Lecture Notes in Chemistry Edited by G. Berthier, M. J. S. Dewar, H. Fischer, K. Fukui, H. Hartmann, H. H. Jaffe, J. Jortner, W. Kutzelnigg, K. Ruedenberg, E. Scrocco, W. Zeil

1

Georges l-lenry Wagniere

Introduction to

Elementary Molecular Orbital Theory and to Semiempirical Methods

Springer-Verlag Berlin' Heidelberg· New York 1976

Author Georges Henry Wagniere Physikalisch-Chemisches Institut der Universitat Zurich RamistraBe 76 CH-8001 Zurich

Library of Congress Calaloging in Publica lion Dala

Wagni~re, Georges Henry, ~933-Introduction to elementary molecular orbi taJ. theory

and to semiempiricaJ. methods.

(Lecture notes in chemistry ; v. 1) Bibliogra.phy: p. Includes index. 1. Molecular orbitaJ.s. I. Title.

Q.D46~. W33 541' .28 76-4000e

ISBN-13: 978-3-540-07865-4 001: 10.1007/978-3-642-93050-8

e-ISBN-13: 978-3-642-93050-8

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re­printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.

Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin . Heidelberg 1976

Softcover reprint of the hardcover 15t edition 1976

Introduction

These notes summarize in part lectures held regularly at the University of Zurich and, in the Summer of 1974, at the Semi­nario Latinoamericano de QUimica Cuantica in Mexico. I am grateful to those who have encouraged me to publish these lec­tures or have contributed to them by their suggestions. In particular, I wish to thank Professor J. Keller of the Univer­sidad Nacional Autonoma in Mexico, Professor H. Labhart and Professor H. Fischer of the University of Zurich, as well as my former students Dr. J. Kuhn, Dr. W. Hug and Dr. R. Geiger.

The aim of these notes is to provtde a summary and concise introduction to elementary molecular orbital theory, with an emphasis on semiempirical methods. Within the last decade the development and refinement of ab initio computations has tended to overshadow the usefulness of semiempirical methods. However, both approaches have their justification. Ab initio methods are designed for accurate predictions, at the expense of greater computational labor. The aim of semiempirical methods mainly lies in a semiquantitative classification of electronic pro­perties and in the search for regularities within given classes of larger molecules.

The reader is supposed to have had some previous basic instruc­tion in quantum mechanics, such as is now offered in many uni­versities to chemists in their third or fourth year of study. The bibliography should encourage the reader to consult other texts, in particular also selected publications in scientific journals.

I wish to express my gratitude to Miss H. B6ckli who has com­petently typed the entire manuscript and to Mr. E. Spalinger for the drawings.

Zurich, May 1976

G. Wagniere

Contents page

I. The hierarchy of approximations 1

1. The Born-Oppenheimer approximation 2. The solution of the electronic problem 3. The subdivision of electrons into different groups

II. Simple Htickel theory of ~ electrons 4

1. The LCAO-MO formalism 2. Further simplifications 3. Some important definitions

III. Many-electron theory of ~ electrons 15 1. Ethylene as two-electron problem 2. The configuration interaction (CI) procedure 3. ~e semiempirical PPP approximation for ~ electrons 4. Benzene as an example 5. Electric-dipole transition probability

IV. Self-consistent-field (SCF) methods 42 1. Simple LCAO-formulation of the closed-shell case 2. Semiempirical simplification (ZDO approximation) 3. More general formulation of the closed-shell case 4. Koopmans' "theorem" 5. Some remarks on localized orbitals 6. Open-shell SCF methods

6.1. The restricted open-shell SCF method 6.2. The unrestricted open-shell SCF method

V. All-valence MO procedures 59 1. The Extended Htickel (EH) method 2. Electronic population analysis 3. Semiempirical all-valence calculations,

including electron interaction

3.1. The CNDO (complete neglect of different~al overlap) method

4. Invariance of the SCF eigenvalue problem to unitary basis transformations

VI. Special topics 71 1. Optical activity 2. Selection rules for electrocyclic reactions

and cycloaddition reactions 3. Molecular orbital theory with periodic (cyclic)

boundary conditions

References

Subject index 100

I. The hierarchy of approximations

The nonrelativistic Hamiltonian for a molecular system com­

posed of many nuclei (indices A,B) and many electrons (in­

dices ~,v) reads (neglecting magnetic interactions):

I (- h2 2) --v 2me ~ +I(-~~AvX)

~ A

I I ZAe 2 + I I

e 2 I I ZAZBe 2

-- + r~A r~'V RAB

~ A ~>'V A >B

which may be more concisely written

Here T stands for the kinetic energy operator, V for the potential energy operator, the subscript e means "elec­

tronic", the subscript n means "nuclear".

1. The Born-Oppenheimer approximation

We seek to solve the time-independent molecular Schrodinger equation

:lC '\f(r,R) '" W • f(r,R)

Due to the great mass difference between electrons and ato­

mic nuclei it proves possible to a satisfactory degree of

approximation (1] to treat the degrees of freedom of the

electrons, designated collectively by r, separately from

those of the nuclei, designated here by R. In this sense

the solution f(r,R) may approximately be written as a pro­

duct of two functions, of which one depends only on the

general nuclear coordinates R:

2

1[1 (r,R) ~'(r,R) • vCR)

By neglecting some terms which in general may be shown to

be small [1,2J, it is thus possible to separate the

Schrodinger equation into:

a) An equation for the motion of the electrons, the nuclei

remaining fixed at frozen positions R':

:J{ 1Ir (r,R') e m

Here m denotes a particular electronic state. The electronic

energy Em(R') depends parametrically on the frozen positions

of the nuclei. Often one holds the nuclei fixed in experi­

mentally known equilibrium positions.

b) An equation for the motion of the nuclei in the field of

the electrons in a given electronic state m:

e . mJ

The electronic energy as a function of nuclear position Em(R)

acts as a potential on the nuclei. For a diatomic molecule

in a bonding electronic state Em(R) + Vnn(R) is generally

described by a Morse potential. vmj(R) represents a vibra­

tional wavefunction j in the electronic state m.

2. The solution of the electronic problem

We consider the electronic equation

= (T + V + V ) 1jJ (r) e en ee m

We no longer explicitly mention the nuclear coordinates R,

once we have stated where they have been fixed. Of course we

still have a many-body problem to deal with, and the solution

of this problem is in general still very cumbersome. In

practice it proves only feasible to obtain approximate

3

solutions, and it is the degree of approximation that is the

crucial question. Even in simplest cases exact solutions re­

quire a quasi-infinite amount of labor. In this sense we

distinguish between:

a) The ab initio procedure. It seeks in principle exact

solutions. All quantities appearing in the calculation are

computed as exactly as numerically possible. If an ab

initio solution is still approximate, which in practice it

always is, this comes from the fact that the form of the

wavefunction has been restricted to facilitate the compu­

tation.

b) The semiempirical procedure. It seeks from the start

only approximate solutions. The simplifications may be quite

drastic, but must always be physically justifiable. One may

in this sense further distinguish between

i) simplification of the electronic Hamiltonian ~ itself

by, for instance, leaving out the electronic repulsion

term Vee and replacing Te+Ven by an effective Hamilto­

nian;

ii) neglect of some intermediate quantities or their em-

pirical calibration on atomic data and on test-molecules.

To study large molecules procedure b) is often the only

tractable one. The more limited reliability of b) as com­

pared to a) is sometimes also compensated by an increased

insight into the interrelation of basic quantities.

3. The subdivision of electrons into different groups

From the chemical point of view the electrons in a molecule

may be subdivided into those which take part in the forma­

tion of chemical bonds, and those which are largely un­

affected by bond formation. The former are generally called

valence electrons, the latter atomic core electrons. If the

4

molecule in its equilibrium conformation (i.e. the equilibrium

geometry of the atomic nuclei) possesses certain elements of

symmetry, such as for instance a plane of symmetry in which

lie all atoms of the molecule, the valence electrons may

appropriately be further subdivided into cr and ~ electrons.

From his experience the chemist knows that. this subdivision

is also physically meaningful. The presence of such ~ electrons

in a molecule influences decisively its reactivity and its

spectroscopic properties.

II. Simple HUckel theory of ~ electrons [3J

The electronic Hamiltonian

may be written

From it we split off a ~ electron Hamiltonian

The cr electrons and the nuclei are assumed frozen into a

molecular core:

(~) (~)

I hcore (tJ.) + I rtJ.\} tJ. >\1

We further average v~~ to obtain an effective ~ Hamiltonian

as sum of pseudo-one-electron parts:

0Uckel ~

(7r)

I heff(tJ.)

5

In this approximation each ~ electron moves in an average

~ of the core and the other ~ electrons.

We now want to solve the one-electron equation

€ • qJ

As we no longer have an explicit Schrodinger equation, the

solutions depend strongly on the form which we impose on them.

1. The LCAO-MO formalism

(and the Ritz variational principle)

We expand our one-electron functions or molecular orbitals

(MO) as linear combinations of basis functions or atomic

orbitals (AO) Xp and write

N

cP I CpXp p=l

In our present case the Xp are 2p~ (E 2pz) atomic orbitals

centered on each atom contributing one (or possibly two)

electron(s) to the ~ system. The total number of such atoms

~ N and the index p also denotes a given atom. The expectation

value for the one-electron energy € then takes the form

SCP*hqxlT €

Jcp*qxi T LL p q

The integrals over the AO's are abbreviated as indicated.

We then make use of the variational principle (without proof):

By making the energy a minimum with respect to the coefficients

c; or equivalently cq , the energy tends towards the lowest

eigenvalue: €min ---. eo. Necessary conditions for a minimum

are:

6

o

(We assume these conditions for our purposes also to be

sufficient. )

We write:

and differentiate with respect to

~:* (I I C;CqSpq) + E: (I Cq8pq ) P P q q

where p = 1,2 ••.• N. Setting the

* c • p'

I q

Cqhpq

derivatives ~ equal to bc*

zero, we obtain the following equations: p

C1 (h11 - eS11 ) + c2(h12 - eS12) +

C1 (h21 - eS21) + c2(h22 e822) +

+ cN(h1 N - eS1 N)

+ cN(h2N- eS2N)

This is a system of N linear homogeneous equations with

o

o

N + 1 unknowns, namely N coefficients c1 ..• cN' and the

eigenvalue e. These equations have non-trivial solutions only

if the determinant of the coefficients vanishes:

o

This determinant is also called the secular determinant, its

polynomial expansion the secular equation.

7

There will, in general, be N solutions for e • To each eigen­

value en there corresponds an eigenfunction ~n' To obtain the

coefficients cnp the condition of normalization must also be

invoked:

1

The solutions then are:

eigenvalues eigenvectors

q:>1 = r C1 pXp P

~2 r C2PXp p

coefficients

Strictly speaking, from a variational point of view only

the lowest solution is physically admissible. If the molecule

of interest has certain elements of symmetry and the solutions

transform according to different irreducible representations,

then the lowest solution of each irreducible representation

is admissible. In general, however, in the frame of the

adopted crude approximations of the RUckel method, all solutions

are considered meaningful.

2. Further simplifications

We write hpp = a p and call it a coulomb integral

hpq = ~pq and call it a resonance integral.

We neglect resonance integrals, except between nearest

neighbors. We neglect overlap integrals; this corresponds

to the zero differential overlap approximation.

Example: Ethylene ----0 0-­~&-8~

2

8

The secular equation is obtained as

I a-£ 13 I 13 a-£

and leads to the

I

" I I

I

----~ a a \ \ , , ,

U

o . • 2 (a-e) = 13

solutions { £1

£2

a-j3 e2 ((l2

a+j3 £1 1:P1

a + 13 a - 13

1 = 'f2' (X1 -X 2 )

1 V2' (X1+x2)

Figure 1 HUckel energy levels in ethylene

;;; 71*

;;; 71

Physically, a may be assumed to correspond to the energy of

an electron in a 2P71 orbital of an sp2-hybridized carbon atom

in its molecular surroundings; it is the negative of the

corresponding atomic valence state ionization potential. 13 is

a measure for the interaction between two such electrons on

different carbon atoms, 1.34 l apart. It may be calibrated

empirically:

Thermochemical calibration of 13:

i) For test molecules the enthalpy of formation AH is d~duced

from measured heats of combustion. It is compared with AH

computed from additive increments for molecular fragments.

The difference is attributed to a resonance energy (see

page 12). The result is 13 - 15 to 20 kcal/mol.

ii) The barrier to internal rotation in ethylene, which is about

25 kcal/mol, is set equal to 213. The result is 13 - 12 to 13 kcal/mol.

9

Spectroscopic calibration of ~:

The longest-wavelength electronic transition in ethylene is

(in part) composed of the ~ ~ ~* transition. It occurs

roughly at 180 nm. A- 1 = 55'000 cm-1

hv = bE S'" 3.5 eV

We note:

Thermochemical predictions require thermochemical calibrations

of S on a test molecule;

spectroscopic predictions require spectroscopic calibrations.

Exercise:

Butadiene; butadiene with symmetry orbitals; analogy with

solutions of the free electron in a box.

Example: Benzene

6

~ ~ 0 0 © '05 2 4

3

Figure 2 Benzene, numbering of atoms, 2p~ orbitals

1 2 3 4 5 6 We divide each

1 a-€ ~ 0 0 0 ~ column by S and

2 ~ a-€ ~ 0 0 0 abbreviate

3 0 S a-€ ~ 0 0 0 £...:.....£ = x

4 0 0 ~ a-€ ~ 0 S

5 0 0 0 ~ ~ We thus get the secular a-€ equation in the form:

6 ~ 0 0 0 ~ a-€

10

x 1 0 0 0 1

1 x 1 0 0 0 One obtains the

solutions: 0 1 x 1 0 0 2k71 0 xk - 2cos b' where 0 0 1 x 1 0

0 0 0 1 1 k 0, ±. 1, ±. 2, 3 x

1 0 0 0 1 x

We then obtain the following energy level scheme and eigen-

functions

3 e3 a -

+2 -2 e±2 a -

-------------+1 n T! -1 et1 a +

H 0 eo a +

~ form of the molecular orbitals:

!J'+2

213

13 tJl+?, CP-2

13 CP+1 CP-1

213 CPo

01

- :: -/ \

/

-$- ~\ ;' l \

/ \ \

6 -$--I

(1 Figure 3 Real benzene MO's

11

Complex rorm or the molecular orbitals:

b k f (-It 'Xp = k L exp (+3p'2'1fi/6) • 'Xp p=l P

N

e:!:2 .1.. L exp (±. 2p·271i/6) • 'Xp These complex orbitals ='{6

p=l are symmetry orbitals

or the subgroup C6 or 1 L exp (±. p'271i/6) . 'Xp D6h. e"!:.1 ='16

p

a .1.. L 'Xp CPo V6 p

The relations between the degenerate real and complex solutions

are

CP+1 ;

1l'-1 ;

Later on we will see that it is more convenient to use complex

orbitals than the real ones. Physically they are or course

equivalent, for any linear combination of two eigenrunctions

belonging to the same eigenvalue is again an eigenrunction to

that eigenvalue.

3. Some important derinitions N

Atomic density:

(Total atomic population)

L bicrrcir i=l

; { occupation number bi = 0, 1, 2

It is a measure of the amount of 'If electrons on atom p. For

all a1ternant hydrocarbons in their ground state

~ = 1 on all atoms.

Bond order:

12

N

I biC~rCiS i=l

0, 1, 2

Within the frame of RUckel theory this is the first-order

density matrix.

Energy of a configuration: It is the sum of the one-electron

energies of the TI electrons in the system of interest. N

E I b i €i

i=l

The lowest configuration possible is the ground configuration.

In it the lowest one-electron levels are all doubly occupied.

It is an approximation to the many-electron ground state. For

benzene we have

6a + 813

Resonance energy: It corresponds to the difference

{EG (TI electrons completely delocalized) -

EG (TI electrons localized in double bonds)}

For benzene this }

may be visualized as l ©] [0] and amounts to

EResonance (6a + 8(3) - (6a + 6(3) 213

Alternant hydrocarbons: They may be divided into alternate

nonneighboring st~d and unst~d carbon atoms

* * * * * ~

t *~* v.v

* 1 *

* *

13

In nonalternant hydrocarbons there

occurs at least one bond between

two starred (or two unstarred) atoms.

In alternant hydrocarbons energy levels are pairwise equidistantly

spaced with respect to a (provided overlap is neglected):

E:i a+xi 13 CDi = I*ciPXp + IO c. ,X I J.p P

p p'

E:(N+l-i) a-xi 13 t'lJ(N+l-i) = I* ci X ~ P P

_ I O ciplXpl

p p'

The corresponding eigenfunctions are characterized by the

fact that for starred atoms the coefficients are the same

and for unstarred atoms they are of equal absolute value,

but opposite sign. The absolute designation of an atom as

"starred" or "unstarred" is of course arbitrary, but such

is also the absolute sign of the molecular orbital.

The so-called "pairing" of electronic states in alternant

hydrocarbons has its physical consequences. For instance, it

follows from the HUckel model that in the radical anion and

radical cation of an alternant hydrocarbon the spin distri­

bution should be identical. The experimental proton hyperfine

splittings in the electron spin resonance spectrum of such an

anion and cation are indeed remarkably similar [4], provided

they are unsubstituted.

The inclusion of heteroatoms:

Much has been written and said on this topic.

In general one may write:

14

Examples [3c] 0 0 Oc=o 0 H

h' N 0.5 - 1.0 hN 1.5 h' 0 1. h" 0

kCN 1. kCN 0.8 kCO 1. kCO

In general one may assume ~Cx ~ ~ . scx, ~ being a proportion­

ality constant. Caution must be exercised in applying this

relation to 3d.-row elements.

Some useful relations:

From E = I biei follows (without proof)

i

1 bE '2 b~sr

Furthermore one defines:

bQs = rrS,q atom-atom polarizability

baq

bPsr = rr bond-bond polarizability b~qt sr,qt

These quantities prove useful in applying perturbation theory

to the HUckel procedure, as they are related to the first and

higher derivatives of the energy with respect to basic

quantities, and as they may be computed from eigenvalues ei

and eigenvectors cip' The pertinent formulae are to be found

in ref. 3f, for instance, and both atom-atom and bond-bond

polarizabilities are tabulated for a large variety of hydro­

carbons in ref. 3e. However, due to increased computer

facilities, the use of perturbation techniques in the frame

of HUckel theory has lost some importance in recent years.

2.

0.8

15

III. Many-electron theory of ~-electrons

We now explicitly consider the interaction between ~-electrons

(5J. Our Hamiltonian has the form

and we no longer consider each ~ electron to merely move in

the inaccurately defined average field of all the others, but

rather to depend more explicitly on their relative positions.

We make the following formal assumptions:

a) Each electron U may in zeroth order of approximation be

described by a spatial one-electron function or orbital

(in our case a molecular orbital, MO) ~(~).

b) Each electron is, with respect to all the others, in a

definite spin-state a(u) or ~(u). We thus associate with

every electron a spinorbital ~(u)'a(u) or ~(u)·~(u). In

the following sections we will often abbreviate ~ • ~,

~ =~.

c) Any many-electron function must be antisymmetric with

respect to the exchange of two electrons, as required

by the Pauli-principle.

Consequently such a function is best represented by a

Slater determinant or by a linear combination of Slater

determinants, each such Slater determinant being an anti­

symmetrized product of one-electron functions, i.e. of

spinorbitals.

1. Ethylene as two-electron problem

We denote the two carbon atoms by a and b and invoke the

cr-~ separability. We thus have a pseudo-two-electron problem.

Starting from our previous one-electron energy level scheme,

16

we can construct 6 configurations consistent with the Pauli-principle.

---1{J2 1 !

H I{J, t t 'Pc.= CPo (t)CP2

1 (2)CP2

1

Of course: 1 Cj)1

V2(1+Sab)'

CP2 1

V2(1-Sab)'

and the ground configuration

1 ~G = V2'

CP1 a ( 1) CP1 t3 ( 1) I CP1 a (2) CP1t3(2)

a b

~ Figure 4

1

! ! (3)CP2

1 (4)CP2

1 cp22

11

Figure 5

{Xa + Xb}

{Xa - Xb}

is wri tten:

v?{CP1a(1)C01t3(2) - Cl'1t3(1)CP1 a (2)}

~~1 (1)7'1 (2) {a(1)t3(2) - t3(1)a(2)}

We abbreviate ~G:; iCP1 CP1 1 , namely as the diagonal part of the Slater

determinant, omitting the normalization factor. The excited

configurations are consequently written:

17

All the singly excited functions are degenerate to zeroth

order.

l2.)~

(.31~~

(4)~~

IC01 q)21

1q)1 q):d

1Ci>1 q)21

ICP1CP21

l2)E~ C~)E~ (4)E~ = a + 13 + a - 13

(in the HUckel approximation)

We assume that a true description of the electrons which we

consider will not be given by ~ configurational function,

but by a linear combination of them:

To determine the correct expansion coefficients AA we apply

the variational principle in formally exactly the same way

as we did in the one-electron case. This leads to a secular

equation

IX~AI - E SAAI I 0

where the indices A and AI run over all configurations. The

dimension of our secular determinant is equal to the number

of configurations of interest.

Now, before we compute the matrix elements of X~ in the

basis of the configurational functions, let us, however,

2a

18

consider the rollowing:

As the operators S2 and Sz ror the total spin commute with

the Hamiltonian (in absence or spin-orbit errects)

o o

matrix elements or KTI will vanish between conrigurational

runctions which are eigenrunctions or S2 and Sz with

dirrerent eigenvalues [6J.

-+ -+ -+ or course S S1 + s2 and Sz s1z + s2z

S2 2 2 -+ -+ s1 + s2 + 2s1 • s2

2 2 + - - + + 2s1 z • s2Z s1 + s2 + S1 • s2 + s1 • s2

where + -s1 = s1 X + iS1 Y , s2 E s2X iS2y etc.

s1Za (1) ~ a(l) S1zfl (1) h We remember that - - fl(l) 2

+ a(l) 0 + fl(l) h a(l) s1 s1

s1 a(l) h fl(l) 51 fl(l) 0

2 a(l) = ih2a(1), 2 fl(l) i h2fl(1) s1 s1

etc.,

consequently:

Sz 4>G 0 and S2 4>G 0

S (1)4> 2 Z 1

l.h. (1)4>2 1

S2 (1).p2 1

2·h2 .(1).p2 1

S (2).p2 z 1 0 S2 (2l.p2

1 h2f2).p~ P).p; }

S (3).p 2 Z 1 0 S2 (3).p 2

1 h2{(2).p~ +(3).p; }

S (4).p2 Z 1

_1·h.(4).p2 1

S2 (4l.p 2 1

2 .h2. (4).p2 1

Sz .p22

11 0 S2 .p22 11 0

19

Exercise: 1) Derive the above relations

Our above functions are already eigenfunctions of SZ. We are

now in a position to write eigenfunctions of both Sz and S2. (2S+l) j We designate these functions by Ms ~i ' where Sand Ms

are total spin and z-component-of-spin quantum numbers.

Singlets Triplets

!~G I q>,. (PI I 3~'l -1 1 1~1cp2\

:~~ = ~ {1~1cp21+1~2~1 I} 3~2 o f = k {1~1qs21-1~2qs11}

~~2,2, o 11 \~2;P21 3~'1

1 1 = \~1 ~21

We now make use of these 6 functions to describe the electronic

states of ethylene. As mentioned, matrix elements of Xn

will vanish between functions of different S or Ms values. So

the triplet functions, as they stand, are already solutions.

We now compute nonvanishing matrix elements of

n e 2 X = hcore(l) + hcore (2) + --­

r12

between the Singlet functions. The matrix elements are ex­

panded into one and two-electron integrals, as indicated. Only

the integration over the spatial variables is explicitly

mentioned. Of course, the orthogonality of a(~) und ~(~) must

also be observed. hcore(~) is just written h(~), where

\..l = 1,2.

+

S\ql1~ \*h(l) \ql1~ \dT1dT2 + id. for h(2) +

SI~1CP1\* ~2 \tP1CP1 IdT1dT 2

Upon expansion of the determinants:

20

For this expression we use the abbreviations

or equivalently

= 2h11 + (11111)

Similarly:

(~~Glx71I~~~) = &Jltp1~11*h(1) {1C(l1~al+I(I)2cp11} d'l"1d'l"2 + id. h(2)

+ k J 1(1)1 q;1 1* ~: 2 {I tp1 q;21+ 1q>2CP1 I} d'l"1 d'l"2

= '12' {(q>1l h lq>2) + (q>1ql11<:p1~2)}

= Y2 {h12 + (11112)} 0

(To ascertain this result, the integrals over molecular orbitals

must be expanded into integrals over atomic orbitals~ see below.)

In the same way we find:

( 1 ~ Ix7111 ~'I.'I.) o G 0 14

(~~~ Ix711 !~~)

(~~~ I X711 ~~!~)

(~~~~ IX711 ~~~~)

(11122)

h~ + h2.~ + (12112) + (12121)

V2' {h •• + (22121)} = 0 to ascertain this, .. ~ see below)

2~2.. + (22122)

From these results the singlet secular equation is obtained.

21

1 1 2.2. 1 ::1 atG at "1 at 1 -----------------------------r-----------------------1

2h1f+ (11\11 )-E (11\22)! 0 I I

(11\22) 2hu+ (22122)-E ! 0 1 -----------------------------:-----------------------

o 0 :hlf+hu+(12Il2)+(1212l)-E -----------------------------~------------------------

We notice that the equation factorizes into a (2~2) equation,

connecting the ground configuration with the doubly excited

configuration,and into a (lxl) equation, of which ~t; is an

eigenfunction with eigenvalue

Consequently, to summarize, our singlet eigenfunctions

have the form:

1 lira

1t2 o t

The coefficients AI and AII have to be numerically determined,

but evidently AI » AII •

The triplet secular equation has no off-diagonal elements

for reasons already discussed. Indeed we have

h11 + h'L'l + (12 112) - (12121)

The three triplet eigenfunctions are given on page 19.

Figure 6 shows at left the relative energies of the various

configurations, neglecting electron interaction, and just

o

22

summing the one-electron energies. Note the degeneracy between

1~~ and 3~~. At right, in a somewhat arbitrary scale, are the

results with electron interaction. A striking feature is the

energy splitting between singlet and triplet .1' no electron interaction c - c;.r with electron interaction

,,,..22 .,.,,,-3cpf

'~-

r~---~A~------~, }2(12 12 T> _

'ep -G

The use of spatial symmetry:

Figure 6

~~ ___ ~y~ ______ ~J

31/1,

(The reader is here assumed to be familiar with elementary

group theory)

The molecule ethylene has the symmetry D2h ,

The MO CP1 transforms according to the irreducible representation blu'

The MO CPa transforms according to the irreducible representation b2g•

(We use small lettering t·o characterize one-electron states and

capital letters to label many-electron states,)

Consequently ~G transforms like Ag blu ® blu ~'l. " " B3u blu ® b2g 1

~·I.'L " i1 " Ag b2g ® b2g

23

As XW transforms like Ag, matrix elements between configura­

tional functions belonging to different irreducible represen­

tations vanish. This explains the factoring of the singlet

secular equation.

The evaluation of integrals:

For numerical computations integrals over molecular orbitals must of course be further expanded into integrals over atomic

orbitals. In this sense we obtain [5J:

a) Two-electron integrals:

(1)(11 (1l(t)

<aalaa> + <ablab> + 2<aalbb> + 4<aalab>

2(1+Sab)2 (1)00 (11l7.J

<aalaa> + <ablab> + 2<aalbb> - 4<aalab>

2(1-Sab)2

(121 12) = J ,'2 ~aalaa~ + ~ablab~ - 2~aalbb~

2(1-S2 ) ab

(12121) = K.'2 ~aalaa~ - ~ab lab~

2(1-sib)

Consistent with our previous notation

S \.l v (ablab) = Xa (1)Xb (2)

(aalbb) = j'Xa (1)xa(2)

(= (aalbb))

(= (ablab»)

The round-bracket notation is used in [5J. These integrals

are either calculated accurately or evaluated semiempirica1ly.

The semiempirical evaluation is important in our context,

i.e. in large molecules, and will therefore be dealt with:

- Zero differential overlap approximation: Wherever the

product Xa(l)Xb(l) occurs it is neglected,unless Xa = Xb·

With this approximation Sab = 0; (aalbb) = 0; (aalab) = O.

24

- We approximate one-center two-electron integrals as

(aalaa) ~ Ia - EAa

where Ia is the potential of ionization of an electron

located in the 2p~ orbital Xa on the Sp2 hybridized carbon

atom a in its molecular surroundings, and EAa is the

corresponding valence state electron affinity. The above equation may be visualized as the transfer of an electron

from one carbon atom in the given valence state to another identical one infinitely far away. The energy required to

carry out that transfer is, on one hand, Ia-EAa, on the other it may be viewed simply as the work required to over­

come the repulsion energy (aalaa).

--------------~~

+ Figure 7

Two-center two-electron integrals of the type (ablab) may be semiempirically approximated by an electrostatic model (for

details, see next section).

b) One-electron core integrals:

<alhla> + <blhlb> + <alhlb> + <blhla> 2(I+Sab)

<alhla> + <alhlb> I+Sab

25

We consider the different terms contained in the core operator

h(l) :: h(Il):

h(l) = T(l) + Ua(l) + Ub(l) + '-----y-----' InI:eractim of the electron w:iih the positive oore at:' cmbon atoms a and b

The matrix element of the

sum of the first two terms

between identical functions

Xa may be viewed as the

negative of the "valence

state" ionization potential

{UH1 (1) + UH2 (1) + UH3 (1) + UH 4 (1)} , ~ .

Relatively small interaction of the electron with the neutral hydrogen atoms. These terms are in general considered negligible

Figure 8

The matrix element of Ub between identical functions Xa may

be considered as the interaction of an electron in Xa with

the positive hole created by the vacancy of an electron in

Xb' and set approximately equal to:

,.., - (ab\ab)

Thus:

The matrix element of h between orbitals on different centers,

Xa and Xb' is difficult to interpret physically term-by-term.

For larger ~ electron systems (see also section III.3) the

core resonance integrals

26

l3ab <a\h\b)

are generally calibrated on a test molecule. The quantities

aa and l3ab, as described here, are not to be confused with the more crudely defined coulomb and resonance integrals of

HUckel theory, in spite of the similarity.

2. The configuration interaction (eI) procedure

The procedure outlined here is, in principle, applicable

to any many-electron system. Our formulation is consequently

not limited to n electrons.

We consider a molecular system with 2N electrons, described

by a Hamiltonian

I h(~) + I I v(~,v) ~ ~>v

e 2 where v(u,v) = --­

- r~v

The wavefunctions for the many-electron system will be linear

combinations of Slater determinants, which in turn are

defined as antisymmetrized products of spinorbitals (spin­

MO's). The MO's are written as linear combinations of basis

orbitals (AO's):

p

We assume that the MO's are ordered with respect to an energy

criterion. They are, in the case of ~ electrons, for instance,

HUckel MO's ordered in the sequence of increasing energy. We

are thus able to define a

ground configuration ~G

singly excited configurations,

27

~~~ = 11:1>1 rp1 ••• rpi epk ••• I:I>i'N1

triplet !~~ = A {II:I>1«>1'" qJi~k'" Cl'lfPNI-1 1:I>1(P1

~~~ 11:1>1 rp1 .•. !:Pi Cflk ••• ~«>N I

doubly excited ~~~, triply excited configurations ~ki~n , etc. ~J Jm

(see Figure 9). In the following we restrict our computations

for simplicity to the ground and singly excited configurations.

The general solutions to the problem

X, E. will accordingly have the form

000 un

'4rG (o)A ~

G G +L L(o)A~ ~k i + ••• '0 ; Eo

i k roc un

{:n= ·n (n)A ~ +L L (n)A~ ~k + ••• *n 1,2 G G i i k

The designations "occupied" orbitals i and "unoccupied"

orbitals k refer to the ground configuration (see Figure 9). The problem now consists in finding the eigenvalues En and expansion coefficients (n)A (n = 0,1,2 ••• ). This is done by

diagona1izing the matrix of X, i.e., in solving the secular

equation, in the basis of the configurational functions ~G k

~i' etc. (see also section III.1). This procedure is called

configuration interaction and is a very general method for treating many-electron problems. If the configurational

functions are in any way reasonably conceived, the coefficient

{o)AG should be large (see also section IV.1 and IV.3) and the coefficients (I1)AG should be small.

28

CPo cp;k cpkl r • 'j

unoccupied { 'PI I I . t I

(virtual) 'Pk k t k t k

orbitals 'PN+1 .

N+1 N+1 N+1

ffJN 1,! N H N H N

occupied ffJJ --4-j H j l j

orbitals ffJ, H i 1 ; ! ; . . . . .

ffJ, H 1 H U 1

Figure 2

We now turn to the necessary evaluation of matrix elements:

To this end we make use of the Slater-Condon rules for matrix

elements between Slater determinants r6]. We find

N

(~GI~I~G) = I 2(~ilhl~i) i=l N N

+ I I r2(~(u)~j(v)lvl~i(U)~j(v» i=1 j=l

- (~i(U)~j(V) Ivl~j(U)~i{V»J

li! I 2(i \h\i) i

+ II {2(ijlij) - (ijlji)} i j

5 I 2hii + I I (2Jij - Kij ) i i j

o \1

til

\1

29

Integrals of the type Jij are called coulomb integrals,

integrals of the type Kij are designated exchange integrals.

The double summations are here taken independently over

spatial orbitals.

We now abbreviate N

Fik = (ilh\k) + I {2(ijlkj) ~ (ijljk)} and find [7]: j=l

(9G \JC 11<i~)

t9~IX11 9~)

(1tilxI19~)

(1 <i~IXI1 t~)

(l<i~IXI f~~)

(3~~IXll~~)

(l~~IXI3~~)

~Fik

(~GIXI~G) - Fii + Fkk - (iklik) + 2(iklki)

- Fmi - (mk\ik) + 2(mklki)

Fkn - (iklin) + 2(ik\ni)

- (mklin) + 2(mklni)

(~GIXI~G) - Fii + Fkk - (iklik)

- Fmi - (mk\ik)

Fkn - (ik lin)

These formulae are exact.

Exercise a) Verify the above expressions for the two-electron

case and compare with section III.l.

b) Verify for the case of the 6-electron problem

III 22 331.

30

3. The semiempirical PPP approximation for ~ electrons [8]

The matrix elements between configurational functions obtained in

section III.2 lead to the evaluation of one- and two-electron in­

tegrals over one-electron functions (MO's). Here we consider the

semiempirical evaluation of these integrals in the frame of ~ electron theory. Consequently we treat in a more general fashion

some points already mentioned in section 111.1., p. 23-26.

a) Two-electron integrals: We write

(Il)

(ijjkt) '-----'

(\I)

and expand

CPi I CipXp P

CD j I CjqXq q

CPk I CkrXr IIII * * CipCjqCkrCts (pqlrs) r p q r s

CPe I CtsXs s

(pqlrs) = J X~(1)X~(2) ;2 Xr(l)Xs(2)dT1dT2 12

= (prlqs) in Pariser-Parr notation [8]

The functions Xp,Xq here. of course, represent 2p~ AO'S on atoms p,q respectively.

Neglect of differential overlap (ZDO-approximation):

leads to the simplification

(pqlpq)

I I c~PcjqCkPCeq (pqlpq) p q

(pplqq) is also often designated ~pq.

31

An estimate of Ypp (see also 11.1, p. 24) is given by the relation

y ""I -EA pp P p

where Ip' EAp are the valence state ionization potential and

electron affinity of atom p, respectively.

For Ypq (p ,; q) Pariser and Parr suggest the "uniformly

charged sphere" approximation (see Figure 10):

Charge density

Slater effective nuclear charge

Figure 10

The radius Rp is given by

R = ~ X 10-8 cm p Zp

R

The number 4.597 is obtained by setting the "electrostatic"

value of Ypp equal to the analytical value.

The simplified form of the electrostatic formula, valid for

R ~ 2.80 A reads:

2 -1/2 R +R 2 -1/2

Ypq = 7.1f75 {[ 1 + (Rp2-:q) ] + [1 + (~) ] } eVe

(For R < 2.80 A, see [8J.)

32

Figure 11 shows the value of Ypq where p and q refer to C atoms as a function of distance

R = 0

~~

" • ~ \. ~ '. ~.-

R~ ~ ~

Fisure 11

b) Core intesrals: The one-electron part of the Hamiltonian,

h(~), refers to the kinetic energy and the core potential. Within the PPP approximation the core consists of the molecule,

fixed in its equilibrium geometry, minu& the ~ electrons. Each

atom contributing one/two ~ electrons carries a positive

charge of one/two in the core. The other atoms of the molecule

(such as H atoms) also are part of the core, but are assumed

neutral (see also III.l, page 25). Upon expansion

(ilhlk) = I I c~pckr (plhlr) p r

we now distinguish different cases

{ p = r, hpp = up ; see below

(plhlr) S hpr p and r are nearest neighbors,

p and r are not n.n., hpr = 0

~pr is best calibrated on the spectroscopic properties of a

test-molecule. It is generally of the order 2-3 eV.

As already stated above and from the point of view of atom p

we subdivide:

hcore(u) T(u)

Kinetic term

+

33

:Ji1ta'1:ction with Intera.cti.on w. lhterootion with atomic cxre p o1her atomic neutral atoms atta:med

(Dres q;ip to the core, sooh as H-atoms

As we have seen previously:

Ip is the "valence state" ionization potential for an electron

in the 2p~ orbital Xp of the sp2-hybridized atom p in the

molecule.

Furthermore:

Uq(U) = U~(u) - z~) J X~(V)Xq(V) ~:V dv

i.e., the interaction with the core atom q is equal to the

interaction with the neutral atom q, Uq(u), minus the inter­

action with the "missing" 71 electrons on that atom, the number

of which being z~). For carbon J71) = 1, for nitrogen i 71)= 1 in

pyridine, but J: = 2 in pyrrhOle; for instance. q

Consequently:

Up = -Ip - I {z~) (pqlpq) - (pIU~IP>} + I (pIU;lp) q;ip r

Neglecting penetration integrals, i.e., integrals corresponding

to the interaction with neutral atoms, or incorporating these

terms implicitly into an effective valence state ionization , potential Ip' the formula simplifies to the "working" ex-

pression

i 71) • 'Y q pq

Let us illustrate this by some examples:

34

-f2 - 'Y12. - 'Y13 - 'Y14 - 'Y15" - 'Y1'

Benzene 1(1) 2 2 - C - 'Y12. - 'Y13 - 'Y14 ,

{ (2)

a., -IN - 'Y12 - 'Y13 -'Y14 - 'Y15

Pyrrhole (2.)

2'Y12. - 2'Y13 • but -IN -

(12 = _f,1) _

2'Y21 - 'Y23 - 'Y24- 'Y25 C

The nitrogen atom in pyrrhole contributes two ~ electrons, 1'1.) thus IN refers to a valence state ionization potential for

double ionization. In the expression for (12' 'Y21 must (~) accordingly be counted twice, as Z1 = 2.

H H H

H H H:O:H + + +

+ + +

H H H N H H H

H

Figure 12, PPP cores

4. Benzene as an example

The ground configuration

of the ~ electron system

of benzene

is represented in Figure 13.

35

b

e+2

e +1 H a

e-2

H e-1

it Figure l~

To discuss the lowest excited states we limit ourselves to con­

sidering only the four degenerate singly excited (singlet)

configurations (see also Section II.2):

The one-electron functions (MO's) in their complex form are

e j =kIw jp Xp j 0, ±.l, ±.2, 3

p

and UJ ;: e 2~i/6

Corresponding to their transformation proper-

ties under the point group Cs we write eo ;: a and e:3 0; b.

Including the ground configuration, but neglecting higher

configurations, the problem would lead to a 5 ><.5 configuration

interaction secular determinant. Full exploitation of symmetry

simplifies the problem decisively, however. In the Dsh point

symmetry of the molecule the ground configuration only mixes

with doubly excited configurations and ~o thus is a relatively

acceptable description of the ground state. (See also Section

IV.3 on the Brillouin theorem.)

36

*0 ~ ~o' transforms like A1g under Deh· From the way the one­electron functions transform under Dsh, we may also deduce how

the configurational functions transform [9]:

.1 1 {ill d2 ~-1 -2 }

- ~+1 transforms like B:m

'2 L {~+2. V2 -1

-2. } + ~+f transforms like B1U

1 {-t _ ~+'2.} } .3 iV'2 ~-1 +1 transform like E1U

1 {-2- ~+2.} and must be degenerate

*4 V2 ~-1 + ~ I

There are no matrix elements between functions transforming

according to different llTeducible representations of the same

group. Consequentl~ the excitation energies are directly

given by the diagonal matrix elements between the above

functions. Taking this into accoun~ and upon expansion:

E1(1B2U) ('1 13(7f I .1 > Eo + h22 - h11 + K02 - K03

E2(1B1U) (t2Ilc7f lh> Eo + h22 - h11 - K02+3Ko3

E3CE1U) (.3I lC7f lt3> Eo + h:?2 - h11 + 2K01 - K03 4 4

where, of course: e 6 6

3(7f I h(\.l) + I I L \l=l \l>v=l

r\lV

Concerning the evaluation of the above matrix elements [5]

A two-electron integral is written

(eieklejee> 5 (iklje> is equal to the expansion

and with the ZDO approximation,

37

(ik I H) 1 \\ (j-i)p+(e-k)q

3b L L llJ • ~pq P q

It may be shown that due to the cyclic symmetry this integral

vanishes, unless

(j-i) (k-e), or (j-i) (k-e) ± 6n, where n = 1,2, •••

All Coulomb integrals (ik\ik) are equal, and the exchange

integrals (ik\ki) reduce to three types, namely K01 , K02 and

K03 , where, for instance:

i k j e (O±ll±l 0)

(±1±2\±2±1)

K01 } = * {'Y11 + 'Y1 2 - 'Y1 3 - 'Y14}

K01

and less obviously:

(-2-1\+2+1) modulo 6 K02 * {'Y11 - 'Y1 2 - 'Y1 3 + 'Y1 4}

(-2-1\+1+2) modulo 6 K03 * {'Y11 - 2'Y1 2 + 2'Y1:3 - 'Y1 4}

In conclusion we obtain the following energy level diagram

(Figure 14).

'E,U ZOeV ",,-2 "'+2 '+'-1 , .... '+'+1 - - - - - 'B,u B.2eV - 'f - I 'B2u -4.geV

If I I I

I I I I I I I L ~'9

Figure 14

38

Listed are the experimental energies above the ground state.

For a comment on the transition probabilities, see the next

section.

Exercise: Verify the energy expression for the 1B2U state.

5. Electric-dipole transition probability

The semiclassical Hamiltonian for a many-electron system in

a radiation field (within the B.O.-approximation) is written

[2]

X = )'.L {p _ ~ Jt (t)}2 + V tr2m \-l c\-l

where t\-l(t) is the vector potential of the field at the site

of electron \-l at time t. V contains all electrostatic terms.

e designates the (negative) charge of the electron.

V I U(\-l) + I I r:: \-l IJ>V

X can be separated into a time-independent part Xo and a time­

dependent part X'(t) containing the perturbation due to the

external field. It can be shown that the time-dependend part

may be written as a mu1tipo1e expansion, convergent if the wave­length of light is much larger than the dimensions of the

molecule (A > 1000 1; L - 10-50 1) [10]:

X' (t) = -E(t).it - H(t).M + .... .. .... E(t) and H(t) are respectively the electric and magnetic

radiation field at some chosen pOint in the molecule. The

probability for a transition from a (ground) state .a to an

excited state 'b is by time-dependent perturbation theory [2] proportional to:

39

In this expression the contribution of the electric dipole

term turns out to be by far the most important one. We conse­quently write. after averaging over randomly oriented molecules:

... Dab is called the dipole strength and R is the electric dipole

operator: ReI e;u . U

The integral <'aIR1'b> is called the transition moment.

For a many-electron (2N) system we find:

and

lib ,.. Bo9?o + I I B~9?~ + i k

where as usual

Consequently

Bo<9?ol~l9?o> + II B~(9?ol~I9?~> • i k

... As Bo is in general small, the first term (where (9?oIRI9?o> within our approximations is the expression for the dipole

moment of the ground state) may be neglected.

Thus to a good approximation

40

'@ e I I B~<CIli \;\CIlk) i k

The problem then boils down to the evaluation of matrix

elements of the operator ~ between one-electron MOls.

I I C~p Ckq <'X.p\~\'X.q) P q

In the frame of PPP calculations on ~ electrons the following

procedure is admissible to estimate orders of magnitude:

.. rop is the position vector of atom p with respect to the

origin of the coordinate system. This expression is exact. The

next expression is only exact between like atoms and orbitals. .. .. .. (rOp+r09) <'X.p\r\'X.q> ~ Spq 2

If one is schematically consistent in neglecting differential

overlap, these cross-terms may even be neglected.

Some symmetry considerations:

A transition a~ is called electric dipole allowed if the

integral <'a\R\'b) = S ': R tb dT fails to vanish. Not to vanish, this integral must transform according to the

totally symmetric irreducible representation of the point

group of the molecule. On the other hand, the integral .. <'a\R\'b) transforms like the triple direct product of the

irreducible representations according to which 'a' Rand

Vb transform respectively. Thus ra®rit®rb must contain the

totally symmetric irreducible representation.

To revert to benzene as an example: ra = A1g, so rit®rb

must contain A1g for the triple product to contain A1g.

This can only be the case if rit = r b • The electric dipole

41

operator R transforms like the vector components x,y,z. The

components x,y in the plane of the molecule transform like

E1U. So rb must be identical with E1U for a transition to

be electric dipole allowed in the plane of the molecule. We

then summarize

Transition A1g ~ E1U electric dipole allowed in the plane of the molecule

Transitions A1g ~ B2U } electric dipole forbidden A1g ~ B1U

If the latter transitions still appear in the spectrum,

this is due to vibronic coupling with normal modes of

appropriate symmetry.

42

IV. Self-consistent-field (SCF)-methods

SCF calculations essentially follow the method proposed by

Hartree and Fock about 40 years ago and applied by these

authors to atoms. It is a variational procedure (analogous

to the simple Ritz method, see section 11.1., page 5) taking

to some extent electron interaction into account explicitly

(l1J.

1. Simple LCAO-formulation of the closed-shell case

We consider a 2N electron system and assume that it can be described by an antisymmetrized function

The Hamiltonian be: 2N 2N

I h(ll) + I

We now seek a set of one electron functions ~1 ••• ~i ••• ~ . such that

J ~~modT J ~:~odT be a minimum.

From the variational principle we may assume that

the minimized energy will approximate the true ground state energy.

We must, of course, impose the constraint that the set of ~i be orthonormal, that is

for all i,j

43

Assuming ~o to be normalized, we find (as on page 28):

Coulomb-term N N N

Exchange-term

L 2(Wi \h\Wi> + I I {2(WiWjIV\~i~j> - (~i~jlvl~j~i>} i=l i=l j=l

The double summations go independently over spatial orbitals.

With the LCAO-expansion:

M

Wi I CipXp p=l

M

~j I CjqXq q=l

M

* I * * ali cirXr r

M

* \' * * ~j L cjsXs

s

We may write:

~rspq

II r p

r s p q

IIII r s q p

[1) (ll (1) (2)

<XrXs \XpXq >

M: number of basis functions

* * CirCjsCipCjq~rspq

* * CirCjsCjqCip~rsqp

(I) (i) (1) C2.)

<XrXs \XqXp>

44

Introducing these expressions into the one for Eo' we see

that Eo becomes a function of the coefficients:

Eo (cip; i=l .•• N, p=l .•• M)

Eo (c~r; i=l .•. N, r=l •.. M).

or equivalently

To minimize this function under the constraints

= I I r q

we obtain the equation below.

'----...... v------'

takes:hID account N orthon:rmali t.v constraints

o r 1 •••• M

Aij is a Lagrangian multiplier. It may be shown (see section

IV.2) that one may set

thereby simplifying our equation to

o

For computational reasons we assume that we consider a par­

ticular value of the indices i and r.

To remember that we consider particular values of i and r let

us set i = i', r=r' and evaluate the above expression. In

differentiating the different terms of Eo we must exercise

some caution. For instance, in the exchange part

* the term in ci'r' will occur twice, once for i=i', r=r', and

once for j=i', s=r', leading to the respective derivatives

45

and

It is easily seen that both terms are equal, leading thus

to a factor of 2 in the general expression. A similar

situation is of course encountered in the coulomb part.

We thus obtain:

2 I ci'phr,p + 4 I Ci'p I I C;sCjq~r'spq p p j sq

Dividing by two and abbreviating N

I C;SC jq == j=l

(==

we find:

I Ci'p {hr,p + 2 I Dsq~r'spq - I Dsq~r'sqp - sr'pe1,} 0 p sq sq

The index i' is now redundant and {P = 1 ... Ml We thus have r' =r=l ••• Mf·

M linear homogeneous equations with M unknowns. The existence of nontrivial solutions requires the determinant of the

coefficients to vanish:

detlhrp + 2 L Dsq~rspq - I Dsq~rsqp - Srpel 0 B:! sq

I

46

p = 1 ... M, r = 1 ... M. The first three terms are customarily

designated by Frp and are the matrix element of the ~

operator F in the basis Xr ' Xp' To solve the problem one

must:

1. Choose an appropriate basis Xp' The bigger M, the better.

M should always be significantly greater than N. For

M=N no energy lowering will be achieved. In this case the

energy will remain constant, as (~oIXI~o) remains in­

variant under a unitary transformation of the ~1 "'~N

among themselves.

2. Compute integrals ~rspq' ~rsqp' hrp ' Srp' However, the factors Dsq couple the equations together and require a

knowledge of the solutions ~i = ; cipXp ' One therefore

proceeds as follows:

3· Guess an approximate form for the CPi and compute approximate

Dsq'

4. Solve the secular problem a first time.

5· Recompute the Dsq with the new eigenvectors.

6. Solve the secular problem a second time.

7. Repeat the procedure until eigenvalues € and eigenvectors

converge (generally about 20 times). Then self-consistency

has been attained.

One thus obtains M SCF-eigenvalues: €1"'€M' corresponding

each to a respective one-electron SCF-function or SCF-MO:

~{C; ~~CF ••• tp~CF. Of these MO' s the N lowest ones are doubly

filled to approximate the many-electron ground state. The

M-N higher ones are virtual orbitals.

If the number of basis functions M becomes quasi-infinite, one

approaches the true SCF energy or Hartree-Fock limit. By

judicious choice of the basis functions one often succeeds in

ab initio calculations to come close to the Hartree-Fock

limit, even if M is finite.

47

2. Semiempirical simplification (ZDO-approximation)

Setting in all integrals X;(~)'Xq(~) 6 pq X; (~) • Xp (~) , the ~ are neglected except for r=p, rspq s=q, and the ~ rsqp are neglected except for q=r, s=p. Remembering that

for a particular matrix element rand p are fixed indices,

sand q running indices, the eigenvalue equation simplifies

to:

det\hrp + 20rp I Dss~rsrs - Dpr~rprp - brp €\ 0 s

Abbreviating ~rsrs !5! ~rs' ~rprp :; ~rp , one obtains:

+ 2 I Dpp~pp .} Dpr is here defined as

F h D ss~ps -pp pp I Cjp Cjr s j

F h - D pr~rp r + p For real orbi tals Dpr=Drp • rp rp

3. More general formulation of the closed-shell case

In the previous sections we optimized the one-electron

functions by varying LCAO-coefficients solely. While this is

very often done in practice, the Hartree-Fock problem may,

however, be more generally formulated without specifying how

the ~i are varied to minimize Eo' We start out from our

energy expression: N

E = Eo I 2<~ilhl~i) + i=l

N N

I I {2<~i~jlvl~i~j) i=lj=l

* and consider the functions ~i' ~j as independent variables.

To minimize this expression with respect to these variables

under imposition of the orthonormality constraints f .. :; lJ

<~ilroj) = 0ij' we may set the following total differential

equal to zero:

\' ~E. 6~i + \' bE * 6 * _ \' \' bf i.i ". 6 _ \' \' bf i.i ,,'" . & * 0 r u~l r bCPi ~i r ~-' b~i lj rt>i r ~bqJ~ iJ tpi

48

This is fulfilled if the factor of any b~i identically vanishes. We thus obtain a series of equations:

o • i 1 ••• N •

which leads to: (To verifY this following expression. write out E and f ij not in terms of brackets. but o£ integrals. Per£orm the differentiation within the integrals and set the sum of integrands equal to zero.)

{2h + 4 I < 'l' j I v I 'l' j >} 'l'i - 2 I < 'l' j I v I 'l'i> cP j - I q> /' j i 0 j j j

We also obtain a set of equivalent complex conjugate equations. 1 Dividing by 2. setting 2 Aji e €ji and abbreviating

•

We use the Coulomb operator J j and exchange operator Kj:

{h + I (2J j -Kj )} 'l'i j

It may now be shown

a) That the matrix of the €ji is Hermitian and may be

brought into diagonal form by a unitary transformation of

the CPi among themselves.

b) That such a unitary transformation leaves the Fock operator

operating on a function q>

{h + I (2J j -K j )} cP = Fer j

invariant.

We thus may assume a priori the CPi to be in the proper form

to write:

{h + I (2J j -Kj )} CPi j

!I

49

These pseudo-eigenvalue equations for the ~i are fulfilled

if the ~i are self-consistent. If not, we must solve the

equation F~ iteratively until the above relation is fulfilled

(see previous section).

From above it follows immediately that

&i' and <~iIFI~j) = &i(~il~j) (fer i+j)

o

This relation of course also holds for matrix elements of F

between filled and eventual virtual orbitals. We now

previously found (section 111.2, page 29):

(IPa 1:lC1IP~) = 'i2' Fik = V2' <~i lF1C11c)

It immediately follows that if the ~i' C11c are obtained by

the same SCF calculation, then these matrix elements vanish.

This situation is summarized as Brillouin's theorem: Matrix

elements between a closed-shell SCF ground state and singly

excited configurations defined within ~he same set of SCF

orbitals vanish.

(This theorem may also be stated in the reverse way. For

instance, we notice that the symmetry orbitals of benzene are

SCF orbitals within the ~ electron approximation, because

all matrix elements between the ground and singly excited con­

figurations vanish.)

From

I (~ilh.+ I (2Jj-Kj)l~i) = I (~iIFI~i) i j i

and as

we immediately find:

N

I ( (i \ F \ i) + (i \ h \ i) )

i=l

50

N

I (€i+hii)

i=l

(This contrasts with the simple HUckel-type one-electron

approximation, where we had N

EHUckel '\ 2€ .• ) o L ~

i=l

The true SCF ground state energy is of course not the true

energy, as the interaction with doubly excited and other

configurations is left out. One generally defines within the

Born-Oppenheimer approximation

Exact nonrelativistic energy minus

Hartree-Fock energy = Correlation energy

4. Koopmans' "theorem"

Monoionization from a neutral closed-shell molecule may, to

an acceptable degree of approximation (with notable and

important exceptions), be pictured as the extraction of an

electron out of a given SCF orbital, the other electrons

remaining unaffected [llJ.

Suppose that the electron comes from a SCF orbital i (see

Figure 16), the energy of the ion will in that approximation

be given by (upper index: 1 singlet, 2 doublet, K Koopmans)

N

1E~CF - [(i\hli)+ I {2(ij\ij) - (ijlji)}]

j=l

Within this approximation it follows immediately that

2E~ =-1 ~on k k + i N + 1

51

1k being the ionization potential. Consequently

(-eN) gives an approximation of the ionization potential to the ground state of the monopositive ion, 11.

( ) { = 12, approximates the ionization potential -eN_l to the first excited state of the monopositive ion,

= 13, approximates the ionization potential to the second excited state, etc.

Koopmans' approximation neglects (Figure 17):

a) the reorganization energy of the electrons in the ion.

This reorganization energy is taken into account in the

restricted open-shell SCF energy of the ion (see Section

1V.6.1).

b) the difference between the electron correlation energy

of the neutral molecule and of the ion.

H eN 1 eN H eN

H H 1 eN-1

H H H

H H H H e, H e, H e,

'EsCF 0 'E sCF e o - N 'E sCF E: o - N-1

... 2E K

... 0 ... 2E K ... 1

Figure 16 Schematic representation of the Koopmans' energy of a monopositive ion 2EK

52

-T- -------I

2£:

R

~ -eN 2[SCF

a2 0

T 2[ 0

'E SCF 0

L1

1[ 0

a, ...L -----------

Figure 17 01: Correlation energy of the neutral closed-shell

molecule. R: Reorganization energy of the ground state of

the monopositive ion. 02: Correlation energy of the ground

state of the monopositive ion. ~: Exact ionization energy

to the ground state of the monopositive ion. Koopmans' "theorem"

states: -€N "" fj,. This implies: R+ 02 "" 01·

5. Some remarks on localized orbitals

The one-electron SCF orbitals ~i obtained by solving molecular

closed-shell Hartree-Fock-problems are called canonical

orbitals. They are the solutions of the pseudo-eigenvalue

equations F~i = €~i. Thus

~SCF o

S ~SCF X ~SCF dT o 0

SCF CPi = CPi

SCF Now it may be shown t~at ~o can be expressed in terms of

any set of orbitals ~i obtained by a unitary transformation

of the (doubly) filled canonical ~~CF among themselves, to

yield the same many-electron energy E~CF [llJ. These new , orbitals ~i of course no longer satisfy the pseudo-one-

electron equations.

It has for instance proven instructive to construct non­canonical orbitals according to certain physical criteria,

such as the criterion of maximization of electrostatic

orbital self-energy D, where [12] N·

D = I <~l~ilvl~l~l) i=l

This corresponds to a minimization of interorbital electron

interactions. Orbitals ~i so chosen can be expected to show nearly minimum interorbital correlation effects. Such orbitals turn out in general to be relatively strongly

localized in certain parts of the molecule and are therefore

called localized orbitals. Their interest lies in the possible

transferability of localized orbitals between different mole­

cules and the eventual transferability of correlation corrections.

There are, of course,other transformation or localization pro­

cedures than the one mentioned, and in considering localized

orbitals it is therefore important to always enquire about the

localization criterion.

6. Open-shell SCF methods

6.1. The restricted open-shell SCF method

Suppose we are interested in computing the lowest triplet

state of a molecule. This may either be done by pure CI or

it may be approached by optimizing one-electron functions in

a manner analogous to the closed-shell HF-method. We thus

need an open-shell SCF procedure. If we require our many­

electron function from the start to correspond to a definite

spin-state, our open shell procedure will, from that point

of view, be called restricted. The restricted open-shell SCF

procedure in its most widely applied form is mainly due to

Roothaan [13].

54

We write our triplet functions as:

_~ ~~ I CP1 ~1 ••.. CPg~g~m~n I

! ~~ /2 {I CP1 ~1 •••• CPg~gCPm~n I - I q>1 ~ .•.• q>g~gq>n~ I }

~~~ Icp1~1 ..• ·cpg~gq>mq>nl

corresponding to a situation as shown in Figure 18.

We want to minimize

We find for the g

energy g g

3E [2 I ~k + I I {2Jke -Kke} 4--- closed shell terms

k=l k=l e=l

g

+ I {2Jkm-Kkm} + k=l

4--- open shell terms

g

I {2Jkn-Kkn}] 4--­

k=l

{closed-open shell

coupling terms

The index g designates the highest doubly filled orbital, the

indices m and n the two singly filled orbitals, the running

indices k and e run over the doubly filled (closed shell)

spatial orbitals only, the indices i and j run over all

spatial orbitals.

The procedure to find a pseudo-one-electron Fock operator

for such an open-shell situation is analogous to the closed­

shell case, with the added complication that it proves diffi­

cult to get rid of the nondiagonal Lagrangian multiplyers

which couple closed and open-shell orbitals. By some clever

but somewhat tedious algebraic manipulations r13] this may

be achieved, leading, for our particular triplet case, to

55

the open-shell Fock operator of the form: (OS a open shell)

FOS h + 2JT - KT + 2M.r -g

I Jk + 1 (Jm+Jn) 2' k=l

g

I Kk + 1 (Km+Kn) 2 k=l

1 (Km+Kn) 2

g

I Mk + ~ (Mm+Mn) , k=l

2KO where

and where

We tnen have the pseudo-eigenvalue equations

The iterative method of solution (for instance within the

frame of an LCAO expansion) is similar to the closed-shell

case. The corresponding eigensolutions then of course satisfy

the relations

. , The g lowest ~i correspond to optimized doubly filled orbitals

and ~m (where m=g+l) and ~n (where n=g+2) to the optimized singly occupied ones. The energy <3~IXI3~> computed from

these orbitals has attained a relative minimum.

56

In the closed-shell SCF case, by Brillouin's theorem, we have

vanishing matrix elements between the ground state (i.e.

ground configuration) and singly excited configurations

defined within the same set of SCF MO's.

Such a general Brillouin theorem does not hold for open

shells. In our triplet SCF case one can prove that matrix

elements vanish between the minimized triplet 3~n function and m other triplet functions with the same number of singly occupied orbitals, one of which must be either m or n: 3~~ • 3~~ , 3~~~, 3~~~. Figure 18 illustrates this in the case of the six-electron

problem. See also ref. [18].

8

5 5 f if 5 t 4

f I f " f H2f " I "

1 3 3 f H if 3 1 H H 1

I H f 2 2 I 2

1! 7 H 11 H H 3cpt 3cp; 3cp: 34>"5

33 34>""

32

Figure 18

Exercise

l. Consider the triplet SCF state 3~" 3' III 22 3 41. Prove

that (3~jlxI3~~> = <4IFOSI5) 0, where 3~f is defined

in terms of the SCF orbitals of 3~j.

57

In the case of a doublet we have

only one singly occupied orbital,

and the SCF equations are accord­

ingly simpler than in the case of

the triplet.

(In reference to Roothaan's paper

[13J: To obtain FOS for the doublet

case requires the same parameters 1 as for the triplet. namely f = 2

a = 1. b = 2; a = 0. ~ = -2.)

t 'Pm

H tpg

H 'Pg-, ,: . H '12 H 'P,

Figure 12 The difference between the Koopmans' energy of the ion (see Section IV.4) and the restricted open-shell SCF energy is

called the reorganization energy.

6.2. The unrestricted open-she'll SCF method

In this method the energy of an arbitrary system of M electrons

with a spin and N electrons with ~ spin is minimized [14J.

The many-electron function ~ is written

~ 1~1~2""~~1~M+2""~M+NI. where the normalization

factor of the Slater determinant of course is {(M+N)!}-1~. The total energy may be ~

E (~I:lCI~>

where

a+~ a+~ a+~ ""

I hii + ~ I I J i j - ~ ( I i i j i

a+~

I i a

I i

means summation over all spin-orbitals with both a; and ~ spin

means summation over all spin-orbitals with only a; spin, etc.

In the double summation the indices take on values independ­ently of each other.

58

Minimization of <~IXI.> leads to separate Fock equations for

orbitals with a and with ~ spin

Fa a IFi

~~

a a £iIFi

£~~

This is accordingly also called the

method of "different orbitals for

different spins".

Although this procedure may take electron correlation to a

higher degree into account than the restricted one, ~ is not

an eigenfunction of S2 and therefore in itself not a physically

~cceptable solution. Once ~ has been optimized, eigenfunctions

of S2 must be projected out of it.

59

V. All-valence MO procedures

As discussed in the introductory chapter, we subdivide the

molecular electrons into groups. From an energy criterion,

we in general merely distinguish between atomic core elec­

trons (for instance, ls electrons of second row atoms) and

valence electrons (2s, 2p electrons of second row atoms,

ls electrons of hydrogen). A further subdivision, into a

and n electrons for instance, is of course only possible

in the presence of an appropriate element of symmetry. The

atomic nuclei and the core electrons are assumed to be

frozen into a static core. The B.O.Hamiltonian of the

valence electrons reads:

:!<Val TVal + VVal-Core + VVal-Val

(Val)

+ I I ~>v

1. The Extended HUckel (EH) Method

This is the all-valence electron analogue of the ordinary

HUckel method. Accordingly, one defines an effective one­

electron Hamiltonian

:KVal eff

(Val)

I heff{~) The first comprehensive application of this procedure to

organic molecules is due to Hoffmann [15], but similar schemes

were followed previously by Wolfsberg and Helmholz [16] and

others to inorganic molecules.

00

For simple hydrocarbons diagonal atomic matrix elements

hqq 5 <XqlheffIXq) are generally set equal to valence state ionization potentials

Hydrogen

Carbon

(lslhlls)

(2slhI2s)

(2plhI2p)

-13.6 eV

-21.4 eV

-11.4 eV

while for nondiagonal elements hqr a variety of modifications

occur in the literature [15]:

hqr k . Sqr(hqq+hrr ) 1 I . 2

hqr -k • ~r(h~·~r)0 II

h ·h hqr k • S . 2 99 rr III qr (hqq+hrr )

hqr (k-ISqrl) • Sqr(hqq+hrr ) 1 IV . 2

In case IV the off-diagonal elements do not automatically

insure invariance of the eigenvalues with respect to a

rotation of the coordinate axis of reference. k is an ad­

justable parameter and is taken to be 1.75 in case I and

between 1.7 and 2.5 in the other cases. As in ordinary HUckel theory, the problem boils down to solving the eigenvalue equation

The overlap integrals Sqr are calculated exactly from Slater

orbitals, with an exponent ~ = 1.625 for carbon and ~H between 1.00 and 1.20 for hydrogen. In ~ electron theory we

have only to consider ~-type overlap (see Figure 20), where­

as in the present case we have all possible combinations of

~ and a-type overlap. The neglect of overlap leads here to

meaningless results. The total electronic energy is just the

sum of one-electron energies: E = t bi £i, bi being the

61

occupation number. For molecules with heteroatoms certain

iterative variants of the EH method have been tested to

improve charge distributions. The diagonal matrix elements

are modified by the atomic charges on the respective atoms

until a (limited) self-consistency is achieved.

Figure 20 ~ and a-type overlap

The EH method has proven very successful in predi-cting most

stable molecular conformations and in studying the local and

overall symmetry of molecular orbitals, in particular in

showing the extent of delocalization of a orbitals. The

success in predicting relative conformational energies is

,difficult to explain in a precise way.

To change the conformation of the nuclei implies (within the

B.O. approximation) not only changing the total electronic

energy, but also the purely classical inter-nuclear (or inter­

core) respulsion. That iS,we have to include the term

I I zAzBe2 (see page 1) in our energy change, where zA

RAB A> B and zB are atomic core charge numbers. We

thus consider the overall Hamiltonian:

62

(Val)

Xovera11 = ~a1 + VCore- = I h(~) Core ~

It appears a posteriori that with a change of conformation

the effective one-electron Hamiltonian, or rather the total

energy computed from it, varies more like Xovera11 than as XVa1 . However, for very small molecules the conformational

predictive power of the EH method becomes questionable. For

instance, according to the EH method the molecule H2 should

have an energy minimum for united atoms.

The EH-method is n2i well suited for spectroscopic predictions:

Electronic transitions have either to be interpreted in a

pure one-electron scheme, or a CI procedure must be added. The

latter possibility does not prove practical here, however.

2. Electronic population analysis [17J

The electronic population analysis gives a systematic procedure

for the interpretation of LCAO-MO data. Consider a simple MO,

such as

; { Xq being on atom A

Xr being on atom B

and suppose the MO is occupied ny N (= b of sections II.3

and V.1; we here use the notation of Mulliken) electrons.

We then have (all functions are assumed real):

N~2 NC~(XqA)2 + 2NCqCrXqA XrB + NC~(XrB)2

Integrating each term over all space gives

N Nc 2 q + 2NCqCr Sqr + Nc 2

r

Netstomb overlap Net atomic p:;u1atlon population popu1aticn on atom A on atom B

63

We also define the gross atomic population on atom

A as N(Cq2 + c c S ) q r qr and on atom

We now generalize for the case of an MO ~i of a polyatomic

molecule, to which every atom contributes more than one AO.

(~) implies summation over all orbitals on a given atom, (~)

implies summation over all atoms. Thus

c 2 i,qA

and consequently

There are no terms between different orbitals on the same

atom because the corresponding overlap integrals vanish.

Based on the above expression we define:

Net atomic population on atom A for orbital i:

Gross atomic population on atom A for 'orbital i:

c 2 i,qA

The sum of this latter expression over all atoms is just Ni •

Summation over the filled levels i of the net and gross

atomic populations then gives the total net atomic and ~

gross atomic populations on atom A respectively.

3. Semiempirical all-valence calculations. including electron interaction

In the all-valence approach a molecule like ethylene,-which

in section III.l we considered as a two-electron problem-,

has twelve electrons. Considering an AO basis consisting of

the 2s,2Px,2Py,2pz orbitals of the two carbon atoms and the

Is orbitals of the four hydrogen atoms, our SCF matrix will

have the dimension 12 x 12. However, we have seen in the ex­

pression for Frp (see page 45) that many-electron integrals

have to be evaluated over all combinations of basis orbitals

which,-in spite of the fact that many integrals will be equal-,

implies the order of 124 integrals. By invoking the ZDO

approximation as defined on page 47, this number is reduced

by a factor of 122. This exemplifies the fact that to keep

calculations on relatively large molecules tractable, approxi­

mations of the ZDO-type may be important.

The following sections are devoted to short descriptions of

such simplified computational procedures.

~l. The CNDO (complete neglect of differential overlap) method

The approximations are f18]:

1) The Xp are treated as if they form an orthonormal set;

thus Spq is set equal to bpq •

2) All two-electron integrals which depend on the overlapping

of charge densities of different basis orbitals are

neglected. This means that

(pqlrs)

3) The electron interaction integrals ~pq are assumed to

depend only on the atoms to which the orbitals Xp and Xq

belong and not to the actual type of orbital. Thus ~pq is

set equal to ~AB' measuring an average repulsion between

65

an electron in a valence atomic orbital on A and another

in a valence orbital on B. (The justification for this

approximation will be given below.)

4) The core matrix element hpp contains the interaction energy of an electron in valence orbital Xp on A with the core of A and with the cores of all other atoms B.

It may be written

h 2 \' hpp = (pl- 2m v 2 - VAlp) - L (pIVB\P) B("A)

and is simplified to

as (pIVB\p) is considered to be the same for all valence atomic orbitals on A.

Upp is essentially an atomic quantity, measuring the energy

of an electron in the atomic orbital Xp on the core of A. (See

p.33: Note the difference with the PPP method in defining the core.)

5) Core matrix elements hpq ' where Xp and Xq are different but both belong to A, may in analogy to 4) be written:

hpq = Upq - I (pIVBlq) B("A)

However, due to the mutual orthogonality of S,Px,Py,Pz' Upq is exactly equal to zer~and the remaining terms are

small, so that one sets

hpq = 0 for p " q , Xp and Xq on A.

6) Core matrix elements hpr ' where Xp is on A and Xr is on

B will for simplicitl be considered proportional to the

overlap integral Spr:

hpr = ~o(A,B,RAB) Spr

~o(A,B,RAB) is a parameter dependent on the nature of

atoms A and B, and eventually on their separation, but

66

not on the form of orbitals Xp and Xr.

With these approximations the Fock matrix elements take on the following form:

(A) (B)

Fpp = Upp - L VAB + 2 L DsslAA + 2 L D SIS Il AB - D pp l AA Sl B(+A) s

or, writing 2 DAA = PAA 2 DBB = PBB (see also page 45):

The expression for Fpq (p+q) applies even if p and q are on

the same atom. Then Spq = 0, and lAB is replaced by lAA.

Parametrization of quantities:

Pople and Segal [18] calculate for second-row atoms:

lAA exactly as (2sA 2sAI2sA 2sA)

lAB exactly as <2sA 2sBI2SA 2SB)

Upp from valence-state ionization potentials, using the com­

puted value of lAA.

where zB is the core charge

number of B

67

~O(A,B) = ~ (~~ + ~~), where ~~ and ~~ are atomic parameters calibrated on small molecules.

- Other approximations for the y:

th (n ) y AB = Y - a exp -11 RAB

yth is the corresponding exact (analytic) quantity. a, 11, and

n are adjustable parameters. For RAB= 0, Y = yth - a. The para­

meter a may be interpreted as a local "correlation term".

- Other approximation for VAB :

VAB = ~ • YAB + P

P refers to the interaction of an electron on A with the

neutral atom B. It is thus a penetration term (and is

eventually neglected). Note the similarity to the PPP

approach (see Section 111.3, p. 33).

4. Invariance of the (exact) SCF eigenvalue problem to unitary basis transformations

We consider a set of closed-shell SCF orbitals ~i' defined in a

basis Xp. In deviation from our usual convention we write

~i I Xp Cpi P

inverting the indices of the coefficients c. In matrix form

(~) (X)(C)

(~) is the row vector of the ~i (i = 1 ••• M), (X) the row vector

of the Xp' and (C) the coefficient matrix of the cpi • This enables us to write the Hartree-Fock equations in a convenient matrix form:

(F)(C) (s)(c)(e)

68

(F) is the matrix Frp of the Fock operator in the basis of the

Xp' (S) the overlap matrix Srp' (e) the diagonal eigenvalue matrix. If {ci } represents the column vector of the coefficients

of ~i' then (F){Ci } = (s){ci}e i , where ei is the (scalar) ith eigenvalue.

We now express our basis (X) in terms of a new basis (X'):

(X) (X' )(u)

Thus (~)= (x')(u)(e) (X' )(e')

The matrices F and S are consequently also to be referred to

the new basis. We find (see below):

(F) (S)

where (U)+ is the conjugate transposed (or adjoint) of (U), and

eFt) and (S') are defined in the basis (X').

For simplicity, we assume both (X) and (X') to form an ortho­

normal basis. Thus (e) then is unitary, and (S) becomes the

unit matrix. (e') must also be unitary and consequently also (U).

Thus:

and we obtain

(U)-1 (F' )(u)(e) (e)(e)

Multiplying both sides by (U) and remembering that (u)(e) we obtain

(F' )(e') or equivalently

(F'){ ci} The SeF equations are now defined in the basis (X') and must

(e' ),

lead to the ~ eigenvalues as the ones defined in the basis (X).

It now remains to be shown that (F') with respect to (X') has

the same form as (F) with respect to (X) (see p. 45):

69

Frp [<Xrlhlxp> + I I I C~j Cqj • S q j

{2<xrxslxpxq> - <xrxslxqxp>}]

With xp I x~ Uap Xq a

X* r Ix'*u* y yr X* s y

and noting that by definition

\' L u~s C~j = Cij s

N

and I Cij C~j .. Di~ (= ~ Pi~) j=l

we obtain

Frp I I Uyr Uap [<X~lhlx~> +

Thus

y a

Frp = I I Uyr Uap F~a y a

which completes our demonstration.

I X~ U~q ~

Ix'*u* o os 0

or

70

In case all the coefficients are real, the word "unitary" in

this section may be replaced by "orthogonal", the word "ad­joint" by the word "transposed".

Invariance of integrals in the CNDO approximation:

If approximation 2) of the CNDO scheme is applied without

further conditions, the integrals do not necessarily transform

to preserve the invariance of the Fock equations with respect

to unitary transformations of the basis.

Consider the integral (PxA sB I PyA sB> which, according to approximation 2), is neglected. Consider the following unitary

basis transformation: Rotate the (local) coordinate system by 450 clockwise around the z axis.

PxA goes into ..l. (p' + p' ) V2 xA yA 1 ( , ,) PyA into i2 - PxA + PyA

and the corresponding integral goes into

1 (p' s' I p' s' > 1 (p' s' I p' s' > "2 + -xA B xA B 2 xA B yA B

1 (p' s' I p' s' > 1 (p' s' I p' s' > "2 + "2 yA B xA B yA B yA B

of which the first and last terms are not neglected and do not

necessarily cancel. The necessary invariance is, however,

restored by adopting an even cruder approximation, i.e. by

setting

(p' s'lp' s'> xA B xA B (p' s' I p , s ' > yA B yA B

Now the transformed integral becomes zero through cancellation.

In this general way we ensure a pseudo-unitary transformation

of the Fock operator, and it may be shown that the SCF eigen­

values remain invariant under the transformation considered.

71

VI. Some special topics

1. Optical activity

A medium is called optically active if the index of refraction

(n) for left (e) circularly polarized light is different from

that for right (r) circularly polarized light:

1'1n ne' - nr + 0

The measurement of this difference as a function of wavelength

A, 1'1n(A), is called optical rotatory dispersion (QfiU). Directly

connected to this effect is the fact that in regions of ab­

sorption the extinction coefficient (e) for left and right

circularly polarized light will also differ:

This latter phenomenon is called circular dichroism (QQ). Inside an absorption band ORD will be anomalous, that is,

there will be an inversion of sign. The combined effect of

CD and anomalous ORD inside a region of absorption is called

a Cotton effect.

Optical activity is a molecular effect. A molecule is optically

active when it cannot be superimposed onto its mirror image.

Such a molecule may ll2l have a rotation-reflection axis

Sn (S1 = cr, S2 = i). Many molecules occurring in living

organisms are optically active.

Every transition a ~ bi in an optically active molecule makes

a certain contribution to ~e and An. A CD/ORD spectrum where

these contributions are clearly resolved may appear as shown

in Figure 21. The transition a ~ b1 leads to a positive

Cotton effect, the transition a ~ b2 to a negative one. While

CD may effectively only be measured in regions of absorption,

ORD curves have long tails outside of regions of absorption

which are the superposition of the contributions of different

transitions.

72

Figure 21

Quantum mechanically it may be shown [2,19] that the contri­

bution which a given transi~ion makes to the CD/ORD spectrum

is proportional to a quantity called the rotatory strengthffiab :

R is the electric dipole operator (see Section 111.5.) and

M the magnetic dipole operator:

L e ~ -e 2mc I-l I-l

For Simplicity we assume here the summation to go only over

all electrons; we neglect vibronic effects due to the nuclei. ~

e and m stand for charge and mass of the electron, el-l = ~ ~

-ih rj..l x vl-l is the angular momentum operator of the I-lth electron.

73

Im { } means that the imaginary part of the quantity in brackets is taken. The rotatory strength is actually a second­

rank tensor, but for a system composed of many identical

randomly oriented molecules one may consider the trace of

this tensor. It is a pseudo-scalar, being the scalar product

of a polar (electric dipole transition moment) and of an

axial vector (magnetic dipole transition moment).

The connection between the rotatory strength and the ex­

perimentally determined quantity 6€(A) is given by the

proportionality (-)

For CD and ORO the rotatory strength plays a role formally

comparable to the one of the dipole strength for ordinary

absorption and dispersion (see Section III.5.):

Dab = Re {<~aIRI~b><~bIRI.a>} - S €~A) dA Band

~

Because of the different transformation properties of Rand ~

M under Sn it may be proven that the rotatory strength always

vanishes for systems containing such symmetry elements.

We now wish to show that even with extremely crude wave­

functions but which correctly reflect local symmetry proper­

ties, a semiquantitative discussion of optical activity is possible.

Case 1: The optical activity of the carbonyl n ~ n* transition

in a ketone (aldehyde). The one-electron energy level scheme

of interest is depicted at left in Figure 22. We assume the

corresponding many-electron states to be well represented by

single-configuration functions. We also consider the highest

filled (n and n) and lowest unfilled (n*) MO's to be markedly

1C'*

n

, I ,-300nm I

-14-0 nm

74

Figure

c o

22

localized on the carbonyl chromophore, as depicted at in Figure 22. In a symmetric ketone, of symmetry C2v' instance, one finds for the transition moments

(nlI!I7r*> 0 (n I iril7r* > + 0 lRn7r* 0

(7rI~I7r*> + 0 (7rliitl7r*> 0 lR7r7r* 0

Thus the n~ 7r* transition, occurring experimentally

right for

at A ~ 300 nm, is magnetic dipole allowed and electric dipole

forbidden, while for the 7r - 7r* transition, occurring at much

shorter wavelength, it is the opposite. The rotatory strength

vanishes in both cases.

Now suppose that we perturb the carbonyl group by introducing

a substituent reducing the overall symmetry to C1 (Figure 23a).

The perturbing substituent will have the effect of slightly

mixing some 7r character to the n orbital:

75

Figure 23a

n' n + An

By first-order perturbation theory

<nIVBln) t>En7l

z

8

8

Figure 23b

where VB is the potential of the substituent R. The rotatory

strength for the n ~ 7l* transition becomes

y

For a given phase of the MO's the sign of Rn7l* will depend on

A. Suppose VR is everywhere positive in space, corresponding

to the potential of the incompletely shielded nuclei of the

substituent (a methyl group, for instance): The sign of the

matrix element <nIVRln) will vary with the position of R as

the sign of the product (n·n), or (y.z), at the position of

R. This leads to a quadrant rule, as depicted in Figure 23b.

If we have just ~ substituent in a given position of cyclo­

pentanone (and if we assume the ring to be planar, which in

fact it is not), this result is trivial: By moving R around

76

the C-O bond we merely go from one enantiomer to the other. However, if we move R from a to ~ position in the same

quadrant, we predict that the sign of Rn~ will remain un­changed. If we have several substituents, we may assume their

influence to be approximately additive, and the relative sign

of their respective contributions to be governed by the

quadrant rule.

The quadrant rule is a special case of a series of group

theoretically deducible sector rules [20J. These sector rules can only predict relative changes in sign of the rotatory

strength •. To predict absolute signs, either empirical cali­

brations, or more elaborate computations are necessary (see

Case 3).

Case 2:

We consider a molecule as composed of two identical monomers

1 and 2. We suppose that these monomers themselves are

optically inactive, but that they are coupled in such a way

that the dimer is optically active. We suppose furthermore

that the splitting of the two degenerate electric dipole­

allowed longest-wavelength transitions may be interpreted

by the dipole-dipole approximation [21J (Figure 24).

---'l~ b_ (A)

--+-~- b+ (8)

_.....1.....1-- a

Figure 24

Then it may be shown that to a degree of approximation which

we presently shall discuss, the rotatory strength of the two

77

longest-wavelength transitions in the dimer is given by

.... .... -+ where R12 = R2 - R1 is the difference of the position vectors of the monomers with respect to some molecule-fixed origin, and -+ .... ~1 and U2 are the electric dipole transition moments of the mono-

-+ .... .... -+ mers: U1 =(a1IRlb1), P2=(a2IRl b 2}. C is a positive constant. As an example we choose 2-2'-diamino-6-6'-dimethyl-biphenyl

(Figure 25). In a simplifying manner we consider the molecule for our purposes to be represented by two coupled aniline

chromophores. We neglect the influence of the methyl substi­

tuants and of the bond connecting the rings. We know from

First transition a - b + Lower energy x,z-polarized Symmetry B under C2

Righthanded Chirality

+ Figure 25

Second transition a-b_ Higher energy y - polarized Symmetry A under C2

experiment that in aniline the longest-wavelength transition is

polarized perpendicularly to the C-N bond [22J. From Figure 25

we find~by the coupled oscillator model applied to the composite

chromophore of C2-symmetry and righthanded chirality, in the

order of decreasing wavelength (see also Figure 24):

First transition B-polarized; Rab+ positive.

Second transition A-polarized; Rab_ negative.

This appears to agree with experiment [22J.

78

In our simplified approach we have localized the transition

moments at the geometric centers of the benzene rings. This

is a point of arbitraryness which must be dealt with. In the

present case this choice appears to be admissible, but in

general, if the dimensions of the monomers are comparable to

the distance betwe~n them, then it is not at all obvious

where we should localize these transition moments. There is,

in fact, only ~ point for each monomer where the correspond­

ing electric transition moment may be localized as a point­

dipole, and for which the formula

JRab C R1 2 • (~2" ~1 )

is exact, otherwise additional terms appear which may be im­

portant and may well even make opposite contributions in sign.

Case 3: The molecule cannot be subdivided into a symmetric

chromophore and an asymmetric surrounding as in Case 1, nor

can it be considered as consisting of interacting subgroups

as in Case 2. Rather, the molecule must be viewed as an in­

herently dissymmetric entity.

An illustrative example is the molecule shown in Figure 26,

which displays a strong optical activity and long-wavelength

Cotton effects of opposite sign at 294 nm and 263 nm [23]. If we apply the procedure of Case 2 indistinctly, localizing

the electric transition moments ~1 and ~2 in the center of

Figure 26

79

the benzene rings 1 and 2, pointing perpendicularly to the

C-N bonds, the computed rotatory strength vanishes, because ~ ~ ~ ~1, ~2 and R12 are coplanar. This implies that the transition moments should be pointed and localized differently. This

must be done cautiously, or it may easily lead to wrong pre­

dictions. In other words, a more accurate assessment of the

quantity 1m {<*aIRI'b><'bIMI*a>} is necessary [24].

A possible procedure, actually leading to correct predictions

with regards to order of magnitude and sign of the Cotton

effects, is the following [10,25]:

1) Compute the SCF ground state (Section IV.I.) and the

lowest excited states by single-excitation CI (Section 111.2.),

assuming local cr-~ separation in all three benzene rings and

invoking the PPP approximation (Section 111.3.). The nitrogen

atoms of the substituents (Figure 26) are considered to be of

the pyrrhole-type and to cintribute two (pseudo-)~ electrons

each. We thus treat the molecule as an inherently dissymetric

(pseudo-) 22~ electron system. The ~pq integrals between all

eighteen carb·on atoms and the two nitrogen atoms enter the

computation. Only the resonance integrals ~pq between nearest

neighbors within a conjugated subunit (N1-benzene ring 1;

N2 -benzene ring 2; benzene ring 3) are taken into account.

The core matrix thus has the aspect given in Figure 27.

2) From the semiempirical wavefunctions the rotatory strength

of the longest wavelength transitions are then evaluated.

In practice it proves necessary to compute the electric dipole

transition moment in the dipole velocity form. For exact

eigenfunctions 'a and $b of the same Hamiltonian we have the ~

identity (see p. 39 for definition of R):

where i = L ~~ . For the rotatory strength we then obtain ~

benzene N/ 1

~ \~ where L ;: L e~

~

80

benzene 2 3

Figure 27

Expressing the wavefunctions in terms of configurational

functions

. •

we find (see Section

(wa lwl1j1b> ... Y2I B~ ik

~ Y2I B~ (tbIL1*a> .... ik

III.5.):

(Cfli I~ I Cflk > and

-+ (Cflkl e 1Cfli>

The configurational coefficients B~ are assumed real, likewise

the SCF-MO's Cfl i • As t;: -ill :;,,,~ and as (Cflkl:;'x~ICfli> =

- <Cflil:;'x~ICflk>' we finally obtain

81

The approximations inherent in this expression originate in

the approximate nature of ta and Wb. k Returning to our example, the ~i and B. are computed as de-~ -+

scribed under 1). The matrix elements <~ilvl~) and -+ ..

<~ilrxvl~) are numerically evaluated without further approximations. They reduce to integrals between atomic

orbitals which, in the case of Slater orbitals, boil down to

the evaluation of linear combinations of overlap integrals

[25]. The matrix elements of ~ A~ depend on the origin of the

coordinate system; the matrix elements of ~ do not. It may be

shown that Rab , as computed above, likewise is origin-inde­

pendent, which of course it should be. However, as mentioned,

the electric dipole transition moment must be computed in the

dipole velocity form [26J.

Using standard PPP parametrization [22J and taking into

account 99 singly excited configurations we obtain for the

longest-wavelength Cotton effects of our triptycene derivative

(Figure 26):

291 nm

285 nm

264 nm ,

R01

R02

-0.44.10- 36 cgs

+0.33.10- 36 cgs

+0.21.10- 36 cgs

This appears to agree with experiment as to order of magnitude

and sign.

2. Selection rules for electrocyc1ic reactions and cyc10-addition reactions

Molecular orbital theory in its simplest form, in particular

the EH approximation [15], has provided a brilliant means of

rationalizing and interpreting the regularities encountered

in concerted organic reactions. The corresponding rules, now

generally called Woodward-Hoffmann-ru1es [27J. grew out of

the necessity to rationalize empirical evidence such as the

82

following:

Thermal (~) ring closure of butadiene proceeds by a conrotatory

movement of the substituents, photochemical (hv) ring closure

goes in a disrotatory way-. For hexatriene it is the opposite.

Already prior to Woodward's and Hoffmann's systematic investi­

gations, Fukui and Oosterhoff had independently [28] suggested

that the course of such reactions might be connected to the symmetry of the highest occupied orbital of the polyene. The

2Pn AO's of the terminal carbon atoms may be thought of as combining with the proper phase to form a bonding a orbital.

In butadiene in the ground state (thermal path) the highest

filled orbital is n2 (Figure 28), while in the first excited

state (photochemical path) it is n3.

From a more general point of view it became clear that such

selection rules could be better interpreted by looking at the

83

Figure 28

totality of participating electrons [29J. Assuming that the

states of the reactant and of the product may be characterized

by a common and relevant symmetry element, correlation dia­

grams can be drawn as shown in Figures 29a,b and 30a,b for

electrocyclic reactions, and 31a and 31b for cycloaddition

reactions. The following points are of importance:

1) During the transformation at least one element of (overall

or local) symmetry is maintained. The states of reactant,

product and most plausible transition state may be

characterized by it.

2) The symmetry element of importance must bisect the bond(s)

which is (are) being formed or broken.

3) In correlating the states of reactant and product the

"noncrossing rule" holds: Lines correlating states of

same symmetry may not cross.

84

4) For simplicity it is assumed that the states of interest

can be described by single configurations, defined in a

set of appropriate one-electron MO's. The one-electron

states are of course also caracterized by the same symmetry

element(s) and correlate in a similar way as the overall

many-electron states. The energetic sequence of the one­

electron states and their occupation determines the

energetic sequence of the many-electron states.

5) There are cases where the correct correlation diagram cannot

be established unambiguously by inspection. In such cases

a series of EH calculations along the reaction path may

illustrate how the molecular orbitals gradually evolve.

Two remarks must immediately be added:

- The present MO theoretical description and interpretation

does not automatically lead to the most compact formulation

of the selection rules, as the practical chemist seeks them.

On this question there also exists a vast amount of

literature [30J.

- The simplicity of the elementary molecular orbital approach

invites further refinement. Until now it appears that more

sophisticated treatments basically confirm the results ob­

tained from the simpler picture [31J.

We now turn to elementary examples:

Figure 29a illustrates the electrocyclic reaction of butadiene

over a conrotatory path. The symmetry element C2 is maintained

throughout and the one-electron states correlate as shown. The

transformation properties of the corresponding many-electron

states, denoted by big letters A (symmetric) or B (anti­

symmetric), is immediately deducible from the symmetry,

designated by small letters a or b, of the MO's occupied by

electrons participating in the reaction. In the thermal case

we see that the ground state (ground configuration) of buta­

diene correlates with the ground state (ground configuration)

Th

erm

al

<l-

--C

>

1C+a

7r3

b

7r2a

T1

\.

1C, b

T±

n:2

_2

1 .I&

.2

A

/'

"-

Bu

tad

ien

e <

I----t

> C

yclo

bu

ten

e

Co

nro

tato

ry

pa

tn

Sym

met

ry C

2

Pho

toch

emic

al

ba*1

a1C*

1

6.

b1C I

aa

I

a2n:

2

A

--/-

f>

I

1C+ a

1C3 b

----r-

/

1C2a

" 1C

, b

Ti /'

n:/n

: 2 :1l

"3

B

rba*

i I a1

C*1 I I I I I

-" t1

b1

C I I

'" t a

a

.

a n:

2a

* B

Fig

ure

2

9a

11;+ a

1C3b

7r2

a 'V

1C, b

n: 1n:

1n:,

B

<r+

-b

att

.......

T

a1C

*

co

C11

'\.

/ '(

b1C

" T

I a

a

a2n:

n:*

B

86

Energy

U7r2U*

Reaction path

Figure 2gb

of cyclobutene. This implies (Figure 29b) that no particular

potential barrier is to be expected on going from one species

to the other. In the photochemical case. on the other hand.

the situation is different: Starting with the first excited

state (configuration) of one species we end with the second

excited state (configuration) of the other (Figure 29a). The

configuration ~f ~2 ~3 of butadiene correlates with cr ~2 cr*

of cyclobutene, and for the reverse photochemical reaction

cr 2 ~ ~ of cyclobutene correlates with ~1 ~~ ~4 of butadiene.

Both paths involve an increase in energy. Now,because all of

these four states are of the same symmetry B with respect to

C2, the adiabatic paths will interact due to the "noncrossing

rule". This implies that if we start out with butadiene in

the configuration ~~ ~2 ~3 we will not end up with cr ~2 cr*

as deduced from Figure 29a, but with cr 2 ~ ~. However, the

Bu

tad

ien

e <

l--C

> C

yc/o

bufe

ne

Dis

rota

tory

pa

th

Sym

met

ry C

s T

herm

al

~

<J-

f----

a"u*

I a"

u *

7C. a"

I

7C. a

"'-

--.

_ .

I I ~

a"1r

* I

a''1

r *

1r 3

a'

/ I

1r3 a'

t I

~

1r2a

" T

I ---

a'7('

7r1a

' ti

----

n a'

u

1I:{1

I:J

A'

q2

11

:*2

A'

7r 2aJ

l -

" I

I a'

7r

7r1 a'

+:l

; ----

fi a'

u

1I:{1

I:f

A'

q211

:2

A'

Fig

ure

30a

Pho

toch

emic

al

<J--

--t>

a"u*

7('

.a"-

--

_....j.

--a"

7r*

1r3a

'-......

. -

7r2a

" T

I

" a'

1r

7r1 a'

fi

----

ti

a' u

2 11

:, 11

:211:3

A"

q211

:11:

*

A"

!S

88

Energy

Reaction path

Figure 30b

reaction will have to proceed over a high potential barrier,

and this course is accordingly very unlikely. For I(t con­

rotatory reaction path the thermal course will consequently

be preferred over the photochemical course.

In a disrotatory reaction path a plane of symmetry is main­

tained. One-electron states and many-electron states are

characterized with respect to it by the respective symbols

a' or A' (symmetric), a" or A" (antisymmetric) (Figures 30a,b).

The correlations are established accordingly. Here we see

that it is the thermal course which leads over a high potential

barrier. In the one-electron picture (Figure 30a) the ground

configuration of one species apparently correlates with a

doubly excited configuration of the other. Although the "non­

crossing rule" comes into play (Figure 30b), it is to be de­

duced that the transition state in the thermal course will

89

nonetheless be of relatively very high energy. On the other

hand, in the photochemical case, the lowest singly excited

configuration of reactant always correlates with the same

singly excited configuration of the product. In the dis­

rotatory reaction path the photochemical course will conse­quently be favored over the thermal course.

We now turn to cycloaddition reactions and consider the

example of the addition of ethylene and butadiene to give

cyclohexene. The course of this reaction proceeds in such a

way that a plane of symmetry is maintained.

~I * I 1{ I

I

q*.(i\, 2~

I

utOo

Figure 3la

1{*

90

In accord with group theoretic usage we characterize the one­

electron states by a ' and a" with respect to the relevant plane

of symmetry. as in the previous example. Figure 31a shows at

left the three highest occupied orbitals of the reactants

butadiene+ethylene in the proper relative energetic sequence

1t1 (a ' ), 1t(a l ), 1t2(a") and the three lowest unoccupied orbitals

in the corresponding relative sequence 1t3 (a '), 1t*(a" ). 1t4 (a").

In Figure 31a at left the three highest occupied and three

lowest unoccupied orbitals of the product cyclohexene are drawn

--a" ut

--- a" a' --- U,*

7C*---a" ~ II ~ a--7C3 -- a'

Figure 31b

91

in a simplified way, also in the proper energetic sequence

a1 (a'), 02(a"), n(a'); n*(a"), aHa'), a~(a"). The correspond­

ing correlation diagram for the one-electron states is shown

in Figure 31b. Considering the occupation of the orbitals,

we see that the ground configuration of the reactant corre­

lates with the ground configuration of the product. making

the reaction thermally allowed.

The photochemical course, starting from the lowest excited

configuration ~f ~2 ~2 ~3 of the reactants, correlates with

the higher excited configuration 0* o~ ~ af of the product

and is therefore energetically unfavorable. The question is of

interest, if the reaction could not proceed from the photo­

excited ethylene, instead of the photoexcited butadiene. The

configuration ~~ ~ ~~ ~ indeed correlates with the lowest

excited configuration of the product, of a~ ~ ~. However,

it is possible that as soon as the reactant molecules inter­

act,the state ~f ~ ~~ ~ internally converts by vibronic

coupling to the lower excited state ~~ ~2 ~2 ~3, thereby

impeding the further course of the reaction.

For a discussion of other concerted organic reactions, such

as sigmatropic or cheletropic reactions, the reader is re­

ferred to the literature [27,30J.

Exercise: Draw the state correlation diagram for the cyclo­

addition reaction ethylene+butadiene, based on Figure 31b.

3. Molecular orbital theory with periodic (cyclic) boundary

conditions

This form of molecular orbital theory provides a means of

studying in the tight-binding approximation the electronic

structure of polymers built from sequentially repeating

subunits [32J. It is assumed that, neglecting end effects,

cyclic boundary conditions may be applied. We consider a

92

polymer consisting of N· monomers and assume that per monomer

v valence orbitals (atomic orbitals) have to be taken into

account. The number of electrons per monomer is ve. Further­

more, we denote the monomers by the indices p, p', q, q',

the AO's within the monomers by s, s', t, t'. We characterize

the polymer MO's by double indices jm, j'm', en, e'n', where

j, j', e, e' designate the symmetry of the MO under the

cyclic point group CN•

The s th atomic orbital within the p th monomer is written

Xps. Let us choose the origin of the molecule-fixed coordinate

system to coincide with the center of a particular AO, Xs = Xs (t). The corresponding AO Xs (t - Er) will be centered in the

point Er, Xs (t - 2Er) will be centered in the point 2Er, etc. If

Er is the primitive translation vector of a linear polymer we

may write

Xs(t) Xos

Xs(t-Er) X1 s

Xs(t-2~) X2S

Xs (t - pit) Xps

Considering the periodic boundary conditions or formal cyclic

symmetry we in general write a molecular orbital, extending

over the whole polymer, as [33J: N v

L j wjpc. X .-' Jm, s ps

p=l s=l

or in bracket notation

I jm) Ips)

p s

where w = exp(2~i/N). In a given polymer the Xps will have to

be appropriately chosen, so as to conform to the particular

(for instance helical) geometry. From the above relations one

93

finds

N \I

I I wjpc X jm,s (p-l)s

p=l s=l

p=l s=l

which, due to the cyclic properties

and consequently

The function must indeed have the same value after N elementary

translations.

The HUckel/Extended HUckel approximation:

See Chapter II and Section V.l.

We define symmetry orbitals (in bracket notation):

N

I js) L wjp Ips)

I j,j' = 1 •••• N

p=l or equivalently, for N even:

N j,j' = O,±l, ... ± (N/2-1), N/2

\j's')= I j 'p' \p's') for N odd:

w j,j' = O,±l,

p'=l ••• ± (N-l)/2

and express the eigenvalue equation in the basis of these

orbitals

\(jslhlj's') - €(jslj's')\ 0

94

This is equal to

II {(PSlhlp's') - e(pslp's')} o

The indices s, s' go from 1 to v. It may be shown [34] that the elements of this secular determinant vanish, unless

j = j'. The eigenvalue problem, originally of order N'v,

thus factorizes into separate equations of order v for

every value of j.

Example: A cyclic chain of N "ethylene" molecules (see

Figure 32). We have s, s' = 1,2

p-1

{J' {1

1 2

(J' 1

p

(J

2

Figure 32

p+1

(J' (J {J' 2

Invoking the ZDO and nearest-neighbor approximation, as in

ordinary HUckel theory, we find

(pllhlpl)

(pllhlp2)

(pllhl (p-l)2)

(p2Ihlp2) a

13 (p2Ihl(P+l)1) = 13'

Adding the contributions for p 1,2 ••• N and then dividing

every element by N we obtain for the secular equation

95

o a - e

Leading to the solutions

{ 27fj}1/2 e a ± ~2 + ~'2 + 2~~' cos N

For ~' = 0, the formula reduces to the HUckel energy levels

of independent ethylene molecules. For ~ = ~' we get

a ± 2~ cos 1t With N=3 we correctly obtain the HUckel energy levels of

benzene (see page 10), j taking on the values 0, ±l.

The SCF formulation:

See Sections IV.l. and IV. 3 ( Attention: The symbol N there and) here has a different meaning.

The Hartree-Fock equation in the basis of symmetry orbitals

may be expressed as

I (jsIFlj's') - e(jslj's') I 0

(jsIFlj's')

N 'Ve/2

(jslhlj's') + L I {2 (js enlj's' en) - (js enlen j'S')} e=l n=l

The summation goes over all occupied polymer orbitals. Here

we assume that there is no overlapping of the energies of

bands for which n $ 'Ve/2 and of bands for Which n > 'Ve/2.

We admit that we have the situation shown in Figure 33a and

not the one illustrated by Figure 33b. A polymer MO len) is written

len) I I weqcen,t Iqt) q t

96

The symmetry index e - like j - takes on the values

e = 1, 2 ••. N or, equivalently, e = 0, ±l, ••• ±(N/2-1), N/2 •

It may be shown [34J that matrix elements of the Fock operator

between symmetry orbitals vanish, unless j' = j. With s,s' =

1, 2 .•. v, the Fock matrix consequently factorizes into N

submatrices of order v. Expanding the Fock operator we find:

(jsIFljs') = [P~, P,~, wj(p'-p) (pslhlp's') +

N N N N

c* c , en,t en,t I I I I wj(p'-p)+e(q'-q) •

e=l n=l t=l t'=l p=l q=l p'=l q'=l

• {2 (ps qtlp's' q't'> - (ps qtlq't' pIS'>}]

This formula is exact within the Hartree-Fock approximation.

It is indeed rather cumbersome to evaluate this expression

for large N, in particular the two-electron part. The fall-off

monomer polymer

1 2 Ve }N

~ Ve -1 ti :::::::::=:l~ } N

Figure 33a

monomer polymer

N

N

Figure 33b

97

of the corresponding integrals with distance may be taken

into account to simplify it, as indicated in [34J.

Within the frame of a semiempirical approach the ZDO approxi­

mation is useful (see Section IV.2). The electron repulsion

integrals (ps qt\p's' q't') are then neglected, except when

ps = p's' and qt = q't'. SimilarlY only integrals of the form

(ps qtlq't' p's') are retained for which ps = q't' and

qt = p's'. We remember that sand s' are fixed indices for

a given matrix element, while t and t' are running indices.

The two-electron part then reduces

for the diagonal element (jsIF\js) to:

N Ve/2

I I {2 I c1n,t Cen,t I L (ps qtlps qt) e=l n=l t p q

- c* c \ \ w(j-e)(q-p) en,s en,s L L (ps qslps qs)}

p q

and for the nondiagonal element (js!F\js') to:

c* en,s c , ';\ w(j-e)(q-p) (psqs'\psqs') en,s '-" L

p q

The computation of optical properties [34J: The CI matrix also

factorizes according to the irreducible representations of the

group CN (for N even):

A, E+1 , E_1 , ••••• E+ j , E_ j ••••. E+(N/2_l)' E_(N/2_l)' B

One starts from a total of N'v polymer MO's, of which

N • ve/2 are filled. Consequently N·.,f possible singly excited

configurations may be constructed, where

j( = N' v2e ('J _ ~e)

98

With respect to their symmetry, these singly excited con­

figurations are evenly distributed over all the irreducible representations, as long as we assume a situation as shown

in Figure 33a. Consequently there should be J(' singly excited

configurations belonging to each of the N representations

A(j=O), B (j=N/2) and sub-representations E+j , E_j' As the

dimension of the polymer grows, the computational labor of

diagonalizing these N large matrices will rapidly become

immense. To limit the expense in calculating optical spectra,

one should then give priority to the excited states to which

transitions from the ground state are electric dipole allowed.

The corresponding selection rules are therefore of immediate

interest. We assume that higher retardation effects may be

neglected. To derive the selection rules it is essential to

distinguish between the formal cyclic symmetry CN - which we

assume always to apply in the sense described - and the actual

geometry.

One finds:

a) For a linear geometry transitions are allowed when j' j.

This implies that there is only one CI submatrix of

dimension Jrbetween singly excited configurations belong­

ing to the irreducible representation A to be considered.

b) For a cyclic geometry transitions parallel to the symmetry

axis of r~tation are allowed when j' = j. In-plane transi­

tions are allowed when j' = j±l. This implies considering

the CI matrices beionging to the irreducible representations

A, E+I and E_I •

c) For a helical geometry the selection rule j' = j holds in

the case of transitions parallel to the helical axis. For

components polarized perpendicularly one obtains j' = j±M,

where M = N/~ and ~ is the number of monomers per helical

turn. In general, ~ will not be an integer. However, M

99

must be an integer. Therefore a clear-cut selection rule

will only hold for values of N which correspond to one or

several translational identity periods along the axis of

the helix. The CI matrices to be considered will belong to

theirreduoible representations A and E+M, E_M•

Some References

[lJ M. Born and R. Oppenheimer, Ann. d. Physik 84. 457 (1927).

[2J H. Eyring, J. Walter and G.E. Kimball, "Quantum Chemistry",

John Wiley, New York 1963.

[3] a) E. HUckel, Z. Physik IQ, 204 (1931).

b) R. Daudel, R. Lefebvre and C. Moser, "Quantum

Chemistry, Methods and Applications", Interscience,

New York 1959.

c) A. Streitwieser, "Molecular Orbital Theory for Organic

Chemists", John Wiley, New York 1962.

d) L. Salem, "The Molecular Orbital Theory of Conjugated

Systems", Benjamin, Inc., 1966.

e) E. Heilbronner and P.A. Straub, Table of HUckel

Molecular Orbitals, Springer-Verlag, 1966.

f) E. Heilbronner and H. Bock, "Das HMO-Modell und seine

Anwendung, Grundlagen und Handhabung", Verlag Chemie,

Weinheim 1968.

[4J A. Carrington and A.D. McLachlan, "Introduction to

Magnetic Resonance", Harper and Row, Hew York 1969, p. 89.

[5J R.G. Parr, "Quantum Theory of Molecular Electronic

Structure", Benjamin, Inc., New York 1963.

[6J a) E.U. Condon and G.H. Shortley, "The Theory of Atomic

Spectra", Cambridge University Press, Cambridge 1963, p. 169-174.

b) J.C. Slater, "Quantum Theory of Atomic Structure",

Vol. I, Mc Graw-Hill, 1960, p. 291-295.

[7J J.A. Pople, Proc. Phys. Soc. A 68, 81 (1955).

[8J R. Pariser and R.G. Parr, J. Chern. Phys. £1, 466, 767 (1953) .

101

[9J W. Moffitt, J. Chern. Phys. 22, 320 (1954).

[10J J. Fiutak, Canad. J. Phys. 41, 12 (1963); see also

R.E. Geiger and G. WagniE'lre, in "Wave Mechanics, the first fifty years", ed. W.C. Price, S.S. Chissick,

T. Ravensdale, Butterworths, London 1973, Chap. 18.

rllJ a) R.S. Mulliken, J. chim. phys. 46, 497, 675 (1949).

b) C.C.J. Roothaan, Rev. Mod. Phys. ~, 69 (1951).

112J C. Edmiston and K. Ruedenberg, J. Chern. Phys. ~, S97

(1965) .

[13J C.C.J. Roothaan, Rev. Mod. Phys. ~, 179 (1960).

[14J J.A. Pople and R.K. Nesbet, J. Chern. Phys. 22, 571 (1954).

[15J R. Hoffmann, J. Chern. Phys. Z2, 1397 (1963); see also W. Hug and G. Wagniere, Tetrahedron~, 631 (1969).

[16J M. Wo1fsberg and L. Helrnho1z, J. Chern. Phys. 20, 837

(1952).

[17J R.S. Mulliken, J. Chern. Phys. ~, 1833,. 1841 (1955).

[18J J.A. Pople, D.P. Santry and G.A. Segal, J. Chern. Phys.

~, S129 (1965); J.A. Pople and G.A. Segal, ibid. ~,

S136 (1965); see also M. Jungen, H. Labhart and

G. Wagniere, Theoret. Chirn. Acta i, 305 (1966);

J.M. Sichel and M.A. Whitehead, Theoret. Ohirn. Acta 1, 32 (1967); R.J. Wratten, Chern. Phys. Letters 1, 667

(1968).

[19J L. Rosenfeld, Z. Phys. 52, 161 (1929); E.U. Condon,

Rev. Mod. Phys. 2, 432 (1937); E.U. Condon, W. Altar

and H. Eyring, J. Chern. Phys. 2, 753 (1937).

[20J W. Moffitt, R.B. Woodward, W. Klyne and C. Djerassi,

J. Am. Chern. Soc. ~, 4013 (1961); J.A. Schellrnan,

J. Chern. Phys. 44, 55 (1966); A. Moscowitz, Adv. Chern.

Phys. 1, 67 (1962); J.A. Schel1man and P. Oriel,

102

J. Chern. Phys. 21, 2114 (1962); G. Wagniere, J. Am.

Chern. Soc. 88, 3937 (1966).

[21J I. Tinoco, Adv. Chern. Phys. ~, 113 (1962); J.A. Sche11-man, Accts. Chern. Res. 1, 144 (1968).

[22J R.E. Geiger and G.H. Wagniere, in "Wave Mechanics, the

first fifty years", ed. W.C. Price, S.S. Chissick,

T. Ravensdale, Butterworths, London 1973; H. Labhart

and G. Wagniere, He1v. Chim. Acta 46, 1314 (1963).

[23J J. Tanaka, F. Ogura, M •. Kuritani and M. Nakagawa,

Chimia 26, 471 (1972).

[24J A.M.F. Hezemans and M.P. Groenewege, Tetrahedron~,

1223 (1973).

[25J W. Hug and G. Wagniere, Theoret. Chim. Acta 18, 57

(1970); G. Wagniere, in "Aromaticity, Pseudo-Aromaticity, Anti-Aromaticity", the Jerusalem Symposia on Quantum

Chemistry and Biochemistry, III. The Israel Academy of

Sciences and Humanities, Jerusalem 1971, p. 127;

G. Blauer and G. Wagniere, J. Am. Chern. Soc. 21, 1949

(1975) •

[26J W. Moffitt, J. Chern. Phys. ~, 467 (1956).

[27J R.B. Woodward and R. Hoffmann, J. Am. Chern. Soc. §I, 395 (1965); R. Hoffmann and R.B. Woodward, J. Am. Chern.

Soc. §I, 2046 (1965); R.B. Woodward and R. Hoffmann,

"Die Erha1tung der Orbita1symmetrie", Verlag Chemie,

Weinheim 1970.

[28J K. Fukui, T. Yonezawa and H. Shingu, J. Chern. Phys. 20,

722 (1952); K. FukUi, T. Yonezawa, C. Nagata and

H. Shingu, J. Chern. Phys. 22, 1433 (1954); L.J. Ooster­hoff, cited in E. Havinga and J.L.M.A. Sch1atmann,

Tetrahedron 16, 151 (1961); see also ref. [27J.

103

[29J H.C. Longuet-Higgins and E.W. Abrahamson, J. Am. Chem.

Soc. §i, 2045 (1965).

[30J N.T. Anh, "Les Regles de Woodward-Hoffmann", Ediscience,

Paris 1970.

[31J J.J. Mulder and L.J. Oosterhoff, Chem. Commun. 121Q, 305; E.B. Wilson and P.S.C. Wang, Chem. Phys. Letters

12, 400 (1972).

[32J J. Koutecky and R. Zahradnik. Collection Czech. Chem.

Commun. ~. 811 (1960); T.A. Hoffmann and J. Ladik,

Advan. Chem. Phys. 1, 84 (1964); J. Ladik and K. Appel,

J. Chem. Phys. 40, 2470 (1964); A. Imamura. J. Chem.

Phys. ~, 3168 (1970); K. Morokuma, Chem. Phys. Letters 2. 129 (1971); J.-M. Andre, G.S. Kapsomenos and G. Leroy,

Chem. Phys. Letters Q, 195 (1971); J. Bacon and D.P.

Santry, J. Chem. Phys. 2§, 2011 (1972).

[33J F. Bloch, Z. Physik ~. 555 (1928).

[34J G. Wagniere and R. Geiger, Helv. Chim. Acta 56, 2706 (1973) •

SUBJECT INDEX

Ab initio calculations 3.45

All-valence calculation. semi-

empirical 59.64 Alternant hydrocarbon 12

Analysis. population 62

Aniline chromophore 77

Approximation. nearest-

neighbor 7.32

-. ZDO 7.30,40,47.64.97 Atom-atom polarizability 14

Atomic density, HUckel 11

-. population 62

Basis function 5 Basis transformation 67.68

Benzene, electronic spectrum

37 -, HUckel model of 9

• PPP model of 34-37 -, transitions 37.41 Bond-bond polarizability 14

Bond order 12

Born-Oppenheimer approximation

1.2

Brillouin theorem 49,56

C2 symmetry 77

Calibration of resonance integrals 8

Canonical SCF orbitals 52

Carbonyl group 74

Charged sphere approximation

31 Chirality 77 Circular dichroism CD 71

Closed shell problem 42

CNDO. Fock matrix 66 method 64

Commutator with spin 18

Complex benzene MO's 10,35

Concerted reaction 81

Configuration. electron 12.16, 84

- interaction. CI 26-29.35.80.

97 - mixing, CI 26-29.35.80.97 Conformation. prediction of 61

Conrotatory electrocyclic re-

action 82/83

Core electrons 59

- integral. PPP 24,32

- matrix elements. CNDO 64

- operator 25 - resonance integral 25 Correlation diagram 84

- energy 50-52

Coulomb integral a. HUckel 7 -, two-electron 29.42

- operator J 48.54 Cotton effect 71

Cyclic symmetry 37.91

Density, atomic 11 - matrix, first order 12,45 Determinant, secular 6,17,45,

47,93,94 -, Slater 15,26-29 Different orbitals for differ-

ent spins 58 Dipole-dipole approximation 76 Dipole strength 39 - vector form of electric

transition moment 79 - velocity form of electric

transition moment 78 Direct product of representa­

tions 40 Disrotatory electrocyclic re­

action 82,83 Doublet state 57

Effective Hamiltonian 59 Electric dipole operator 38,

39,78,79 - dipole transition moment

38,39,78,79 - dipole transitions,

selection rules 40, 97 Electrocyclic reaction 81 Electron affinity, atomic 24 - configuration 12,16,84 - repulsion integral, CNDO 64 - repulsion integral, PPP

30-32 Electronic population analy­

sis, Mulliken 62

105

Electrostatic orbital self-

energy 53 Energy levels of benzene 37 - of ionization 50 - of reorganization 51 Equation, secular 6,17,45,47,

93,94 Ethylene, HUckel 7 -, PPP 15 Exchange integral 29,42 - operator K 48,54 Extended HUckel approximation

59,93

Fock matrix elements, general

closed shell 45,59 - matrix elements, general

restricted open shell 55,56 - matrix elements, CNDO 66 - matrix elements, with cyclic

symmetry 96 - matrix elements, ZDO 47,97 - operator F, general closed

shell 45 - operator, general restricted

open shell 55,56 - operator, unrestricted open

shell 57,58 - operator, invariance 48

Gross atomic population,

Mulliken 63

Hamiltonian, nonrelatiVistic,

Born-Oppenheimer 1

, semiclassical radiation 38

Hartree-Fock limit 45

- method, see also under Fock

42,95

Hermitian matrix 48

Heteroatoms, HUckel parameters

for 13

HUckel approximation 4,59,93

- method 4 - method, extended 59

Inherent dissymmetry 78

Integrals,

semiempirical evaluation of

- CNDO 64-67

106

- extended HUckel 60

- HUckel 7-9,13,14

- PPP 30-34 Invariance, to basis trans-

formation 67-70

Ionization energy 50

- potential, . valence state 24

Irreducible representation of

point groups 22,40,97

J, see Coulomb operator 48-54

K, see Exchange operator 48,54

Koopmans 1 "theorem" 50

Lagrangian multipliers 44,48,

54

LCAO-MO, Ritz variational

method 5

Limit, Hartree-Fock 45

Linear combination of atomic

orbitals, see LCAO-MO 5

- combination of configura­

tions, see Configuration

interaction 26-29

Localization of transition

moments 78

Localized orbitals 52

Magnetic transition moment 72

Many-electron theory 15,42,59

Matrix-elements, CI 29,36,49,

56,97

- elements, Fock 47,66,97

Molecular orbitals

-, analysis of 62,63

CNDO 64-67

-, Cyclic 10,35,91-97

-, Extended HUckel 59-62

-, HUckel 7-14

, MO see LCAO-MO 5

, PPP 30-38

, SCF 42-46

-, of benzene 10,35

of ethylene 8,16

-, in concerted reactions 81-91 Moment, electric transition

39,72

Nearest-neighbor approximation

7,32

Net atomic population 62,63

Noncrossing rule 86

n~*-transition 73

Off-diagonal Lagrangian

multipliers 44,48,54

One-electron integrals

- approximation of 24-26

- CNDO 64-67

- Extended HUckel 60

- HUckel 7-9, 13,14

- PPP 30-34 Open-shell SCF method,

restricted 53

unrestricted 57 Optical activity 71

Orbital

-, symmetry- 11,93 -, symmetry of 22,81-91

Orbitals

-, see LCAO

-, see Molecular orbitals

-, canonical SCF 46,52

-, localized 52

ORD, see Optical activity 71

Orthogonal transformation of

basis 67-70 Overlap population 62

Pairing of electronic states

13

107

Parametrization, see Integrals

Pauli-principle 15

Periodic boundary conditions

91-99 Photochemical cycloaddition

reaction 91

- electrocyclic reaction 87,88

Pi-electron approximation 4,30

- separation, overall 4,15,30

- separation, local 79

n~*-transitions

-, in benzene 35-41

-, optical activity of 76-81

Point group symmetry 22,40,84,97

Polarizability atom-atom 14

- bond-bond 14 Population analysis, Mulliken

62

-, atomic 62

-, overlap 62

PPP approximation 30

- core 34 Pseudo eigenvalue equation, SCF

49,55 Pyrrhole 34

Quadrant rule 75

Reactions, concerted 81-91

Real form of benzene orbitals

10

Reorganization energy 51,57 Representation, see

Irreducible representation

Resonance energy 12

Resonance integral ~,

- HUckel 7 - PPP 25,26 Restricted open-shell SCF

method 53 Ritz variational principle 5 Rotational strength, see

Rotatory strength 72 Rotatory strength 72

SCF method, see self con­sistent field

- equations, open shell 53-58 - equations, closed shell

42-53 Sector rule 76

108

Secular equation,

see Equation

Determinant

Selection rule for electric

dipole transitions 40,97-99 Self consistent field SCF,

see Hartree-Fock method Fock matrix

Molecular orbitals SCF

Semiclassical radiation theory

38,72 Semiempirical procedures,

see Approximations

Integrals

Molecular orbitals

Separation, Born-Oppenheimer

1,2 , a-n 4,15,30,79

a electrons 4,59

Singlet function 19,26 Singly excited configurations

see Configuration interaction

Brillouin theorem

Slater-Condon rules Slater-Condon rules 28 Slater determinant 15,26-29 Spectroscopic calibration of ~

9 Spectrum, computed of benzene

37 , circular dichroism 71-81

-, computed of ethylene 22 Spin eigenfunctions of Sz, S2

18 Spinorbital 15 Strength, dipole 39 Substituent effect 74 Symmetry

-, spatial see Point group

symmetry

- of excited benzene states

37,41 - orbitals 11,93 - rules for concerted re-

actions 81-91

Thermal reaction, concerted

84,88 Thermochemical calibration of

resonance integral 8 Total gross atomic population

63 - net atomic population 63 - SCF energy 49

Transition moment

-, electric dipole 39,72,79

-, magnetic dipole 72

Transition probability,

electric dipole

see Dipole strength 39

Triplet function 19,27,54

Two-electron integrals 19,23,

30

see also Coulomb integral

Exchange integral

Uniformly charged sphere

approximation 31

Unitary transformation of

basis 48,67-70

109

Unrestricted open shell SCF

method 57

Valence, all-, procedure 59

- electrons 3 - state ionization potential

24,33

Variational principle

-, Ritz 5,17

-, SCF 42

Vibrational wavefunction 2

Zero differential overlap

approximation ZDO 7,23,30,

36,40,47,64,97