Fundamentals

308 Pages·1973·5.9 MB·English

by  Jerry Goodisman (Eds.)

DIATOMIC INTERACTION
POTENTIAL THEORY
Jerry Goodisman
DEPARTMENT OF CHEMISTRY
SYRACUSE UNIVERSITY
SYRACUSE, NEW YORK
Volume 1
Fundamentals
ACADEMIC PRESS New York and London 1973
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1973, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
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ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
Goodisman, Jerry.
Diatomic interaction potential theory.
(Physical chemistry, a series of monographs)
Includes bibliographies.
CONTENTS: v. 1. Fundamentals.-v. 2. Applications.
1. Quantum chemistry. I. Title. II. Series.
QD462.G65 54Γ.28 72-9985
ISBN 0-12-290201-7 (v. 1)
PRINTED IN THE UNITED STATES OF AMERICA
Preface
The calculation of the energy of a diatomic system as a function of inter-
nuclear separation is a problem which has a long history and has generated
an enormous amount of literature. Due in part to advances in computational
hardware and software in the past few years, quantum chemists can now
produce reliable interaction potentials for diatomic systems in their ground
states. The situation for excited states and for polyatomic systems is less
satisfactory, but there is hope that it will shortly improve.
These two volumes cover the theoretical material involved in calculations
for diatomic systems in their ground states, with attention given to the
variety of the approaches one may use. The first volume contains mostly
basic and general material; the second includes more in the way of specific
descriptions of modern calculations.
The problem is defined in Chap. I, Vol. 1. A discussion is given of the
nature of an interatomic interaction potential or potential energy curve,
including its relation to reality (experiment). Chapter II presents a general
discussion of its shape. Chapter III treats the main approaches to schemes
of calculation: variation theory, perturbation theory, the virial and Hell-
mann-Feynman theorems, local energy principles, and quantum statistical
theories. In Chapter I of Volume 2, the calculation of the interaction poten­
tial for large and small values of the internuclear distance R (separated
and united atom limits) is considered. Chapter II treats the methods used
for intermediate values of R, which in principle means any values of R.
The Hartree-Fock and configuration interaction schemes described here
have been the most used of all the methods. Semiempirical theories and
methods constitute the subject of the last chapter of Volume 2.
The level of treatment throughout, it is hoped, is sufficiently elementary
for the material to be understood after an introductory quantum mechanics
course. By means of this book, the reader should be able to go from that
degree of preparation to the current literature.
Vll
viii Preface
Work on this book started about five years ago; its proximate cause was
my participation in a special topics graduate course, with Prof. D. Secrest
and Prof. J. P. Toennies, at the University of Illinois. The course largely
dealt with scattering experiments and their relation to potential energy
curves. At that time I was struck by the fact that there was much material
which was common knowledge among those involved in quantum chemical
calculations but unfamiliar to students, even those with good course back­
grounds. Thus, one goal of the present book is to make that material con­
veniently available to students and others interested in the subject, and to
introduce them to the current literature.
For those interested in the theory of quantum chemical calculations, I
want to provide in one place as much information as I can on the varied
methods which are available. For those interested in potential curves or
in quantum chemistry, but not particularly interested in calculating poten­
tial curves, I hope this book will be a guide to what has been going on, as
well as an aid in reading the literature.
The subject has been limited to diatomic interactions, and, still further,
to diatomic ground-state interactions. Of course, the limitation on the
calculations discussed does not mean that the methods of calculation have
no other applicability. I hope that the general discussions, particularly in
the first volume, will be of interest to those who care about other systems.
Some of the methods may even find their greatest applicability to those
other systems. However, only by severely limiting my subject could I hope
to attain some measure of completeness of coverage. Even so, I have had
to give very limited space to certain topics. I have not discussed calculations
specifically applicable to one- and two-electron systems (another book
should be written on these, as was done for the corresponding atoms);
I have slighted relativistic and magnetic effects; I have given unjustly brief
coverage to many-body theory. There are undoubtedly other sins of omis­
sion.
Nevertheless, I believe that this book gives a balanced picture of the
enormous amount of work that has been done on ground state diatomic
potential curves. While limitations of certain methods have been discussed,
I hope the main point is still clear: after many years of effort, reliable poten­
tial curves can now be generated for most systems of interest.
Notes on Notation and Coordinate Systems
As an aid in keeping formulas more legible, a Dirac-like bracket nota­
tion is employed frequently, without necessarily implying notions of states,
representations, and so on. The triangular bracket (0J0 y means the
k
product of Φι* and 0 , integrated over the entire configuration space,
k
which must be the same for the two functions. This means integration
over all spatial coordinates and sums over spin coordinates. The arguments
of Φι and 0 need not be stated, although sometimes they are. The "factors"
k
in this "scalar product" may be considered separately. Thus | Φ> and Φ
are equivalent, both being wave functions.
The adjoint wavefunction is written <Φ |, but this is also used to denote
an operator in the following sense: <Φ | multiplying | Ψ} gives the bracket
(0 | Ψ}, a number which is computed by multiplying Φ* by Ψ and inte­
grating over the configuration space. Thus, <Φ | may be interpreted as the
operation of multiplication by Φ*, followed by integration. If the functions
ψι form a complete set,
ΣI ψΐΧψι Φ) = \Φ>
so that the operator in the square bracket is the identity operator.
Italics are used for operators, e.g., For h. We use (Ψ \ F \ X} equivalently
to (Ψ | FXy: thus F operates on the function X and the scalar product
of the result with Ψ is then taken. Operators are assumed to operate to the
right in all bracket expressions. Thus,
(0 | P | ψ} = <Ρ+φ I ψ) = (ψ I Ρ+φ>*
where Pf is the adjoint of P.
In general, we use bold face for matrices, e.g., M is the matrix with
elements M^. The determinant of the matrix M is written | M | or det[M]
IX
X Notes on Notation and Coordinate Systems
and the trace is written Tr[M]. The dagger (f) is used to indicate the
adjoint of matrices, so (M+) , the (i j) element of the matrix Mf, is M*,.
i:? 9
The transpose is similarly denoted by T: (MT)^· = M^. The transpose or
adjoint of a column vector is a row vector and vice versa.
The expression "matrix element of the operator P between states (or
functions) Ψι and X" means the integral
^Ψ*ΡΧάτ = (Ψ\Ρ\Χ\
ι ί
The "scalar product" of X and X (X \ A^>, is sometimes referred to as
t j9 {
an element of the overlap matrix. In these integrals or brackets an integra­
tion, or summation where appropriate, is implied over all coordinates, un­
less there is a specific remark to the contrary.
"Real part of" is denoted by Re, e.g., Re(</>). Curly brackets are used to
refer to all the members of a set of functions, as in: "we orthonormaHze
the {φι} among themselves."
Several different coordinate systems are used in our discussions. Con­
sider the two nuclei separated by a distance R, and a Cartesian coordinate
system located at the midpoint, with the Z axis along the internuclear axis.
The position of an electron can be specified by its distances from the nuclei,
plus the angle ψ between the plane containing the ABe triangle and a ref­
erence plane containing the Z axis. Alternatively, one can use the angles
Θ and Θ together with φ, or one of the sets (r θ ,φ) and (r ,θ ,φ).
Α Β A9 Α B Β
The last two are of course just spherical polar coordinates centered on one
nucleus or the other.
It is convenient for many purposes to use coordinates involving r and
A
r , together with φ. Most often, one uses rather than r and r themselves,
B A B
the confocal ellipsoidal coordinates
f = {r + r )/R
A B
V = (r - r )/R.
A B
Notes on Notation and Coordinate Systesm XI
ξ and η are dimensionless. The surfaces of constant ξ are ellipsoids of
revolution with foci at the nuclei. The value of ξ can run from 1 (correspond­
ing to the line between the nuclei) to oo. The surfaces of constant η are
hyperboloids of revolution with foci at the nuclei. The value of η is between
— 1 (corresponding to the extension of the internuclear axis to the left of A)
and +1 (corresponding to the extension of the axis to the right of B). η = 0
is the plane bisecting the internuclear axis.
The volume element for this coordinate system is
dx = ("f-) V " V2) άξ άη άφ
and the Laplacian is
I2 - η* d* 1
All distances are proportional to R for given values of the coordinates ξ
and η. Thus
f x [(f2 1)(1 η2)Υ 2 C S ψ
r = 4" < + ri> = 4" ~ ~ ' ° '
A
r = ~^-r,); y = 4 K£2 - 00 - V2)]1'2 sin ψ,
B
R t
xii Notes on Notation and Coordinate Systems
and the distance between two points (ξ η ) and (ξ , η ) is
ΐ9 χ 2 2
2 2 2 2 1 2 2
ri = (-^) {(teil - D(i - v )r - m - DO - v. )] ' )
Chapter I Introduction to Potential Curves
A. Introduction
This book discusses the theoretical approaches to the following problem:
Calculate the wave function and energy for the lowest state of a system
of N electrons moving in the field of two fixed point charges (the nuclei
of a diatomic system) separated by a distance R. The energy, which we
denote here as E \, depends parametrically on R. From a series of calcula­
e
tions for different values of R, we produce a function E\(R), to which we
e
generally add the internuclear repulsion Z Z /R (Z and Z denote the
A B A B
nuclear charges) to produce what we shall refer to as a potential energy
curve or interaction potential. For large R, it approaches the energy of the
separated atoms or ions; for small R it becomes infinite because of the
internuclear repulsion.
A first interpretation of the potential energy curve is that it represents
the potential energy that the nuclei experience as R changes. Then the
description of the nuclear motion reduces to the problem of two mass
points interacting with this potential energy. This picture is approximately
correct, and a useful first approximation to which corrections may be
calculated. It would be exact if the nuclear masses were infinite, rather
than simply very large (ratio of 104 or higher) compared to electronic
masses. In reality, the nuclei are never fixed, and their motion must be
discussed simultaneously with that of the electrons. It can be said that
1
2 I. Introduction to Potential Curves
they move slowly compared with the electrons, or crudely, that the electrons
can adjust to changes of nuclear position, so that we may consider the
electronic problem for various values of R. To the extent that this is true,
the notion of a potential energy curve is reasonable and useful. This model
will be expressed more correctly later in terms of the adiabatic theorem.
In order to show in more detail the relation between the clamped-nucleus
potential curve, with whose calculation we are concerned, and the true
nuclear-electronic wave function, we shall present the treatment of Born
and Oppenheimer and a more general treatment, sometimes referred to as
the Born separation. Then some discussion of the experimental measure­
ment of interaction potentials will be given to help clarify the relation of
the subject matter of this book to reality.
In Chapter II, we consider some simple ideas related to the form of the
potential curve, which we denote by U(R). Most of what we have to say
here was known in the earliest years of the application of quantum me­
chanics to chemistry, and is semiquantitative, involving little or no calcula­
tion. This is in contrast to the main subject matter of this book—the recent
progress made in calculating accurate potential curves. However, the quali­
tative discussions in Sections A, B, and C of Chapter II give the frame­
work around which the more detailed work was done. Section D of Chap­
ter II presents most of the functional forms that have been used for U(R),
based in part on the ideas discussed in the preceding sections. The title of
Chapter II, Section A, "Very Large Internuclear Distances," refers to values
of R large enough for neglect of overlap between the electronic wave func­
tions of the interacting atoms (we do not discuss relativistic effects in this
book). The calculation of U(R) for such values of R is discussed in detail
in Section A of Chapter I (Volume 2). Similarly, details of the calculations
corresponding to the qualitative discussions of Section B of Chapter II are
found in Chapter I, Section B of Vol. 2. In the latter two sections, we con­
sider values of R too small for the neglect of interatomic overlap, so that
whe are concerned with the formation of a diatomic system from the sep­
arated atoms. The ideas associated with the valence bond theory are most
useful here. The concepts of the molecular orbital approach are discussed in
Chapter II, Section C. As the valence bond picture is related to the separated
atom (large R) limit, the molecular orbital approach has a kinship to the
united atom (small R) limit. Calculations based on the united atom concepts
are discussed in Chapter I, Section C (Vol. 2). However, calculations in the
molecular orbital framework require in addition most of Chapter II (Vol.
2) for discussion, since they make up the bulk of the work done on the cal­
culation of diatomic interactions.