Theory and practice of MO calculations on organic molecules (Progress in theoretical organic chemistry) 0444414681, 9780444414687

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PROGRESS IN THEORETICAL ORGANIC CHEMISTRY VOLUME 1

THEORY AN PRACTICE OF MO CALCULATIONS ON OLECULES ORGAN I. G. Csizmadia

iplHp |BpBl p f .1111 Elsevier

PROGRESS IN THEORETICAL ORGANIC CHEMISTRY VOLUME 1

THEORY AND PRACTICE OF MO CALCULATIONS ON ORGANIC MOLECULES

PROGRESS IN THEORETICAL ORGANIC CHEMISTRY VOLUME 1

THEORY AND PRACTICE OF MO CALCULATIONS ON ORGANIC MOLECULES

I.G. CSIZMADIA Department of Chemistry University of Toronto Toronto, Ontario Canada M5S 1A1

ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam — Oxford — New York 1976

ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O. Box 211, Amsterdam, The Netherlands AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue New York, New York 10017

The Library University of Regina

ISBN: 0-444-41468-1 Copyright © 1976 by Elsevier Scientific Publishing Company, Amsterdam All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, Jan van Galenstraat 335, Amsterdam Printed in The Netherlands

PREFACE

!>

There it.

is

This means

only one way to learn something that

on organic molecules.

begin.

by doing

the road to knowledge in theoretical

organic chemistry is built

organic chemist has

....

from quantum chemical computations

It is hoped that after an experimental

finished this book he will be ready to

Even if he feels he

is not ready,

he should st6rt and

after some computational experience has been gained the con¬ tent ~f

this

introductory book will mean considerably more.

V

TO MY PARENTS

ACKNOWLEDGEMENTS

This book has lectures

emerged from a series of seminars,

and graduate courses

the period 1965-1975

the author has delivered during

at three Canadian Universities:

University of Alberta, Queen's

University,

Edmonton

Kingston

University of Toronto,

The and S.

author is

Wolfe

as well

Toronto

indebted to Professors 0.

P.

Strausz

as

G.

Mezey for

to Drs.

E.

Lown and P.

reading parts of the manuscript during the early period of its preparation. The assistance of John D.

Goddard during the

period of manuscript preparation is Without his

conscientious

"final"

gratefully acknowledged.

efforts it would have been impossible

to finish this manuscript on time. Thanks

are also due to Frank Safian for the art work,

to John Glover for the photography and to Patti Epstein for typing the manuscript. Most of the various projects

computed results,

applications of MO theory,

used to illustrate the

emerged from research

financially supported by the National Research

Council of Canada.

4

*

'

TABLE OF CONTENTS

A. INTRODUCTION

B.

I

Introductory Remarks

II

Mathematical Introduction

III

Quantum Mechanical Background

1

’

6 45

THEORY OF CLOSED ELECTRONIC SHELLS

77

77

IV

Non-Empirical or Hartree-Fock MO Theory

V

Semi-Empirical MO Theories

VI

Excited and Ionized States

117 in the Framework

of Closed Shell MO Theories VII

Hybrid Atomic Orbitals

(HAO)

Localized Molecular Orbitals VIII IX

150 and (LMO)

Limitations of Molecular Orbital Theories Applications

187

of MO Theory to Closed Shell

Problems

212

C. THEORY OF OPEN ELECTRONIC SHELLS

X

Open Shell SCF Theories

XI

Limitations

269

269

and Applications of Open Shell

SCF Theories

D. PRACTICAL ASPECTS

Sets

(jD 158

284

OF MO COMPUTATIONS

for Molecular Orbital Calculations

307

XII

Basis

XIII

Information on Selected Computer Programs

325

XIV

Closing Remarks

352

E. APPENDIX

XV

307

365

Detailed Formalisms of Roothaan's SCF Theories

365

\

V

V

SECTION A

INTRODUCTION

CHAPTER I

INTRODUCTORY REMARKS

/ 1

The Role of Theories

and Models

/

2 1.

The

Role of Theories

The way all

and Molecules

following scheme encompasses

in a very general

scientific activity

\

MODEL

THEORETICAL

EXPERIMENTAL

TECHNIQUE

TECHNIQUE

Figure 1-1. A schematic illustration of the interrelationship of experiment, theory and modelling.

It should be noted that THEORY and EXPERIMENT are equally fundamental a rigorous

stage.

in any branch of science

that has reached

MODELS on the other hand are built either on

theoretical and/or experimental observations and are expected to provide a conceptual explanation for

the phenomenon

investigated. When identify the including all

its

implications

"THEORETICAL TECHNIQUE" with its various branches.

tified with all the "Selection Rules", The

chemical

"Rules" we have the

The

are

sought we can

"Quantum Theory" "MODEL"

may be

iden-

in chemistry

such as

the

"Woodward-Hoffmann Rules'"

and the

like.

application of any THEORETICAL TECHNIQUE to a particular

chemical problem is turn produces

carried out through computation which in

the Theoretical

very much the same way as TECHNIQUE tative

leads

(i.e.

numerical)

Observations

the application of an EXPERIMENTAL

to qualitative

numerical)

(i.e.

(i.e.

results.

non-numeric)

or quanti¬

The application of a MODEL

V

in

3 to the

same chemical problem produces qualitative

numerical)

that we have

and possessing them we usually declare

some understanding of the problem.

There are at that the

tative

least three things

to be noticed.

results obtained from theory are

from models

always qualitative,

thing to notice

is

while experimental

intellectual

constructs while

is

that there exists a v/hole spectrum of

not be trivially obvious this

above viewpoint,

the

in which all of our scientific activities are

confessed opinions 1.

fits.

stage it is necessary to point out that

by no means

the chemical community.

of the

and it need

into which category a construct

unified on an equal basis, in

substances.

constructs between theory and models

At

second

in any chemical

is working with actual chemical

The third point

The

that in both theory and' models we are

dealing with intellectual experiment one

One

always quanti¬

results may be either qualitative or quantitative.

ance

non-

results which may be regarded as concepts or con¬

ceptual explanations

is

(i.e.

Only MODELS

into

In

enjoys uniform accept¬

fact,

we may classify most

the following four categories.

and therefore Conceptual

Explanations

are of any importance and therefore of any real relevance to chemistry because THEORY is incomprehensible 2.

THEORY is

largely

and consequently irrelevant.

important in

so far as

it supports

a

useful MODEL. 3.

THEORY

is of primary importance and a MODEL is

acceptable only in so far as

it is

in close agree¬

ment with the Theoretical Observations. 4.

THEORY

is

the only scientifically acceptable and

therefore relevant method that complements EXPERIMENT.

Consequently all MODELS are more or

less useless

and thus

irrelevant.

Perhaps most experimentalists' fitted in the views

first three categories while most theoreticians'

could be accommodated by the

chemist,

however,

the

last three.

To the average

the relatively moderate views of

are most appealing and the extreme;

point of view could be

(2)

first and last statements

and

(3)

appear

first one being at the ultra conservative extreme

t *

4 while the last is be appropriate

at the radical

extreme.

At this

stage it may¬

to make an operational distinction between

theory and a model.

Such a distinction may be made in terms

their applicability.

If we apply a Theory to investigate

of

a

question that asks WHAT we may compute an answer within any quantitative theory. geometry of NH^?

For example:

We may obtain r(N-H)

Alternatively we may ask: inversion in NH,?

WHAT is

We may compute

theory can answer a question: is

WHAT is the most stable =

1.01 8 and

since

S. .

^ 0

n

n

E

E

i=l

j=l

=

n

n

E

E

a . 6. .b .

i ID

=

E

a.b.

li

3

= a,b,

+ a„b_

11

2 2

+

i=l

a b n n {11-28}

i=l

j=l

The

last part of this equation is the usual expression

one defines the i.e.

n

inner product in terms of vector components,

in terms of the coefficients of the If the vectors

are not real but complex then the

appropriate complex-conjugate {11-26-28}

as

linear combination.

should be used in equations

indicated below.

n

n

HII 1—1

i=l

a.*S..bj l i] 3

{II-27a}

n vector space are real

some

the Euclidean vector The basis vectors of the

and the basis vectors of the

Hermitian vector space are complex so that the Euclidean vector

17 space

is

a special case of the Hermitian vector space.

characteristics of the two spaces

The

are best illustrated in terms

of the properties of the inner product of two vectors: Euclidean Vector Space

Hermitian Vector Space

a)

a)

The

inner product is real:

The inner product is com¬

plex

=

t a2^2

• • • •

=

(a^ *b

+ a2*b2 +

, b)

The

metric

inner product is (i.e.

it is

b)

sym¬

{II-29a,b)

The inner product is

Hermitian

equal to

(i.e.

it is

its own transpose)

to its own adjoint)

‘!’ = =

....

equal

= lx> {ll-30a,b}

c)

The

c)

inner product is

The

inner product is

bilinear

bilinear

+

=

= +

+

{II-31a,b) d) The norm is positive

d) The norm is positive

N =

N =

0

following figure

)>

>

{II-32a,b}

0

illustrates by 3 dimensional geometrical

analogues the Euclidean

(real)

vector space.

X3

^3

An orthogonal 3D- Euclidean vector space

A general

3D-Euclidean vector space

l

Figure

4.

II.7 Geometrical illustrations Euclidean vector spaces.

Matrices When real

or complex scalars

tangular array the resulting object matrix consisting of m-rows The

for

i,

jth.

are arranged is

called a matrix.

and n-columns

element of the matrix is

in a rec¬ A

is an m x n matrix.

found at the intersection

18 . th

A

., and the

=

. th column. j

11

a12

a13

a14

21

a22

a23

a24

31

a32

a33

a34 /

A square matrix has the

1

b12

b13

b21

b22

b23

b31

b32

b33

\

case there

is

x 4 matrix

{11-33}

and columns.

a 3 x 3 matrix

{11-34}

a unique

off diagonal elements.

The diagonal elements

fall on the principal diagonal of the all other elements

a 3

same number of rows

/bll

=

^

(i.e.

b^,

i^j)

square

are those that (i.e.

b.

n

)

while

are called off-diagonal

elements. The

sum of the diagonal elements of a square matrix is

usually referred to as the trace of the matrix tr)

and it is of

some

importance

(abbreviated as

in quantum chemistry

n tr B

=

b. .

E

{11-35}

li

i=l

The multiplication of matrices in computational quantum chemistry.

is

of great importance

Suppose we wish to multi¬

ply matrix B with matrix A defined in equations {11-33}

{11-34}

and

respectively to obtain a product matrix C

2

=

I 5

{11-36}

The elements of C are defined as:

\

19 n c . . ID

£

b .. a, . lk k;j

{11-37}

k=l

This means that the elements of the

ibb row in B

plied by the corresponding elements of the j the sum of these corresponds to the matrix £.

This

is

th

are multi-

column in A and

i,jbb element of the product

illustrated for C2

3 in the next two

equations

C13

ciA

C21

C22

C23

C24

C32

C33

C3V

=

—

fh 11

b12

1—1

c 12

1 1 CO

/cll

b21

b22

b23

\b31

b32

b33/

/all

a12

a13

a14\

a21

a22

a23

a24

\a31

a32

a33

a3 4 / {11-38}

C2 3 From the

b21a13

+

b22a23

+

{11-39}

b23a33

foregoing it should be evident that not any two matrices

may be multiplied together and even two matrices which may be multiplied together in one particular order might not be multi¬ plied in the reverse order. plied but B many columns

.

For example A

£ may be multiplied. as many rows

are

This

A

(3x4)

(3x3)

(3x4)

(nxm)

(nxk)

(kxm)

=

B cannot be multi¬

is because B has

so

in A.

S

C

.

{11-40}

Even if one deals with square matrices

the order which they are

multiplied together is quite

since A.B is not necess¬

arily the

important

same as B.£:

A. B

1

B A

{11-41}

20 There are some

special matrices

used in computational quantum chemistry.

that are frequently These

are the

following. A Diagonal Matrix is

a square matrix with non-zero diagonal

elements and with all other elements being zero.

{11-42}

A Unit Matrix is elements.

a special case of D with unit diagonal

Naturally all off-diagonal elements

are

zero

{11-43}

Multiplication of a matrix with unit matrix leaves the matrix unchanged.

Also in this

case the order of multiplication

is

immaterial.

B

.

1

=

A Null Matrix is a matrix

1

.

B

=

B

{11-44}

(may or may not be a square matrix)

which has only zero elements

0

O

0

{11-45}

0

The

addition and multiplication that

must obey the

following relationships

involves

a null matrix

21

£

+

O

h

k

•

2

2*i

=

2

(for any dimensions)

{11-46}

(for square matrices)

{11-47}

There are certain matrix operations that are used frequently in computational quantum chemistry The Complex Conjugate.

If the elements of Matrix A are complex

then the complex conjugate of the matrix

(£*)’ may be generated

by taking the complex conjugate of the elements.

(i*)ij

=

aij *

{11-48}

If the elements of || are real then

/ A*

=

£

.

/

{11-49}

Consequently in this particular case A is a real matrix.

The

transpose of a matrix is obtained by interchanging the rows

and

columns

11 A

a12

21

a22

/all

a13 \

= a23

J

A'

{II-50a,b}

= a?t

\a13

matrix is changed to a matrix

A'

=

(nxn)

a21

a22

a23

3x2 matrix.

the dimension of the matrix remains the same.

/all

a12

a13 ^

a21

a22

a23

\ a31

a32

a33 /

/ =

\

all

a21

a31

a12

a22

a32

a13

a23

a33

\ {11-51}

Note that the diagonal elements remained unchanged and the i,j indices a32 ’

interchanged in such a way that the

2,3 element of A' is

22 In the

case of a

symmetric arrangement when

aij

the transpose of a matrix is

-

aji

{11-52}

the same as the original matrix

A'

=

A

{11-53} V

This type of matrix is usually called a symmetric matrix. the elements of the matrix are real then the matrix referred to as a real

symmetric matrix which has

If

is normally

the

following

property

(A*)'

=

A

or

{II-54a,b} a ij

This matrix is very frequently used in quantum chemistry.

The

transpose complex conjugate is usually called the adjoint and denoted by a dagger:

A+

The

=

(A*) '

Inverse of a Matrix is defined by the

following identity:

{11-55} For example:

£

{II-56a,b}

i

{11-57}

23 The method of matrix inversion will not be discussed here since those matrices whose very special unitary

(g)

inverse are normally required fall into a

category.

These are

the orthogonal

(0)

and the

matrices whose inverse are the transpose and the

adjoint respectively

{11-58}

{11-59}

The

following relationships

should be observed in connection

with the transpose and the inverse of the product of

two

1

can one

=

A~1

/

.B’

.B-1

{11-61}

can be used to discover under what conditions

interchange the order of matrix multiplication

Usually

However,

A*

1—1

These properties

=

a

(k

1)

sin2oc

-

2

a

^k

coscc

since

(k-1)

+ q

COS

CC

{11-116}

2

qrs(k)

= qsr(k_1)

=

tqss(k_1>

- c!rr(k_1)]

COS'c

sin"

+ qrs(k_1)

[cos °=

-

2

sin *

{11-117}

As noted above the purpose of diagonalization is to eliminate the off-diagonal elements must be set equal to

[qs£/k 1')

so that the

last equation

zero

- qrr^k

cos* sin* + qrs^k_1^

[cos2*

-

sin2*]

=

{11-118}

By noting the

following trigonometric relationships

cos* sin*

=

(1/2)sin 2* {II-119a,b}

cos

we obtain the

- (1/2 Iq^^

2

.

2

* - sm *

=

cos

2*

following simplified expression

- qss^k 1^]

sin 2*

- q^ ^k_1^ rs

.

Cos

2* =

0

{11-120}

or,

combining the appropriate terms,

0

39

2q tan 2“

=

(k-1) rs {11-121}

(k-1)

- q ss

rr It is customary to choose the

limiting cases

(k-1)

« within the range of + 45° since in

2« = +90°.

The

45° limit is

appropriate whenever the denominator becomes grr as

- ^ss

‘

The

si