-HartreeFock energies in analytical form. The trajectory calculations for the reaction

+

HeH+

+

H2-+He

+ Hi +

2.6 eV

constrained to follow reaction pathways of CzVsymmetry, were performed for various values of the relative translation energies of approach and for internal vibrational W. A. Lester, J. Chem. Phys., 1970, 53, 1511. A. Lester, J. Chem. Phys., 1971, 54, 3171. log R. A. White and E. F. Hayes, J. Chem. Phys., 1972, 57, 2985. no M. D. Gordon and D. Secrest, J. Chern. Phys., 1970, 52, 120. 111 D. R. McLaughlin and D. L. Thompson, J. Chem. Phys., 1973,59,4393. lo7

lo8W.

24

Theoretical Chemistry

states of the reactants with quantum numbers 0 and 2. A detailed analysis of the partitioning of the available energy among the relative translational motion and the vibrational modes of the products is given, It is found that most of this energy ends up as vibrational energy in Hi, more than one-half of the collisions producing greater than 90 % vibrational energy conversion. HIi‘ Surfaces for A: + B.1nteractioms.There is another general class of diatomic AB systems for which the HartreeFock wavefunction for large internuclear separation passes properly into the product of the Hartree-Fock wavefunctions for A and B. This occurs when A possesses a closed shell and B has a half-filled orbital. Again, as Wahl has documented,63the Hartree-Fock SCF method yields ‘semi-quantitative’ potential curves for such systems. In the diatomic examples of such systems, no electron pair initially present is broken and no new one is formed, for example, Na(2S) He(%’) to form NaHe(2Z+) or He(4S’) + F(2P) to form HeF(211). While the same proper asymptotic behaviour of the Hartree-Fock wavefunction is also obtained in those cases where A and/or B are diatomic or polyatomic systems, e.g. H2 C1 to yield H + HCl, the Hartree-Fock method, in general, does not yield a ‘semi-quantitative’energy surface for such systems. In these cases the bonded pair and the unpaired electron exchange centres between reactants and products. The change in the correlation energy along the reaction pathway which is responsible for the loss of the parallel behaviour of the Hartree-Fock and correlated surfaces is frequently very large in the portion of the surface describing the transition state of the reaction. For example, in the reaction H + H2, the height of the barrier to reaction predicted at the single determinantal SCF level is approximately twice that obtained with a properly correlated function.112 The reaction of H2 + Cl to yield H HCl has been studied by Rothenberg and Schaeferll3 in a single determinant SCF calculation using a relatively large basis set of GTO’s with added polarizing functions. A linear reaction pathway is predicted, with a barrier height of 110 kJ mol -l and a saddle point geometry of R(H-H) = 1.78 a.u. and R(H-CI) = 2.76 ax., i.e. at a point where both the H2 and HCI equilibrium distances are expanded by ca. 0.37 a.u. The experimental Arrhenius activation energy is known for this system and is reported to be 23 kJ mol-1.114 While a calculated classical barrier height and the Arrhenius activation energy are not one and the same quantity,l12the calculated SCF barrier is obviously far too large. Another example of this type of behaviour is provided by the reaction H + F2-+HF + F for which a potential surface has been calculated by O’Neil and co-workers115 at the single-determinant SCF level of approximation and at a level which includes a significant fraction of the correlation energy. The transition state is again found to be linear with R(H-F) = 3.87 a.u. and R(F-F) = 2.97 a.u. These values correspond to more than a doubling of the H-F equilibrium separation and to an increase of 0.06 a.u. in the F-F equilibrium separation. This study is referred to in more detail later in this review. Here we wish to note that the height of the reaction barrier calculated for the above saddle point geometry was 51 kJmol-l at

+

+

+

112 113 114 115

I. Shavitt, R. M. Stevens, F. L. Miim, and M. Karplus, J. Chem. Phys., 1968,48, 2700. S. Rothenberg and H. F. Schaefer, Chem. Phys. Letters, 1971, 10, 565. P. G. Ashmore and J. Chanmugan, Trans. Faraday SOC.,1953,49, 254. S. V. O’Neil, P. K. Pearson, H. F. Schaefer and C . F. Bender, J. Chem. Phys., 1973, 58, 1126.

25

Ab initio Calculation of Potential Energy Surfaces

the SCF level and only 4.2 kJ mol-l in the calculation which included electron correlation. The experimental activation energy is 5.0 kJ mol -1. Edmiston et al.116 have calculated the potential energy surface for the linear He%+ system at the Hartree-Fock SCF level and beyond this approximation. It was found that both the SCF and the correlated wavefunctions for HeHH+ predicted a potential energy minimum ca. 20 kJ mol -1 deep for R(J3e-H) N 2.0 a.u. and R(H-H) = 2.0 a.u. The correlation energy changed by ca. 7.1 kJmol-l when R(He-H) changed from 7.0 to 2.0 a.u. with R(H--H) held fixed at 2.0 a.u. However, the correlation energy changes by ca. 30 kJ mol -1 along the pathway for the formation of the linear symmetrical intermediate HHeH +. The above results indicate that the Hartree-Fock SCF method is, in general, not a suitable one for calculating the potential surface of a system in which a bonded pair and an unshared electron are exchanged between reactants and products, i.e. reactions of the type

-4-B

+

C.

A.

+B-4

In general, a significant change in the correlationenergy occurs between the reactants and the transition state and this appears to be true whether or not there is a correspondingly large change in the correlation energy between reactants and products. Thus, the reaction of H-H with C1*is predicted to be endothermic by 13 kJ mol-1 and the near Hartree-Fock estimate is 10 kJmol-1.113 Still, the barrier height for the reaction was approximately three times too large in the single-determinant SCF calculation. In the F-F + H-reaction the experimental and SCF values for the exothermicity of the reaction are very different, 429 and 554 kJmol-l, respe~tively.~15 and here again the barrier height predicted by the SCF calculation is many times too large. The only known exception to this inadequate description at the SCF level for this type of system appears to be that obtained for linear H e m + . This is understandable as the only strongly correlated pair in this system is the ls2 pair on He and this pair is not broken but rather only polarized in the formation of the intermediate HeHH + from H i and He. HF Surfaces for s N 2 Reactions.-We now return to the review of the reactions which have been studied at the single-determinantal SCF level of approximation. The majority of these studies have concentrated on mapping out the minimum energy pathways between reactants and products. Such reaction pathways for a number of s N 2 displacement reactions have been determined. The systems which have been most thoroughIy investigated are the following:

H-

+ CH4

3

[H-CHS-HI-

+ CH3F + [F--cE-IrF]H- + CH3F + [H-CH2F-H]H - + CH3F + [H--CH3-F]CN- + CH3F mC--CH3-F]F-

3

116

+ HCH3F + F+ CH3F + H + CH4 + F--+ CH3CN + F -

+ 4

CH4

(A) (B)

(C)

@I (E)

C.Edmiston, J. Doolittle, K. Murphy, K. C . Tang, and W. Wilson, J. Chern. Phys., 1970, 52, 3419.

26

Theoretical Chemistry

Van der Lugt and R o s ~ ~have ‘ investigated the geometry of the transition state for reaction (A). Mulder and Wright 118 have reported an energy barrier of 289 kJ mol-1 for (A) using an STO double-zeta basis set. They found that the minimum energy geometry for C q is of D3h symmetry, the transition state geometry required to explain the Walden inversion which is characteristic of s N 2 reactions. Other geometries for CH; were found to lie at least 230 kJ mol-1 above that for the D3h species. (The same authorsllS however, found that various geometrical configurationsof CH; differed very little in energy.) Ritchie and Chappell 119 have determined the pathway for reaction (A) using a (Ss, 3p/3s) set of GTO’s. The most thorough and detailed investigations of these reactions have been given by Veillard and co-workers86s87~120for reactions (A)-*@) and by Duke and Bader 88 for reactions (B) and (E). Both sets of workers used large GTO basis sets with polarizingp-type functions on the hydrogens and d-type functions on the carbon, nitrogen, and fluorine. For example, Duke and Bader used a basis set of above double-zeta quality to perform the geometry searches. (As mentioned before, geometrical parameters are better predicted employing a basis set short of that required for the Hartree-Fock limit.) The final energy calculations employed Huzinaga’s121 6s set for hydrogen contracted to three groups of two plus a set of p-type functions and Dunning’s (lOs, 6p) set for the first-row atoms contracted to [5s,3p]. Because of the diffuse nature of the charge distribution on CN- and F-, the basis sets for these species were further augmented by (optimized) diffuse s- and p-functions. Finally, a full set of d-functions were added to the basis sets for all first-row atoms. Without polarizing functions, both Veillard et aZ.86 and Duke and Baderas predict an intermediate rather than a transition state for reaction @). With polarizing functions, Veillard et al. obtain a reaction barrier of 33 kJ mol-1 for reaction (B) while with their slightly larger set, Duke and Bader obtain a value of 31 kJ mol-l. Both sets of authors, through thorough geometry searches, find the linear [FCH3F] complex of Dan symmetry to be the point of highest energy on the minimum energy path. Duke and Bader 88 repeated the determination of this energy barrier using two complete sets of polarizing d-type functions on all the first-row centres. Only a small change (- 0.8 kJ mol-1) in the activation energy was obtained, indicating that the basis sets containing the singte d functions were yielding results close to the s, p, d-type saturated description of the binding. Energy barriers were found for all five of the above reactions. Values of ca. 250 kJ mol-1 were calculated for reactions (A) and (C), 92 kJ mol -l for reaction (E), 29-33 kJ mol -l for reaction (B) and 17 kJ mol -l for reaction (D). Bohme et aLS9have experimentally measured the activation energies for a number of sN2 reactions in the gas phase using a flowing afterglow system.122They found the reactions of H -,NH,, OH -,F -, and C2H - with CH3C1 to proceed rapidly in the apparent absence of any significant activation barrier. In contrast, the reaction of C N - with CH3CI was surprisingly slow with an activation energy in excess of 21 117

118 119 l20

121

122

W, Th. A. M. Van der Lugt and P. Ros, Chem. Phys. Letters, 1969, 4, 389. J. J. C.Mulder and J. S. Wright, Chem. Phys. Letters, 1969, 5, 445. C. D. Ritchie and G. A. Chappell, J. Amer. Chem. SOC.,1970, 92, 1819. A. Dedieu and A. Veillard, J. Amer. Chem. SOC.,1972, 94, 6730. S. Huzinaga, J. Chem. Phys., 1965, 42, 1293. D. K. Bohme, R. S. Hemsworth, H. W.Rundle, and H. I. Schiff, J. Chem. Phys., 1973, 58,

3504.

Ab initio Calculation of Potential Energy Surfaces

27

kJ mol-l. Also surprising were the observations that the reactions of €3-, NH,, OH-, C2H-,and CN- with CHsF were observed to be at least 100 times slower than the corresponding reactions with CH3Cl. For reaction @) above, Bohme et al. measured an activation energy of 15 kJ mol-1 compared to the value of 16 kJ mol -l calculated by Dedieu and Veillard.120 This experimental verification of the existence and even, in one case, the magnitude of activation barriers to gas phase s N 2 displacementson CHSFis very encouraging. While the reactions are all closed-shell systems with overall retention in the number of electron pairs, the transition states do represent a significantly different arrangement of two electron pairs. In gross terms, one lone pair of the nucleophile and the pair binding the leaving group in the substrate are both transformed into extended bonded pairs in the transition state. The experimental agreement with the theoretical barrier for reaction (D) suggests that no significant change in the correlation energy is encountered in the formation of the [I-I-CH3-F]- transition state. The reader is asked to recall that the magnitudes of simple inversion barriers are well accounted for in the Hartree-Fock method. Dedieu and Veillard120 have, in fact, reported a limited calculation of the correlation corrections to the energies of the reactants and transition state for reaction (B). The contributions of the correlation energy to the reactants and transition state differed by Zess than 4 kJ mol -l. However, Dyczmons and Kutzelnigg123 have also performed SCF calculations and calculations of part of the correlation energy for CH4, H - and the CH; transition state of Dan symmetry. They found an SCF activation barrier of 293 kJ mol-1 for reaction (A) (compared with ca. 250 kJ mol-1 obtained by Veillard et al.120),a value which is reduced to 230 kJ mol -1 with the inclusion of correlation into the wavefunctions. It is clear, however, that neither set of authors has approached the Hartree-Fock limit in their single-determinant SCF calculations. Thus, Dyczmons and Kutzelnigg obtain a somewhat better energy than Veillard et al. for CH4 + H-, but a somewhat poorer one for CH,. A striking feature of all five of these reactions is the delayed departure of the leaving group. Dedieu and Veillard120 have given a detailed discussion of the energetics and changes in charge density along the reaction pathways for reactions (At-@), including comparisons with the empirical rules for s N 2 reactions of Hammond,l2* Th0rnton,l2~ and Bunton.126 Bader, Duke, and Messer127 have presented an analysis of the changes in the energy and charge density for reactions (B) and (E) using the virial partitioning procedure. This is a partitioning method determined by the topological properties of the charge distribution, which yields a spatial partitioning of the charge and energy for a molecular system. The kinetic and potential energies of the molecular fragments obtained in this method obey the quantum mechanical virial and hypervirial (for the operator r- V) theorems.128 The fragment properties and changes in the fragment properties thus obtained possess direct physical significance. 123

lZ4 la5

l27 128

V. Dyczmons and W. Kutzelnigg, Theor. Chim. Acta, 1974,33, 239. G. S. Hammond, J. Amer. Chem. SOC.,1955, 77, 334. E. R. Thornton, J. Amer. Chem. SOC.,1967, 89, 2915. C. A. Bunton, ‘Reaction Mechanisms in Organic Chemistry’, ed. J. H. a d d , Elsevier, Amsterdam, 1963, vol. 1. R. F. W. Bader, A. J. Duke, and R. R. Messer, J. Amer. Chem. Soe., 1973, 95, 7715. S. Srebrenik and R. F. W. Bader, J. Chem. Phys., 1974, 61,2536.

28

%oretical

Chemistry

Other Closed-shell HF Results.-Biirgi, Lehn, and Wipff 12Q have calculated the minimum energy pathway for the addition of H - to formaldehyde using an s-p GTO basis set. The system was chosen as a simple model for the nucleophilic addihave determined the reaction pathtion to a carbonyl group. Schaefer et ~21.~309131 way for the isomerization of CH3NC to CH3CN using a GTO basis set of doublezeta quality. This work was prompted by the prediction of Harris and Bunker132 that the isomerization of CH3NC may not follow RRKM theory of unimolecular reactions. Of particular importance to this argument was the nature of the barrier to the internal rotation of the methyl group as a function of the angle governing the extent of the swing of the isocyano group from the configuration -NC to -CN. In addition, Dewar and Kohn,I33 on the basis of semi-empirical MIND0 calculations, had predicted the existence of a stable triangular n-complexed intermediate along the reaction co-ordinate. Schaefer et aZ. have determined the potential energy surface as a function of the methyl rotational angle as well as a function of the swing angle of the isocyano group. They did not find a minimum in the potential path corresponding to the intermediate predicted by Dewar and Kohn. Yarkony et aL30 have calculated the potential surface for the interaction of two rigid HF molecules in an SCF calculation using a GTO basis set equivalent to double-zeta plus polarizing functions. The surface is to serve as the potentialfunction governing the pure rotational excitation and relaxation of H F gas. Because of the significant dipole-dipole interaction which leads to the formation of hydrogen bonded dimers in hydrogen fluoride, the inability of the Hartree-Fock method to account for dispersion forces is of only minor importance in these calculations. There is now a considerable body of evidence which suggests that the single-configuration SCF theory provides a qualitatively correct description of hydrogen bonding.134 A complete tabulation of their final energies for this HF-HF surface is presented. The authors experienced considerable difficulty in their attempt to cast their results into a convenient analytical form. Bader, Novaro, and Beltr6n-LopCz 135 have calculated the potential surface for various geometries of three and four helium atoms near to the Hartree-Fock limit to determine the deviation of the HartreeFock energies of interaction from pair-wise additivity. L i ~ c h k a has l ~ ~performed SCF calculations, using an s-p GTO basis set with polarizing functions, of the interaction potentials for the following pairs of systems; He-HF, H e H 2 0 , H2-HF, and H2-H20 all at large and intermediate separations. A calculation of the energy arising from the cross correlation of the electrons between the two systems was also performed. For these non-polar-polar molecule interactions at large distances, the SCF energies and the correlation corrections were found to be of almost equal magnitude in determining the potential energy minima. 129 130 131

l33

134 135 136

H.B. Biirgi, J. M. Lehn, and G. Wipff, J. Amer. Chem. SOC.,1974, 96, 1956. D. H.Liskow, C. F. Bender, and H. F. Schaefer, J. Amer. Chem. SOC.,1972, 94, 5178. D. H. Liskow, C. F. Bender, and H. F. Schaefer, J. Chem. Phys., 1972, 57,4509. H. H. Harris and D. L. Bunker, Chem. Phys. Letters, 1971, 11, 433; D. L. Bunker, J. Chem. Phys., 1972, 57, 332. M.J. S. Dewar and M. C. Kohn, J. Am. Chem. SOC.,1972,94,2705. P. A. Kollman and L. C. Allen, Chem. Rev., 1972,72,283. R. F. W. Bader, 0. A. Novaro, and V. BeltrAn-Lopez, Chem. Phys. Letters, 1971, 8, 568; 0. A. Novaro, and V. Beltrb-LopCz,J. Chem. Phys., 1972,56, 815. H.Lischka, Chem. Phys., 1973, 2, 191.

Ab initio Calculation of Potential Energy Surfaces

29

Correctionsfor ImproperHF AsymptoticBehaviour.-There are two techniques which m a y be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single deteminantal approximation goes to the incorrect asymptotic limit. These techniques are: (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one);137 and (ii) to employ a single detenninantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3Pand 1D states with a hydrogen molecule,138and we illustrate them through a discussion of this work. In general,63the correlation error in the Hartree-Fock method may be associated with two distinct causes: (a) the increase in correlation energy caused by forming new electron pairs and bringing more electrons into closer proximity with one another; and (b) the inability (in some cases) to describe correctly the dissociation of the molecule into fragments. As discussed in the next section, the correlation error m a y be eliminated by expanding the wavefunction in some complete set of determinantal wavefunctions, each of which represents a different arrangement or configuration of the electrons among the set of spin orbitals for the system. Hence the method is called configuration interaction. The interesting point for the present discussion is that in general the configurations which must be added to correct for the two possible deficiencies of the Hartree-Fock method, (a) and (6) above, are usually quite distinct. Thus, one may add some minimum number of configurations to obtain the correct Hartree-Fock description of the dissociation products. The contributions of such configurationsare, however, generally not important for those ~ ~ internuclear separations requiring close separations where bonding O C C U T S , ~ i.e. the type of correlation in (a) above. For such regions, one must add additional configurations which exist for the molecule, but which vanish formally in the atoms. They are termed molecular configurations and their contribution to the energy lowering of the system is called extra-molecular correlation energy.63 There is a long and continuing interest in the differences which exist between the chemistry of the singlet and triplet forms of one reactant, e.g. O(P,lD); S(3P,1D,1S), and C H Z ( ~ B ~ , ~In A~ particular ). is the differing abilities of the singlet and triplet forms to undergo insertion and abstraction reactions. To gain an understanding of these systems, potential surfaces were determined for the model system of O(3P,1D) with Ha(lX;) to yield HzO(3B1,lAl) on insertion or OH(2II) H(2S)on abstraction. In the study of the insertion of O(1D)into H2(lZ;) to yield ground state HiO(lA1) it is possible to augment the Hartree-Fock single determinantal wavefunction for this system with just two determinantal wavefunctions such that one obtains the correct limiting Hartree-Fock description of O(1D) and H2(1Xl). At the same time, however, these two configurations make a negligible contribution to the description of the water molecule in the region of its equilibrium geometry. Thus, one has what

+

137 188

180

A. Pullman and G. N. J. Port, Theor. Chim. Acta, 1973, 32, 77. R. F. W. Bader and R. A. Gangi, Chern. Phys. Letters, 1970, 6, 312; R. F. W. Bader and R. A. Gangi, J. Amer. Chem. SOC.,1971, 93, 1831. E. R. Davidson, Rev. Mod. Phys., 1972,44,451.

Theoretical Chemistry

30

is an essentially Hartree-Fock description of the total dissociation process of HzO(lA1) into O(1D) + Hz(lZi). The two added configurations become increasingly important as the separation of the products increases and provides a smooth and continuous correlation of the energy surface with that of the Hartree-Fock energies of the products. One must avoid the indiscriminate mixing of additional configurations in configuration interaction calculations, if one is to avoid spurious results because of an 'unbalanced' ~ e t . Thus, ~ 3 it is best either to add the minimum number of configurations to just correct the asymptotic difficulties, or to add a very large number of configurations using selection criteria, as discussed below. The ground-state electronic configuration of H2O is given by (lai)2(~i)2(lbz)2(3a1)2(lbi)2 'A1, @ I

In a minimal basis set description, la1 corresponds to the Is on oxygen, 2a1 to 2s on oxygen plus some mixing in of H' 1s orbitals, 162 is the 2py on oxygen and its overlap with the H 1s orbitals, 3al is primarily 2pz on oxygen plus some H 1s orbitals and lbl is the 2pz orbital, a lone pair on oxygen perpendicular to the molecular plane. In the neighbourhood of the equilibrium geometry, the first virtual orbital is 4a1, a diffuse s-like (3s) Rydberg orbital centred on oxygen. At the limit of separation of H2 from O(lD), the 3al and 1b2 orbitals interchange their energy ordering and 3a1 passes smoothly into the 1offorbital of HZand lbz becomes a pure 2pu orbital on oxygen. At this same limit, the 4al orbital becomes the 2pz orbital on oxygen. The orbital description of one component of the open-shell H2O system at the H2(lX;) O ( l 0 ) limit is given by

+

where @pz and

Ip3

are determinantal wavefunctions derived from the configurations (la1)z(2a1)2(3a1>2(1b2)2(4a1)2lAi, @ 2

(1 a1)2(2a1)2(3a1)2( 1b ~ ) ~ ( k'iA)1,~@ 3

All three component wavefunctions involve double occupation of the 3ai (= 1og of H2) orbital. The description of O(1D) is given by the linear combination Pwith the 3a1 orbital omitted. Hence, Y as written above corresponds to an (antisymmetrized) product of the HartreeFock wavefunctions for H2 and O(l0). It is generally found that if one increases the flexibility of a single-determinantal wavefunction by allowing each space orbital to assume an independent form (rather than insisting on double occupation by an ~1 and a electron for those orbitals which would otherwise be so occupied as dictated by the electronic configuration) that the asymptotic difficulties of the wavefunction are removed. Thus, the unrestricted Hartree-Fock method usually predicts the correct dissociation products of a molecular system.140 The symmetrical (CZ,)insertion of O(V) into H2 yields the 3B1 state of the HzO system. The electronic configuration of this state expressed in terms of the unrestricted set of orbitals is

A. W.Salotto and L. Burnelle, Chem. Phys. Letters, 1969, 3, 80.

31

Ab initio Calculation of Potential Energy Surfaces

where the bar denotes a @ spin function and its absence an oc spin function. The asymptotic forms of the orbitals are the same as the restricted counterparts for the lA1 state and the single determinantal wavefunction composed of the unrestricted orbitals correctly yields a product wavefunction for ground state Ha and O(3P). The singlet insertion reaction was determined using the limited configuration interaction treatment outlined above.l3* The singlet abstraction reaction O(1D)

+ H2('2;)

+

OHH(2U)

OH(2II)

+ H(2S)

and the triplet insertion and abstraction surfaces were calculated using the unrestricted Hartree-Fock method. The surfaces were first mapped out using a (5s3p/2s)STO basis set for the singlets and a (6s3pl2s) set for the triplets, the extra (low exponent) s function in the triplet case being required to describe the very diffuse nature of the triplet charge distribution. Calculations were then performed for selected points along the minimum energy paths using a large basis set which consisted of (lOs,5p,2d/4s,lp)+ [5s,3p,2d/2s,lp],for the singlets and the same set but with 11s functions contracted to 6s functions for the triplet surfaces. The minimum energy pathways for the singlet and triplet insertion reactions with geometries determined by the small basis set but with energies determined at the corresponding points using the large basis sets are shown in Figure 1 . After a small initial increase in energy at large separations, the symmetrical insertion of O(1D)

-75-7

1 1 (2.76) .

-75.8

--

(3.43)

(1.55) .

J

-*

~

----- - - - - __ _ _

/

(1.41) i-

O('m H,+(IZg+)

. 0(3P) + Hd'&

Figure 1 The minimum energy paths for the singlet and triplet symmetrical insertion reactions to produce the IAI, 3 B ~and , IBI states of HaQ. The bracketed values are the optimum H-H internuclear separations for each value of ro, the distance of the oxygen from the midpoint of H-H axis (Reproduced by permission from J. Chem. Phys., 1971,55,5369)

32

Theoretical Chemistry

into H2(lZi) is an energy releasing process to yield ground-state H20(1A1). The continued approach of oxygen along the Cz2)co-ordinate yields the linear form of ground-state water, which is predicted to lie 138 kJ mol -l above that of the equilibrium geometry. Further calculationsl41indicate that the CzUapproach is the minimum energy one until the oxygen is ca. 3 a.u. from H2. When the oxygen is in the region between 3 and 2 a.u. from the midpoint of the H2 molecule, the potential surface assumes a double minimum character with respect to motion of the oxygen off the C Zsymmetry axis. In this region of the surface, the minimum energy approach corresponds to the oxygen lying ca. 30" off the bisector. For distances less than 2 a.u. the preferred approach of oxygen is again along the C2 axis, The symmetrical insertion of O(3P) into H2 results in a barrier of approximately 335 kJ mol-1 and yields a linear metastable intermediate. For distances of approach of O(3P)to H2 less than 3.0 a.u., the triplet insertion path is unstable to an unsymmetrical motion of the nuclei which leads to OH(211) and H(2S). The energy minimum on the symmetric triplet path is therefore, a saddle point and does not represent a stable nuclear configuration for triplet water. The linear abstraction reaction possesses an energy of activation of 103 kJ mol-l.13* However, at an acute angle of approach of O(lD) towards Hz, the activation energy for the formation of OH(2II) and H(2S) decreases to 70 kJmol-1. This energy is small enough to account for the formation of OH radicals which are also found as products in O(1D) insertion reactions. The linear triplet abstraction reaction requires an activation barrier of 147 kJ mol-l. All points on the triplet surface were found to be connected by downhill paths to either the reactants or H(2S) OH(2II). One can summarize the chemistry contained in these potential surfaces by stating that singlet oxygen inserts by choice and abstracts by chance, while triplet oxygen abstracts by necessity. The lB1 surface of the H2O system has also been calculated and a general discussion of the correlation of the states produced when singlet and triplet oxygen are added to H2(1Zi) has been given.141 From these latter considerations, the existence of a conical intersection, of the type first postulated by Teller,142was predicted to occur as a result of the crossing of the 1 C l and lIT surfaces which result when the ground and first excited 1A1 states assume linear geometries. This conical intersection yields a pathway for the dissociation of ground-state water into H(2S)and OH(2Z f) as opposed to OH(211). Also, if the B ~ A state I of water is linear in its most stable configuration, as now seems certain,l43$144 and if there is an energy barrier to dissociation from this state when the molecule is bent, but not when it is linear, then one would expect that dissociation into OH(211) H(2S) would occur when the molecule is excited into the BIA1 state. Flouquet and Horsley145 have recently published a study of the potential energy surface of the B1A1 surface of H2O in the SCF approximation and with some limited configuration interaction. They found the form of the surface for this state

+

+

R. A. Gang and R. F. W. Bader, J. Chem. Phys., 1971,55, 5369. E. Teller, J. Phys. Chem., 1937, 41, 109. 143J. A. Horsley and W. H. Fink, J. Chem. Phys., 1969,50, 750. 144 J. W. Moskowitz and M. C. Harrison, J. Chem. Phys., 1965,43,3550; C. R. Claydon, G. A. Segal, and H. S. Taylor, ibid., 1971, 54, 3799. 146 F. Flouquet and J. A. Horsley, J. Chem. Phys., 1974, 60, 3767. 141 141

Ab initio Calculation of Potential Energy Surfaces

33

to be dominated by the conical intersection with the ground-state surface in the linear geometry, as postulated above. The dissociation of H2O into the BlAl state to give the OH(2E+) fragment with a highly abnormal rotational distribution is discussed in the light of the calculated surface. Murrell, Pedley, and Durmazl46 in a study of the potential surface for the symmetrical (C2,) insertion of singlet and triplet CH2 into H2(1C,'), obtain results similar in their principal features to those shown in Figure 1 for the corresponding insertion of O(l0) and O(3P) into H2. Methylene and the oxygen atom are isoelectronic and the three lowest states of linear (or bent) CH2 correlate in terms of symmetry and in terms of the states produced in reaction with H2 on C2, approach with those of the O(3P,lD) H2(lCi) system. Thus, the ground state of CH2 is 3C,(3B1) which gives a 3B1 surface on combination with H2, corresponding to the lowest 3B1surface of H20 from O(3P).The first excited singletstateis 1A&A1 + 1B1) and as in the reaction of O(1D) with H2, gives rise to lA1 and 1B1 states for the adduct with H2. Their calculations were performed using a minimum set of STO's without configuration interaction or the use of the unrestricted Hartree-Fock method. Consequently, their computed surfaces and reaction paths are not continuous from reactants to products. Their overall results, however, parallel those found for the H2O system. (Compare their Figure 7 with Figure 1 here.) The insertion of the singlet form of CR2 into H ' 2 to form ground state CH4 is found to be energetically favourable with little or no activation barrier when the system is allowed to distort from CzVsymmetry. The insertion of triplet methylene into Ha requires passage of the system over a large energy barrier and into a small well to form the triplet state of CH4, the most stable geometry of which is found to have a square-planar structure. This behaviour is again analogous to the O(3P) H2(lc,') CzVinsertion path. Baskin et aZ.147 have calculated the reaction pathway for the abstraction of H from H2 by methylene at both the SCF level and at a level which yielded a considerable fraction of the correlation energy. A doublezeta basis set of contracted GTO's was employed in the calculation. The surface was calculated as a function of three geometrical parameters: R,the distance of the carbon atom from the closer of the two hydrogen atoms of H2; r, the H-H separation of H2 and 8, the methylene bond angle. The study was restricted to an end-on approach of H-H along the CzVaxis of CH2. The reaction exothermicity was predicted to be 22.5 kJ mol -1, compared to the experimental value148 of 19 kJ mol-l. The saddle point was found to OCCUT at R = 2.640 a.u., r = 1.702 a.u., and 6 = 126.5", a geometry intermediate between that of reactants and products. An activation barrier of 65 kJ mol-l was calculated for this abstraction, a value which is consistent with the observation that no abstraction of H from H2 by CH2 could be detected at 300 K.149 The geometry determined at the SCF level of approximation was very similar to that obtained with the correlated wavefunction, but the activation barrier predicted by the SCF procedure was much higher, 105 kJ mol -l. The authors concluded their

+

+

N. Murrell, J. B. Pedley, and S. Durmaz, J.C.S. Faraday ZI, 1973, 69, 1370. C. P. Baskin, C. F. Bender, C. W. Bauschlicher, jun., and H. F. Schaefer, J. Amer. Chem. SOC., 1974,96,2709. R. W. Carr, J. Phys. Chem., 1972,76, 1581. W. Braun, A. M. Bass, and M. Pilling, J. Chem. Pltys., 1970, 52, 5131.

1413 J. 147 148 149

34

Theoretical Chemistry

paper with a discussion of possible definitions of ‘reaction pathways’ or ‘reaction co-ordinates’. Guest, Murrell, and Pedley150have used a minimum basis set of STO’s in an SCF study of the potential surface of CH,+and its formation from H2 and CHg. The approach of H2 to CH$ was found to proceed with no barrier to activation to yield CH; whose most stable geometry was found to be of Cssymmetry. Dixon,l51 using a GTO basis set of double-zeta quality has determined a potential surface for CH; to study the Jahn-Teller distortions which occur in the 2T2 state of tetrahedral geometry. The most stablizing distortion is found to be a tetragonal one, leading to a geometry of D2a symmetry. This leads to an energy lowering of 1.41 eV as compared with the next most favourable motion which yields a species of CsV symmetry stablized by 1.22 eV. Vibronic perturbation parameters are derived from this surface and employed in a discussion of the 2Tz_1A~ photoelectron transition. Single determinant unrestricted Hartree-Fock calculations have been carried out for the H CH4 system in a study of the reaction pathways for hydrogen exchange and abstraction reactions.152 The same technique has been used to determine the is predicted to have a pyramidal possible dissociative paths of H ~ 0 . This l ~ radical ~ equilibrium geometry with a low inversion barrier of ca. 8 kJ mol-l. The barrier to dissociation into H20(lA1) + H(2S) is found to be 28 kJ mol-1. The NH4 system, again studied by the unrestricted Hartree-Fock method, was found to have a tetrahedral equilibrium geometry, a 2A1 electronic state, and exist in a potential well isolated from the products NH3(lA1) + H(2S) by a minimum barrier (for a C3e dissociative pathway) of ca. 92 kJ mol-1.154 The work of Ehrenson and Newton152has shown that the most stable geometry of CH5, a structure of D3h symmetry, corresponds to a saddle point on the reaction pathway for hydrogen exchange. Thus, of the three radicals, A H n +I, CH5 is unstable, OH3 possesses only a slight stability, but NH4 is predicted to require at least 84 kJ mol-1 for its decomposition and should be an isolatable species. €IFDescription of Addition to Double Bonds.-Another interesting aspect of the chemistry of the singlet and triplet species of 0, S, and CH2 concerns the stereospecificity or its absence in their addition to substituted ethylenes. Strausz et aZ.155 have used as s-p GTO basis set to study the addition of S(3P) and S(l0) to ethylene in an SCF calculation. The addition of S(3P) to an ethylene is a stereospecific one in contradistinction to the additions of O(3P)or C H Z ( ~ B ~Strausz ). et al. find that the lower triplets of the addition product of S(3P) and ethylene, thiirane, all possess a ring-opened geometry in which the terminal methylene is in a plane perpendicular to the CCS plane and that a considerable energy barrier exists with respect to rotation of the terminal methylene group. This description of the adduct nicely accounts for the stereospecificity of the addition reaction with S(3P). In the addition of O(3P) to ethylene, a ring-opened geometry is again ~ r e d i c t e d . 1However, ~~ the barrier to

+

150 151 152 153 154

155 156

M. F. Guest, J. N. Murrell, and J. B. Pedley, Mol. Phys., 1971, 20, 81. R. N. Dixon, Mof. Phys., 1971, 20, 113. S. Ehrenson and M. D. Newton, Chem. Phys. Letters, 1972, 13, 24. R. A. Gangi and R. F. W. Bader, Chem. Phys. Letters, 1971, 11, 216. J. Pelletier and R. F. W. Bader, unpublished results. 0. P. Strausz, H. E. Gunning, A. S. Denes, and I. G. Csizmadia, J. Amer. Chem. Soc., 1972, 94, 8317. M. E. Stephens, R. A. Gangi, and R. F. W. Bader, unpublished results.

Ab initio Calculation of Potential Energy Surfaces

35

the rotation of the end methylene group is, in this instance, calculated to be only ca. 17 kJ mol -l. Thus, theory predicts a non-stereospecific reaction in the addition of O(3P)to substituted olefins as is indeed found to be true experimentally. Haines and C~izmadia15~ have investigated possible modes of addition of NH to ethylene to form aziridine in an SCF calculation using an s-p GTO basis set. Aziridine is predicted to be formed in its first excited triplet state, a state which again corresponds to a ring-opened intermediate. Shih et d 1 5 * have performed SCF and SCF plus configuration interaction calculations of the reaction surface for the addition of NH2 to ethylene. They found a high energy of activation (150-170 kJmol-l) for this addition, which again is predicted to yield a ring-opened intermediate. Electrocyclic Reactions.-Peyerimhoff, Buenker, and co-workers have carried out very detailed studies of the electrocyclictransformations159between cyclic and openchain hydrocarbons. The calculations employ a large GTO set of s-and p-type basis functions. In every study, the necessity of including limited configuration interaction was carefullyinvestigated. The prototype electrocyclictransformation of cyclobutene to cis-butadiene via the thermochemical process has been studied in detail.l6O9161 The same authors also give an analysis of the qualitative theories for such reactions based on their ab initio calculations.16a A similar study of the electrocyclic transformations of cyclopropyl and alIyl systems has also been made.163 Horsley et al.164 have determined a reaction pathway for the geometrical isomerization of cyclopropane by a calculation and search of the full 21-dimensional hypersurface! The calculations were performed using a minimal basis set of STO’s in a modified SCF calculation using the restricted Hartree-Fock formalism suggested by Nesbet.165 In the region of the trimethylene diradical, a 3 x 3 configuration interaction calculation was included. Ion Solvatiom-Clementi and co-workers have embarked on an ambitious study of the hydration of simple ions and the structure of liquid water and ionic solutions. The studies are primarily at the SCF single determinantal level using relatively large basis sets of GTO’s with added polarizing functions. Potential energy surfaces have been determined for a water molecule in the fields of a Li + ion,f66 Na + and K ions 167 and F- and Cl- ions.168 The geometry searches in all of these studies were wide ranged and very detailed. In two further publicationsl69g.170a very detailed Hartree-Fock computation of the potential surface for the water dimer (with rigid +

I57 158

W. J. Haines and I. G. Csizmadia, Theor. Chim. Acta, 1973, 31, 283. Shingkuo Shih, R. J. Buenker, S. D. Peyerimhoff, and C. J. Michejda, J. Amer. Chem. SOC., 1972,94,7620.

R. B. Woodward and R. Hoffmann, J. Amer. Chem. SOC.,1965, 87, 395. 160 K. HSU,R. J. Buenker, and S. D. Peyerimhoff, J. Amer. Chem. SOC.,1971,93, 2117. 161 K. HSU,R. J. Buenker, and S. D. Peyerimhoff, J. Amer. Chem. SOC.,1972,94, 5639. l a a R. J. Buenker, S. D. Peyerimhoff, and K. Hsu, J. Amer. Chem. Soc., 1971, 93, 5005. 163 P. Merlet, S. D. Peyerimhoff, R. J. Buenker, and S. Shih, J. Amer. Chem. SOC.,1974,96, 959. 164 J. A. Horsley, Y.Jean, C. Moser, L. Salem, R. M. Stevens, and J. S. Wright, J . Amer. Chem. SOC.,1972, 94, 279. 165

166 167

16* 169

R. K. Nesbet, Rev. Mod. Phys., 1963, 35, 552. E. Clementi and H. Popkie, J. Chem. Phys., 1972, 57, 1077. H. Kistenmacher, H. Popkie, and E. Clementi, J. Chem. Phys., 1973, 58, 1689. H. Kistenmacher, H. Popkie, and E. Clementi, J. Chem. Phys., 1973, 58, 5627. H. Popkie, H. Kistenmacher, and E. Clementi, J. Chem. Phys., 1973, 59, 1325. H. Kistenmacher, H. Popkie, E. Clementi, and R. 0. Watts, J. Chem. Phys., 1974, 60, 4455.

36

Theoretical Chemistry

internal geometries) was performed and subsequently used in a Monte Carlo simulation of the liquid state of water. This was done by fitting an analytical expression to the computed interaction potential and then using this analytical expression as an effective pair potential for the description of liquid water. The Monte Carlo calculations were performed with the SCF potentia1169.170 and with the SCF potential modified by a semi-empirical correction for the correlation energy. The Hartree-Fock effective pair potential gave a realistic representation of the structure of liquid water but predicted poor thermodynamic properties. The discrepancies were reduced with the empirically correlated pair potential. Schuster and Preuss 171 have also carried out calculations, near to the Hartree-Fock limit for the systems Li +-H20 and Be2+-H20. Pullman and co-workers172.173 have initiated a study of the hydration of molecules of biochemical interest at the SCF level of approximation. The hydration of formamide by a single water molecule has been studied at the SCF level using a (7s,3p/3s) set of GTO’s contracted to one of minimal size.174 Various possible configurations of the H20-NH2CHO complex are considered - hydrogen bonding with the carbonyl group, to the amino group, in the C-H region and finally the possibility of n hydrogen bonding. The relative conformations of the H2O and formamide molecules were also carefully studied for each of these configurations. A similar study on the hydration of methyl and ethyl ammonium ions has also been carried out 175 using a minimal STO basis set, with each STO approximated by three GTO functions.l76 Alkylammonium groups occur frequently in pharmacologically active molecules and changes in the interaction of such groups with water on modification of the alkyl groups is thought to account for the accompanying changes in the drug’s activity.

5 Beyond Hartr-Fock The motion of each electron in the HartreeFock approximation is solved for in the presence of the average potential of all the remaining electrons in the system. Because of this, the Hartree-Fock approximation, as discussed earlier, does not provide an adequate description of the repulsion between pairs of electrons. If the electrons have parallel spin, they are effectively kept apart in the Hartree-Fock method by the antisymmetric nature of the wavefunction, producing what is commonly known as the Fermi hole. Electrons of opposite spin, on the other hand, should also avoid each other, but this is not adequately allowed for in the HartreeFock method. The avoidance in this latter case is called the Coulomb hole. Thus, if a proper account of electronic interactions is to be made, the motion of all the electrons must be correlated so that the position of one depends on the instantaneous position of the other, rather than on the average position of the other

174

P. Schuster and H. W. Preuss, Chem. Phys. Letters, 1971, 11, 35. A. Pullman and M. J. Port, Compt. rend., 1973, 277,2269. M. Dreyfus, B. Maigret, and A. Pullman, Theor. Chim. Acta, 1970, 17, 109. G.Alagona, A. Pullman, E. Scrocco, and J. Tomasi, Internat. J. Peptide Protein Res., 1973,

176

5, 251. G. N. J. Port and A. Pullman, Theor. Chim. Acta, 1973, 31, 231. W. G. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys., 1969,51, 2657.

171 172 179

37

Ab initio Calculation of Potential Energy Surfaces

electrons. The method of configuration interaction first discussed by Hylleraasf77 in 1928, is one of the oldest techniques used to surmount this problem. In configuration interaction (a), one mixes states arising from different spinorbital configurations.In theory, one may expand the exact solution to Schrodinger’s electronic equation (equation 5 ) in terms of the complete (infinite) set of determinantal wavefunctions, which in turn are constructed from some complete set of oneelectron spin orbitals.33 This is obviously not a practical solution, and one must find instead, a smaller number of determinantal functions formed from the complete set which will give a close approximation to the true solution. A CI state function Yi is expressed as a linear combination of the determinantal wavefunctions @5 formed from configurations of identical symmetry and spin,

vt

= CCS@5,

= 1,2,3,...

(24)

j

The coefficients of mixing, Ccp, and the energies E(CI)r of the resulting CI states are found by solving the set of secular equations:sa [H-E(CI)bqC = 0

A restricted Hartree-Fock calculation on a closed-shell n-electron system using a basis set of N orbitals will produce n/2 doubly-occupied molecular orbitals and N-n/2 vacant or virtual orbitals. In a standard CI calculation, the excited-state determinants are formed by systematically promoting electrons from the occupied orbitals of the ground-statedeterminant to the vacant or virtual orbitals. The number of configurations which can be formed in this way from N electrons and n basis functions17*is of the order of nN. Thus, even with today’s high speed computers, a ‘full’ CI is possible only for very small systems. Minimal Co&guration Interaction.-A minimum requirement of a potential energy surface is that the wavefunction continuously describe the system as the reaction takes place, producing reactants and products in their correct states. In order for this situation to be obtained, the correlation error must either be taken fully into account or at the very least, be approximately constant over that portion of the potential energy surface under consideration. As discussed and illustrated in the previous section, one should at least perform a limited configuration interaction in which those configurations needed to provide the correct description of the dissociation products are included. Truncated CI Wavefunctions.-Between the limits of a minimal and a ‘full’ configuration interaction one is faced with the problem of choosing how many configurations and more importantly, which configurations are to be included. A computational problem arises because of the very slow convergence, generally 1’7 17*

E. A. Hylleraas, Z. Physik, 1928, 48,469. Z. Gershgorn and I. Shavitt, Internat. J. Quantum Chem., 1968, 2, 751.

Theoretical Chemistry

38

obtained in the CI expansion. The convergence is slow in a standard CI calculation because the virtual orbitals used to form the excited configurations do not occupy the same physical space as do the occupied orbitals. Thus, the electrons in the virtual orbitals move in the field of all N electrons and not in the field of N- 1 electrons as do those in occupied orbitals. Within the standard CL framework, there are three methods of selecting the ‘most important’ configurations to be included. The first is achieved by disregarding configurations which, because of Brillouin’s theorem,42a have vanishing coefficients. For closed-shell systems, this occurs when the ground state differs from the excited state by only one spin-orbital. The matrix element between two configurations p in q, can be written as and q differing in just one spin-orbital, ua in p and Hpq =

+ j #i,m C (

rzI -l)udk)uj(l)>

- ) This can be rewritten in terms of the Hartree-Fock operator (equations 17 and 18) as H P 4

= (ui(k)lA(k)

+

CG(k)lum(M) j

From equation (17) w p q

= Em

The orthogonality relationship given in equation (12) then leads us to H p q =

0

It is important to note that Brillouin’s theorem does not state that singly-excited states do not contribute to the total energy of the system since matrix elements between this state and states other than the ground state, doubly-excited states for example, will in general be non-zero. What is stated is that they will make no firstorder contribution to the total energy. An important implication of Brillouin’s theorem is the guarantee of no first-order errors for all one-electron operators provided the exact Hartree-Fock wavefunctions are used.42b This follows because the first-order correction to the correlation effect is derived from two-electron excitations. The second approximation often used to truncate CI wavefunctions is to neglect configurations formed by more than two-electron excitations. It is easily seen that they cannot mix into the CI wavefunction in first order. The third method comes from perturbation theory. If one takes the Hartree-Fock energy as the zeroth-order approximation to the energy Eo =

(281

and assumes that the Hartree-Fock function 00dominates the wavefunction, then the energy correct to second order is

Ab initio Calculation of Potential Energy Surfaces

39

The denominator of the second term on the right is merely the ‘excitation energy’ between the ground state and the excited state k. Therefore, if the state k is one of very high energy, it will contribute relatively little to the total energy of the system. In obtaining an approximate CI wavefunction, one usually neglects states of this type with small introduction of error. The above methods of truncating the ‘full’ CI wavefunction can be greatly improved upon by transforming the molecular orbitals into natural orbitals - a method which will now be discussed. The generalized oneelectron density function is defined as

s

P(’)(xI;x~= N @(XI,~

..

2 , . XN)@*(X;,

.

~ 2 , . .XN) dx2..

.dxnr

(30)

Using the determinantal wavefunction given in equation (11) one can expand p(l)(xr,xi) in terms of spin-orbitals

where the &/ are numerical coefficients. For the CI wavefunction given in equation (24) the density matrix can be written as

where pgL is the first-order transition matrix defmed as in equation (30)where @ and @* are replaced by @ K and ,@: determinants which differ only in one spinorbital (uk:in @ K and uzin @L) and the spin-orbitals are assumed to be orthonormal. The first-order density matrix can now be written in terms of these spin-orbitals as

Since p(l)(xl;x;) is Hermitian, i.e.,

the coefficients form a Hermitian matrix so that a unitary transformation exists which diagonalizes the y. UfyU

=

n

If we now define a new set of spin-orbitals by Xk

= XuZUZk E

the first-order density matrix can be written in terms of the transformed orbitals P(’)(xI;xi) = Cxm(xl)xXxhn 112

The Xm are called the natural spin-orbitals (NSO) of @ and the rtm are the occupation numbers. The occupation numbers indicate the importance of each orbital Xm to the total CI wavefunction.

40

Theoretical Chemistry

The natural spin-orbitals, which were first introduced by Lowdin179in 1955, were shown to produce the most rapidly convergent expansion of the first-order density matrix.lsOThus, the number of determinants needed for any required degree of accuracy in the wavefunction can be greatly reduced if the determinantal wavefunctions are constructed from natural orbitals at each stage in the calculation. The most dramatic indication of the convergence property of NSO’s can be seen in the work of Hagstrom and Shulll*l on H2. Using a basis set of 15 functions, a full configuration interaction (33 configurations) was performed. The first-order density matrix was then diagonalized and the occupation numbers of all but 15 of the natural spin-orbitals were found to be zero. The energy of - 1.1731 a.u. accounts for 97 % of the correlation energy. The energy was then evaluated after eliminating those NSO’s with small but non-zero occupation numbers. The energies computed with the three and five spin-orbitals with highest occupation numbers were - 1.1607 a.u. (66% of the correlation energy) and - 1.1704 a.u. (90% of the correlation energy), respectively. The effectiveness of NSO’s in reducing the expansion size in systems with more than two electrons is not as great and, in fact, for larger systems, their use is not practical. The loss in practicality is immediately obvious when one realizes that in order to obtain them, one must diagonalize the first-order density matrix of the ‘exact’ wavefunction, i.e. a ‘full’ configuration interaction must first be performed. Two methods have been introduced in order to regain the initial usefulness of natural orbitals: the pseudonatural orbital method and the approximate or iterative natural orbital method. In the pseudo-natural orbital (PSNO) method,l82 a natural orbital calculation is performed on selected pairs of electrons in the Hartree-Fock field of the n-2 electron core. These orbitals are then used as a basis for a CI calculation. In their work on the HeH+ system, for which the orbital occupancy is l
1B0 181

Ab initio Calculation of Potential Energy Surfaces

41

repeated until successive iterations yield no change. In their work on LiH, Bender and Davidsonlss began with 50 configurations. Their final 45-configuration wavefunction contains ca. 87% of the correlation energy. It is of interest to compare these results with a calculation done later by the same investigators,l87 in which they included all single and double excitations. This latter work resulted in an energy 0.0002 a.u. higher than the IN0 calculation. Although as many as four or five CI calculations may be needed to achieve convergence using this method, the scope of each CI calculation is much smaller than that needed for one extensive conventiond CI including many configurations (938 in the LiH case). In contrast to the relationship between virtual and occupied orbitals, natural orbitals are localized in the same regions of physical space. The strong coupling thus produced by the overlap of ground-state and excited orbitals effectively lowers the energy of the system. Another method which achieves the same purpose and therefore improves the convergence of a CI expansion is the multiconfiguration selfconsistent field (MCSCF) method. The method can be described as one which ensures that both the CI coefficients [Ctj in equation (24)] and the molecular orbital coefficients [crj in equation (22)J have been varied to minimize the energy. A doublyiterative process is used which, upon convergence, yields the optimum orbitals and the optimum configurationalmixing coefficientsfor the basis sets used. The optimization of the orbitals ensures that the virtual orbitals are in the same physical space as the occupied orbitals. This means that generally, many fewer configurations are needed in the MCSCF method as compared to a conventional CI calculation in order to achieve the same results. This MCSCF method was first used on the oxygen atom by Hartree, Hartree, and Swirles in 1939.lS8A recent example of the application of the MCSCF method is found in the work of Das and Wahllsga on the F2 molecule. As in a conventional CI calculation, the problem of selecting important configurations still exists in the MCSCF method. Das and Wahl considered three classes of configurations in what they call the optimized valence configuration (OVC) method. These are: (i) those necessary for proper dissociation of the molecule to the Hartree-Fock atoms; (ii) those that exist in the molecule but vanish formally in the dissociated atoms; and (iii) those that describe mainly atomic correlation but whose contribution changeswithin a numerically defined threshold of significance as the molecule forms. The F a potential energy curve thus produced is very similar to the experimentally derived one and the spectroscopic constants, dissociation energy, vibrational frequency, and F-F equilibrium distance of 1.67 eV, 942 cm-l, and 2.67 a.u. compare very favourably with the experimental values of 1.68 eV, 932 cm-1, and 2.68 a.u. Hosteny, Hinds, and Wahlls9 b have recently calculated a portion of the surface for the 3B1 state of H2O by the MCSCF method. Unrestricted Wave€iinctions.-Up to this point, little has been said about the form of the molecular orbitals used in the determinantal wavefunction. Since the choice

188 189

C. F. Bender and E. R. Davidson, Phys. Rev., 1969,183,23. D. R. Hartree, W. Hartree, and B. Swirles, Phil. Trans. Roy. SOC.,1939, A238, 229. (a) G. Das and A. C. Wahl, Phys. Rev. Letters, 1970, 24, 440; Adv. Quantum Chem., 1970, 5, 261; J. Chern. Phys,, 1972, 56, 3532; (b) R. P. Hosteny, A. R. Hinds, and A. C. Wahl. Chem. Phys. Letters, 1973,23, 9.

42

Theoretical Chemistry

of the orbitals can have an effect on the correlation problem, we will discuss the two commonly used restrictions placed on the orbitals. In the restricted Hartree-Fock (RHF) method, two restrictions are placed on the molecular orbitals ug in equation (11). The first is chat each uztransform according to one of the irreducible representations of the point group of the molecule. The second restriction is that the space functions $i, come in identical pairs one with spin function a and the other with spin function /I.These are called, respectively, the symmetry and equivalence restricti0ns.1~0 When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction lgl (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. The obvious advantage of this method is the retention of the single-determinant wavefunction. The drawback is that the resulting wavefunction is no longer an eigenfunction of the spin operator <S2>.There are cases where the eigenvalue equation S2Y = S(S + 1)Y is almost satisfied. This was found to be the case in the calculations on the insertion of O(3P)into the ground state hydrogen molecule reported by Bader and Gangi.138 Table 4 shows the values of <S2> along the reaction path which, for a spin-pure

Table 4 Values of <S2> along the Czvpath of the insertion of O(3P) into H2(1X,+)138 r 0la.u.

rHH/a.u.

<s2>

3.00 2.00 1.84 1.11 0.00

1.56 2.66 2.99 3.72 4.61

2.01 2.01 2.01 2.01 2.04

tripiet, should be 2. If <S2>is found to diverge from the anticipated value, one must apply a projection operator in order to recover a spin-pure or nearly spin-pure state. This can be done in several ways. Amos and Snyder193 have proposed a method which annihilates only the contaminant of next higher spin multiplicity (i.e. the triplet for a singlet state, quartet for a doublet, etc.) under the assumption that this is the major contaminant. A method of complete projection194 has also been proposed but since in this and the previous method the variational principle is applied lgoR. K. Nesbet, Proc. Roy. SOC.,1955, A230, 312. 191 J. A. Pople and R. K. Nesbet, J. Chem. Phys., 1954, 22, 571. lQ2 P.-0. Lowdin, Adv. Chem. Phys., 1959, 2, 207. 193 T. Amos and L. C. Snyder, J. Chem. Phys., 1964,41, 1773. 194

P.-0. Lowdin, Rev. Mod. Phys., 1960,32,328; 1964,36,966; A. Hardisson and J. E. Harriman, J. Chem. Phys., 1966,46,3639; J. E. Harriman, J. C'hern. Phys., 1964,40,287.

Ab initio Calculation of Potential Energy Surfaces

43

before projection, the final wavefunction suffers from the lack of proper opt*tion. The spin-extendedHartree-Fock (EHF) method corrects for this by projecting before optimization on each iteration. In their work on the insertion of triplet oxygen to the hydrogen molecule, Bader and Gangi138found it necessary to use the UHF method in order to investigate the changes of polarization of electron spins at various points on the potential energy surface. The spin density distribution function a(r) is defined as

In other words, o(r) is the number of excess a[a(r)>O] or B[o(r)
4r) =

2cdr)c;o - XMM;(d i j

where the sums run over only the singly occupied orbitals containing the unpaired a- or &electrons. In the UHF approximation, which allows for different spatial orbitals with different spins, all the orbitals are singly occupied and hence all of the orbital densities contribute to a(r) at every point in space. The results of this work show that there is an unpairing mechanism corresponding to the localization of charge density with identical spin components on both centres of the substrate bond to yield an unstable triplet insertion product, and an uncoupling mechanism corresponding to a separate localization of the a and B spin distributions on both centres of the substrate bond to yield abstraction products of OH(2rr) H(2S). These two principal forms of spin polarization account nicely for the observed differences in the chemistry of O(3P) and O(1D) in both insertion and abstraction. For example, Figure 2 illustrates the total charge distribution p(r) of the H2O system at various points along the minimum energy CzU insertion pathways, (ro = distance of oxygen from midpoint of H-H and rH is H-H separation). Shown on these diagrams are the 'zero-flux' surfaces S(r), i.e. those surfaces satisfying Vp(r)-n = 0 VrES(r)

+

which partition the system into two hydrogen fragments and an oxygen fragment.l2', 128 The quantum mechanical virial theorem, and the hypervirial theorem for the operator - r . V are both obeyed over the individual fragment volumes defined by these surfaces. In addition, all properties of such fragments (a) are uniquely defined, including a total energy for each fragment (Q), E ( n ) which, because of the satisfaction of the virial theorem, has the property of equalling minus the kinetic energy of the fragment, E ( a ) = -T(Q), when no external forces are exerted on the system. The populations and kinetic energies of the (El) fragments during the course of these two insertion reactions are listed in Table 5, together with the net charge C O ,where C(H) = Z H - N ( H ) . The population figuresmake quantitative the observationsmade from the changes in the charge distributions, Figure 2. At ro = 4.00 a.u. (not shown in the Figure)

44

Theoretical Chemistry SINGLET

TRIPLET

ro

= 3.00

1.58

1.54

Figure 2 Contour maps of the total charge distributions in the plane of the nuclei for the symmetrical singlet and triplet insertion reactions to form IAI and 3B1 states of HaO, respectively. The contours increase in value from the outermost member, in the order 2 x lo-", 4 x lo-", 8 x a.u. for decreasing values of n, beginning with n = 3

45

Ab initio Calculation of Potential Energy Surfaces

Table 5 Kinetic energies andpopulationsof (H) in singlet and triplet insertion reactions r0la.u.

3.0 2.0 1.1 0.0

Singlet surface T(H)/a.u. N(H) 0.5013 0.893 0.3644 0.575 0.3474 0.398 0.2614 0.259

C(H) 0.107 0.425 0.602 0.741

Triplet surface T(H)/a.u. N(H) 0.5496 0.953 0.3756 0.748 0.3506 0.6621 0.761 0.4290

C(H) 0.047 0.252 0.339 0.239

the perturbations are quite small and the charge distributions resemble closeIy those of the isolated reactants. At ro = 3.00 a.u. the effect of the increasing H-H separation is apparent in the pinching up of the high-density (0.2 a.u.) contour encircling the protons in both the singlet and triplet systems. For values of ro < 3.00 a.u. the energy of the singlet path decreases while that of the triplet beings a rapid increase. Correspondingly,marked differencesappear in the charge distributions of the singlet and triplet systems. After an initial scission of the 0.2-a.u. contour encompassing the protons due to the further increase in rH in both systems (ro = 2.00 a.u.), the singlet system is characterized by a complete merging of the hydrogen and oxygen charge distributions into a singIe, compact distribution with contours of high value encompassingand binding all three nuclei. The result (ro = 1.1 1 a.u.) is the stable, low-energy insertion product, H20(1A1). In the triplet system, motion along the minimum energy path leads to a marked expansion of the charge density into the nonbonded regions of the protons (roG2.00 a.u.). This expansion results in a very diffuse distribution in the regions of the protons and in low concentrations of charge density between each pair of nuclei. These effects are maximized at ro = 0.00 a.u. and in addition there is a separate localization of the charge density in the region of each nucleus. In contrast, even when the singlet system is forced into the linear geometry at a cost of ca. 138 kJ mo1-1, the charge distribution is still compact with all three nuclei encompassed by contours up to ca. 0.3 a.u. in value. We now attempt to account for the gross differences in the charge distributions of the singlet and triplet systems and hence for their different reaction co-ordinates in terms of the spatial distribution of spin density u(r). Spin-density distributions in the plane of the nuclei for configurations along the minimum energy triplet path are illustrated in Figure 3. The contour maps of c(r) are for the Ms = 1 spin component of the triplet state, i.e. with two unpaired a electrons which initially are localized on the oxygen atom. One of the unpaired spins remains almost entirely localized in a distribution which has its maximum values in a plane perpendicular to the plane of the nuclei shown in the diagram. Even at ro = 4.00 a.u. a spin polarization is evident in the system, corresponding to the creation of excess a spin density in the charge distribution of the hydrogen molecule. Simultaneous with this, is the appearance of excess B spin density in the region of the oxygen. Both of these effects are enhanced as ro is decreased, with the excess a spin population forming diffuse, expanded distributions of increasing extent in the regions of the protons. A comparison of the contour values in the p(r) and a(r) distributions for ro = 2.00 a.u. and 1.11 a.u. indicates that almost 100% of the outermost non-bonded charge density on the protons represented by the 0.002, 0.004, and 0.008 a.u. contours in the p(r) maps is composed of a spin density. The

46

Theoretical Chemistry

,

, ,

---

1 0.00

u ro = 3.00

ro = 4.00

f

ro = 2.00

_-------.. \

ro= 1.11

ro = 0.00 Figure 3 Contour maps for the spin-density distributions for the symmetrical triplet insertion reaction. Excess cc spin density is denoted by solid contours, and excess /3 spin density by dashed contours. The solid and dashed contours increase (excess a) or decrease (excess /3), respectively,from the zero contour in the order & 2 x 10 -*, 4 x 10 -n, +_ 8 x 10 -n a.u. for decreasing values of n beginning with n = 3

Ab initio Calculation of Potential Enepgy Surfaces

47

separation, localization, and enhancement of positive spin density increases until, in the linear configuration, the a spin density accounts for SCr-lOo% of the total charge density in the spatial region of each proton. Although the integral of a(r) over all space will yield two electrons (or spins), the appearance of negative spin density in the o(r) distribution indicates that charge density previously paired has been unpaired. That is, integration of just the positive contours of a(r) over all space will yield more than two electrons. The behaviour of the spin-density distribution along the reaction co-ordinate does account for the major differences in the charge distributions and chemistry of the singlet and triplet species. At ro = 2.00 and 1.11 a.u. the extreme localization of charge density with an identical spin component on each proton results in a great reduction in the total charge density in the region between the protons. This follows as a consequence of the requirements of the Pauli principle on the two-electron charge density33 that Pr(I-1, r2)-*O for r1+r2. The spin polarization very effectivelyweakens the bond in the hydrogen molecule. In effect the spin polarization caused by the approach of O(sP) corresponds to a quasi-excitation of H2 from its singlet ground state to its unbound 3Ci state, a state characterized by a localization of charge density on each nucleus, each distribution being polarized away from the other into its non-bonded region. The two localized, almost pure CY spin distributions on each proton are most widely separated and the energy minimized in the linear configuration. Thus the mechanism of triplet insertion corresponds to an inducement of identical spin distributions on each centre of the substrate bond, an effect which weakens the bond and leads to the formation of a loosely bound metastable intermediate. Thus, along the singlet path there is considerable charge transfer from the hydrogen fragments to (0)to yield an overall tightly bound distribution. In the triplet reaction the net charge transfer from (H) to (0)is much less and the final distribution is diffuse and separately localized on all three centres. The UHF method was used in 1970 by Salotto and Burnellel95 in their investigation of the potential energy surfaces of HNO and H2NO. More recent work includes that of Ehrenson and Newton152 on CH5, referred to above. It is noteworthy that obtained in their calculation on CH5, range from 0.795 to 0.846 the values of

(for a doublet <S2>= 0.75) and therefore the wavefunction is appreciably contaminated by states of higher multiplicity. Brown and Williams196 performed a singleannihilation on their UHF wavefunction in their work on the potential energy surfaces of some small free radicals. Nevertheless, they find that in the dissociation of NH2 in its ground and first-excited states (2B1and aA1) the values of (S2>deviate sharply from 0.75 as the N-€€ bond lengths are increased. States of multiplicity greater than quartets are evidently mixing-in and the necessity for a complete projection is indicated. A similar situation is found by these authors in the BH2 and CH + dissociation calculations. It appears that one must approach the use of the UHF method warily, being prepared to l95

A. W. Salotto and L. Burnelle, J. Chem. Phys., 1970,52,2936;J. Chem. Phys., 1970,53, 335. D. Brown and G. R. Williams, Mol. Phys., 1973, 25, 673.

lQ6R.

48

Theoretical Chemistry

undergo the additional computational effort required to use spin projection to whatever extent is needed to ensure that the final wavefunction is spin-pure. The IEPA Method.-Another method of including electron correlation has found growing use in the calculation of potential energy surfaces. In this technique, called the ‘independent electron pair approach’ (IEPA), the total correlation energy in a system is expressed as a sum of the correlation energies of pairs of electrons in the field of the other electr0ns.19~To a good approximation, it can be shown1828198 that the total energy of the system can be written as follows: E

N

EHF +

zeta i

+ i< zeal j

where ~ i isg the energy associated with the correlation between electrons in the same orbital (intrapair correlation) and represents the energy resulting from the correlation of electrons in different orbitals (interpair correlation). Examples of this method are the recent work of Driessler, Ahlrichs, Staemmler, and KutzelnigglgQon the equilibrium geometries and other properties of CH3, C H t , the inversion barrier in CH, and the potential surface for LiHi referred to previously.lo5 6 H

+

A Benchmark Reaction Progress in the calculation of complete potential energy surfaces can best be followed by considering in detail the work that has been done on one of the simplest of exchange reactions, namely H H2-+H2 H. Semi-empiricalCalculations.-Many groups have used both semi-empirical and ab initio methods to produce potential energy surfaces for the ground state of Ha. A thorough bibliography of the work prior to 1967 is given by Hayes and Parr200 and an excellent review of the earlier results in this field can be found in reference 201. Because much of the earlier work involved the use of semi-empirical methods and the fact that the authors have chosen to restrict this work to a discussion of ab initiu calculations, we will describe, for the purpose of comparison, only one semi-empirical potential energy surface, the one which Porter and Karplus 22 calculated for H3 in 1964. The Porter and Karplus (PK) H3 surface is based on a modification of the London-Eyring-Polanyi (LEP) method.19 All overlap and three-centre terms were included. Empirical values were introduced for two-centre integrals by a modified Sat0 technique,20 and analytical approximations were used for the more complex integrals. Linear and non-linear configurations for H3 were considered. A linear symmetrical col or saddle-point configuration was found at an H-H separation of R = 1.701 a.u. with a barrier height of 38.2 kJ mol -l. [The barrier height E b = Ecalc(H H2)-Ecalc(H3 at saddle point), the energies of all species being calculated the same way.] When the zero-point vibrational energy is included, this becomes 36.8 kJ mol-l. As we will discuss below, this value is in very good agreement with the cu. 38 kJ mol-l predicted from experiment. Earlier semiempirical calculations predicted the presence of a basin or well in the saddle-point H2:

+

+

+

197 198 109 200 201

For a review of this method, see; R. K. Nesbet, Adv. Chem. Phys., 1969, 14, 237. 0. Sinanoglu, J. Chem. Phys., 1962, 36, 706, 3198. F. Driessler, R. Ahlrichs, V. Staemmler, and W. Kutzelnigg, Theor. Chim. A d a , 1973,30, 3 15. E. F. Hayes and R. G. Pam, J. Chem. Phys., 1967, 47, 3961. K. J. Laidler, ‘Theories of Chemical Reaction Rates’, McGraw-Hill, New York, 1969.

Ab initio Calculation of Potential Energy Surfaces

49

12.0

-

8.0

-E"mu

'

4.0

0.0 0.0

1.o

2.0

3.0

R1a.u.

Figure 4 Comparison of energy profiles along the reaction path for linear Ha. (-Shavitt, Stevens, Minn, and Karplus, ref. 112; ---- Liu, ref. 210; Porter and Karplus, ref. 22; - - - - Conroy and Bruner, ref. 207)

--

9

9

+

region of the surface. No well was found in the barrier region of the PK surface. It is interesting to note that this is the thinnest of the theoretical surfaces computed for Hs to-date (see Figure 4). Ab Imzio Calculations.--The first ab initio calculation on H3 was performed by Hirschfelder, Eyring, and Rosen202on the linear symmetric system in 1936. The valence bond method with the inclusion of ionic states was used. The wavefunctions were composed of 1s atomic orbitals. A barrier height (Eb) greater than 40kcd mo1-1 was obtained, a value which was clearly much too high. In 1954, Barker et aZ.203 performed similar calculations on linear H3, but with independent variation of the effective charges on the middle and outer atoms. The barrier height found was in close agreement with that of Hirschfelder et al. Although the next few years show some activity in the field 205 improvements in results had to await the perfection of high-speed digital computers. An example is the variational calculation made on a few configurations of the H3 system by Boys and Shavitt205 in 1959. The barrier height of 62 kJ mol-1, while an improvement over previous ab initio work, still did not compare with the experimental or the semi-empirical results given by the PK surface. 204p

20%

*03 *04

205

J. 0. Hirschfelder, H. Eyring, and H. Rosen, jun., J. Chem. Phys., 1936, 4, 121, 130. R. S. Barker, H. Eyring, C. J. Throne, and D. A. Baker, J. Chem. Phys., 1955, 22, 699; R. S. Barker and H. Eyring, J. Chem. Phys., 1957, 26, 971. B. J. Ransil, J. Chem. Phys., 1957, 26, 971; R. Snow and H. Eyring, J. Phys. Chem., 1957, 6 1 , l ; G . E. Kimball and J. G. TruIio,J. Chem. Phys., 1958,28,493; V . Griffing, J. L. Jackson, and B. J. Ransil, J. Chem. Phys., 1959, 30, 1066. S. F. Boys and I. Shavitt, University of Wisconsin Naval Research Laboratory Tech. Rept. WIS-AF-13 (1959).

50

Theoretical Chemistry

In 1967, Hayes and Parr200did studies on linear symmetric H3 at separations of 1.6, 1.7, and 1.8 a.u. between adjacent protons using a 70-configuration CI calculation. The basis set was composed of single-centreSTO’s. The energy of - 1.6358 a.u. for the symmetrical H3 complex with an H-H separation R = 1.80 a.u. was an improvement over previous ab initio calculations. Hayes and Parr did not perform calculations for the separated species H2 + H and therefore a barrier height Eb as defined above, cannot be ascertained for their work. We can, however, define another barrier height Eb in the following way:

&xact(H

+

H2) = Eexact(H) = 0.5000

+

+

Eexact(H2)

1.1745 = 1.6745 a.u.

The value for H2 is taken from the work of Kolos and Wolniewicz.206 Hayes and Pam’s results yield a value of E’, of 101.6 kJ mol-1, again too high for agreement with experiment. But 1967 was a year of great activity in the ab initio calculation of potential energy surfaces of H3, for three other groupsl12~18*~ 207 had also entered the field. We will discuss the three other computations in the following paragraphs. In 1964, Conroy had developed a method207a for solving the molecular Schrodinger equation using variational trial functions that explicitly included the interelectronicco-ordinates and thus took account of the correlation effect. He had applied the method to various small systems. In 1967, Conroy and B r ~ n e r ~ O ~ applied this method to the H3 surface. The potential energies of 41 different linear arrangements and 27 triangular arrangements of H3 were calculated. The minimum energy path was found to pass through a maximum 32.4kJmol-1 above the reactants at a linear symmetric configuration with R = 1.76 a.u. The energy at this geometry was reported as - 1.6621 a.u. The data in Table 6, which summarize the results of the most recent calculations on the H3 surface, indicate that to-date this is the lowest energy found at the saddle-point geometry of H3. It is important to note, however, that Conroy’s method of calculation does not necessarily yield Table 6 Calculations on the linear H3 molecule R/a.u. E1a.u. &/kJ mol-1 E2/kJmol-1 Force constants1a.u. symmetric stretch bending asymmetric stretch a

Ref. 22.

206

207

@

PKa HPb 1.701 1 .so - 1.6600 - 1.6358 38.2 38.1 101.6 0.36 0.024 -0.124

Ref. 200. Ref. 207.

-

-

CBc 1.76 - 1.6622 32.4 32.4 0.32 0.026 ca. 0.00

Refs. 184 and 209.

EKd 1.80 -1.6493 56.5 66.1

-

-

SSMKe 1.764 -1.6521 46.0 58.6

LIUf 1.757 -1.6581 41 .O 43 .O

0.31 0.024 -0.061

0.320 - 0.058

Ref. 112. f Ref. 210.

W. Kolos and L. Wolniewicz, J. Chem. Phys., 1965, 43, 2429. (a)H.Conroy and B. L. Bruner, J. Chem. Phys., 1967,47,921; (b) H. Conroy,J. Chem. Phys., 1964,41, 1327, 1331, 1336, 1341.

Ab initio Calculation of Potential Energy Surfaces

51

an upper bound to the energy.* Thus, the usual criteria for comparing calculations (Le. how low is the energy?) cannot be applied here. The method appears to produce results whose accuracy is inconsistent from point-to-point on the surface. This leads to two unexpected results, one being a very flat barrier top and the other being the H2 ca. 2 a.u. away from the presence of an energy minimum with respect to H col (see Figure 4). At about the same time, pseudonatural orbitals were used by Edmiston and Krauss184~209 as a basis for a 65-configuration CI calculation on the linear H3 surface. Five s, two pa, three pnx, and two pn, type GTO’s were centred on each of the three nuclei and the energies at 28 geometries were computed. At the linear symmetricgeometry, a minimum energy of - 1.6493 a.u. was found at R = 1.8 a.u. This value was further reduced to - 1.65492 a.u. by extrapolating the values of the computed correlationerror. This corresponded to a barrier height Eb of 50 kJ mol -l. The energy of the dominant contiguration in the CI expansion is - 1.5922 a.u., which closely approaches the HartreFock limit. It is interesting to note the change in the correlation energy along the reaction path computed in this work. The correlation H, is ca. 0.04 a.u. and this increases steadily to a value error of the reactants, Hz of ca. 0.06 a.u. at the saddle point. This significant change is to be expected in the light of the previous discussion on the importance of including correlation effects in a system of this type. The fourth group to work on the theoretical calculation of the H3 surface in 1967 was Shavitt, Stevens, Minn, and Karp1us.ll2 An STO basis set composed of Is, ls’, 2pz, 2pV,2pz on each of the three hydrogens was used in calculations on both the linear and triangular geometries of the system. All possible determinants were used in a configuration interaction calculation (200 in the linear symmetric and 402 in the linear asymmetric). In this work, as in all previous work, the linear geometries were found to yield the reaction path. For the linear geometries, potential energies at 158 points were calculated and at 26 of these, the orbital exponentswere optimized. The saddle point was found at R = 1.765 a.u. (again no well appeared at the top of the barrier) with a potential energy of - 1.6521 ax. This corresponds to a barrier height (4%) of 46 kJ mol -l. The most thorough work to-date on the H3 surface is that reported in 1973 by Liu.210 An STO basis set was used consisting of Is, ls’, 2s, W ,2p, 2p’, 3p’, 36 and 4d functions on each hydrogen atom. Here we see the first inclusion of d-type functions in H3 calculations. The exponents were optimized for the ground state of Hz at a separation of 1.4 a.u. This optimization was conducted by minimizing the complete configuration interaction (126 configurations). The SCF energy obtained with the major configuration is within 5.0 x 10-5 a.u. of the Hartree-Fock limit. Comparing his CI curve of H2(X1Z3 with the essentially exact results of Kdos and Wolniewicz,206 Liu found that the CI curve is very nearly parallel to the exact curve near the equilibrium separation. The energy difference between the two results at

+

+

*

See ref. 208 and footnote 30 in ref. 112.

208

209 210

H. Conroy and B. L.Bruner, J. Chem. Phys., 1965, 42,4047. C. Edmiston and M. Krauss, J. G e m . Phys., 1965, 42, 1119. B. Liu, J. Chem. Phys., 1973, 58, 1925

.

52

Theoretical Chemistry

a nuclear separation of 1.4 a.u. is 7.7 x 10 -4 a.u., evidence of a sufficiently adequate wavefunction, at least for Ha. Considering the large basis set involved (27 o-type, 15 n-type, and six 6-type functions) a complete CI, while within the capabilities of available computers, is certainly using these capabilities to their limits. In this work, Liu tests yet another method for truncating a full CI wavefunction. The method begins with an MCSCF calculation on a severely truncated CI wavefunction. Configurations are then systematically added in order to establish a convergence pattern which could be extrapolated to the complete CI limit. The nparticle space is partitioned into a set of configurations including all dominant terms in the full CI expansion for all nuclear geometries considered, and a set containing the remainder of the n-particle space. The first of these is composed of what Liu calls the internal approximate natural orbitals (ANO) which are those associated with large occupation numbers. The second set is further divided into a finite number of sub-spaces according to the following rule: the ith-order sub-space contains those and only those n-particle functions which are orthogonal to all lower-order subspaces and have non-zero Hamiltonian matrix elements with at least one function in the (i- l)th-order sub-space. In the case O f €33, Liu's first-order sub-space contains all configurationswith one or two electrons in internal orbitals and the other electron or electrons in the remaining or external orbitals. The second-order sub-space contains configurations with all three electrons occupying external orbitals. Using this method, Liu extrapolates his results to produce a minimum energy of - 1.6581 a.u. on the linear reaction path at a saddle-point geometry of R = 1.757 a.u. This yields H. a barrier which is 41 kJ mol-1 above the calculated energy of H2 Rate constant measurements211 and quasi-classical trajectory calculations2 1 2 s 213 indicate a barrier height of ca. 38 kJ mol-1 for the H + H2 exchange reaction. The 38.2kJmol-1 barrier height of the Porter-Karplus surface and the 41 kJ mol-1 value obtained by Liu, therefore, are in good agreement with the experimental results. The situation is much different, however, regarding the reported hydrogen-hydrogen distances at the col. Note from Table 6, for example, that three of the a6 initio calculations, those of Conroy and Br~ner,~o'Shavitt and coworkers,ll2 and Liu,210 agree on a value of 1.76 a.u., whereas two others, Hayes and Parrz*Oand Edmiston and Krauss18* report a higher value of 1.80 ax. The PorterKarplus surface, on the other hand, predicts an H-H distance of only 1.701 a.u. The preponderance of theoretical evidence, then, seems to be on the side of the larger values, and the Porter-Karplus value is probably too small. Table 6 shows that the variation in the reported symmetric stretch and bending force constants is relatively small. For the asymmetric stretching force constant only the Shavitt et al. and Liu values of -0.061 a.u. and - 0.058 a.u., respectively, are in substantial agreement, the PK surface yielding a value of -0.124 a.u. which is more than a factor of two greater. Figure 5 shows the minimum energy path for three of the computed linear H3 surfaces considered here. Shavitt et al. who gave a similar plot112noted a peculiar

+

211

W. R. Schulz and D. J. LeRoy, J. Chem. Phys., 1965, 42, 3869; R. E. Weston, jun., ibid.,

212

1959, 31, 892; I. Shavitt, ibid., 1959, 31, 1359. M. Karplus, R. N. Porter, and R. D. Sharma, J. Chem. Phys., 1964,40, 2033. M. Karplus, R. N. Porter, and R. D. Sharma, J. Chem. Phys., 1965,43, 3259.

213

Ab initio Calculation of Potential Energy Surfaces

1.o

2.0

53

3.0

R,/a.u.

Figure 5 Comparison of reactionpaths for linear Hs surface. See Figure 4 for details feature which is found in all semi-empirical surfaces. This is the 'kink' which is found in the saddle-point region of the Porter-Karplus plot. The minimum energy path for the Conroy and Bruner207surface (not shown in Figure 5 ) very closely parallels that of Liu except for what appears to be a very small 'kink' at the col. Note that in both Figures 4 and 5 the Liu210 surface is very nearly parallel to the surface of Shavitt et In comparing these latter two sets of calculations, one notes that there are just two major differences: the Liu surface approaches the equilibrium H2 geometry more quickly and its barrier height is lower. The H3 potential surface which has been most widely used in transition stateY2l4 classical dynamicaly213,215 and quantum mechanical dynamical216 -218 calculations has been the semi-empirical surface of Porter and Karplus.22 Because of the thin potential barrier of the PK surface, one would expect a larger amount of quantum mechanical tunnelling to be predicted at room temperature (this has been found to be the case in calculations performed by Johnston219 on barriers in the very similar LEPS surfaces). However, Karplus et aL2139215 compared classical and quantum mechanical calculations on the PK surface and found that reaction cross-sections for both are very similar, and therefore that the tunnelling effect in the Hs system is small. Transition-statetheory was applied to the SSMK surface1l2by Shavitt22* in 1968. The plots of the log of the rate constant against the reciprocal of the temperature calculated by Shavitt are in excellent agreement with experimental results on 214

216 216

217 d18

219 220

3

K. Morokuma, B. C. Eu, and M. Karplus, J. Chem. Phys., 1969,51, 5193. M. Karplus and K. T. Tang, Discuss. Faraday Soc., 1967, 44, 56. J. W. Duff and D. G. Truhlar, Chem. Phys. Letters, 1973, 23, 327; D. J. Diestler, J. G e m . Phys., 1971, 54, 4547; S. Wu and R. D. Levine, Mol. Phys., 1971, 22, 881; B. R. Johnson, Chem. Phys. Letters, 1972, 13, 172. D. J. Diestler and M. Karplus, J. Chem. Phys., 1971, 55, 5832. S.-F. W u , B. R. Johnson, and R. D. Levine, Mol. Phys., 1973, 25, 609 H. S. Johnston, Ado. Chem. Phys., 1960, 3, 131. I. Shavitt, J. Chem. Phys., 1968, 49, 4048.

Theoretical Chemistry

54

+

+

H Hz, H D2, D + H2, and D + D2,221although it was necessary for him to scale his surface to a barrier height of 41 kJ mol-1. This 11% reduction in the barrier height was applied uniformly along the total reaction path. The asymmetric force constant given in Table 1 has been scaled down by the same amount. Shavitt includes the quantum-mechanical tunnelling effect by fitting an Eckart 222 barrier to the calculated SSMK barrier. Kupperman et al.223 -226 compared transition-state, classical, and quantum mechanical thermal rate constants using the SSMK surface.l12In ab initio potential energy surface calculations, the potential energy is known only as a list of values at selected geometries of the system. It becomes necessary, then, if trajectory calculations are to be made, to fit the calculated points to a smooth and continuous ‘map’. In their calculations, Kupperman et al. fit the collinear SSMK surface by the rotating Morse function procedure of Wall and Porter.227 Koeppl228 has applied transition-state theory to the Liu surface. Again, a plot of log k 1:s 1/ T agrees with the experimental results cited above.221Since the barrier height was computed by Liu to be 41 kJmol-l, no barrier height scaling was necessary in this case. Truhlar and Kupperman229have shown that tunnelling calculations from Eckart potential fits to the potential energy barrier lead to systematic errors and that agreement with experiments must be considered the result of fortuitous cancellation of errors. For this reason, Koeppl takes no account of tunnelling in his work. More recently, Yates and Lester230 fitted Liu’s surface with a slightly modified form of the Porter-Karplus formulas after first fitting Liu’s H2 potential to a simple Morse function. They then use the resulting surface to calculate the three-dimensional classical trajectory of the system. Their empirical fit very closely duplicates Liu’s saddle-point properties. Reaction probabilities on this surface are compared with those on the PK surface. All things considered, it appears that the Liu surface probably approaches very closely to the true potential energy surface of H3. In addition, Liu’s method of selecting configurations is certainly one which will find wider application in future CI calculations. It is uncertain at present, however, how large a molecular system can thus be accommodated.

7 Potential Energy Surfaces from Correlated Wavefunctions H;.--In 1970, Csizmadia et aZ.231 carried out an extensive SCF-CI calculation on the potential energy surface of H l . Four or five s-type GTO’s and at least onep-type (a) B. A. Ridley, W. R. Schulz, and D. J. LeRoy, J. Chem. Phys., 1966, 44,3344; (b) A. A. Westenberg and N. deHaas, ibid., 1967, 47, 1393; (c) D. N. Mitchell and D. J. LeRoy, ibid., 1973,58, 3449; ( d ) W. R. Schulz and D. J. LeRoy, Canad. J. Chem., 1964,42, 2480. 222 C. Eckart, Phys. Rev.,1930, 35, 1303. 223 J. M. Bowman and A. Kuppermann, J. Chem. Phys., 1973,59, 6524. 224 J. M. Bowman and A. Kuppermann, Chem. Phys. Letters, 1971, 12, 1. 225 D. G. Truhlar and A. Kuppermann, Chem. Phys. Letters, 1971, 9, 269. *26 D. G. Truhlar and A. Kuppermann, J. Chem. Phys., 1970, 52, 3841. 2 2 7 F. T. Wall and R. N. Porter, J. Chem. Phys., 1962, 36, 3256. 208 G. W. Koeppl, J. Chem. Phys., 1973, 59, 3425. 229 D. G. Truhlar and A. Kuppermann, J. Amer. Chem. Soc., 1971, 93, 1840. 230 A. C. Yates and W.A. Lester, jun., Chem. Phys. Letters, 1974, 24, 305. 231 I. G. Csizmadia, R. E. Karl, J. C. Polanyi, A. C. Roach, and M. A. Robb, J. Chem. Phys., 1970, 52, 6205. 221

Ab initio Calculation of Potential Energy Surfaces

55

GTO was centred on each atom in the SCF calculation. A CI calculation was then performed using 120 configurations involving 15 of the molecular orbitals. The excitations were to the virtual orbitals produced by the SCF method. The equilibrium geometry was found to be an equilateral triangle with bond lengths of 1.66 a.u. which is in general agreement with previous ~ o r k . A ~ study 3 ~ of the effect of changes in the basis set showed that the addition of a singlep-type function resulted in a reduction in the energy of the system by cn. 0.02 a.u. at the equilibrium geometry before CI. The effect is even greater with the CI wavefunction. The SCF energy at the equilibrium geometry (with a 5s,2p GTO set) is reported to be - 1.29930 a.u. which, after CI, is reduced to -1.3397 a.u. Similar CI calculations performed at collinear symmetric geometries produced a minimum energy of - 1.2765 a.u. at an H-H separation of 1.54 a.u. Csizmadia and co-workers233 then used a smaller basis set (5sJp) in a CI calculation at 250 points on the H: potential surface. Threedimensional classical trajectory calculations for the reaction D + + H2+H + + HD were then carried out. In a subsequent work, Preston and TullylS discuss the limited value of these latter calculations due to the neglect of a non-adiabatic interaction of the ground state, with a second state which describes the dissociation of the system into the products H + HD+. More recently, Carney and P 0 r t e r ~ 3used ~ a GTO basis set of 21 functions, six of which have centres outside of the molecule, to calculate the potential energy for 42 equilateral geometries of H:. A limited optimization of the exponents was performed. The SCF energy at a calculated internuclear distance of 1.64 a.u. was found to be - 1.299 273 a.u. A second calculation involving a 309-configuration CI gave an energy of - 1.339 358 a.u. at R = 1.649 984 a.u. The close agreement with the work of Csizmadia et al.231 is obvious even to a comparison of the cuts through the two potential surfaces which are shown in ref. 234. There is also good agreement with the energy and equilibrium bond lengths predicted by Salmon and P o ~ h u s t a . ~ ~ ~ The avoided intersection of the two potential surfaces in the H i system was studied in detail in a full CI calculation by Bauschlicher et aZ.236 An uncontracted 5s and 2p GTO basis set was used. The lowest surface was found to be attractive, with a binding energy of 444 kJ mol -1 for relative to H and €12. The second surface which represents H H;f is reported to be repulsive in nature. He + H;-+HeH+ + H.-Two different approaches of He to the H i ion were investigated by Edmiston et aZ.116 using the same method they applied to H3.184. 209 An uncontracted GTO basis set with polarization functions ( 6 4 2pz, 3pz,von helium and 4s, lp,, 3px,tlon each hydrogen) was used and pseudonatural orbitals were calculated for the 102 pair. A 65-configuration CI followed in which ca. 90 % of the correlation energy was accounted for. The energies of 22 different linear He-H-H + geometries were calculated. A minimum which lies about 21 kJ mol-1 below the separated reactants is found at He-H and H-H+ bond distances of ca. 2.0 a.u.

+

232

z33 234

235 236

Hl

+

R. E. Christoffersen, J. Chem. Phys., 1964, 41, 960; W. Kutzelnigg, R. Ahlrichs, I. LabibIskander, and W. A. Bingel, Chem. Phys. Letters, 1967, 1, 447. I. G. Csizmadia, J. C. Polanyi, A. C. Roach, and W. H. Wong, Canad. J. Chem., 1969, 47, 4097. G . D. Carney and R. N. Porter, J. Chem. Phys., 1974, 60, 4251. C. Salmon and R. D. Poshusta, J. Chem. Phys., 1973, 59, 3497. C . W. Bauschlicher, jun., S. V. O'Neil, R. K. Preston, H. F. Schaefer, and C. F. Bender, J. Chem. Phys.,:1973, 59, 1286.

56

Theoretical Chemistry

The barrier height is about 75 kJ mol-l. The authors consider this to be a type of hydrogen-bonding arising from the polarization of the He atom by the HZ. A potential energy barrier of 105 kJ mol-1 is found for the dissociation of the symmetric H at H-H and He-H distances of ca. 2.1 a.u. complex H-He-H+ into HeH+ (12 points were calculated for this geometric configuration). Comparing this to the value for the linear approach, forces one to rule out the insertion of He into the H-H+ bond along a CzVpath as a reaction path. Note that an electron pair is neither made nor broken along the reaction path. As discussed earlier in this chapter, one would expect that in this case a HartreeFock wavefunction would be sufficient to describe the system. The results of Edmiston et al. show that this is indeed the case, for the correlation energy is found to be approximately constant along the reaction path. Thus, the HF and CT paths are nearly parallel. With this in mind, a non-CI calculation was performed at a large number of points (82) in the He-H-H+ system by Brown and Hayes.185 The results are in general agreement with Edmiston et al. An He-H-H bond angle variation at four values of the angle at the minimum energy bond lengths indicates that the linear form is most stable. KuntzZ3'used the diatomics-in-molecules(DIM) approximation to fit a function to the surface of Brown and Hayes.ls5 The resulting collinear potential surface is in qualitative agreement. However, extrapolation to non-linear geometries appears to indicate that the isosceles geometries are energetically as favourable as the linear ones, which is in disagreement with Brown and Hayes. Clearly, further ab initio work on the non-linear geometries of this system is needed. HeH2.4ordon and SecrestllO used various numbers of STO's in an SCF-CI calculation on the HeH2 system. The smallest basis set used was composed of two s-type functions on He and one s-type function on each of the hydrogens. This set was sequentially built up to a set of two s-type and threep-type functions on He and one s-type and threep-type functions on each hydrogen. The combined energy of the separated species, helium atom plus hydrogen molecule, was calculated with a.u. from the Hartree-Fock limit. At a the largest basis set to be within 6 x fixed H-H distance of 1.40 a m , the distance of the He atom to the centre of the H-H bond (X) and the angle 8 between X and the H-H internuclear axis was varied from 2.8 to 7.0 a.u. and from 0 to go", respectively. Eighty configurations were mixed in a CI. This latter wavefunction gave a total energy of the separated products of -4.04921 a.u. which is 99.3 % of the experimental energy for He H2. The effect of orbital optimization was studied with the smaller basis set at X = 2.80 and 3.80 a.u. and with 8 at 90 and 0".Only at the shorter distance at the linear geometry is there a change in interaction energy, the effect at the other three geometries being negligible. This effect is seen with and without CI. Gordon and Secrest used an analytical function in the form proposed by Roberts2S8 and Krauss and Mies239to approximate the interaction potential as a function of X, 8, and the H-H internuclear distance. A plot of their results closely approximates

+

+

237

238 239

P. J. Kuntz, Chem. Phys. Letters, 1972, 16, 581. C . S . Roberts, Phys. Rev., 1963, 131, 203. M. Krauss and F. N. Mies, J. Chem. Phys., 1965, 42, 2703.

Ab initio Calculation of Potential Energy Surfaces

57

the experimental results240 in the range 2.8 < X <5.2 a.u. It is also found that at the small X values, the interaction energies found with and without CI are very similar. Note that this is a situation in which the products of dissociation are both closedshell systems and therefore another case in which an SCF potential energy surface properly described the dissociation. The interaction potentials calculated by Gordon and Secrest have subsequently been used in scattering calculations by Eastes and Secrest 241 and Alexander and Berard.242 Electron density maps at various points on the reaction path were generated by Gordon, Secrest, and L l a g u n ~using , ~ ~ the ~ Gordon-Secrest SCF wavefunctions. Schaefer et al.244investigated the interaction potential between ground-state He and the B state of HZusing a (6sY2p)GTO basis on helium contracted to a [3sY2p]set. For the hydrogen molecule, a (4s) set was contracted to [2s]and then an additional s function was added and optimized for the B state of H2 at R = 2.43 a.u. in order to describe properly the very diffuse nature of the 1cu orbital. As many as 706 configurations were included in a CI calculation. A large van der Waals interaction between He and the B state of H2 is found at R = 3.72 a.u. with a depth of 0.05 eV. H2 + D2-+2HD.-The theoretical search for the probable shape of the transition state in the four-centre exchange reaction Ha D2+2HD has been under way for many years (a review of the earlier work in this field may be found in ref. 245). The controversy is centred on the inability of thorough ab initio calculations to find a mechanism for the reaction with a barrier height as low as that suggested on the basis of shock-tube experiments.246~247 Three studies were done recently, two with ~~ de full CI 245,248 and one explicitly using interparticle c o - o r d i n a t e ~ ,a~method scribed above in conjunction with the work of Conroy and Bruner207on H3. Wilson and Goddard248 used a minimal Slater basis set (one ls-type function on each hydrogen) in a full (36 configuration) CI. A doublezeta basis set (two ls-type Slater functions on each hydrogen) was used in the CI calculations (up to 176 configurations) of Rubenstein and S h a ~ i t t . ~Although 4~ the energies of the latter work are lower, both groups report results which are essentially the same. Table 7 shows that the barrier heights reported by Rubenstein and Shavitt relative to two H2 molecules at infinite separation are, except for the linear system, all above 400 kJ mol-1. Wilson and Goddard 248 investigated three additional configurations, a regular trapezoid, centred equilateral triangle and a kite-shaped molecule. The barrier heights for these systems were also over 400 kJ mol -l. The problem that arises in these works is that the shock-tube experiments indicate that the activation energy for the reaction is only 176 kJ mol -l. As can be seen

+

240

241 242

243 244

z45 246

247 248 249

I. Amdur and A. P. Malinauskas, J. Chem. Phys., 1965, 42, 3355. W. Eastes and D. Secrest, J. Chem. Phys., 1972, 56, 640. M. A. Alexander and E. V. Berard, J. Chem. Phys., 1974, 60,3950. M. D. Gordon, D. Secrest, and C. Llaguno, J . Chem. Phys., 1971, 55, 1046. H. F. Schaefer, D. Wallach, and C. F. Bender, J. Chem. Phys., 1972, 56, 1219. M. Rubinstein and I. Shavitt, J. Chem. Phys., 1969, 51, 2014. S. H. Bauer and E. Ossa, J. Chem. Phys., 1966, 45, 434. A. Burcat and A. Lifshitz, J. Chem. Phys., 1967,47,3079; D. Lewis and S. H. Bauer, J. Amer. Chem. SOC.,1968,90, 5390. C. W. Wilson, jun. and W. A. Goddard, J. Chem. Phys., 1969, 51, 716; 1972,56, 5913. H. Conroy and G . Malli, J. Chem. Phys., 1969, 50, 5049.

58

l'koretical Chemistry

Table 7 Barrier heights calculated by Rubinstein and S h a ~ i t for t ~ various ~~ transition states for the reaction H2 -+ D2 -+ HD System

Linear equidistant Rectangular (4 : 5) Square Rhombic (70")

Tetrahedral

Rmin/a.u. 1.7 2.09; 2.61 2.47 2.5 3.8

AE/kJ mol-1 180 456 594 632 757

from Table 7, the linear path has approximately the right barrier height. Note, however, that this path cannot serve as a transition state for the four-centre exchange reaction. The other systems produce barrier heights over twice as great as the experimental results. In fact, from the ab initio results, one would be led to the conclusion that the breaking of an H2 bond (ca. 448 kJ mol-l) followed by the reaction H + D2+D + HD would be energetically more favourable. Transition states which correspond to two unequal-length HZmolecules are consistent with the mechanism proposed by Bauer and co-workers 246 which involves a vibrationally excited Hz molecule reacting with a normal one. Wilson and Goddard's investigation of systems with unequal H-H bond distances, however, proved to be equally unfruitful. As stated before, one cannot directly equate the barrier heights to experimental activation energies; trajectory calculations must be performed on the surface in order to arrive at a theoretical activation energy. One can make an order-of-magnitude comparison, however, by noting that in the H2 + H system the activation energy was calculated213to be lower than the barrier height by about 8 kJ mol-l. A lowering of this order of magnitude will certainly not alter the situation. Tapia and Bessis250 investigated four geometric conformations in a CI study of H4 in which polarization functions were added, Although their results do not shed further light on the problem of the mechanism of the reaction (a very small set of geometric variations were investigated) they did find a minimum in the van der Waals region for the linear configuration close to the one predicted by Gordon and Cashion.251 Bender and Schaefer252 performed a set of full CI calculations on the linear symmetric H4 system using a GTO basis set which included p-functions. More recently, Silver and Stevens253extended the work even further by optimizing the orbital exponents and including more geometrical configurations in the study (the perpendicular approach, an alternative kite, and a parallelogram). Although a path was still not found in these latter two works which allowed a bimolecular mechanism for exchangerequiring less energy than a single H Zdissociation,Silver and Stevens253 found a path which requires c 25 kJ mol -1 of energy above this limit. This path is one which leads from trapezoidal to linear structures. In calculating portions of potential energy surfaces, it is only natural, considering the computer time used for ab initio calculations with CI, for investigators to choose 250

251

2s2 253

D. Tapia and G . Bessis, Theor. Chim. Acta, 1972, 25, 130. R. G. Gordon and J. J. Cashion, J. Chem. Phys., 1966, 44, 1190. C. F. Bender and H. F. Schaefer, J. Chem. Phys., 1972, 57, 217. D. M. Silver and R. M. Stevens, J. Chem. Phys., 1973, 59, 3378.

Ab initio Calculation of Potential Energy Surfaces

59

to study first those portions of the surface with high symmetry. In the present case, however, it appears that a complete hypersurface for H4 may have to be calculated before the question of the mechanism of the exchange reaction is fully resolved. HeH+ + Ha+He + H$.-If the problems of the potential energy surface of the tetra-atomic H4 system are difficult to surmount, the HeHf system which, because it is heteronuclear lacks the symmetry of the former case, could have proven to be intractable. Nevertheless, three groups in recent years have attacked the problem of the reaction HeH+ + H2+He + H l using ab initio rnethods.104~254~255 The most recent of these is the work of Benson and McLaughlin.104 While this work was discussed earlier, details of their calculation are given here. An uncontracted GTO basis set composed of 9s- and lp-type function on He, and 4s- and lp-type function on each hydrogen (a total of 33 functions) was used. An estimate of the completeness of this set was obtained by calculating the single-determinant SCF energies of the four dominant fragments in the reaction: HeH+, Hz, He, and Hi. The results approached within 0.003 a.u. (8 kJ inol-1) of the Hartree-Fock values. As in the H Icase, the major problem that had to be resolved was the geometry of the reaction path. For this purpose the following four paths were investigated:

czw:

HeH+ approaching perpendicularly (with the H leading) to the Hz bond on the approach, and He leaving the triangular H8+ on the same path in the reverse direction. He brought into the centre of the H i triangle along a line 30" off Cs(planar): the altitude of the triangle and at various points the H2 was made to recede from HeH+. Cs(non-planar): Similar to CzVexcept that the He atom was tilted out of the H,f plane. c3u: He leaving perpendicularly from the centroid of the H,+triangle.

It was found that the CzUpaths were the lowest in energy. On the CzWpaths, three co-ordinates are needed to define the system. Benson and McLaughlin defined the following: the H-He distance R ; the distance r between the hydrogens of what originally is the H2 molecule; and the distance h from the hydrogen which was originally bonded to He to the centre of the axis of the other two hydrogens. Ten values of R and h (nine between 1.4 and 5.0 a.u. and one at 50.0 a.u.) were considered and three values of r (the calculated Hz equilibrium distance of 1.3950 a.u., the caIculated H: equilibrium distance of 1.6405 ax., and an intermediate value of 1.5000 a.u.). Thus, 300 points on the potential energy surface were computed. The energies at three points on the surface (see Table 8) were then recalculated. after optimization of the Gaussian orbital exponents. Point A represents conditions near the beginning of the reaction. Point C is near the end of the reaction and point El is near the middle of the reaction. The total energy was found to change by no more than 1% and since the lowering is uniform at these three points, it was assumed that it would remain uniform over the entire surface. 254

255

I. Funke, H. Preuss, and G. Diercksen, Mol. Phys., 1967, 13, 517. R. D. Poshusta, J. A. Haugen, and D. F. Zetik, J. Chem. Phys., 1969,51, 3343.

60

Theoretical Chemistry

Table 8 Co-ordinates (1a.u.) of three points on the Czv surface of HeH: selected for detailed investigation by Benson and McLaughlinl O4 Point

A B

c

a

R(HeH +) 1.4650a 2.oooo 4.oooo

Equilibrium value for HeH+.

h 4.94717 1.94717 1 .42075b

Equilibrium value for H + .

1.375OC 1SO00 1.640S’ Equilibrium value for H2.

+

The reaction HeH+ + H w H e + H$, like that of He + H$-+HeH+ H, does not involve the making or breaking of electron pairs. As such, the HartreeFock results should parallel those of a CI calculation. To verify this, Benson and McLaughlin performed a CI using the iterative natural orbital method for the reactants at infinite separation, the products at infinite separation and at the points defined in Table 8. The five helium s functions with the largest exponents were contracted to a single s function using helium atom orbital coefficients. The computed correlation energies, which accounted for 80% of the total amount, were found to range from 0.0730 to 0.0767 a.u. indicating, as noted before, that the Hartree-Fock results are indeed nearly parallel to the correlated results over the entire surface. This reaction was found to be exothermic by 2.63 eV (254 kJ mol-1) in general agreement with the 2.3 eV found by Funke, Preuss, and Diercksen 254 using a restricted basis set. These latter workers predicted a path in which the H2 molecule lies initially on the HeH+ axis but swings perpendicular to this axis as the reaction proceeds, i.e. ending up with the geometrical configuration which Benson and McLaughlin use throughout the path. The results of Benson and McEaughlin104 indicate that no barrier to reaction exists, nor do they show a minimum, which is in contrast to the stable HeH; complex bound by 0.44 eV reported by Poshusta, Haugen, and Zetik255 using a minimal valence-bond CI technique. Hez-+He; + He.-Vauge and Whitten%6using a GTO basis set with polarization functions did single and double excitation CI calculations on the He: system and determined that the linear symmetric structure was more stable than the equilateral triangle. The minimum energy with respect to symmetric dissociation was found to be -7.83198 a.u. at an internuclear distance of 2.340 a.u. The dissociation energy of 0.07 eV which, although low compared with the experimental value reported by Patterson,267 (0.17+. 0.03 eV) verifies that the ion is loosely bound. F + H2+FH + H.-Bender et ~21.~5~ recently reported the results of two different sets of calculations on the potential energy surface for the reaction F + H2-’ FH H. This work is of interest from three points of view. First, it is truly a rigorous quantum mechanical calculation typifying the current state of the art for systems of this size: a GTO basis set of double-zeta quality is used (9s5p/4s)for a total of 32 GTO’s, and this is contracted according to the Dunning method51 to

+

256

257 258

C. Vauge and J. L. Whitten, Chem. Pfiys. Letters, 1972, 13, 541. P. L. Patterson, J. Chem. Phys., 1968, 48, 3625. C. F. Bender, P. K. Pearson, S. V. O’Neil, and H. F. Schaefer, J. Chem. Phys., 1972,56,4626; C. F. Bender, S. V. O’Neil, P. K. Pearson, and H. F. Schaefer, Science, 1972,176, 1412.

Ab initio Calculation of Potential Energy Surfaces

61

[4s2p/2s]:polarization functions are added (d’s on the fluorine andp’s on hydrogens) yielding a basis set of 48 GTO’s - (9sSp2d/4slp)contracted to [4s2pld/2slp]:and the iterative natural orbital (INO) technique is used on 214 configurations in a CI calculation - typically four iterations were needed to obtain convergence. These calculations were performed at each of 150 linear geometries using both basis sets and 200 non-linear geometries using the [4s2p/2s] basis set for the FH2 system. Secondly, the final potential surface produced should be of high enough accuracy to allow for dynamical studies of the reaction. Thirdly, the values of the properties of the surface computed in four different ways can be compared.

Table 9 Summary of the results of Bender et al.258on the reaction F in diferent ways Without polarizationfunction SCF CI

Barrier height/ kJ mol-1 Exothermicity/ kJ mol-1 Saddle-point geometry/A F-H H-H Total energy/a.u. F H2 Total energy/a.u. FH + H

+

With polarizationfunction SCF CI

14.4

24

123

-2.4

85

55

1.06 0.81

1.37 0.81

+ Ha culculuted

1.18 0.836

6.9

144 1.54 0.767

Experimental 7.1s

133b L

- 100.5208

- 100.5424

- 100.5257

- 100.5620

-

- 100.5191

-

100.5726

- 100.5470

- 100.6169

-

A. F. Trotman-Dickenson,J . Chem. SOC.,1960, 1064.) J. Berkowitz, W. A. Chupka, P. M. Guton, J. M. Holloway, and R. Spohr, J. Chem. Phys., 1971,54, 5165. a Experimental activation energy (G. C. Fettis, J. H. Knox, and

The minimum energy path was found to be the one for the linear arrangement F-H-H. Table 9 shows the results of the four different calculations: the basis set lacking polarization functions with and without CI, and the same basis set with polarization functions with and without CI. If we compare the exothermicities we see that one needs the added flexibility of the d functions on F and the p functions on the H’s, plus a rather extensive CI in order to approach the experimental value. Mu~kermanz5~ suggests that the barrier height should be cu. 2.9 kJ mol-1 less than the activation energy. This would give a barrier height of ca. 4 kJ mol-1 and again the CI calculation with polarization functions gives the best value. The great improvement in the total energies of the system F + H2 with the F at infinite separation and the system FH H with the H at infinite separation further illustrates this point. Note in particular the difference in the effect of the polarization functions on the reactants and the products. The energy is lowered by only 0.0195 a.u. for F H2 but the decrease is 0.0443 a.u. for FH + H. This, of course, accounts for the greater exothermicity reported for the final calculation, a value very close to the reported experimental value.

+

+

x59

J. T. Muckerman, J. Chem. Phys., 1971, 54, 1155.

62

Theoretical Chemistry

The saddle point occurs early in the inlet channel with and without CI but earlier in the former case, corresponding to a small displacement from the H-H equilibrium bond distance (ca. 0.03 a.u.) while the H-F distance is ca. 1.1 a.u. greater than that in HF. H F2-+HF F.-After their success with the 11-electron H2F system, Bender and co-workers1f5then attacked the 19-electron HF2 problem. A double-zeta GTO basis set is again used, (9s5p on F and 4s on each H contracted to a [4s2p/2s]set, a total of 52 basis functions in this case). The addition of the same polarization functions as were used in the H2F surface would increase the basis set to 75 GTO’s and would make the calculation of as many points needed to define the potential energy surface prohibitively time-consuming. Noting that the non-polarization function results of H2F were at least of qualitative accuracy, Bender et al. decided to do what they called a ‘first study’ on this system leaving the polarization function out. Using the I N 0 method, 555 carefully selected configurations were mixed in a CI calculation. Linear and non-linear approaches of the H atom were investigated and, as in the H2F case, the results predicted a linear approach. The saddle point in this case again occurs early in the inlet channel, the F-F bond having separated by only 0.06 a.u. while the F-H distance is more than twice that for the isolated diatomic. Table 10 summarizes the results of this work. The exothermicity predicted by the

+

+

Table 10 Summary of results of Bender et al.l15on the reaction H Barrier height/kJ mol-l Exothermicity/kJ mol-l Saddle-point geometry/A H-F F-F a

Experimental activation energy (ref. 260).

SCF

CI

51

4.2 369

554 1.56

1.49

+

Fa

Exper-intental lo? l a 429b

2.05 1.57

I

-

See footnote 12 in ref. 115.

CI calculation is found to be smaller than the experimental value by 60 kJ mol --I. A glance at Table 9 will show that without polarization functions, the exothermicity in the I: Hz was too small by the same order of magnitude (48 kJ mol-l). The activation energy reported in Table 10 is, of course, not exactly comparable to the barrier heights reported here. Bender et al. report the results of an unpublished work by Muckerman wherein he reports that the barrier height in this case should be 5 kJ mol-I less than the activation energy for this system. The experimental activation energy is reported to be lo+ 1 kJ mol -1,260 and this correction gives an experimental barrier height of ca. 5 kJmol-1 which is in excellent agreement with the barrier height computed with the CI wavefunction. Comparison of the poor results obtained without polarization functions in the H2F work with these results leads one to believe that the good agreement with experiment may just be fortuitous. On the other hand, one may also conclude that the basis set produces an entrance

+

260

R. G. Albright, A. F. Dodonov, G. K. Lavronskaya, I. I. Morosov, and V. L. Tal’rose, J. Chem. Phys., 1969,50, 3632.

Ab initio Calculation of Potential Energy Surfues

63

+

channel H F2 -+ H-F-F which is more closely parallel to the true entrance HF to the true exit channel than is the calculated exit channel H-F-F -+ H channel. This would account for good agreement with the experimental barrier height in spite of poor agreement with the experimental exothermicity. Surfaces for X + M2 and M + Xs.-The potential energy surfaces for the two exothermic reactions considered immediately above, F H2 and H F2, are repulsive in the nomenclature of Polanyi,l4915 i.e. a surface in which most of the exothermicity is released as the products separate. The other possible extreme of behaviour of a potential energy surface for an exothermic reaction is denoted by characterizing the surface as being of the ‘early downhill’ or ‘attractive’ type.l4~l5 In these systems, most of the exothermicity of reactions appears as vibrational excitation of the newly formed bond. The alkali metal (M), halogen (X) reactions of the type X M2 + MX M and X2 MX X M

+

+

+

+

+

+

+

both possess this type of potential surface, and both have been the subject of experimental studies by molecular beam techniques. Struve, Kitagawa, and Herschbach261have investigated the reactions of C1 atoms with Na2 and Kz, in which they observed electronic excitation of alkali atoms. Zare and co-workersz62 have performed crossed molecular beam experiments on the I K2 system, in which they detected a chemiluminescence spectrum which may be due to a bound IK2 species. Miller, Safron, and H e r s c h b a ~ h ,in~ ~crossed-beam ~ studies of the reaction M2 M1 + M2X + MlX

+

+

have established the importance of a long-lived collision complex XMlMZ in the interpretation of the dynamics of these simple reactions. The family of reactions, XZ -+ MX X, has been extensively studied by crossed molecular beam M techniques.264 Potential energy surfaces for both types of reactions have now been calculated by ab initio techniques; for the system Li + F2 -+ LiF + F by Balint-Kurti and Liz -+ LiF + Li by Pearson et aZ.267 KarplusZ65p266and for F Pearson et aZ.267 have carried out a single configuration SCF and a twoconfiguration SCF calculation of the potential surface for the FLi2 system. They point out that the single configuration SCF approximation itself, in general, provides a relatively good description of ionic species, such as are involved in this reaction. The added configuration, for most regions of the surface, insured the correct

+

+

+

W. S. Struve, T.Kitigawa, and D. R. Herschbach, J. Chem. Phys., 1971, 54, 2759. R. N. Zare, unpublished work referred to in ref. 267. 26-W. S. Struve, MoI. Phys., 1973, 25, 777. 263 W. B. Miller, S. A. Safron, and D. R. Herschbach, Discuss. Faraday Soc., 1967, 44, 108. 264 J. H. Birely, R. R. Herm, I S.R. Wilson, and D. R. Herschbach, Discuss. Faraduy SOC.,1967, 44, 176; R. Grice and P. B. Empedocles, J. Chern. Phys., 1968,48, 5352. 265 G. G. Balint-Kurti and M. Karplus, Chern. Phys. Letters, 1971, 11, 203. G. G. Balint-Kurti, MoI. Phys., 1973, 25, 393. 267 P. K. Pearson, W. J. Hunt, C . F. Bender, and H. F. Schaefer, J . Chern. Phys., 1973,58,5358. 261 262

64

Theoretical Chemistry

asymptotic description of the system for infinite separation of all three nuclei. Although the added configuration does not properly describe this separated atom limit for the portions of the surface describing CzUgeometries of LiFLi, it does provide the most important single correction to the wavefunction in the region of the predicted equilibrium geometry for this species. A large s-p basis set (doublezeta quality) of GTO’s was employed with added polarizing d-type functions on F. In addition, a diffuse set of p-type functions was added to the F basis set. Duke and Baderas have pointed out that without this added diffuse function, Dunning’s (9~,5p)-+[4~,2p] set51 for F provides a much better description of the neutral F atom than it does of the F - ion, with the consequence that even the Hartree-Fock estimate of the electron affinity of fluorine is not obtainable with the use of the unaugmented set for the description of both F and F-. Use of the augmented set yields 97 % of the Hartree-Fock estimate of the electron affinity of fluorine.88 A minimum energy path was obtained for the collinear reaction. However, a point of particular interest in these calculations was a search of the potential surface to verify the postulated existence of a stable collision complex263or even a stable molecule262 for the system FLi2. Both the single- and double-configurations predict a potential well of ca. 17 kJ mol -l depth with respect to LiF + Li for the linear F-Li-Li species. A much deeper well with a depth of ca. 34 kcal mol -l was found for the LiFLi speciesof CZ2, symmetry. Struve,262aby employing the pseudopotential method of Child and Roach,23 has calculated a potential energy surface for the FLiz system. He obtains a collinear well of ca. 38 kJ mol -1 for F-Li-Li and an isosceles triangle well depth of ca. 84 kJ mol -l. While the reliability of the ab initio results are difficult to assess in this case, the authors267 conclude that certain salient features of their surface findings could be included in a LEPS-type formulation of the surface and employed in a dynamical study of the F + Liz and/or LiF Li reactions to obtain information concerning the importance of complex formation in such systems. In the theoretical determination of the potential surface for the LiFz system by Baht-Kurti and Karplus2659 266 a multistructure valence-bond approach was used. This method of calculation was augmented by the so-called orthogonalized Moffit method. This latter technique was applied to correct for the relatively large errors made in the ab initiu description of F - relative to F. Their calculation provided the first fully theoretical prediction that the potential surfaces for the M XZsystems are indeed of the ‘attractive’ or ‘early downhill’ type. Model studiesl4*I5had indicated that the observed high vibrational excitation of the newly formed M-X bond was a characteristic feature of such surfaces, a point which has now been theoretically substantiated. The dynamic consequences of this calculated surface are X2 reactive systems. The compared266with those of previous models for the M possibility of non-adiabatic contributions to the reactive scattering is also considered.266 CH2 + CH2-+C2H4.-The coplanar approach of two methylenes to form ethylene was investigated by Basch268 using the MCSCF method. The states of bent methylene that correlate with the ground state of ethylene are the triplet states. It is found that for two closed-shell singlet-state methylenes,the reaction path is purely repulsive.

+

+

+

a*8

H. Basch, J. Chem. Phys., 1971, 55, 1700.

Ab initio Calculation of Potential Energy Surfaces

65

The Valence Bond Approach.-In two papers on the potential energy surfaces of BeHl and H4+,Poshusta and c 0 - w o r k e r s ~ ~ ~used ~ 2 ~a0 combination of SCF and valence bond configuration interaction (VBCI) methods. For the reaction Ha Be++BeH,+,2G9a large portion of the potential energy surface was evaluated with the SCF method and then the VBCI technique was applied along the reaction path thus defined. For the lowest state (2A1symmetry) a minimum is found on the Cavpath with H-Be-H angle of 20.5" and BeH distance of 4.17 a.u. The binding energy is reported as 0.11eV. In these calculations, ca. 40 % of the correlation energy was accounted for. For the reaction Ha + Ha+-+Hs+ H,270 VBCI calculations were performed on all the points considered. Five attitudes of approach of H2 and H i and three attitudes of retreat of H i and H were considered, while the geometries of H2, H i , and H,+were fixed at their respective calculated equilibrium values. The steepest approach potential is found to be one in which the H i axis bisects the H2 axis. The steepest retreat potential is one in which the H atom leaves the triangular H,+ ion on the same axis on which it entered. With this general reaction path defined, the calculations were repeated with the optimization of the H-H bond distance. A minimum was found at H-H and H-H + distances of 1.66 a.u. and 4.1 1 a.u., respectively, while the distance from the leading H atom of the H i ion to the bisector of the H2 molecule is 1.44 a.u. The binding energy is found to be 3.6 kJ mol-l. Close examination of the studies reviewed in this chapter reveals a very important fact. Although the general approaches used in all the problems discussed have their similaritiesand although general rules can be stated, the field of ab initio calculations of potential energy surfaces has not yet reached the point where the computations can be routinely carried out. Although it is true that if one picks a large enough basis set of either GTO's or STO's and performs a full CI at a large enough number of points on the hypersurface, one will be reasonably assured of obtaining meaningful potential energy surfaces, the problem of time expended posed by even the largest and fastest computers available today requires that each system be treated differently. Until the day either when newer and faster computational methods are devised which will attain the same degree of accuracy, or when computers are made with speeds many orders of magnitude greater than those in existence today, the quantum chemist who attempts to produce meaningful potential surfaces will have to study his problem very carefully. It is in this respect that one can call this area of theoretical chemistry an art.

+

+

269

270

R. D. Poshusta, D. W. Klint, and A. Liberles, J. Chem. Phys., 1971, 55,252. R. D. Poshusta and D. F. Zetik, J. Chem. Phys., 1973, 58, 118.

2 Intermolecular Forces BY J.

G. STAMPER

1 Introduction The theory of intermolecularforces poses the same problem as the theory of valence, but in a slightly different form. The fundamental difficulty in calculating bond energies is that they appear as small differences between large numbers (the total energies of the molecules and atoms concerned), so that a much greater percentage accuracy is needed in the calculations than is aimed at in the dissociation energy itself. In the case of intermolecular forces this problem is even more serious. Typical binding energies at the van der Waals minimum between non-polar molecules are of the order lo-* ax., so that an attempt to calculate such a quantity by direct comparison of the calculated energies of the two separated molecules and of the ‘supermolecule’ made up of the two together requires an accuracy of at least 1 part in lo7 even for quite small systems. Nevertheless, such calculations have been and are being made and some examples are referred to in Section 3. In principle, however, the intermolecular force problem has a compensating advantage. Because of the very small changes introduced by the interaction, at least until the repulsive part of the potential surface is reached, this is an ideal situation for the application of perturbation theory which has not, in general, been a particularly powerful tool in valence problems. Even so, except at large distance where the overlap of the wavefunctions of the two molecules is negligible, the application of perturbation theory raises some problems. In the main these were sorted out in the years around 1970, but work in this area continues and some of it is discussed in Section 2. Actual calculations that have been carried out are still confined almost entirely to atoms, so that the title ‘Intermolecular Forces’ is almost a misnomer, ‘Interatomic Forces’ being much closer to the state of the art. Some calculations are aimed primarily at understanding the short-range, repulsive region of interaction. These are discussed in Section 3, while Section 4 covers recent results on the long-rangeregion. A number of other topics are collected together in Section 5. These include systems containing more than two molecules, the hydrogen bond, the eRect of intermolecular forces on the properties of molecules, and a brief survey of recent work o n empirical potentials and on van der Waals molecules, which are likely in the future to provide more and more detailed information for the theoretician to interpret, 2 General Theory Almost all the work in this area recently has been concerned with the difficultiesof applying perturbation theory in the region in which overlap is non-negligible. The

66

Intermolecular Forces

67

origin of this difficulty can be thought of in several different ways, but it is perhaps most easily seen by considering the symmetry of the unperturbed Hamiltonian Ho. The natural choice for this Hamiltonian is that for the separated molecules, so that it will contain only that part of the total potential which is due to interactions of electrons with their ‘own’ nuclei and with the other electrons in their ‘own’ molecule. Such a Hamiltonian will not be invariant to the antisymmetrizing operator A which permutes all the electrons, including those belonging to different molecules. The total Hamiltonian H is of course invariant to this operator, so that we may write

AH-HA = 0 However, when we write

H = Ho+ V it follows that

AHo-HoA = VA-AV of which neither side is zero. Here we have an equality between an apparently zeroorder expression and an apparently first-order expression which shows that any simple perturbation theory in which the various terms are considered order by order is bound to fail. The various methods of surmounting this difficulty which have been developed in the past few yearsl-5 have been brought within a common framework by Chipman, Bowman, and Hirschfelder.6 They start by writing the exact wavefunction for the pair of interacting molecules which has a definite symmetry as vy = ~ A @ ( Y ) where v labels the particular irreducible representation to which yy must belong, and vA is the corresponding projection operator. The equation which @(v) must satisfy is

which only defines the component of O(v) in the space of functions of The authors express this by writing

Y

symmetry.

( H - E ) @ ( y ) = (1 - ’A)F(v) which must hold for any F(v) whatever. They show that the various kinds of perturbation theory correspond to different choices of F(v) and develop a formalism for choosing an optimum F(v), i.e. an optimum form of perturbation theory. In the succeeding paper they apply this formalism to the case of He * OH+and provide a detailed discussion of the relative success of the various theories, i.e. the various possible choices of F(v). None of them seems to be conspicuously preferable and the reader is referred to the paper for details. By choosing a variationally 1 2 8

4 6 6 7

J. 0. Hirschfelder and R. Silbey, J. Chem. Phys., 1966, 45, 2188. J. N. Murrell and G. Shaw, J. Chem. Phys., 1967,46, 1768. J. I. Musher and A. T. Amos, Phys. Rev., 1967,164, 31. A. van der Avoird, Chem. Phys. Letters, 1967, 1, 24, 41 1. J. 0. Hirschfelder, Chem. Phys. Letters, 1967, 1, 325, 363. D. M. Chipman, J. D. Bowman, and J. 0. Hirschfelder, J. Chem. Phys., 1973,59,2830. D. M. Chipman and J. 0. Hirschfelder, J. Chem. Phys., 1973,59, 2838.

68

Theoretical Chemistry

optimized form for F(v), Chipman and Hirschfelder obtain an energy to first order which is within 6 kJ mol-1 of the exact value at an internuclear distance of 1 a.u., which is quite remarkable for a perturbation theory. In addition to this very general theory, two other ways of dealing with the problem have appeared in the past year. Berrondos points out that the eigenvalues of the antisymmetrizing operator (A) are highly degenerate and that there are, therefore, an infinite number of orthonormal transformations that can be applied to a basis set of simple product functions to diagonalize the representation of A. He suggests ways in which a particular choice of transformation may be made before carrying out any standard kind of perturbation theory such as Rayleigh-Schrodinger in the transformed basis. It is not easy to see how practical an approach this is likely to be; certainly the transformations that the author chooses as examples are cornplicatedlooking ones which involve the continuum. Gouyet 9 has applied the formalism of second quantization and diagrammatic perturbation theory to the problem. In a basis of non-orthogonal orbitals, diagrammatic perturbation theory becomes more difficult; Gouyet follows previous authorslo~llin transforming to a new basis of ‘biorthogonal’ orbitals in which the Hamiltonian is no longer Hermitian. He is able to show that certain classes of diagrams corresponding to intramolecular terms cancel exactly, but it is not clear whether the formalism is likely to be computationally attractive. In a second paper l2 he extends his work to situations with degeneracy. A number of other formalisms of various kinds have appeared. Kirtman and his co-workers have extended their ‘distinguishable electron method’13 to the inter.~~ mediate distance region14 and also to a discussion of quadrupole t e r m ~ChanglS has applied this method to the case of nuclear permutation symmetry and used it for the repulsive regions of the H: and HHe2+ potential. A combined variational and perturbation method which is applicable at all internuclear distances has been described.17 The method is, however, limited to one-electron systems, and the authors themselves say that it will probably be difficult to extend to polyelectronic cases. Daudey et al.18 choose to develop perturbation expansions for the isolated molecules and for the two molecules together and compare terms in the two expansions to find which are intermolecular terms. The approach does not seem to be too promising since they find that for the model case of two helium atoms l9 they need to go as far as the fifth order of perturbation to get even a qualitatively correct form for the potential curve. This appears to be partly because of difficulties with intraatomic correlation effects, and partly because of the orthogonalization procedure that they employ. They also apply their formalism to the water dimer.20

*

M. Berrondo, Mol. Phys., 1973, 26, 329. J. F. Gouyet, J. Chem. Phys., 1973,59, 4637. D. J. Newman, J. Phys. and Chem. Solids, 1969, 30, 1709; 1970, 31, 1143. l1 M. Moshinsky and T. H. Seligman, Amer. Phys., 1971, 66, 31 1. l 2 J. F. Gouyet, J. Chem. Phys., 1974, 60, 3690. l 3 B. Kirtman and R. L. Mowery, J. Chem. Phys., 1971,55, 1447. l 4 T. J. Venanzi and B. Kirtman, J. Chem. Phys., 1973,59, 523. l5 T. J. Venanzi and B. Kirtman, J. Chem. Phys., 1973,59, 2158. l 6 S . - Y . Chang, J. Chem. Phys., 1973, 59, 1790. l7 T.-S. Nee, R. G . Parr, and S.-Y. Chang, J. Chem. Phys., 1973,59, 4911. J. P. Daudey, P. Claverie, and J. P. Malrieu, Internat. J. Quantum Chem., 1974, 8, 1. l9 J. P. Daudey, J. P. Malrieu, and 0. Rojas, Internat. J. Quantum Chem., 1974, 8, 17. 2 o J. P. Daudey, Internat. J. Quantum Chem., 1974, 8, 29. lo

Intermolecular Forces

69

The intermolecular potential, which is the objective of all the work mentioned so far, has several applications. One of these is in the understanding of the properties of gases, typically by the calculation of virial coefficients. Baer and his co-workers have been concerned with the possibility of going directly from the Hamiltonian of the system to the virial coefficients without calculating a potential function explicitly. Initially21 they confined themselves to the Coulomb forces, but they have now22 extended their formalism to include exchange effects. Their method consists essentially in working in an interaction representation, and making a Dyson expansion for the perturbation. They have also been able23 to obtain an expression for the virial coefficients in the zero-overlap approximation in terms of the polarizabilities of the isolated molecules at imaginary frequencies analogous to the familiar corresponding results for the dispersion f0rces.2~This work is very elegant, but whether it has any significant computational use remains to be seen. A few ‘loose ends’ have been tidied up. It has been generally believed that at the level of a restricted HartreeFock calculation, the interaction between pairs of closed-shell atoms must be asymptotically repulsive at large distances. This was shown to be true for a pair of helium atoms some years agoZ5 but has now been proved to be generally true.26 Steinera? has explained an apparent paradox in the treatment of the long-range interaction of two hydrogen atoms by means of the Hellman-Feynman theorem in which different orders of perturbation appear to be mixed. There continue to be calculations on model systems in order to look at the nature of the various terms which arise in the perturbation expansions. The work of Himwelder et al.? on H. -H+ is of this kind. Chdasiliski and Jeziorski have particularly concerned themselves with various second-order terms. They have obtained28 a multipole expansion for the exchangepolarization energy for H- H+, and they also give some approximations and limiting forms which give 99% of the exact value at the van der Waals minimum. In a further paper,29 they have investigated the second-order induction energy for the He OHsystem. This energy is, of course, asymptotically zero since the potential due to a neutral atom is zero at long distances. At short range, however, there is an exponentially decreasing potential due to the detailed structure of the atom, and it is the effect of this that the authors have considered. They have obtained an ‘exact’ ht-order wavefunction and secondorder energy using a Green’s function method. They find that most of the change in the wavefunction of one atom occurs near the nucleus of the other. At distances of the order of the chemical minimum, this part of the second-order Coulombic energy is much more important than the dispersion part, as is shown in Table 1. The authors also note that the combination of the Heitler-London energy and EEL at the chemical minimum is very close to the exact value, showing that E!t\ contains most

-

-

S. Baer and A. Ben-Shaul, J. Chem. Phys., 1972,56, 3773. Baer and E. Bergmann, J. Chem. Phys., 1974,60,440. 23 S . Baer and A. Ben-Shad, J. Chem. Phys., 1973,59,2229. 24 A. Dalgarno, Adv. Chern. Phys., 1967, 12, 143. 25 N. R. Kestner, J. Chem. Phys., 1968, 48, 252. aa A. M. Lesk, J. Chem. Phys., 1973, 59, 44. 27 E. Steiner, J. Chem. Phys., 1973, 59, 2427. 28 G. Cha€asidskiand B. Jeziorski, Internat. J. Quantum Chem., 1973, 7 , 63, 745. z9 G. Chalasihki and B. Jeziorski, Mol. Phys., 1974,27, 649. 21

a2 S.

70

Theoretical Chemistry

Table 1 Relative importance of induction and dispersion in Ha * * H29 Internuclear distance/a.u. EGblE&,

8 0.087

4

6 0.06

0.6

ca.

2 3.5

of the effects due to ‘ionic’ terms, and that exchange polarization and higher-order terms are relatively unimportant. 3 Short- and Intermediate-range Forces At long range, a perturbation method is the obvious choice for the calculation of intermolecular interactions. At short ranges the standard methods of valence calculation are an equally obvious choice, and quite a lot of work of this kind is being reported. This type of calculation is also being carried out at intermediate distances at which perturbation theories of the type mentioned in Section 2 are also appropriate. Straight self-consistent field calculations have been carried out on the interaction of water with neon and arg0n.~0Here it is possible to obtain some of the attractive contribution to the intermolecular force since there will be an inductive secondorder interaction caused by the large dipole of the water molecule. Such interactions appear at the SCF level and the authors find a minimum in the potential at an 0.- *Ne distance of 3.63 A, with a binding energy of 0.71 kJ mol-1. There is a shallower minimum in the case of argon. Calculations at a similar level of sophistication have been carried out on the Hz. -He system,sl primarily, however, from the viewpoint of the hydrogen atom recombination reaction. The authors comment on the fact that the potential they obtain can be fitted to within ca. 10%as the sum of pairwise interactions. An SCF calculation has also been carried out32 on the He. * -Be2+system. The repulsive potential agrees quite well with scattering data down to an internuclear distance of ca. 0.4& after which the results disagree. * Ne and Be Ar, while Kaufman 33 has reported similar calculations on Be Poshusta and Zetik in a valence-bond cal~ulation3~ find a small binding (3.6 kJ mol-l) between HZand HZ. The majority of calculations of this kind, however, go beyond the SCF limit, either by ordinary configuration interaction, or by a fully fledged multiconfiguration SCF calculation. Both of these methods are capable of including dispersion effects and are therefore to be expected to be useful at larger distances than the straight SCF calculation. Two provisos must be made at this point, both of which have been commented on by authors cited below. One is that the kind of excited configuration needed to include dispersion forces is rather different from that appropriate for intramolecular correlation, so that it is necessary, in calculations using relatively modest basis sets, to include additional functions of appropriate symmetry with rather small and carefully chosen exponents. The second is that, particularly in the region where interaction is small, a spurious binding effect may arise because the

-

--

-

3O 3l 32

39 34

-

M. Losonczy, J. W. Moskowitz, and F. H. Stillinger, J. Chem. Phys., 1973, 59, 32.64. C. W. Wilson, jun., R. Kapral, and G. Bums, Chem. Phys. Letters, 1974, 24, 488. S. W. Hamson, L. J. Massa, and P. Solomon, J. Chem. Phys., 1973, 59, 263. J. J. Kaufman, J. Chem. Phys., 1973,58,4880. R. D. Poshusta and D. F. Zetik, J. Chern. Phys., 1973, 58, 118.

Intermolecular Forces

71

basis functionscentred in one atom or molecule can provide additional flexibility for the wavefunction of the other atom or molecule, resulting in a lowering of the intramolecular or intra-atomic energy. Liu and McLean,35 taking considerable care to avoid the overcomplete basis set problem, have carried out a large configuration interaction calculation on He He. Their stated aim was an accuracy of 0.1 K and they calculate the depth of the van der Waals minimum as 9.23 K at an internuclear distance of 3.02 A. Tsapline and Kutzelnigg36 have made somewhat less accurate calculations on the H2- -He system around its van der Waals minimum. They attempt to avoid the problems outlined above by transforming their SCF orbitals to a localized basis and then calculating intermolecular pair correlation energies. They ignore the variation of intramolecular correlation with distance and, what is probably more important, the coupling between inter- and intra-molecular correlation. For the linear configuration there is a minimum 21 K in depth, and for the CZ,configuration there is a saddle point with a binding energy of 14 K. The result averaged over all orientations is slightly too low, while the anisotropy is slightly larger than the asymptoticvalue, which is supported by the experimentalevidence. A somewhat similar calculation 37 has been carried out on Hz- *HZ in four configurations. In this case a rather smaller basis set which included polarization orbitals with exponents adjusted to obtain the deepest van der Waals minimum was used. The authors believe that they have not eliminated all the spurious binding energy. All the data on the H2 - - - H2 potential at short range obtained by ab initio calculations have been reviewed by McMahon et R Z . , ~ who ~ discuss them from the point of view of scattering experiments. SCF-CI calculations have also been carried out for Ne - - * -Ne39in much the same way as in ref. 37, attempting to extend the calculations to quite large distances and including a d-orbital in the basis. The results at long distances cannot quite match the semi-empiricalmultipole ones. SCF-CI calculations40similar to those of ref. 36 have also been carried out on the systems He. * -HzO and He. * H Fto see how this type of calculation performs in a polar situation. The more difficult multi-configuration SCF method, which has already been applied to He. .He,41has now been applied to Ne. * *Ne42and to Ar. .H.43In each case only an approximation to the exact solution within the chosen basis set has been obtained in order to avoid the problem of spurious energies. The authors have been particularly interested to see how their results behave at the larger distances, They define a quantity AEvan der Waals as the difference between their total energy and the SCF energy, and find that it can be well represented by an expression of the c1o/R1O. In the case of Ne, their AEvan der wsals values are form C6/R6 + Cs/R8 within a few K of the calculations of Starkschall and Gordon.44 The c6 and c8

-

-

+

35 36 37 38 39 40

41 42 43 44

B. Liu and A. D. McLean, J. Chem. Phys., 1973,59,4557. B. Tsapline and W. Kutzelnigg, Chem. Phys. Letters, 1973, 23, 173. E. Kochanski, B. ROOS,P. Siegbahn, and M. H. Wood, Theor. Chim. A d a , 1973, 32, 296. A. K. McMahan, H. Beck, and J. A. Krumhansl, Phys. Reu. (A), 1974,9, 1852. E. Kochanski, Chem. Phys. Letters, 1974, 25, 381. H. Lischka, Chem. Phys. Letters, 1973, 21, 448. P. Bertoncini and A. C. Wahl, Phys. Rev. Letters, 1970, 25, 991. W. J. Stevens, A. C. Wahl, M. A. Gardner, and A. M. Karo, J. Chem. Phys., 1974, 60,2195. A. F. Wagner, G. Das, and A. C. Wahl, J. Chem. Phys., 1974,60, 1885. G. Starkschall and R. G. Gordon, J. Chem. Phys., 1971,54,663; 1972,56,2801.

72

Theoretical Chemistry

values for Ar * - * H agree with earlier ~ a l u e s . 245~The ~ He - . - - He calculation has also been refined46 to allow for the variation in intra-atomic correlation energy with distance. This has the effect of reducing the well depth by ca. 1.2 K. The well depths and positions are given in Table 2. Table 2 Potential minima calculated by multiconfguration SCF

Ne.

Well depth/K

Rmin/a.u.

10.79 39.20 48.22

5.60 5.824 6.75

He. *He46 sNe42

&. ..H43

The most extensive calculation reported recently on a simple system is that of Kolos and W o l n i e w i c ~on ~ ~the intermediate and long-range parts of the hydrogenhydrogen potential for the lowest singlet and triplet states. Using a 60 term variational function, they calculate the energies of the two states for internuclear distances between 6 and 12 a.u. By comparing their curves for the two states they can separate the exchange and polarization contributions to the interaction energy and compare their magnitudes and how well they are given by asymptotic approximations. Tables 3 and 4 present some of their results. It can be seen that the exchange energy Table 3 Analysis of the interaction energy (cm-1) for two H atoms ( X l C ; R1a.u. 6 8 10 12

Exchange energy - 112.6 -3.87 -0.118 -0.0035

Polarization energy -71.5 -8.30 - 1.800 -0.556

Asymptotica exchange -97.63 -3.671 -0.117 - 0.0034

Variationalb polarization - 71.47 -8.27 - 1.80 -

Total energy 183.088 12.168 1.918 0.559

Calculated from the expression490.821 x R5/2exp (-2R). Calculated in the same way as the total energy but leaving out all exchange terms in the wavefunction.

Table 4 Analysis of the interaction energy (cm-1) of two H atoms (b3Cfstate)47 Asymptotic exchangea Variational polarization 26.2 -4.60 -- 1.68

+

R1a.u. 6 8 10 a

See notes to Table 4.

b

First-order exchangeb f Variationul polarization 27.6 -4.84 - 1.70

First-order exchange Asymptotic second-order polarization 33.2

+

-5.6

- 1.74

Total calciilated energy 41.26 -4.42 - 1.68

Heitler-London wavefunction.

can be approximated well by the asymptotic formula in the singlet case and by a simple Heitler-London first-order calculation in the triplet state, but that the asymptotic approximation to the polarization interaction is unsatisfactory in the 45 46 47

W. D. Davison, J. Phys. (B)., 1969,2, 1110. P. J. Bertoncini and A. C. Wahl, J. Chem. Phys., 1973, 58, 1259. W.Kolos and L. Wolniewicz, Chem. Phys. Letters, 1974, 24, 457.

Intermolecular Forces

73

region of the van der Waals minimum for the triplet state (R = 7.85 a.u.). Taking only the leading term in the R-l expansion gives even poorer agreement. In significantly larger systems, the SCF level is all that is at present attainable. Clementi and his co-workers have been carrying out a series of SCF studies48of the interaction between singly-charged ions and a water molecule. Some of their results are presented in Table 5. Where comparison is available it can be seen that the agree-

Table 5 SCF data on the interaction of an ion with a water molecule48 Ion Li+ Na+

a

AHcalc.a/kJmol-l 140.6

K+

97.1 65.7

Fc1-

A Ug calca/kcal mol-1 98.5 49.6

For the process M=t,H,O = M=t

+

H,O.

AHexpt. 01b / k J mol-1 142.3 100.4

b

Re(M-0)la.u. 3.58

4.25

74.9

5.08

L F -H -0 4.5" 14.6"

4.75

6.26

Ref. 50.

ment with experiment is remarkably good. The investigation of the potential surfaces is very much more extensive in this work than is represented by the data in the table. Calculations for halide ions surrounded by a whole co-ordination sphere of water molecules have also been carried out51but only at the CNDO level. On this basis, the optimum co-ordination number is found to be 8. The system H2CO - Li+ has also been the subject of an SCF cal~ulation.~~ The binding energy is found to be 181 kJ mol-1 and the equilibrium 0 - * .Li distance 1.77 A. Both molecule and ion are extensively polarized. Calculations which do not involve the SCF approach are also being made. Conway and Murrell53have calculated the exchange energy for Nee *Nearound the van der Waals minimum by 'first-order' perturbation theory. They find that a very accurate SCF function for the atom is needed for a satisfactory result. 'Double-zeta' wavefunctions, for example, are not sufficient. They compare their results with the analytic fits to other kinds of calculation and find that they agree well with an SCF cal~ulation5~ but poorly with a Thomas-Fermi-Dirac calculation.55 This type of approach does not appear to be adequate near the potential minimum. Combining their calculation with the dispersion results of Starkschalland Gordon,44 Conway and Murrell obtain a potential curve which is in agreement with the experimental one to within the somewhat large experimental error. KochanskiSO has made similar calculations for the H2 - . H2 system, while Brumer and Karplus57 have applied the same kind of formalism to the interaction of alkali and halide ions, obtaining a

-

-

--

H. Kistenmacher, H. Popkie, and E. Clementi, J. Chem. Phys., 1973,58,1689,5627;59,5842. C. Herring and M. Flicker, Phys. Rev., 1964,134A, 362. 60 I. Diidit and P. Kebarle, J. Phys. Chem., 1970,74, 1466. 61 B. Kh. Bunyatan, D. A. Zhogolev, and F. Ritschl, Chem. Phys. Letters, 1974,24, 520. 6% P. Russegger and P. Schuster, Chem. Phys. Letters, 1973,19, 254. 6% A. Conway and J. N. Murrell, Mol. Phys., 1974,27, 873. 54 T. L. Gilbert and A. C. Wahl, J. Chem. Phys., 1967,47, 3425. 66 A. A. Abrahamson, Phys. Rev., 1969,178,76. 66 E . Kochanski, J. Chem. Phys., 1973,58,5823. 67 P . Brumer and M. Karplus, J. Chem. Phys., 1973,58, 3903.

48

49

74

Theoretical Chemistry

formal derivation of the Rittner58 potential. At a much more semi-empirical level, Schug and Dyson5Q have used this formalism to investigate the interaction of benzene with the halogens. Their approximations give significantly too small a binding energy, but they do suggest that there are many configurations of comparable stability for the complex. Only two kinds of geometry, in fact, give no binding at all. An alternative to the perturbation theory approach is the approximate method of Gordon and JSim.60 In this method the electron density is first calculated as the s u m of the densities of the separate atoms and the energy is then obtained as the s u m of a Coulomb term calculated exactly, and kinetic energy, exchange, and correlation terms calculated from the free electron gas model. Though it worked well for larger systems, this calculation was not very satisfactoryfor He. * .He. Rae61 has pointed out that, as originally employed, the method of Gordon and Kim included an additional spurious electron self-energy. While this is unimportant for heavier atoms, it becomes very significant in a four-electron system like He. - *He. He shows how this error may be corrected and also adds dispersion terms to the corrected potential, obtaining potential curves for all the homonuclear inert gas pairs from He* * .He to Kr- .Kr. A further improvement has been suggested by Lloyd and Pugh,62 who apply the electron-gas expressions to a density due to the valence electrons only. This leads to an improvement over the results obtained by Rae for both A r e *Ar and Kr - - - *Kr. In the former case there is almost exact agreement with experiment, but the latter one is less satisfactory, though possibly within the experimental error. The method of Gordon and Kim has also been applied to the system Ar - w H C ~ . ~ Here it gives moderate agreement with the experimental data including the anisotropic part, especially at the shorter distances, but the potential well is too shallow, probably because of the neglect of dispersion terms. Schneidere4has also used this method for the interaction of Xe with itself and the other rare gases. Except in the case of Xe. .He where, for the reasons already mentioned, good agreement is not to be expected, he obtains values for the well depth, the position of the potential minimum, and the point at which the potential curve recrosses the zero-energy line which are within 10%of the experimental values, where these are known. +

-

-

-

-

4 Long-range Forces The theoretical understanding of the interaction between molecules at distances where the overlap is negligible has been well established for some years. The application of perturbation theory is relatively straightforward, and the recent work in this area has consisted in the main of the application of well-known techniques. In the case of neutral molecules, the first non-zero terms appear in the second order of perturbation, so that some method of obtaining the first-order wavefunction, or of approximating the infinite sum in the traditional form of the second-order energy expression, is needed. 58 59

Bo 61

62 6s

64

E. S . Rittner, J . Chem. Phys., 1951, 19, 1030. J. C. Schug and M. C. Dyson, J. Chem. Phys., 1973,58,297. R. G. Gordon and Y.S. Kim, J. Chem. Phys., 1972,56,3122. A. I. M. Rae, Chem. Phys. Letters, 1973, 18, 574. J. Lloyd and D. Pugh, Chem. Phys. Lerters, 1974,26, 281. S. Green, J. Clzem. Phys., 1974, 60,2654. B. Schneider, J. Chem. Phys., 1973,58,4447.

75

lntermolecular Forces

The oldest approximation, in which the energy denominators are approximated by a single average excitation energy, allowing the summation to be evaluated in closed form, is still of use. Alvarez-Riggatti and Mason,65 for example, have used this approximation to obtain some relations between the coefficients in the asymptotic expressions. They give two expressionsfor the ratio cS/c6for a pair of unlike atoms in terms of properties of the isolated atoms and the corresponding pairs of like atoms. One of them had been obtained previously,s6 but the other, derived variationally, is new, and does not involve the average excitation energies explicitly. The authors state that they can estimate values of CSto 10-15 % accuracy. An alternative, and more recent, idea is the pseudo-spectral expansion, 67 in which Ho is diagonalized in some large basis of convenient functions and the summation in the second-order energy is evaluated exactly over the resulting eigenfunctions. This method has been used most recently by Bukta and Meath, who use it to obtain thirdand fourth-order terms intheH. * * -Hinteraction,e*andalso to obtain the asymptotic form of the interaction between a hydrogen atom in its ground state and one in its first excited ~tate.6~ They point out that in this case there is a resonant interaction which gives rise to a term in R-9 which has no other contribution. They also show that the whole expansion is of dubious value in this case since it appears to be strongly divergent. The same pseudo-spectral expansion has also been applied to the case of two lithium at0ms.70 In this, owing to the more complicated nature of Ho, difficulties arise from the finite size of the basis set. The authors use an extrapolation procedure to surmount this difficulty. Because the summation in the second-order energy is dominated by the first term, whose magnitude can be obtained experimentally from the excitation energy and oscillator strength of the corresponding electronic transition, the authors plot the calculated value of C6 against the calculated first term for various sizes of basis set, and choose for c6 that value which corresponds to the experimental value of the first term. In this way they obtain Cs(Li- - - .Li) =

-

- 1458

8 a.u.

0.711 and Ce(Li - - *He) They also obtain values for Cs(Li- * .H) [ -(67.4 [-(22.7 0.2)] which are not, however, corrected by an extrapolation procedure since in this case the second-order expansion is not dominated by a single term. Both these problems have been attacked recently by other authors, the groundstate-excited-state hydrogen case by Deal and Young,71 the lithium-lithium case by Caves,72who gives values for Ct3 and CSfor two ground-state atoms, and also c6 values for the interaction of a ground-state atom with atoms in the 3s, 45,5s, and 6s states. Another approach to the evaluation of higher-order energies is to obtain the perturbed wavefunction, or some approximation to it, explicitly. This method has

+

65 O7 68

69 70

71 7a

+

M. Alvarez-Rizzatti and E. A. Mason, J. Chem. Phys., 1973, 59, 518. H. L. Kramer and D. R. Herschbach, J. Chem. Phys., 1970, 53, 2792. R. E. Johnson, S. T. Epstein, and W. J. Meath, J. Chem. Phys., 1967, 47, 1271. J. F. Bukta and W. J. Meath, Mol. Phys., 1974,27, 1235. J. F. Bukta and W. J. Meath, Mol. Phys., 1973,25, 1203. J. E. Kouba and W. J. Meath, MoI. Phys., 1973,26, 1397. W. J. Deal and R. H. Young, Infernat. J. Quantum Chem., 1973, 7, 877. T. Caves, J. Chem. Phys., 1973,59,6177.

76

rneoretical Chemistry

been used in several recent calculations, usually by finding the wavefunction variationally. Thus Arrighini et aZ.73have calculated the third-order dispersion energy for the interaction of a pair of hydrogen atoms, obtaining EF&, = 3.475x 103R-ll + 2.914x 105R-13

+

2.305 x 107R-15 (a.u.)

Their leading term agrees well with the even more accurate value obtained in the pseudo-spectral calculation cited above 68

c:)

=

3474.898 a.u.

-

Stewart and Webster74have solved the first-orderequation for the H * H case by a finite difference method. A much simpler series of calculations of this kind has been the investigation of the use of very simple trial functions as approximations to the first-order wavefunction by Teixeira-Dias and Varandas. Using a single Slater-type p-orbital or d-orbital they have calculated dipole and quadrupole polarizabilities at imaginary frequencies for hydrogen75 and heli~rn,~e and using a singlef-orbital they have obtained the octupole terms also for hydrogen.77 A few long-range calculations have also been carried out for larger atoms. These have, as usual, concentrated on obtaining the imaginary frequency polarizabilities and hence the various expansion coefficients,Doran 78 has extended his earlier work 79 on neon to argon, krypton, and xenon, using Brueckner-Goldstone perturbation theory. Table 6 lists the values for the leading coefficients of the dipole (D), quadru-

Table 6 Leading multipole dispersion coeflcients (in a.u.) for pairs of like inert-gas atoms 79 Gas Ne

C(DD) 6.882

Ar

66.89 135.11 281.15

Kr Xe

CWQ) 73.87 1176.4

2580.9 7033.5

C(QQ ) 379.68 10180 24275 83101

C(D0) 585.36 19116.9 41 191 157146

pole (Q), and octupole (0)interactions that he derives. The imaginary frequency polarizabilities are also accessible experimentally, in principle, via the oscillatorstrength distribution. Saxon 80 has reviewed the available evidence for neon and derived new values for the polarizabilities. Her resulting value for the CScoefficient is 6.352 a.u., which seems to be significantlydifferent from the value quoted in Table 6. Kestner 81 has extended his calculations using double-perturbation theory and the interchange theorem to the imaginary frequency polarizabilities of Ne, and gives He - *Neand Ne . .Ne asymptotic coefficients.These values differ from those just

-

73 74

75 76 77 i8

79 80

81

G . P. Arrighini, F. Biondi, and C. Guidotti, MoZ. Phys., 1973, 26, 1137. R. F. Stewart and B. C. Webster, Chem. Phys. Letters, 1973, 19, 462. J. J. C. Teixeira-Dias and A. J. C. Varandas, Mot. Phys., 1973, 25, 1 1 85. J. J. C. Teixeira-Dias and A. J. C. Varandas, Chem. Phys. Letters, 1974, 26, 197. A. 3. C. Varandas and 3. 3. C. Teixeira-Dias, Mol. Phys., 1973, 26, 241. M. B. Doran, J. Phys. ( B ) , 1974, 7 , 558. M. B. Doran, J . Phys. (B), 1972, 5, L151. R. P. Saxon, J. Chem. Phys., 1973,59, 1539. J. T. Broussard and N. R. Kestner, J. C b m . Phys., 1973, 58, 3593.

77

Intermolecular Forces

quoted by up to 20%, showing that higher orders of, particularly, the correlation perturbation need to be considered. The problem of the effect of an inaccurate zeroth-order wavefunction on dispersion calculations has been looked at again by Magnasco82 using double perturbation theory for the He * He case. Using hydrogenic, screened hydrogenic, and approximate Hartree wavefunctions he obtained for cf3the values 0.702,1.057,and 1.304a.u., compared with the exact value 1.47. Riera and Meaths3 have done almost the only long-range calculations of intermolecular forces between molecules as opposed to atoms that have appeared recently. They have been interested in what they call the ‘non-diagonal’ terms in the secondorder energy which have both odd and even terms in the R-1 expansion. They give expressions for such terms and point out that they vanish on averaging over the various possible angular positions of the two molecules. Abdulnurg4 has given upper and lower bounds to the various Cn coefficients in terms of various sum rules that are available either experimentally,semi-empirically, or theoretically. Similar rules have previously been given, but the present ones give narrower bounds than are obtained by the application of correspondingmethods to the imaginary frequency polarizabilities. McRury and Linder85 have given a general formulation of the long-range interaction between a number of molecules, obtaining a diagrammatic expansion of the free energy of the system and also reformulating their results to emphasize collective behaviour. Mackrodt 86 has used diagrammaticmethods to investigate the effect on multipole forces of an external electric field.

- -

5 Miscellaneous Topics

Systems of more than Two Molecules.-Although most calculations in the field of intermolecular forces are still concerned with pairwise interactions, a number of calculations are being made on larger systems. The interest is usually in the departures from additivity which may be found, as in the SCF calculations on He3.87988 These calculations are restricted to the short-range region and show that for a linear configuration the non-additive part of the energy is positive and very nearly a function of the sum of the two short intermolecular distances, while for isosceles triangle configurations it is negative except at very short distances, when it increases rapidly. This system, and also He4, has been looked at by Magnasco and his co-workers though with the use of a less flexiblewavefunction.They obtain quite good agreement with the results just mentioned for He3, which suggests that their He4 results should also be at least qualitatively reliable. Perhaps their most interesting observation is that if they expand their energy terms in powers of the overlap integral (S),then 893

82

83 84 85 86

87 88 89 90

V. Magnasco, Chem. Phys. Letters, 1974, 26, 192. A. Reira and W. J. Meath, Internat. J. Quantum Chem., 1973, 7 , 959. S. F. Abdulnur, J. Chem. Phys., 1973,58,4835. T. B. McRury and B. Linder, J. Chem. Phys., 1973,58, 5388,5398. W. C. Mackrodt, Mol. Phys., 1971, 27, 933. 0. A. Novaro and V. BeltrBn-Lbpez, J. Chem. Phys., 1972,56, 815. L. GonzBlez-Tovany, V. BeltrBn-LSpez, and 0. Novaro, J. Chem. Phys., 1974, 60, 1694. V. Magnasco and G. F. Musso, J. Chem. Phys., 1974,60, 3744. G. F. MUSSO, V. Magnasco, and M. P. Giardina, J . Chem. Phys., 1974, 60, 3749.

Theoretical Chemistry

78

truncating the expansion after terms of order S2 gives the wrong sign for the nonadditive part of the energy even when S is as small as 0.03. In the region of negligible overlap, the three-body interactions can be calculated, just like the two-body ones, from imaginary frequency polarizabilities. This has been done, for example, by Doran 78 for interaction of triplets of noble-gas atoms. He gives the coefficients for dipole-dipole-dipole, dipoledipole-quadrupole, dipole-quadrupole-quadrupole, quadrupole-quadrupoleequadrupole, and dipole-dipoleoctupole terms. Most of the other calculations which involve more than two molecules raise the question of what is meant by a molecule. A number of SCF calculations have been carried out on assemblies of small atoms and ions which do not form familiar chemical species but which nonetheless appear to have larger dissociation energies than would normally be thought of as intermolecular. These include HeH3+,91 XeH+,92HenH, and HenH+ where n = l-4kg3 It is concluded that linear symmetric He2H+ is stable while HesH+ and He4H+ are not. Another calculation that is of interest, but difficult to classify, is that of Wilson et al.94 on the Hz- - .He system. They are primarily interested in obtaining a potential surface on which to study the role of He as a third body in hydrogen atom recombination reactions, so that they include in their study configurations in which the Ha - H distance is as large as 5 a.u., where the H-H interaction is quite small. They find that even in a calculation which does not include intermolecular correlation there are quite long-range contributions to non-additivity. Properties.-In the past few years there has begun to be an emphasis on calculating other properties of weakly interacting systems beside the energy. For example, systems of interacting identical non-polar molecules can develop a dipole moment which is responsible for the appearance of pressure-induced i.r. spectra. In the zerooverlap region the dipole can be expanded in powers of R-l just like the energy, and the leading term turns out to be of order R-7. The coefficient of this term, which is usually denoted by D7,has been calculated95by Whisnant and Byers Brown for the systems He. * - H and He. .He exactly, and they also suggest an approximate method which could be used for larger systems. They note that the dispersion contribution to the dipole at the collision diameter is of opposite sign to the exchange contribution, which the same group has also calculatedg6for a variety of inert-gas pairs. In this case they find that it is adequate to consider only the leading term in the expansion in terms of the square of the overlap. They also find, as has previously been noted for energy calculations (see e.g. ref. 53), that it is necessary to use accurate atomic wavefunctions; double-zeta functions, for example, are not adequate. Extrapolation from molecular SCF calculationsat shorter distances, as was done, for example, by Matcha and Nesbet, 97 is also unsatisfactory. Earlier, Byers Brown and Whisnant98 considered the calculation of D7 for pairs of H atoms, and they have

-

91 92

93 94 95 96 97 98

R. D. Poshusta and V. P. Agrawal, J. Chem. Phys., 1973, 59, 2477. C. Kubach and V. Sidis, J. Phys. ( B ) , 1973, 6, L289. M. B. Milleur, R. L. Matcha, and E. F. Hayes, J. Chem. Phys., 1974, 60,674. C. W. Wilson, jun., R. Kapral, and G. Burns, Chem. Phys. Letters, 1974, 24, 488.

D. M. Whisnant and W. Byers Brown, Mol. Phys., 1973,26, 1105. A. J. Lacey and W. Byers Brown, Mol. Phys., 1974, 27, 1013. R. L. Matcha and R. K. Nesbet, Phys. Rev., 1967,160,72. W. Byers Brown and D. M. Whisnant, Mol. Phys., 1973,25, 1385.

Intermolecular Forces

79

given an approximate formula that is accurate to 1%. D7 can also be evaluated in terms of sum rules, and the necessary expressionshave been given by Abdulnur 99 for dissimilar S-state atoms. Unfortunately, many of the sum rules needed are not easily available, and the c8 coefficients from the energy expressions are also needed. Abdulnur has given some results for H, He, and Ne, however. Dipole moments for various configurations of Ha** * Hand H2* *H2have been calculated by ab initio methods.lO0 Martin101 has calculated the long-range dipole moment for systems of three identical atoms. For the H . .He O Hsystem he has used a pseudo-spectral perturbation method, while for He...He...He he has used the closure approximation. These quantities could be experimentally accessible via a pressure-induced i.r. absorption whose intensity varied as the cube of the pressure. Polarizabilities of pairs of interacting atoms have also continued to be studied. In this case the leading term at large distances is A&-6 and there are parallel and perpendicular componentsto be calculated. Buckingham and his co-workerslo2 have studied H - - .H and obtained both components of A6 and As. They find them to be larger than previous estimates. They have also studied He - -He, using both coupled Hartree-Fock and finite perturbation theories (which agreed).l03 They fmd that the limiting calculation is inaccurate even at large distances and they suspect that the polarizability changes sign twice as R increases. Another property which has been considered is the hyperfine shift in hydrogen. Fortune and Certain lo4have cakulated the effect on this quantity of the interaction with a helium atom at large distances and give an estimate of the R-*coefficientof the change. Hydrogen-bonded Systems.-The hydrogen bond may be looked upon as the weakest chemical bond, or as the strongest intermolecular interaction, but however it be classified, it continues to attract the theoretician; the number of SCF calculations continues to increase rapidly. The earlier calculations on water polymers have been summarized and discussed critically in terms of the size of basis set used.105 All the calculations give a similar description of the general features of the system. The nature of the hydrogen bond in water dimers has also been reinvestigated using bond orbitals by Bowers and Pitzer106 and using the formulation discussed earlier18 by Daudey.2O Both groups come to the familiar conclusion that there are three contributions to the interaction of approximately equal magnitude, two bonding (the electrostatic and charge transfer contributions) and one antibonding (the exchange contribution), the net energy of the hydrogen bond being equal in magnitude to all of them. The water dimer has also been studied experimentally in the molecular beam,107 the results agreeing well with the earlier calculations discussed in ref. 105. A new and very extensivecalculationhas been made for the water dimer by Clementi S. F. Abdulnur, Chem. Phys. Letters, 1973, 23, 355. R. W.Patch, J. Chem. Phys., 1973,59, 6468. 101 P. H.Martin, Mol. Phys., 1974, 27, 129. loa A. D. Buckingham, P. H. Martin, and R. S. Watts, Chem. Phys. Letters, 1973, 21, 186. loS A. D. Buckingham and R. S. Watts, Mol. Phys., 1973, 26, 7 . 104 P. J. Fortune and P. R. Certain, J. Phys. ( B ) , 1973, 6, L39. 105 J. E. Del Bene and J. A. Pople, J. Chem. Phys., 1973, 58, 3605. 106 M. J. T. Bowers and R. M. Pitzer, J. Chem. Phys., 1973, 59, 163. l o 7 T. R. Dyke and J. S, Muenter, J. Chern. P h p , 1974,60,2929,

loo

80

17teoretical Chemistry

and his co-workers, logwho go on to use their potential-energy surface in a MonteCarlo calculation for liquid water. They find good agreement with ‘experiment’, except that no second neighbour peak appears in their oxygen-oxygen radial distribution function. They attribute this to their use of a potential calculated for the dimer only. Water polymers have also been considered by Lentz and Scheraga.lo9 They were particularly interested to see if there were any additional non-additive effects in cyclic polymers in comparison with linear ones. Considering systems up to the tetramer, they found no evidence for such effects. They also used their results to calculate the lattice energy of Ice I. SCF calculations have also been made on the dimers of HFllO and formaldehyde.11l In the latter case two potential minima were found of comparable energy, one hydrogen-bonded and the other held together by the electrostatic interactions of the two carbonyl dipoles. The barrier between the two is, however, very low, and the two forms could almost certainly not be isolated. The situation in HCN is similar.l12 Other systems that have been studied in a similar way include : HF * * HCN, 113 in which the equilibrium structure is with the HF hydrogen-bonded to the HCN CHz0,114with R = H, nitrogen and there are no other potential minima; ROH. CH3, NH2, OH, or F, where the strength of the H bond parallels the o-withdrawing power of the substituent; ROH. -NH3,l15where the situation is similar; the intramolecular hydrogen bond in HOCH~CHZCH~OH, 116for which a strength of only 3.9 kJ mol-1 was found ; and CH3CONHCH3 -H20,117 a model system for proteins. The calculations on H2O- - aC1-48 have already been mentioned. Calculationshave been made on the interaction of a water molecule with formaldehyde in its lowest triplet and first excited singlet states.118-120The two calculations agree in locating the hydrogen bond between the water oxygen and the formaldehyde carbon because of the shift of charge in the carbonyl group on excitation. Reference 119 reports two rotational isomers in the triplet complex (separated by only a very small barrier) but only one in the singlet. Reference 118 reports blue shifts of 1100 cm-1 for the singlet and 1420 cm-1 for the triplet. The singlet value is confirmed in ref. 120, which also gives shifts for a number of other hydrogen-bonded formaldehyde complexes. In addition to these ab initio calculations, CNDO calculationscontinue to be made on hydrogen-bonded systems. The influence of substituents on acetic acid dimers, for example, has been investigated at this level of approximation.lZ1

-

0

H. Popkie, H. Kistenmacher and E. Clementi, J. Chem. Phys., 1973, 59, 1325. B. R. Lentz and H. A. Scheraga, J. Chem. Phys., 1973,58, 5296. 110 D. R. Yarkony, S. V. O’Neil, H. F. Schaefer, C. P. Baskin, and C. F. Bender, J. Chem. Phys., 1974, 60, 855. 111 J. E. Del Bene, J. Chem. Phys., 1974, 60, 3812. 113 J. E. Del Bene, Chem. Phys. Letters, 1974, 24, 203. J. E. Del Bene and F. T. Marchese, J. Chem. Phys., 1973, 58, 926. 11* J. E. Del Bene, J. Chem. Phys., 1973, 58, 3139. J. E. Del Bene, J. Amer. Chem. SOC.,1973, 95, 5460. 116 A. Johansson, P. Kollrnan, and S. Rothenberg, Chem. Phys. Letters, 1973, 18, 276. 11‘ A. Pullman, G.Alagona, and J. Tomasi, Theor. Chim. Acta, 1974, 33, 87. 118 S. Iwatu and K. Morokuma, Chem. Phys. Letters, 1973, 19, 94. ll9 J. E. Del Bene, Chem. Phys. Letters, 1973, 23, 287. l a o J. E. Del Bene, J. Amer. Chem. Suc., 1973, 95, 6517. 121 B. M.Rode, A. Engelbrecht, and W. Jakubetz, Chem. Phys. Letters, 1973, 18, 285. lo8

lo9

Intermolecular Forces

81

Empirical Potentials and Related Topics.-Although this review mainly deals with the calculation of intermolecular potentials either ab initio, or at least from parameters such as sum rules which are derived from experiments not directly concerned with molecular interactions, nevertheless, there have recently been so many attempts to derive more accurate empirical potentials (with which any calculation must be compared) from new experimental data that it seems appropriate to mention some of this work. The noble gases have received most attention. New evidence which has been obtained from the spectra of the van der Waals molecules involving these atoms has been reported.122, 123 From this and other data, potential curves have been obtained by Maitland et al. f o r k . * .Ar,124Ne* .Ne,125Kr** .Kr,126andXe. * .Xe.12'They give numerical potentials and also an empirical form with a variable exponent in the repulsive part, i.e.

--

where n = 13.0

+

r* =

y(r*-l)

r/Ymin

-

which they suggest 128 is superior, at least for Ar - .Ar, to both that of Barker e f al.lZg and of Smith and Thal~kar.1~0 These latter authors have recently refined their potential, in which they use Hermite interpolation between a Morse potential at short distances and an R-n expansion at large distances, in the light of the recent evidence for all the noble-gas pairs.131 Taylor and Weis~man,13~ however, suggest that none of the empirical potentials that they have tested, which includes that of Maitland et al., as well as the traditional Lennard-Jones and Buckingham ones, has sufficient flexibilityto fit their thermal diffusion data on 2ONe. .22Ne.The sensitivity of thermal diffusion has also been discussed by Nenfeld and A~iz.1~3 The Buckingham (exp - 6) potential has been modified by Toennies, 134 who has added theoretical estimates44of R-* and R-l0 terms and obtains what are probably fortuitously accurate positions for the potential minima in the noble-gas dimers. Present 135 has considered the derivation of the A r e .Ar potential and suggests that a revised value should be used for the dissociation energy. Spectral data on a wide variety of van der Waals molecules are now becoming available. The noble-gas dimers have already been mentioned, but many others are now yielding detailed data on, especially, the angular dependenceof potentials, which

-

lZ2 lZ3 124

lZ5 126 lZ7

lZ8 129 130 131 132 133 l34 135

Y.Tanaka, K. Yoshimo, and D. E. Freeman, J. Chem. Phys., 1973, 59, 564. M. C. Castex and N. Damany, Chem. Phys. Letters, 1974, 24, 437. G. C. Maitland and E. B. Smith, Chem. Phys. Letters, 1973, 22, 443. G. C. Maitland, MoZ. Phys., 1973, 26, 513. D. W. Gough, E. B. Smith, and G. C. Maitland, Mol. Phys., 1974, 27, 867. D. W. Gough, E. B. Smith, and G. C. Maitland, Mol. Phys., 1973, 25, 1433. A. N. Dufty, G. P. Matthews, and E. B. Smith, Chem. Phys. Letters, 1974, 26, 108. J. A. Barker, W. Fock, and F. Smith, Phys. Fluids, 1964, 7 , 897. V. H. Smith, jun. and A. J. Thakkar, Chem. Phys. Letters, 1972, 17, 274. A. J. Thakkar and V. H. Smith, jun., Mol. Phys., 1974,27, 191 (Ar, Kr,Xe), 593 (He, Ne). W. L. Taylor and S. Weissman, J. Chem. Phys., 1974, 60, 3684. P. D. Neufeld and P. A. Aziz, J. Chem. Phys., 1973, 59,2234. J. P. Toennies, Chem. Phys. Letters, 1973, 20, 238. R. D. Present, J. Chem. Phys., 1973, 58, 2659.

Theoretical Chemistry

82

represents something of a challenge to the theoretician. The subject has been reviewed to 1973 in a paper on the i.r. spectrum of 0 2 - .Ar by Henderson and Ewing, 136 who mention, beside the noble-gas pairs, (02)2, (HF)2, (C02)2, (N2)2, (H2)2, and H2- * -Ar. A list of preliminary observations by molecular beam methods is given by Novick et aZ.137 More detailed results have recently been given on Ar- * .HCl and Ar * * DCI 138 and Ar - HF,139 from molecular beam measurements, and on Ar...H214* from the i.r. spectrum. The i.r. spectrum of hydrogen in an argon matrix has been interpreted in terms of a van der Waals molecule Ar - - .H2,141while evidence on the anisotropy of the He. .H2 and Ne. .H2 potentials has been obtained by n.m.r. relaxation-time measurements.142 The long-range part of the halogen-halogen potential correlating with the B3n[ouf excited state of the diatomic molecule has been obtained by the RKR method, yielding c5, CS, and c8 coefficients for I - - .I,143Br - - - Br, and C1-

-

-

136 lS7

G . Henderson and G. E. Ewing, J. Chem. Phys., 1973, 59, 2280. S. E. Novick, P. B. Davies, T. R. Dyke, and W. Klemperer, J . Amer. Chem. SOC.,1973, 95, 8547.

13*

139 140 141 142

143 144

S. E. Novick, P. B. Davies, S. J. Harris, and W. Klemperer, J. Chem. Phys., 1973, 59, 2273. S. J. Harris, S. E. Novick, and W. Klemperer, J. Chem. Phys., 1974, 60, 3208. G. Henderson and G. E. Ewing, Mol. Phys., 1974, 27, 903. S. S. Cohen, Chem. Phys. Letters, 1973, 18, 369. J. W. Riehl, C. J. Fisher, J. D. Baloga, and J. L. Kinsey, J. Chem. Phys., 1973, 58, 4571. K. K. Yee, Chem. Phys. Letters, 1973, 21, 334. K. K. Yee and T. J. Stone, Mul. Phys., 1973, 26, 1169.

3 Quantum Mechanical Calculations on Small Molecules BY C. THOMSON

1 Introduction This Report deals with recent theoretical work on small molecules, and covers the literature to 1 July 1974. Discussion is restricted to molecules containing up to four atoms, and work on molecules containing approximately 5-12 atoms will be reviewed in Vol. 3. This Report does not aim to be in any sense comprehensive, especially since the literature on quantum chemistry is expanding so rapidly, and discussion is restricted to those results in the field of calculations on small molecules which are judged by this Reporter to be of particular importance. Both non-empirical (ab initio) and semi-empirical calculations will be dealt with, but since most work on small molecules in recent years has used ab initio methods, this type of calculation will be emphasized. Most of the papers cited refer to work published during the period 1972-74, but there are also references to work of particular importance prior to 1972 if necessary. Developments in the theoretical methods as such will not be dealt with in any detail, the emphasis being on the results of calculations. Results are usually quoted in atomic units* @istances/bohr, energies/hartree), but there are occasional exceptions to this. A number of books and review articles dealing with the subject matter of this chapter should first be mentioned. The most comprehensive introduction to recent work is by Schaefer,l whilst a more recent book on ab initio calculations, by Cook, is also recommended.2A fairly comprehensive bibliography (up to 1974) on a b initio calculations is provided by Richards et aL3 Volume 1 of the MTP Review of Science4 deals with theoretical chemistry and contains several articles of interest, particularly an article by Wahl on diatomic molecules, and a review of the work of Pople’s group. There have also been several books dealing with semi-empirical methods in

H. F. Schaefer, tert., ‘ElectronicStructure of Atoms and Molecules’, Addison Wesley, Boston, 1972. D. B. Cook, ‘Ab-Initio Valence Calculations in Chemistry’, Wiley, London, 1974. W. G. Richards, T. E. H. Walker, and R. Hinkley, ‘Bibliography of Ab-Initio Molecular Wave Functions’, Oxford University Press, 1971, 1974. ‘Theoretical Chemistry’, ed. W. Byers-Brown, ‘MTP International Review of Science: Physical Chemistry, Series One’, 1972, Vol. 1.

* 1 bohr = 0.529 18 A; 1 hartree

= 27.21 eV.

83

84

Theoretical Chemistry

quantum chemistry, and of particular note are those by Beveridge and Pople,5 and by Murrell and Harget.6 Among many recent review articles, Browne and Matsen’ have given a thorough review of recent work on three- and four-electron molecules, particularly He2, He:, LiH, and LiH+, and a series of articles dealing with many aspects of ab initiu work have appeared in two recent books which collect together the proceedings of two conferences.8. A detailed review on electron correlation surveys recent work and defines some of the problems in this areasloThere has been an excellent review of Walsh diagrams and molecular geometry from the point of view of csb initio calculations,ll and also a survey of ab initiu methods in chemistry.12 The recent developments in generalized Valence Bond (GVB) theory have been reviewed by Goddard and co-w0rkers,~3and also the use of natural orbitals in theoretical chemistry,l4,15 and the accuracy of computed one-electron properties.ls The XCC method has been reviewed by Johnson,17 and Hurley has discussed highaccuracy calculations on small molecules.18 Several other reviews of interest have appeared in Advances in Quantum Chemistry.l7 Localized orbital theory has been reviewed by England, Salmon, and Ruedenberg,lg and the bonding in transitionmetal complexes discussed by Brown et aL2O Finally, the recent developments in computational quantum chemistry have been reviewed by Ha1LZ1 2 Diatomic Molecules containing up to Four Electrons It is appropriate to review first the recent work on HZ and Hz. A. H i and H2.-Hl and H2 are particularly important molecules, which have been used both for testing new methods in quantum chemistry and also for investigation of the inclusion of small terms in the molecular Hamiltonian. The extensive earlier work has been reviewed by Kolos.22s23 Most ab initio calculations on molecules with more than three electrons have involved the assumption of the Born-Oppenheimer ( S O ) appr0ximation.l HowD. L. Beveridge and J. A. Pople, ‘Approximate Molecular Orbital Theory’, McGraw-Hill, New York, 1970. 6 J. N. Murrell and A. J. Harget, ‘Semi-empirical Self-consistent Molecules’, Wiley, London, 1972. 7 J. C. Browne and F. A. Matsen, Ado. Chem. Phys., 1973, 23, 161. a ‘Energy Structure and Reactivity’, ed. D. W. Smith and W. B. McRae, John Wiley, New York, 1973. 9 ‘Computational Methods for Large Molecules and Localized States in Solids’, ed. F. Herman, A. D. McLean, and R. K. Nesbet, Plenum Press, New York, 1973. 10 W. Kutzelnigg, Fortschr. Chem. Forsch., 1973, 41, 31. 11 R. J. Buenker and S. D. Peyerimhoff, Chem. Rev., 1974, 74, 127. 12 J. L. Whitten, Accounts Chem. Res., 1973, 6, 238. 13 W. A. Goddard, tert, T. H. Dunning, jun., W. J. Hunt, and P. J. Hay, Accounts Chem. Res., 1973, 6, 383. 14 E. R. Davidson, Adv. Quantum Chem., 1972, 6, 235. 15 E. R. Davidson, Rev. Mod. Phys., 1972, 44, 451. 16 S. Green, Adv. Chem. Phys., 1974,25, 179. 17 K. H. Johnson, Adv. Quantum Chem., 1973, 7 , 143. 18 A. C. Hurley, Ado. Quantum Chem., 1973,7, 315. 19 W. England, L. S. Salmon, and K. Ruedenberg, Fortschr. Chem. Forsch., 1971, 23, 31. 20 D. A. Brown, W. J. Chambers, and N. J. Fitzpatrick, Inorg. Chim. Acta, 1972, 6, 7. 21 G. G. Hall, Chem. SOC.Rev., 1973, 2, 21. 22 W. Kolos, Internat. J. Quantum Chem., 1968, 2, 471. 23 W. Kolos, Ado. Quantum Chem., 1970, 5, 99. i)

Quantum Mechanical Calculations on Small Molecules

85

ever, a recent series of papers by Bishop and co-workers has examined the use of a higher level approximation (the adiabatic approximation), which is based on the Born expansion of the total wavefunction, with neglect of non-diagonal nuclear terms. This approximation still allows the concept of a single potential-energy curve for a given electronic state. Bishop and Wetmore24p25 have calculated the adiabatic energy curves, dissociation energies, and vibrational energy levels for Hi, HD, and Daf.26 The non-adiabatic corrections were calculated using second-order perturbation theory. Most of the discrepancy between theory and experiment was shown to be due to the non-adiabatic terms, which amount to 1 cm-1. Bunker27 has also discussed the adiabatic correctionsto the B-0approximation for diatomic molecules. In a later paper, Bishopz8 has derived values for the equilibrium internuclear distances and force constants from the adiabatic calculations for Hl, E D + , and Dl, and these are the most accurate yet calculated without a full non-adiabatic treatment. The highly accurate James-type wavefunction for HZ has been improved using perturbation theory, yielding energies which agree with the exact solutions over the range of R from 0 to 5 A variety of calculations on the excited states of H l have appeared, covering a wide range of internuclear separations. Spectroscopic properties of the 2pnu(211u)and 3dag(2X3 states were computed by numerical integration of the Schrodinger e q u a t i ~ n . ~ The * ~ ~2pnu l and 3dng states were also studied by the one-centre method.32 The use of different types of basis functions in molecular calculations continues. The ‘0s’ or Hulthkn-type function has been used in calculations on Ha+and H2,33 and is more effective than the 1s-typefunction. Hi-type elliptical orbitals have also been employed in variational calculations on Hi, H2, He:’, and Hi.34 The H i orbitals were also used as basis orbitals for SCF calculations. A four-function basis set with two (non-linear) variational parameters yields 99% of the Hartree-Fock energy. The floating spherical Gaussian orbital method (FSGO), used so successfully by Frost et 41.,35-37 has been extended to deal with open-shell molecule~,~7 and, among other diatomics, Hzf and H2 were studied (we refer to other calculations by this method throughout this article). A generalization of the FSGO idea, in which the orbitals are ellipsoidal (FEGO) has been proposed and tested on Hz and a variety of small molecules.38 The method seems to give more accurate energies and bond angles but bond lengths are longer than those obtained in FSGO calculations.

-

D. M. Bishop and R. W. Wetmore, Mol. Phys., 1973,26, 145. D. M. Bishop and R. W. Wetmore, MoI. Phys., 1974, 27, 279. 26 D. M. Bishop and R. W. Wetmore, J. Mol. Spectroscopy, 1973, 46, 502. 27 P. R. Bunker, J. Mol. Spectroscopy, 1972, 42, 478. 28 D. M. Bishop, J. Mol. Spectroscopy, 1974, 51, 422. 29 R. P. McEachran and M. Cohen, Chem. Phys. Letters, 1973,20,298. 80 C . L. Beckel, M. Shafi, and J. M. Peck, J. Chem. Phys., 1973,59, 5288. 31 M. Shafi and C. L. Beckel, J . Chem. Phys., 1973,59, 5294. 32 R. F. Stewart, MoZ. Phys., 1973, 25, 1451. S . Ishimaru, K. Tanaka, and T. Yamabe, Bull. Chem. SOC.Japan, 1973,46, 3577. 34 C. T. Llaguno, S. K. Gupta, and S. M. Rothenstein, Internat. J. Quantum Chem., 1973,7, 819. 95 A. A. Frost, J. Chem. Phys., 1967,47, 3707. 38 A. A. Frost, Ado. Chem. Phys., 1971,21, 65. 87 P. H. Blustin and J. W. Linnett, J.C.S. Faraday 11, 1974,70, 327. 38 G. Simons and A. K. Schwartz, J. Chem. Phys., 1974,60,2272. 24

25

4

Theoretical Chemistry

86

The discrepancies between theory and experiment for HZ have been studied in detail over the past few years, and Kari and co-workers39 have presented a complete variational of,H2 using uthe full electronic Hamiltonian. In this work, the --- -------treatment " r,e,gy I . . , . 1 " . " . , . . ~ U . * "..

AJLS,,&+&,G.IJ

L.1

)I

kJmol-l. Also surprising were the observations that the reactions of H-, NH;j ,OH-, C a H , and CN- with CHsF were observed to be at least 100 times slower, than the corresponding reactions with CH3Cl. For reaction @) above, Bohme et al: measured an activation energy of 15 kJ mol-1 compared to the value of 16 IkJ mol -l calculated by Dedieu and Veillard.120 r This experimentalverification of the existence and even, in one case, the magni-: tude of activation barriers to gas phase s N 2 displacementson CHsF is very encour-) aging. While the reactions are all closed-shell systems with overall retention in the: number of electron pairs, the transition states do represent a significantly different: arrangement of two electron pairs. In gross terms, one lone pair of the nucleophile. and the pair binding the leaving group in the substrate are both transformed into extended bonded pairs in the transition state. The experimental agreement with the suggests that no significant change in the cord theoretical barrier for reaction 0) relation energy is encountered in the formation of the [H-CHpF]- transition1 ]state. The reader is asked to recall that the magnitudes of simple inversion barriers :are well accounted for in the HartreeFock method. Dedieu and Veillard120 have,\ ]infact, reported a limited calculation of the correlation corrections to the energiies: 'of the reactants and transition state for reaction (B). The contributions of the: !correlation energy to the reactants and transition state differed by Zess than 4 kJ: jmol -l. However, Dyczmons and Kutzelnigg123have also performed SCF calcula-;tions and calculations of part of the correlation energy for CH4, H - and the C H ~ I ]transitionstate of Dan symmetry. They found an SCF activation barrier of 293 kJ: Imol-l for reaction (A) (compared with ca. 250 kJ mol-1 obtained by Veillardi let U Z . ~ ~ a~ value ) , which is reduced to 230 kJ mol -1 with the inclusion of correlation ,intothe wavefunctions. It is clear, however, that neither set of authors has approached lthe Hartree-Fock limit in their single-determinant SCF calculations. Thus, DyczImons and Kutzelnigg obtain a somewhat better energy than Veillard et al. for (CH4 H-, but a somewhat poorer one for CH,, A striking feature of all five of these reactions is the delayed departure of the: cleaving group. Dedieu and Veillard120 have given a detailed discussion of the energetics and changes in charge density along the reaction pathways for reactions I(A)-(D), including comparisons with the empirical rules for sN2 reactions of Hammond,l24 Thornton,l25 and Bunton.126 Bader, Duke, and Messe1-127 have presented an analysis of the changes in the energy and charge density for reactions ((B)and (E) using the virial partitioning procedure. This is a partitioning method fdeterminedby the topological properties of the charge distribution, which yields a ppatial partitioning of the charge and energy for a molecular system. The kinetic ~4m~q~k1~ wPrFie%crfLtbkznrlfireltsLf.;lg-n&ry'-htnirAhir3bt- mafFnr' phnxl$he t

+

39 41 42

43 44

R. E. Kari, A. C. H. Chan, G . Hunter, and H. 0. Pritchard, Canad. J. Chem., 1973,51,2055. W. Kolos and L. Wolniewicz, J . Chem. Phys., 1964, 41, 3663. T. Orlikowski and L. Wolniewicz, Chem. Phys. Letters, 1974, 24, 461. V. Staemmler and M. Jungen, Theor. Chim.Acta, 1972,24, 152. IS. Helfrich, Theor. Chim. Acta, 1973, 30, 169. G . Das and A. C. Wahl, J. Chem. Phys., 19615~44,87.

Quantum Mechanical Calculations on Small Molecules

87

determine exact solutions to the molecular Schrodinger equation by first solving for a wavefunction which is least distorted from a product of atomic wave function^.^^ An application of the method to H2 has since been successfully carried out. Using a Slater-type orbital (STO) basis, the energies obtained were better than HF, and differed from the exact values mainly through the omission of angular correlations. This novel approach would be very useful if it could be extended to larger molecules. One of the major advances in recent years in attempts to calculate more exact wavefunctions, including electron correlation, has been the implementation by Boys and Hand~,469~~ of a computational scheme based on the method of moments, called the transcorrelated wavefunction method. In this method, a correlation factor is built into the wavefunction, which is written in the form (l), where @ is a Slater

Y

=

c@

determinant and C is of the form (2) (for n-electrons); fcan depend explicitly on ~ t j ,

and very accurate wavefunctions and energies may be obtained through the use of the operator C-lS'C. Details of the method are to be found in the series of papers by Boys and 47 Handy has recently extended this w0rk.~8The earlier calculations on He, Be, Ne, and LiH47 used numerical integration to obtain the necessary six-dimensional integrals. However, following a suggestion of Boys, Handy showed that if thef(rt, r j ) are written as linear combinations of gaussians such as exp ( -ar;) and exp (-b&) then all the integrals in the full transcorrelated equations can be evaluated analytically. Handy has described the results of calculations on He, LiH, H2, and H20,m using a small gaussian basis set (which of course makes @ a bad approximation to the best determinant).The results for the SCF and correlation energies together with the exact values are given in Table 1. The high accuracy

Table 1 Computed SCF and correlation energies by the transcorrelated wav mth0d.48 Exact correlation energy in parentheses Molecule He H2 LiH Ha0

SCF

Total energy Exact SCF

-2.710 -0.976 -7.589 -64.23

-2.862 -1.133 -7.987 -76.07

Correlationenergy -0.0399 (- 0.0420) -0,0419 (-0.0405) -0.0759 (-0.082)

-0.254 (-0.364)

achieved with a small number of functions both for Re and also at R = 2. in the case of H2 is encouraging, and leads one to hope that accurate molecular correlation energies might be obtained by this method when larger basis sets are used. 45 46 47

48

W. H. Adams, Chem. Phys. Letters, 1971, 11, 4 4 1 . S . F. Boys and N. C. Handy, Proc. Roy. Soc., 1969, A309,209; ibid., A310,43, 63. S. F. Boys and N. C. Handy, Proc. Roy. SOC.,1969, A311, 309. N. C. Handy, Mot. Phys., 1972,33, 1.

88

Theoretical Chemistry

Bernardi and B0ys4~have examined the problem of the accuracy of the energy and other variables in this method, and give explicit formulae for improving the calculations. The original formulation of the method to cover the calculation of expectation values was given by Handy and Epstein in 1970.5OArmour 51has examined the method of moments and the transcorrelated wavefunction method (which is a particular form of the method of moments) in some detail. Several expectation values were evaluated in the course of applications of the former method to H2, and in general fairly accurate results were obtained, but numerical problems can occur, and further study is needed. In a later paper, h o u r s 2 examined the accuracy of previous transcorrelated wavefunctions for H2 by evaluating the variance of the wavefunction numerically, and comparing it with the variance of conventional variational wavefunctions. Despite the high accuracy of the energy, the transcorrelated wavefunction is only of the same accuracy as the variational function using the same basis, but both are much less accurate than the Kolos-Roothaan (KR)function.53It is, however, rather easy to improve the transcorrelated function to give a wavefunction which, though better than the variational, is still inferior to the KR wavefunction. In a later paper, Armour evaluated expectation values and compared these with the KR and variational results.54 In this case the transcorrelated wavefunctionsare substantially worse than variational functions, and in addition, the expectation values are much more difficult to evaluate. The calculation of molecular properties is as important as the evaluation of the energy, and a recent paper on spin-rotation interaction and magnetic shielding in 1X molecules, using the Kolos-Wolniewicz (KW) wavef~nction,~~ shows that theory and experiment are in good agreement, if an old approximation in the spin-rotation interaction theory is removed.55 Calculations of the dependence of the nuclear shielding on bond length in H2 appeared some years ag0,56 and the authors have recently calculated the magnetic susceptibility using their wavefunction and compared it with the KW function.57 Several important predictions of this work remain to be experimentally verified. The polarizability 0: is an important second-order molecular property. Its variation with internuclear separation has been investigated for H i and Hz by Zeroka,58 using the method of Das and Bersohn. Spectroscopic constants calculated from the BO and adiabatic KW functions for HZwere also studied by Wu and B e ~ k e l . ~ ~ Nuclear spin-spin coupling constants (JAR) are among the most difficult of molecular properties to compute by ab initio methods, because of the need to include the excited states in the expression for J. In recent years, a variety of calculations have appeared using ground-state SCF-MO’s, and expressing the coupling constant 49 50

51

52

53 54 56 57

58 59

F. Bernardi and S. F. Boys, Mol. Phys., 1973, 25, 35. N. C. Handy and S. T. Epstein, J. Chem. Phys., 1970, 53, 1392. E. A. G. Armour, Mol. Phys., 1973, 25, 993. E. A. G . Armour, Mol. Phys., 1973, 26, 1093. W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys., 1960,32, 205. E. A. G. Armour, Chem. Phys. Letters, 1974, 25, 614. R. V. Reid, jun., and A. H.-M. Chu, Phys. Rev., 1974, 9A, 609. D. B. Cook, A. M. Davies, and W. T. Raynes, Mol. Phys., 1971,21, 113. W. T. Raynes, J. P. Riley, A. M. Davies, and D. B. Cook, Chem. Phys. Letters, 1974,24, 139. D. Zeroka, Internat. J. Quantum Chem., 1974, 8, 91. F. M. Wu and C. L. Beckel, Internat. J. Quantum Chem., 1973, 57, 135.

Quantum Mechanical Calculations on Small Molecules

89

in terms of the usual second-order perturbation theory sum-over-states formula. Apart from problems inherent in the latter approximation, one other reason for the poor agreement with experiment usually obtained is the use of an uncorrelated zeroth-order function and an incomplete basis set. Kowalewski et aZ.60 have studied the removal of these restrictions in calculations on HD. The zeroth-order wavefunction is substantiallyimproved via a large CI, and the first-order correction to the wavefunction may be computed by expansion in all singly and doubly excited triplets, with the expansion coefficients being found iteratively. The computed value of JHD = 43.48 Hz is in good agreement with the experimental value of 42.94i-0.1 Hz.With the exception of earlier calculationsby Moulson and Lowe,61 Meyer,g2 and Dutta et this represents a significant improvement, especially since the method should be readily extendible to other larger systems. The above calculations on the ground state have been supplemented by calculations on a variety of excited states of H2. The IT states have been very thoroughly investigated by McLean and co-workers."-66 These workers used double-configuration wavefunctions of the type (3), where @HF = (q,ztu), @HF' = (cung)for the

Y = A@HF + B@HF' (3) states, and @HF = (cgzg), @HF' = ( g u n U ) for the 1~3n,states. Alimited basis set was used (Is, 23, 2pc, 2pn, 3dz) and the exponents were separately optimized for all four states at various R values. The exponents were found to be strongly dependent on state and distance. Various approximatewavefunctionswere also studied and P.E. curves constructed. For the best wavefunction, the shapes were in agreement with earlier, more extensive calculations.409 g79 68 One-electron expectation values were calculated, and the effects of a-n separation studied.66 The results of the latter study are of relevance for semiempirical 0-n theories. Colbo~1-1169has also investigated the I l l u and 31Tu states, using a larger basis set than Jug et al.,64-66 derived from the smaller basis set for which exponents were optimized over a range of internuclear separations. The determinationof singlet-triplet energy differencewas the main aim of this study, using ST0 basis functions and CI calculations. Less accurate calculations at the SCF level for the 311u state have been presented by Truhlar, using a STO basis set.70 This work was carried out in order to be used for calculations of electron-impact cross-sections, and P.E. curves and some one-electron properties were calculated. The 1#3C+excited states of H2 have been discussed by Huestis and Goddard 71 in an important paper dealing with the general construction of excited-statewavefunctions, and the well-known Hund-Mulliken correlation diagrams. One problem with SCF 13311u

60

62

63 64

65 66 67 68

69

70

71

J. Kowalewski, B. Roos, P. Siegbahn, and R. Vestin, Chem. Physics, 1974, 3, 70. T. Moulson and J. P. Lowe, Mol. Phys., 1971, 22, 723. W. Meyer, 2.Physik., 1969, 239,452. C. M. Dutta, N. C. Dutta, and T. P. Das, Phys. Rev. Letters, 1970, 25, 1695. K. Jug, P. G . Lykos, and A. D. McLean, Theor. Chim. Acta, 1972, 25, 10. K. Jug, P. G. Lykos, and A. D. McLean, Theor. Chim. Acta, 1972, 25, 17. K. Jug, A. D. McLean, and P. G. Lykos, Theor. Chim. Acta, 1972, 25,41. W. T. Zemke, P. G . Lykos, and A. C. Wahl, J. Chem. Phys., 1969,51,5635. W. M. Wright and E. R. Davidson, J. Chem. Phys., 1965,43, 840. E. A. Colbourne, J. Phys. (B), 1973, 6, 2618. D. G. Truhlar, Internat. J. Quantum Chem., 1973, 7 , 1175. D. L. Huestis and W. A. Goddard, tert., Chem. Phys. Letters, 1972, 16, 157,

90

%oretical Chemistry

methods is that the wavefunction often incorrectly describes the dissociation of the rn~lecule.~ The generalized valence-bond wavefunctions (GVB) extensively studied by Goddard and co-workersl3 (see Section 3A for further details) do not have this disadvantage, and it is shown that a qualitative discussion of the energies and wavefunctions and correlation diagrams can be easily obtained. Quantitative calculations on H2 were reported in this paper. Extensions of this work to other molecules have been reviewed recently.l3 Mulliken has also investigated the a l l s state of H2.72 The inclusion of correlation effects in the calculation of second-order properties, such as the polarizability, has been examined by Bartlett and co-workers, and an application to H2 reported.73 A very accurate CI wavefunction due to Liu has been used to calculate the Compton profile for molecular Hz, with results in good agreement with experiment.74 Rather few papers have dealt with the computation of thermodynamic functions from the results of ab initio calculations, but for H2, where the latter are of spectroscopic accuracy, Kosloff, Levine, and Bernstein have computed the thermodynamic properties of H2, D2, and HD, using the best theoretical results.75 This work represents the first example of an accurate determination of a bulk, macroscopic property from first principles. The slow-convergence problem in CI calculations is well known, and several attempts to circumvent this have been proposed. A recent example is the suggested use of natural orbitals76 determined using first-order perturbation theory. This procedure results .in substantial improvements in speed for less than complete CI calculations, and the results for HZand LiH were almost identical to those obtained in more conventional CI treatments, Finally, we should mention some approximate calculations on H2. Jug 77 has developed a semi-empiricalversion of the multiconfiguration SCF (MCSCF) method, using CNDO- and INDO-type approximations, and has reported the results of a double-configuration approach to H2. It was shown that the eigenvalues of the EHF operator have physically interpretable characteristics and follow dissociation properly. Further results of this method should be very interesting. Roby78 has given details of an approximate non-empirical method in which a minimal basis of symmetrically orthonormal atomic orbitals is used, and all integrals are evaluated non-empirically. Extending this approach to include complete CT, Koster and Ruttink79 have examined H2 and H3 by this method (essentially a non-empirical NDDO method). Results are in good agreement with ordinary ab initio calculations, but CI has almost no effect on the total energies at Re. Bond lengths are well predicted, however, and optimization of the orbital exponent is possible in this method, unlike other semi-empirical methods, e.g. CND0/2. B. HeH, HeH+, and Hez.-These small molecules have also been studied by a variety of methods. Calculations on HeH+ by the FSGO method were reported in 72

73

74

75

76 77 c8

79

R. S. Mulliken, Chem. Letters, 1972, 14, 141. R. J. Bartlett, J. C. Bellum, and E. J. Brtindas, Internat. J . Quantz4m Chem., 1973, 7,449. R. E. Brown and V. H. Smith, jun., Phys. Rev., 1972, A5, 140. R. Kosloff, R. D. Levine, and R. B. Bernstein, MoZ. Phys., 1974, 27, 981.

A. K. Q. Siu and E. F. Hayes, J. Chern. Phys., 1974,61, 37.

K. Jug, Thew. Chim. Acta, 1973, 30, 231. K. R. Roby, Chem. Phys. Letters, 1971, 11, 6 ; ibid., 1972, 12, 579. J. L. Koster and P. J. A. Ruttink, Chem. Phys. Letters, 1972, 17,419.

Quantum Mechanical Calculations on Small Molecules

91

ref. 37, and the HeH2+system has been used as a test case for a method for calculating potential-energy surfaces for complex values of the nuclear co-ordinates,sO a procedure which is useful in the theoretical investigation of collision phenomena. The dissociation of HeH+ under electron impact has been studied, using an elliptic orbital basis, assuming the Born-oppenheimer approximation.81 Banyard and coworkers 82 have re-formulated extensive one-centre CI wavefunctions as a natural expansion and examined electron correlation in this species as a function of R. The effects on the P.E.curve and the spectroscopic constants are fairly small. Bartlett and Brhdas have examined the ground states of HeH+ and Hz, and the excited states (fX+)of HeH+, using a new method, called the reduced partitioning procedure, in extensive CI studies.83, 84 This procedure, which makes use of Lowdin's partitioning technique, provides a set of systematic corrections to a Hartree-Fock or truncated CI wavefunction, and yields a series of dramatic energy improvements as converging upper bounds to the full CI result. Computationally attractive, the results give accurate ground-state energies in low orders. For the excited states of HeH+, potential curves were obtained,84 and again rapid convergencewas achieved, i.e. an eighth- or lower-order solution yields at least 99.9 % of the full CI result. The use of this procedure in investigating second-order properties has already been mentioned-73 Model potential calculations on HeH+ have been reportedySsusing techniques developed earlier.86 Results were in good agreement with more rigorous ab initio results. He: has been the subject of three papers, the first being concerned with a onecentre calculation of the direct ionization of He by He+ impact, which is considered to arise from a molecular autoionization of the 22; state of He: as it crosses the boundary of the continuum @Id++ e-).S7 A second paper describes a groundstate calculation with a Slater basis,88and the third a FSGO study.37 A number of papers, several years ago, dealt with the species HeH (see ref. 37), and recently it has been observed e~perimentally.~~ FSGO calculations by Blustin and Linnett37 gave a small binding energy for a two-determinant wavefunction, and predict the HeH system to be more stable than Hz...H, in agreement with experiment. C. He~.-Work on He2 and He: has been thoroughly reviewed recently.7 The repulsive curve has been extensively studied in the past, and the various components have been analysed. The recent calculations by Bertoncini and Wahl90 and McLaughlin and Schaefer,91 who studied the long-range attraction, have been supplemented by two recent studies not reviewed in ref. 7. Liu and McLeanQ2have K. Morukuma and T. F. George, J. Chem. Phys., 1973,59, 1959. K. Mullick, T. K. D. Pai, and A. K. Barna, J. Phys. ( B ) , 1974,7, 288. 82 K. E. Banyard, M. Dixon, and A. D. Tait, J. Phys. ( B ) , 1972,5,2160. 83 R. J. Bartlett and E. J. Brbdas, J. Chem. Phys., 1972, 56, 5467. a4 R. J. Bartlett and E. J. Brlndas, J. Chem. Phys., 1973, 59, 2032. 85 C. Botcher, J. Phys. (B), 1973, 6, 2368. 86 C. Botcher, Chem. Phys. Letters, 1973, 18, 457. 87 V. Sidis, J. Phys. (B), 1973, 6, 1188. 88 Z . Cirule and A. B. Bologin, Lief. Fiz. Rinkinys, 1973, 13, 63. 89 J. Gray and R. H. Tomlinson, Chem. Phys. Letters, 1969, 4, 251. 90 P. Bertoncini and A. C. Wahl, Phys. Letters, 1970, 25, 991. 91 D. R. McLaughlin and H. F. Schaefer, tert., Chem. Phys. Letters, 1971, 12, 244. 92 B. Liu and A. D. McLean, J. Chem. Phys., 1973,59,4557. 81

92

Theoretical Chemistry

reported an extremely large CI calculation which should be within 0.1 K (3 x 10-7 hartree) of the exact clamped-nuclei interaction energy [AE(R) = E(R)- E(co)]. The discrepancy between the best calculation and experiment is 1.5 K. This discrepancy is currently under investigation. It should be stressed, however, that the calculation is of a very small quantity, and the accuracy attained is impressive. It should also be noted that the RHF approximation cannot predict any asymptotic attractive interaction for systems consisting of two like rare-gas atoms.93 A recent perturbation procedure for the calculation of intermolecular energies94 has been applied to the above problem.95The resulting well depth is only about twothirds of that observed, but the calculations throw some light on the energetics of interacting atoms. The second-order calculation does not predict binding, and the best calculation giving a qualitatively correct curve necessitated a fifth-order calculation. The work on the excited states is of importance in interpreting the rich molecular spectrum of helium, and there have been a number of papers not reviewed by Browne and Matsen.7 Calculations of the potential maximum for the AIZG state by Mukamel and Kaldor,96 using MO-CI wavefunctions with a large number of configurations, gave a potential curve that was quite accurate near its minimum. Agreement for the higher vibrational levels was not so good. This calculated potential was subsequently used to calculate the AlX: t X1Z,f absorption spectrum.97 Agreement was quite good for most quantities which could be compared with experiment. Similar calculations on the A1Zi and ClCgf states of He2 were carried out 98 using Goddard’s GVB method.13 Results were in excellent agreement with experiment. The repulsive nature of these states at large R results from non-bonding interactions between the singlet pairs of orbitals located on different centres. The explanation of the humps in the PE curves does not agree with some earlier proposals. The potential curves for the ‘Cb, 3Zi, and 52; states arising from two triplet metastable He atoms are important in the study of Penning and associative ionization. These have recently been studied by Garrison et aZ.,99using a large CI expansion, and ionization cross-sections have been calculated. Coupled Hartree-Fock perturbation theory has been used to calculate the polarizability of a pair of He atoms as a function of R. However, the authors conclude that further work is needed, including electron correlation.1° O Two rather more unusual calculations on He2 have appeared. Carrington and DoggettlOl have used a non-orthogonal pair theory (see for example ref. 1) in which the wavefunction Y has the form (4). Such functions have received relatively N

Y

=

dA1(l, 2)A2(3, 4).

. . AN(2N- 1 , 2 N )

(4)

little attention in the past because of difficultiesin working with non-orthogonal pair functions. The authors present a version in which there are no inter-pair orthoA. M. Lesk, J. Chem. Phys., 1973, 59, 44. J. P. Daudey, P. Claverie, and J. P. Malrieu, Iirternat. J. Quantum Cheni., 1974, 8 , 1. 9 5 J. P. Daudey, J. P. Malrieu, and 0. Rojas, Internat. J. Quantum Chem., 1974, 8, 17. ‘JG S. Mukamel and U. Kaldor, Mol. Phys., 1971, 22, 1107. ~ 3 7 S. Mukamel and U. Kaldor, Mol. Phys., 1973, 26, 2. 8 8 S. L. Guberman and W. A. Goddard, tert., Chem. Phys. Letters, 1972, 14, 460. Qg B. J. Garrison, W. H. Miller, and H. F. Schaefer, tert., J. Chem. Phys., 1973,59,3193. loo A. D. Buckingham and R. S. Watts, Mol. Phys., 1973, 26, 7. lol P. J. Carrington and G. Doggett, Mol. Phys., 1973, 26, 641. 93 94

Quantum Mechanical Calculations on Small Molecules

93

gonality constraints, and the method was applied to H2 and He. - -He. Calculations with different pair functions were carried out. It turns out that the shape of the interaction potential does not come out well, but the general results should encourage applications to bigger molecules. Within the Hartree-Fock approximation, calculations on molecules have almost all used the matrix SCF method, in which the HF orbitals are expanded in terms of a finite basis set of functions. Direct numerical solution of the H F equations, routine for atoms, has, however, been thought too difficult, but McCullough has shown that, for diatomic molecules, a partial numerical integration procedure will yield very good results.102 In particular, the He2 results agree well with the usual calculations, and it is claimed that the orbitals are likely to be of more nearly uniform accuracy than in the matrix HF calculations. Extensions to larger molecules should be very interesting; so far, published results are available for He, He2, and LiH. 3 Diatomic Hydrides of First-row Atoms, AH A. LiH and LiH*.-LiH has been a favourite molecule of theoreticians, and many calculations have been reported in recent years. Early work has been listed in Richards' bibliography,3 and Browne and Matsen' have discussed more recent work. Calculations of molecular properties for a variety of small molecules, including LiH and various other hydrides, have been reviewed by Lipscomb.1°3 The bulk of previous work has concentrated on the X I P ground state, but there is now an increasing number of studies on the excited states, particularly with wavefunctions beyond HartreeFock, and therefore including electron correlation. For comparison with this work, the near-HF calculation of Cade and Hu0104 remains a useful comparison point. Docken and Hinzelos have presented a 'very detailed study of the potential-energy curves for five valence excited states of LiH by the MCSCF method. In this type of calculation, the wavefunction, expressed in the form (3, is variationally optimized

Y

= ~ C I @ I I

(5)

with respect to both the Cr and the orbitals in terms of which the @ I are expanded. Docken and Hinze considered only valence-electron correlation, and 15 configurations were included for the 1X+ states, the orbitals being expanded in terms of a large STO basis set with exponents optimized in the molecule at Re.This basis yielded the Hartree-Fock limit in the single-configuration approximation. The dependence of the wavefunction on the internuclearseparation R is well shown in these calculations. The computed De of 0.089 hartree is in good agreement with the experimental value of 0.092 hartree. The PE curves were used to calculate the dipole and quadrupole moments, field gradients at the nuclei, and other properties.loSSome of these were averaged over the appropriate rotational-vibrat ional wavefunctions, and various electronic transition moments were also computed. In general, the agreement with experiment was very good. The best energy obtained at R = 3.015 was -8.02131, 108 1°3 lo4 105 106

E. A. McCullough, jun., Chem. Phys. Letters, 1974, 24, 55. W. N. Lipscomb, in ref. 4, Ch. 5, p. 167. P. E. Cade and W. M. Huo, J. Chem. Phys., 1967, 47, 614. K. K. Docken and J. Hinze, J Chem. Phys., 1972,57,4928. K. K. Docken and J. Hinze, J. Chem. Phys., 1972,57,4936.

94

nteoretical Chemistry

which is still above the best previous CI calculation of Bender and Davidson,lo7 who obtained - 8.0606 hartree. The transcorrelated wavefunction calculation of Boys and Handy47 still gave the lowest energy yet calculated for LiH, and Handy48 has obtained the same energy ( E = - 8.063 hartree) in calculations using gaussian basis functions such as were described above for H2.48 Although the emphasis in ab initio calculations in recent years has been on calculations using the MO approach, alternative methods have not been neglected, particularly Valence Bond (VB) methods of various types. Particularly promising has been the work of Goddard and co-workers with the so-called Generalized Valence Bond (GVB) method.13 The GVB method avoids a major deficiency of the HF method, namely the incorrect dissociation behaviour of the wavefunction, by taking the wavefunction to be of is a group operator that simultaneously takes care of the form (6),108 where Pauli's principle and of spin symmetry. @ is a product of spatial orbitals (7) and x is a @ =

41a(l)42a(2).

. . $ns(n)$lb(n

f

1).

- . +m b ( N )

(7)

product of one-electron spin functions @).Awavefunction of the above form has no

x

= a(l)a(2).

. . a(n)@(n

+

1).

. . B(fV>

(8)

restrictions on the form of the 0rbitals,10~~ l10 which are solutions of a set of coupled integro-differential q u at ions

2ida

= 8icja

i

=

la, l b , .

..

(9)

The orbitals may be self-consistently optimized,lll and this leads to a set of two one-particle equations to be solved, the resulting orbitals being called the GVB (or G1 or GI) orbitals. This method differs from the Hartree-Fock method in that one obtains two orbitals, one for each electron, rather than a doubly occupied orbital. If 4la = 4 l b the GI wavefunction reduces to the HF function. The fact that the orbitals are not constrained to be doubly occupied means that such wavefunctions can correctly describe molecular dissociation,and do include a certain amount of electron correlation. The different GVB methods differ in the spin-coupling schemes used.112 Calculations on LiH by Palke and Goddard (Gl),lOQand by Ladner and Goddard,ll2 in which the spin coupling was optimized (SOGI method), give essentially the same results apart from spin-dependent properties. Calculations on the ground statelog and the form of the orbitals are described in several papers. These calculations give an energy of E = - 8.01605 hartree, which is substantially less than the SCF result. The excited-state results are considered below. It should be emphasized that although this method has been implemented and many results have been described for small molecules, its extension to larger molecules at the present time seems likely to be computationally expensive. There is no doubt, however, that the insight obtained 107

C. F. Bender and E. R. Davidson, J. Phys. Chem., 1966,70,2675. A. Goddard, tert., Phys. Rev., 1967, 157, 81. W. E. Palke and W. A. Goddard, tert., J. Chem. Phys., 1969,50,4524. W. A. Goddard, tert., and R. C. Ladner, J. Amer. Chem. SOC.,1971,93, 6750. W. J. Hunt, P. J. Hay, and W. A. Goddard, tert., J. Chem. Phys., 1972, 57, 738. R. C. Ladner and W. A. Goddard, tert., J. Chem. Phys., 1969, 51, 1073.

lo8 W.

110 l11

Quantum Mechanical Calculations on Small Molecules

95

into molecular bonding is considerable, and for details the many papers of Goddard and co-workers should be consulted.11S The relationship of the method to others, such as the MCSCF and I N 0 4 methods, has been discussed by Hunt, Hay, and Goddard,lll who also examined the effect of the various possible restrictions on the energy and other quantities for LiH and various other small molecules. Another important use of these calculations has been an investigation of the component parts of the total energy, particularly those parts responsible for binding. Wilson and Goddardll4~ have reported the results of such an analysis for Hz, LiH, and several other diatomic molecules. The original VB method has been difScdt to use in practice because of the nonorthogonality of the atomic orbital basis, but there has been a revival of interest in it recently. Gallup and co-workers116-118 have described a new technique for carrying out such calculations, and the results of applications to the first-row hydrides. Calculations were carried out using a minimal basis set constructed from gaussian lobe orbitals. The orbitals were scaled to their best atom value and also optimally scaled in the molecule. Atomic populations were also computed.118 Calculations based on the many-electrontheory of Sinanoglu119 have been carried out by Ahlrichs and co-workers.120s121 Their method, called the IndependentElectron-Pair Approximation (IEPA), starts with an SCF calculation, followed by a transformation to localized orbitals. Then for any two electrons of the system, the energy increment Bit or 8 t j ( i # t j ) is calculated by an algorithm which is equivalent to admixing all possible doubly substituted functions with respect to QHF. The pair functions are calculated in terms of their natural orbital expansion. Using a basis of gaussian lobe functions, the relative importance of intra- and inter-pair correlations was studied for a variety of hydrides, including LiH,120Sl21where all the valence correlation is intra-pair. About 85% of the correlation energy was obtained in this calculation. Kirtman and co-workers,122 in an interesting series of papers, have described a method related to both the VB and the Atoms in Molecules method, and also to the GVB methods developed by Goddard et al.13 The method, called the Distinguishable Electron Method @EM), involves the use of a variation perturbation treatment to correct an unrestricted orbital-product approximation to the unsymmetrized molecular wavefunction. This method avoids the problems of using a Hartree-Fock Y 0, and uses an initial approximation to the wavefunction which is not symmetric with respect to electron exchange. Consequently, each electron moves in a different potential field, and the orbitals are obtained as solutions of one-electron perturbation-type equations. The success of this method hinges on the initial orbital approximation. If the best orbitals are determined self-consistently, the method reduces to that of Goddard (GVB). However, in a perturbation treatment, this is not necessary, For extensive references, see ref. 13. C. W. Wilson, jun., and W. A. Goddard, tert., Chem. Phy3. Letters, 1970, 5, 45. 116 C. W.Wilson, jun. and W. A. Goddard, tert., Theor. Chlm. Acta, 1972, 26, 195, 211. 116 G. A. Gallup, Internat. J. Quantum. Chem., 1972, 6, 899. 1 1 7 G.A. Gallup, Adv. Quantum Chem., 1973, 7 , 1 13. 118 J. M. Norbeck and G. A. Gallup, Internat. J. Quantum Chem., 1973, 7S, 161. 119 0. SinanGglu, Ado. Chem. Phys., 1964, 6, 315; ibid., 1969, 14, 237. 120 R. Ahlrichs and W. Kutzelnigg, J. Chem. Phys., 1968,48, 1819. 1 2 1 M. Jungen and R. Ahlrichs, Theor. Chim. Actu, 1970, 17, 339. l P 8 T. J. Venanzi and B. Kirtman, J. Chem. Phys., 1973,59, 523, and references therein.

118

114

96

Theoretical Chemistry

and a simpler orbital model based on the distorted atoms-in-moleculesidea has been pr0posed.12~Each orbital consists of a pure atomic function, which is rescaled in the molecule, and then polarized by a variable effective internal field. Thus the MO’s are expressed in terms of known atomic polarization functions. Illustrative calculations on H2, LiH, and BeH+were presented at Re. The energies obtained lie about halfway between the RHF and GVB values. Properties were computed in fairly good agreement with experiment. Further calculations of the static polarizability for LiH have given encouraging resu1ts.l2* It seems likely that this method can easily be used on small molecules (8-10 electrons), but for larger molecules it may become timeconsuming. Calculationsby the separated-electron-pairmethod, described above,lO1 have been reported for LiH. The energy obtained ( E = - 8.0109 hartree) was very good, and the calculationsare not too difficult to carry out. The semi-numericalSCF method102 has also been tested on LiH, with results in excellent agreement with the Cade-Huo calculation. Of particular note is the improvement in the cusp, and it is likely therefore that the overall wavefunction is more accurate. The success of calculations including correlation, described above, should not obscure the fact that such calculations are (i) relatively expensive, and (ii) not easy to apply to much larger molecules. Therefore, attempts to compute the correlation energy semi-empiricallythat can be applied to larger molecules are of considerable interest. Clementi has for many years advocated such an approach, and in a recent series of papers, Lie and Clementi 125 have demonstrated one such method for firstrow hydrides (AH) and diatomics (Az). The authors 1z5 propose that the correlation energy be calculated semi-empirically in terms of a functional of the Hartree-Fock density p, since H F wavefunctions are relatively easy to obtain. This type of approach, of which earlier examples have been given by Gornbas,lZ6expressed Ec in the form (10). EC =

1p&(p)du

(10)

Clementi and Lie introduce an empirical parametrization of &(p) such that Ec is given by (ll), where pm, the modified density, is given by (12) and ni is the orbital Ec =

10.02096(1.2 +

p%)-lp% dv

+

s

0.02096ln(l

+

2.39p2)pmdv (11)

occupation number. This form ensures that the computed values of the atomic correlation energies are in good agreement with experiment for both closed- and open-shell states. This functional was then used in calculations of Ec for first-row hydrides, with p obtained either from H F calculations or MCSCF calculations. In the latter the calculations were carried out including linear combinations of enough determinants to ensure that the molecule dissociates correctly (up to three in most cases). The 123 124

125 128

B. Kirtman, S. Y.Chang, and W. R. Scott, J. Chem. Phys., 1973, 58, 3304. D. P. Chong, W. R. Scott, C. P. Yue, P. S. C. Wang, M. L. Benston, and W. E. Palke, Internat. J. Quantum. Chem., 1974, 8, 137. G. C . Lie and E. Clementi, J. Chem. Phys., 1974,60, 1275. P. Gombas, ‘Pseudopotentials’, Springer-Verlag, New York, 1967.

Quantum Mechanical Calculations on Small Molecules

97

resultant PE curves for the HFPD (Hartree-Fock with Proper Dissociation) functions were obtained and the values of Ec computed. Agreement with experiment for the binding energies was much improved, and in particular, the incorrect prediction of HF calculations (or HF plus Ec) of a gradual increase in binding along the series LiH, BeH, and BH was corrected. The HFPD results show a different trend, i.e. LiH is nearly as binding as BeH, and BH is N 1 eV more strongly bound. Spectroscopic constants were also improved with this treatment, particularly De, which was in error by only a few per cent. The method looks promising for calculations on larger molecules. Table 2 gives the computed spectroscopic constants for the first-row hydrides computed by Lie and Clementi.

Table 2 Computed spectroscopic constantsa for first-row hydrides by HF and HFPD methods,l25 and also including correlation semi-empirically Molecule Method WeXe me Be ae Re De LiH XIC+ HF 23.26 7.426 1433 0.1945 3.034 1.49 22.46 3.077 1.91 HFPD 7.156 1306.1 0.216 Eqn. (11) 19.61 7.394 1359.2 0.203 2.999 2.36 23.20 Exptl. 7.513 1405.6 3.015 2.52 0.213 BeH P I C + HF 34.60 10.392 2147 0.2647 2.528 2.18 HFPD 27.61 2137.6 2.532 2.10 10.358 0.249 Eqn. (11) 0.258 28.17 10.607 2197.5 2.502 2.32 Exptl. 10.316 2060.78 36.31 0.3030 2.538 (2.2; 2.5) BH XIC+ HF 49.04 2.305 2.78 12.273 0.3726 2499 HFPD 48.56 11.693 2281.9 2.361 3.18 0.418 Eqn. (11) 47.36 11.992 2378.6 2.326 3.76 0.396 Exptl. 49 12.016 0.408 2367.5 2.336 3.58 HF 14.882 0.4712 CH X211 55.50 3053 2.086 2.47 HFPD 70.13 2766.0 2.139 2.97 14.156 0.576 Eqn. (11) 14.550 2881.4 65.70 0.554 3.60 2.110 Exptl. 63.0 2858.5 2.116 3.65 14.457 0.534 NH XSZ- HF 66.78 17.319 3556 1.923 2.10 0.5715 HFPD 28.805 95.13 1.352 3176.5 1.976 2.73 85.46 3305.2 Eqn. (11) 29.495 1.951 3.41 1.250 Exptl. 73 (3125.6) 0.646 16.668 1.9614 3.40 74.54 1.795 3.03 HF 0.6501 4062 19.712 OH X21T HFPD 99.19 3677.9 0.809 18.809 1.838 3.62 91.83 3801.6 Eqn. (11) 0.763 19.198 1.818 4.37 Exptl. 82.81 3735.2 18.871 1.8342 4.63 0.714 21.868 1.696 4.38 HF XIX+ HF 80.34 4469 0.7693 21.063 97.78 HFPD 4128.3 0.904 1.730 4.95 21.205 1.713 5.77 Eqn. (11) 93.39 4244.6 0.775 1.7328 6.12 Exptl. 90.44 4139.0 0.797 20.949 All units are given in inverse centimetres, except Re (in atomic units) and De (in electron volts).

-

N

The atttractive prospect of treating only the valence electrons in ab initio calculations, by devising an effective potential formulation for the core electrons, is not new, and the pseudopotential idea in Hartree-Fock calculations has been extensively explored.127 Kahn and Goddard128,129 have, however, shown how a unique and 127

J. D. Weeks, A. Hazi, and S. A. Rice, Adv. Chem. Phys., 1969, 16, 283.

128

L. R. Kahn and W. A. Goddard, tert., Chem. Phys. Letters, 1968,2, 667. L. R. Kahn and W. A. Goddard, tert., J. Chem. Phys., 1972,56, 2685.

la9

98

Theoretical Chemistry

local effective potential on atoms can be obtained from GVB wavefunctions. This is possible because, unlike the HF orbitals, the GVB orbitals are non-orthogonal, nodeless, and unique. Consequently an ab initio effective potential can be obtained from calculations on atoms, and used to replace the core orbitals in molecules. Comparison with full ab initio GVB calculations on LiH and various other molecules shows that these two wavefunctions are in excellent agreement. Bond lengths and dipole moments are well predicted. This procedure appears to hold some promise for applications to larger molecules. Applications to excited states are discussed below. A model potential approach in MO calculationshas been described by Schwartz.,l30 and comparative calculations with full ab initio results have been presented for LiH and 132 Excellent agreement between the two sets of calculations of the total valence energy was found. The basic atomic model potential consists of a core-charge coulomb potential modified by the addition of a short-range gaussian screened coulomb potential. A simple model that has been shown to give accurate oneelectron properties is the point-charge model due to Tait and Hall,133e134which has been applied to LiH. Other papers dealing with molecular properties of LiH are those by Slepukhin et al. on polarizabilities,l35where agreement with experiment was not very good, and discussionsof dipole-moment calculations by Green.16 A detailed discussion of the electron pair in diatomic molecules has been given by Daudel et al. 136 with respect to first-row hydrides, using the concept of loges.137 The authors find, from an analysis of near-HF wavefunctions, that the most probable partitioning of the system is one in which electrons are localized in well-defined spatial regions or loges. Although calculations on the XIC+ ground state have been the most common, two important papers have dealt with various excited states of LiH. Melius and Goddard have computed GVB138wavefunctionsfor a variety of lX+, 3C+,ill,and 3II states of LiH at R = 3.015 bohr. Comparison with the Bender and Davidson CI results107 shows that most energy values are -0.02 hartree lower, primarily because the (1s)Z core double occupancy is relaxed in the GVB method. It is shown that the molecular excitations giving rise to these states can be interpreted as excitationsfrom only the most loosely bound orbital, 42a. A detached discussion of the excited states in terms of orbital-correlation diagrams was very revealing. It should be noted, however, that the R value is in fact quite far from Re for the excited states. The effective-potential method129used for the excited states gives excellent agreement with the full GVB treatment . A more comprehensive study of the ME+, BII1,3C+ and 311 excited states was reported by Docken and Hinze,lo5who computed ab initio PE curves and a variety of properties by the MCSCF procedure. The calculated values of Re of 5.12,4.175, and 3.76 bohr for AIX+, BlII, and 311 were much longer than the value of R used by 130 131

M.E. Schwartz and J. D. Switalski, J. Chem. Phys., 1972, 57, 4125, 4132.

M. E. Schwartz, Discuss. Faraday Soc., 1972, 54, 21. M. E. Schwartz, Chem. Phys. Lerrers, 1973, 21, 314. l33 G.G. Hall, Chem. Phys. Letters, 1973, 6, 501. 134 A. D. Tait and G. G. Hall, Theor. Chim. Acta, 1973, 31, 311. 135 A. Yu. Slepukhin, M. A. Kovner, and K. I. Gur'ev, Teor. i eksp. Khim., 1973, 9, 799. 1 3 6 R. Daudel, R. F. W. Bader, M. E. Stephens, and D. S. Barrett, Canud.J. Chem., 1974,52, 1310. IR7 C. Aslangul, R. Constanciel, R. Daudel, and P. Kottis, Ado. Quantum Chem., 1972, 6, 93. 138 C. F. Melius and W. A. Goddard, tert., J. Chem. Phys., 1972, 56, 3348. 132

Quantum Mechanical Calculations on Small Molecules

99

Goddard, and compare with the experimental values of 4.9064 and 4.494 bohr for A1C+ and BlJJ. The calculated values of De, one-electron properties, and transition moments were in reasonable agreement with experimental values where these were known. The open-shell molecule LiH+ has been rather less studied. Goddard’s GVB calculation 138 gave an energy of - 7.74309 hartree, leading to a predicted ionization potential (IP) of LiH of 7.600eVYwhich is similar to that obtained by Browne.139 The bond length for LiH+ was studied also by Linnett et al. in FSGO cal~ulations.~7 this species seems to be rather uncertain, different calculationsgiving values between 3.0 and 4.25 bohr. SCF calculations using a GTO basis but with a set of s-GTO placed in the bond region have been carried out on LiH+.140 The calculations give almost the same energy as obtained by Cade and H~0.104 B. BeH, BeHf, and BH.-There have been fewer calculations on BeH and its ions than on most other hydrides AH. Cade and Huo’s104 H F calculation was supplemented by Mulliken, who extended the calculations to larger values of R.141 One interesting conclusion is the simultaneous existence between 4 and 5 bohr of two MO solutions of different forms, energies, and populations. The results also indicated the presence of a barrier in the H F potential curve. Bender and Davidson,l42 and Jungen and Ahlrichs,143 computed wavefunctions beyond HF, by including CT and by the IEPA method, respectively, for the experimental values of Re. Excited states were discussed by Chan and D a ~ i d s o n ,using l ~ ~ a large CI based on natural orbitals, and these were also used by Popkiel45 to calculate spectroscopicconstants and transition probabilities between the A211 and X2E+states. In an attempt to clarify the experimental results, Bagus et al.146 have presented the results of complete-valence-shell CI calculations of the PE curves of the A2JJ and XaZ+ states, including all single, double, and triple replacements in the HartreeFock reference function. This large calculation yields results in excellent agreement with experiment, and with the qualitative features of the X2X+state obtained by Mulliken at the SCF 1e~el.l~’ However, the potential maximum disappears when all configurations are included in the CI.The predicted De(BeH X2Z+) of 3.1 15 eV is in good agreement with recent experimental work, and these calculations show that precise values of molecular properties can be computed even if the core electrons are not correlated. The calculated properties in Lie and Clementi’s work on BeH125 are in good agreement with the CI results; however, the HFPD wavefunction can be obtained with much less computational effort. Absorption spectra of BeH and BeD have been observed recently, and Colin et al. have computed the H F energy of BeH+ with the same basis set as used by Bagus.14’ Combined with an estimate of the molecular correlation energy, the authors derive IP(BeH) = 8.19 5 0.06 eV, which is in excellent agreement with the experimental value of 8.21 f 0.09 eV. The BeH+ ground state was also studied by the DEM method, as recently modified and 139 140 141 142 143 144 145 148 l4’

J. C. Browne, J. Chem. Phys., 1964, 41, 3495. S. Y . Chu, Thew. Chim. Actu, 1972, 25, 200. R. S. Mulliken, Internat. J. Quantum Chem., 1971, 5 , 951. C. F. Bender and E. R. Davidson, Phys. Rev., 1969, 183, 23. M.Jungen and R. Ahlrichs, Thew. Chim. Acta, 1970, 17, 339. A. C. H.Chan and E. R. Davidson, J. Chem. Phys., 1968,49, 727. H.E. Popkie, J. Chem. Phys., 1971, 54, 4597. P. S. Bagus, C. M. Moser, P. Goethals, and G. Verhaegen, J. Chem. Phys., 1973, 58, 1886. R. Colin, D.De Greef, P. Goethals, and G. Verhaegen, Chem. Phys. Letters, 1974, 25, 70.

100

Theoretical Chemistry

described above.123 In a slightly different type of calculation, Hinkley148 et al. have used the Cade and Huo SCF wavefunctions to calculate the A doubling constants in the 2II states of a variety of hydrides, including BeH, where excellent agreement was found if Van Vleck’s hypothesis of ‘pure precession’ was assumed.l** Calculations on BeH and BeD for other than the lowest rotational state were also reported in a later paper.149 BeH and BeH+ were also studied, and are described in Daudel’s paper.136 Several groups have investigated BH including electron correlation. Lie and Clementi investigated BH in the paper dealing with HFPD wave function^,^^^ and several groups have included correlation in a strictly ab initio fashion. The IEPA method, previously described,l20)121has been used to study the variation of the correlation energy in BH with R, using a gaussian lobe basis The best calculated E of -25.26318 hartree at R = 2.28 bohr was not quite as low as in Bender and Davidson’s best calculation,l42but the dependence of the different pair-correlation energies on R could be determined. Although the inter-pair correlation energy is only 17% of the total, its variation with R is much more pronounced than that of the intra-pair correlation energy. A detailed semi-empirical study of effective paircorrelation energies has been given by Pamuk, with applications to BeH, CH, NH, OH, and HF.151The method is based on the Many Electron Theory of S i n a n o g l ~ , ~ ~ ~ and full details are to be found in ref. 151. Houlden and Csizmadia152 have carried out a variety of CI calculations on the ground and various excited states of BH. Their contracted gaussian (CGTO) basis set was augmented by orbitals which increase the electron density in the bonding region. Valence-shell CI, both limited and full, was carried out, and the results were analysed in terms of the average natural orbitals (ANO) which are formed by averaging the density matrices of a set of wavefunctions resulting from intermediate calculations (double substitutions in the CI). These ANO, used as basis orbitals for a more extensive valence-shell CJ, give results comparable in accuracy to more extensive CI calculations. BH has been extensively studied by Goddard during the development of the GVB methods. Early calculations were reported both with (SOGI) and without spin optimization (GI).153,Is* Contributions from other spin functions than the usual VB function (13) are negligible, and the SOGI and GVB calculations give very similar

-

XI =

4 w - P 4 a P - Pa)

(1 3) energies and orbitals. A later paper154 devoted to a discussion of methods of optimizing the orbitals of the GVB wavefunction compared this method with other related work. In particular, the strong orthogonality and perfect pairing restrictions were investigated, both for the ground state and the A3II and AlII excited states. In the case of the latter there are significant changes in the spin coupling as R changes. A detailed discussion of the orbitals, geometries, symmetry, and form of the PE surfaces for the low-lying states was also presented by Goddard and BIint.155 R. K. Hinkley, J. A. Hall, T. E. H. Walker, and W. G. Richards, J. Phys. ( B ) , 1972, 5, 204. R. K. Hinkley, T. E. H. Walker, and W. G. Richards, J. Phys. (B), 1972, 5, 2016. l50 M. GBlus, R. Alilrichs, V. Staemmler, and W. Kutzelnigg, Theor. Chim. Acta, 1971, 21, 63. l5I H. 0. Pamuk, Theor. Chim. Acta, 1972, 28, 85. 152 S. A. Houlden and I. G. Csimadia, Theor. Chim. Acta, 1973, 30, 209. 153 R. J. Blint and W. A. Goddard, tert., J. Chem. Phys., 1972, 57, 5296. 154 W.J. Hunt, P. J. Hay, and W. A. Goddard, tert., J. Chem. Phys., 1972, 57, 738. ’ 5 5 W. A. Goddard, tert., and R. J. Blint, Chem. Phys. Letters, 1972, 14, 616. 148

149

Quantum Mechanical Calculations on Small Molecules

101

A very recent paper by Blint and Goddard,l58 using the SOGI method, has given more details of the wavefunctions and properties of the XIX+, a3H, All& and 3X+ states of BH. The PE curve for the N I T state was found to have a hump at N 3.9 bohr, due to the recoupling of the orbitals which must occur as R decreases from R = 00 to Re. The dipole and quadrupole moments, and the electric field gradient, were calculated as a function of R. BH was also studied using the GVB effective-potential method referred to ab0ve.1~~ A method of calculating the correlation energy, intermediate between the IEPA approximation and the total pair excited variational function, has been described.157 The results of an application to lZ+BH are in good agreement with other calculations. Mulliken158 has carried out near-HF calculations on the 3 X + and 1X+ excited states of BH. Convergence problems for the lZ+ state at R > 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

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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

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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.

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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,

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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.

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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

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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.

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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.

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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 <200 configurations based on perturbation theory was developed,316 and the computed dipole moment was then within 0.01 D of the experimental value. It should be recalled that the RHF value has the wrong sign. In later papers the open-shell a311state of CO was investigated,317 particularly the effect of the quality of the basis set.318It was shown that a CI function will contain essentiallythe same error due to the choice of basis set as the SCF function upon which it is based. The best calculation gave an error for this state of 0.06D. More recently, the a’3X+ state was investigated.31g Results were 0.4D differentfrom experimental ones, and the possible reason for this was given. An interesting recent study of the perturbations of the a311 and AlII states of CO due to spin-orbit and rotational-orbit interactions has a~peared.3~0 By including CI functions built from HF orbitals optimized for each state, good agreement was obtained between theory and experiment in most cases, but the single-configuration approximation is seriously in error. Gaussian basis calculationson diatomics have been rather rare, but Vladimiroff321 312 313 314 315

316

317 319 s20

821

P. S. Bagus and H. J. T. Preston, J. Chern. Phys., 1973, 59, 2986. S. V. O’Neil and H. F. Schaefer, tert., J. Chem Phys., 1970, 53, 3994. A. K. Q. Siu and E. R. Davidson, Internat. J. Quantum Chem., 1970, 4, 223. S. Green, J. Chem. Phys., 1970, 52, 3100. S. Green, J. Chem. Phys., 1971, 54, 827. S. Green, J. Chem. Phys., 1972, 56, 739. S. Green, J. Chem. Phys., 1972, 57, 2830. S . Green, Ado. Chem. Phys., 1974, 25, 179. J. A. Hall, J. Schamps, J. M. Robbe, and H. Lefebvre-Brion, J . Chem. Phys., 1973, 59, 3271. T. Vladimiroff, Chem. Phys. Letters, 1974, 24, 340.

Quantum Mechanical Calculations on Small Molecules

121

has reported the results of a (9s5p)+ [4s3p] calculation on CO using Is, 2pz, and 2py functions centred within the bonding region. By optimization of the exponents and the position of the centre, an energy lower than that obtained in an earlier calculation with a more extensive basis322 was obtained, but one-electron properties were not quite so accurate. Green317 has also reported calculations on CS, both for the ground and A l I I excited states. Agreement between theory and experiment for p was very good for both states. Spin-spin and spin-orbit constants for CO have been investigated using semiempirical methods, for states which can be described by a single configuration.323 SCF calculations on CO and CO+ have been reported by Certain et aZ.,324who computed a dipole moment of 2.5 f0.5 D and discussed the differences between the value in CO, CO+, and CN. GVB calculations on CO have been reported.325 SO has been studied relatively little, although a near-HF wavefunction was obtained in 1967.326 The PE curves for 72 states have been investigated by Heil and Schaefer.327 In these a minimal STO basis was used with full CI, with results very similar to those described above for CO. The limitations of minimal-basis full CI have been summarized by Schaefer,l but the method does yield reliable PE curves. The predicted ordering of the state is X1X+, a3H7d 3 C + ,3A, $2, PC-, AlH, 1A, 6E+, and 511.This paper illustrates how useful such calculationscan be in analysingspectra. NO, NO+,and PO. Until quite recently there had been few theoretical studies of the X2n state of NO, although unpublished calculations by Cade have been quoted. Using these near-HF wavefunctions, Politzer and Harris328 have computed the electron-densitydistribution in NO and NO+ and analysed the bonding in these molecules. The NO difference density was also computed. Comparison with the isoelectronic 0; shows there to be less charge in the internuclear region in NO. In the case of CO the increased charge build-up in this region and the localization of charge in lone pairs also explains the higher value of De. NO+ has a substantially altered charge distribution, with much greater concentration in the internuclear region, and this is in accordance with its stronger bond. Kouba and ohm have gone beyond the SCF approximation and calculated a limited CI wavefunction, using NS0.329The lowest energy obtained was - 129.2599 hartree. This is still above the estimated HF limit. The computed dipole moment of 0.121 D is in fair agreement with the experimental value of 0.148 D. The spin density at the two nuclei was also computed. A more extensive study was carried out by Green.s3O Using a large STO basis, both ground X 2 n and excited A2X+ states were studied. The calculated energy is -0.05 hartree above the RHF limit, and computed properties were in reasonable agreement with experiment (within 10%). However, a later limited CI study331 gave better spin 322

323 324 325 326

327

328 329

330 881

D. B. Neumann and J. W. Moskowitz, J. Chem. Phys., 1969,50, 2216. R. W. Field and H. Lefebvre-Brion, Acta Phys., 1974, 35, 51. P. R. Certain and R. C. Woods, J. Chem. Phys., 1973, 58, 5837. A. P. Mortola and W. A. Goddard, tert., J. Amer. Chem. SOC.,1974, 96, 1. A. D. McLean and M. Yoshimine, Internat. J. Quantum Chem., 1967, lS, 313. T. G. Heil and H. F. Schaefer, tert., J. Chem. Phys., 1972, 56, 958. P. Politzer and R. R. Harris, J. Amer. Chem. SOC.,1970, 92, 1834. J. E. Kouba and Y . Ohm, Internat. J. Quantum Chem., 1971, 5, 539. S. Green, Chem. Phys. Letters, 1972, 13, 552. S. Green, Chem. Phys. Letters, 1973, 23, 115.

122

fieoretical Chemistry

densities for the 2C+ state, but the dipole moment and quadrupole coupling constant are not in good agreement with experiment. NO+ is also a molecule that was relatively little studied until recently. An SCF calculation of the quadrupole moment giving a value in reasonable agreement with experiment has been reported;3s2and a more detailed OVC wavefunction was computed by B i l l i n g ~ l e y in~order ~ ~ to calculate the absolute i.r. intensities for the 0-1, 0-2, and 1-2 vibration-rotation transitions. Thulstrup and have computed full PE curves for NO and NO+, using a minimal basis set and full CI. Assignments were presented for the photoelectron spectra, and were in reasonable agreement with experiment. Theoretical studies of PO have been carried out recently by Mulliken and Liu,2*3 who obtained a wavefunctionclose to the Hartree-Fock limit. An investigation of the Rydberg states has also been rep0rted.3~~ The first calculations using CI were those of Tseng and who have studied a variety of PE curves for the low-lying 211, 4l7, 2X+, 2C-, 4E-, 2A, and 2@ states. A minimal STO basis and full valence-shell CI were used. The relative positions of the states agree well with experiment, and several predictions were made for as yet unobserved states. Roche and Lefebvre-Brion 337 have also recently investigated PO, performing valence-shell CI calculations for several states, including several also studied by Tseng and Grein.336 Spin-orbit coupling constants for the 2IT states showed a larger variation with R than does the experimental constant for the first excited 2n state. Schaefer et al. also investigated PO in their work on the representation of valence orbitals using contracted C T O ’ S . ~ ~ ~ C10 and FO. Interest in these species has stimulated near-HF calculations by O’Hare and Wah1.338~339 Using previously developed methods for estimating the correlation energy, the binding energy of FO was estimated to be 3.0 eV, in fair agreement with experimental estimates. Calculations on OF+(3C), OF(211), and OF-(lC) allowed estimates of the VIP and EA.338 Similarcalculationsfor ClO and its ions were reported in a later paper.339 Calculated properties were in good agreement with experimental data where available: however, in common with many recent calculations on unstable species, a great many of the properties stand as predictions. E. Halides.-Metal Halides. E.s.r. spectra of various diatomic fluorides in 2C states have been observed, such as MgF, CaF, SrF, and BaF.340Ab initio calculations on these species should be of interest. The alkali-metal fluoride wavefunctions previously rep0rted3~69341 have been used to calculate the electric field gradient in these and other halides by Laplante and Bandrauk.342 It was found that simple models can give unreliable predictions, and a detailed analysis of the contributing terms was

-

332 333 334 335 33fi

337 338 339

340 341 342

Ch. Jungen and H. Lefebvre-Brion, J. Mol. Spectroscopy, 1970, 33, 520. F. P. Billingsley, tert., Chem. Phys. Letters, 1973, 23, 160. E. W. Thulstrup and Y . Ohm, J. Chem. Phys., 1972, 57, 3716. F. Ackermann, H. Lefebvre-Brion, and A. L. Roche, Canad. J. Phys., 1972, 50, 692. T. J. Tseng and F. Grein, J. Chem. Phys., 1973, 59, 6563. A. L. Roche and H. Lefebvre-Brion, J. Chem. Phys., 1973, 59, 1914. P. A. G. O’Hare and A. C. Wahl, J. Chem. Phys., 1970, 53, 2469. P. A. G . O’Hare and A. C. Wahl, J. Chem. Phys., 1971, 54, 3770. L. B. Knight, jun., W. C . Easley, W. Weltner, jun., and M. Wilson, J. Chem. Phys., 1971, 54, 322. A. D. McLean and M. Yoshimine, supplement to I.B.M. J. Res. Develop., 1968, 12, 206. J.-P. Laplante and A. D. Bandrauk, Canad. J. Chem., 1974, 52, 2143.

Quantum Mechanical Calculations on Small Molecules

123

made. There is a differencebetween the behaviour of c- and n-electrons, exchange distortions of the o-electrons are important, and also polarization of the n-electrons. The photoelectron spectra of AlCl and GaCl were interpreted in terms of NO'S computed both by ab initiu and semi-empiricalmethods.343 LiF, being the simplest fluoride, has been extensivelystudied, and in a recent paper Williams and Streitweiser344 have examined the effect of variation in the size of the basis set on the charge distribution and molecular properties in SCF calculations. The basis sets used ranged from a minimal (STO4G) to a Ss4p GTO set on F, 8s on Li, to which was added lithium 2po and 2pn functions and a set of 3d functions on F. It was shown that the minimal basis set overweights Li and gives unrealistic bond properties. In extended basis sets, the inclusion of 3d functions polarizes the electron density from behind F towards Li. It is encouraging to note that the basis without 3d functions on F gives reliable values for the bond length, force constant, and dipole moment. Comparison with the near-HF wavefunction was made. Classical VB theory has been relatively neglected because of the orthogonality problem until quite recently, However, computational methods have been recently proposed for evaluating the relative matrix elements more efficiently,and Carrington and Walton have investigated the molecule TiF3+ using a limited number of chemically significant structures.345 It was necessary to devise and use a pseudopotentialfor the core electrons in order to handle these, and to use a STO-GTO expansion to simplify the integral computations. Comparison of the results with conventionalSCF calculations shows that the method is competitiveregarding computing time and has an additional advantage that the excited triplet- and singlet-state wavefunctions are also obtained. Minimum-basis-set (STO) calculations on GaF were reported in 1970 by Stevenson and Lipscomb.346 Halides of B, C, and Si. Recent work has been confined to the fluorides of C and Si. CF has been the subject of a near-HF calculation by O'Hare and Wah1,347together with CF+ and CF-. Spectroscopic constants were evaluated and compared with experiment. One interesting result is that the dipole moment of CF was computed to be 0.48 D, with polarity C-F+. This surprising result needs confirming by MCSCF calculations, because although near-HF dipole moments can be in error if p is small, i.e. CO, it is estimated that an error of 0.5 D in these calculations is unlikely. The sign is not determined experimentally. It is noteworthy that BF is also predicted to be B-F+, with ,uz 1 D. The correlation energies of the first-row diatomic fluorides were also deduced in this study. LCAO-MO-SCF calculations on the ground state and several low-lying states were carried out later by Hall and Richards.348 The basis set used was not as large, but the sign of the dipole moment was confirmed. The computed spin-orbit coupling in the ground state was in good agreement with experiment. The related molecule SiF was also studied by Wahl and co-workers, and its properties were computed.347 In this case p = 1.20 D (Si+F-), which is substantially less than the value for SiO (3.40 D) despite the fact that R(SiO) < R(SiF). A later 343

s44 s45 346 347

848

J. Berkowitz and J. L. Dehmer, J. Chem. Phys., 1972, 57, 3194. J. E. Williams and A. Streitweiser, jun., Chem. Phys. Letters, 1974, 25, 507.

P. J. Carrington and P. G . Walton, Mol. Phys., 1973, 26, 705. p. E. Stevenson and W. N. Lipscomb, J. Chem. Phys., 1970, 52, 5343. P. A. G. O'Hare and A. C. Wahl, J. Chem. Phys., 1971, 55, 666. J. A. Hall and W. G. Richards, Mof. Phys., 1972, 23, 331.

124

Theoretical Chemistry

paper examined the computed PE curves obtained using a faster computer.349The computed dipole moments d8er by several per cent for the computed Re, compared with those at RFP. Goddard has also computed GVB wavefunctions for CF, but details are not a~ai1able.l~ However, the GVB wavefunctionswere compared with those of CH, and it was shown that additional repulsive interactions with the lone pairs on F should make the state of CF much more strongly bound than the *Z- state. (The calculated AEs 2.8-2.9 eV.) Fluorides of N, P, S, Se, and As. Extended-basis-set near-HF calculations on the X 3 Z - , alA, and blX+ states of NF and the 211states of NF+ and NF- were reported by O’Hare and Wahl in 1971.350N F is isoelectronic with 0 2 and has similar lowlying excited states. The calculations were carried out at e P for the neutral molecules. Values of De derived with the use of semi-empirical correlation energies were in excellent agreement with experiment. The dipole moments of the low-lying states are different, unlike the case of NH, because of variations in bond length. The NF bond is slightly more ionic than the NH bond. Results for PF and its ions were also discussed.350Little is known about this species experimentally. CI calculations on NF and NF+ have been reported by Anderson and Full PE curves were computed, using a full CI with a minimal STO basis set. The X3Z- energy was substantially higher than the near-HF result, despite the CI. The computed value of De (3.6 eV) obtained by Wahl et a2.850 is also different from that obtained by Anderson (2.2 ev). The predicted IP of 9.7 eV is much lower than the experimental value or the value obtained by Wahl(l3 ev). Several predictions were made, but since the wavefunctionis not as good as the near-HF function, one would tend to trust the latter. Very similar calculations on NF and NF- were also reported by Ellis et uZ.267 However, the above 3 sets of workers used slightly different values of Re for computing properties. The energies attained by Anderson et aZ.351 and by Ellis et aZ.,2e7 as well as the properties, are, however, almost the same. The bond length of NF- is slightly longer than in NF. However, only Anderson and Ellis have so far studied the low-lying excited states. PF was also studied, and the PE curves were calculated, together with the electron affinity and dipole moment, in the calculations of ref. 349.

SF, SeF, and AsF. These diatomics have only recently been synthesized, and there have been no calculations prior to 1970, when Wahl and co-workers reported nearHF wave function^.^^^ In the light of the known accuracy of these, it is suggested that the experimental appearance potentials are too high by 2 eV. Thermodynamic calculations suggest that SF- and SeF- might be moderately stable. A later, more extensive study of the PE curves of SF has appeared.349 A large number of predictions were made for the spectroscopic constants. A very recent PE curve has been computed for SeF and its ions.353 The value of D Ois estimated to be 3.2k0.1 eV. The computed p is significantly different from the experimental value. The AsF mole-

-

349

350 952 355

P. A. G. O’Hare, J. Chem. Phys., 1973, 59, 3842. P. A. G. O’Hare and A. C. Wahl, J. Chem. Phys., 1971,54,4563. A. Anderson and Y. Ohm, J. Mol. Spectroscopy, 1973,45, 358. P. A. G. O’Hare and A. C. Wahl, J. Chem. Phys., 1970,53, 2834. P. A. G. O’Hare, J. Chem. Phys., 1974, 60,4084.

Quantum Mechanical Calculations on Small Molecules

125

cule has also been the subject of recent near-HF calculation~.~5~ The wavefunctions were used to computeideal-gas thermodynamicfunctions for AsF in this investigation. F. Nitrides and Carbides.-A number of recent papers have dealt with BN, which was studied some years ago by Verhaegen et al.355 The lZ+, lIT, 3rI, and 3Zf states were investigated, and computed bond lengths obtained. The 3rI state is the ground state. The 1C+ excited state was studied by Moffat using three GTO basis sets of different quality.356 The computed bond length decreases as the size of the basis set goes up, the computed value being in good agreement with experiment for the (9,5) basis, although the energy is -0.08 hartree above the HF limit. Two papers by MeIrose and Russell have dealt with the excited s t a t e ~ . ~ 358 ~7~ Extensive calculations of the PE curves of a large number of states of CN have been carried out by O’Neil and Schaefer.359 A number of as yet unobserved states were predicted, using a minimal basis set and full CT. Configurations differing by up to 7 orbitals from the reference configuration were included. The calculated order of the eight known doublet states agrees with experiment, and it is thus likely that the positions of the unobserved states are correctly predicted. It is interesting that recent experimental work has determined the hyperfine, p-doubling, and rotational constants from observation of the CN molecule in an interstellar cloud.360 The dipole moment of the a2rI state has also recently been measured.861 CN- has been studied by Doggett and M~Kendrick,3~~ and one-electronproperties have been computed and listed in a later paper.363 The structure of CN- was also discussed in a review by Clementi.364 NS is an unstable radical, which has recently been investigated by O’Hare, together with its ions.365The EA was also computed in this work. The diatomic BC has been studied by Kouba and h n . 3 6 6 The ground state is *Z-, according to calculations using full-valence-shell CI. 54 low-lying states were investigated. G. InterhaIogens.-ClF is the most extensively studied of these molecules, and several calculations have been reported recently. Guest et al.,367 using a STO-GTO expansion basis, calculated Re = 1.67 a (expt.: 1.628 A) but gave no total energy. Breeze et al.36salso carried out an SCF calculation at RYP, using a moderatssize basis. They obtained a total energy of - 557.306 hartree in their most extensive calculation. 354 355 356 357 358 359 860

381 864

363 3G4 365 366 3G7 368

P. A. G.O’Hare, A. Batana, and A. C. Wahl, J. Chem. Phys., 1973, 59, 6495. G. Verhaegen, W.G. Richards, and C. M. Moser, J. Chem. Phys., 1967,46,160. J. B. Moffat, J. Mol. Structure, 1973, 16, 307. M.P. Melrose and D. Russell, J. Chem. Phys., 1971, 55, 470. M. P. Melrose and D. Russell, J. Chem. Phys., 1972, 57, 2586. T. G. O’Neil and H. F. Schaefer, tert., J. Chem. Phys., 1971, 54, 2573. A. A. Penzias, R. W. Wilson, and K. B. Jefferts, Phys. Rev. Letters, 1974, 32, 701. T. J. Cook and D. H. Levy,J. Chern. Phys., 1973,59,2387. G . Doggett and A. McKendrick, J . Chem. Soc. ( A ) , 1970, 825. M. Dixon and G. Doggett, J.C.S. Fnraday 11, 1973, 69,298. E. Clementi, in ‘ElectronicStructure of Molecules, Especially in Cyano Compounds’, Ch. 1 of the book ‘C=N’, Interscience, New York, 1970. P. A. G.O’Hare, J. Chem. Phys., 1971, 54, 4124, J. E. Kouba and Y . Ohrn, J. Chem. Phys., 1970,53, 3923. M. F. Guest, M. B. Hall, and I. H. Hillier, J.C.S. Furuduy 11, 1973, 69, 182. A. Breeze, D. W. J. Cruickshank, and D. R. Armstrong, J.C.S. Faraday II, 1972, 68, 2144.

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Theoretical Chemistry

The dipole moment is very sensitive to the basis set, but the largest basis gives polarity CI+F-. The 3d-orbital exponent was optimized in these calculations. Green,36ghowever, carried out a much bigger calculation, since experimentally the direction of the dipole was found to be Cl-Ff. The computed value in Green’s calculation was 1 .OW, with polarity Cl+F-, and other spectroscopic properties were in moderate agreement with experiment. It was concluded that the experimental sign is in error. In his review of errors in dipole-moment calculation,l6 Green has reported CI calculations of p, including single and double excitations. The value p = 0.84 D was obtained, with a positive value, i.e. Cl+F-. Evidence that delta orbitals are important was presented. The 200-configuration CI wavefunction gave an energy of - 559.1112 hartree, which is 2 hartree lower than in Breeze’s calculation. A calcufation using a larger basis set (30o18n)gave an SCF error of 0.0017 hartree and changed p by only 0.1 D. Hence it is clear that the experimental sign of p(CIF) is incorrect. An approximate SCF method treating only the bonding electrons but allowing split shell orbitals, and using approximate integrals, has been reported and applied to the interhal0gens.3~~ Rather good agreement with experiment was found, using only one adjustable parameter. A much more extensive study of diatomic halides, interhalogens, and halogen hydrides has been described by Straub and McLean.571 DZ + P basis sets were used for all molecules except IBr and 12, where single zeta + polarization basis sets were used. The expected errors in basis sets of this kind for computations on molecules containing heavy atoms were described. Total energies were used to compute dissociation energies and enthalpies. The computed Re values were in satisfactory agreement with experiment. Several properties were computed with fairly high accuracy. This series of calculations again emphasized the usefulness of near-HF SCF wavefunctions. The disagreement between theory and experiment over p (hence the charge distribution) has been further studied by measurement of the core binding energies in ClF,372 where the results are consistent with the charge distribution Cl+F-. H. Miscellaneous Calculations on Diatomic Molecules.-A variety of more approximate treatments have been tested on diatomics. Naray-Szabo has tested a method for approximate H F calculations373 (in which the integrals are approximated) in applications to various first-row diatomics. The results were reasonable for non-polar molecules but the method fails for molecules where the atoms have very different 2, i.e. Be0 and BeF, amongst others, did not give a minimum in the PE curve. An interesting molecule that has been little studied is SiS lZ+,which should be similar to the isovalent CO, CS, and SiQ. BA P calculations at Re by GreenI6 give a value of /A that is -0.5 D larger than the experimental value. The diatomic molecules PN and PC were also studied by Bauschlicher and S~haefer.~ls In concluding this section, we draw attention to a paper by Roby,374 who has

-

+

ntiq

370 371 372

373 37*

S. Green, J. Cliem. Phys., 1973, 58, 3117. A. B. Sannigrahi and S. Noor Mohamrnad, Zitternat. J. Quantum Cliem., 1973, 7 , 1183. P. A. Straub and A. D. McLean, Theor. Chim. Acta, 1972, 32, 227.

T. X. Carroll and T. D. Thomas, J. Chem. Phys., 1974,60, 2186. G.NAray-Szab6, Internat. J. Quantum Chem., 1973,7 , 569. K. R. Roby, Mol. Phys., 1974, 27, 81.

Quantum Mechanical Calculations on Small Molecules

127

examined in detail the question of the definition of a charge in a molecule, and has examined several diatomic molecules using the results of HF calculations. Extensive references to earlier work on this topic are given in this paper. 8 Linear Triatomic Molecules The past three years have seen considerable advances in the study of the electronic structure of triatomic molecules, and calculations of comparable accuracy to those reported on diatomics are now feasible. Calculations of PE surfaces are dealt with elsewhere in this volume, and in this section we deal with calculations on isolated molecules. An extensive compilation of the wavefunctions of linear molecules was published several years and a great many recent calculations have attempted to extend the list of molecules for which near-HF wavefunctions are available. Furthermore, in a certain number of cases, wavefunctions including electron correlation have been obtained. We will consider linear and non-linear molecules separately. A. He/H2 and &.-It is convenient to consider first these species. Tsapline and Kutzelnigg375 have applied the IEPA-PNO method, previously described, to the ground state of the He/H2 system. The van der Waals minimum was computed, using a gaussian lobe basis set with carefully optimized exponents. The collinear arrangement with a depth of 21 K was found for the van der Waals minimum, with a saddle point of -14K for the CzVgeometry. The computed surface was compared with experiment and with the term. The anisotropy of the potential is larger than that predicted asymptotically. Knowledge of an accurate H3 potential surface is of considerable importance, and accurate wavefunctions have been obtained during the past few years. Goddard, in 1969,112reported SOGI calculations for several nuclear configurations, and showed H+H H2 reaction. that the spin coupling changes significantly during the H2 Changes in the orbitals were studied, and it is apparent that the orbitals change continuously during the course of the reaction.l1° Approximate non-empirical calculations on H3 were in good agreement with early ab initio calculations.79 The most accurate calculations reported to date are those of Liu.376, 377 In 1973, a very large CI calculation was reported using a large STO basis set for the linear species. The calculated surface was believed to lie between 0.8 and 3.4 kJ mol-1 above the exact surface. These calculations also are a landmark in computational work. B. Hydrides AH2.-HeHi has been investigated at SCF level, using a CGTO basis set. The ground-state PE surface was computed in this work.378 The semi-empirical diatomics-in-moleculesmethod 379 has been used to study the PE surface of linear HeHl,380 and later it was further used in studies of a variety of fist-row hydrides, including linear B ~ H ZBeH2 . ~ ~has ~ been rather extensively

+

375 376 =?? 378 379

380

881

B. Tsapline and W. Kutzelnigg, Chern. Phys. Letters, 1974, 23, 173. B. Liu, Internat. J. Quantum Chem., 1971, 5, 123. B. Liu, J. Chem. Phys., 1973, 58, 1925. P. J. Brown and E. F. Hayes, J. Chem. Phys., 1971, 55, 922. F. 0. Ellison, J. Amer. Chem. Soc., 1963, 85, 3540. P. J. Kuntz, Chem. Phys. Letters, 1972, 16, 581. J. C . Tully, J. Chem. Phys., 1963, 58, 1396.

+

128

Theoretical Chemistry

studied recently, following an extended GTO basis-set calculation in 1968.382The ground state was also investigated with the IEPA method by Ahlrichs and Kutzelnigg,383who investigated the physical properties and predicted experimental conditions under which it might be observed. A later paper dealt with the inter- and intrapair correlation energies.121 There have been a few VB calculations on this molecule, particularly since the recent advances in computer capability make the problem of non-orthogonality of the orbitals less serious. Maclagan and Schnuelle384 have reported the results of a minimal-basis STO calculation, including all configurations which do not involve excitation of the Be core electrons. Comparison with a MO calculation was made. A VB equivalent to the CND0/2 method was proposed. Gallup e f d 3 8 5 have also carried out a similar calculation and proposed a population analysis scheme for VB wavefunctions. Other calculations on BeH2 by these workers have appeared.1l6-1l8 Further comparative calculations by both VB and MO methods on BeH2 have been reported, using a minimal STO basis, with exponents optimized for various R.386With choice of particular VB configurations, it is shown that a model based on the resonance valence state, including ionic configurations, is particularly useful. Both methods show a localized-pair bond description to be useful. The most recent large calculation on BeH2 is a large CI calculation by Hosteny and Hagstrom,387 who obtained a total energy of - 15.8478 hartree, which accounts for 55% of the correlation energy. An STO basis set was used and a discussion of the wavefunction was given. LiH2 has been discussed by Kahn and Goddard, using GVB w a v e f ~ n c t i o n s . ~ ~ ~ C. Compounds of the Inert Gases.-Several groups have examined theoretically whether species such as HeO2, HeO:, KrF2, XeF2, and Be2+He2are expected to be stable. Matrix-isolation spectroscopy prompted theoretical studies of ion-induced dipole clusters, and Be2+He2in particular is predicted to have a stable linear configurati0n.~8*These calculations, using a large GTO basis set, gave a near-HF wavefunction, and D e z 74.3 kJ mol-l. It should be interesting to see if this prediction is borne out experimentally. The compound KrF2 is of great importance in the chemistry of krypton, and the nature of the binding is a challenge to the theoretical chemist. Early calculations at the minimal-basis-set level were reported by Collins et a p 9 ,390 using STO-3G expansions. Extra valence orbitals were added to the basis set. Electron-density plots showed the three-centre nature of the bond. A much more extensive set of calculationswere carried out by Bagus et a1.,s91 who calculated both SCF and first-order wavefunctions, the latter with up to 993 configurations. The best energy was considerably lower than that obtained by Collins 382

383 384 385 386 387 388

389 390 391

J. J. Kaufman, L. M. Sachs, and M. Geller, J. Chem. Phys., 1968, 49, 4369. R. Ahlrichs and W. Kutzelnigg, Theor. Chim. Acta, 1968, 10, 377. R. G. A. R. Maclagan and G . W. Schnuelle, J. Chem. Phys., 1971, 55, 5431. G . A. Gallup and J. M. Norbeck, Chem. Phys. Letters, 1973, 21, 495. K. A. R. Mitchell and T. Thirunamachandran, Mol. Phys., 1972, 23, 947. R. P. Hosteny and S. A. Hagstrom, J. Chem. Phys., 1973, 58, 4396. S. W. Harrison, L. J. Massa, and P. Solomon, Chem. Phys. Letters. 1972, 16, 57. G . A. D. Collins, D. W. J. Cruickshank, and A. Breeze, Chem. Comm., 1970, 884. G. A. D. Collins, D. W.J. Cruickshank, and A. Breeze, J.C.S. Faraday 11, 1974,70, 393. P. S. Bagus, B. Liu, and H. F. Schaefer, tert., J. Amer. Chem. SOC.,1972, 94, 6635.

129

Quantum Mechanical Calculations on Small Molecules

et al. It was also shown, as expected, that polarization functions are much less important than suggested by Collins. The SCF result shows that KrF2 is unstable with respect to dissociation into Kr F2, but the CI calculation corrects this. The best calculation gives reasonable values for R(Kr-F), but De is only ca. 3 of the experimental value. The most interesting feature of the PE curve is the prediction of a maximum, which results from avoided crossing of an attractive ionic and a repulsive covalent curve. The details of the binding were very dependent on electron-correlation effects. This calculation illustrates very well the usefulness of ab initio calculations on unusual molecules. XeF2 was first investigated by ab initio SCF calculations in 1971, using a limited basis ~et.89~ Population analyses of the wavefunction were compared with experimental ESCA binding energies in a paper by Carroll ef a1.393 There are several puzzling features, since the derived fluorine charges seem too small compared with calculations. Calculations using an extended basis set are needed here. D. The Bihalide Ions .-Spectroscopic observations on these species have been made, but there are a number of interesting and unresolved questions which theoretical calculations should be able to clarify. Matrix-isolation studies are in disagreement as to whether vacuum-u.v. photolysis of HCl in an argon matrix produces ClHCl or the ClHCl- anion.394 Semi-empirical studies of hydrogen bonding have been 396 but recently some extensive ab initio calculations have appeared. Almlof 397 has investigated using a large GTO basis for the linear configuration. Calculated properties were in good agreement with experiment, and the symmetricconfiguration was found to be the most stable. A more extensive basis set was used by Noble and K o r t z e b ~ r n who , ~ ~ ~also investigated linear HF2 and HeF2. The calculations predicted a weak (- 13 kJ mol-l) bond in HF2 and a long F a * H Fbond. HeF2 was predicted to be an unstable repulsive state. A more recent study of ClHCl- has appeared, by JanoshekY3g9 who varied both C1. * *C1and the position of H. He calculated the energy levels, frequencies,and various other observables. Results were in reasonable agreement with experiment. A more extensive investigation recently by Clark et al.40° has attempted to resolve this question. Calculationswith a DZ P STO basis set show ClHCl- to be bound, unlike ClHCl, with a nearly symmetric hydrogen bond. E. HAB Molecules.-The most common and extensively studied of these is HCN. Near-HF wavefunctions were computed some years agoY34lbut Doggett and coworkers 362s 363 have reported DZ basis calculations on HCN, using approximations for some of the integrals. Various one-electron properties were computed, and

+

m-

-

+

a02

s93

H. Basch, J. W. Moskowitz, C. Hollister, and D. Hankin, J. Chem. Phys., 1971, 55, 1922. T. X. Carroll, R. W. Shaw,jun., T. D. Thomas, C. Kindle, and N. Bartlett, J . Amer. Chem. SOC.,

1974,96, 1989. For references see A. J. Downs and S. C. Peake in ‘Molecular Spectroscopy’, ed. R. F. Barrow, D. A. Long, and D. J. Millen (Specialist Periodical Reports), The Chemical Society, London, 1973, Vol. 1, Ch. 9. s95 G. J. Jiang and G. R. Anderson, J. Phys. Chem., 1973, 77, 1764. s96 G. J. Jiang and G . R. Anderson, J. Chem. Phys., 1974, 60, 3258. 397 J. Almlof, Chem. Phys. Letters, 1972, 17, 49. Sg8 P. N. Noble and R. N. Kortzeborn, J. Chem. Phys., 1970,52,5375. 399 R. Janoschek, Theor. Chim. A d a , 1973, 29, 57. 400 D. A. Clark. C. Thomson, and T. C. Waddington, to be published. 394

130

Theoretical Chemistry

except for the electric field gradient they were in reasonable agreement with experiment. Absar and McEwen 401 have also employed all-valence-electroncalculations to study various non-linear excited states of HCN. The isomer of HNC has been rather neglected until recently, apart from semiempirical studies,*02but, following the reported detection of an emission from HNC from a galactic source, Pearson et aZ.403 carried out ab initia calculations with a DZ + P quality GTO basis set. The predicted geometries and properties of both HCN and HNC were given. Furthermore, first-order wavefunctions (6343 configurations) were also computed. The computed HCN geometry was in excellent agreement with experiment, and therefore the predicted bond lengths in HNC of R(H-N) = 0.995 A and R ( N - 4 ) = 1.169 A are believed to be reliable to within -0.003 A. The calculated value of Be of 90.48 GHz was within 0.2 % of the ‘experimental’value. Hyperfine structure of the line was also computed. A new method for obtaining SCF orbitals has been described by Davids~n.~O~ In this msthod, a particular form of the orbitals is derived, known as the internally consistent SCF orbitals (ICSCF). In contrast to the usual canonical SCF orbitals, the sum of the orbital energies bt is equal to the total energy, and the values of bt - &j are expected to be close to the excitation energies for transitions between valence orbitals. Stenkamp and Davidson investigated4o5HCN and a variety of other triatomics by this method, and also AH2 species, studying particularly the equilibrium bond angle. The resulting orbital-energy diagrams were similar in appearance to those derived from canonical orbitals, and also the ICSCF valence-orbital energy sum correlates better with the equilibrium bond angle than the sum of the canonical orbital energies. A recent paper by Schwenzeret aL4o6has outlined a novel approach to calculations on a variety of excited states that is applicable to several states of the same symmetry. After selecting a physically meaningful set of orbitals appropriate to the various states, and a single-configuration reference function for the state in question, CI is performed, including all configurations differing by one orbital from any one of the selected reference states. Detailed predictions for 12 low-lying states of HCN were made, and compared with experiment and the Walsh diagrams. It was concluded that the experimental assignment of the @A” state was wrong. Where the predictions disagree with those from the Walsh diagram, this is usually due to the inadequacy of the singleconfiguration approximation. Two isoelectronic molecules that have recently been studied spectroscopically have been the subject of near-HF calculations. HCP is the phosphorus analogue of HCN, and a study of the computed wavefunction407indicates the existence of a G P bond; this is the only known example of this bond. A variety of properties 401

*02 403 404 *05

406 407

I. Absar and K. L. McEwen, Canad. J. Chem., 1972, 50, 653. G . Loew, Theor. Chim. Acta, 1971, 20, 203. P. K. Pearson, G. L. Blackman, H. F. Schaefer, tert., B. ROOS,and U. Wahlgren, Astrophys. J., 1973, 184, L19. E. R. Davidson, J. Chem. Phys., 1972. 57, 1999. L. Z. Stenkamp and E. R. Davidson, rheor. Chim. Acta, 1973, 30, 283. G. M. Schwenzer, S. V. O’Neil, H. F. Schaefer, tert., C. P. Baskin, and C. F. Bender, J . Chem. Phys., 1974, 60, 2787. C. Thomson, Theor. Chim. Acta, 1974, 35, 237.

Quantum Mechanical Calculationson Small Molecules

131

were computed and compared with experiment and with earlier calculations with a GTO basis,408 which gave a poorer energy and especially a poor value of p. HBS has also been studied by Thom~on.~O~ In both sets of calculations, basis sets of BA + P quality were used. Computed bond lengths were in good agreement with experiment, and in both cases a large number of one-electron properties were computed, and these stand as predictions.409 The HCC (ethynyl) radical has apparently been detected in stars,410 and also trapped in argon matrices at 4 K.411 The ground state is 2Z+, and this was confirmed by SCF calculations.410 The computed rotational J = 1 4 0 line was in good agreement with thzt observed,412 and it is thus likely that CCH has been observed. The HBO molecule is a further example of an unstable intermediate which has only recently been detected in matrix-isolation e~periments.~l~ SCF calculations by Thomson and Wishart 414 have yielded reliable values for the bond lengths, and the authors have discussed the bonding and one-electron properties. It is hoped that such calculations will stimulate further experimental work on this and similar unstable triatomic molecules. F. C02,CS2, Li20, and Al2O.-Although C02 has been much studied in the past, its normal vibrational modes have not been studied theoretically, and these have been the subject of a recent SCF calculation by Ohrn and c o - ~ o r k e r s . ~The ~ 5 basis set used was a 7s3p GTO basis with added 3d-functions. This must be considered a near-minimal basis, not a DZ + P, as quoted by the authors. The quadrupole moment was calculated in this work. The generally good agreement for the force constants is gratifying. Theoretical work on the low-lying excited states has been confined to rather small 417 but a recent very detailed study using a GTO DZ P calculations in the basis set has appeared.418 Open-shell SCF calculations were carried out for 13 low-lying valence states, including five Rydberg states, and in addition a pi-electron CI calculation was carried out, including all single and double excitations for eight of these states. The calculated transition energies provided definitive assignments for various states observed by optical and electror,-impact studies. There are four states (lAl, 3B2, 3 A ~ , lA2) which are bound relative to CO(lC+) O(3P). The accuracy of the calculations was impressive. CS2 had onIy been studied by CNDO previous to a 1971 calculation by Fischer and K e m m e ~ , who ~ l ~ used a large GTO basis with 3d-functions. Comparison calculations on C02 were also carried out. The charge distribution indicates transfer of

+

+

408

409 410

*l2

413 414

415 416 417 418 *19

J.-B. Robert, H. Marsmann, 1. Absar, and J. R. Van Wazer, J. Amer. Chem. SOC.,1971, 93 3320. C. Thomson, Chem. Phys. Letters, 1974, 25, 59. J. Barsuhn, Astrophys. Letters, 1972, 12, 169. W. R. M. Graham, K. I. Dismuhe, and W. Weltner, jun., J. Chem. Phys., 1974, 60,3817. D. Buhl and L. E. Snyder, Nature, 1970, 228, 267. R. F. Porter and E. R. Lory, J. Amer. Chem. SOC.,1971, 93, 6301. C. Thomson and B. W. Wishart, Theor. Chim. Acta, 1974,35, 267. M. VuEelid, Y.Ohm, and J. R. Sabin, J. Chem. Phys., 1973, 59, 3003. M. Krauss, S. R. Mielczanck, D. Neumann, and C. E. Kuyatt, J. Geophys. Res., 1971,76,3733. P. S. Julienne, D. Neumann, and M. Krauss, J. Atmos. Sci., 1971, 28, 833. N. W. Winter, C. F. Bender, and W. A. Goddard, tert., Chem. Phys. Letters, 1973, 20, 489, C. R. Fischer and P. J. Kemmey, Mol. Phys., 1971, 22, 1133.

132

Theoretical Chemistry

charge to S from C. The molecular quadrupole moment was computed and found to be negative. Two interesting linear molecules whose geometries were slightly unexpected are Liz0 and Al2O. These are highly ionic species, and the linear structure of Li2O was predicted several years ag0.420 In the case of AhO, the experimental results are ambiguous. Wagner has recently published a study of the PE surfaces of these mole~ules,4~1 for which semi-empirical methods predict bond structures with acute apex angles and long bonds. Both molecules are predicted to be linear. Large contracted GTO basis sets were used (without polarization functions). The computed vibrational frequencies were in reasonable agreement with experiment. Liz0 has also been studied by the separated-electronpair method.422 A STO basis set of DZ quality was used. Doubly occupied geminals were used in constructing the wavefunction, and hence electron correlation is included. The molecule was again predicted to be linear. The best energy was -0.15 hartree lower than in the above SCF calculations. A detailed analysis of the MO’s was given in this paper. G. CCO and TiC0.-These two species have little in common, but it is convenient to examine them together. OCC425 has been the subject of a near-HF study.It is an unstable species, as yet characterized only in matrix-isolation experiment~.~~4 The computed bond lengths were R(0-C) = 2.121 bohr, R(C-C) = 2.58 bohr, and several one-electron properties were predicted. The computed force constants were in fair agreement with experiment. The simplest carbonyls to be studied theoretically are T i c 0 and TiCO+.325The GVB method has been applied to these molecules, using a minimal STO basis set with added Ti(4p) orbitals. The orbitals were found to have basically the same shape as in CO. Both c-and z-bonding occur, but not to such an extent that the orbitals of separated Ti and CO change much. The back-bonding is considerably weaker in TiCO than in TiCO+, and the former is less stable than the latter. Extension of the qualitative ideas gives a useful discussion of the bonding in more complex carbonyls.

H. Molecules containing B, C, and N.-An extensive set of calculations, in many cases of near-HF accuracy, on various linear molecules containing B, C,and N has been reported by Thomson.424 This study was prompted by the observation of several species such as B2C, BC2, and BCN (or BNC) in mass spectra, but little was known about their structure prior to this work. Calculations approaching HF accuracy showed that BNC should be more stable than BCN, BCC more stable than CBC,and BCB more stable than BBC.424 Combined with estimates of the correlation energy, the atomization energy of BNC was predicted to be -0.48 hartree, in good agreement with experiment. The groundstate X2X+configuration of BCC is consistent with the results of e.s.r. measurements, which show BCC to be a a-radical. A similar study of the molecules NCN, NNC, CNC, and C C N 4 S 5 was carried out 420 421 422 423

424 425

R. J. Buenker, and S. D. Peyerirnhoff, J. Chem. Phys., 1966,45, 3682. E. L. Wagner, Theor. Chim. Acta, 1974,32, 295. T.K.Liu and D. D. Ebbing, Internat. J. Quantum Chem., 1972, 6, 297. C.Thomson and B. J. Wishart, Theor. Chim. Acta, 1973,31, 347. C . Thomson, J. Chem. Phys., 1973,58,216. C.Thomson, J. Chem. Phys., 1973,58, 841.

133

Quantum Mechanical Calculations on Small Molecules

later. These molecules (except for NNC) are all known from Herzberg's spectral inve~tigations,~~s and calculations at the DZ + P level gave bond lengths in reasonable agreement with experiment, and for NNC the predicted bond lengths of R(N-N) = 2.53 bohr, R(N-C) = 2.19 bohr should be reasonably accurate. One particularly important point emphasized in this series of papers concerns the use of Mulliken population analyses to interpret the charge distribution and the contributions of d-functions to the bonding. Many papers have discussed this question, and in particular, workers on larger molecules with minimal or DZ basis sets often use these quantities. The results of Thomson and c o - ~ o r k e r s , ~ ~425 ~ - who have computed wavefunctions for many linear molecules at DZ, DZ P, BA P, and near-HF accuracy, show quite clearly that computed population analyses are very sensitive to the wavefunction, and even at BA + P level they should be used with caution. Smaller basis sets than DZ + P are likely to give serious errors in populations, and the sign of the charge may be incorrect. As an example we show in Table 6 the results for six of the molecules considered. It is clear that variations can be quite large, and hence conclusions based on small populations computed with basis sets of less than DZ + P quality may be quite wrong. 4239

+

+

Table 6 Total atomic population Gt on atom i in some linear X Y Z molecules with different basis sets 424 XYZ

BCN BNC

BCC CBC BBC BCB

Basis set DZ DZ P BA + P DZ DZ P BA + P DZ DZ + P BA P DZ DZ P BA P DZ

+

+

+ + + DZ + P BA + P DZ DZ + P BA + P

GX

GY

4.58 4. a4 4.71 4.57 4.85 4.65 4.40 4.59 4.56 5.91 6.10 4.80

6.18 5.99 6.05 7.76 7.14 7.58 6.68 6.27 6.47 5.18 4.80

GZ

-

7.23 7.16 7.23 5.67 6.01 5.76 5.92 6.14 5.97 5.91 6.10 -

4.79

5.23 5.05

6.15

4.61 4.74

6.77 6.52

4.61 4.74

5.97

A few other studies on these molecules have appeared. BCN was the subject of a calculation with a small GTO basis set,427 and the spin-spin interaction in the ground states of NCN, CNN, and CCO has been computed by Williams, using a STO-4G and an extended 4-31G basis set.428 Results were not in very good agreement with experiment. Although spin-orbit contributions were believed to be the main reason 426

G. Herzberg, 'Electronic Spectra of Polyatomic Molecules', Van Nostrand, New York, 1966.

427

J. B. Moffat, J. Mol. Structure, 1971,7, 474.

4aa

G. R. Williams, Chem. Phys. Letters, 1974, 25, 602.

Theoretical Chemistry

134

for the discrepancy, it seems likely that the wavefunctions are themselves not accurate enough to compute this small quantity accurately. There have been many other papers dealing with various aspects of theory relevant to linear molecules, but space does not permit discussion of these. 9 Non-linear Triatomic Molecules A. A H 2 Molecules, including H,f.-The simplest of these species is H3+,which has been the subject of a number of recent studies. Work previous to 1970 is listed by B ~ r k m a nThe . ~ ~ground ~ state is triangular (D3h) and it has been studied using a basis set of elliptical orbitals. Contracted GTO basis sets were used in near-HF calculations by Harrison et a1.,430 and correlated wavefunctions were computed by Salmon and Poshusta,431and by Handy.4a2 In the latter calculation a new form for the correlation factor was used, and the good results suggest that it might be useful for other small molecules. The ion BeH; should be linear, like BeH2, according to Walsh's rules. However, explicit calculations,433both SCF and VB-CI, show that the ground state is an electrostatically bound complex with 2A1 symmetry, and the nearby excited state is a metastable bound state of 2B2symmetry. The HBeH angle is 20"and R(Be-H) = 4.43 bohr, whereas the excited state has an angle of -70". Explicit orbital-energy diagrams show a crossing of the 2a1 and lbz energies near loo", but in the ground state the H-H distance is nearly that in H2, suggesting a description of the molecule as an ionic complex. The results strongly indicate that the two states should be observable mass-spectroscopically. BH2 has been less studied than most of the first-row AH2 molecules, although BH; was studied some years ago by Kaufman et a1.434 However, with the recent advances in computing power, more accurate calculations on open-shell systems are now feasible, and three separate studies of BH2 have appeared. Bender and Schaefer435 have computed first-order wavefunctions for CH,' and BH2, using a contracted GTO (9s5pl4s) basis, and for BH2, the 2Ai and 2B1 states were investigated. The computed bond angle is within 2" of experiment, and the prediction for CH; is that the ground state is bent and the first excited state linear, a conclusion at variance with earlier semi-empirical calculations436 which predicted the ground 2A1 state to be linear. Goddard and co-workers155 have computed GVB wavefunctions for BH2 with similar results to Schaefer, and have discussed the H in terms of the GVB orbitals. It is interesting that formation of BH2 from BH the tervalent nature of B is already apparent from the 2 P ground-state boron orbitals. Staemmler and Jungen437 have obtained the PE surfaces for these two states using the IEPA method and obtained a similar but somewhat lower total energy. A large number of properties were computed for both states. It is claimed

-

+

419 430 431 432

433 434

435 436 437

R. F. Borkman, J. Chem. Phys., 1970,53, 3153. S. W. Harrison, L. J. Massa, and P. Soloman, Nature Phys. Sci.,1973, 245, 31. L. Salmon and R. D. Poshusta, J. Chem. Phys., 1973,59, 3497. N. C. Handy, MoI. Phys., 1973, 26, 169. R. D. Poshusta, D. W. Klint, and A. Liberles, J. Chem. Phys., 1971, 55, 252. L. M. Sachs, M. Geller, and J. J. Kaufman, J. Clzem. Phys., 1970, 52, 974. C. F. Bender and H. F. Schaefer, tert., J. MoI. Spectroscopy, 1971, 37, 423. P. C. H. Jordan and H. C. Longuet-Higgins, MuZ. Phys., 1962, 5 , 121. V. Staemmler and M. Jungen, Chem. Phys. Letters, 1972, 16, 187.

135

Quantum Mechanical Calculations on Small Molecules

that the experimental value of the differencein zero-point energies of the two states is incorrect. Unpublished calculations by Meyer on BH2 have been quoted by Staemmler,437 but no details have appeared except the total energy. The BH,f cation, which is linear, has been studied by Jungen by the IEPA method.438 A discussion of Walsh’s rules and the first-row hydrides has been given by Wasserman now that quantitative calculations have been carried These rules have also been reviewed by Buenkerll and further studied by Davidson et al.405 CH2, the next hydride in the series, is the classic example of the power of a6 initio calculations in that the clear prediction was of a non-linear geometry, and this led to a consequent revision of the experimental data by Herzberg. Several recent calculations on this species should be mentioned. The calculations prior to 1971 have been summarized in Schaefer’s book, and some of the results of later work are given in Table 7. There are two particularly important questions: (i) the grsund-state geometry, and (ii) the ~ A I + - + ~energy BI separation. Successive calculations by Lathan et al.,440,441 del Bene,4*2 O’Neil Table 7 Results of ab initio calculationson CH2, energy of 3B1groundstate, geometry, 3B1-1A1 energy separation AE3-11

Elhartree

hartree

Method SCF SCF SCF SCF SCF SCF SCF SCF SCF

STO-3G ST04-31G Gaussian lobe GTO GTO GTOa GTO GTOa C-H bond function

- 38.4362 - 38.8696 - 38.892 - 38.9078 -38.9123 - 38.9136 - 38.9202 -38.921

1.11 1.11 1.06 1.07 1.06 1.11 1.075 1.11

SCF SCF SCF VB-CI CI CI VB-CI CI GVB-CI CI CI CI CI IEPA

GTO GTO GTO minSTO GTO GTO Gaussian lobe GTO GTOa GTO GTO GTO a GTO GTOa

- 38.9264 - 38.9308 - 38.9327 - 38.877 -38.904 - 38.908 - 38.915 - 38.9248 -38.9598 -38.9696 - 38.9826 - 39.0121 - 39.0319 - 39.0754

1.096 1.069 1.072 1.11 1.12 1.08 1.06 1.06 1.11 1.092 1.095 1.088 1.095 1.081

a

Basis

-

N

100.5 105.4 133 130 130 126 130.4 135

-

135 128.5 129.5 127.5 129 136 140 130 135 131.7 133.3 134 134 134.2

Re1.73 440 1.60 440 1.76 446 1.45 442 1.50 450 445 1.08 1.39 443 1.03 449 1.17 Meyer 1968 (see 451) 1.40 455 1.09 451

-

444

1.71 1.06 1.04 1.39 0.88 0.50 0.97

448 452 447 446 450 449 443

0.40

455 451

-

444

444

Including d-functions on C.

438 439 440

441 443

M. Jungen, Chem. Phys. Letters, 1970, 5, 241. E. Wasserman, Chem. Phys. Letters, 1974, 24, 18. W. A. Lathan, W. Hehre, and J. A. Pople, J. Amer. Chem. SOC.,1971, 93, 808. W. A. Lathan, W. Hehre, L. A. Curtiss, and J. A. Pople, J. Amer. Chem. SOC.,1971,93, 6377. J. del Bene, Chem. Phys. Letters, 1971, 9, 68.

136

Theoretical Chemistry

et a1.,443McLaughlin et and Bender and co-workersY445 using SCF calculations with large basis sets, or SCF plus large CI, predict the 3B1 state to be bent, and the most extensive calculation of ref. 444 predicts R(C-H) = 1.088 A, 8 = 134", in good agreement with experimental estimates. There have also been a few VB calculations. An early calculation by Harrison and Allen used a small gaussian lobe basis ~et,~46 and included 20 primary structures. The resulting lowest energy of -38.915 hartree and bond angle of 140" were similar to the values obtained by MO cal~ulations.4~~ Another VB calculation by Tandardini and co-workers44* used a minimal STO basis set, and a h e d CH distance. Various excited states were studied with up to 150 VB structures. Although the total energies were not particularly good, owing to the limitation of the basis set, the computed properties were very similar to those of O'Neil et al., who used a much more extensive MO calculation. GVB calculations on CH2 have also been reported,449so a value of 134 _+ 2"seems well established. Other SCF-CI calculations on the low-lying states have been reported,450 and it now seems clear that the HCH angle in the ground state is 134k2". However, the size of the singlet-triplet gap [AE(1Al++3B1)]has been much harder to compute. A survey of previous work on this question has been included in refs. 445 and 451, where the following conclusions are drawn. All SCF calculations without d-functions predict A E to be between 134 and 168 kJ mol-l, but inclusion of dfunctions lowers this to 105 kJ mol-l. However, the inclusion of electron correlation is even more important. CI without d-functions gives values of 84-105 kJ mol-1, which is reduced to 50 kJ mol-1 with polarization functions. The extensive calculations by Bender et aZ.443-445and S t a e m n ~ l e r ~ (who ~ l used the IEFA method) are in general agreement in that A E is 38 k 12 kJ mol-l. This value should be more accurate than current experimental estimates, but it is of interest that the small value is primarily the result of a differential lowering of the energy of the lA1 state by the d-polarization functions. A large number of molecular properties for CX32were computed by Staemn~ler,~~l and in particular the binding energy of 187 kcal mol-1 is in fair agreement with experiment. A recent MIND0/3 semi-empirica1453calculation gives a value of A E that is in agreement with the ab initio results, and which is superior to MIND0/2. The spin-spin splitting parameters D and E have also been the subject of several recent ab initio calculations. Limited CI calculations by Harrison gave D in good agreement with experiment, but E was too large.454 More extensive CI calculations

-

N

443 *44 445

4J6

447 448 449

450 451

452 453 454

S. V. O'Neil, H. F. Schaefer, tert., and C. F. Bender, J. Chem. Phys., 1971, 55, 162. D. R. McLaughlin, C. F. Bender, and H. F. Schaefer, tert., Theor. Chim. Acta, 1972, 25, 362. C. F. Bender, H. F. Schaefer, tert., D. R. Franceschetti, and L. C. Allen, J. Amer. Chem. SOC., 1972,94, 6888. J. F. Harrison and L. C. Allen, J. Amer. Chem. Sor., 1969, 91, 807. J. F. Harrison, J. Amer. Chem. SOC.,1971, 93, 4112. G . F. Tantardini, M. Raimondi, and M. Simonetta, Internat. J. Quantum Chem., 1973, 7 , 893. P.J. Hay, W. J. Hunt, and W. A. Goddard, tert., Chem. Phys. Letters, 1972, 13, 30. S. Y. Chu, A. K. Q. Siu, and E. F. Hayes, J. Amer. Chem. SOC.,1972, 94, 2969. V. Staemmler, Theor. Chim. Acta, 1973, 31, 49. J. M. Foster and S. F. Boys, Reu. Mod. Phys., 1960, 32, 305. M. J. S. Dewar, R. C. Haddon, and P. K. Weiner, J. Amer. Chem. SOC.,1974, 96, 253. J. F. Harrison, J. Chem. Phys., 1971, 54, 5413.

Quantum Mechanical Calculations on Small Molecules

137

have been reported recently by Langhoff and Davidson,455 using a contracted gaussian lobe basis, and also by Harrison and Liedtke.456 The best wavefunction was comparable to that of McLaughlin et a2.444 and gave D = 0.781 cm-l, E = 0.050 cm-l. These values give, if combined with estimates of the spin-orbit contribution, a value of D of 0.9 k 0.1 cm-l, which is higher than the experimental value; this latter is, however, subject to some uncertainty (see ref. 456 for details). The second-order effect of spin-orbit coupling on the angular dependence of the zero-field splitting has been investigated by Hall and Han1eka.~5' NHS, NH;, and NH;. The radical NH2 was the fist polyatomic free radical to be observed experimentally, and it has since been extensivelystudied spectroscopically. Bender and Schaefer458 have referenced calculations on NH2 prior to 1970, and have presented results for the most accurate calculations yet carried out, to which we return below. However, del Bene,459at about the same time, reported the results of an SCF study, using small (STO-3G) or (ST04-31G) basis sets. The predicted 2B1 ground state of the radical has 8 = 108", R(N-H) = 1.01 A, and for the 2A1state 8 = 144", R(N-H) = 0.98 A. The more extensivecalculations of Bender and Schaefer,458 using a DZ P basis set, evaluated both SCF and first-order wavefunctions. 252 configurations were included for the 2B1 state, and 387 for the 2A1 state. The computed bond lengths and other properties are given in Table 8. The ~BI-~AI splitting is predicted to be 1.67 eV from the best CI calculation. The SCF wavefunction is a rather good approximation to the total wavefunction.

+

Table 8 Computed geometries and force constants for N H 2 (experimental values in parentheses)458 aAi & FO SCF FO 0.997 1.010 (0.975) 1.055 (1.024) 141.9 144.7 (144+5) 102.7 (103.3) -55.6799 -55.5237 -55.6185

2Bl 7

Bond length/A Bond angle/" Interpolated total E/ hartree Stretching force constant/ mdyn A-1 Bending force constant/ mdyn A-1 a

SCF 1.019 105.4 -55.5757 7.77

6.50

7.94

7.05

0.71

0.64

0.48

0.31

First-order wavefunction.

Pople and c o - ~ o r k e r shave ~ ~ l also studied NH2 in various states and geometries, but with rather smaller basis sets. However, a very recent paper has used more extensive basis sets460 (ST0431G), with results close to the limit for singledeterminant wavefunctions. However, R(N--H) is slightly lower than observed. 455

S. R. Langhoff and E. R. Davidson, Internat. J. Quantum Chem., 1973, 7 , 759. J. F. Harrison and R. C. Liedtke, J. Chem. Phys., 1973, 58, 3106. W. R. Hall and H. F. Hameka, J. Chem. Phys., 1973, 58, 226; ibid., 1974, 60, 4104. C. F. Bender and H. F. Schaefer, tert., J . Chem. Phys., 1971, 55, 4798. J. del Bene, J. Chem. Phys., 1971, 54, 3487. P. C. Hariharan and J. A. Pople, Mol. Phys,, 1974, 27, 209.

4 5 ~

457

458 459 460

meoretical Chemistry

138

A number of papers have dealt with the calculation of the isotropic hyperfine coupling constants in NH2, but these are reviewed e1~ewhere.l~~ NH;, which is isoelectronic with CH2, has been studied again relatively recently, first by Chu, Siu, and Hayes,450 then by Pople and co-w0rkers,~*1and finally by Harrison and E a k e r ~ .The ~ ~ l3B1, I&, 1.41, and 1AT states were investigated by Chu et aZ.,450using a gaussian lobe basis: both SCF and CI calculations were reported. The computed PE curves show that the ground3B1statehas a bond angle of 130".Of particular interest is the fact that the singlet-triplet separation (~AI-~BI) for NH; is predicted to be about twice as big as in CH2. The results obtained by the other workers are in reasonable agreement generally, but Lee and M ~ r u k u m a in , ~an ~ ~earlier CI study with a larger basis set, found the ground state to be linear. However, since the barrier to linearity is small, these calculations cannot yet have completely resolved the question. A recent study of some other excited states of NH2 by Brown and Williams463 uses the UHF method. Of particular interest is the 2B2 state, with R(N-H) = 3.04bohr, 0 = 26.6".Thomson and Brotchie4G4have repeated the calculations with a basis set giving near-HF accuracy and confirm these results. However, the species is not bound, and it seems to correspond to a H2 + N* system. However, it is also possible for the SCF to converge to another apparently bound state, which population analysis shows to be an erroneous conclusion. However, some analogous states of BF2 are bound (see below). OH2 and OH:. The water molecule is of paramount importance, and its structure and chemistry have recently been extensivelyre~iewed.~~5 Numerous recent theoretical studies have appeared, and only a few of these will be mentioned. Considering first the SCF calculations,early work has been discussed by Schaefer,l and a recent paper by Goddard and Hunt 466 gives a complete list of references up to 1974. Pople and co-workers reported STO-3G + ST04-31G calculations of the equilibrium geometry in 1971.*40 Davidson, in an interesting paper, has explored the use of the internally consistent orbitals (ICSCF) in the case of H20,467and he found these to be better for describing spectra than the traditional canonical orbitals. The most comprehensivenear-HF calculation on H2O was carried out by Dunning, Pitzer, and Aung,46*who examined the use of a variety of GTO and STO basis sets in a study of a large number of one-electron properties of the ground state. About 70 % of the dissociation energy was obtained, with the energy 0.003 & 0.002 hartree above the H F limit. The computed VIP's were N 1-1.5 eV too low, and the force constants were in error by 15-20 %. Kern and co-workers have made an extensive st~dy46~-472 of zero-point vibrational corrections to one-electron properties.

-

4131 J.

-

F. Harrison and C. W. Eakers, J. Amer. Chem. Soc., 1973, 95, 3467. S. T. Lee and K. Morukuma, J. Amer. Chem. Soc., 1971, 93, 6863. 46s R. D. Brown and G. R. Williams, Mol. Phys., 1973, 25, 673. 4154 C. Thomson and D. A. Brotchie, Mol. Phys., 1974, 28, 301. 465 C. W. Kern and M. Karplus in 'Water: A ComprehensiveTreatise', Plenum Press, New York, 1972. 466 W. A. Goddard, tert., and W. J. Hunt, Chem. Phys. Letters, 1974, 24, 464. 467 S. T. Elbert, S. P. Langhoff, and E. R. Davidson, J. Chem. Phys., 1972, 57, 2005. 4 6 8 T. H. Dunning, jun., R. M. Pitzer, and S. Aung, J. Chem. Phys., 1972, 57, 5044. 469 W. C. Ermler and C. W. Kern, J. Chem. Phys., 1971, 55,4851. 47* W. C. Ermler and C. W. Kern, J. Chem. Phys., 1971,55,4851. 471 L. L. Sprandel and C. W. Kern, Mol, Phys., 1972, 24, 1383. 47a B. J. Krohn, W. C. Ermler, and C. W. Kern, J. Chem. Phys., 1974, 60,22. 462

Quantum Mechanical Calculations on Small Molecules

139

The coupled Hartree-Fock method has been reviewed and applications to the computation of one-electron properties of H20 have been described by Thomsen and were generally in good agreement with Swanstrsm in two p a p e r ~ . ~474 7 ~Results * experiment when a near-HF wavefunction was used, particularly second-order force constants. However, the electric polarizability was not well reproduced. Bendazzoli et aZ.475have also computed a variety of properties using the Pople STO-3G method, and also a method in which only two-electron integrals were approximated, with all one-electron integrals evaluated exactly using STO’s. Total energies were better in the second method owing to a better description of the 1s orbitals. Afzal and Frost476have recently made a detailed analysis of the energy terms which contribute to the total energy in FSGO calculations. A point-charge model of H20 was also found to be capable of yielding accurate values of various one-electron properties.477 The MsX, method has been shown to predict incorrect bond angles in H2O and N H 3 . 4 7 8 Antoci et al.479 have shown that this is due to the way the one-electron molecular potential is treated. Turning now to calculations by other than MO methods, there have been a number of recent VB calculations. Peterson and Pfeif€er4Sohave implemented a conventional multi-structure VB scheme, using gaussian lobe basis functions. Results on the ground state of H2O included calculation of the dissociation energies for H20+ HO + H (DI = 316kJ mol-l)and OH+ 0 + H (D2 = 230 kJ mol-l), in reasonable agreement with experiment. In a second paper,481 several valencestate energies were computed. The GVB methods have also been applied to H20,1g6 and detailed electron-density maps have been presented. The GF energy obtained was quite low, and an informative discussion of how the bonding changes in the series 0- OH-+ H2O was given. One result of particular note is the invalidation of the interpretation of the bond angle as being due to H-H repulsions: it turns out that the GVB orbitals are approximately sps hybrids. The orbitals are distributed in space in a manner consistent with Linnett’s double-quartet theory.482 A later paper 154 compared the GVB orbitals computed with and without the strong orthogonality description and with other group-function calculations. The VB calculationsreported above by Gallup for LiH have also been applied to BeH2, H20, and HF.l18 More effort has recently gone into including electron correlation by MCSCF or SCF-CI methods, and the first definitivestudy was by Schaeferand Bender in 1971,483 who used avariety of basis sets and the IN0 procedure. An important conclusion from these calculations was that minimal basis set plus full CI essentially duplicates the correlation effects observed with larger basis sets and less than full CI, as was found 473 474 475 476

477 478 479 480 481 482 483

K. Thomsen and P. Swanstrsm, Mol. Phys., 1973, 26, 735. K. Thomsen and P. Swanstrsm, Mol. Phys., 1973, 26, 751. G. L. Bendazzoli, M. Dixon, and P. Palmieri, Internat. J . Quantum Chem., 1973, 7 , 223. M. Afzal and A. A. Frost, Internat. J. Quantum Chem., 1973, 7 , 51. A. D. Tait and G. G. Hall, Theor. Chim. Acta, 1973, 31, 311. J. W. D. Connolly and J. R. Sabin, J. Chem. Phys., 1972, 56, 5529. S. Antoci, L. Mihich, and G. F. Nardelli, J. Chem. Pkys., 1974, 61, 1245. C. Peterson and G. V. Pfeiffer, Theor. Chim. Acta, 1972, 26, 321. C. Peterson and G. V. Pfeiffer, Theor. Chim. A d a , 1974, 33, 115. J. W. Linnett, ‘The Electronic Structure of Molecules’, Wiley, New York, 1964, H. F. Schaefer, tert., and C. F. Bender, J. Chem. Phys., 1971,55, 1721.

140

Theoretical Chemistry

also for diatomics. A later paper by Schaefer and co-workers444 employed a larger basis set but a less extensive CI, and studied CH2 and H20. Results were generally very good. At present the best wavefunction for HzO using conventional methods is that obtained by Meyer, and reported in 1971.484 The results of this extensive calculation have been thoroughly discussed by Schaefer:1 here we just note that 84 % of the correlation energy was obtained ! Lishka,l93however, in a very recent paper, using the TEPA method, has almost obtained as low an energy, and has discussed theoretically the proton affinity of OH-. The best results for the stretching force constant are within 5 % of the experimental value. However, R(0-H) is now too long by about the same amount as it is too short in SCF calculations. A rather different calculation on H20 using many-body perturbation theory (MBPT) has been described by Miller and Kelly.4g5 Details of the method are not given here, but a total energy of - 76.48 k 0.07 hartree was obtained, giving a computed binding energy of -0.42 If: 0.07 hartree (experimental value - 0.371 hartree). Applications to other molecules should be interesting, and this calculation gives nearly the same energy as Handy’s transcorrelated wavefunction calc~lation.~8 The excited states have also been the subject of a large number of recent papers. The use of improved virtual orbitals by Hunt and G 0 d d a r d 4 ~is~ discussed by Schaefer,l and in a recent paper487 these authors have investigated 32 excited states of HzO which lie at < 11.7 eV. Agreement with experiment for the eight states known is within 0.1 eV, and the calculations are accurate enough to expect that they will be useful in helping experimentalists who are trying to assign bands to specific states. MCSCF calculations on the 3B1 state have been reported by Hosteny et al.,488who found this state to be unbound, in contradiction to recent experimental interpretations of electron-impact work. Flouquet and Horsley489 have also studied the PE surface of the @ A ( l A ’ ) state of HzO with a limited CI treatment. A more novel calculation of excitation energies by Truhlar490 uses excitation operator methods and other methods that are described in detail by McKoy and coworkers.491 Comparison with the results obtained by other workers shows that the singlet excited states cannot be adequately represented by a valence-like basis set :it is necessary to include diffuse orbitals in the basis, as were used by Hunt et aZ.486,487 Finally, we should mention calculations on several molecules, including H20, using localized orbitals, both SCF and SCF-CI.49a H20+has only recently been studied by ab initio methods. The PE curves in some UHF calculations show the ground state to be ~ B Iwith , a bond angle of 112.5°.49s Meyer also found the same result.484 However, the 2A1 state was predicted to be bent, N

JR5 486

487 488

4y9 499

dgl 492

443

W. Meyer, Internat. J . Quantum Chem., 1971, 55, 341. J. H. Miller and H. P. Kelly, Phys. Rev., 1971, 4A, 480. W. J. Hunt and W. A. Goddard, tert., Chem. Phys. Letters, 1969, 3,414. W. A. Goddard, tert., and W. J. Hunt, Chem. Phys. Letters, 1974, 24, 464. R. P. Hosteny, A. R. Hinds, A. C. Wahl, and M. Krauss, Chem. Phys. Letters, 1973, 23, 9. F. mouquet and J. A. Horsley, J. Chem. Phys., 1974, 60, 3767. D. G. Truhlar, Internat. J. Quantum Chem., 1973, 7 , 807. T. H. Dunning and V. McKoy, J. Chem. Phys., 1968,48, 5263. D. L. Wilhite and J. L. Whitten, J. Chem. Phys., 1973, 58, 948. H. Sakai, S. Yamabe, T. Yamabe, K. Fukui, and H. Kato, Chem. Pliys. Letters, 1974,25, 541.

Quantum Mechanical Calculations on Small Molecules

141

although the barrier is very small. However, Meyer's results are likely to be more reliable, as they employed a much larger basis set. Cederbaum et a2.494 have discussed the calculation of the ionization potential of H2O and also the vibrational structure in the photoelectron spectrum.495 H2F+. This species has only recently been directly observed, but it was investigated theoretically by Csizmadia and co-workers using various GTO basis sets in 1968.496 Diercksen et a1.497computed the optimum geometry recently, using a very large CGTO basis. The computed values of R(F-H) = 0.95 A and 6 = 114.75' were obtained. The proton binding energy of 504kJmol-1 compares well with the values in NHf and S O + and indicates this to be a fairly stable species. Hydrogen bonding between H2F+ and FH was also investigated in this work. In a later paper, Lishkalg3went beyond the H F approximation via IEPA calculations and obtained a similar geometry, and also a similar proton affinity. Leibovici498 has also studied this species at the SCF level, but with a similar basis set, as have Pople and co-workersY441using a much smaller basis. Leibovici's energy is slightly lower than that obtained by Diercksen et aZ.497 There have also been a number of studies of hydrides of second-row elements, the most common being H2S, for which a near-HF wavefunction was reported in 1970, together with a variety of properties, by Rothenberg and co-workers.499 Similar quality calculations were reported by Roos and Siegbahn in 1971,50° and the two sets of results were in good agreement. 3d-Functions were not found to be very important in H2S. The ion HzS+ had not been studied until recently, when an ab initio UHF wavefunction was reported.493 The computed angle and bond length are in good agreement with experiment, and the radical is very similar to H2O+. Spin-density contour maps were presented in this paper. It is also interesting that the 2B2 state is similar to that observed for NH2 in having a very small bond angle, and the interpretation of this result is similar to that proposed by Brown et aZ.463 and Thomson and B r o t ~ h i e . ~ ~ ~ SiH2 has been the subject of an extensive study by Wirsam.501 The ground state is predicted to be a singlet state, but the triplet state (3Bl) is only 0.2 eV higher. These calculations included CI. B. HAB Molecules.-The HO2 radical has been considered an important intermediate in chemical reactions for many years, but the geometry was not established accurately until some recent theoretical work by Liskow and co-workers,502 and Gole and Ha~es.50~ Liskow 502 et al. carried out SCF and CI calculations, using their first-order wavefunction technique, with an essentially DZ P basis set. The computed values of

+

494

4s5

496

L. S. Cederbaum, G . Hohlneicher, and W. von Niessen, Mol. Phys., 1973, 26, 1405. L. S. Cederbaum and W. Domcke, Chem. Phys. Letters, 1974, 25, 357. A. C. Hopkinson, N. K. Holbrook, K. Yates, and I. G. Csizmadia, J. Chem. Phys., 1968, 49, 3596.

497 498 4s8

501 503

G. H. F. Diercksen, W. von Niessen, and W. P. Kraemer, Theor. Chim. Acta, 1973, 31, 205. C. Leibovici, Internat. J. Quantum Chem., 1974, 8, 193. S. Rothenberg, R. H. Young, and H. F. Schaefer, tert., J. Amer. Chem. SOC.,1970, 92, 3243. B. Roos and P. Siegbahn, Theor. Chim. Acta, 1971, 21, 368. B. Wirsam, Chem. Phys. Letters, 1972, 14, 214. D. H. Liskow,H. F. Schaefer, tert., and C. F. Bender, J. Amer. Chem. SOC.,1971,93, 6734. J. L. Gole and E. F. Hayes, J. Chem. Phys., 1972,57, 360.

I42

Theoretical Chemistry

the geometrical parameters obtained for the CI calculation were R(0-H) = 0.973 A, R(0-0) = 1.458 A, and 8 = 104.6'. The parameters are not in agreement with INDO results.504Calculations on 0 2 were also reported in this paper. The bond angle is consistent with Walsh's predictions. At about the same time, Gole and Hayes 503 also reported ab initio investigations on the 2A" and 2A' excited states, but with a smaller basis set. However, fixed values of R(0-H) = 0.96 A were used throughout: the 0-0 distance is thus too short in these calculations. Thus the computed ground-state angle of 116" is too large. A limited CI treatment found that the 2A' state has about the same angle as the 2A" ground state. The energy separation was predicted to be 67 kJ mol-1. The correlation diagram for the reaction H + 02+OH 0 was discussed in this paper. A more recent investigation of interoxygenbonding in a variety of small molecules, including HO2, has been reported by Blint and Newton.505Various GTO basis sets were used in SCF calculations, and the force constants, in particular, were examined. The 2A"and 2A' states were studied, and the energy gap was computed to be < 1 eV, in agreement with Gole et al. The formyl radical HCO has been the subject of very detailed experimental investigations in recent years, but surprisingly few theoretical studies have appeared. However, Thomson and Brotchie506 have recently reported the results of a variety of SCF calculations. The cation HCO+ has been suggested as being responsible for an unidentified interstellar microwave line at 89.190+ 0.002 G H z . ~ OHowever, ~ Barsuhn has suggested that CCH is responsible410 (see the discussion above). In order to see if Klemperer's suggestion was correct, Wahlgren et al. 508 have carried out an extensive GI calculation on HCO+ (6343 configurations) with a GTO basis, and also a large SGF calculation with an STO basis, which gives a wavefunction with R(H--C) = 1.09 A, R(C-0) = 1.07 A, and rather similar results to the CI results. The predicted rotational line would be at 89.9 GHz, which is 0.7 GHz too high. However, Barsuhn independently410 carried out a less accurate calculation on HCO+, with results very close to those obtained in his CCH calculation and with AE ( J = 0 - 1 ) somewhat lower in value than that obtained by Wahlgren.598 It seems clear that this question is not yet fully answered. More extensive calculations including CI are needed on HCO+, HNC, and CCH to try to reduce the err or. The free radicals HNF and HBF are related to NHz and NF2 and BMz and BF2, and they have been investigated by Thomson and B r o t ~ h i e SCF . ~ ~ ~calculations using various contracted GTO basis sets were carried out, and the minimum energy configuration corresponded to R(N-H) = 2.20 bohr, R(N-F) = 2.66 bohr, and 0 = 101", in good agreement with experiment. Using a rather small basis set, the low-lying 2A' state was found to be 0.095 hartree above the ground state. HBF has not yet been observed, and the calculations give predicted geometries for the ground and several low-lying excited states which are similar to those of HCQ. Spin densities

+

-

N

N

504 505

507 508 509

M. S. Gordon and J. A. Pople, J. Chem. Phys., 1968,49,4643. R. J. Blint and M. D. Newton, J. Chem. Phys., 1973, 59, 6220. C. Thomson and D. A. Brotchie, Internat. J . Quantum Chem., 1974, 8S,277. W. Klemperer, Nature, 1970, 227, 1230. U. Wahlgren, B. Liu, P. K. Pearson, and H. F. Schaefer, tert., Nature Phys. Sci., 1973,246,4. D. A. Brotchie and C. Thomson, Chem. Phys. Letters, 1973, 22, 338.

Quantum Mechanical Calculations on Small Molecules

143

were computed for this species. The related carbene CHF was studied by Harrison a few years a g 0 . ~ ~ 7 The HNO molecule was investigated spectroscopicallyseveral years ag0,d26 and a UHF calculation was reported in 1970.510A more recent study of HNO and HON with a small GTO basis has been reported in which the bond lengths and angles were optimized.511 The HON molecule is not known, however. Detailed electron-density maps were discussed in this work. The nitrenium species NHF+was studied by Harrison et al. in the paper cited above.461 The singlet and triplet states are essentially degenerate in NHF+,the triplet lying only 17 kJ mol-1 below the singlet state. One interesting species, unknown until recently, is HOF, which was studied by Peslak et a1.,511 and electron-density contour maps were described. More recently, it was studied by Kim and Sabin.512 The computed bond lengths with a near-minimal GTO basis were R(H-0) = l.O80A, R(0-F) = 1.450& and 8 = 100.8”, in fair agreement with the results of a microwave investigation. Force constants were computed and the population analysis was reported. A more extensive basis set was used in a similar calculation by Ha, who computed a dipole moment of 2.72 D.513 Since the photoelectron spectrum of HOF has been recently observed, Chong et al. have computed the VIP’s using perturbation corrections to Koopman’s theorem.514 These were then used to assign the spectra. A very recent paper has also presented ab initio calculations on HOCl,515 using a (9, 5 ) basis on 0, a (12,9) basis for Cl, and a 5s basis on H, all of which were contracted. Several electrical properties were computed and compared with microwave data. The interaction energies between He * HF, He H20, H2 - - HF, and H2 * H20 have been investigated by Li~chka,~ls and clusters of rare gases with H and H+, i.e. HenH and HenH+(n = 1-4) have been studied by Milleur et aL517

-

C. A B 2 Molecules.-In the case of AB2 molecules, the number of electrons in the system is now 30-40 for some of the molecules containing second-row atoms; consequently these species present a much greater computational problem if high accuracy is to be achieved. Nevertheless, much useful information can be obtained from wavefunctionsat the SCF level, or including limited CI, and the past four years have seen a large amount of effort in this area. An extensive discussion of Walsh’s rules, including their application to AB2 molecules, has already been referred to.ll It is convenient to divide the molecules into the following groups: (a) A02, (b) AF2, (c) miscellaneous molecules. A02. We include in this series ozone, 0 3 , which has been the subject of a number of detailed studies, and several unexpected and important conclusions relevant to experimental investigations have been drawn from these theoretical calculations. 610 511 512

513 514

515 516

517

A. W. Salotto and L. Burnelle, J. Chem. Phys., 1970, 52, 2936. J. Peslak, jun., D. S. Klett, and C. W. David, J. Amer. Chem. Soc., 1971, 93, 5001. H. Kim and J. R. Sabin, Chem. Phys. Letters, 1973, 20, 215. T.-K. Ha, J. Mol. Structure, 1973, 18, 486. D. P. Cliong, F. G. Herring, and D. McWilliams, Chem. Phys. Letters, 1974, 25, 568. G. L. Bendazzoli, D. G. Lister, and P. Palmieri, J.C.S. Faraduy 11, 1973, 69, 791. H. Lishka, Chem. Phys. Letters, 1973, 20, 448. M. B. Milleur, R. L. Matcha, and E. F. Hayes, J. Chem. Phys., 1974, 60, 674.

144

Theoretical Chemistry

Rothenberg and Schaefer, in 197lY5l8 published a near-HF wavefunction for O8 and computed a large variety of properties at the experimental geometry. With the exception of the molecular quadrupole moment, these were in good agreement with experiment (where known), and a large number of predictions were made. It is noteworthy that d-functions are much less important in 0 3 than in S02. At about the same time, Hillier and Saunders519also published a minimal STO-3G study, but carried out geometry optimization, obtaining reasonably good agreement with experiment. It should be noted that their best energy of -221.26535 hartree was considerably higher than that obtained by McCain and Palke520in another minimal STO calculation (- 223.4798 hartree), and almost 3 hartree above the near-HF value. Thus conclusions based on these poorer quality wavefunctions should be treated with some caution. Even the near-HF calculation does not predict 0 3 to be bound with respect to three oxygen atoms. All of the above calculations assumed a lA1 ground state, but during the course of a study of the excited states Goddard and c o - ~ o r k e r s ~have ~ l demonstrated how poor the HF approximation is for this molecule, since it turns out that the 3B2 state is predicted to be of lower energy at the experimental geometry (by 2.16 eV) in the case of a RHF calculation employing a DZ basis set. This fact emphasizes the need for CI studies in this case, and the first of these were reported by H e a t ~ et n al. ~ ~in~an investigation of 0 3 and 0;.A large basis set and extensive CI for the lA1 state of 0 3 and the 2B1 state of 0; gave a reliable value for the computed EA of 0 3 . In addition, a geometry search showed these species to have almost the same geometry (in disagreement with minima1 STO calculations). Several recent papers have examined a variety of excited states of ozone, in order to clarify the complicated experimental observations. The first such study was by Hay and Goddard,6a1 using GVB and GVB-CI, and a minimal CGTO basis set. Eight excited states with vertical excitation energies < 7 eV were found. All have minima with 100"< 8 < 130" except the 21A1 state. This has a minimum for 8 = 60", with a computed energy 1.5 eV above the llA1 ground state (using a larger GTO basis), but the minimal basis predicts the two lA1 states to be very close in energy. Wright 523 has also investigated 0 3 with a minimal STO basis plus limited CI, and also finds a cyclic state with a minimum energy. CI makes the energies comparable (as found by Goddard et al.). Clearly, more extensive calculations were needed, and these have since appeared, but we first mention a minimal STO-3G calculation by Devaquet 524 et al., who found the 1Bz(m*) state to be more stable for unequal bond lengths. The double-minimum PE surface of this state results from an avoided crossing involving (i) the singlet (nn*)state itself, which is stabilized by an unsymmetrical distortion, and (ii) a lower-lying nn*(lAl) state, which is destabilized by the same distortion. A later paper by Grimbert and Devaquet 5z5 investigated six low-lying states with a h18

j19 52* 521

522 623

524

525

S. Rothenberg and H. F. Schaefer, tert., Mol. Phys., 1971, 21, 317.

I. H. Hillier and V. R. Saunders, Mol. Phys., 1971, 22, 193. D. C. McCain and W. E. Palke, J. Chem. Phys., 1972, 56, 4957. P. J. Hay and W. A. Goddard, tert., Chem. Phys. Letters, 1972, 14, 46. M. H. Heaton, A. Pipano, and J. J. Kaufman, Internof. J. Quantum Chem., 1972, 6 S , 181. J. S. Wright, Canad. J. Chem., 1973, 51, 139. A. Devaquet and J. Ryan, Chem. Phys. Letters, 1973, 22, 269. D. Grimbert and A. Devaquet, Mol. Phys., 1974, 27, 831.

Quantum Mechanical Calculations on Small Molecules

145

minimal STO basis and limited CI. Both singlet and triplet A2 states are predicted to be sharply bent, with angles of 9 9 O . This result is slightly different from that found by Goddard and co-workers, who calculated angles of 100-101" for these states in the calculations cited below. Extensive GVB-CI calculations with both minimal and extended bases were reported recently by Hay et aZ.,526 with variation of bond length and bond angle. Despite the large basis and extensive CI, the bond length in the llA1 state was 0.2 b o b too large, probably because of the absence of 3d-functions in the basis set. However, the excitation energies should be within 0.2 eV of the exact values. Space does not permit a detailed discussion of the 14 excited states, (of which four are bound). However, since the energies are so small, these results could not be said definitely to predict the existence of bound excited states, but their values should stimulate a lot of further work. Unlike Wright's conclusions, the calculations predict the ring state to be 1.5 eV above the ground state, and the authors do not think therefore that it is a good candidate for the 'ozone precursor'. Wadt and G0ddard5~' have very recently summarized the previous work and compared the wavefunctions with INDO-type approximate calculations to see how the latter compare with correlated wavefunctions. INDO turns out to be biased towards closed geometries (even for the ground state) and too strong mbonds, but the use of INDO approximations with correlated t,b certainly seems to be a fruitful source of further study. Analysis of the GVB orbitals for 0 3 is presented in the review of ref. 13. Siu et al. also reported SCF calculations on the ground, ring, and open forms of 03.528 Table 9 summarizes some of the best estimates of spectroscopic properties of bound states of 0 3 from the most extensive calculations526on various states of this molecule. However, more extensive geometry optimization, with large basis sets and extensive CI,is still needed on this molecule. N

N

Table 9 Theoretical estimates of spectroscopic properties of bound states of 0 BondFirst strong Excitation energy dissociation absorption from A1 ground state energy to , -A- . , ,-*--, Od3x;) Upper State Vertical Adiabatic + OPP) state AEa 1A14n 0 0 (1.13)b l l B z ( h ) 4.7 3 B 4n ~ 1.47 -0.7 -0.4 23Ai(4~) 5.8 3 A 2 5 ~ 1.80 -0.8 -0.3 21Bi(53t) 4.0 lA2 5 ? ~ 1.98 -1.0 -0.1 2lB1(5n) 3.8 -0.0 _*Bi 5n 1.74 -1.1

* Calculated using optimum R at

0 =

116.8".

b

3 526

Lowest vertical ionization potential

I Ion state 2Ai(4?Z) 12.9 2A2(3n) -12.8 2Ai(4n) -12.1 2 ~ i ( 4 ? Z ) -11.9 -

Experimental value (see ref. 526).

An interesting mixed-basis-set method for use in SCF calculations has been described by Billingsley and Trindle,529 with application to Li02. One-centre and most two-centre integrals are evaluated analytically, whilst less tractable integrals are approximated by a gaussian expansion of the STO's. Examination of portions of the 626 527 528 529

P. J. Hay, T. H. Dunning, jun., and W. A. Goddard, tert., Chem. Phys. Letters, 1973,23,457. W. R. Wadt and W. A. Goddard, tert., J. Amer. Chem. SOC.,1974, 96, 1689. A. K. Q. Siu and E. F. Hayes, Chem. Phys. Letters, 1973, 21, 573. F. P. Billingsley, jun., and C. Trindle, J. Phys. Chem., 1972, 76, 2995.

146

Theoretical Chemistry

LiOz potential surface indicates that the CzVand Cssymmetries are of comparable stability. A more extensive set of calculations, with approximately a DZ basis set, are described by O'Neil et aZ.530 Using a large CGTO basis, the ground 2Az state was predicted to be an isosceles triangle; R(Li-0) = 1.82 A, 8 = 44.5'. The 2B2 state, with a similar geometry, was only 59 kJ higher in energy. There appears to be little or no barrier between the CzVand CmVforms. There have been several previous calculations on NOz, and a near-HF wavefunction for the ground state has been obtained by Rothenberg and Schaefer, who give references to earlier work.51s The excited states have also been studied, though less extensively. Fink studied some 15 states, at the experimental ground-state geometry, some years ag0,531 and A more recent GVB-CI similar studies were reported by Gangi and B~rnelle.53~ study has appeared of the 2B1 state, and various observed transitions were a ~ s i g n e d . ~ ~ 3 The possibility of unequal bond lengths in some states of NO2 such as the 2B2 state has been studied in detail by Hinze et aZ.,534 who have proposed a mechanism for such effects. It remains to be seen if accurate calculations bear these conclusions out. FO2 has been relatively little studied, although an early SCF study was reported by Gole and Hayes.535 Recently, some open-shell CNDO calculationshave a ~ p e a r e d , ~ ~ 6 and in an unpublished series of calculations Brotchie537 has examined the ground state with three different gaussian basis sets. The computed 0-0 bond length is 2.50 bohr, which is quite long, as is also the case in H02.502The unpaired electron is strongly localized on the terminal oxygen. Calculationswere also carried out on the low-lying 2A' state, which lies -0.5-0.8 eV above the ground state. The FOO angle in this state is 101",compared with the value of 107" in the ground state. A minimal-basis-set STO calculation on this species by McCain and Palke520 predicted the hyperfine coupling constants. However, geometry optimization was not carried out, and results were not in very good agreement with experiment. These authors also reported calculations on (2102, which was also studied by Golc and Hayes.535 In this case, the ground state of the Csconfiguration is only slightly lower in energy than the Cz, ground-state configuration. There have also been a few studies of S02, including a near-HF calculation of the properties.538 The influence of polarization functions was found to be significant by Roos and Siegbahn.500 S02, SO;, and SO:- were also studied by Dacre and Elder,539 using a small GTO basis, with a view to understanding the complexing properties of SOz. A variety of calculations on SO2 by Hillier's group have been reported. Early references are given in a paper by Guest et al.,540 who reported calculations with N

N

530 531 532 533 534 535

536 537 538 539 540

-

S. V. O'Neil, H. F. Schaefer, tert., and C. F. Bender, J. Chem. Phys., 1973, 59, 3608. W. H. Fink, J. Chem. Phys., 1971, 54, 2911. R. A. Gangi and L. Burnelle, J. Chem. Phys., 1971, 55, 843. P. J. Hay, J. Chem. Phys., 1973, 58, 4706. J. Hinze, R. Solarz, and D. H. Levy, Chem. Phys. Letters, 1974, 25, 284. J. L. Gole and E. F. Hayes, Internat. J. Quantum. Chem., 1970, 35,519. P. &sky, M.Machiicek, and R. Zahradnik, Call. Czech. Chem. Comm., 1973, 38, 3067. D. A. Brotchie, Ph.D. Thesis, 1973, University of St Andrews. S. Rothenberg and H. F. Schaefer, tert., J. Chem. Phys., 1970, 53, 3014. P. D. Dacre and M. Elder, Theor. Chim. Acta, 1972, 25, 254. M. F. Guest, I. H. Hillier, and V. R. Saunders, J.C.S. Faraday II, 1972, 68, 114.

Quantum Mechanical Calculations on Small Molecules

147

larger basis sets on SO2 and calculations of localized orbitals. A rather large d-orbital population obtained in this study was probably the result of the basis set not being very well balanced, but more accurate calculations on this species are still needed. A F 2 . The fluorides AF2 form an interesting series whose structures have been extensivelystudied during the past few years. However, the large number of electrons has of necessity made most authors restrict the calculations to single-configuration wavefunctions. BeF2, BF2, CF2, N F 2 , and OF2 have been thoroughly studied with a CGTO basis set of essentially DZ P quality in a definitive SCF study of all the first-row dfiuorides.641 The experimental geometry was assumed for BeF2. Polarization functions were least important for this molecule of the series. A large number of molecular properties were computed, most of which are predictions. Most properties show a periodic trend across the series, or a behaviour like that of a potential curve, with a maximum or minimum at CF2. The agreement with the known properties of OF2 is quite good. Independently, Thomson and Brotchie have studied BF2,542y 543 but in addition to using five different GTO basis sets, have optimized the geometry, which was uncertain (Schaefer assumed the BF bond length to be as in BF3 and the angle 120"). The best wavefunction gave an energy of -223.64617 hartree, slightly higher than obtained by Schaefer and R ~ t h e n b e r gwho , ~ ~included ~ polarization functions in the basis set. The predicted angle was 120"and the bond length 2.495 bohr. The angle is larger than that estimated from e.s.r. work. The authors also computed the energies of the A2&, 2A2,and 2B1excited states with a minimal CGTO basis set. The population analysis, as expected, was very sensitive to the basis set. The authors also calculated the isotropic hyperfine coupling constants from the spin density at the nucleus. Although RHF spin densities are difficult to compute accurately, the agreement with experiment was reasonable for the boron nucleus, but only half the observed value for fluorine. This is typical of other RHF spin-density calculations (see ref. 176 for details of this work). The low-lying 2B2state has been referred to above. However, unlike NH2, population analysis indicates this state to be bound, and it remains to be seen if this conclusion is correct. N F 2 has also been investigated in a similar manner by the same authors,537.542 who carried out calculations OD the ground 2B1 state and also on the 2A1 excited state, the latter at the minimal-basis-set level. The excited state is predicted to be bent, with an angle of 148".In the case of NF2, unlike BF2, none of the calculations predict N F 2 to be bound, and the RHF approximation in this case is rather seriously in error. Population analysis shows no significant amount of pn-pn bonding, as earlier postulated in semi-empirical studies. N F 2 has also been investigated by Del Bene,442 who obtained similar results for the geometry of the ground state. The most extensive calculations predict R(N-F) = 2.97 bohr, L F N F = 148" for the 2A1 state, and R(N-F) = 2.61 bohr, L F N F = 103" for the X2B1 ground state. Calculations on HNF and HBF have also been reported.509 Most calculations reported above have used various forms of the RHF method,

+

541 542

543

S. Rothenberg and H. F. Schaefer, tert., J. Amer. Chem. Sac., 1973, 95, 2095. C. Thomson and D. A. Brotchie, Chem. Phys. Letters, 1972, 16, 573. C. Thomson and D. A. Brotchie, Theor. Chim. Ada, 1973,32, 101.

148

Theoretical Chemistry

with or without CI, but Brown et aZ.544have examined NF2 with four different basis sets and the UHF method. Computation of several one-electron properties was carried out, with reliable results being obtained for the DZ quality basis, where the 2p function is more accurately represented. Computation of the isotropic and anisotropic hyperfine coupling constants showed only the latter to be in good agreement with experiment, particularly if single annihilation was carried out. Calculations by HinchcliRe and C ~ b busing , ~ a~GTO ~ basis with d-orbital optimization, gave better agreement with experiment for this molecule. Turning now to the difluorides of Group IIA elements, a very extensive study of MgF2 and CaF2 together with BeF2 has been reported by Gole, Siu, and Hayes.546 GTO basis sets of roughly DZ accuracy were used. BeF2 is predicted to be linear, and d-orbitals in the basis have little effect on this prediction. MgF2 is also predicted to be linear, but the d-orbital influence is greater here. Results of CI studies (96 configurations)do not change the predicted geometry, but the energy required to change the angle is rather small. CaF2 is also predicted to be linear without including d-orbitals in the basis, but has a computed angle of -145" when d-orbitals are included. The BeF2 excited states were studied in the RHF approximation in a later paper.547 Of some interest is the prediction that the lowest energy transitions should be different from those of the isoelectronic C02. A study of MgF2 by both ab initio and semi-empirical methods548 shows the latter to favour a bent structure, but the former predicts the linear structure to be stable, in agreement with Gole et aZ.546 The most accurate calculations on MgF2 were those of Astier et ~ 1 . who ~ 5used ~ ~ several basis sets of better than DZ quality. The SCF calculations gave a linear geometry, and with evaluation of the second-order correlation correction by perturbation theory, the linear geometry is still the most stable, but the barrier to deformation is very small, and thus these results agree with those of Gole et aZ. Coulson has recently discussed the bond angles in these molecules in a review paper,550 using two simple models. An ionic model would predict these molecules to be linear, but in a second model, in which valence states arising from s2+spys2+ sd excitations were permitted, the first would lead to linear, the second to bent structures. From a consideration of the excitation energies, it was shown that when the central atom is heavy and the attached halogens are as electronegative as possible, the bent situation is favoured, in agreement with the experimental results. The predictive value of the Walsh diagram is well known, and for a group of molecules ScF2 through ZnF2, where experimental information is limited, ab initio studies might be expected to be of considerable value. ZnF2 has been studied recently by Yarkony and Schaefer551 to obtain information on the orbital-energy diagram for this molecule. Using Zn (14s9p5d) and F (9s5p) basis sets with a [9s5p2d] and [4s2p] contraction, SCF calculations were carried out for both variations of R(Zn-F) and 8, and population analyses were also presented. The molecule R. D. Brown, F. R. Burden, B. T. Hart, and G. R. Williams, Theor. Chim. Acta, 1973,28,339. A. Hinchcliffe and J. C. Cobb, Chem. Physics, 1974, 3, 271. 546 J. L. Gole, A. K. Q. Siu, and E. F. Hayes, J. Chem. Phys., 1973, 58, 857. 547 J. L. Gole, J. Chem. Phys., 1973, 58, 869. 548 M. Allevena and S. Besnainou, J. Mol. Structure, 1972, 11, 439. 549 M. Astier, G. Berthier, and P. Millie, J. Chem. Phys., 1972, 57, 5008. j50 C. A. Coulson, IsraelJ. Chem., 1973, 11, 683. 551 D. R. Yarkony and H. F. Schaefer, tert., Chem. Phys. Letters, 1973, 15, 514. 544

545

Quantum Mechanical Calculations on Small Molecules

149

is predicted to be linear, with R(Zn-F) = 1.75 A. The orbital-energy diagrams show that these are dominated by the relative positions, within the molecule, of the metal 3d- and fluorine 2porbitals. It is predicted that for SCFZthe Walsh diagram will be different from that of ZnFz. A detailed study of the carbenes CH2, CHF, and CF2447has been referred to above, and the geometries of the lA1, 3B1, and lB1 excited states of CF2 have been reported. Goddard and co-workers have discussed the GVB orbitals and bonding in CFz.13 Turning now to non-metal difluorides, there have been a number of semi-empirical studies, and two ab initio calculations. T h ~ m s o has n ~ investigated ~~ SiF2 with two gaussian basis sets, the largest with Si(lOs6pld)-, [6s4pld] and F(7s3p)- [4s2p], which is roughly of DZ quality and similar to that used in a study of the SO2 mole cule50° (which is isoelectronic with SiF2). Of particular note in this work was the optimization of the 3d-orbital exponent, whose optimized value was 0.3. The computed geometry was in good agreement with experiment, and the 3d-orbitals were found to be much less important than in S02. The effect of the orbital-exponent optimization on the populations was found to be quite significant. Wirsam553 independently published the results of both SCF and CI calculations, using a gaussian lobe basis set. The minimum SCF energy of - 487.74057 hartree is substantially lower than that obtained by Th0mson5~2 (- 487.503539 hartree). Various low-lying excited states were investigated and excitation energies computed. Results were generally in reasonable agreement with experiment. PF2 has been studied using the UHF methodY5S4 but with an assumed geometry, and e.s.r. hyperfine couplings were computed. Approximate treatments involving only valence electrons have been used to study the electronic structure of SF2, with reasonable results.555CND0/2 studies on SF2 and other small sulphur molecules have been rep0rted.~5~ Finally, we mention a large ab initio calculation on S C ~ Zincluding , ~ ~ ~d-functions. The basis set consisted of 102 uncontracted GTO’s, contracted to 75 CGTO, and comparison was made with the photoelectron spectrum. d-Functions are more important on S than on C1. D. ABC Molecules.-There have been relatively few investigations of this type of molecule. LiNO and LiON were studied by Peslak, Klett, and Da~id,~11 together with FNO and FON. The lithium species had almost identical energies. Clearly, studies using more extensive basis sets are needed to settle the question of the relative stability of the various isomers. FNO is predicted to be more stable than FON, and a description of the bonding via the population analysis was presented. NSF has been studied at the minimal-basis-set level, and the results were correlated with the photoelectron spectrum.558 Several recent papers, dealt with elsewhere, have detailed the investigations on metal carbonyls, usually via SCF calculations. An exception to this is a GVB study of the simple carbonyls TiCO and TiCO+, using a minimum ST0 basis set plus Ti 4p652 653 554 655

558 555

568

6

C. Thomson, Theor. Chim.Acta, 1973, 32, 93. B. Wirsam, Chem. Phys. Letters, 1973. 22, 360. J. C. Cobb and A. Hinchliffe, Chem. Phys. Letters, 1974, 24, 75. R. G. Hyde, J. B. Peel, and K. Terauds, J.C.S. Faraday ZZ, 1973.69, 1563. V. B. Koutecky and J. I. Musher, Theor. Chim. A d a , 1974, 33, 227. B. Solouki, P. Rosmus,and H. Bock, Chem. Phys. Letters, 1974, 26, 20. R.L. Dekock, D. Lloyd, A. Breeze, G. A. D. Collins, D. W. J. Cruickshank, and H. J. Lempka, Chem. Phys. Letters, 1972, 14, 525.

Theoretical Chemistry

150

orbitals.s25 One interesting conclusion of these studies is that the description of the bonding and vertical excitation energies is very similar for both minimal and extended basis sets, although this statement has not been extensively examined for other than first-row atoms. The results show that the CO bonds through a 0donating, n-accepting interaction, the n-back-bonding being much weaker in TiCO than in TiCO+. These conclusions prove helpful in a discussion of bonding in other carbonyls. In concluding this section, we note the results of a series of calculations on ONF, using a small basis set, by Pulay and co-workers,559 who evaluated the force constants and showed that reliable results may be obtained even with a (5s2p)basis set. 10 Tetra-atomic Molecules Included in this section are AH3, AB3, HAB2, H2AB molecules, and also dimers, and interactions between diatomics or triatomics and atoms or ions. A. A H 3 Molecules.-The ion HeHi has not been observed experimentally, but early work as to its stability was not clear-cut. Poshusta and A g r a ~ a l , ~using ~ O a GTO basis set with polarized orbitals, carried out VB-CI calculationswith various amounts of ionic, covalent, and correlation effects. The molecule is predicted to be stable with respect to He + H;, but only by 100 cm-1. However, a different conclusion was reached by Benson and McLaughlin,561 using more extensive SCF-CI. Clearly, more accurate work and definitive experiments are needed to settle this question. BH3 is an important small molecule for which experimental data are lacking, since it readily dimerizes to B2Ha. Several earlier papers were devoted to this question, particularly studies by Kutzelnigg and co-workers, who 562 estimated from IEPA calculations that the dimerization energy is 151 f21 kJ mol-1. BH3 was also studied in SCF calculations some years ago by Schwartz and Allen,563 who considered the inner-shell and valence-shell ionization energies. More recently, Goddard and Blint have discussed GVB wavefunctions for BH3.155 Gelus and Kutzelnigg56*have more recently computed the equilibrium geometry uia the IEPA method with a gaussian lobe basis. R(B-H) was found to be 1.192 A, and by studying the in-plane and out-of-plane deformations, the force constants were obtained. The effect of correlation on these was rather small, and the zero-point energy of BH3 was calculated to be 73 kJ mol-1. A more unusual calculation on BH3, including electron correlation, has been described by Paldus e f aZ.,565who have used the coupled-pair many-electron theory and compared the results with early full CI calculations with a minimal basis set. The agreement between the two sets of calculations was very good. The methyl radical CH3 has also been extensively studied, particularly with respect to the hyperfine coupling constants, to which we refer elsewhere.176 However, more recent work with various large GTO basis sets has appeared, with energy minimizaN

559 560 561 562

563 564 565

W. Sawodny and P. Pulay, J. Mol. Specfroscopy, 1974, 51, 135. R. D. Poshusta and V. P. Agrawal, J . Chem. Phys., 1973, 59, 2477. M. J. Benson and D. R. McLaughlin, J. Chem. Phys., 1972, 56, 1322. M. Gelus, A. Ahlrichs, and W. Kutzelnigg, Chem. Phys. Lefters, 1970, 7 , 503. M. E. Schwartz and L. C. Allen, J. Amer. Chem. SOC.,1970, 92, 1466. M. GClus and W. Kutzelnigg, Theor. Chim. Acfa, 1973, 28, 103. J. Paldus, J. ciiek, and I. Shavitt, Phys. Rev., 1972, 5A, 50.

Quantum Mechanical Calculations on Small Molecules

151

tion.566 The population analysis was described and the various one-electron properties were computed. Konishi and Morukuma567 investigated this molecule, with a slightly smaller basis set both at the SCF level and with CI calculations, and the IEPA calculations of Driessler et al. are also of about the same accuracy.568The ions CH3+and C&- were also studied. CH, is nearly tetrahedral, with an inversion barrier of -8 kJ mol-1. A similar barrier was found by Duke in SCF calculations with a large GTO basis with diffusep-functions added.S6QHe ascribed the decrease in the barrier as being due to the inclusion of these functions, whereas Driessler's results indicate that correlation effects may be responsible. Pople and co-workers have also reported calculations on CH3f.570 NH3 and P& may be discussed together, and several calculations have recently appeared on both molecules, and some on the related ions. Some earlier results on N H 3 were discussed by Schaefer,l particularly the definitive near-HF calculation by Clementi and ~ o - w o r k e r swho , ~ ~computed ~ a value for the inversion barrier in good agreement with experiment, using a large GTO basis. The conclusion that HF calculations can predict the barrier accurately was substantiated by Stevens, using a large STO basis ~ e t . Pipano 5 ~ ~ et al., however, with a smaller basis, had to include CI to get agreement with e~periment.5~~9 574 Since these earlier papers, Laws et a1.575 have computed a large number of oneelectron properties, using a near-HF wavefunction, and investigated their dependence on the basis set. All computed properties were within 5 % of experiment for the largest near-HF basis set. Pulay and Meyer5'6 have calculated the force constants of N H 3 with a variety of GTO basis sets, all of which yield excellent agreement with experiment. One problem in carrying out near-HF calculation even with GTO is the complexity of d-orbital integrals. In an attempt to reduce this by representing the d-functions by other than lobes or Cartesian gaussians, symmetry-equivalent d-orbitals, needing only threegaussian lobe functionsper d-orbital, have been used by Gerloff et al. in a study of NH3.577Good values for the barrier were found with only one set of d-functions on N and one set of p-functions on the H atoms. Studies beyond SCF have been carried out recently by several groups. An MCSCF wavefunction has been obtained by Dejardin and co-workers,578 who Iist earlier references on NH3. 51 configurations were included, and a large basis set was used. The largest correlation energy obtained was 0.071 hartree, compared with an estimated value of 0.329 hartree. Electron-density maps showed the angular and leftright correlations. A detailed discussion of this work was given in a more recent E. Kari and I. G. Csizmadia, Internat. J. Quantum Chem., 1972, 6, 401. H. Kohnishi and K. Morukuma, J. Amer. Chem. SOC.,1972,94, 5603. F. Driessler, R. Ahlrichs, V. Staemmler, and W. Kutzelnigg, Theor. Chim. A d a , 1973,30,315. A. J. Duke, Chem. Phys. Letters, 1973, 21, 275. P. C. Hariharan, W. A. Lathan, and J. A. Pople, Chem. Phys. Letters, 1972, 14, 385. A. Rauk, L. C. Allen, and E. Clementi, J. Chem. Phys., 1970,52,4133. R. M. Stevens, J. Chem. Phys., 1971, 55, 1725. A. Pipano, R. R. Gilman, C. F. Bender, and I. Shavitt, Chem. Phys. Letters, 1970, 4, 583. A. Pipano, R. R. Gilman, and I. Shavitt, Chem. Phys. Letters, 1970, 5, 285. E. A. Laws, R. M. Stevens, and W. N. Lipscomb, J. Chem. Phys., 1972,56,2029. P. Pulay and W. Meyer, J. Chem. Phys., 1972, 57, 3337. M. Gerloff, E. Ady, and J. Brickmann, Mol. Phys., 1973, 26, 561. P. Dejardin, E. Kochanski, and A. Veillard, Chem. Phys. Letters, 1972, 15, 248.

5613 R. 567 568 569 670

57l 572 578 574 575 576 577 578

152

Theoretical Chemistry

~aper.5'~ Wilhite and Whitten580 have determined localized orbitals for N H 3 for both SCF and CI wavefunctions, and have also presented the results of a detailed comparisonof SCF and CI calculations for NH3and PH3.581Propertiescomputed for NIG were in good agreement with experiment, even without CI. For PH3, it is evident that d-orbitals contribute to the o-bonding but have little influence on the internuclear angle, whereas the opposite is true for NH3. Property values demand the use of large basis sets for good agreement with experiment. Lehn and Munsch 582 have also reported SCF calculations on PF3 using extensive basis sets, and have compared the results with work on NH3. Walker 583 has recently computed force constants for PH3, using a large basis set, with d-orbital-exponent optimization. Table 10 summarizes the recent work on the NH3 molecule. Table 10 Calculated and experimental values of the total energy ET and molecular dipole moment ,uzfrom some recent calculations on NH2

Method SCF SCF SCF SCF-CI SCF MCSCF SCFa CIa

Experimental a

E~lhartrt~ -56.22191 - 56.221 13 -56.19525 - 56.35637 -56.2105 - 56.4500 -56.1681 - 56.2675 - 56.573

P D

1.66 1. S O -

1.848 -

1.48

Ref. 57 1 575 573 574 577 578, 579 580 5 80

-

Localized orbital.

There have been various theoretical studies of H30+ in view of its importance in solution chemistry, and earlier work has been cited by Kollman and Bender,5*4who have attempted to resolve conflicting views of the structure. Calculations without polarization functions showed H30+ to be planar, but inclusion of polarization functions makes the non-planar configuration more stable. Kollman and Bender have carried out near-HF calculations with a large CGTO basis set, including d-functions on 0 andp-functions on H. The lowest energy computed was - 76.33260 hartree for R(0-H) = 0.963 A and 8 = 112.5". Comparison with similar calculations on H2O and NH3 indicates that 8 should be within 2" and the bond length be shorter by 0.014.02 than the HF values. The inversion barrier was found to be -8 kJ mol-l. The earlier results of Almlof and W a h l g ~ - e ndisagree ~ ~ ~ with those mentioned above, particularly the angle. Lishka and Dycmons have gone beyond the SCF calculations, with larger and more flexible basis sets, and have included correlation by the IEPA method.586The equilibrium geometry is pyramidal in both sets of calculations, the correlation results N

N

579

P. Dejardin, E. Kochanski, A. Veillard, B. Roos, and P. Siegbahn, J. Chem. Phys., 1973, 59, 5546.

580 5RL 588

583

584 585 586

D. L. Wilhite and J. L. Whitten, J. Chem. Phys., 1973, 58, 948. J. D. Petke and J. L. Whitten, J. Chem. Phys., 1973, 59, 4855. J. M. Lehn and B. Munsch, Mol. Phys., 1972, 23, 91. W. Walker, J. Chem. Phys., 1973, 59, 1537. P. A. Kollman and C. F. Bender, Chem. Phys. Letters, 1973, 21, 271. J. Almlof and U . Wahlgren, Theor. Chim. Acta, 1973, 28, 161. H. Lishka and V. Dycmons, Chem. Phys. Lerters, 1973, 23, 167.

Quantum Mechanical Calculations on Small Molecules

153

showing R(O-€3) = 1.84 b o b , the out-of-plane angle being 18’. The inversion barrier is quite significantly larger when correlation is included, and is now N 18 kJ mol-1. The effect of correlation here is much more marked than in the case of NHS. The unusual systems F& and F3H, (for which there is no experimental evidence), have been studied by Shillady and Trindle.587 Using a very small basis set, FH3 has a T-shaped geometry and is computed to be stable, This type of molecule has been termed ‘hypervalent’ by Musher.588 Second-rowhydrides have been studied recently, particularly SiH3 and PH3. SiH3, SiHt, and Si&- were investigated, with particular emphasis on the low-lying excited states, by W i r ~ a m . 6The ~ ~ ground-state geometry of SiH3,computed using a GTO lobe basis set, was of CsVsymmetry, like that of SiH;, SiH: being D37t. CI calculations confirm this conclusion. The PE curves of several low-lying states were also presented. Aarons, Hillier, and Guest have also reported calculations on SiH3 and various other species,59owith the computed geometry in reasonable agreement with that found by Wirsam. B. AB3 Molecules.-Most molecules of this type have B as a halogen, and a large number of calculationson these species have been reported. BF3 has been investigated recently by several authors. An S C F calculation with a medium-size basis set (7s3p) without polarization functions gave energy levels that were compared with photoelectron results using Koopman’s the0rem.5~1BCk, not previously investigated by ab initio methods, was also studied, with optimization of bond length. Population analyses show in BCl3 a considerable amount of n-bonding. A rather more accurate wavefunction was reported by Perkins and co-workers in papers dealing with adducts of BF3.592,593 An earlier paper by Cadioli et al. investigated various excited states with a minimal GTO basis.594 BFk was studied by Walker and Horsley595 a few years ago. Ab initio calculationson CF3 at various levels have been reported by Guest et a2.,596 Kohnishi and Morukuma,56’ and McCain and Palke.520The various calculations agree that CF3 is pyramidal, and calculated geometrieswere in reasonable agreement with experiment. cC13 was studied by the same authors.596 However, in these calculations the basis sets used are rather small. SiF3 and Sic13 have geometriesshowing similar trends to those of the CX3 radicals. PF$ and PCIQ were also investigatedin this work. PF3 itself has been the subject of a few calculation~,5~6-5~~ the most recent being by Guest et aZ,,596 who carried out D. D. Shillady and C. Trindle, Internat. J. Quantum Chem., 1972, 6S, 187. J. I. Musher, Angew. Chem. Infernat. Edn., 1969, 8, 54. 689 B. Wirsam, Chem. Phys. Letters, 1973, 18, 578. 590 L. J. Aarons, I. H. Hillier, and M. F. Guest, J.C.S. Fumday II, 1974, 70, 167. 591D. Goutier and L. A. Burnelle, Chem. Phys. Letters, 1973, 18, 460. s92 R. M. Archibald, D. R. Armstrong, and P. G. Perkins, J.C.S. Faruday 11, 1973, 69, 1793. 5g3 D. R. Armstrong and P. G. Perkins, Inorg. Chim. A d a , 1974, 10, 77. K94 B. Cadioli, U. Pincelli, E. Tosatti, U. Fano, and J. L. Dehmer, Chem. Phys. Lefrers, 1972,17, 587

588

15. 5~35 596

59’ 5gg

6Q9

T. E. H. Walker and J. A. Horsley, MoI. Phys., 1971, 21, 939. M. F. Guest, I. H. Hillier, and V. R. Saunders, J.C.S. Faruday 11, 1972, 68, 867. I. H. Hillier and V. R. Saunders, Chem. Comm., 1970, 316. A. Serafini, J.-F. Labarre, A. Veillard, and G. Vinot, Chem. Comm., 1971, 996. I. H. Hillier and V. R. Saunders, J.C.S.Dalton, 1972, 21.

1 54

meoretical Chemistry

localized orbital calculations, with a minimal STO-3G basis. This paper dealt in some detail with the bonding. PCb was discussed in another paper.599 ClFs is an important interhalogen compound and has been studied by Breeze et al.,368with a small basis set. It can be regarded as derived from CIF by the addition of two fluorine atoms by way of a three-centre bond. Finally, a very large calculation on FeF3600should be noted, The authors point out the deficienciesof the calculations using a small basis set on this type of molecule, and they note that only in the cases601 of Nil?$- and NiFNQ+ have basis sets of DZ quality been used. The authors investigated the Walsh diagram of FeF3, using a DZ basis set. The planar ground state is 6A;, which lies 743 kJ mol-l lower than the low-spin 2A1 state. A variety of molecular properties were computed and the population analysis was discussed. This paper is also a landmark in such calculations since FeF3 has 53 electrons and is a large system quantum-mechanically. There are not many studies of MO3 species, except for ClO;, CO$-, and Co3.602 Probably of most interest is the C03 molecule,603 which was determined to be of CzV symmetry with a bond angle of 66". The unique 0 is of carbonyl type, and there is a weak bond between the symmetry-equivalent oxygens. C. A4 Molecules.-There are two cases of interest here; stable species such as P4 and interactions between homonuclear diatomic molecules. HI is the simplest of the species, and it has been the subject of a great deal of theoretical study because of its relevance to the H2 + D2+2HD exchange reaction. Early work has been carefully discussed by Schaefer,l and Bender and Schaefer have carried out more extensive calculationssince then on the linear form.604It is predicted that two H2 moleculesmay approach to within 1.6 bohr with an energy only 181 kJ above that of the separated molecules. A van der Waals attraction of 22K is predicted at a separation of the centre of mass of H2-H2 of 7.1 bohr. The results of other studies have failed to find a transition state lying less than 458 kJ above HZ + Dz. The calculations of Bender et al. used a DZ basis plus 2p-functions, larger than that used in earlier work by Wilson and Goddard, and by Rubinstein and Shavitt (see ref. 1). Full CI with 2172 configurations was carried out, and the van der Waals minimum predicted was not found by the earlier workers. Kochanski 605 has also investigated the van der Waals minimum, using a perturbative procedure, Results are in good agreement with other calculations, and with experiment. A calculation using a more extended basis set, also investigating four different geometries, was carried out by Kochanski and co-workers.606Results are improved with respect to those from the use of smaller basis sets. Finally, a very detailed search for the reaction path in the above reaction with a DZ STO basis and CJ did not reveal an appropriate path.607 However, a path was found, in which the two HZmolecules go through a trapezoidal to a linear arrangement, which requires N

R. W. Land, J. W. Hunt, and H. F. Schaefer, tert., J. Amer. Chem. SOC.,1973, 95, 4517. A. J. H. Wachters and W. E. Nieuwpoort, 'Selected Topics in Molecular Physics', ed. E. Clementi, Verlag Chemie, Berlin, 1972, p. 135. C o 2 J. A. Connor, I. H. Hillier, V. R. Saunders, and M. Barber, Mol. Phys., 1972, 23, 81. eo3 J. R. Sabin and H. Kim, Chem. Plzys. Letters, 1971, 11, 593. 604 C. F. Bender and H. F. Schaefer, tert., J. Chem. Phys., 1972, 57, 217. 605 E. Kochanski, J. Chem. Phys., 1973, 58, 5823. G O 6 E. Kochanski, B. ROOS, P. Siegbahn, and M. H. Wood, Theor. Chim. A d a , 1973, 32, 151. Ro7 D. M. Silver and R. M. Stevens, J. Chem. Phys., 1973, 59, 3378, 6oo 601

Quantum Mechanical Calculationson Small Molecules

155

less than 25 kJ of energy above the dissociation limit. Clearly, resolution of this important question is not easy, and it is possible that a re-analysis of the experimental values is called for. The N4 and P4 species have been studied by Guest et a1.608A localized-orbital description was used. The results for P4 are consistent with PES results. N4 is not predicted to be stable, unlike P4. Erundle609 et al. have also computed the IP of P4 with a more extensive basis set, and they obtained good agreement with experiment. D. AzB2 and A2H2 Molecules.-The most common A2H2 molecule is H202, which was studied several years ago, but recently a more extensive investigation of the dependence on basis set has been carried out by Ranck and Johansen.610Four different basis sets were used, and the results were compared with Veillards earlier ones obtained with a very large GTO basis.611 d-Functions on 0 are essential for computing rotational barriers, but a (7,3/4,1) basis gives good values for geometrical parameters H2S2 has also been studied by Veillard612and by S ~ h w a r t z . 6 ~ ~ N2H2 has been investigated several times before,614usually under the assumption that the ground state is lAg. This is confirmed by Wagnikre615for the cis and trans geometries of H " H , but the 3 A state ~ of HZNZis lower in energy than either of the above: it should be noted that geometry optimization was not carried out, however, and this could conceivably change the order. Further studies are needed, but with this size of basis set, the conclusions may well be valid. Duben et a1.616 have carried out extensive calculations of the correlation energy in the ground and ionized states of acetylene, C2H2. SizHz is not known experimentally, but SCF and SCF-CI calculations on it have a~peared.~~~~ Very few A2B2 molecules have been studied. Strausz et aL619 have examined F2C=C in its three forms (1)-(3). It is clear that the acetylene structure (2) is much more stable, as is also the case for the C2H2 species.

.

C202 and (2202-have been studied by Beebe and Sabin.620 (2202is a 3 2 - state and is bound, as is the ion. It is predicted that (2202 should be observable, and several M. F. Guest, I. H. Hillier, and V. R. Saunders, J.C.S. Faraday 11, 1972, 68, 2070. C. R. Brundle, N. A. Kuebler, M. B. Robin, and H. Basch, Inorg. Chem., 1972, 11, 20. J. P. Ranck and H. Johansen, Theor. Chim. Acta, 1972, 24, 334. 611 A. Veillard, Theor. Chim. Acta, 1970, 18, 21. 612 A. Veillard and H. Demuynck, Chem. Phys. Letters, 1970, 4, 476. 613 M. Schwartz, J. Chem. Phys., 1969, 51, 4182. 614 D. P. Wong, W. H. Fink, and L. C. Allen, J. Chem. Phys., 1970,52, 6291. 615 G. Wagnikre, Theor. Chim. Acta, 1973, 31, 269. 616 A. J. Duben, L. Goodman, H. 0. Pamuk, and 0. Sinanoclu, Theor. Chim. Acta, 1973,30, 177. 61' S. Y.Chu, I. Ozkan, and L. Goodman, J. Chem. Phys., 1974, 60, 1268. 618 B. Wirsam, Theor. Chim. Acta, 1972, 25, 169. 619 0. P. Strausz, R. Norstrom, A. C. Hopkinson, M. Schoenborn, and I. G. Csizmadia, Theor. Chim. Acta, 1973, 29, 183. G 2 0 N. F. Beebe and J. R. Sabin, Chem. Phys. Letters, 1973, 24, 389. 608 609

Theoretical Chemistry

156

properties have been calculated. Semi-empirical calculationson this molecule621 have been reported (CND0/2). E. Dimersof Diatomic Molecules,A B - . - - AB.-TheHF dimer has beeninvestigated by Diercksen and Kramer,622 who found a structure with a single linear hydrogen bond to be the most stable, with an HFH bond angle of 140".The computed binding energy was in fair agreement with experiment. Similar results were found by del Bene623and by Kollman and AIlen.624 However, the most extensive calculations on this problem are those using the IEPA, CEPA, and PNO-CI approache~,6~~ i.e. explicitly including correlation. The results of all three methods were in good agreement, and the effect of electron correlation is rather small for this system. There have been several other papers on hydrogen bonding, and Van Niessen626 has also studied the HF dimer. The NO dimer has been studied, with conflicting results. Williams and Murrell 627 found the trans-conformation (4) to be more stable. Single-zeta GTO studies,628 however, show that a cyclic form (5) is marginally more stable than the cis, and that

R0 0//N-N

N=N

I I

O---O

the trans is higher in energy, Skancke and Boggs629have also studied this system, getting different results again, but with a larger basis, and they have concluded that the cis-isomer is more stable. However, none of the basis sets used was large enough to resolve this question properly. A more realistic basis set was used by Kollman and co-workers in an investigation of ( L ~ H ) z . ~The ~ O existence, properties, and thermodynamic functions for this species were predicted, using both SCF and CI wavefunctions.It will be interesting to see if this species is detectable. A similar study of (LiF)2 and (NaH)2 was subsequently reported.631The cyclic geometry is lowest in energy, and (NaH)2 should be harder to observe than (LiH)z. Parameters for the known (LiF)2 were in good agreement with experiment. The HCI. * .HF system was also studied by Kollman et aE., and also ( H c l ) ~ . ~ ~ ~ F. Miscellaneous Tetra-atomic Molecules.-Formaldehyde, HCHB, has received a great deal of attention from theoreticians. Space does not permit more than a brief J. FIeischhauer, M. Bechers, and H. D. Scharf, Tetrahedron Letters, 1973, 4275. G. H. F. Diercksen and W. P. Kraemer, Chem. Phys. Letters, 1970, 6, 419. 623 J. E. del Bene and J. A. Pople, J. Chem. Phys., 1971, 55, 2296. 624 P. A. Kollman and L. C. Allen, J. Chem. Phys., 1970, 52, 5085. 625 H. Lishka, J. Amer. Chem. SOC.,1974, 96, 4761. 6 2 6 W. von Niessen, Theor. Chim. Acta, 1973, 31, 297. eZ7 J. E. Williams and J. N. Murrell, J. Amer. Chem. Soc., 1971, 93, 7149. T. Vladimiroff, J. Amer. Chem. Sac., 1972, 94, 8250. 629 P. N. Skancke and J. E. Boggs, Chem. Phys. Letters, 1973, 21, 316. 630 P. Kollman, C. F. Bender, and S . Rothenberg, J. Amer. Chem. SOC.,1972,94, 8016. 631 C. P. Baskin, C. F. Bender, and P. A. Kollman, J. Amer. Chem. SOC.,1973, 95, 5868. 688 P. Kollman, A. Johansson, and S . Rothenberg, Chem. Phys. Letters, 1974, 24, 199. GZ2

Quantum Mechanical Calculations on Small Molecules

157

reference to some of the work. Several auth0rs~33-63~ have shown that SCF-CI treatments are capable of giving a good account of the experimental electronic spectrum, including the Rydberg transitions. It is necessary for the excited states to use the appropriate excited-state SCF orbitals. The description of the l(z--n*)state is more difficult, since u-+ u* excitations have also to be included, Fink6361637 has computed PE surfaces for the lowest two singlet and triplet states with a smaller basis set in order to examine the dissociation process. Hayes and Morukuma carried out similar investigations of the pr0cess.~3~ Excitation energies have been computed by the equations-of-motionmethod for the vacuum-u.v. part of the Finally, the complete set of harmonic force constants have been computed from HF wavefunctions by Pulay and M e ~ e r . ~Of ~ Oparticular interest is the conclusion that accuracy is attainable even with a (7/3/3/1) basis set. The large CI treatment of Langhoff G41 et al. has given the most accurate ground-state wavefunction yet for this molecule. Computed values of the dipole moment and the dipole-dipole parameters were in reasonable agreement with experiment. The H2CS molecule is of interest, particularly its excited states, compared with H2C0, and it has recently been studied in a minimal STO-3G basis set.642Ground and s(n,n*)and 3(n, n*)states were studied, the results being in good agreement with microwave data. The Q and n bonds are predicted to be polarized in opposite directions compared with H2CO. A more extensive set of calculations on HzCS has been reported by Bruna et aZ.,643who used a DZ P basis set and carried out extensive CI. The CS bond is substantially longer than the CO bond in H2CO in the corresponding states. A detailed comparison was made with the spectrum, and between HZCOand HKS. The boron-nitrogen compounds such as iminoborane have attracted the attention of theoreticians recently, and HBNH is predicted to be linear, with B-N bonds comparable in strength to C=C and which thus may be truly represented as a triple bond, albeit strongly polarized.644 The corresponding analogue of cyclobutadiene (obtained by dimerization of iminoborane)has been investigated by Baird 645 and by Armstrong and Clark.646 Finally, we mention a few isolated calculations, such as those on the aminonitrenes HzNN in their lowest singlet and triplet sfates,647 and calculations on

+

633 634

R. J. Buenker and S. D. Peyerimhoff, J. Chem. Phys., 1970,53, 1368. S. D.Peyerimhoff, R. J. Buenker, W. E. Kammer, and H. Hsu, Chem. Phys. Letters, 1971,8

635 636 637 638 639

J. L. Whitten J. Chem. Phys., 1972, 56, 5458. W.H.Fink, J. Amer. Chem. SOC.,1972,94, 1073. W. H.Fink, J. Amer. Chern. SOC.,1972,94, 1078.

129.

D. M. Hayes and K. Morukuma, Chem. Phys. Letters, 1972, 12, 539. D. L. Yeager and V. McKoy, J. Chem. Phys., 1974, 60,2714. 1340 W.Meyer and P. Pulay, Theor. Chim. Acra, 1974, 32, 253. 641 S. R. Langhoff, S. T. Elbert, and E. R. Davidson, Internat. J. Quantum Chem., 1973, 7,999 643 N.C. Baird and J. R. Swenson, J. Phys. Chem., 1973,77, 277. 643 P. J. Bruna, S. D. Peyerimhoff, R. J. Buenker, and P. Rosmus, Chem. Physics, 1974, 3, 35. 644 N. C. Baird and R. K. Datta, Inorg. Chem., 1972, 11 I?. 645 N. C. Baird, Inorg. Chem., 1973, 12, 473. 646 D. R. Armstrong and D. T. Clark, Theor. Chim. A d a , 1972, 24, 307. 647 N. C. Baird and R. F. Barr, Canad. J. Chem., 1973,51, 3303.

158

Theoretical Chemistry

HzSQ648 PFZH,"~and a comparison of the role of d-orbitals in HNCO and HNCS.650 It is hoped that subsequent reports will deal with molecules containing five or more atoms.

648

649 fi50

J. M. Howell, I. Absar, and J. R. van Wazer, J . Chem. Phys., 1973, 59, 5895. I. Absar and J. R. van Wazer, J. Amer. Chem. SOC.,1972,94, 6294. J. M. Howell, I. Absar, and J. R. van Wazer, J. Cheni. Phys., 1973, 59, 5895.

4 Electronic Calculations on Large Molecules BY B.

J. DUKE

1 Introduction The past few years have seen a welcome closing of the gap between the meaning given to the term ‘large molecule’ by a quantum chemist and that given by an organic or inorganic chemist. The improvement in computer hardware and software, as well as the rapid development of new methods, has resulted in the application of many powerful methods of quantum chemistry to molecules that we would all recognize as large. To the quantum chemist ‘large’ may be defined in terms of the number of atoms or of electrons or by some parameter more directly related to the difficulty of obtaining the simplest wavefunction for the whole system. The final choice for dividing large molecules from small ones is arbitrary. For the purpose of this Report we d e h e as large a molecule which would require a minimum of 40 atomic orbitals or basis functions at the minimum basis set level of molecular orbital calculations which treat all the electrons. We exclude any diatomic or triatomic molecule. The dividing line is thus somewhere between benzene, with 42 electrons, requiring 36 basis functions, and cyclohexane, with 48 electrons, requiring 42 basis functions. Progress in the past ten years can be clearly seen by comparing the present position with Pople’sl two-dimensional chart of quantum chemistry Here a measure of sophistication of the method used is plotted against the number of electrons in the molecules for which the method applies Large molecules by the definition which now seems appropriate could, at that time, only be treated by the most simple semiempirical methods Progress since that date has been charted in a number of reviews.2-7 This Report is concerned with progress in the past two to three years, during which a number of significant developmentshave taken place. In the Reporter’s opinion these are: (a) The extension of cab initio molecular orbital (MO) calculations to large mole-

cules. Such calculations treat all electrons in the molecule and make no approxi1 8

* 4 5

6

7

J. A. Pople, J. Chem. Phys., 1965, 43, S229. W. G . Richards, T. E. H. Walker, and R. K. Hinkley, ‘A Bibliography of Ab initiu Molecular Wave Functions’, Clarendon Press, Oxford, 1971. (a) G. G. Hall, Chem. SOC.Rev., 1973, 2, 21; (b) B. J. Duke, Ann. Reports (A), 1971, 68, 3. L. Radom and J. A. Pople, in ‘Theoretical Chemistry’, ed. W. Byers Brown, MTP International Review of Science,Physical Chemistry Series One, Volume 1, Butterworths, London, 1972, p. 71. ‘Computational Methods for Large Molecules and Localized States in Solids’, ed. F. Herman, A. D. McLean, and R. K. Nesbet, Plenum Press, New York, 1973. ‘Wave Mechanics - the First Fifty Years’, ed. W. C. Price, S. S. Chissick, and T. Ravensdale, Butterworths, London, 1973. D. A. Brown, W. S. Chambers, and N. J. Fitzpatrick, Inorg. Chim. Ada, Rev., 1972, 6, 7.

159

160

Theoretical Chemistry

mations other than the molecular orbital form of the wavefunction and the truncation of the basis set employed. The evaluation of good wavefunctions within the ab initio MO scheme to be close to the Hartree-Fock limit. This limit is reached when the basis set is extended sufficiently that the necessary truncation is insignificant. This development has been achieved only for small molecules, but has almost been achieved for benzene,8 which is close to our definition of the dividing line between large and small molecules. The knowledge gained from these calculations and less accurate ones on larger systems has considerably clarified our understanding of the weaknesses of the ab initio MO approach. An increasing clarification of the possibilities and limitations of semi-empirical MO methods and the development of methods which bridge the gap between semi-empirical and ab initio approaches. ( d ) The MO approach in the linear combination of atomic orbitals (LCAO) form no longer has complete dominance of the large-molecule calculation field. Several methods go beyond the MO method although semi-empirical parameters are necessary for large molecules. The X , exchange approximation provides alternatives to the LCAO approach which have had some highly successful applications. (e) An increased interest in calculations on inorganic molecules. These developments will be the main subject of this Report.

2 The Linear Combination of Atomic Orbitals-Molecular Orbital Approach Introduction.-The LCAO-MO method remains the most important approach for evaluating wavefunctions for large molecules in spite of its known defects. The details of the method are well known. The main outlines are given here only to establish the nomenclature. The method attempts to describe the wavefunction as a single Slater determinant comprised of oneelectron space-spin functions or spin orbitals :

Each spin orbital @pz is a product of a space function q5i and a spin function a or In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a spin function and then with the ,8 spin function. For openshell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ,8 spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator s.2 The three cases are illustrated by the examples below. 8

W. C. Ermler and C.W. Kern, J. Chem. Phys., 1973,58, 3458.

Electronic Calculations on Large Molecules

161

Closed shell: 2n electrons

In the RHF case + 1 is the singly occupied orbital. In the UHF case the orbitals ds, i = 1 , 2 . . n are different from the orbitals d i , i = n + 1, n + 2 , . . 2n + 1. In both cases the number of electrons of ,8 spin is one more than the number of electrons of a spin. The molecular orbitals $i are expanded in terms of a set of simpler functions xp:

.

.

m

The coefficientsCpzare the parameters to be determined. The set of functions 2, is known as the basis set and often consists of atom-centred functions obtained from the solution of the Schrodinger equation with some central field potential. The coefficients Cpt are obtained by minimizing the expectation value of the Hamiltonian for the wavefunction Y from equation (2). In the more commonly occurring closed-shell case this gives the secular equation

FC = SCA

(6)

where C is the matrix of coefficients Cpg,S the overlap matrix spv =

<x,lxv>

(7)

I a diagonal matrix of terms ili known as orbital energies, and F, the Fock matrix, has eIements

Hp, is the matrix element over the one-electron components in the Hamiltonian and the remaining integrals depend on the two-electron repulsion term, e.g.

A full expansion of the MO’s in terms of the basis set gives

162

Theoretical Chemistry

where Dnv is an element of the reduced density matrix in the basis set representation

The total electronic energy, the expectation value of the Hamiltonian for Y, is given by a number of equivalent expressions: n

E =

c

cn

+

2&p

i=l n

==

C

n

i=l

(2Jij-Kij)

i,j=l

Ta

c

2

2&-

i=l

=

(2Jij-K.ijj

i,j=l

(At

+

&pj

where

Kip =

<$Z+jl4i4j)

and %I is the one-electron component of the Hamiltonian. Equation (6) does not give a unique solution. The choice of it as a diagonal matrix, however, does give a unique solution, the canonical orbitals. If off-diagonal elements of I are retained, the MO’s & can have particular properties such as being localized in regions of the molecule rather than fully delocalized as are the canonical orbitals. The use of these localized molecular orbitals has been the subject of two recent reviews.9asb So far only one approximation has been made after the initial choice of equation (2) for Y. This is the necessary truncation of the basis set expansion in equation (5). We chose the set xp to be m elements of a complete set. If we increase the value of m until addition of further elements gives no significant change in the result we have reached a well-defined point - the Hartree-Fock limit. This limit has been reached for small molecules using a variety of different choices for the basis set xp. The value of m,however, is so large that such calculations for large molecules are still prohibitively expensive. Although the above discussion and equations apply to the closed shell, a similar analysis for the open-shell case can be made. Ab initio LCAO-MO Methods.-The so called ab initio MO method is defined by the discussion so far. The two approximations made are the MO form of the wavefunction and the necessary truncation of the basis set. It is important to realize that all the ab initio calculations on large molecules are far from the Hartree-Fock limit and that even this limit may be inadequate to describe a particular feature of the molecule. An excellent introduction to a6 initio calculations is the recent book by @

(a) H. Weinstein, R. Pauncz, and M. Cohen, Adv. Atomic Mol. Phys., 1971, 7,97; (b) W. England, L. S. Salmon, and K. Ruedenberg, Fortschr. Chern. Forsch. (Topics Current Chem.), 1971. 23, 31; (c) D. B. Cook, ‘Ab Itzitio Valence Calculations in Chemistry’, Butterworths, London, 1974; ( d ) H. F. Schaeffer, tert., ‘The Electronic Structure of Atoms and Molecules A Survey of Rigorous Quantum Mechanical Results’, Addison-Wesley Publ. Co., Reading, Mass., 1972.

Electronic Calculations on Large Molecules

163

COO^.^^ A critical discussion of calculations mainly on small molecules is given in Schaeffer’s book.9d The inadequacies of Hartree-Fock theory and small basis sets are clearly discussed here, and several calculationson large molecules are discussed in detail. Ab initio calculations on large molecules have become feasible only in recent years. Indeed less than ten are listed in the bibliography of ab initio calculationsby Richards, Walker, and Hinkley2 which covers the literature to December 1969. There is now, however, a considerable literature of such calculations. An excellent review, complete to the end of 1971, of calculations on organic molecules is given by Radom and Pople.4 ChristoffersenlOlists calculations on large molecules up to 1970. Himella gives an excellent discussion of theory, but no complete bibliography. A full bibliography of ab initio calculations supplementing the earlier work by Richards has recently appeared.llb This claims to be complete to the end of 1973 and is a most worthwhile addition to the literature. The published proceedings of two recent conferences129l3should also be consulted. These reviews, along with other papers in the literature, should be consulted for a discussion of choice of basis sets. Only a brief guide will be given here. The simplest basis set of atom-centred functions is the minimum basis set of one function for each orbital in the isolated atoms. Hydrocarbons, for example, would give a basis set of Is-, 2s-, 2 p ~2pT, , and 2pz-orbitalson each carbon and 1s-orbitals on each hydrogen atom. The simplest choice of these orbitals are the Slater-type orbitals (STOs), consisting of a simple exponential function multiplied by a spherical harmonic (or some real combination of spherical harmonics). This basis set is popular, but leads to difficulties in evaluating the multi-centre integrals < x ~ x , , ~ x A x ~ > . An alternative is the use of Gaussian-type orbitals (GTOs) which replace the simple exponential exp ( - v r ) by exp (- w2).Integrals are now easy to evaluate, but many more basis functions are required. It is usual to take fixed combinations of GTOs to describe a single basis function. These are known as contracted Gaussians (CGTOs). The choice may be made to fit STOs or Hartree-Fock atomic orbitals in some sense or by experiencefrom calculation on small molecules. Improved basis sets are obtained if each STO (or CGTO) is replaced by two such orbitals. This is the double-zeta basis. A compromise is to use single STOs (or CGTOs) for inner-shellelectronsand two for outer-shell electrons. Even better basis sets result if more STOs (or CGTOs) centred on each atom are used, but partial double-zeta-quality basis sets [designated here as min (+) STO or min (+) CGTO] are about the feasible limit for large calculations. We designatebasis sets close to minimum-basis-set quality as rnin STO or min CGTO, the latter not distinguishing the various methods for choosing the contraction of GTOs. Another widely used choice of basis set is a small set of spherical GTOs centred in bond and lonepair regions as well as in atoms. Such a basis set can give remarkably R. E. Christoffersen, Ado. Quantum Chem., 1972, 6, 333. (a)J. Hinze, Adu. Chern. Phys., 1974,26,213; (b) W. G. Richards, T. E. H. Walker, L. Farnell, and P. R. Scott, ‘Bibliography of Ab initio Molecular Wave Functions’, suppIement for 19701973, Clarendon Press, Oxford, 1974. 12 ‘Quantum Chemistry - the State of the Art’, Volume 3, ‘Hartree-Fock Theory’, Proceedings of S.R.C. Atlas Symposium No. 4, ed. V. R. Saunders and J. Brown, Atlas Computer Laboratory, Chilton, near Didcot, Oxford, 1974. 1s ‘Energy, Structure, and Reactivity’, Proceedings of the 1972 BouIder Summer Research Conference on Theoretical Chemistry, ed. D. W. Smith and W. B. McRae, Wiley, New York, 1973. 10 11

Table Ab initio calculations on large molecules Ref. Molecule Alicyclic molecules Vinylcyclobutane Bicyclo[2,2,llhexane Cyclohexene

Formula

NO.

(see p. 176)

Author

Cf3HlO C6HlO C&lO

46 46 46

Hehre Lehn Leroy

rnin CGTO rnin CGTO min CGTO

aa ab

1-Methylcyclobutylcarbinyl Homotropylium

c6Hk

46

Pople

rnin CGTO

ac, ad

C7Ht

48

Hehre

rnin CGTO

ae

Cyclohexane

C6Hl2

48

Hoyland Preuss Leroy

rnin CGTO min CLGTO min CGTO

af bi

bi

n-Hexane

C6Hl4

48

Leroy

min CLGTO

bi

Norbornadiene 1- and ZCyclopropylvinyl cyanide 1- and 2-CycIobutylAuoroethylene Silylcyclopentadiene Bicyclo[2,2,l]heptane (norbornane)

C7H8 C5H7CN

50 50

Palmer Hehre

rnin CGTO min CGTO

ah aa

CeHgF

54

Hehre

min CGTO

aa

C5H5SiH3 C7H12

52 54

Palmer Lehn

min (+) CGTO min CGTO

ai a.

p-Benzoquinone

CsH40e

56

Nieuwpoort

min CGTO

ak

Comment

Comparison with transferability method CircumambuIatory degenerate rearrangement Boat-chair conversion Comparison with transferability method Comparison with transferability method ESCA spectrum

8 Photoelectron spectra Involvement of bridgehead C-H bonds Ground state and n+n* excited state

R'

n

ir $

.z

Cubane

CsHs

56

Methylcyclohexane Fluorocyclohexane Bicyclo[4,2,0]octa2,4,7-triene

C7H I 4 CsHiiF C8H8

56 56 56

Preuss Lehn Eilers Eilers Wipff

min CLGTO min CGTO min CGTO min CGTO rnin CGTO

aJ a1

am am an

Cyclo-octatetraene

C8HS

56

Wipff

min CGTO

an

Cyclo-octatetraene dianion

QH$-

58

Wipff

min CGTO

an

1- and 2-Cyclobutyl-

CsHgCN

58

Hehre

min CGTO

aa

vinyl cyanide Bicyclo[2,2,2]octane

C8H14

62

Lehn

rnin CGTO

ai

C9H16

70

O'Leary

min CGTO

ao

Aromatic hydrocarbons Naphthalene ClOH8

68

min CLGTO Preuss Christoffersen FSGO min CLGTO Buenker

aP

Tiberghien

at

1,3,3-Trimethylcyclohex-1,Zene

a99 ar as

Geometry optimization. Predicted to be less stable than cyclooctatetraene (opposite prediction by CNDO) Geometry optimization. Good barrier to inversion Geometry Optimization. Found to be planar (D8h)

R' cl

k

2

s g. 2 0 3

1;

i6$ 2

Involvement of bridgehead C-H bond Required for SAM0 calculation on alltruns-retinal All give poor energy difference between naphthalene and azulene. FSGO gives poor first ionization potential Triplet exchange integrals

-

E

CI

ForrnuIa

No.

Molecule Fulvalene

CioHa

68

Azulene

CiaHR

68

Author Type Christoffersen FSGO

Ref. (see p. 176) a9

Ant hracene

C14H10

94

Buenker min CLGTO Christoffersen FSGO Christoffersen FSGO

Phenanthrene

C14H10

94

Christoffersen FSGO

au

Benzene derivatives Benzyl radical

CsHsCHz

49

Preuss Hinchcliffe

af uw

Phenoxyl radical Anilino radical

C6H50 CsH5NH

49 49

Hinchcliffe Hinchcliffe

rnin CGTO rnin CGTO double zeta CGTO double zeta CGTO double zeta CGTO

Toluene

C7HS

50

CsH5F

50

min CGTO FSGO min CGTO rnin CLGTO min CGTO rnin CGTO rnin CLGTO

ay, az ba

Fluorobenzene

Pople Brailsford Pople von Niessen Almlof Pople von Niessen

50

a cn

Comment Poor energy comparison with naphthalene

CIS

aq

au, UL'

,

UHF with limited spin annihilation. Calculation of e.s.r. proton coupling constants. Improved basis set gives better energy, but no significant improvement in coupling constants

ax

ax ax

UY,

bb bc az bb

Comparison with phenanthrene. Discussion of ionization potentials (comparison with semi-empirical methods)

a*

)Barriers to rotation

3

85. s

Aniline

CeHsNHa

50

Pople

min CGTO

az,

bd Phenyl cyanide

CaH5CN

Styrene

CsH5CHCH2

Phenylethyl CsH6 CeH& X = C=CH, NC, CHNH, CHO, NCH2, NNH, NO, OF, Et, OOH, NHOH, NHNH2, CH2F, CHF2, CF3, CH20H, CH2NH2, NHMe, NHF, NO2, ONH2, OMe, COF, COzH, CONH2, or COMe p-C6H4(NH2)X X = NH2, Me, NO2, F, or OH CeH4(OH)X X = Me, F, OH, NO2, CHO, or CN (p-) X = F or Me (0-,m-,orp-) 0-, m-,andp-C6H4(0H)X (X = OH, F, or CN) 0-,m-,andp-CgH4FX (X = F or CN) 0-,m-,and p-CaH4(CN)z

54

56 56

0-,m-,and p-C6H4Me2

Pople von Niessen Pople Almlof Hehre Pople

min CGTO

az

min CLGTO rnin CGTO min CGTO min CGTO min CGTO

bb az be bf

Pople

min CGTO

bd

Pople

min CGTO

Radom von Niessen von Niessen von Niessen

min CGTO rnin CLGTO rnin CLGTO rnin CLGTO

bh bb, bj bb, bj bb, bj

Radom

min CGTO

bh

Heterocyclic molecules Maleic anhydride C4HzOs

50

Roos

rnin CGTO

Succinic anhydride 3-Methylsydnone

52 52

Roos Palmer

min CGTO min CGTO

C4H403 C3NzOzH4

az

I

rn-Derivative always most stable. Comparison with 'molecules-inmolecules' method

Comparison with photoelectron spectrum bk bl

c3.

Molecule Urazole

Phosphazene (phosphorin)

Formula C2H5N302

C5H5P

NO. 54

56

Thymine

56

Thymine anions ‘1HT-’ and ‘3HT-’ Cytosine C4H5N30

60

5-Azauracil Thiophen dioxide Histamine Proline Imidazolin-Zamide 6a,l,6-Thiadioxapentalene

C3N302H~ c4w4so2

C5HlON28 C4HgN30 C5H404S

58

58 60 60 62 62 66

Ref. (see p. 176) Comment bm Also calculation on cation and anion radicals. Considerable reorganization of both 0-and nframework on ionization bn, bo Comparison with pyridine and arsabenzene ionization potentials &Orbitals included bP but not found to be important bq, br, bs bt CC Excited states bw

Author Kramling

The rnin CGTO

Clark

rnin (+) CGTO

Palmer

rnin (+) CGTO

Pullman Clementi Snyder Clark Snyder

min CGTO rnin CGTO rnin CGTO rnin CGTO

cc

Pullman Clementi Ahrlichs Clark Clark Palmer Pullman Christoffersen Christoffersen Palmer

rnin CGTO rnin CGTO min CLGTO

bq, br, bs

rnin CGTO rnin (+) CGTO min CGTO FSGO FSGO rnin (+) CGTO

bt, bu bv bw bx bY bz ca, cb ca bP

o\

00

Y

d-Orbital involvement found to be small

g 3 g c

Indole Isoindole Benzofuran Isobenzofuran Arsabenzene Adenine 1,6a,&Dithiaoxapentalene 1,6a,6-Dithiaazapentalene Benzothiophen Guanine 3-Propylproline N-(2-hydroxyethyl)proline 1,6,6a-Trithiapentalene Carbazole Serotonin

68 68 68 68 68

C90H6 C90H6 C~H~AS C5H5N5

70

CsH4Sz0

74

Palmer Palmer Palmer Palmer Clark Pullman Clementi Clark Palmer

74

Palmer

rnin (+) CGTO

76 78 78 78

Palmer Clementi Christoffersen Christoffersen

rnin (+) CGTO rnin CGTO FSGO FSGO

bY bt, bu ca, cb cb

82

Palmer

rnin (+ ) CGTO

bP, cf

88

Clementi

rnin CGTO

bu, cg, ch, ci

94

Pullman

rnin CGTO

ce

94

Christoffersen FSGO Nieuwpoort min CGTO

ca, cb

104 105 106 110

Nieuwpoort Nieuwpoort Pullman

ci ci

C9NH7 c9m7

rnin (+) CGTO rnin (+) CGTO rnin (+) CGTO min (+) CGTO rnin (+) CGTO rnin CGTO min CGTO rnin (+) CGTO

Ionization potentials

bP

-&

CsH4Ss

CioHi3N20

+

3-Pentylproline TetracyanoquinoC6Ha[C(CN)a]s dimethane, TCNQ TCNQTCNQaBufotenine Ci2Hi7Nz0+

rnin CGTO rnin CGTO rnin CGTO

cj

ce

2 Ionization potentials. Comparison with Xa method Conformationalstudy - in general agreement with PCILO results Excited states and n.m.r. coupling constants Conformational study and comparison with PCILO results

L-r

s

Molecule Cytosine-guanine pair

2,4,6-Trinitro-9fluorenone

FurmuIa

Ci3N307H5

2,4,6-TNF-carbazole adduct

No. 136

Author Clementi

Type rnin CGTO

160

Clementi

rnin CGTO

248

Clementi

rnin CGTO

rnin CGTO rnin CGTO rnin CGTO rnin CGTO

Other organic molecules HC rC(CH2)3CH Me3C-C=CH Diazocyclopentadiene C5H4Na

44 46 48

Pople Radom Hillier Clark

Formylglycinamide

HCONHCHzCONHz

54

Christolfersen FSGO

MeCN,BF3 EtNHz,BF3 Lithium oxalate hydrate Acetylcholine

LiHC204,HzO

54 58 58

Hillier Hillier Almlof

MeC02(CH&NMea

62

Sodium oxalate hydrate

NaHCz04,H~0

66

Pullman rnin CGTO Christoffersen FSGO Almlof CGTO

+

Ref. (see p. 176) Comment buy ck No double H bond well found. Comparison with semi-empirical methods Calculation of electron cg, ch, ci, bu affinity. TNF, like carbazole, is an important component of the IBM photocopier! bu A good example of a large molecule treated by merging integrals and use of an adjoined basis set ad bh el Cl

cm,cn, co

rnin CGTO rnin CGTO CGTO

ESCA and photoelectron spectra studied Conformational study. Fully optimized FSGOs

Hydrogen-bonding crycs av, C J I , ct CY

>C::Eational

study

w

4

0

Hexafiuoroethane

C2F6

66

Pople

rnin CGTO

cu

Glycylglycine All-trans-pentaene Amphetamine Alkylammonium hydrates

+NHsCH2CONHCH2CO2 CioH12 CaH5CH2CH&H3)Me

70 72

Whit ten Christoffersen Brailsford Pullman Pullman Pullman Pullman Christoffersen

rnin CLGTO FSGO FSGO min CGTO min CGTO min CGTO rnin CGTO FSGO

cv cb ba

Clementi

min CGTO

ex, CY

Poly glycines

Sugar phosphate

INEt4l+

INMeEts]+, H2O WMe2EtzI+,(H20)~ [NMe41f,(Ha0)4 HCONH(CH2CONH)nCHzCONHa (n = 0, 1, 2, or 3) C10~1908P and c1oHi8osP-

74 74 76 78 82 54, 84, 114, 144

158

cw

cw cw cw

Comparison with PDDO method Some C.I. Prototype for alkylammonium groups in cholinergic and adrenergic drugs

av, cn, co

Barrier to rotation

sugar

Transition-metal compounds Technetium hydride r]rcHg]2anion Silver-ethylene [Ag(C2H4)lf complex Nickel tetrafluoride [NiFd2Bis-n-allylnickel Ni(CaH5h

54

Basch

min (+I CGTO

CZ

62

Basch

rnin (+) CGTO

da

66 74

Basch Veillard

min (+) CGTO rnin (+) CGTO

db dc, dd

n-Cyclopen tadienylnitrosylnickel

78

Hillier

rnin (+) CGTO

de, 4

(n-C5H5)NiNO

Geometry search ASCF calculation because of breakdown of Koopmans’ theorem Koopmans’ theorem breakdown. ASCF calculation gives good agreement with photoelectron spectrum

Y

4

N

Molecule Tetracyanonickelate dianion

Formula [Ni(CN)412-

Nu. 82

Author Veillard

Type

min (+) CGTO

Hillier

min (+) CGTO min (+) CGTO min (+) CGTO min (+) CGTO

Tetranitrogenylnickel Ni(N&

84

Veillard Hillier Ros Ros

Dicarbonyldinitrosyliron

Fe(CO)2(NO)z

84

Hillier

min (+) CGTO

Tricarbonylnitrosylcobalt

Co(C0)sNO

84

Hillier

min (+) CGTO

Nickel hexafluoride

“&I4-

86

Basch

min (+) CGTO

Ros

one-centre STO

Veillard

min (+) CGTO

Nieuwpoort

min (+ ) CGTO

Tetracarbonylnickel

Ni(CO)J

84

Ref. (see p. 176) Comment RHF calculation. dg, Breakdown of de Koopmans’ theorem. Viellard used crystal-lattice point charge potential dg di, dj, dk charge on nickel dl Similar bonding to dl carbonyl dk Comparison with photoelectron spectrum. ASCF calculation dk Comparison with photoelectron spectrum. ASCF calculation dm U H F and RHF -good comparison with experiment dn UHF - poor agreement with experiment do RHF - poor 2. agreement with experiment RHF - similar to 0 Cr’P Basch‘s study

? 9

&

H5

fs

Cupric hydrates

(n = 2, 4, or 6) 47,67,87 Veillard [C~(H20)~]2+

Peroxychromate anion [CrOs]3Mn(C0)sX (X = H, Me, CN, or Cl)

91 96, 104, 108, 112

rnin

min (+) CGTO rnin (+) CGTO rnin (+) CGTO

Fe(CO)5

96

Ferrocene

Fe(C5Hsh

96

Veillard Hillier Veillard

98

Hillier

min (+) CGTO

99

Veillard

min (+) CGTO

AmminepentaCr(C0)5NHa carbonylchromium Hexacarbonylcr(C0)~ chromium (benzene)Cr(CO)a (benzene)zCr PentacarbonylCr(C0)5PH3 phosphinechromium Copper pentachloride [CuC15I3Co(acacen) Co(acacen)Oz Co(acacen)OzCN Co(acacen)Oz(imidazo1e)

104

Hillier

min (+) CGTO

108

Hillier

min (+) CGTO

108 108 112

Hillier Hillier Hillier

min (+) CGTO min (+ ) CGTO min (+) CGTO

117

Veillard

min (+) CGTO

101

Veillard Veillard Veillard Veillard

min (+) CGTO min (+ ) CGTO min (+) CGTO min (+ ) CGTO

117 131 153

solvation

CGTO rnin (+) CGTO min (+) CGTO

Dacre Veillard Hillier

Pentacarbony1manganese derivatives Pentacarbonyliron

Butadienetricarbonyl- Fe(C0)3(C4Hs) iron Copper tetrachloride [CuC14]2-

Mechanism of

min (+) CGTO

$

3

3

Photoelectron spectrum assigned

Breakdown of Koopmans’theorem. ASCF calculation

RHF calculation. Geometry study. ASCF calculation. Excited states. Example of JahnTeller distortion

1

Q5

i 5 g.

z Q

5

s$

F

Photoelectron spectrum, discussed using Koopmans’ theorem

i

and C4v structures examined

D3h

1 /various geometries

J

d

w 4

-L

Molecule Boron compounds 1,5-CzB3H5

Formula

iV0 .

Ref. (see p. 176)

Author

&H2;

32 38

Hillier Lipscomb

rnin (+) CGTO rnin STO

ed ee

1,2- and 1,6-CzB4Hs

3%

Hillier

rnin (+) CGTO

ed

Lipscomb

niin STQ

ef, e.Y

Lipscomb Lipscomb Lipscomb Hillier

rnin STO rnin STO rnin STQ rnin (+) CGTO

eh ei eh ed

4,S-CzBdHs Hexaborane 2,4-dicarbaheptaborane(7)

40 40 44

Octaborane( 12) Enneaborane Deca borane( 10) dianion Decaborane

52 60 62

Lipscomb Lipscomb Lipscomb

rnin STO rnin STO min STO

ee ee ee

64 64 66

min STO min CGTO min CGTO rnin STO min CGTO rnin (i) CGTO

ej

BioHiz12Decaborane( 14) dianion Tetraboron tetrahalides

Lipscomb Hillier Hillier Lipscomb Hillier

BlXt (X = Fori‘i

56, 88

ek ek ee ek

z!

2 Cornment

Comparison with PRDDO Minimum basis inadequate for predicting 1,6 more stable than 1, 2. Larger basis set required Minimum basis set gives 1,6 more stable than 1,2

Comparison with INDO

g.

fa Photoelectron spectrum assigned 0 correctly by 8 Koopmans’ theorem

$, -b

2

Other main-group nzolecules Difluorophosphine P2F4

@ 56

58

60

Phosphorus pentafluoride and its derivatives

(PF4X)

Water polymers

(H2O)n (n = 2-6) (HF)6

60

ClF5 ClF, ClFt

62 54 52

€IFpolymers

Chlorine pentafluoride and other chlorine fluorides

Wagner

Hillier Veillard

min (+) CGTO

min (+ ) CGTO rilin (-1- ) CGTO

Rotational barriers, comparison with CNDO. d-Orbitals do not affect barrier Role of d-orbitals in determining geometry Geometry optimization. Mechanism of ligand exchange

e9

er

es

(X = H, F, or NH2)

60

Pople Allen Pople Allen Hillier

rnin CGTO min CGTO rnin CGTO rnin CGTO rnin (+) CGTO

et eu et eu

ci’

?8 -b ij.

8‘

ii

0

!?

3

%

ev

Geometry optimization. d-Orbitals important for accurate bond lengths ESCA spectrum. d-Orbitals important for S 2p shifts

Sulphur hexafluoride SFs

78

Roos Gianturco Bendazzoli

min (+) CGTO rnin (+ ) CGTO rnin STO

ew ex, eY ez

Sodium hydrates

Na(Mz0)G

71

Pho sphonyl derivatives HCN polymers Xenon fluorides

PC130

74

Clementi Ahrlichs Hillier

min CGTO min CLGTO min (+) CGTO

(HCN)G XeF2 XeF4 XeFa

84

Johansson Basch Basch Basch

rnin CGTO rnin ( + ) CGTO rnin (+ ) CGTO rnin (+) CGT

bu bu em, en, eo, ep Importance of d-orbitals fa,fb Poor description of fc photoelectron fc spectrum fc

72 90 108

5a

J

% 0 9 %

2

Y

cn 4

References for Table W. J. Hehre, J. Amer. Chem. SOC.,1972, 94, 6592. J. M. Lehn and G . Wipff, Theor. Chim. Acta, 1973, 20, 223. ac L. Radom, J. A. Pople, V. BUSS,and P. von R. Schleyer, J. Amer. Chem. SOC.,1970,92,6380. ad L. Radom, J. A. Pople, and P. von R. Schleyer, J. Amer. Chem. SOC.,1972,94, 5935. ae W. J. Hehre, J. Amer. Chem. Soc., 1972, 94, 8908. af H. Preuss and R. Janoschek, J. Mol. Structure, 1969, 3, 423. au J. R. Hoyland, J. Chem. Phys., 1969, 50, 2775. dh M. H. Palmer and R. H. Findlay, Chem. Phys. Letters, 1972, 15, 416. at S. Cradock, R. H. Findlay, and M. H. Palmer, J.C.S. Dalton, 1974, 1650. a5 J. M. Lehn and G . Wipff, Theor. Chim. Acta, 1974, 33, 43. ak H. T. Jonkman, G. A. van der Veble, and W. C. Nieuwpoort, in ref. 12. at J. M. Lehn and G. Wipff, J.C.S. Chem. Comm., 1973, 747. a m J. E. Eilers, personal communication, 1973. a n G. Wipff, U. Wahlgren, E. Kochanski, and J. M. Lehn, Chem. Phys. Letters, 1971, 11, 350. ao B. O’Leary, B. J. Duke, J. E. Eilers, and E. W. Abrahamson, Nature, 1973, 246, 166. ap H. Preuss, Internat, J. Quantum Chem., 1968, 2, 651. aq R. E. Christoffersen, J. Amer. Chem. SOC.,1971, 93, 4104. ar G. M. Maggiora, D. W. Genson, R. E. Christoffersen, and B. V. Cheney, Theor. Chim. Acta, 1971, 22, 337. a8 R. J. Buenker and S . D. Peyerimhoff, Chem. Phys. Letters, 1969, 3, 37. at A. Tiberghien, Ph. Devaux, and G. Delacbte, Chem. Phys. Letters, 1971,9, 642. au R. E. Christoffersen, Internat. J. Quantum Chem., Symp., 1973, 7, 169. au L. L. Shipman and R. E. Christoffersen, Chem. Phys. Letters, 1972, 15, 469. aw A. Hinchliffe, Chem. Phys. Letters, 1972, 13, 594. ax A. Hinchliffe, Chem. Phys. Letters, 1974, 27, 454. a v W. J. Hehre and J. A. Pople, J. Amer. Chem. SOC.,1970, 92, 2191. aa W. J. Hehre, L. Radom, and J. A. Pople, J. Amer. Chem. SOC., 1972, 94, 1496. ba D. F. Brailsford, in ref. 12. bb W. von Niessen, Theor. Chim. Acta, 1974, 33, 185. bc J. Almlof, A. Henriksson-Enflo, J. Kowelewski, and M. Sundbom, Cheri. Phys. Letters, 1973, 21, 560. bd W. J. Hehre, L. Radom, and J. A. Pople, J.C.S. Chem. Comm., 1972,669. be J. E. AlmlQf, P. U. Isacsson, P. J. Mjoberg, and W. M. Ralowski, Chem. Phys. Letters, 1974, 26, 215. bf W. J. Hehre, J. Amer. Chem. SOC.,1972, 94, 5919. bg L. Radom, W. J. Hehre, J. A. Pople, G . L. Carlson, and W. G. Fateley, J.C.S. Chem. Comm., 1972, 308. bh L. Radom, J.C.S. Chem. Comm., 1974, 403. b. G. Leroy and D. Peeters, Theor. Chim. Acta, 1974, 36, 11. bj W. von Niessen, Theor. Chim. Acta, 1974, 33, 7. bk M. Almemark, J. E. Backvall, C. Moberg, B. Akermark, L. Asbrink, and B. Roos, Tetrahedron, 1974, 30, 2503. bl M. H. Palmer, A. J. Gaskell, and M. S. Barber, J. Mol. Struchrre, 1972, 12, 197. bm R. W. Kramling and E. L. Wagner, Theor. Chim. Acta, 1969, 15, 43. aa ab

bn

bo bp

bq br bs bt

bu bw bW

bx

by br ca

cb cc Cd

ce Cf

Cg

ch ci

ck

CR

co Cp

cq Cr Cp

Ct cu cu CW CX

CV

cz

C. Batish, E. Heilbronner, V. Hornung, A. J. Ashe, D. T. Clark, U. T. Cobley, D. Kilcast, and I. Scanlan, J. Amer. Chem. Soc., 1973,95, 928. D. T. Clark and I. W. Scanlan, J.C.S. Furuday 11, 1974, 70, 1222. M. H. Palmer, R. H. Findlay, and A. J. Gaskell, J.C.S. Perlcin 11, 1974, 420. B. Mely and A. Pullman, Theor. Chim. Actu, 1969,13, 278. A. Pullman, M. Dreyfus, and B. Mely, Theor. Chim. Actu, 1970, 17, 85. R. Bonaccorsi, A. Pullman, E. Scrocco, and J. Tomasi, Theor. Chim. Acta, 1972, 24, 51. E. Clementi, J. M. Andre, M. Cl. AndrB, D. Klint, and D. Hahn, Actu Phys., 1969, 27, 493. E. Clementi, Proc. Nat. Acad. Sci. U.S.A., 1972, 69, 2942. R. Ahrlichs, Theor. Chim. Acta, 1974, 33, 157. M. Barber and D. T. Clark, Chem. Comm., 1970, 23. D. T. Clark and D. M. J. Lilley, unpublished work; D. T. Clark, I. Scanlan, J. Muller, and D. B. Adams, in ref. 12. M. H. Palmer, in ref. 12. B. Pullman and G. N. J. Port, Mol. Phurmucol., 1974, 10, 360. R. E. Christoffersen, ref. 13, p. 357. R. E. Christoffersen, in ref. 10. L. C. Snyder, R. G. Schulman, and D. B. Neumann, J. Chem. Phys., 1970,53, 256. M . Barber and D. T. Clark, Chem. Comm., 1970, 24. G. N. J. Port and B. Pullman, Theor. Chim. Acta, 1974, 33, 275. M. H. Palmer and R. H. Findlay, Tetrahedron Letters, 1972, 4165. I. P. Batra, P. S. Bagus, E. Clementi, and H. Seki, Theor. Chim. Acta, 1974, 32, 279. P. S. Bagus, 1. P. Batra, and E. Clementi, Chem. Phys. Letters, 1973, 23, 305. I. P. Batra and H. Seki, ref. 5, p. 229. H. T. Jonkman, G. A. van der Velde, and W. C. Nieuwpoort, Chem. Phys. Letters, 1974, 25, 62. E. Clementi, J. Mehl, and W. von Niessen, J. Chem. Phys., 1971, 54, 508. L. J, Aarons, J. A. Connor, I. H. Hillier, M. Schwarz, and D. R. Lloyd, J.C.S. Faruduy 11, 1974,70, 1106; D. T. Clark, D. B. Adams, I. W. Scanlan, and I. S. Woolsey, Chem. Phys. Letters, 1974, 25, 263. L. L. Shipman and R. E. Christoffersen, J. Amer. Chem. Suc., 1973, 95, 1408. L. L. Shipman and R. E. Christoffersen, Proc. Nut. Acud. Sci. U.S.A., 1972, 69, 3301. R. E. Christoffersen, J. Amer. Chem. Soc., 1973, 95, 4733. M. Barber, J. A. Connor, M. F. Guest, I. H. Hillier, M. Schwartz, and M. Stacey, J.C.S. Furuduy 11, 1973, 69, 551. J. Almlof, J. Lindgren, and J. Tegenfeldt, J. Mol. Structure, 1972, 14, 427. A. Pullman and G. N. J, Port, Theor. Chim. Acta, 1973, 32, 77. G. N. J. Port and A. Pullman, J. Amer. Chem. SOC., 1973,95,4059. D. W. Genson and R. E. Christoffersen, J. Amer. Chem. Soc., 1973, 95, 362. M. D. Newton, W. A. Lathan, W. J. Hehre, and J. A. Pople, J. Chem. Phys., 1969, 51, 3927. J. A. Ryan and J. L. Whitten, J. Amer. Chem. SOC.,1972, 94, 2396. G. N. J. Port and A. Pullman, Theor. Chim. Acta, 1973, 31, 231. E. Clementi and H. Popkie, Chem. Phys. Letters, 1973, 20, 1. G. C. Lie and E. Clementi, J. Chem. Phys., 1974, 60,3005. H. Basch and A. P. Ginsberg, J. Phys. Chem., 1969, 73, 854.

9 2 52 22 E3 2

s

5:

(3

s iP c,

5 2:

da

db dc ali

de

uf dh

dt

d3 dk

dl li7’l

dn d*

(Ep *q

dr dt

d‘

du

dm

dz dg dz

ea eb

ec ed

ee

ef eg

eh e* ek

References for Table H. Basch, J. Chem. Phys., 1972, 56, 441. H. Basch, C. Hollister, and J. W. Moskowitz, in ‘Sigma Molecular Orbital Theory’, ed. 0. Sinanoglu and K. Wiberg, Yale University Press, New Haven, 1970, p. 449. A. Veillard, Chem. Comm., 1969, 1022, 1427. M.-M. Romer and A. Veillard, J.C.S. Chem. Comm., 1973, 250. I. H. Hillier and V. R. Saunders, Mol. Phys., 1972, 23, 449. S. Evans, M. F. Guest, I. H. Hillier, and A. F. Orchard, J.C.S. Faraday 11, 1974, 70, 417. J. Demuynck and A. Veillard, Theor. Chim. Acta, 1972, 28, 241. J. Demuynck, A. Veillard, and G. Vinot, Chem. Phys. Letters, 1971, lo? 522. I. H. Hillier and V. R. Saunders, Chem. Comm., 1971, 642. M. Barber, J. A. Connor, I. H. Hillier, and V. R. Saunders, Chem. Comm., 1971, 682. I. H. Hillier, M. F. Guest, B. R. Higginson, and D. R. Lloyd, Mol. Phys., 1974, 27, 215. H. B. Jansen and P. Ros, Theor. Chim. Acta, 1974, 34, 85. C . Hollister, J. W. Moskowitz, and H. Basch, Chem. Phys. Letters, 1969, 3, 185, 728; J. W. Moskowitz, C. Hollister, C. J. Hornback, and H. Basch, J. Chem. Phys., 1970, 53, 2570. D. E. Ellis, A. J. Freeman, and P. Ros, Phys. Rev., 1968, 176, 688. H. M. Gladney and A. Veillard, Phys. Rev., 1969, 180, 385. A. J. H. Wachters and W. C . Nieuwpoort, iizternat. J . Quantum Chem., Symp., 1971, 5, 391. A. Veillard, in ref. 12. P. D. Dacre and M. Elder, J.C.S. Dalton, 1972, 1426. J. Fischer, A. Veillard, and R. Wejss, Tlieor. Chim. Acta, 1972, 24, 317. M. B. Hall, M. F. Guest, and I. Hillier, Chem. Phys. Letters, 1972, 15, 592. M. F. Guest, M. B. Hall, and I. H. Hillier, Mol. Phys., 1973, 25, 629. J. A. Connor, L. M. R. Derrick, M, B. Hall, I. H. Hillier, M. F. Guest, B. R. Higginson, and D. R. Lloyd, Mol. PAYS., 1974, 28, 1193. M.-M. Coutikre, J. Demuynck, and A. Veillard, Theor. Chim. Acta, 1972, 27, 281. M.-M. Rohmer, A. Veillard, and M. H. Wood, Chem. Phys. Letters, 1974, 29, 466. J. Demuynck and A. Veillard, Chem. Phys. Letters, 1970, 6, 204. J. Demuynck, A, Veillard, and U. Wahlgren, J . Amer. Chem. SOC.,1973, 95, 5563. B. R. Higginson, D. R. Lloyd, J. A. Connor, and I. H. Hillier, J.C.S. Faraday 11, 1974, 70, 1418. 1. H. Hillier and V. R. Saunders, Mol. Phys., 1971, 22, 1025. M. F. Guest, I. H. Hillier, B. R. Higginson, and D. R. Lloyd, Mol. Phys., 1975, 29, 113. M. F. Guest and I. H. Hillier, Mol. Phys., 1973, 26, 435. J. H. Hall, D. S. Marynick, and W. N. Lipscomb, J. Amer. Cliem. SOC.,1974, 96, 770. I. R. Epstein, T . F. Koetzle, R. M. Stevens, and W. N. Lipscomb, J. Amer. Chem. Soc., 1970, 92, 7019. I. R. Epstein, D. S . Marynick, and W. N. Lipscomb, J. Amer. Chem. SOC., 1973, 95, 1760. D. S. Marynick and W. N. Lipscomb, J. Amer. Chem. SOC., 1972, 94, 8692. I. R. Epstein, J. A. Tossell, E. Switkes, R. M. Stevens, and W. N. Lipscomb, Inorg. Chem., 1971, 10, 171. E. A. Laws, R. M. Stevens, and W. N. Lipscomb, J. Amer. Chem. SOC.,1972, 94,4467. M. F. Guest and I. H. Hillier, J.C.S. Furaday 11, 1974, 70, 2004.

el

am en eo

ev eq

er es et

eu ev

ew ez

ev ez

fa fb

fc

M. F. Guest and I. H. Hillier, J.C.S. Faraday ZZ, 1974,70, 398. I. H. Hillier and V. R. Saunders, Chem. Comm., 1970, 1510. M. Barber, J. A. Connor, M. F. Guest, I. H. Hillier, and V. R. Saunders, Chem. Comm., 1971,943. M. F. Guest, I. H. Hillier, and V. R. Saunders, J.C.S. Faraday 11,1972, 68, 867. I. H. Hillier and V. R. Saunders, J.C.S. Dalton, 1972, 21. E. L. Wagner, Theor. Chim. Acta, 1971, 23, 127. P. Baybutt, M. F. Guest, and I. H. Hillier, Proc. Roy. Sac., 1973, A333, 225. A. Strich and A. Veillard, J. Amer. Chem. SOC.,1973, 95, 5574. J. Del Bene and J. A. Pople, J. Chem. Phys., 1970,52,4858; 1971,55, 2296. L. C. Allen and P. A. Kollman, J. Amer. Chem. SOC.,1970,92, 4108. M. F. Guest, M. B. Hall, and I. H. Hillier, J.C.S. Faraday 11, 1973, 69, 1829. U. Gelius, B. ROOS,and P. Siegbahn, Chem. Phys. Letters, 1970, 4, 471. F. A. Gianturco, G. Guidotti, U. Lamanna, and R. Moccia, Chem. Phys. Letfers, 1971, 10, 269. F. A. Gianturco, Chem. Phys. Letters, 1972, 17, 127. G. L. Bendazzoli, P. Palmieri, B. Cadioli, and U. Pincelli, MoZ. Phys., 1970, 19, 865. A. Johansson, Acta Acad. Aboensis Math. Phys., 1973, 33, No. 2, 1. A. Johansson, P. Kollman, and S . Rothenburg, Theor. Chim. Acta, 1972, 26, 97. H. Basch, J. W. Moskowitz, C. Hollister, and D. Hankm, J. Chem. Phys., 1971, 55, 1922.

03

tr

R

3

5F Pa

180

Theoretical Chemistry

good results for some features, particularly if the exponents and positions of the STOs are all accurately optimized. This, however, is a non-linear optimization and is expensive. For large molecules a viable technique due to Christoffersenlo is to transfer GTOs whose exponents and positions have been optimized for small fragment molecules to the large molecule. This method we designate FSGO. An alternative use of spherical GTOs is to build the lobes of p , d functions from a fixed linear combination of spherical GTOs. This is the so-called lobe GTO method (designated here as CLGTO). In view of the cost of ab initio calculationson large molecules, it seems worthwhile to attempt a full bibliography of such calculations to date. This is given in the Table. The basis set is described only roughly as indicated above. N is the number of electrons in the molecule. Under Author is listed either the fist author or an author whose name allows the work to be related to similar work from the same group. Some smaller molecules are listed if they appear in comparative studies or if they are of particular interest. All simple six-membered-ring organic molecules are excluded, as are the simple oxyanions such as MnO,, Cloy, SO:-, etc., although these have been extensively studied.14 Ab initiu calculations of this type can now be made almost as a routine matter, but they still require large amounts of computer time. The majority of calculations are therefore for well chosen systems where it is believed that an ab initio calculation will resolve differing conclusions from simpler methods of calculation. The calculations on acetylcholinel5~16for example were carried out in an attempt to resolve differing predictions of geometry. Port and Pullman, using an STO-3G (each STO fitted by three GTOs) minimum basis set, predict the gauche-form to be the preferred conformer, in contradiction to some semi-empirical methods and the FSGO results of Christoffersen. They argue that the STO-3G basis set result will be superior to the FSGO one, but in a later survey paper Christoffersen et aZ.17 suggest reasons why this may not be so. This very difficult conformational problem remains unresolved. An attempt to study the effect of solvation on this conformational problem has recently been made18 but only the semi-empirical INDO method was employed. Other calculations on drug molecules have been completed on representative molecules to test semi-empirical conclusions. Inspection of the Table confirms the view that simple organic molecules remain the most popular area for ab initiu calculations. The number of calculations on smaller molecules is of course considerably greater than the number on large molecules. Particularly signiscant is the work of Pople and his co-worker~.~ The majority of such calculations employ a rather small basis set, the so-called STO-3G basis, where each STO in a minimum basis set calculation is fitted to three GTOs. Such calculal4

15

17

(a) H. Johansen, Theor. Chim. Actu, 1974, 32, 273; (6) J. A. Connor, I. H. Hillier, M. H. Wood, and M. Barber, J.C.S. Furuday 11, 1974, 70, 1040; (c) U. Gelius, P. Siegbahn, and B. Roos, Theor. Chim. Acta, 1971, 23, 59; ( d ) A. P. Mortola, H. Basch, and J. W. Moskowitz, Internat. J. Quantum Chem., 1973, 7 , 725. (a) A. Pullman and G . N. J. Port, Theor. Chim. Actu, 1973, 32, 77; (b) G.N. J. Port and A. Pullman, J . Amer. Chem. Soc., 1973,95, 4059. (a) L. L. Shipman and R. E. Christoffersen, Proc. Nut. Acud. Sci. U.S.A., 1972, 69, 3301; (b) D. W. Genson and R. E. Christoffersen, J. Amer. Chem. Soc., 1973,95, 362. R. E. Christoffersen, D. Spangler, G. G. Hall, and G. M. Maggiora,J. Amer. Chem. SOC.,1973,

95, 8526. 18

D. L. Beveridge, M. M. Kelly, and R. J. Radna, J. Amer. Chem. Soc., 1974,96, 3769.

Electronic Calcuhtions on Large Molecules

181

tions are relatively cheap and can be improved by augmenting the basis set to double zeta quality or by adding polarization functions. Surprisingly, calculations at the ab initio MO level can give reliable predictions of small energy differences such as internal rotations. The large errors in such calculations arising from neglect of correlation and relativistic energies and the inadequacies of the basis set must cancel if good results are to be obtained. This appears to happen for methyl rotations. For rotations similar to the rotation in hydrogen peroxide and for inversionsof ammonialike systems more care is required. However, good results with a large basis set have been obtained without the necessity of going beyond the Hartree-Fock approach. This important area will not be reviewed in detail here as a review on ab initio calculations of barrier heights has recently appearedlg and includes a table of such results. More recently attention has been given to energy differences involving bond breaking and bond formation. Typical of this work is a discussion of the acidity of alcohols, substituted acetylenes, phenols, and xylenes by Radom.20 The computed heats, for example, of the reactions

PH ‘X

?H ‘X

X = ForMe agree remarkably well with the experimental values. Many calculations, particularly on transition-metal complexes, have been made by theoretical chemists working in collaboration with experimental photoelectron or ESCA spectroscopists. The aim here is to assist in the identification of the ion states and to test the validity of theoretical models for predicting ionization potentials. The result has been an understanding of these processes which could not have been achieved by semi-empirical methods. The simplest molecular orbital predictor of ionization potentials is given by Koopmans’ theorem,21 which states that the ionization potentials are the negatives of the orbital energies if the ion can be described by the same orbitals as the parent system. This condition, of course, does not hold; the orbitals ‘relax’ on ionization, leading to a correction to the orbital energy known as the relaxation energy. In addition, there may be a contribution from the difference in correlation energy (the error of the Hartree-Fock energy) between the ion and the parent molecule. For heavy atoms and particularly for their coreelectron ionization there will be a term arising from relativistic corrections. The ionization potential for removal of an electron in orbital q5t is thus given by (I.p.)g = - & + AEfrelax) + AEPorr) + AE(’e1) (13) Finally there will be an error in k if the wavefunction is not at the Hartree-Fock limit. If the correction terms are roughly constant for different i, Koopmans’ theorem may still predict the correct ordering of the photoelectron peaks. If it fails, l9

A. Veillard, in ‘Internal Rotation in Molecules’, ed. W. J. Orville-Thomas, Wiley, London,

2o

L. Radom, J.C.S. Chem. Comm., 1974, 403. T. Koopmans, Physicu, 1933, 1, 104.

1974, p. 385. 21

7

182

Theoretical Chemistry

this is called a Koopmans’ breakdown. The correction terms are sometimescalled the Koopmans’ defect. The relaxation energy term can be calculated directly by separate MO calculations on the ion and the parent molecule. The RHF method is used for the ion. This ‘ASCF’ method gives by definition

where Ei and E are the energies of the ion and parent molecules respectively. For many transition-metal compounds a large breakdown in Koopmans’ theorem occurs. Use of the ASCF method leads to an important conclusion. Ionization of an electron from an MO of mainly d-orbital character leads to a substantially larger relaxation energy than ionization of an electron from an MO of mainly ligand character. For example, the relaxation energy of the 6bi and lOai MOs of Fe(C0)2(N0)2 is ca. 1 eV. These MOs have considerable NO character. The relaxation energy of largely Fe d-orbitals, the 6b2 and 3a2, MOs, is 5-6 eV.22 Similar results arise in other work by Hillier and in Veillard’s calculation^.^^ Use of the ASCF allows a correct assignment of the photoelectron peaks when this is not possible using Koopmans’ theorem. The relaxation energy appears to be the most important correction for transitionmetal compounds. Work on small molecules, however, suggests that the problem may be even more complex. All calculations on transition-metal complexes are far from the Hartree-Fock limit. For N2 a breakdown of Koopmans’ theorem occurs at the Hartree-Fock limit, but not at minimum-basis-setleve1.24 The ASCF method also gives the incorrect ordering of the ion states. The correlation energy correction here is the most important term, although in magnitude they are comparable and opposite in sign to the relaxation terms. This problem has been treated by Cederbaum et aZ.25-27 using a many-body perturbation theory method. It would be valuable if this approach could be extended to large molecules, although this may require the evaluation of accurate Hartree-Fock wavefunctions for a few representativesystems. The work of Hillier and Veillard is a good example of what can be achieved by ab initio calculations. Semi-empirical methods, particularly those parametrized to fit experimental data, not only fail to give the understanding achieved from ab initio results, but actually disguise it. Semi-empiricalmethods will be unreliable predictors unless they are firmly based on accurate calculations. It is more difficult to evaluate core ionization potentials accurate in an absolute sense, but the ASCF method appears to give good estimates of the change in I.P. (the ESCA chemical shift) for a range of similar molecules.28 The calculations on higher boranes by Lipscomb and co-workers29-31 are interesting for several reasons. They are the only major group of calculations on large 22

23 24

25 26 27

28 29

30

31

I. H. Hillier, M. F. Guest, B. R. Higginson, and D. R. Lloyd, MoZ. Phys., 1974, 27, 215. M. M. Couti;re, J. Demuynck, and A. Veillard, Theor. Chim. Acta, 1972, 27, 281. P. E. Cade, K. D. Sales, and A. C. Wahl, J. Chem. Phys., 1966, 44, 1973. L. S. Cederbaum, G. Hohlneicher, and W. von Niessen, Chem. Pliys. Letters, 1973, 18, 503. L. S. Cederbaum, Chem. Phys. Letters, 1974, 25, 562. L. S. Cederbaum, G . Hohlneicher, and W. von Niessen, Mol. Phys., 1973, 26, 1405. L. J. Aarons, M. F. Guest, M. B. Hall, and 1. H. Hillier, J.C.S. Furaduy ZZ, 1973, 69, 563. (a) J. H. Hall, D. S. Marynick, and W. N . Lipscomb, J. Amer. Chem. SOC.,1974, 96, 770; (h) 1. R. Epstein, J. A. Tossell, E. Switkes, R. M. Stevens, and W. N. Lipscomb, lnorg. Chem., 1971, 10, 171; (c) D. S. Marynick and W. N. Lipscomb, J . Amer. Chem. SOC.,1972, 94, 8692. (a)I. R. Epstein, T. F. Koetzle, R. M. Stevens, and W. N. Lipscomb, J . Amer. Chem. Soc., 1970, 92, 7019; (6) I. R. Epstein, D. S. Marynick, and W. N. Lipscomb, ibid., 1973, 95, 761. E. A. Laws, R. M. Stevens, and W. N. Lipscomb, J. Amer. Chem. SOC.,1972,94,4467.

Electronic Calculations on Large Molecules

183

molecules employing an STO minimum basis, a number of one-electron properties are calculated, and the delocalized canonical orbitals are transformed into localized orbitals. The latter essentially confirms the ideas on three-centre bonding put forward by Lipscomb many years ago, although some refinement is necessary. The ab initio calculations using the FSGO approach are clearly less accurate than those using large basis sets. They give a poor total energy owing to an inadequate representation of the core orbitals. However, the use of carefully optimized FSGOs in bond regions may account for the generally good performance of this method for answering conformational questions and, because of its simplicity, calculations on large molecules are economic. Recent work with all a&initio methods has strengthened the conclusion that great care needs to be taken with the choice of basis set. Minimum basis sets need to be augmented by the use of double-zeta representations of valence orbitals and the use of polarization functions. d-Orbitals for sulphur and phosphorus atoms are generally found to act as polarization functions. They simply improve some of the deficiencies of the sp basis.32When sulphur or phosphorus is bonded to a more electronegativeatom, the d-orbitals are stabilized and play a more effective part in the bonding.33~34 These calculations have recently become increasingly more feasible because of the ready availability of efficient program packages and of large computers. The available programs now incorporate several recent developments leading to important increases in efficiency. The major problem is handling the large number of electronrepulsion integrals, which is of the order of N 4 , where N is the number of basis functions employed. Several workers have investigated systematic ways of neglecting very small integrals and approximating other small integrals.35-38 Clementi,37 for example, neglects all integrals of value less than 10-7 a.u. and uses a single GTO to approximate integrals in the range 10-7-10-5 a.u. The result is that the time for the calculation on the cytosine-adenine pair is reduced from 200 h to 2 h with no significant loss in accuracy. The use of STOs is now competitive with expansions in terms of GTOs since the problem of evaluating three- and four-centre integrals over STOs has been effectivelysolved. Methods for such integrals are reviewed by Br0wne.5~ Most program packages, however, still use GTOs. The use of symmetry to reduce the number of independent integrals needing evaluation is now common. One aspect of this is discussed by Brailsford and Hylton.40 The evaluation of integrals over GTOs in an efficientmanner is discussed by Ahlrichs,38 Raffenetti,4l and A r e n t ~A . ~care~ ful use of a lobe function basis can give accurate results with a small number of functions (see ref. 43 and references therein). At the SCF stage, improvements are also still being made44in methods for setting 32 33

34

s5 96

37 38 s9

40 41 42 43

44

U.Gelius, B. Roos, and P. Siegbahn, Theor. Chim. Acta, 1972, 27, 171. B. Roos and P. Siegbahn, Theor. Chim. Acta, 1971, 21, 368. U. Gelius. B. Roos, and P. Siegbahn, Theor. Chim. Acta, 1971, 23, 590. V. Dyczmons, Theor. Chim. Acta, 1973, 28, 307. E. Yurtsever and D. Shillady, Chem. Phys. Letters, 1974, 25, 605. E. Clementi, Proc. Nat. Acad. Sci. U.S.A., 1972, 69, 2942. R. Ahrlichs, Theor. Chim. Acta, 1974, 33, 157. J. C. Browne, Ado. Atomic Mol. Phys., 1971, 7 , 47. D. F. Brailsford and J. Hylton, Chem. Phys. Letters, 1973, 18, 595. R. C. Raffenetti, J. Chem. Phys., 1973, 58, 4452. J. Arents, Chem. Phys. Letters, 1972, 12, 489: 14, 292. P. W. Deutsch and A. B. Kunz, J. Chem. Phys., 1973, 58, 1779. A. J. Duke, Chem. Phys. Letters, 1972, 13, 76.

184

Theoretical Chemistry

up the Fock matrix. Convergence problems can be removed by the use of level s h i f t e r ~and , ~ ~several workers have suggested good starting points to cut down the number of iteration^.^^ The time for the SCF stage of an ab initio calculation is now only a small fraction of the integral evaluation time. It is clear that further savings are still possible, so ab initio calculations on large molecules are likely to be even more common in the future. Approximate LCAO Methods.-In view of the expense and generally high call on computer resources of ab initio methods, approximate or semi-empirical methods play a major role in the understanding of large molecules. No attempt, however, will be made here to cover the extensive literature on such applications. In terms of number of articles published the most common methods are those such as CNDO (Complete Neglect of Differential Overlap), INDO (Intermediate Neglect of Differential Overlap), and MINDO (Modified INDO), which are all based on the neglect of integrals in the LCAO-MO scheme. These methods, and some others, are fully covered by the review of Klopman and O’Leary4’ and the books by Pople and Be~eridge,~s Murrell and Harget,49 and D e ~ a r . ~A O recent review by Zerner provides a stimulating discussion of these methods. Approximate methods for inorganic compounds are covered in the book by Doggett 52 and the review by Brown et al.‘ Other older methods are based on the Mulliken or related approximations. Nicholson 53 gives a most stimulating critique for the Mulliken and differential overlap approximations which should be, but regrettably clearly is not, essential reading for anyone devising a new semi-empirical method. It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further 45 46

47

48

49 50

61 52

53

V. R. Saunders and I. H. Hillier, Internat. J. Quantum Chem., 1973, 7, 699. ( a ) E. Clementi, H. Kistenmacher, and H. Popkie, J. Chem. Phys., 1973, 58, 4699; (b) J. H. Letcher, I. Absar, and J. R. van Wazer, Internat. J. Quantum Chem., Symp., 1972, 6, 451; ( c ) B. J. Duke, Chem. Phys. Letters, 1974, 28, 437. G. Klopman and B. O’Leary, Fortschr. Chem. Forsch, (Topics Current Chem.), 1970,15,445. J. A. Pople and D. L. Beveridge, ‘Approximate Molecular Orbital Theory’, McGraw-Hill, New York, 1970. J. N. Murrell and A. J. Harget, ‘Semi-empirical Self-consistent Molecular Orbital Theory of Molecules’, Wiley, London, 1972. M. J. S. Dewar, ‘The Molecular Orbital Theory of Organic Chemistry’, McGraw-Hill, New York, 1969. M. C. Zerner, ref. 5 , p. 117. G. Doggett, ‘The Electronic Structure of Molecules: Theory and Application to Inorganic Molecules’, Pergamon, Oxford, 1972. B. J. Nicholson, Adv. Chem. Phys., 1970, 18, 249.

Electronic Calculations on Large Molecules

185

approximations are made. The success of such methods as MIND0 and some parametrized versions of CNDO and INDO forces us to examine the claims of the fully semi-empirical workers. Unfortunately these workers have not been concerned with a full theoretical analysis of their methods. They are satisfied when good agreement with experiment can be obtained. The result is that very promising results can be obtained for a range of similar molecules to those used for the parametrization. The danger is that there is no theoretical analysis to tell us the limits beyond which the method will give poor predictions. A further danger is that these methods may give us an incorrect or inadequate understanding of experimental phenomena even though they may give good quantitative results. There can be no denying that these methods work in an empirical sense. The theoretical analysis of why this is so is, in the Reporter’s opinion, long overdue, but it is unlikely to be easy. Even the theoretical analysis of methods which aim to mimic ab initio results is incomplete. However, a most interesting and promising start on this problem has been made by Freed.54s55 He attempts to calculate parameters in an ab initio way using the theory of exact sohtions to the Schrodinger equation. His results appear to indicate that such parameters are more complex than they appear to be in methods such as MINDO. What is taken as a single parameter in such theories is, in reality, a range of parameters each of which should be evaluated separately. An extension of this analysis to consider methods actually in use and to devise new methods would be most welcome. Of those methods that aim to mimic ab initio results some are very accurate indeed. The disadvantage is that they may be only marginally more economical than the full ab initio calculation. Such methods are PDDO (Projectors of Diatomic Differential Overlap), developed by Newton and CO-worker~~56-58 and LED0 (Limited Expansion of Diatomic Overlap), devised by Billingsley and Bloor.59 Both methods are essentially ways of accurately approximating some of the multi-centre integrals. Both approximate a two-centre distribution by one-centre distribution according to

dx;

Qi

where QiB is a member of the set and and Q: are members of the sets and $2; respectively. This equation is a generalization of the Mulliken approximation 61

f&p

6oe

or the Lawdin approximationG2

K. F. Freed, (a)J. Chem. Phys., 1974,60, 1765; (b) ref. 13, p. 374; (c) Chem. Phys. Letters, 1974, 24, 275. 55 K. F. Freed, Chem. Phys. Letters, 1972, 13, 181; 15, 331. 56 M. D. Newton, N. S. Ostlund, and J. A. Pople, J. Chem. Phys., 1968, 49, 5192. 57 M. D. Newton, J. Chem. Pftys., 1969, 51, 3917. 58 M. D. Newton, W. A. Lathan, W. J. Hehre, and J. A. Pople, J. Chem. Phys., 1969, 51, 3927. 5 9 F. P. Billingsley, tert. and J. E. Bloor, J. Chem. Phys., 1971, 55, 5178, 60 R. S. Mulliken, J. Chim.phys., 1949, 46, 497. 61 K. Ruedenburg, J. Chem. Phys., 1951, 19, 1433, 6% P.-0. Liiwdin, J, Chem, Phys., 1953, 21, 374, 54

Theoretical Chemistry

186

A similar expansion, originated by Sklar, 63 uses one-centre distributions centred between the atoms A and B. Body64 has developed ideas similar to PDDO and LEDO using this approach. PDDO uses a least-squares criterion for finding the coefficients G Pand Gqand augments the sets Qiand "9" by using basis functions on A and B not required for the molecular orbitals. LED0 uses only the basis functions required for the MOs and finds the coefficients by the condition that twocentre hybrid integrals of type (Qil 1/r121ntB)must be given exactly. These conditions are sufficient to find the coefficients as solutions of a set of linear equations which, however, can be ill-conditioned. Both methods are quite accurate but have many disadvantages. They are difficult to apply to large basis sets since the number of coefficients to be determined is too large. They are both only marginally cheaper than full ab initio methods where the integrals are consistently determined by expanding STOs by a limited expansion of GTOs. The latter have the advantage of being variational and are thus classified as ab initio,unlike PDDO and LEDO. There are few applications to large molecules. Of comparable or better accuracy, but much faster, is the PRDDO (Partial Retention of Diatomic Differential Overlap) method proposed by Halgren and Lipscomb.65 This method uses the exact one-electron matrix over STOs transformed to an orthogonal atomic orbital (OAO) basis.66 Electron-repulsion integrals are systematically approximated using exact values of some integrals over the original STQ basis. The method recognizes three basis sets: B

s-112

@(STQ)+$(AO) +z(OA8)

(18)

The matrix B transforms the STO basis to an A 8 basis. The 2s-functions are Schmidt orthogonalized to the 1s-functions, and 2pfunctions are aligned along the local atomic principal axis. S-l/z (S is the overlap matrix) is the usual Lowdin orthogonalization. The following approximations are made: (a)

($I$) (x;lx3

is evaluated using the identity =

(4;M:) + cx;-c:l4:)

+ C4;Ix;-QZy) + Cx",4;lx;-K) (19)

but neglecting the last term and approximating the second and third terms by as appropriate: replacing #2 by a local spherical average, $ 2 = or

@is

This equation is used for x, and xv on the same or different centres. The terms involving x and 0 occur below. (b) ( x p ~ , l x ~ for ) , x,, %?and X A on same or different centres, are all retained and evaluated as (x,J,]@;). Defining (xclxylm:) and (@,0J~~) as the pv terms of the matrices Qx(Si)and Q Q @ ~we ) have

Q,@;) 63 64 85 86

= RQd$:)RT

(21)

A, L. Sklar, J. Chem. Phys., 1939, 7 , 984. R. G. Body, Theor. Chim. Acta, 1970, 18, 107.

T. A. Haigren and W. N. Lipscomb, Proc. Nat. Acad. Sci. U S A . , 1972,69,652; J. Chem. Phys., 1973,58, 1569. P.-0. Lowdin, J. Chem. Phys., 1950, 18, 365,

Electronic Calculations on Large Molecules

187

where

R

= S-l/zB

(22)

The diagonal elements of Qx(T5i) are required for integrals of the first category. (c) Other integrals ~ ~ , , I x A xare U )neglected except for one-centre and two-centre terms where x, = XA and x,, = xu. Several expressions are used for these integrals. The original referenceG5 should be consulted for details. The method has been employed in both a non-parametrized form and in a parametrized version. The parametrization involves scaling of some integrals. A similar idea was imployed originally by Cooka7 and later used in a calculation on sF6.68 The errors in PRDDO have been carefully considered in a large number of comparisons for small molecules against ub initio results.66 Results are significantly better than for other semi-empirical schemes. The method has been compared with ab initio results for a series of large boranes 29 and applied to very large boranes such as BlS&O, Bl8H2ZY i-Bd22, B20Hi8, photo-BzoHq;, and B20H16.~~ This method appears to combine well the conflicting problem of accuracy and economy for reproducing the results of ub initio LCAO-MO calculationsfor molecules containing first-row atoms and hydrogen. Accurate mimicking of ab initio results has been achieved by some workers using mixed or merged basis sets6', 70, 7 l but the method does not appear to have been developed since an earlier review.3b The essential idea here is to use an accurate basis for one and two-centre integrals and a less accurate basis for three- and four-centre integrals. The method clearly merges into the use of adjoined basis sets as discussed earlier.37The latter is truly ub initio since the errors are kept well under control. The former may give results substantially in error and is not variational. The range of zero differentialoverlap methods such as CNDO, INDO, and NDDO originally suggested by Pople et aL72continues to have wide use in spite of the fact that they are less accurate than the methods just discussed. Of these NDDO might be expected to be the most accurate but there has been surprisingly little use of this approach. NDDO neglects integrals by the use of The method has been used to mimic ub initio results and by parametrization to fit experimental values. Roby 7 3 has given the method some theoreticaljustifications in the former sense. He shows that if a complete set of functions is used on each centre, retaining only the NDDO integrals over an A 0 basis is equivalent to using all the integrals over an OAO basis. One-electron integrals can be evaluated over the OAO basis to give, in principal, a consistent exact method. The basis set, however, cannot be complete and it is not clear what errors are introduced by truncation. NDDO D. B. Cook, P. C. Hollis, and R. McWeeny, Mol. Phys., 1967, 13, 553. G. L. Bendazzoli, P. Palmieri, B. Cadioli, and U. Pincelli, Mol. Phys., 1970, 19, 865. eQ D. A. Dixon, D. A. Kleier, T. A. Halgren, and W. N. Lipscomb, J. Amer. Chem. SOC.,1974,96,

67

68

2293. 70

71

7z 73

(a) D. B. Cook and P. Palmieri, Mol. Phys., 1969, 17, 271 ; (b) D. B. Cook, P. D. Dacre, J. L. Dodds, and M. Elder, Theor. Chim. Actu, 1971, 22, 167. (a) R. D. Brown, F. R. Burden, and B. T. Hart, Theor. Chim. Acra, 1971, 22, 214; (b) F. R. Burden and B. T. Hart, Austral. J. Chem., 1973, 26, 1395. J. A. Pople, D. P. Santry, and G. A. Segal, J. Chem. Phys., 1965,43, S129. K.R. Roby, Chem. Phys. Letters, 1971, 11, 6.

188

Theoretical Chemistry

remains a family of closely related methods and no particular approach is clearly superior to the others. The majority of calculations74-7gare for small molecules. The study of reactivity of ambenzenes79 concludes that CNDO is unsuccessful for understanding phenomena depending on lone pairs owing to the spherical averaging of integrals. NDDO is more successful. Similar conclusions were reached by Jesaitis and Streitweiser,78 who also developed a simpler method, IRDO (Intermediate Retention of Differential Overlap), in which NDDO terms are retained for directly bonded atoms and CNDO terms used for other atom pairs. There has been an NDDO study80 of CIFs and an interesting comparative study of NDDO and other methods. 81 The CNDO and INDO methods, which are computationally cheaper than NDDO, remain very popular methods, perhaps because of the wide availability of computer programs for CNDO/INDO and M"D0. No attempt will be made to cover the wide range of applications of these approaches. The CNDO/S method of Del Bene and Jaff6,S2which is parametrized to deal with electronic spectra and uses limited configuration interaction for excited states, has had several extensions, to boron-, fluorine-, and chlorine-containing molecules,83 to doublet states,*4 and to triplet states.85 There have been many applications, including substituted benzenes,86-88 porphyrins,sg and n a ~ h t h a l e n eThere . ~ ~ has also been a study of its use for interpreting photoelectron spectra.g1 CNDO and INDO are no longer restricted to molecules containing first-row atoms. Sulphur- and phosphorus-containing heterocycles have been studied by Schulte and S ~ h w e i gParameters .~~ for Ge, As, Se, and Br have been given.93 Both methods have had applications to transition-metal complexes.94-g6 Voigt 97 has proposed a new method intermediate between INDO (a) R. B. Davidson, W. L. Jorgensen, and L. C. Allen, J. Amer. Chem. SOC.,1970, 92, 749; (b) P. A. Kollman and L. C. Allen, ibid., p. 753; ( c ) R. Sustmann, J. E. Williams, M. J. S. Dewar, L. C. Allen, and P. von R. Schleyer, ibid., 1969, 91, 5350. 7 5 (a) H. J. Koehler and F. Birnstock, 2. Chem., 1973, 13, 29; (b) H. J. Koehler, ibid., p. 157; (c) M. J. S. Dewar and C. A. Ramsden, J.C.S. Chem. Comm., 1973, 688. 713 (a) R. D. Brown, F. R. Burden, and G. R. Williams, Theor. Chim. Actu, 1970,18,98; (b) R. D. Brown, F. R. Burden, G. R. Williams, and L. F. Phillips, ibid., 1971, 21, 205. 7 7 R. D. Brown and K. R. Roby, Theor. Chim. Actu, 1970, 16, 175. 78 R. G. Jesaitis and A. Streitweiser, jun., Theor. Chim. Acta, 1970, 17, 165. 79 S. Birner, H. J. Kohler, and C. Weiss, Chem. Phys. Letters, 1974, 27, 165. so A. A. Bugatur'yants, N. M. Limenko, and M. E. Dyatkina, Zhur. strukt. Khim., 1973,14, 757. 81 A. Golebiewski, J. Mrozek, and R. Nalewajski, Theor. Chim. Actu, 1973, 28, 169. s2 J. Del Bene and H. H. Jaff6, J. Chem. Phys., 1968, 48, 1807. 83 G . Kuehnlenz and H. H. Jaffk, J. Chem. Phys., 1973, 58, 2238. g4 H. M. Chang and H. H. Jaffk, Chem. Phys. Letters, 1973,23, 146. 85 H. H. Jaff6, H. M. Chang, and C. A. Masmanidis, J. Comput. Phys., 1974, 14, 180. 86 C. Sieiro and J. I. Fernandez-Alonso, Chem. Phys. Letters, 1973, 18, 557. 87 G . W. King and A. A. G. Van Putters, J. Mol. Spectroscopy, 1972, 42, 514. 88 J. S. Yadav, P. C. Mishra, and D. K. Rai, Mol. Phys., 1973,26, 193; Chem. Phys. Letters, 1973, 19, 445. 89 G. M. Maggiora and L. J. Weimann, Chem. Phys. Letters, 1973, 22, 297. R. L. Ellis and H. H. Jaffk, J. Mol. Spectroscopy, 1974, 50, 474. 91 R. L. Ellis, H. H. Jaffk, and C. A. Masmanidis, J. Amer. Chem. SOC.,1974, 96, 2623. Q2 K. W. Schulte and A. Schweig, Theor. Chim. Acta, 1974, 33, 19, 93 H. L. Hase and A. Schweig, Theor. Chim. Acta, 1973, 31, 215. 94 (a) D. W. Clack, Mol. Phys., 1974, 27, 1513; (b) D. W. Clack, N. S. Hush, and J. R. Yandle, J. Chem. Phys., 1972,57, 3503. 95 (a) D. A. Copeland and C. J. Ballhausen, Theor. Chim. Acra, 1970,20, 317; (b) D. A. Copeland, ibid., 1973, 32, 41. T. Ziegler, Acta Chem. S c a d . , 1974, A28, 29. Q7 B. Voigt, Theor. Chim. Acta, 1973, 31, 289. 74

Electronic Calculations on Large Molecules

189

and NDDO based on the decomposition of integrals in multipole-multipole-type interactions. The MIND0 approach of Dewar and co-workers has been further refined as the MIND0/3 method:* although full details have not as yet been published. This method is, of course, firmly of the type that aims to give results of ‘chemical accuracy’, superior to Hartree-Fock results even though only a single-determinant wavefunction is apparently employed. It is regrettable that this approach is either enthusiastically welcomed or scorned almost entirely as a pure matter of opinion. Experimental organic chemistsfall largely into the former camp while many theorists are in the latter. MIND0/3, even more so than MIND0/2, appears to contain so many parameters that it is impossible to explain its undoubted successes or be clear where it will fail. This Reporter agrees with a nicely worded criticism of MIND0/3 by Das Gupta and Huzinaga? ‘The method (MIND0/3) contains a large number of parameters and ad hoc conventions, and is a far cry from the original concept of the semi-empirical theory as proposed by Pariser and Pam, and Pople. Rather, the present authors are tempted to regard the approach typified by the MIND0/3 as a pseudo-quantum-mechanicalmodel exquisitely built to simulate closely certain selected aspects of real molecular systems. Within the safe range of the applicability, such a method can be very useful and attractive for experimental chemists. It is to be remembered, however, that through a calculation by the method one is looking into a model world which would hopefully behave just like a real world for one’s chosen interest’. It is of course true that Hartree-Fock theory is also a model world, unlike the real world, but its defects are becoming increasingly well understood. This can not yet be said about MIND0/3, and theorists will continue to be suspicious until a theoretical analysis supports its undoubted success in predicting experimental values. One attractive aspect of the MIND0/3 work is that the method is closely associated with efficient routines for full geometry optimization. A study of annulene,Qsa for example, carries out a full geometry optimization of 120 internal co-ordinatesand predicts the molecule to be planar with Dm symmetry and alternating bonds. The routine use of such techniques with other semi-empirical methods would be welcome. A recent development of CNDO theory99 gives some possibility that the dilemma posed by Klopman and O’Leary47 of having to choose between methods which give good heats of formation and poor bond distances or good bond distances and poor heats of formation may be solved by a fairly simple semi-empirical method. The authors point out that since the one-electron and two-electron contributions to the energy are of opposite sign, approximations in the two-electron integrals need to be carefully compensated by approximations in the one-electron terms. A new, versatile formula for computation of Coulomb integrals is presented and encouraging results are found for hydrocarbons. The idea of transferring data from ub initio calculations on small molecules for use in approximating the wavefunction on large molecules has been revived. This idea 98

99

(a) M. J. S. Dewar, R. C. Haddon, and P. J. Student, J.C.S. Chem. Comm., 1974, 569; (b) M. J. S . Dewar, R. C. Haddon, and S. Ho Huck, ibid., p. 611 ;(c) M. J. S.Dewar, R. C. Haddon, and R. K. Weiner, J. Amer. Chem. Soc., 1974,96,253; (d) M. J. S. Dewar and R. C. Haddon, ibid., p. 255; (e) M. J. S. Dewar and S. Kirschner, ibid., pp. 5246,6809; (f)M. J. S. Dewar and A. C. Griffin, ibid., p. 6225; ( g ) M. J. S. Dewar and W. K. Li, ibid., p. 5569. A. DasGupta and S. Huzinaga, Theor. Chim. Acta, 1974, 35, 329.

190

Theoretical Chemistry

goes back to the n-electron work of Orloff and FittslOO who approximated PariserParr-Pople wavefunctions for large molecules by transferring matrix elements over the effective one-electron Hamiltonian from small molecules. The non-empirical molecular orbital (NEMO) method of Lipscomb and co-workerslol applied similar ideas to all-electron calculations by transferring diagonal Fock matrix elements from ab initio calculations on small molecules. Recent methods differ in two important respects. Firstly both diagonal and off-diagonal matrix elements are transferred. Secondlythe small molecules, designated ‘pattern’molecules, are chosen much morecarefullyto fit the environment and geometryof a particular regionof the large molecule or ‘target’ molecule. For this reason the methods have to date been restricted to organic molecules. The whole of this area of transferability in molecular orbital theory is the subject of a forthcoming review102 and will therefore only be dealt with briefly in this Report. Two separate but related approaches have been used. The simulated ab initio (SAMO) methodlo3uses a basis set of hybrid orbitals. Both diagonal and off-diagonal Fock matrix elements are transferred directly from a set of pattern molecules which have been carefully chosen to fit regions of the target molecule. Pattern molecules must be large enough to give matrix elements between orbitals which are up to four atoms (C, N, 0, etc.) apart in a chain. Butane, for example, is sufficient to simulate the n-alkyl chains in molecules. A normal alcohol would require butane and propanol. Matrix elements not present in any pattern molecule are neglected. The basis set is non-orthogonal so the overlap matrix is evaluated directly. The total electronic energy can be evaluated without recourse to the two-electron integrals by use of the third expression in equation (12). The SAMO method has had several applications to organic molecule^.^^^-^^^ It has been applied to simple polymersl06 and extended to certain open-shell radicals in both the UHF107and RHFl** approaches. Leroy and co-workers utilize the idea of transferring both matrix elements and localized orbitals. The MOs are expanded as a linear combination of orthogonal localized orbitals. Fock matrix elements over this basis set are not transferred directly from particular pattern molecules. Each type of interaction is obtained as an average value of this type of matrix element in a range of small ‘pattern’ molecules. This method has been applied so far to a number of alkanes109 and alkenes.ll0 The major difficulty in both the SAMO method and Leroy’s approach is collecting together the necessary matrix elements. For large molecules the number of such terms is not insignificant when it is considered that a separate decision on each is required. What type of interaction? Which pattern molecule? What value? In the M. K. Orloff and D. R. Fitts, J. Amer. Chem. Suc., 1963, 85, 3721. (a) M. D. Newton, F. P. Boer, and W. N. Lipscomb, J. Amer. Chem. Soc., 1966, 88, 2353, 2367; (b) F. P. Boer, M. D. Newton, and W. N. Lipscomb, ibid., p. 2361 ; (c) D. B. Boyd and W. N. Lipscomb, J. Chem. Phys., 1968, 48, 4955; ( d ) J. A. Tossell and W. N. Lipscomb, J. Amer. Chem. SOC.,1972, 94, 1505. 102 B. O’Leary, B. J. Duke, and J. E. Eilers, Ado. Quantum Chem., 1975, 9, 1. 1 0 3 J. E. Eilers and D. R. Whitman, J . Amer. Chem. SOC.,1973, 95, 2067. I o 4 B. J. Duke and B. O’Leary, in ref. 12. 105 B. O’Leary, B. J. Duke, J. E. Eilers, and E. W. Abrahamson, Nature, 1973, 246, 166. 106 B. J. Duke and B. O’Leary, Chem. Phys. Letters, 1973, 20, 459. lo’ B. J. Duke, J. E. Eilers, and B. O’Leary, J.C.S. Faraday 11, 1974, 70, 386. 108 B. J. Duke, M. Collins, J. E. Eilers, and B. O’Leary, unpublished work. 109 Ph. Degand, G. Leroy, and D. Peeters, Theor. Chim. Acta, 1973, 30, 243. 110 G. Leroy and D. Peeters, Theor. Chirn. Acta, 1974, 36, 11.

100 101

Electronic Calculations on Large Molecules

191

SAMO method some success has been achieved in making this process automatic.111 This problem in Leroy’s method is less severe as simple lists of matrix-element parameters for particular types of interaction can be made. The number of separate values is small. To compensate for this advantage, however, Leroy’s method is less accurate than the SAMO technique. Both Leroy and co-workers and the SAMO workers have studied a related approach. Here a basis set of hybrid orbitals is first transformed by a Lowdin symmetric orthogonalization. Matrix elements over the new basis set are transferred. Duke and 0’Learylo4hoped that by this transformation the small matrix elements which have to be neglected in the normal SAMO method would be even smaller. Surprisingly this is incorrect, and the use of the orthogonal basis set gives results which are in most respects slightly worse than the SAMO method itself. Leroy’s mFthod,112 again using a simple list of averaged terms, gives results comparable to his method using a localized orbital basis set. Transferability of localized orbitals has been studied and utilized by several groups. Perhaps the most important is the ‘molecules-in-molecules’ (MIM) method devised by von Niessen.l13 This approach aims to construct the wavefunction for a large molecule by using the localized orbitals of separate fragments. It is particularly well adapted to dealing with simple dimers such as hydrogen-bonded systems. The wavefunction is the usual Slater determinant of one-electron orbitals, some of which are localized orbitals transferred directly from the separate fragments and others are MOs which are to be optimized in the new environment of the dimer. The problem is that this optimization requires the use of most, if not all, of the two-electron repulsion integrals. The saving of time over the full ab initio treatment, of which the MIM method is an approximation, is disappointing. The method does, however, give some interesting information about the changes in orbitals on dimer formation. 3 The X, Exchange Approximation The SCF-X, Scattered-waveMethod.-An alternative to the LCAO-MO method for molecules is the X , scattered wave or multiple-scattering X , method, which was suggested less than 10 years ag0114 and now has an extensive literative. The basic theory has been refined in several directions115and there are several excellent reviews.11e-120 The method is based on the scattered-wave method in solids121 and aims to solve the Fock equation, F@t = A& 111 112

113

114

115 116 117 118 119 12* 121

(24)

B. O’Leary, B.J. Duke, and J. E. Eilers, Program 263 (SAMOS), Q.C.P.E., Bloomington,Indiana. A. Deplus, G.Leroy, and D. Peeters, Theor. Chirn. Acta, 1974, 36, 109. W.von Niessen, J. Chem. Phys., 1971,55, 1948; Theor. Chim. Actu, 1973,31, 111,297; 1973, 32, 13 ; 1974, 33, 7. (a) J. C. Siater, J. Chem. Phys., 1965, 43, S228; (b) K. H. Johnson, ibid., 1966, 45, 3085. (a) J. C. Slater and K. H. Johnson, Phys. Rev. (B), 1972,5, 844; (b) K. H. Johnson and F. C. Smith, jun., ibid., p. 831. K.H.Johnson, J. G. Norman, and J. W. D. ConnoIly, ref. 5, p. 161. J. C. Slater, Adv. Quantum Chem., 1972, 6, 1. K. H.Johnson, Adv. Quantum Chem., 1973, 7, 143. K. H.Johnson, ref. 6, p. 332. K. H. Johnson, Internat. J. Quantum Chem. Symp., 1973, 7 , 347. (a) J. Korringa, Physica, 1947, 13, 392; (b) W . Kohn and N. Rostoker, Phys. Rev., 1954,94, 1111.

192

Theoretical Chemistry

directly but using an approximate potential in F. The Fock operator F can be expressed as

P = ? + - t c + v E

(25)

where ?,‘ VC, and ?E are the kinetic energy, Coulomb potential, and exchange potential respectively. $‘c, evaluated classically, is the potential due to the nuclei charge and the electronic density p, given by

where Ni is the occupation of the ith spin orbital @i. p(1) can be divided into terms due to spin orbitals of a and /3 spin, p 7 (1) and p 1 (1). The exchange term is given by the X, approximation

with a similar expression for the spin-down term. The value of the parameter cc can be chosen in a number of ways. In order to solve the Fock equations, the molecule is divided into a number of regions, typically: (a) atomic regions of non-overlapping spheres; (21) interatomic region between the atomic spheres and an outer sphere; (c) extramolecular region outside the outer sphere. The potential terms are spherically averaged inside each atomic region, assumed constant in the interatomic region equal to the volume average, and spherically averaged in the extramolecular region. With these approximations the spin orbitals can readily be found using an iterative scheme starting from Herman-Skillman 122 free-atom potentials. This method has some important differences from the usual Hartree-Fock scheme. The total energy is given by the Slater statistical total energy EX,. The eigenvalues of the Fock equation are the first derivative of EX, with respect to occupation numbers and not, as in the Hartree-Fock scheme, negative ionization potentials. Koopmans’ theorem therefore does not hold, and ionization potentials are evaluated by separate calculations of eigenvalues for states with occupation numbers of 3,the so-called transition states. This procedure takes relaxation into account. A further improvement of this method over the usual Hartree-Fock scheme is that the statistical energy EX, dissociates correctly into the energy of separate atoms. The eigenvalues of virtual orbitals are, like the occupied orbitals, calculated for a field of (n- 1) electrons. They thus differ from the Hartree-Fock orbital energies for virtual orbitals, which are appropriate to a field of n electrons, and they are therefore more suitable for describing excited states. Because of these features the X , method has had many successful applications to large molecules, such as complex ions, metal clusters, and biologically interesting molecules. The review by Johnson et aZ.116 should be consulted for details and references. A recent application to transition-metal complexes123 gives good agreement with experimental F. Herman and S. Skillman, ‘Atomic Structure Calculations’, Prentice-Hall, Englewood Cliffs, N.J., 1963. S. Larsson and J. W. D. Connolly, J. Chern. Phys., 1974, 60, 1514.

laz lZ3

Electronic Calculations on Large Molecules

193

e.p.r. and n.m.r. measurements. Waber and Averill124 have illustrated the potentiality of the method for heavy atoms by a comparative calculation on PtFs and the unknown hexafluoride of element 110. Relativistic effects are discussed but not introduced into the method. The main advantages of the X , method are (a) it is economical, since there is no N4 problem of molecular integrals, (6) it gives a good description of charge densities, orbital orderings, ionization energies, optical excitation, and other one-electron properties, (c) it is particularly appropriate for studying impurity clusters in a solid environment and the effect of surrounding ions in solution can readily be incorporated, and (d)it appears possible to introduce approximate relativistic effects for large molecules. The method does, however, have some disadvantages. For many excitation processes the method only gives an average energy of states of differing multiplicity. It gives poor molecular geometries where lone pairs are involved. Water, for example, is predicted to be linear 125 unless additional ‘atomic’regions are introduced to describe the lone pairs. Similarly,a rather poor description is given for the barriers in ammonia and hydrogen peroxide.126 Its advantage of economy has been challenged by Clementi et aZ.127They show that a highly optimized LCAO-MO ab initio calculation can give a prediction of the barrier to rotation in ethane comparable to that given by the X , method128using less computer time. The real question, however, is a comparison for large molecules, where the Xu method does appear to be more efficient. A calculation on SF6 for example used only 6 min of IBM 360165 t h e in contrast to 20-30 h on a similar machine for an ab initio calculation. It must also be doubted whether the Xu computer programs have yet reached the high level of sophisticated optimization present in Clementi’s IBMOL 5 program for ab initio calculations. Indeed Johnson 116has already reported a reduction by a factor of two in the time for the above calculation. For small molecules, where detailed tests are possible, the X , method has some significant failures. For N2, 02, and F2, for example, the predicted dissociation energies are too small and the predicted internuclear distances too large.129However, for F2 the X , method does, unlike the Hartree-Fock method, predict a bound molecule. The ab iititio bond lengths for these molecules are substantially better than the X, results. These deficiencies may be due to the use of spherical regions - the socalled ‘muffin-tin’approximation. There have been several attempts to remove this restriction. The use of ellipsoids in place of spheres has been suggested by Kaufmann.130She also suggests the use of charge-densitycontour maps to calibrate sphere sizes and to transfer these from ab initio LCAO-MO calculations on small molecules for use in X , calculations on large molecules. Non-muffin-tin corrections have been discussed by Danese and Conn01ly.l~~ The use of overlapping spheres has been considered132 and found to give improved results for n-electron compounds such as 1Z4

125 126 127 128

1Z9

I3O 131

J. T. Waber and F. W. Averill, J. Chem. Phys., 1974, 60,4466. J. W. D. Connolly and J. R. Sabin, J. Chem. Phys., 1972, 56, 5529. U. Wahlgren, Chern. Phys. Letters, 1973, 20, 246. E. Clementi, H. Kistenmacher, and H. Popkie, J. Chem. Phys., 1973, 58, 4699. U. Wahlgren and K. H. Johnson, J. Chem. Phys., 1972,56, 3715. P. Weinberger and D. D. Konowalow, Internat. J. Quantum Chem., Symp., 1973, 7 , 353. J. J. Kaufmann, Internat. J. Quantum Chem., Symp., 1973, 7 , 369. J. B. Danese and J. W. D. Connolly, Internat. J. Quantum Chem., Symp., 1973, 7 , 279. N. Rosch, W. G. Klemperer, and K. H. Johnson, Chem. Phys. Letters, 1973, 23, 149.

194

Theoretical Chemistry

benzene. In a generalization of the basic theory, Johnson 133 had already considered an alternative approach to overlapping potentials. The overlapping-spheres method is used in a calculation on ferrocene.134 Recently the use of arbitrarily shaped volumes has been ~onsidered.l3~ The weaknesses of the X , approximation itself have been considered by Slater.136 He gives reasons why the present method should be poorer for small atoms than for larger ones and offers suggestions for improvement. Diamond 137 derives the secular equations in symmetrized form. Other Related Methods.-Baerends and Ros have developed a method suitable for large molecules in which the LCAO form of the wavefunction is combined with the use of the X, approximation for the exchange potential. The method makes use of the discrete variational method originally proposed by Ellis and Painter.l3* The oneelectron orbitals are expanded in the usual LCAO form and the mean error function is minimized.

with

~ ( ris) a positive weight function and the summation is over a discrete set of sample points pk. This leads to a secular equation,

HC

=

ESC

(29)

where the matrix elements are now sums in place of integrals:

The method is obviously a numerical approximation to the usual integrals. The Hamiltonian K, given by

is derived in the usual way by applying the variation principle to the statistical total energy. It is the use of the exchange potential ^VX, which simplifiesthe method and yet requires the above numerical integration since analytical integration over the 133

13* 135

13’

138

K. H. Johnson. Internat. J. Quantum Chem., Symp., 1971, 4, 153. N. Rosch and K. H. Johnson, Chem. Phys. Letters, 1974, 24, 179. L. Scheire and P. Phariseau, Chem. Phys. Letters, 1974, 26, 149. J. C. Slater, Internat. J. Quantum Chem., Syrnp., 1973, 7 , 533. J. B. Diamond, Chem. Phys. Letters, 1973, 20, 63. (a) D. E. Ellis and G . S. Painter, Phys. Rev. ( B ) , 1970,2, 2887; in ‘Computational Methods in Band Theory’, Plenum Press, New York, 1971, pp. 271, 277; (b) G. S. Painter and D. E. Ellis, Phys. Rev. ( B ) , 1970,1,4747.

Electronic Calculations on Large MolecuIes

195

+power of the density leads to difficulties. The method is described in detail by Baerends, Ellis, and R O S . ~ ~ ~ Applications140on Tic14 and vCl4 use a non-SCF procedure. The Coulomb and exchange potentials are determined from a superposition of neutral atom densities, The application141 to TiF5(C2H4) and NiFs(C2H4) uses an approximate SCF procedure. The complete method is tested on a number of small systems by Baerends and Ros.142, 143 One very interesting result reported here is worth mentioning, although for the very small molecule N2. The transition-state ionization energies predict the correct order of the highest nu and ogion states. As mentioned previously, both Koopmans’ theorem and the ASCF method at the ab initio Hartree-Fock level predict the incorrect order of these states. The use of the X , potential appears to be introducing some correlation effects in addition to the relaxation terms introduced through the transition states. Results obtained by this method have been compared with the X, scattered-wave results for FeC14 ani0ns.1~~ Results for ferrocene compare well with ab initio results and experiment.145 They appear to give better ionization potentials than the ab initio results of Veillard.23 This method has several clear advantages. The use of the LCAO form avoids the muffin-tin potentials and may make the method more easily visualizable for chemists. The numerical integration imposes no restriction on the form of the basic functions. STOs can be used as readily as CGTOs. The N4 problem is avoided since only N2 matrix elements need be stored. Anderson and Woolley146 have proposed a linear combination of muffin-tin orbitals method for molecules based on the earlier cellular rnode1.14’ No applications to large molecules have been reported. All these methods, based on methods used for the band theory of solids, appear to have a most welcome and useful application to large molecules. At present they probably give, for transition-metal complexes, the most worthwhile method other than the ab initio approach. Even this latter qualification is disputed by some workers.

4 Beyond the Molecular Orbital Approach Introduction.-In principle an exact solution to the non-relativistic Schrodinger equations for a molecule can be achieved by the configuration interaction technique. A complete set of one-electron spin orbitals $t is used to form a complete set of Slater determinants by choosing all possible ordered sets of n elements of the set of 4i’s. A linear combination of these determinants is then used: =

c.

jl&.

where tpjl,

.j,

c j l ,3 2 ,

*

--

j n y j 1 , j2,

--

3n

(32)

. . . f n is the Slater determinant formed from 4j1,$j2,. ..4jn.

j2,

E. J. Baerends, D. E. Ellis, and P. Ros, Ckem. Phys., 1973, 2, 41. 140 D. E. Ellis and T. Parameswaran, Internat. J. Quantum Chem., Symp., 1971, 5, 443. 1 4 1 T. Parameswaran and D. E. Ellis, J. Chem. Phys., 1973, 58, 2088. 142 E. J. Baerends, D. E. Ellis, and P. ROS, Theor. Chim. Acta, 1972, 27, 339. 143 E. J. Baerends and P. Ros, Chem. Phys., 1973, 2, 52. 144 D. E. Ellis and F. W. Averill, J. Chem. Phys., 1974, 60, 2856. 145 E. J. Baerends and P. Ros, Chem. Phys. Letters, 1971, 23, 391. 1413 0. K. Anderson and R. G. Woolley, Nol. Phys., 1973, 26, 905. 147 (a) H. S. Fricker and P. W.Anderson, J. Chem. Phys., 1971,55, 5028; (b) H.S. Fricker, ibid., 139

p. 5034.

Theoretical Chemistry

196

In practice, of course this equation must be truncated. It is then convenient to describe the expansion as

where yo is the Hartree-Fock determinant, y({) is the determinant formed by replacing the Hartree-Fock occupied orbital $8 by a new orbital $ j , y@) is formed by replacing the Hartree-Fock orbitals $t and $5 by two new orbitals $k and $z,and so on. In practice even this scheme in the ab initiu approach is limited to small molecules. There have been few applications to large molecules, benzene148 being the largest molecule for which an extensive expansion has been used. Semi-empirical methods can, however, be employed. The PCILO Method.-Of all semi-empirical methods, the PCILO (Perturbative Configuration Interaction over Localized Orbitals) method is the most important. Indeed it can claim to be the single most important tool of the quantum biochemist. It is faster than CNDO and appears in general to be more accurate. It is therefore surprising that the method has been virtually neglected in recent general reviews of the field of large-moleculecalculations. The original theory is due to Diner, Malrieu, and Claveriel49 whose paper references the main background work. It was then applied to n-electron systems150 using the Pariser-Parr-Pople approximations, but its main applications have been in the all-valence-electron framework using the CNDO approximation.151 The starting point is a single determinant built not from molecule orbitals but from localized bonding orbitals. These, in turn, are constructed from hybrid orbitals taken in pairs. A set of localized antibonding orbitals which are optimized are used to form excited-state configurations. This set of co&gurations is then used as a basis for configuration interaction. Rayleigh-Schrodinger perturbation theory is used up to third order to determine the energy and wavefunctions. The zero-order function is the single determinant yo,

where 1, . . . i, . . . n are the spin orbitals qhcc, . . . . . . &a and T, . . . f, . . . i i are the spin orbitals $I@, . . . 4$, . . . #&. A set of excited configurations Y I are formed (a) by replacing i byj*, 1

(35) wherej" is a member of the set of antibonding localized orbitals, or (b) by replacing i byj* and k by I*

14* 149 150 151

P. J. Hay and I. Shavitt, J. Chern. Phys., 1974, 60,2865. S. Diner, J. P. Malrieu, and P. Claverie, Theor.Chirn. Acta, 1969, 13, 1. J. P. Malrieu, P. Claverie, and S. Diner, Theor.Chim.Acta, 1969, 13, 18. S. Diner, J. P. Malrieu, F. Jordan, and M. Gilbert, Theor.Chirn. Acta, 1969, 15, 100.

Electronic Calculations on Large Molecules

197

The energy and wavefunction are then given by E = Eo

+ E2 + Es

Izo

= <~olXlI~o>

I<1polHl WI>

-k

I

Eo-EI

where

Py=yo+y1+

yz=wo+

yI c <WOIf;rlYI> Eo-Er

If0

IZJZO

There is a zero first-order contribution to the energy. The use of the CNDO approximation considerably simplifies the terms arising in these expansions and gives terms with a clear physical significance. These may be summarized as follows: Energy to second order (a) y~ is a mono-excited configuration: (i) Mono-excitation in a bond

This term, the polarization energy, corrects for a poor choice of bond polarity in forming the bonding orbitals. (ii) Mono-excitation between two bonds 1pI

= y(7)

This term allows for delocalization and is therefore called the delocalization energy. (b) ' p is ~ a di-excited configuration. The CNDO approximation limits these terms to only two. (i) Di-excitation in one bond yI =

yqr)

This term is the intra-bond correlation energy. (ii) Di-excitation in two bonds

This term is the inter-bond correlation energy.

198

Theoretical Chemistry

Energy to third order (a) y~ and W J are mono-excited configurations:

(0

yI =

(ii)

yx = y ( r )

YJ =

'dr>

(iii)

yI =

w(f*)

WJ =

wc>

Yd")

Y J = y(7)

(6) y r is a mono-excited and YJ a di-excited configuration: Yr

=

Y6*>

(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

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