Equivalent orbital theory of polymers

Synthetic Metals, 1 7 (1987) 123- 128

I23

EQUIVALENT ORBITAL THEORY OF POLYMERS

G.G.HALL Division of Molecular Engineering,

Kyoto University,

Kyoto 606 (Japan)

ABSTRACT The

feasibility

of using

structure of polymers

equivalent

orbital

theory

to study

the electronic

is discussed. The bands of trans-polyacetylene

are derived

as an example. The effect of end groups is considered briefly. In some, "surface states" are found and s o m e show a d d i t i o n a l

states in the band gap. The theory

can also be used as a convenient method for ab initio calculations.

INTRODUCTION At the present

time the quantum chemical

methods used for the discussion of

the electronic structure of polymers range from the simple HHckel method through the extended HHckel method and the CNDO methods calculate suggest

s o m e of

the Bloch

that e q u i v a l e n t

orbitals

orbital

to the ab initio methods which

[I]. The

purpose

ideas could extend

of this

paper

these m e t h o d s

is to

and w o u l d

relate them better to quantum physical methods. Equivalent orbitals

(or localized orbitals) were

of relating the chemical description of a molecule etc. to the m o l e c u l a r o r b i t a l w a v e f u n c t i o n unitary

transformation

self-energy,

of the occupied

localized

orbitals

which

first introduced as a means in terms of bonds,

lone pairs

(for an early r e v i e w see [2]). By a

molecular quantify

orbitals these

which

chemical

maximizes

concepts

the

can be

defined. The idea was later extended to valence crystals (an account is given in [3]).

Another

advantage

inclusion of correlation

of

this

effects

localization

since

is

that

it

the largest of these

facilitates

the

is the correlation

of the two electrons within the same localized orbital. Two types of equivalent orbital treatment are discussed. The simplest equivalent

orbital

0379-6779/87/$3.50

method

uses

existing

calculations

of

form of

the band structure

to

© Elsevier Sequoia/Printed in The Netherlands

124 fit a model Hamiltonian whose parameters should be called a s e m i - t h e o r e t i c a l methods.

In the next

method

section an example

such a theory is further

illustrated

are integrals over the orbitals. by a n a l o g y

with

of this is presented.

This

semi-empirical The advantage of

in the third section where

the effects

of

different end groups on the polymer are considered. In

its

ab

optimization

initio

form,

the

localized

orbitals

are

found

by

of the total e n e r g y per unit cell u s i n g trial functions

direct for the

orbitals. Such a calculation was performed for the diamond crystal by Coulson et al. many years ago [4]. The functions have to be orthogonalized while remaining equivalent solve.

i.e. translating

into one another.

There are no orbital

equations

to

The procedure has the major advantage over conventional Bloch methods of

i n c l u d i n g all the o c c u p i e d bands in the c a l c u l a t i o n numerical suggested

integration over the band to calculate in the final s e c t i o n

at once and of a v o i d i n g a

the effective

that a c a l c u l a t i o n

potential.

It is

of this type on a p o l y m e r

would now be feasible.

TRANS-POLYACETYLENE This

prototype

separate

polymer

has,

in theory,

into pi and s i g m a o r b i t a l s exactly. The s i g m a o r b i t a l s

cell c o n s i s t of two e q u i v a l e n t longer

a flat form so that its e l e c t r o n s

than the other.

T(2,1/2;2;~)

This d i s t o r t i o n

to T(I;2;~)

in

in each unit

CH bonds and two CC bonds, one w e a k e r

the

from

notation

screw

axis

proposed

for

symmetry, polymer

and so

i.e. from

groups

[5],

produces the gap between occupied and conduction bands. The p a r a m e t e r s neighbours

r e q u i r e d to d e s c r i b e

the e l e c t r o n i c

s t r u c t u r e up to second

are:

c- the energy of a short CC bond d- the energy of a long CC bond h- the energy of a CH bond p- interaction of adjacent CH and CC bonds q- interaction of trans CH bonds r- interaction of gauche CH and CC bonds x- interaction of adjacent CC bonds y- interaction of trans short CC bonds z- interaction of trans long CC bonds. It

is clear

from

the

results

of a s i m i l a r

Heilbronner [6] that, while the n e a r e s t form of the bands,

calculation

neighbour

on p o l y e t h y l e n e

interactions

determine

by the

the second and perhaps some third neighbour terms are needed

to obtain close a g r e e m e n t

with

the p h o t o e l e c t r o n

spectra.

The v a l u e s of the

125 parameters

were

determined

[7]. The n u m e r i c a l c - -32.6,

values obtained

d = -]7.4,

r = 0.29, x = -4.2, The

equation

difference

h s -28.75, y = -3.225,

determining

between

h- E

cells,

p+re -if

l

p+re if

by a fitting were

z = 0.605

the

band

of the calculated

bands

(in eV):

p = -4.5,

q = -4.425 .

energy

E

as a f u n c t i o n

of

f,

the

phase

is:

q(l+e -if)

c+2y cos f- E

of the edges

r+pe -if

p+re -if

lq(l+eif)

p+re if

h-

Ir+pe if

x(]+e if)

p+re if

,~0.

x(1+e -if)

E

p+re -if d+2z cos f - E

10--

0- 1 0 ~

E

~ L U HO

-20

-30

.°.. .......... -°... o. . . . . . . . . . . . . . .

-

°°'°

°°°--

..-40

-

..°.°°*

--'"

°°o~

-50

-

,

,

J

0

Fig.].Bands

The band lines

i

f

1"I-

for trans-polyacetylene

structure

thus

obtained

using equivalent

orbitals.

for the sigma electrons

is shown using dotted

in Fig.].

The pi e l e c t r o n s theory

,

the pi atomic

require orbital

a different is defined

treatment. using

In the e q u i v a l e n t

the standard excited

orbital

state of the

126 molecule

since this allows

together. orbital

The o c c u p i e d

equations

the occupied and unoccupied orbitals to be localized

orbitals

similar

for an a l t e r n a n t

are then

in form to those of PPP theory.

the s o l u t i o n s

For this polymer

of the

orbital energy equation is

~

+26eos

f-C

~ + ~ e -if

+ y e if

~ +2 6cos

f - K

so that the bands are: E "~+26cos

f +

(~2+~2+2~co

s f).

The parameters are also fitted and have the values: =-

-5.675,

~ ~ -5.625,

V = -1.425,

6"

-0.738

.

The bands are shown in Fig. l with solid lines.

THE EFFECT OF END GROUPS The u t i l i t y of this theory can be a p p r e c i a t e d by c o n s i d e r i n g terminating

the effect of

the p o l y m e r w i t h ring s y s t e m s of v a r i o u s sizes. The i n t e r e s t i n g

effects are in the pi orbitals and include the location of the occupied orbital energies in relation to the bands and the charge distribution. The calculations have the same

form initially as in HHckel theory. Along the

chain the orbitals are governed by the same equations whether or not is finite. The orbitals are waves of constant amplitude the end group is a cyclic p o l y e n e junction

atom

imposes

a new

its o r b i t a l s w i l l have a s i m i l a r

structure

on these initial

the wave from the chain meets the cyclic orbitals and, phase change

and a reflection.

Thus

a five-membered

If

form. The

forms. At the junction in general,

it suffers a

If the group contains an odd number of atoms

the ring the a l t e r n a t i o n of the p o l y m e r charge.

the chain

and phase difference.

in

is lost and the groups m a y have a net

ring w i l l

have a n e g a t i v e

c h a r g e and a seven-

membered ring a positive charge. One of these at each end will give the polymer a strong rectifying character.

With a four- membered

a l t e r n a n t and the charge d i s t r i b u t i o n singlet

ring the molecule

remains

in s o m e states w i l l be u n i f o r m but the

ground state may not be so. The molecule will distort with alternating

w e a k and strong bonds, to achieve greater s t a b i l i t y and a u n i f o r m charge, and this w i l l

extend

into

the ring.

At

the

junction

there w i l l

be an o c c u p i e d

"surface" orbital which has most charge on the junction and decays exponentially along the chain. This and its u n o c c u p i e d a l t e r n a n t p a r t n e r w i l l have e n e r g i e s o u t s i d e the band system, one above and one b e l o w before distortion,

it. The f o u r - m e m b e r e d

ring,

also has two o r b i t a l s w i t h zero energy. One of these has a

node at the junction and so does not interact with the chain. The other couples directly

to one

zero

energy

orbital

of

the

chain

i.e. in the

gap b e t w e e n

127

occupied

and

delocalized,

empty

bands.

Two

can be highly

the p o l y m e r

degenerate

significant

since a v i b r a t i o n a l

states,

in giving

distortion

one

localized

semi-conductor

and

one

properties

to

of the ring can m i x the states and

induce conduction.

AB INITIO CALCULATIONS The e q u i v a l e n t calculation.

orbital

Since

corresponding

ON POLYMERS

bond

method

each

bond

is a practical

in

unit

in other cells,

this energy must be constrained

the trial

fully occupied,

functions.

the C o u l o m b

These must be associated contribution techniques cannot problem

may sometimes

there

in terms of the d e n s i t y

translation respect

be needed.

symmetry

is most

procedure of L~wdin [8]

terms

neutral.

For the pi electrons,

to obtain a

Ewald

because

summation

the potential

The f o r m u l a t i o n

(or the bond orders)

of the d e n s i t y

of

since all the bands are

are some problems.

to be fully exploited.

to the e l e m e n t s

is easily

are sums over all the cells.

is e l e c t r i c a l l y

matrix

the

minimization

This c o n d i t i o n

correctly with the electron-nuclear

localized,

to

of all the basis functions

partners.

terms

form

per unit cell

orthogonalization

and e x c h a n g e

in

The direct

For the sigma system,

from each cell w h i c h

be fully

energy

by the orthogonality

their t r a n s l a t i o n

of doing an ab initio

identical

calculation.

readily applied by using the symmetric to m o d i f y

means

is

the total

found and can be used in a v a r i a t i o n

inside the cell w i t h all

cell

of the

[9] a l l o w s

The energy can be m i n i m i z e d

matrix

provided

that

this

the with

remains

idempotent. A major theoretical correlation

advantage work

of u s i n g

localized

that

correlation

of two electrons

these are excited wavefunction

is

the

as

effects

a starting are

for this.

localized

orbitals

In particular

can be added

REFERENCES J.M.Andr~ and J.Ladik (eds.), Electronic and Molecular

Crystals,

Structure

Plenum Press, New York,

2

G.G.HalI,

Rep.Prosress

3

A.A.Levin,

Quantum Chemistry

]977.

by

for the

in Ph~s.~ 22 (]959) of Solids,

to the SCF

the energy per unit cell remains

meaningful.

i

point

dominated

in the same localized orbital and CI terms in which

into anti-bonding

to allow

orbitals

of Polymers ]975.

].

McGraw-Hill,

New York,

128

4

C.A.Coulson, L.B.Redei and D.Stocker, Proc.Roy.Soc.,A270 (]962) 357.

5

G.G.HalI, Applied Group Theory, Longman, London, 1967.

6

E.Heilbronner, Helv.Chim.Actar60 (]977) 2248.

7

T.Yamabe, K.Tanaka, H.Teramae, K.Fukui, A.Imamura, H.Shirakawa

8

P.O.L~wdin, J.Chem.Phys., 18 (1950) 365.

9

G.G.HalI, Proc. Roy. Soc. A, 229 (1955) 251.

and S°Ikeda, J.Phys.C., ]2 (1979) L257.