Theoretical study of simultaneous few-electron transfer in polymeric systems

Synthetic Metals, 41-43 (1991) 3505-3508

THEORETICAL

STUDY

OF

3505

SIMULTANEOUS

FEW-ELECTRON

TRANSFER

IN P O L Y M E R I C

SYSTEMS

V.

MAY

Institute

of

Molecular

Biology,

R.-Roessle-Str.

i0,

Berlin,

1115

(F.R.G.)

ABSTRACT The

simultaneous

polymeric

chain

interaction. localized their

is

If

the

Such

a

studied

the

state

available

during

motion

of

by

two

taking

corresponding the

amount

excess of

or

more

into

excess

account

interaction

electrons

kinetic

in

mutual

Coulomb

the

energy

overcomes

energy,

electrons

the

of

the

state

an

a

initially

upper

remains

limit

of

localized

time-evolution. localization

electronic

is

wavefunction

intra-monomer

weakened

and

is

characteristics

of

state

a closed

the

by

only

vibrational

open

to

of

dephasing

partly

modes.

chain with

effects

modified

The

display

acting

charge

the to

input-output

mechanism

input

on

a coupling

charge

a switching

increasing

by

from

an

rate.

INTRODUCTION The

study

strongly

of

influenced

polyacetylene. electron The

chains

appropriate

transfer

all

charge

studies

0379-6779/91/$3.50

the

of

motion

a

in

long

charge

polymeric

large

studies

have

of

with

in

observed

these

phenomena

systems

e n e r g y as w e l l The

by

investigation

polymeric

excess

charge

Besides

transfer

synthesized

for

excess

systems

type

transfer

has

converting

light

been

of

polymeric

doped

systems,

in c h e m i s t r y .

reactions

complexes

been

has

conductivity

extended

tradition

donator-acceptor

different

systems

in

divided

stimulated

energy

by

into

newly

by

the

finite search

electrical

as by t h e p r o p o s a l s coming f r o m m o l e c u l a r e l e c t r o n i c s . event injected

of

interest

from

concentrate

the

on

in

such

donator

the

systems

side

transfer

into

of

is

the

the

chain.

single

motion

of

However~

charges

per

© Elsevier Sequoia/Printed in The Netherlands

3506 intervening discuss

polymeric

those

electrons

move

To

providing excess

a

allowing assumed

electron

highest

as

the

the

matrix

the

the

of

from

of

at

the

of

for

the

couplings

to

the

large can

the

are

Coulomb

neglected

restrict

further

the

other

on-site

be

To

LUMO

what

ourselves

assume

polaron

that

shifts

level

are

and

the

This

of

the

of

investigated

in

chain,

beginning of

the

remain

kinetic

localized

inequality

the

potential

is

interaction strength

the

two to

what

the

Coulomb

the limit

transfer

Coulomb i n t e r a c t i o n

as a s t a n d i n g

of

since

the upper

e n e r g y each t i m e .

instead

a

remain

one w o u l d e x p e c t

twice

energy of

the

changes from

vanishes

(roughly

second of

electrons

somewhat weakened i f

of

and

strength

H o w e v e r , due t o

either

density

two e l e c t r o n s

(first

the

localization

the chain

well

matrix

input-output

two e l e c t r o n s

contrast

into

charge

the chain

in

energy in

two-particle

the motion of

the

as

density

CHAIN

When i n c r e a s i n g

c a s e where t h e

equation

[1-4].

the

behaviour is

Coulomb

Increasing

Schroedinger

IN A R I G I D

dynamics of

been

one-electron

which

be t r a n s f e r r e d

two e l e c t r o n s

has

in

T per e l e c t r o n )

[2].

monomer

chain

transfer

we

a motion

V

long-ranged

(extended

all

interaction)

details

the

coupling

the

Coulomb

move away f r o m one a n o t h e r .

moving p a i r .

chain

a

energetic

a

between

the

polymeric

to

This

kinetic

cannot

Coulomb consider

same

as

motion

for

MOTION

the

motion

classical

the

well

decoupling

studied.

Coulomb i n t e r a c t i o n ,

electrons

the

the

been

[1].

will

the

LUMO's

time-dependent

the

located

localized

mutual we

per

molecule

electrons as

to

excess

have

electrons.

distance

motion

TWO-ELECTRON

has

dispersive

excess

interaction

Hartree-Fock

characteristics initially

the

the

few-electron

understand

monomer)

of

of

paper

more

level.

resulting

To

in

V

few-particle

SIMULTANEOUS

Supposing

energetic

equations

(applying

T.

injected

simultaneous

solving

from

index

occupied

The

move

occupation

spin

(LUMO)

should

one

or

We t a k e a p o l y m e r i c

values

the

the

modes,

Additionally,

double

Coulomb

than

to

identica]

levels

present two

molecular

orbital

LUMO

monomers.

externally

mutual

smaller

a finite

molecular

the if

of

excess electrons).

These

of

simultaneously.

effects

of

the have

chain

the

model for

all

to d r o p

the

their

polymeric

aim

expected

vibrational

for

to

the be

to

single

interaction, allows

the

is to

coupling

model

electron.

position

to

the

simple

Peirls-Hubbard

it

have

demonstrate

and

sufficiently

which

through

clearly

interaction

chain,

effects

If

V

>

4 T

or

a

slowly

one c o n s i d e r s

a nearest-neighbour interaction

(e.g-

a

one in

3507 considering reduced. three

different The

or

Note

types

particle

of

chains)

localization

excess

that

the

approximative

density

the

localization

of

the

such

a description

is

necessary

Nevertheless, coupling

to

a dissipative

instead

velocity appears

more

overestimates

electrons

the

also of

environment

two

the

pair

one

is

considers

[I].

matrix

approach

excess

as

of if

somewhat

electron to

well

take

as

pair

into

charge

[4].

account

a

input

and

(of

the

output.

COUPLING If

TO

VIBRATIONAL

the

whole

coupling

chain

to c o n s i d e r parts The

of

as

remaining

one

can

assume

the

presence

trapping long

as

~)

of

.

the

LUMO

an

excess

the

no

drastic

type

to

But,

an

if

V

/ ~

a

it

of

introducing

to

energy

remains

changes

in

the

the

the

the [5].

excess

type.

monomer

Then, during

so-called

equations. below

the

(of

monomers)

strongly

matrix

suffices

a dephasing

high-frequency

density

(Hartree

closed

becomes

self-

However,

as

the

Coulombic

few-electron

dynamics

chains

is

electrons

i.

reached

results

shift)

occupation

of

the to

produces

probability

e.,

to for

with

from

according

what

the

and

above

mentioned

electronic

comparable

transfer,

behaviour

the

occupation dephasing

dephasing

nonexponential electrostatic

the

Coulomb

makes

> T time

shift

of

interaction

hopping-rates what

rate

rate

depending

the

on

probability

[3].

CHARACTERISTICS

the

discussed

physical

input-output

characteristics

a

a chain

of

always

occupied

states).

Fig.

last

modes

weak,

different

polarization

of

incoherent

nonlinear

INPUT-OUTPUT

the

the

equipartition

equipartition

different

model

is

realizing on

coupling

electron

into

vibrational

intra-monomer

instantaneous

Such

electronic

All

bath

modes of

an

of

levels

diffusion

heat

self-trapping

region

between the

assumed

leads

dependence. the

a

of

monomers)

wavefunction

considered

probability

/

as

set

single

EFFECTS

[4].

the

the

modes

energy

dephasing

(in

be

the

appear

In

the

contribution

interaction will

a definite of

vibrational

will

DEPHASING

to

electronic

electrons

AND

as

well

these

the

MODES

attached

states)

1 displays

centre

in

the

mechanisms

of at

and

the

the its

ends

a particle

electronic

chain

manifest

chain

in

the

which to

a

themselves

have

been

particle

sink

occupation stationary

(with

source

the in (with

always

probability case

in

studied

empty Pdo

versus

of the

3508 rescaled ranged

charge type

As

a

state

of

V is

main

with

appears

result

the

Now,

at

in

rate.

difference

charge

flow

mean

number

short

of of

f r o m an open t o

through of

interaction beginning

a Pauli-blocking

between

and

long-

importance).

we o b s e r v e a s w i t c h

total

the

(The

minor

to

Coulomb

localization results

of

respect

when

two.

input

the

electrons

starts

to

chain

(at

the

charge

chain.

input

Such

N reaches work,

small

a

closed

a

switch

the

and

value

particle

dephasing

rates)

[2,5].

N ~

Fig.l.

P40 a n d

rescaled

charge

rate ~

R;~

rate ~

Ro~/T

variation

/T

/S

Z , ~,,, •

input

'1

(output

-

....-'-'7-

= i);

2

of O<, = V / T ,

rescaled rate

3

N versus

~--'~ . . . .

dephasing

r

=

~RE~/T

sel f t r a p p i n g

and

constant

~" = '~ /S; 0.01

curve

~

Y"

:)'/

0.I

0

l

0

2

5

I

0

3

5

0.i

0

4

i0

0.I

3

I I I I I |

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o.o';

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,tr,.R,;,,/T

REFERENCES i

V.

May, P h y s i c a D,

2

V.

May,

Proc.

Int.

40

(1989)

Conf.

Moscow,

1989,

accepted for

3

V.

May,

to

be p u b l i s h e d

4

V.

May,

to

be p u b l i s h e d

5

V.

May, P h y s i c a A,

152

in

173.

Molecular publication Phys.

(1988)

Lett.

127.

Electronics in

O. M o l .

And B i o c o m p u t e r s , Electr.