Determination of the Coulombic Interaction Parameters of the Extended Hubbard Model in the Organic Conductors

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Determination of the Coulombic Interaction Parameters of the Extended Hubbard Model in the Organic

Conductors

F. Castet, Agnès Fritsch, L. Ducasse

To cite this version:

F. Castet, Agnès Fritsch, L. Ducasse. Determination of the Coulombic Interaction Parameters of the

Extended Hubbard Model in the Organic Conductors. Journal de Physique I, EDP Sciences, 1996, 6

(4), pp.583-597. �10.1051/jp1:1996231�. �jpa-00247204�

Determination of the Coulombic Interaction Parameters of the Extended Hubbard Model in the Organic Conductors

F.

Castet,

A. Fritsch

(*)

and L.

Ducasse (*)

Laboratoire de

Physico-Chimie Th60rique (**),

Universit6 de Bordeaux1, 33405 Talence Cedex, France

(Received

23 October 1995, received in final form and

accepted

20 December

1995)

PACS.31.20.Rx Valence bond calculations

jab

initio _or

not)

PACS.71.10.+x General theories and

computational techniques

PACS.71.28.+d Narrow-band systems:

heavy-fermion solids;

intermediate-valence solids

Abstract. A mixed

Valence-Bond/Hartree-Fock

method

applied

to all valence electrons calculations _on finite size clusters _was derived in order to size up the relative

magnitude

of the coulombic interactions between conduction electrons in organic conductors. An effective

Hamiltonian built _{upon a}

single

site orbital _per molecule

reproduces

the whole set of VB energy levels. This extraction _is most consistent when molecular orbitals

optimized

_{on monomers are} used. Coulombic parameters

describing

the interactions in clusters of organic molecules derived of TTF

(tetrathiafulvalene)

_may be thus extracted. The

magnitude

of the coulombic terms

(U

m 4 eV V

(nearest-neighbour)

= 2 3

eV)

differs

qualitatively

from previous estimations and evidences the

long

_range part of the

potential.

The calculated

charge

transfer excitations

are still in

good

agreement with the experimental results.

R4sum4. Une m4thode mixte

Valence"Bond/Hartree-Fock appliqu4e

1des calculs tous Alec-

trons de valence _sur des

agr4gats

de taille finie _a 4t6

d6velopp6e

afin de d4terminer l'ordre de

grandeur

des interactions coulombiennes entre 41ectrons de conduction dans les conducteurs _or-

ganiques.

Un Hamiltonien effectif construit dans la base d'une seule orbitale de site par mo14cule

reproduit

l'ensemble des niveaux

d'4nergie

VB. Cette extraction est la

plus

coh6rente

lorsque

l'on utilise des orbitales mo16culaires

optimis6es

_sur les monomAres. Les

parambtres

de Coulomb qui d6crivent les interactions dans les

agr6gats

de mo16cules

organiques

^{d6riv6es du TTF}

(t6- trathiofulvalbne)

ont ainsi _pu Atre obtenus. Les termes coulombiens

(U

_m 4 eV V

(premiers voisins)

= 2 3

eV)

diffbrent

qualitativement

des estimations ant4rieures et mettent _en 4vidence la partie

longue port4e

du

potentiel.

Les excitations I transfert de

charge

calcu14es demeurent

cependant

_en bon accord _avec

l'exp4nence.

1. Introduction

Since the

discovery

of

organic superconductivity

in the

Bechgaard

salts

TMTSF2X (TMTSF

=

tetramethyltetraselenafulvalene;

X~

anion) iii,

the _maximum critical

temperature Tc

was

raised to 12.8 K

[2]. Meanwhile,

the true

origin

of this

property (BCS

_or non-BCS

mechanism, microscopic

nature of the screening of the electron-electron

repulsion)

remains unclear. The

(*)

Authors for correspondence

(e-mail: ducasse©lpct.u-bordeaux.fr) (**)

^URA 503 CNRS

©

Les

#ditions

_de

Physique

1996

superconductivity

is not the

single property

to which _a

large

literature has been devoted in the

past

few _years

[3j.

^The

phase diagram

of monodimensional

(1D)

_or

quasi-1D salts,

_rep-

resented

by

the

Bechgaard

salts

TMTSF2X

_[4] and their sulfur

analogs TMTTF2X (TMTTF

=

tetramethyl-tetrathiafulvalene) _[5],

is

complex:

insulator

(antiferromagnetic, AF;

_{non mag-}

netic, spin-Peierls, SP),

metallic _or

superconducting

states have been observed. More 2D

salts,

_as those obtained from the derivatives of the BEDTTTF molecule

(bisethylenedithia- tetrathiafulvalene),

also show _a

competition

between

magnetic

and

superconducting ground

states: this is for

example

met in the _~z

(BEDTTTF)2CuN(CN)2Cl

salt [6].

The

difficulty

in _a theoretical

analysis

of these salts lies in their

complex microscopic

structure

(one,

two or more

large

molecules _per unit

cell, partial filling

of the one-electron band

resulting

from the transfer between the anion and the

cation, .)

^and on the

possible interplay

between various terms

(kinetic _part

of the

Hamiltonian,

Coulomb

interaction, electron-phonon coupling,

disorder of the

anion). Among

these

factors,

the electronic correlations

certainly

_appear to be

a

leading

term. For

example,

the limitation of the one-electron model is

clearly

_seen in the

band structure results _on the

M2X

salts. This

approach

leads to a metallic behavior for every

kind of salt based _on the unit formula

M)X~, _although

the

conductivity

measurements show metallic _or

semi-conducting

behaviors for salts

exhibiting

_very similar

crystal

structures

iii.

The effects of the electron-electron correlations have been

investigated using

^different the- oretical models. The first _one is the Fermi gas model also known _as the

g-ology

model

[8,9],

derived in the ~veak

coupling

limit where the correlation terms are considered _{as a}

perturbation

to the one-electron formalism.

Low-energy properties

_may be

analyzed

within this

approach:

phase transitions,

localization _processes and NMR relaxation times. In the

opposite

limit of

strong coupling,

the Hubbard Hamiltonian involves transfer

integrals

and

potential

terms which

are

large

_energy

parameters. Moreover,

Mila and Zotos [10] have shown

recently

that _even the

low-energy properties

_may be described ~vithin the framework of this Hamiltonian.

Local methods have been

developed

in _an attempt to solve the Hubbard _or extended Hubbard models for clusters of finite _size

using

Monte Carlo simulations and Lanczos

technique [10-12].

In the domain of

organic salts,

the solutions of such Hamiltonians have been

investigated

as a

function of the delocahzation

it)

and correlation

parameters (U, V) [10,12].

The one-electron transfer _{terms may} be obtained from

tight-binding

band _structure calculations [13] ^but very few

attempts

have been

reported

for the correlation terms. In the domain of

high-Tc oxides,

such calculations have been

reported

_[14] and it is also

possible

to extract useful information from the combination of

photoemission

and

Auger spectrocopies.

In the domain of _organic

conductors,

the situation _is not so favorable

because,

_{on one}

hand,

the

photoemission

spectra

is controversial

and,

_on another

hand,

the fact that the active

(molecular)

site contains _a

large

number of atoms constitutes _a

strong

limitation for theoretical

approaches.

Recently,

_we

analyzed

the solutions of the extended Hubbard Hamiltonian

by using

crude estimates of U and V terms

describing

the

Bechgaard

salts

[12].

^This paper is aimed to

present

a

more accurate and consistent model to determine these terms from cluster calculations

involving

all valence electrons of the

organic

molecules.

2. Method

We

adopt

a local

point

of view m the

spirit

of the Valence Bond

(VB) approach.

Within this

model,

the wavefunctions _are

developed

_{on a} basis of multielectronic functions that _can be associated with distinct

charge

transfer states. These

configurations

_are defined

by ascribing

a

given

number of electrons _on each molecule. This _is achieved

by imposing

to each orbital which is involved in the electronic

configuration

to be

developed

_on the atomic orbitals of _a

specific

molecule.

This

approach

is restricted to the valence electrons of the

organic

molecule and _we used the

semi-empirical

AMI Hamiltonian

[15].

^{The aim of this work is} to obtain _some information

on the relative

magnitude

^{of the} electronic interactions _so that it is not

really

_necessary to

perform sophisticated

ab initio calculations.

Furthermore, Demiralp

and Goddard

[16]

^have shown that the

semi-empirical

MNDO

energies

and the ionization

potential

of the molecule and the cation of BEDT-TTF _are similar to the

corresponding

ab initio results.

Let _us consider _a cluster of N molecules. In _a

given

electronic

configuration,

the molecule I bears _n~ electrons: _n~ takes

only

3 values if _one takes

only

into account the

charged species M°, M+

and

M~+.

The cluster electronic

configuration

is then built _{on a} set of

orthogonal

molecular

orbitals

(MO) doubly

_or

singly occupied.

Each molecule may therefore be described

by

two electronic shells: the closed shell

collecting

all the

doubly occupied

MO's and the _open shell which contains at most _one

MO,

which is assumed to be the

highest occupied

MO

(HOMO).

In such _a

configuration,

the total electronic _energy is thus

expressed following

the Restricted

Open

Shell Hartree-Fock

(ROHF)

formulation

[17j:

E

= 2 Tr

Ri(h

+

Gi)j

+ Tr

R2(h

+

~G2)1(1)

2 2

where the index 1

(2)

refers to the closed

(open) shell,

_R~ is the first order

density matrix,

h is the one-electron Hamiltonian matrix

(involving

kinetic term _as well _as attractive terms between the _core and the valence electrons

).

G~ is the two-electron interaction matrix

involving

the Coulomb J and

exchange

K matrices. All matrices may be

expressed

in the atomic orbital basis. With M AO'S _per

molecule,

_one is lead to

(N

_*

M)~

matrices. One has

(in

atomic

units)

Gi

=

G(2Ri

+

G(R2

closed shell

G2

=

G(2Ri

₊

G'(R2)

_open shell

GIR)

=

JIR) )KIR) ^G'IR)

=

JIR) KIR)

J(R)rs

=

~Rutlrt~i/r-~sU) K(R)rs

=

~Rutlrt~i/r-~USI (2)

All atomic _one and two electron

integrals

_are evaluated from the

crystallographic

coordinates of the

atoms, following

the AMI

parametrization.

This

part

^{of the} calculation

prevents

us from

using

cyclic boundary

conditions which would avoid

possible edge

effects. We shall _see below that _a

thorough analysis

of the results _on chain clusters nevertheless

provides

parameters ^which do not

depend

_on the finite size of the clusters.

The restriction to

strictly

molecular orbitals combined with the AMI

approximations

has several consequences :

the

density

matrix is

block-diagonal

_m the atomic orbital

basis,

had the AO'S been clas- sified

according

to the molecules' label. Such _a

bloc-diagonal

form allows _us to consider

only

the molecular blocks of all interaction matrices. Each molecule _may thus be

fully

characterized

by

M _* M

matrices,

the total electronic _{energy is} the _sum of molecular

energies,

a convenient

simplification

furthermore _arises from the

semi-empirical

nature of the AMI

integrals

and the local nature of the

MOS,

that lead to the

vanishing

of allintermolecular

exchange

effects _so that _any

configuration

_can be cast into _a

single

Slater determinant

with all

unpaired

electrons

bearing

the _same

spin

without _any effect _on the _energy. Dis-

criminating

between the different

spin

states would

only

arise from the interaction be-

tween the various

configurations.

Such _a

simplification

also _appears in the _{more approx-}

imate and

phenomenological

_quantum cell

(frontier orbital)

models such _as the

Hfickel,

Hubbard _or

Pariser-Parr-Pople

models.

2.I. OPTIMIzATION _oF EACH ELECTRONIC CONFIGURATION. It is

important

to notice

that the

neglect

of differential

overlap

involved in the AMI

approximation together

with the

purely

molecular _nature of the orbitals _may also lead to _a viable

optimization procedure

of the MOS _m the

spirit

of the Hartree-Fock method. The energy functional _may be

optimized

with

respect

to the atomic orbital coefficients

subject

to the normalization constraints upon the MOS. Two

st.ationnary

conditions _are involved in this

procedure:

_one for the closed shell

MOS,

the second _one for the open shell _one. This draws _us to two distinct Fock

equations

relative to each shell:

where the F~ are the Fock matrices for the closed and open shells

respectively,

the C~ are the

eigenvectors

and _e~ the &IO'S

energies. Only

the molecular blocks _are considered here _so that

we have in fact two

equations

_per

molecule,

I.e. at most 2N

equations.

These relations _are valid

only

^if we can insure that the

orthogonality

constraints between the closed and the _open shell MOS _are fulfilled. This is

accomplished by conducting

_a

two-step

self-consistent field iterative

process.

First, starting

from _a set of _guess

MOS,

the closed shell

equation

^is

solved, producing

a new set of orbitals

represented by

the

Ci

matrix.

Secondly,

the _open shell Fock

equation

is

projected

into the virtual

subspace spanned by

the

empty

^orbitals obtained from the closed shell

equation.

In this way, this

projected equation produces

_{new open} shell orbitals

(occupied

or

empty)

which

belong

to the

orthogonal complement

of the closed shell. The

procedure

_is

repeated

until convergence of the total electronic energy.

Such _a calculation therefore

provides

_us with

optimized

sets of MOS for each

charge

transfer

configuration.

In this way, ~ve allow for the relaxation of the valence MOS of the molecules under the influence of their _own

charge,

but also under that of the

neighbounng

molecules.

2.2. EXTENDED HUBBARD HAMILTONIAN. The relative

energies

of the VB

configurations

used in the calculations may be

tentatively expressed

in terms of the electrostatic parameters

entering

the definition of the

following

extended Hubbard Hamiltoman:

H =

~ _{tv (C$~Ci+1,~}

₊

_h-c-j

₊

j £

_U~n~(Y~~

_ii

i,j,a 1

~) ~ _{KJ(~l zl)l~J zJ) 14)}

~~J

where

t~j

are the transfer

integrals

restricted to first

neighbours,

_{U~ are} the on-site

repulsion energies

and

l~j

are the electrostatic interaction terms

(in

^the

following,

I

j

defines the

Ii j)~~ neighbour); c(~(c~, a)

^is a creation

(annihilation)

_operator for _a site orbital located at site I with _{spin a, n~,~}

=

(C£~c~,~)

^{is the}

operator

^number of electrons. The

Z(s

are core

charges,

so that non zero terms

oily

occur between M+ and

for ^M~+

cations. It is

noteworthy

to

remark that the total energies related to the Hamiltonian

(4)

_are not

equivalent

to the energies deduced from the resolution of

equation (3).

Our aim _is to look for _a

possible relationship

between _energy differences calculated within the two

approaches.

Table I. Results _on the neutral

TTF°,

monocationic TTF+ and dicationic

TTF~+ species:

E _are the total

energies, Iit(TTF° )

and

IIK(TTF° )

_are the calculated

first

ionization

potentials for ^TTF° using

total energy

differences

and

Koopman's

theorem

respectively,

and

Iit(TTF+ )

and

IIK(TTF+ )

_are the

corresponding

data

for

the

charged

molecule TTF+ U is relative to

equation (5) (all

entries in

eV).

total

energies

and ionization

potential

E -1599.767

E -1592.806

E -1581.200

11.95

U 4.64

3. Results

In order to check the

reliability

^of the

method,

_we first calculated the

energies

^of model TTF

(tetrathiafulvalene) regular

chains

(the interplanar

^{distance between}

neighboring

TTF is fixed to 3.5

Ji).

3.1. MONOMERS. Table I shows the total _energies obtained for the neutral

TTF°,

_mono-

catiomc

TTF+

and dicationic

TTF~+ species.

The monocation _was treated

by

_means of the ROHF

approach.

The calculated first ionization

potentials

for

TTF°, _using

total _energy differences

(Iit (TTF°

6.97

eV)

and

Koopman's

theorem

(IIK(TTF°)

= 7.28

eV)

_are in

good agreement,

as

usually

found for closed-shell

system.

For the

charged

molecule

TTF+,

the _same difference of 0.3 eV appears between

Iit(TTF+

= 11.60 eV and

IIK(TTF+)

= 11.95 eV. This

agreement

^results from the

adequacy

of the method

employed.

This

equality

is indeed

implicit

in the true Re-

stricted

Open

shell Hartree-Fock formalism which is not true for the

widely

used half-electron

approximation

for open shells

systems.

Let _{us now} consider the fictitious

disproportionation

^reaction:

M+

₊

M+

_~

M~+

₊

M° (5)

where both molecules _are isolated from each other. In _a quantum cell model with _a

single

site orbital

(immaterial)

_per molecule with similar site

energies

whatever the

charge

of the

molecule,

such _a process ^creates a

doubly occupied

site orbital _on

M°,

and it is clear that the energy

change

for this reaction

only

reflects the electrostatic interaction between two electrons

on the site orbital of

M°.

We thus write the energy

change

/lE related to this reaction _as:

/lE=U

where U is the _on-site

repulsion.

The value obtained for U from the difference between the total energies of the _{monomer m} the distinct

charge

states _are given m the last line of Table I. This calculated U is 4.64 eV. On

the other

hand,

had U be

formally

defined _as the coulombic

repulsion integral

for two electrons

on the _same site orbital _xp of

TTF°,

and

identifying

_xp to the molecular HOMO:

U = <

xpxp~i/r~~~xpxp

>

16)

we would

get

U ₌ 5.32 eV. This difference _{is a} clear evidence of the

impact

of the orbital relaxation:

extracting

U from

is)

allows _us to take into _account the self-consistent

adaptation

of the whole electronic cloud to the

change

in the molecular

charge,

whereas

(6)

does not.

3.2. DIMERS. Let _{us now} consider two

interacting neighbouring

TTFS in _a

crystal.

The two

configurations

involved in _our calculation _are related to _a

doubly charged

dimer:

TTF+TTF+

and

TTF~+TTF°.

_{These are involved in}

a

charge

transfer

(CT)

_process that

strongly

modifies the electrostatic

potential

and should intervene in the

high

_energy CT bands. The

disprorpor-

tionation reaction

is)

_{may now}

effectively

be _{seen as a}

charge

transfer _process between first

neigbhour

molecules. In _a similar way to that used for the _monomers, the energy difference between the two

configurations

in

is)

_may be related to

changes

in the electrostatic _energy of the

configurations

in the _now extended Hubbard model.

Introducing

the

repulsion

V between two electrons

singly

_occupying

neighbounng

site

orbitals,

_we

get:

/lE=U-V.

ii)

From the /lE calculated from total _energy differences

involving

all valence electrons of the dimer and

using

the U value of Table

I,

_we then obtain V

= 3.05 eV. The V value determined without

allowing

for _an in-situ relaxation of the molecular orbitals

ii-e- by injecting directly

the MO's obtained _on the

monomers)

is V

= 2.95

eV,

which is within

4%

of the

previous

value.

3.3. TETRAMERS. It is of

uppermost importance

^of course to check whether the cluster size has any sizeable effect _on these results. We

consequently investigated

4 TTF clusters. In

this _case, it is

possible

to

study

the main transfer _processes with _a total electron

population

corresponding

to the _mean band

filling,

_{i e.}

TTF(+.

_There

is an inversion center in the middle of this

cluster,

_so that the relevant

symmetry ineqmvalent optimized configurations

_are:

~A > =

M+M°M°M+

_{~B >}

=

M°M°M+M+

~C ^> =

M+M°M+M°

~D ^> =

M°M+M+M°

~E ^> =

M°M°M~+M°

_{~F >}

=

M°M°M°M~+

We

give

in Table II the

corresponding

total

energies

for each

configuration.

We also _give in

parentheses

the values determined without

allowing

for _an in-sit~ relaxation of the molecular orbitals. We also give in the _same table the

corresponding

_energies

expressed

in _an extended

(to

all

neighbours)

Hubbard model

(Eq. (4)).

Considering configurations

_~E > and

~F

>, which have the _same Hubbard _energy

3U,

but

clearly

different VB

/HF

total

energies,

evidences the

inadequacy

of the Hubbard model to relate the

configurations

_energies to _a proper combination of U and V terms in _{a consistent manner.}

The _same conclusion _is reached in _an

attempt

to extract the

Vi

term: within the Hubbard

Hamiltonian,

it should be identical

by

_using the energies of the

configurations _~E

> and ~D ^>

or the

energies

of the

configurations ~F

> and ~B >. The results

definitively

show an effect of the environment: the

configuration

_energies

depend

on the local environment because the

VB/HF

_energy evaluation

procedure

takes into account

implicitly

the

charge

of the molecules

surrounding

the dimer under

investigation.

^The

Vi

term extracted from the

configurations

~D > and ~E ^{> is}

equal

to 3.07 eV: _it is thus _very close from the dimer values because the

M+

M+ dimer _m ~D > is

symmetrically

surrounded

by

neutral molecules. On the contrary,

Table II. Results _on the tetramers:

TTIi~+:

total

energies

E

(in eV) of

the

configurations

~A > to

~F

>.

EH give

the

corresponding expressions for

the extended Hubbard Hamiltonian

(see Eq. (/)). Ej give

the

corresponding expressions for

the Hamiltonian

defined

in

equation (8).

The mtersite terms

l~j

and

/l~j of equation ($ )

and

(8)

_are noted

by

Vk and

/lk respectively

with k

= ~i

j~

^{denotes the k~~}

neighbour.

The total

energies

^calculated without relaxation

of

the MO _are

given

in

parenthesis.

E

E~

> ₌ -6382.946 2U ₊ V3 2U ₊ V3 +

20/l1

+12/l2

₊

4/l3

> ₌ -6381.537 2U ₊

(

2U +

Vi

+

18/l1

+12/l2

+

6/l3

> = -6382.630 + V2 + V2 +

-6382.

547) +12/l2

₊

6/l3

> ₌ -6381.763 2U +

(

2U ₊

Vi

₊

(-6381.596) +12/l2

+

8/l3

> ₌ -6380.197 3U 3U +

+12/l2

+

8/l3

> = +

+12/l2

+

4/l3

the dimer

M+ M+

is located _on the cluster

edge

_{of ~B} > and the _energy is

higher

^{than in the}

preceding

_case and the

corresponding Vi

is

significantly

smaller with _a value of 2.83 eV.

To

summarize,

_an electrostatic model of

repulsion

between net

charges

is not sufficient to describe the

change

in the total

energies

^when a

charge (one charge

2+ _{on one} site _or two

charges

+ on

neighbounng sites)

_moves

along

_a neutral chain. This

phenomenon

_is well known

m

quantum chemistry

and is related to what is called

"penetration"

effect

[18].

We then define

/l~j

_as the

penetration integral

between molecule _I and molecule

j.

This term embodies the difference between the electron-electron

repulsion

and the electron-core

attraction. It is

expected

to be rather small and to decrease

steeply

with the intermolecular

distance,

since for

long

distance

interactions,

both electron-electron and core-electron

repulsions

reduce to point

charge

interactions. The Hubbard Hamiltoman modified

along

these lines then reads:

H =

~ _{tv (C(~C~+1,~}

₊

_h-c-)

₊

j £ _{U~n~(n~ 1)}

~,j,a ~

+( ^£ Kj _In~ zi)Inj zj)

+

£ _{~v In~z~}

₊

njzj) ₁₈₎

~#J ~#J

We have assumed that the terms

l~j

_are identical for electron-electron and _core-core in-

teractions,

which may ^not be true but the _core-core interaction _{is a} constant in this

type

^of

Harniltonian, merely inducing

_a

global

shift of the energy

spectrum.

The last column of Table II collects the

corresponding

formulations

E~ taking

into account the different terms U~,

l~j

and

/l~j

of

equation (8).

The extracted

l~j

values _are

~nique

what-

ever the

configurations

considered. The

edge

effects _{are now}

properly

taken into account and have _no influence _on the values of the parameters.

Assuming

that the

penetration integrals

are

negligible beyond

second

neighbours (/l3

# 0 in

Eq. (8)),

the _comparison of AMI _energy

differences

(relative

to the lowest total _energy

EA)

leads to the

following

values of the param- eters

(in eV):

Vi

₌

3.08;

V2 =

1.97;

_V3

=

1.42; /l1

= 0.12

while the

corresponding

values determined without

allowing

^for _an in-situ relaxation of the molecular orbitals _are:

Vi

=

2.95;

V2 ₌

1.82;

V3

=1.29; /l1

₌ 0.09

The

(

values determined in both _{ways are} identical to the dimer values of 3.05 eV

(2.95 eV)

and this result is _a

strong

^{evidence of} the

transferability

of the

parameters.

3.4. HEXAMERS. The extension of the results to 6 TTF clusters

TTF(+

is

straightforward

and it is not detailed here. The parameters ^issued from such _a calculation _are the

following

ones:

Vi

₌

3.10;

V2 "

2.02;

V3

=1.49;

V4 "

1.18;

V5 "

0.97; /l1

=

0.13; /l2

= 0.035 while the

corresponding

values determined without

allowing

for _an in situ relaxation of the

molecular orbitals _are:

Vi

₌

2.95;

V2

=1.82;

_V3

=1.29;

V4 =

0.99; l§

=

0.81; /l1

=

0.10; /l2

= 0.025 These values confirm the conclusions reached in the

previous paragraph

about the

adequacy

of the method to extract reliable

parameters

from chain clusters calculations.

It is

noteworthy

to remark that these results

imply

that the

polarization

of the MOS induces

a rather sizeable effect which is almost

independent

of the type of

neighbour

and which raise to about 0.2 eV.

We

present

ⁱⁿ

Figure

the variation of V _{as a} function of the distance R between _two TTF molecules. The calculation of the V terms _was

performed following

different

approaches.

Using

the above results _on

TTF(+,

it is

possible

to extract V _up to the 5~~

neighbour (the

limitation to

larger

distances is

directly

related to the size of the cluster that _can be handled

by

AMPAC and to our

computer facilities).

A

comparison

is made in

Figure

1 between the

parameters ^{obtained from} the

non-optimized

MOS and from the

optimized

MOS. It is also

possible

and fruitful to extract V for all

neighbours interactions,

from dimer

only

calculations:

the molecules _on the dimer _are then located at _a distance R.

Finally,

for

large R,

V may be

simply

calculated _as the interaction between

point charges.

Were the V terms

expressing

_an

electrostatic

repulsion, they

should follow the Coulomb law for

large

distances. The Coulomb

potential,

noted

Vc,

is then

(Vc

_is

expressed

in eV and R in

Ji)

14.394

Vc

₌

~

Figure

1 shows that the calculated Vk ^do not all behave at

long

distance _as

Vc

^The

agreement

stands for the values deduced from dimer calculations and for the values obtained from hexamer results using

non-optimized

MOS. This result _means that the parameter ^extraction using the Hamiltonian described

by equation (8)

is

fully

size consistent when MOS

optimized

_on

monomer are used. There thus exists _a model Hamiltonian which

properly

describes the full

set of Valence Bond

configurations. Therefore,

_m the Hilbert space

spanned by

these VB

functions,

the

diagonal

elements of the Hamiltonian matrix may be

properly

calculated within

a

simple

quantum cell model reminiscent of the

Panser-Parr-Pople model,

extended to include

5

4

~s 3

j iS

i

o

0 5 Jo 15 20 25 30

R

(A)

Fig.

1. Variation of V

(eV)

_{as a} function of the distance R

(A)

between two TTF molecules.

The _squares and the circles

correspond

to the results extracted from calculations _on

TTF(+

with

and without MO

optimization respectively.

The _crosses

correspond

to dimer calculation with MO

optimization.

The solid line represents ^the Coulomb

potential.

penetration

interactions. It is also remarkable that coulombic

repulsion

terms may be evaluated from dimer calculations.

On the contrary, ^{the V values} deduced from hexamer results

using optimized

MOS show _a small

departure

from the Coulomb law at

long

distance

(the

difference is about 0.15

eV).

The effective Hamiltonian _is thus not suited for _{a proper} extraction of electrostatic interactions _m

this _case. The full

optimization

of the MOS _m each

configuration yields

_an extra

energetic

term

which

depends

on the nature of the

configuration

itself. It results in _a decrease of the total cluster _energy which is _maximum for the

configurations ~E

> and

~F

> of Table

II,

I.e. 1&~hen there _are

doubly charged

molecules in the cluster: the relaxation of the molecular orbitals is

more effective _m the _more ionic

configuration.

3.5. PARAMETERS _oF ORGANIC SALTS. In order to discuss _more

precisely

the absolute values of the electrostatic

parameters,

^these were calculated for different molecules all derived from the tetrathiafulvalene molecule

(TTF): tetramethyl

TTF

(TMTTF),

Bis

Ethylene-dithia

TTF

(BEDTTTF)

and

Dimethyl Ethylene-dithia

TTF

(DIMET).

The atomic coordinates

are taken from

crystallographic

structures:

TMTTF2PF6

measured at 4 K and atm

[19], fl BEDTTTF2I3

at 4.5 K under _{a pressure} of P

= 1.5 kbar

[20], DIMET2SbF6

at 300 K and 1 atm

[21].

^{These salts} are

representative

^of

quasi-one

and

quasi-two

dimensional structures for

TMTTF2PF6

and

fl BEDTTF213 respectively,

while the

DIMET2SBF6

salt evidences _a

quasi-1D

structure but with _a

large

dimenzation of the intrastack transfer

integrals

_as discussed

previously [12].

3.6. MONOMERS. Table III shows the total

energies

obtained for the neutral

M°,

monoca-

tionic M+ and dicatiomc

M~+

_species which _are used to calculate U. The total

energies

_were calculated

by

_means of the AMI

semi-empirical approximation.

The

corresponding

Ii values

(see Eq. (5))

_~re

given

_m ^the

penultimate

^line of Table III.

They

_range around 4.1 eV and

expectedly

decrease with the _size of the molecule.

They

_are twice _as

large

as previous

results,

obtained using the half-electron CNDO calculation

[12].

Table III. Same entries _as Table I

for

molec~le M. U

(ab imtio)

_were determined

from

total energy

differences

calculated

by

means

of

the GAUSSIAN 92

package (Gaussian 92,

Revision

D-e,

Frisch

M.J.,

Trucks

G.W.,

Head-Gordon

M.,

Gill P.

M.W., Wong M.W.,

Foresman

J-B-,

Johnson

B-G-, Schlegel H.B.,

Robb

M-A-, Replogle

E.

S., Gomperts R.,

Andres

J.L.,

Raghavachari K., Binkley J-S-,

Gonzalez

C.,

Martin

R.L.,

Fox

D.J., Defrees D.J.,

Baker

J.,

Stewart

J.J.P.,

and

Pople J.A., Gaussian, Inc., Pittsburgh PA, 1992j.

M TMTTF BEDTTTF DIMET

-2222.951 -2944.761 -2583.398

-2216.216 -2937.951 -2576.576

6.81 6.82

7.12 7.27 7.23

11.15 10.71 10.94

11.57 11.23 11.41

U 4.99 4.61 4.78

I _{+a+h +c}

f+~_+~ ^I+h

~

~

I

~~

~

-

/~

Sl

~

~

j_~

I+2h +c

~

I +h +c

Fig.

2. Schematic representation ^of the TMTTF

packing

_m the

(a,

b)

plane. Symmetry operations

from the central TMTTF

(labelled I)

_are indicated. The zig-zag structure of the chain _imposes that both I and

I

_{+ b +}

c molecules

belong

to the unit cell _so that the intrachain interactions, noted >£1 and

52,

_are

inequivalent.

The interchain interactions _are noted Ii, 12 and 13.

Equivalent

pictures also

apply

to the definition of the interactions _in

fl

BEDTTF2I3 and DIMET2SbF6.

The last line of the table collects the values of U obtained from ab initio Hartree-Fock calculations

using

_a double-zeta 4-31G basis _set. The ab initio U range around 4.8

eV,

which

is somehow

larger

than the AMI results but _not

qualitatively

different. The variation of U with the size of the molecule sho1&>s the _same trend.

&Ioreover,

the ratio U

(A&11) /

U

(ab

initio

)

is the _same for each molecule. These ab initio results sustain the _use of _a

semi-empirical approach

for

determining

the correlation parameters.

The AMI _or ab initio

purely

molecular U terms may appear far too

large compared

to estimations from

experimental

data

[22-24].

This point ^{will be} clarified in

following

sections.

3.6.1. Dimers. Dimer calculations _were

performed

for the three structures:

TMTTF2PF6, fl BEDTTTF2I3

and

DIMET2SbF6,

and all

types

^{of first}

neighbour

interaction. in

Figure 2,

we

figure

out the distinct labels associated with the first

neighbour

interactions. S~ or

[

stand for the intrastack and interstack interactions

respectively.

The V values which _are collected in

Table IV _are obtained from

equation (4)

_using the U values of Table llI.

Table IV. Results _on the dimers

Ml

_: /lE _are the

differences

between the total

energies of

the

fM+

M+

j configurations

and the

fJP+ M°j configurations,

and V _is the interaction between

first neighbours (all

entries in

eV).

interaction

52 Ii

12 13

1.50 1.53 2.49 2.20 2.46

V 2.91 2.88 1.92 2.21 1.95

V 2.41 2.11 2.11 2.08 2.16

V 2.80 2.24 2.06 1.96 2.16

~

~~

Sl 52

~ ~

52

Sl

Type

I

- -

Type

11

Fig.

3. Schematic representation of the TMTTF chain

showing

the two

possible geometries

for _a tetramer.

The calculated V values _range

roughly

between 2 and 3

eV,

I.e. of the order of

U/2. They

are

closely

related to the

crystal

structure, but in _a different _way than the transfer

integrals

do. The V terms do

reproduce

the

dimensionality

of the

crystal:

for the

quasi-1D TMTTF2PF6 salt,

the intradimer

Vsi

and interdimer

Vs2

terms are I eV

larger

than _any other interstack

Vii

term. The 2D BEDTTTF results show that the intradimer _term

Vsi

is the

largest

_one, but it is

only

0.3 eV

higher

than the other terms. The _case of DIMET is intermediate:

Vsi

is the

largest

_one, the Vj~ _are

equivalent

and similar _to the TMTTF _case, while the interdimer

interaction, Vs2,

is intermediate between the two sets.

The _average value /lE is _now around 1.8 eV which is of the _same order of

magnitude

_as the visible CT bands observed

experimentally [22-24]. Obviously,

the calculated values do _not

correspond exactly

to the _energy differences between

stationnary

states that _are in fact VB mixtures of the different

configurations.

The interaction energy between the two

configurations

which _we could call the transfer

integral

_is therefore also involved in the CT band.

Nevertheless it is admitted that such transfer corrections _are rather weak in molecular

crystals,

due to the weak

overlap

between the MO's _on

neighbounng

molecules

[13],

^and our calculated

transition energies thus should _not be _very different from the true theoretical

absorption

band.

3.6.2. Tetramers. The tetramers calculations _were

performed only

m the _case of the intra- chain interaction

corresponding

to the

crystal _geometry

of

TMTTF2PF6. Taking

into account the real

crystal

structure leads to two

possible

choices for the tetramer

TMTTF(+,

illustrated

by

the two different interaction sequences shown in

Figure

3.

We

give

_m Table

V,

the

corresponding

total

energies

for each

configuration. Again,

the total _{energies using} the molecular orbitals

optimized

_on the isolated _{monomers are}

given

in

Table V. Results _on the tetramers TMTTF

(+:

total

energies

E

of

the

configurations

_~A >

to

~F

>

(the

values

corresponding

to

non-optimized

MOS _are

given

ⁱⁿ

parenthesis). E~ give

the ezpressions

for

the model Hamiltonian

(see Eq. (8); /l3

terms _were

omitted).

For each

configuration,

the

first

line

corresponds

to the

type

^I

cluster,

and the second line _to the

type

II cluster

of Figure

3.

> -8875.835 2U + +

M+M°M°M+

-8875.785 2U + +

20/l1

+

12/l2

> + + +

M°M°M+M+

_-8874.579

(-8874.407)

2U +

Vsi

+

18/l1

+

12/l2

> + + +

l/1+M°M+M°

_{-8875.526 2U + V2 +}

18/l1 +12/l2

> + + +

M°M+M+M°

-8874.843 2U +

Vs2

+

16/l1

+

12/l2

> + +

M°M°M~+M°

-8873.357 3U ₊

16~i

₊

12/l2

> -8872.880 3U +

20/l1

+

12/l2

M°J£I°&I°M~+

_{-8872.845 3U +}

20/l1

+

12/l2

parentheses;

in this _{way we can} estimate the influence of the environment _on the _energy and orbital relaxation. We also _give in the _same table the

corresponding energies expressed

in _an extended

(to

all

neighbours)

Hubbard model. The calculations _were

performed

for the two clusters I and it defined in

Figure

3. With U

= 4.41

eV,

the V parameters take the

following

values

(in eV):

Vsi

=

2.96; Vs2

_"

2.92; % "1.98; V3(1) =1.46; V3(II)

=

1.47; /l1

= 0.13

while the

corresponding

values determined without

allowing

for _an in-situ relaxation of the molecular orbitals _are:

Vsi

_"

2.77; Vs2

_"

2.74;

V2

=1.79; V3(1) =1.26; V3(II)

=

1.28; /l1

₌ 0.09 As

expected

from the above discussion _on TTF

clusters,

the

Vsi

values _are close to the

ones extracted from dimer calculations. Their decrease relative to the TTF values reflects the difference _m the

corresponding

U values. The variation of V ~vith the intermolecular distance follo~vs the Coulomb

potential

la~v _as

already

discussed. As _an

illustration,

the

only slight

dimerization of the stack in

TMTTF2PF6

does not induce _a

significative

difference between

V3(1)

^and

V3(II).

From the

energies

of the tetramer

configurations,

_{one may}

get

some

insight

into the

charge

transfer

(CT) _spectrum by looking

at the

possible

CT excitations

involving neighbounng

molecules. The

corresponding

excitations energies may be also

expressed

_{as a} function of the electrostatic terms

entering

the definition of

equation (8). Figure

4 collects the results

obtained

along

these lines. We do not have _access

yet

to the true theoretical CT bands. _since

we are

only considering

in this work VB structures instead of exact

stationary

states. The intensities of the band _are also

beyond

the _scope of this paper.

Two _main groups of

spectral

lines _are found: the first _one involves CT _energies around 4000

cm~~

_and

corresponds roughly

to the V

terms;

the second group involves both U and V and is around 13000

cm~~

These results

qualitatively

_agree with the

experimental spectra

of

CT bands (cm-1)

~ ~~~

l6000

~ U- Vs

3 0

~ 12000

2 0 U Vs 8000

)

^Vs V2 2A1 B

i o D

4000

vs v2- 2A1 V2 V3 2A1

C V3 2A1

0 0 A 0

Fig.

4. Schematic

representation

of the CT bands _m TMTTF chains. The left part of the

figure

shows the

energies

of the Valence Bond

configurations

_~A > to ~F > relative to the energy of ~A ^>

(see

Tab. V for

definition).

The

possible charge

transfers between these states _are also indicated in the framework of the Hamiltonian described in

equation (8).

The

right

_part of the

figure

_{is a} crude

representation of the theoretical spectrum. The effect of stack dimerization _on

Vs,

which leads to _a

doubling

of the CT bands at U Vsi

(Vs2)

and Vsi

(Vs2)

V2 2/h1 is not

represented

for

clarity.

organic

salts which evidence _a band called "B" whose _energy is

larger

than 1 eV and _a "A"

band in the domain 0.25 to 0.40 eV

[23].

4. Conclusion

in order to size _up the relative

magnitude

of the coulombic interactions bet1&~een conduction electrons _{m organic}

conductors,

_we set up a mixed Valence-Bond

/Hartree-Fock

method

applied

to all valence electrons calculations _on finite size clusters. The

hierarchy

of _energy levels associated to the relevant set of electronic

configurations

of the clusters _was

analysed1&~ith

an effective Hamiltonian built upon a

single

site orbital _per molecule and thus

comparable

to model Hamiltomans such _as the

(extended)

Hubbard _or

Pariser-Parr-Pople

Hamiltomans. The extraction of this effective Hamiltomaii

yields

_a

Pariser-Parr-Pople

like

Hamiltoman,

extended to include

penetration

interactions.

Moreover,

it appears that it is most consistent when &IOS

optimized

_{on monomer are}

used;

the effective Hamiltonian

reproducing

the whole _set of VB energy levels.

Therefore,

_m the Hilbert _space

spanned by

these VB

functions,

the

diagonal

elements of the Hamiltoman matrix may be

properly

calculated within _a

simple

quantum cell model.

The effective on-site

repulsion

U extracted in this _{way are} about 4 eV while the intersite V

repulsion

terms range between 2 and 3 elf for

first-neighbour

interactions and

they

follow _a Coulomb law for

large

intermolecular distances.

Although

^these terms may each _appear too

large compared

to previous estimations

[22-24],

the

roughly

calculated CT excitations _are still in

good

_agreement with the

experimental

results which

only rely

on energy differences. Recent

proposals

for _a determination of these parameters on the basis of

i-effectivity

measurements

[10b]

would

essentially

lead to the _same range of parameters as m

[22- 24],

because the

analysis

of the

experimental

results

depends

_on the

type

^of the model _{one uses,} I.e. the extended

Hubbard model restricted _to

first-neighbour

interactions. ive must state here that there is no

inconsistency

^between our results and the values

commonly quoted

in the literature.

As mentioned

above,

this work concentrated _on the extraction of the

diagonal part

^of the effective Hamiltonian. It _remains to set up a similar

analysis

for

deriving

the

off-diagonal

_part

of the effective Hamiltonian which

essentially

relies _on the

charge

transfer _processes between

neighbouring

molecules. Matrix elements between the various VB structures _now involve _non

orthogonal

molecular orbitals: the MOS _are

orthogonal

in each VB

structure,

^{but the MOS} of _a

given

molecule _{m a}

given charge

state _are not

orthogonal

_anymore to those of the _same

molecule in another

charge

state. The off

diagonal part

of the Hamiltoman matrix will therefore

require

much _more efforts than the

tight-binding techniques

for

calculating

transfer

integrals.

It must also be

emphasized

that _one should take

great

care of the counter-anions

potential,

which for

symmetry

_reasons

might strongly

contribute to the transfer terms. We assumed in this work that the anions could be discarded for

comparing

the various

charge

distributions in the organic stacks.

indeed,

if the electronic clouds of the anions and

organic

molecules _were

optimized

_{on monomers,} the anion-cation

interactions,

for

symmetry

_reasons, would be

simply

additive and give

essentially

the _same contribution for each VB

configuration.

The difference

m the cation ~fOs would

merely

induce _a

slight

correction in the coulombic

terms,

similar _to the D correction. The anions should nevertheless be taken into account if _one has to deal with

possible

effective interaction mechanisms

involving

the differential

polarization

of the molecules and anions in distinct electronic

configurations.

Such considerations raise many

methodological questions

about the _most

appropriate

_way of

setting

_up calculation schemes that would give the clearest

interpretation

of the relevant

physical phenomena

in these materials. VB

inspired

_or

not,

variational _or

perturbational,

such schemes

always rely

_{on some} zeroth order

stage. Any

theoretical

analysis

is therefore

strongly dependent

_on the choice of this zeroth order reference. For

example,

had _we

imposed

a

unique

set of l/IOS for all VB

configurations,

_we would have obtained _a different effective Hamiltonian with

larger

interaction

parameters

an extensive

Configuration

interaction

(Cl)

or

perturbation

scheme

being

_necessary to reach the _same level of _{accuracy as} the

present

^VB calculation. in the _same way, one should ask whether the MOS should be

optimized

in-situ

or not, ^the second alternative

being obviously

less accurate but much _more

simply analysable.

In the

prospect

^of

dealing

with sizeable two dimensional

clusters,

it seems more reasonable to

employ

monomer

optimized MOS;

the results of this work

showing

_no great differences _m

terms of the order of

magnitude

of the coulombic

parameters.

^The

understanding

of the effect

of the full

optimization

of the MOS

(projected

in _our zeroth order

space)

would allow for the inclusion of _new

polarization

terms in the effective Hamiltonian.

The

magnitude

of the coulombic _terms is found _to differ

qualitatively

from

previous

estima- tions and evidences the

long

_range

part

of the

potential,

which has sometimes been advocated

m the literature

[25].

^{It is} moreover of first

importance

to

analyze

the solutions of extended

Panser-Parr-Pople Hamiltoman,

within the determined

parameter

_range, and to check whether

or not these solutions may

alternatively

be attained

by

_means of _{a more} usual

(extended)

Hub- bard Hamiltonian.

References

[1] JArome

D.,

Mazaud

A.,

Ribault M. and

Bechgaard K.,

J.

Phys.

Lett. France 41

(1980)

L95-98.

[2]

Wang H.H.,

Carlson

K.D.,

Geiser

U., Kini, A.M.,

Schultz

A-J.,

William

J.M., Montgomery L-K-,

Kwok W.K.,

Welp U.,

Vandervoort

K.G.,

Schriber

J.E., Overmyer D.L., Jung D.,

Novoa J.J. and

Whangbo M.-H., Synth.

Metals 42

(1991)

1983-1990.