Approximate Molecular Orbital Theory

221 Pages·2016·2.26 MB·English

by  Pople John A.

McGRAW-HILL SERIES IN ADVANCED CHEMISTRY
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John A. Pople
AMDUR AND HAMMES Chemical Kinetics: Principles and Selected Topics
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BENSON The Foundations of Chemical Kinetics
BIEMANN Mass Spectrometry (Organic Chemical Applications)
DAVIDSON Statistical Mechanics
DAVYDOV (Trans. Kasha and Oppenheimer) Theory of Molecular Excitons
DEAN Flame Photometry
DEWAR The Molecular Orbital Theory of Organic Chemistry
ELIEL Stereochemistry of Carbon Compounds
FITTS Nonequilibrium Thermodynamics
FRISTROM AND WESTENBERG Flame Structure
HAMMETT Physical Organic Chemistry
HELFFERICH Ion Exchange
HILL Statistical Mechanics
HINE Physical Organic Chemistry
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KOSOWER Molecular Biochemistry
LAIDLER Theories of Chemical Reaction Rates
LAITINEN Chemical Analysis
McDOWELL Mass Spectrometry
MANDELKERN Crystallization of Polymers
MARCH Advanced Organic Chemistry: Reactions, Mechanisms, and Structure
MEMORY Quantum Theory of Magnetic Resonance Parameters
PITZER AND BREWER (Revision of Lewis and Randall) Thermodynamics
POPLE AND BEVERIDGE Approximate Molecular Orbital Theory
POPLE, SCHNEIDER, AND BERNSTEIN High-resolution Nuclear Mag
netic Resonance
PRYOR . Free Radicals
RAAEN, ROPP, AND RAAEN Carbon-14
ROBERTS Nuclear Magnetic Resonance
ROSSOTTI AND ROSSOTTI The Determination of Stability Constants
SIGGIA Survey of Analytical Chemistry
WIBERG Laboratory Technique in Organic Chemistry
Approximate
Molecular Orbital
Theory
JOHN A. POPLE
Carnegie Professor of Chemical Physics
Carnegie-Mellon University
DAVID L. BEVERIDGE
Associate Professor of Chemistry
Hunter College
City University of New York
McGRAW-HILL BOOK COMPANY
NEW YORK ST. LOUIS SAN FRANCISCO DUSSELDORF
LONDON MEXICO PANAMA SYDNEY TORONTO
APPROXIMATE MOLECULAR ORBITAL THEORY
Copyright © 1970 by McGraw-Hill, Inc. All rights
reserved. Printed in the United States of America.
No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photo
copying, recording, or otherwise, without the prior
written permission of the publisher.
Library of Congress Catalog Card Number 70-95820
07-050512-8
45 678910 KPKP 78321098
Preface
Since its inception in the early days of quantum mechanics, molecular
orbital theory has become a powerful method for studying the electronic struc
ture of molecules, illuminating many areas of chemistry. In quantitative
form, it has developed both as an ab initio method for computing molecular
wavefunctions directly from the fundamental equations of quantum mechanics
and also as a semiempirical technique for interrelating various physical proper
ties of atoms and molecules using a simplified formalism as a framework for
parameterization. Until recently, ab initio calculations dealt mainly with
very small systems while the semiempirical methods were oriented toward
the 7r electrons of larger planar molecules. In the last few years, however,
both approaches have become more concerned with general polyatomic mole
cules and they now overlap somewhat in their areas of application.
This book has the limited objective of presenting the background of
self-consistent molecular orbital theory and following this with a description
of certain elementary semiempirical schemes which use the general theory
as a basic framework. These are methods based on zero-differential overlap
(complete neglect of differential overlap, or CNDO, and intermediate neglect
of differential overlap, or INDO) which are simple enough to be applied to a
wide range of chemical problems without major computational effort. The
necessary general theory is covered in Chaps. 1 and 2 leading up to simple
examples of molecular orbital calculations for diatomics. In Chap. 3, the
approximations involved in the semiempirical schemes and the corresponding
parameterizations are discussed in detail. In Chap. 4 we survey applications
of the methods which have been made to date, including studies of electronic
charge distributions in molecules, dipole moments, equilibrium geometries,
nuclear hyperfine structure in the electron spin resonance spectroscopy of
organic free radicals and the spin coupling constants measured by nuclear
magnetic resonance.
Many of the conclusions based on the simple methods described in this
book will undoubtedly be modified by larger and more sophisticated calcula
tions which are rapidly becoming possible. Nevertheless, we believe that
theoretical studies at this simple level do provide a first approximation which
is realistic, informative, and direct enough to allow widespread application.
11- is to facilitate such applications that we have collected the material in this
volume.
Much of the work described herein has been the result of a collaborative
rITort with a number of colleagues at Carnegie-Mellon University. These
include David P. Santry, Gerald Segal, Mark S. Gordon, Paul A. Dobosh,
Neil S. Ostlund, and James W. Mclver, Jr. Helpful discussions with Herbert
V
vi PREFACE
Fischer and Keith Miller are also acknowledged. The efforts of Kathryn
Severn in preparing the typescript are greatly appreciated. Permission to
reproduce material has been granted by the Journal of Chemical Physics and
the Journal of the American Chemical Society.
The support of the U.S. Public Health Service, Grant 1-F2-CA-21,281-01
is gratefully acknowledged by David L. Beveridge.
JOHN A. POPLE
DAVID L. BEVERIDGE
Contents
PREFACE v
Chapter 1
QUANTUM-MECHANICAL BACKGROUND
1.1 Introduction 1
1.2 The Schroedinger Equation 3
1.3 General Properties of Operators and Wavefunctions 7
1.4 The Variational Method 11
1.5 The Orbital Approximation 12
1.6 Electron Spin 14
1.7 The Antisymmetry Principle and Determinantal
Wavefunctions 16
1.8 Electronic Configurations and Electronic States 19
1.9 Atomic Orbitals in Molecular Orbital Theory 22
Chapter 2
SELF-CONSISTENT FIELD MOLECULAR ORBITAL THEORY
2.1 Introduction 31
2.2 The Energy Expression for a Closed-shell Configuration 32
2.3 The Hartree-Fock Equations for Molecular Orbitals 37
2.4 LCAO Molecular Orbitals for Closed-shell Systems 41
2.5 An LCAOSCF Example: Hydrogen Fluoride 46
2.6 Molecular Orbitals for Open-shell Systems 51
Chapter 3
APPROXIMATE MOLECULAR ORBITAL THEORIES
3.1 Introduction 57
3.2 Invariant Levels of Approximation 60
3.3 Complete Neglect of Differential Overlap (CNDO) 62
3.4 The CNDO/1 Parameterization 69
3.6 The CNDO/2 Parameterization 75
3.6 Intermediate Neglect of Differential Overlap (INDO) 80
3.7 Neglect of Diatomic Differential Overlap (NDDO) 83
Chapter 4
APPLICATIONS OF APPROXIMATE MOLECULAR ORBITAL THEORY
4.1 Introduction 85
4.2 Molecular Geometries and Electronic Charge Distributions 85
vll
vlii CONTENTS
4.3 Electron-spin-Nuclear-spin Interactions 128
4.4 Nuclear-spin-Nuclear-spin Interactions 149
4.6 Further Applications of Approximate Molecular Orbital
Theory 159
Appendix A
A FORTRAN-IV COMPUTER PROGRAM FOR CNDO AND INDO
CALCULATIONS 163
Appendix B
EVALUATION OF ONE- AND TWO-CENTER INTEGRALS 194
B.l Basis Functions 194
B.2 Coordinate Systems 195
B.3 The Reduced Overlap Integral 197
B.4 Overlap Integrals 199
B.5 Two-center Coulomb Integrals Involving s Functions 200
B.6 One-center Coulomb Integrals Involving s Functions 203
B.7 Implementation of Integral Evaluations in CNDO and INDO
Molecular Orbital Calculations 204
NAME INDEX 207
SUBJECT INDEX 213
1
Quantum-mechanical
Background
1.1 INTRODUCTION
The main objective of any theory of molecular structure is to provide
some insight into the various physical laws governing the chemical
constitution of molecules in terms of the more fundamental universal
physical laws governing the motions and interactions of the constituent
atomic nuclei and electrons. In principle such theories can aim at a
precise quantitative description of the structure of molecules and
their chemical properties, since the underlying physical laws are now
well understood in terms of quantum theory based on the Schroedinger
equation. However, in practice mathematical and computational
complexities make this goal difficult to attain, and one must usually
resort to approximate methods.
The principal approximate methods considered in molecular
quantum mechanics are valence bond theory and molecular orbital
theory [1]. Valence bond theory originated in the work of Heitler
l
2 APPROXIMATE MOLECULAR ORBITAL THEORY
and London and was developed extensively by Pauling. Molecular
orbital theory has its origins in the early research work in band spec
troscopy of diatomic molecules and has been widely used to describe
many aspects of molecular structure and diverse molecular properties
such as electronic dipole moments, optical absorption spectra, and
electron and nuclear magnetic resonance. Among those involved in
the original works were Hund, Mulliken, Lennard-Jones, and Slater.
We are concerned herein exclusively with molecular orbital theory,
and particularly with the theories and problems encountered in carry
ing out the calculation of molecular orbitals for large molecules.
Molecular orbital theory provides a precise description of molec
ular electronic structure only for one-electron molecules, but for
many-electron molecules it provides a sufficiently good approximate
description to be generally useful. The full analytical calculation of
the molecular orbitals for most systems of interest may be reduced to a
purely mathematical problem [2], the central feature of which is the
calculation and diagonalization of an effective interaction energy
matrix for the system. The digital computer programs that have
been prepared to carry out these calculations have been mostly the
result of extensive work by highly coordinated research groups. A
number of these groups have generously made their programs available
to the scientific community at large [3], but even with the programs
in hand the computer time involved in carrying out sufficiently accurate
calculations is often prohibitively large, even for diatomic molecules.
On the other hand, many applications of molecular orbital theory do
not necessarily require accurate molecular orbitals for the system.
In many chemical and physical problems, a qualitative or semi
quantitative knowledge of the form of the molecular orbitals is suffi
cient to extract the necessary information. Thus there is considerable
interest in the development of good approximate molecular orbital
theories to serve this purpose, and this constitutes the subject of the
present book.
Approximate molecular orbital theories are based on schemes
developed within the mathematical framework of molecular orbital
theory, but with a number of simplifications introduced in the compu
tational procedure. Often experimental data on atoms and prototype
molecular systems are used to estimate values for quantities entering
into the calculations as parameters, and for this reason the procedures
are widely known as semiempirical methods.
Approximate molecular orbital theory may be approached from
two basically different points of view. One approach involves choosing
appropriate values for the elements of the aforementioned interaction
QUANTUM-MECHANICAL BACKGROUND 3
energy matrix from essentially empirical considerations, and is char
acteristic of the so-called Huckel [4] and extended Huckel [5] methods.
The other approach is based explicitly on the mathematical formalism,
and involves introducing approximations for the atomic and molecular
integrals entering the expression for the elements of the energy inter
action matrix. The latter approach is referred to as approximate
self-consistent field theory [6]. Both Huckel theory and approximate
self-consistent field theory were originally developed within the frame
work of the 7r electron approximation, treating the w electrons of planar
unsaturated organic molecules explicitly with the remaining a electrons
and atomic nuclei considered as part of a nonpolarizable core. Huckel
T electron theory has been given a most definitive treatment by
Streitweiser [4], and likewise ir electron self-consistent field theory is
developed in considerable detail in the recent books by Salem [7] and
Murrell [8]. We thus restrict our consideration to more recent approxi
mate molecular orbital theories applicable to all valence electrons of a
general three-dimensional molecule.
In the following presentation, we have attempted to give the
essentials of quantum mechanics and molecular orbital theory pertinent
to the understanding and application of approximate molecular orbital
calculations to chemical problems. The remainder of this chapter is
a cursory and informal discussion of certain quantum-mechanical
principles and an introduction to the orbital description of electronic
structure. In Chap. 2, the methods of molecular orbital theory are
introduced and illustrated in some detail. Chapters 3 and 4 are
concerned with approximate molecular orbital theory, presenting first
the formalism of acceptable approximation schemes followed by a
discussion of applications reported to date. Appendix A contains a
description and listing of a digital computer program for carrying out
calculations by some of the more extensively tested approximate
molecular orbital methods.
1.2 THE SCHROEDINGER EQUATION [9]
According to classical mechanics, the energy E of a system of inter
acting particles is the sum of a kinetic-energy contribution T and a
potential-energy function V,
T + V = E (1.1)
Hchroedinger suggested that the proper way to describe the wave char
acter of particles was to replace the classical kinetic- and potential-