Foundation of Heisenberg Hamiltonians for systems with several active electrons per centre : acetylene and polyines

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Foundation of Heisenberg Hamiltonians for systems with

several active electrons per centre : acetylene and

polyines

Daniel Maynau, M.A. Garcia-Bach, Jean-Paul Malrieu

To cite this version:

Daniel Maynau, M.A. Garcia-Bach, Jean-Paul Malrieu. Foundation of Heisenberg Hamiltonians for

systems with several active electrons per centre : acetylene and polyines. Journal de Physique, 1986,

47 (2), pp.207-216. �10.1051/jphys:01986004702020700�. �jpa-00210197�

Foundation

of

_Heisenberg

Hamiltonians

for

_systems

with several active

electrons per

centre :

_acetylene

and

_polyines

D.

_Maynau,

M. A. Garcia-Bach

₍₊₎

and J. P. Malrieu

Laboratoire de

_{Physique Quantique,}

Université Paul Sabatier,

Unité Associée au C.N.R.S. n° _505,118, route de Narbonne, 31062 Toulouse Cedex, France

(+)

Universitad de _Barcelona, Facultad de Fisica,

Fisica de l’Esta Solid, Diagonal 645, Barcelona, 28 _Spain

(Reçu le 27 mars 1985,

_accepté

sous

_forme

_difinitive le 4 octobre ₁₉₈₅₎

Résumé. 2014 Un Hamiltonien de

Heisenberg non

_{empirique, proposé}

_{pour des systèmes} 03C0

_conjugués

à _partir de

calculs sur

_{l’éthylène,}

s’était révélé très

_performant.

Son fondement

_théorique

découlait de _façon évidente de l’Hamiltonien total, grâce à la Théorie des Perturbations Quasi

Dégénérées (QDPT),

comme l’a montré Anderson.

Si on veut, de la même façon, définir un Hamiltonien de

_Heisenberg

_{pour des}

_systèmes

comportant deux électrons actifs dans deux Orbitales

_{Atomiques orthogonales}

_par centre comme dans les

_polyines,

on _peut extraire

l’infor-mation d’un calcul

_poussé

sur

_{l’acetylène.}

Le

_{développement}

_QDPT ne _{converge pas,} et la

_{simple application}

des définitions de Bloch et des Cloizeaux donne des Hamiltoniens effectifs mal conditionnés, très

_éloignés

de la structure

de _Heisenberg; leur transfert à

_{l’homologue C3H2}

donne de mauvais résultats. Nous définissons un autre Hamil-tonien effectif _qui a bien l’allure d’un Hamiltonien de

_Heisenberg

et _qui conduit à un _spectre raisonnable _pour

C3H2.

Abstract. - A

_{non-empirical}

_Heisenberg Hamiltonian had been

_proposed

for

_{conjugated 03C0}

_systems from ab initio calculations on

_ethylene,

and

_proved

to _{be very efficient. Its theoretical} foundation from the full Hamiltonian was

evident from the _concept of effective Hamiltonian and the use of

_{Quasi-Degenerate-Perturbation Theory}

_(QDPT), as shown

_by

Anderson. When one wants to define a

_Heisenberg

Hamiltonian for _systems

_bearing

two active

elec-trons in two

_orthogonal

Atomic Orbitals _per centre, as in

_polyines,

one may try to derive it from an accurate cal-culation on _acetylene. The _QDPT

_procedure

does not _{converge, and it is shown that the direct} use of the Bloch and des Cloizeaux definitions lead to ill-behaved effective Hamiltonians, which deviate _strongly from a _Heisenberg

structure, and give erroneous results on

_C3H2.

A modified definition of the effective Hamiltonian is

_proposed,

which recovers the form of a _Heisenberg Hamiltonian and proves to

_give

a reasonable _spectrum of

_C3H2.

J.

_Physique

47

₍₁₉₈₆₎

207-216 FTVRIER _1986,

Classification

Physics Abstracts

31.20G - 31.20R - 75.10 - 75.1OJ

Introduction.

The

_Heisenberg

Hamiltonians

_[1]

have been

_essentially

developed

in solid state

_physics

for Mott insulators.

They

have been

_applied

in coordination

_chemistry

to rationalize the lower

part

of the

_spectrum

of mole-cules with several

_unpaired

electrons. A series of

previous

papers

[2-5]

have shown however that the

Heisenberg

Hamiltonians

_might

be

_successfully

applied

to the 7r _systems of

_conjugated

molecules

usually

described

_through

Molecular Orbital

Theory.

The

_Heisenberg

Hamiltonians are not

_only

an

elegant

and correlated

_description

of the rules

governing

the electronic distribution in the

molecules,

_they

are able

to

_predict

accurate

_geometries

of

_ground

and excited

states, rotational

barriers,

spectroscopic

features and

photochemical

behaviours

_[5].

An extension of this

approach

to more

_general

chemical _systems seems an attractive idea.

In the above mentioned studies the neutral

confi-gurations

(with

one n electron and one n AO _per

carbon

atom)

span a model space on which the

magnetic

Heisenberg

Hamiltonian is

_expressed.

This Hamiltonian is obtained

_{perturbatively using}

the

Quasi

Degenerate

Perturbation

Theory

[6] following

the

_{procedure proposed by}

Anderson

_[7]

and Bran-dow

_[8].

It _may also be obtained from the canonic

definitions of effective Hamiltonians

_{proposed by}

Bloch

_[9]

or des Cloizeaux

_[10]

from the

knowledge

of

accurate wave functions and

energies,

obtained for

instance from

_{high quality}

molecular

calculations,

with

_large

basis sets and extensive

_{Configuration}

Interaction

_(CI).

The

_practical

successes mentioned

above were obtained after extraction of such a

senberg

Hamiltonian from accurate MO-CI calcula-tion on

_ethylene.

When one wants to

_generalize

this

_approach

one

must remember that the

_Heisenberg

Hamiltonians are

restricted to half

_{filled-bands,}

i.e.

_systems

where each

atom

_brings

_p electrons in _p Atomic Orbitals

(AO).

The purpose of this paper is to determine whether it is

possible, following

the _way

_{opened by}

Anderson,

to

obtain

_Heisenberg

Hamiltonians for half-filled bands with several active electrons _per atom. The

_simplest

physical problem

deals with the

acetylene

molecule and the

_polyines

treated as n

_systems.

The J core is

kept

frozen and each carbon atom

_brings

two

electrons and two

_orthogonal

_px and _py AO’s. The conclusions obtained from this

_{precise study}

will be

general

and valid for other half-filled bands.

The

_{qualitative problem}

_may be formulated as

follows : for

_systems

with one electron and one AO

per centre

_(for

instance n

conjugated

molecules,

or

alkali

clusters)

the model _{space involves all the neutral}

determinants

_(in

the sense of Valence Bond

(VB)

Theory).

The effective Hamiltonian does not

_give

_any

information on the

_{predominantly}

ionic

_eigenstates,

but delivers all the neutral valence

eigenstates, .

which

represent most of the lower

_part

of the

_spectrum.

In the case

of p e-, p

AO _{per centre, the}

Heisenberg

Hamiltonian does not exclude

_only

the ionic deter-minants

(and

ionic

_{eigenstates),}

it also excludes a whole category of neutral

determinants;

the

Heisen-berg

Hamiltonian _{model space is}

_composed

of deter-minants where each active AO bears one

_electron;

but other neutral determinants

exist,

where one AO

bears 2

electrons,

while another one on the same atom

is

_empty.

Hereafter the stars _represent a

_spins

and the

_{circles p}

spins;

these

_symbols

are easier to read than the

usual T

and!

arrows.

belong

to the model space while

do not

_belong.

These situations will be called

_{«pseudo-ionic}

».

The occurrence of such situations is

_general

for

_systems

having

more than one electron per centre. These

neutral situations are

_{slightly higher}

in energy than

those where each

AO

_only

bears one

_electron,

because

of the excess

_repulsion

of the two electrons in the same

AO,

but

_they

are much lower in energy than the ionic

situations,

and their

_proximity

will result in

_practical

and

_{conceptual problems.}

One _may either

i) keep

a _strong restriction of the model _space

to the

_{configurations}

with one electron _per

_AO;

ii)

or

_enlarge

the model _space to all neutral

configu-rations,

including

those with 0 or 2 e- in the same AO.

The

_Heisenberg

formalism

_{necessarily belongs}

to

the first case; however the

present

paper demonstrates that the usual foundations of

_Heisenberg

Hamilto-nians based on the effective Hamiltonian _concepts

and

_{Quasi Degenerate}

Perturbation

_Theory

are not

applicable

and that a new _type of effective

Hamilto-nians must be defined to obtain

_Heisenberg

Hamil-tonians in such

_problems.

The second

_{hypothesis (enlarged}

model

_space)

which _goes out of the

_Heisenberg

formalism will be treated in paper II.

2. The second order

_Heisenberg

Hamiltonian.

2.1 THE MODEL SPACE. - The effective Hamiltonian must

_satisfy

some

_consistency

rules. The basis set must involve all the determinants of the chosen

type

(here

one electron _per

_AO).

When

_Sz

has its maximum

value ± n for n _atoms, all

spins

are

parallel,

and there

is

_only

one determinant in the basis set. The energy of this determinant will be

_kept

as the zero of energy

in the rest _{of the paper. When}

_Sz

decreases,

the dimen-sion of the basis increases to reach the value

_C2n

when

_Sz

= 0. For the

acetylene

molecule,

there are

four

_equivalent

determinants for

_S.

= _± 1.

For

_Sz

=

0,

there _are six determinants of three different _types

The other three are obtained

by exchanging

stars

and circles. One may notice that the determinants of

type

(1) and (2)

have

_Sz

= 0 in both _7c

bonds,

allowing

for electron delocalization in both

bonds,

while

electron delocalization is forbidden in determinants

of

_type

_(3).

The determinants

_(D

also

_respect

the Hund’s rule on both atoms,

having Sz

= ± 1 on

each atom, while

Sz

= 0 in determinants

(2)

and

@.

2.2 THE HEISENBERG HAMILTONIAN AS A

SECOND-ORDER EFFECTIVE HAMILTONIAN. -

Figure

1

_gives

a

schematic view of the full

_S.

= 0 n Hamiltonian. The J core is

_supposed

to be frozen

(for

instance at

the SCF

_level),

the 7c _{AO’s Xa, ya, xb, Yb} are

supposed

to be

_{orthogonal(ized),}

and the 7c VB determinants

are divided into neutral 1

e-/AO

determinants,

neutral

pseudo

ionic determinants with 0 or 2

_{e-/A0, singly}

ionic

_{(A -}

_B+)

and

_doubly

ionic

_{(A - -}

_{B+ +)}

deter-minants.

209

Fig.

la. - Definition of the VB determinants.

Fig.

lb. - Schematic

picture

of the full n electron VB Hamiltonian matrix. K and F _represent

_respectively

the intra-atomic

exchange and

_{hopping integrals.}

Fig.

lc. - Structure of the

Heisenberg Hamiltonian for the

acetylene

molecule.

intra atomic

exchange

operators

between

_orthogonal

AO’s,

which are far from

negligible,

and will contribute in first order to the

_Heisenberg

Hamiltonian. The model space

(1 e/AO)

determinants have the lower energy,

they

are

_{coupled through}

F

_hopping

bicentric

_integrals

to the

_singly

ionic

determinants,

of

_higher

_energy.

Among

the

_pseudo

ionic determinants

_(0,

2

_e-/AO)

the determinants

_X2

₂

are _very low in _energy,

they

are not

_{directly coupled}

_{with the model space, but}

they

are obtained from the

_singly

ionic determinants

through

a second interatomic electron

jump.

Following

the _way which led to a rationalization of

the

_Heisenberg

Hamiltonian for 1

_e-/AO

_per atom

problems,

one _may establish the structure of a second

order effective Hamiltonian built in the model space

through

the

_{Quasi Degenerate}

Perturbation

_Theory.

The

_perturbation

_operator is defined

_by

and in

_practice

The second order

_expression

of the effective Hamil-tonian matrix elements is

The

_singly

ionic determinants a will

_perturb

the

diagonal

energy of the determinants

presenting

two

electrons of

opposite

spins

in each n bond

_(i.e.

the first four

_{determinants)}

and

_they

will

_couple

them

Calling g

= 2

F 2/ ð.E,

where ð.E is the _energy difference between the neutral and

_singly

ionic

deter-minants,

one obtains the

Heisenberg

Hamiltonian

pictured

in

_figure

_lb,

and

_expressed

as

where i

_{and j}

_represent

_px or _py

AO’s,

a and b

symbo-lizing

the A and B atoms

_{respectively.}

The interatomic effective

_{exchange - g only couples}

_px or _py

AO’s,

but the

_coupling

is the same in both 1t

subsystems.

To this _stage, the derivation leads to a

_Heisenberg

Hamiltonian. However two

_problems

will

_evidently

occur

i)

The third order

_expression

is non Hermitian if there exist some matrix elements

of the

_perturbation

operator V in the model

_space,

as occurs for the K matrix elements in the model _space.

ii)

The fourth order

_expression

_{of , I}

_{I H(4)}

_{I J >}

involves terms of the

_type

where

_{the p}

determinants _may be the

_pseudo-ionic

neutral states which have a _very low energy.

_They

will

lead to a

_divergent

behaviour of the

perturbation

series.

This is a

_{major problem}

and one must refer to the

basic definitions of the effective Hamiltonians which the

_QDPT

is

_supposed

to obtain.

3. Features of exact effective Hamiltonians

3 .1 DEGREES OF FREEDOM IN THE DEFINITION OF EXACT EFFECTIVE HAMILTONIANS.

3.1.1 The Bloch non-Hermitian

_effective

Hamilto-nian. - The

Quasi Degenerate

Perturbation

_Theory

leads,

when it _converges, to an effective Hamiltonian built on the model _{space. If}

_Po

is the

_projector

on the

model _space

S,

of dimension _ns

the effective Hamiltonian H

_{has ns}

exact

_eigenvalues

s.

equal

to

_eigenvalues

of the exact

_Hamiltonians

_H,

and the

_{corresponding}

_eigenvectors

_t/lm

of H are the

_projections

of the exact

_eigenvectors

into the model _{space. There exist ns} roots of H such that

This is the Bloch definition of the effective

Hamilto-nian,

and the usual

_QDPT

series

_expansion

leads

to this effective Hamiltonian

ns

-where

the vector

ti;,

is the

_biorthogonal

transform

of 0.

in S

where S

is the

_overlap

matrix of the

_T..,s

_(Smn

=

t/J m I t/J n > ).

One should notice that this effective Hamiltonian exists even when the

_QDPT

_{expansion diverges,}

if _{the ns} vectors

_t/Jm’s

and

_energies

_s. are known.

When one has solved the

_{problem Hgi = 8t/J,}

it is

always possible

to select

_{the ns}

_{eigenvectors having}

the

_largest

_components in the model _space i.e.

_{the ns}

vectors

_ql.

such that

The existence of the effective Hamiltonian is

inde-pendent

on the _convergence

properties

of the series

expansion.

One _may thus

_pick

_up all the

_{eigenfunctions}

of H

_having

the

_largest

_components

in S and build H from

_{equation (1).}

In our

_problem

these states are a part of the so-called neutral states of VB

_theory,

those which are

essentially spanned by

neutral

deter-minants with one electron _per AO.

As

_apparent

from

_equation

(1),

_{due to}

the

non-orthogonality

of the

_projected

functions

_4/.,

the Bloch effective _operator is not Hermitian. From a

practical

point

of

view,

obtaining

an effective Hamiltonian

for the

_acetylene

molecule _may

_proceed

as follows.

- Perform accurate MO-CI calculations of the

lowest

_part

of the

_acetylene

molecule _spectrum. One should obtain all the neutral states of interest.

The

_highest

of them is the

_quintuplet

with one active

electron per

AO,

since no delocalization takes

place

in this state. The other relevant states are selected from evident

_symmetry

considerations and in view of their components in the model space.

- The

corresponding

wave functions are

_projected

211

and

_{applying equation}

₍₁₎

_gives

the Bloch effective Hamiltonian to which the usual

_QDPT

_expansion

converges within the radius of convergence.

Of course this

procedure

is

only interesting

for

transferability

purposes; one

_hopes

that the _operators. so obtained _may be transferred to

_larger

molecular

systems, as occurred for the one-electron one AO _per

centre

_problems

_(alkali

and n

conjugated

molecules).

3.1.2 The des Cloizeaux Hermitian

_effective

Hamil-tonian. - To have

a Hermitian effective Hamiltonian

one _may

_{simply orthogonalize}

in a

symmetric

_way,

through

an

S-112

_transform,

_{the n.}

vectors

-as

_{proposed by}

des Cloizeaux

_[10]

Then

one defines an Hermitian effective Hamiltonian

Hdc

such that

admitting

the ns

orthogonalized projected eigenvectors

4/’ m

as

_{eigenvectors.}

The construction of a Bloch effective Hamiltonian

represents

the smallest loss of information.

_Going

to

the des Cloizeaux definition one introduces a further

distorsion of the

_{eigenvectors.}

This effective Hamiltonian may be obtained

through

a variant of the

_QDP

Theory,

proposed by

Brandow.

This variant is not

_supposed

to have better _convergence

properties

than the usual one.

3.1.3 New

_{proposal : a}

_{hierarchically}

Hermitized

effective

Hamiltonian. - The des

_Cloizeaux

_procedure

alters

_equally

the lowest

_vectors

_{I tiim)’}

which are of

main

_{physical importance,}

and the

_highest

vectors.

As will be shown on the

acetylene problem,

the

ground

state vector _may be

strongly

distorted

_by

the

S - 1/2

transform,

loosing

much of its

_{physical significance.}

One _{may propose,} a hierarchized

orthogonalization

procedure

of

_the

_{I t/lm}

_),

_dividing

them into two classes. The first class M

_(of

dimension

_nM)

defines a

_projector,

where

_Sm

is the

_overlap

matrix of the _nM vectors of the first class. The

_{complementary}

_part

_{(class I)}

is defined

by

the

_projector

In

_practice

the

_procedure

consists in a

projection

of the

I

Vii >

which do not

_belong

to the first class into

_{PI by}

subtracting

their

_components

in the first class vectors

Then

I 1/17)

are then

_orthogonal

to

_then

_IV/.>;

_if

the

(major)

elements of the S matrix between the

_I/Im

are

of the

_type

_Vim

_{I I/Ii ),}

the

_operator

built on the set

1/1"

=

{tiim’

gi§’ )

is

(nearly)

Hermitian.

If

_the

_{11jÎ", )’s}

are not

orthogonal,

an

SM 1/2

_symmetrical

orthogonalization

within the class M _may be

_applied,

and

_SI-1/2

_{symmetrical orthogonalization}

_may also be

_applied

in the class I to obtain

_{perfect hermiticity.}

3.2 STRUCTURES OF THREE « EXACT » EFFECTIVE

HAMIL-TONIANS FOR THE ACETYLENE MOLECULE. - Table I

gives

the

_eigenvalues,

the truncated

_eigenvectors

and the _symmetry of the six

_S,,

= 0 neutral _states

having

the

_largest

_components on the model space of the six

determinants with 1 e-

_per

n AO. The

energies

are

obtained

_using

a double-zeta valence basis set and extensive CI’s. The zero of energy has been taken

equal

to _{the energy of the}

_quintet

_{5 E g+}

state. The table also

_gives

the two states

_lying

in the same _{energy range}

which have weak or zero _components in the model

space, and the

components

of the

_eigenvectors

on the neutral determinants with zero or two electrons in the n AO’s

_(i.e.

pseudo-ionic determinants).

The two

1 Eg+

vectors which must be

_given

_by

the effective Hamiltonian have

large

components on these

pseudo-ionic

determinants,

and these

_large

_components will be

_responsible

for some

major problems.

The

_overlap

of the 1 and 6

_{(1 Eg+)}

_eigenvectors

when

projected

in the _{model space is very}

_{large : (}

_T,

_{I ’" 6 ) =}

0.47. The

Bloch,

des Cloizeaux and

_{Hierarchically}

Hermitized effective Hamiltonians will therefore be very different.

One should notice that the

ground

state determinant has its

_largest

coefficients on the determinants

I xa ya xb .vb

I or

xa Ya xb Yb

I

which

_satisfy

both the atomic Hund’s rule and the

_interatomic

_{ferromagnetic}

coupling.

The

S-1/2

transformed

_4/’

vector is

_deeply

modified,

having

coefficients of 0.45 and 0.54 on the

determinants

xa ya xbyb

I

(Hund) and

I xaYa XbYb

I

(non Hund)

instead of 0.52 and 0.47

_respectively

in

_ViI.

There are four

_Sz

= 1 neutral determinants of

equal

energy. The four

corresponding eigenstates

(3Eu+,

3 Au,

_{5,E +}

and

_{3,A )}

have

_equal

_components

on these four

determinants

and different

_{phases; being}

of

different

_{symmetries, they}

are

_orthogonal;

the

_Sz

= 1 effective Hamiltonians are therefore identical for the

Bloch,

des Cloizeaux and hierarchic Hermitization

definitions

_(and

of course

_Hermitian).

This effective

Hamiltonian is

_given

in table

II;

it is clear that it does not deviate much from a

_Heisenberg

Hamiltonian

structure. The intraatomic

_{ferromagnetic exchange}

integral

K is + 0.029 _a.u., the

_{antiferromagnetic}

coupling - g

between two AO’s

belonging

to the

same n

_system

is - 0.098

Table I. - Neutral

eigenvectors

in the basis set

_{of neutral}

and

_pseudo

ionic determinants. There is one line

_for

two

determinants

_differing

_by

_{a full}

_spin

inversion. The

_symbol

_:t means that the two

_{coefficients of}

the two

_spin-inverse

determinants have

_opposite

signs.

The

_{starred ’A.}

state is neutral but

_essentially

_pseudo

ionic

₍₀

or 2 e- _per

_AO).

Notice the occurrence

of an

ionic 1 E:

state _among the neutral

_eigenstates.

Table II. -

Sz

= 1

effective

Hamiltonian

for

acetylene

molecule

_resulting

_from

an ab initio MO-CI calculation

The

_symbols

between

_parenthesis

_represent the matrix elements

_of

a

Heisenberg

Hamiltonian.

the

_diagonal

matrix element should

be - g

+ K =

- 0.069

instead of the exact value - 0.066. The

diffe-rence is due to terms which involve more than two

AO’s. One may notice the existence of a small

non-diagonal

term

_(-

0.003

_{a.u.) representing}

an

_exchange

between

_{the px}

_{and pB}

_{AO’s ;}

this term _appears

_through

third order contributions in

_{QDPT, through}

the sequences

corresponding

to the third order contributions to the

(II HI

I>

term

where

L, J,

I

_belong

to _{the model space,} a

belonging

to the outer _space. This third order contribution

( -

2

_KF2/åE2)

is

_negative.

A third order correction

to the monocentric

_exchange

has an

_opposite

_value,

as seen in the above scheme

_(i.e.

0.003

_a.u),

so that

the _proper

_{two-body exchange}

is K = 0.032 _a.u. The third order corrections are electron

three-AO contributions which

_might

be written in second

quantization

or

_symbolically

Notice that in the first

_operator

_{the I Ya > Ya}

_I

(i.e.

_a;’

_ay.)

product

is a

_simple

counter. The same

is true for the

_product

_{I Yb > Yb}

_I

in the second term.

Table IIIA

_gives

the Bloch Hamiltonian for the

S.,

= 0 model space. The _structure which _one would

expect

from a

_Heisenberg

Hamiltonian _{appears in}

terms

of g

and K matrix

elements,

together

with the deviations of the Bloch effective Hamiltonian from the

_Heisenberg

structure

_assuming

the values of

_{the g}

and K

_integrals

extracted from the

_Sz

= 1 effective Hamiltonian. The results deserve the

_following

com-ments :

i)

the

_{non-hermiticity}

of the Bloch effective

Hamil-tonian is

_{remarkable, the}

difference between

trans-posed elements

_{I Hij - Hji}

I

is about + 0.06 a.u.

i.e.

_larger

than + 2 K !

ii)

very

large

matrix elements appear between determinants which are not

_{coupled by}

a

Heisenberg

213

Table III. -

Sz

= 0

effective Hamiltonians for acetylene

in _a.u. The

g and K

integrals

are those

deduced from

the

Sz

= 1 matrix

(0.098

and 0.032 _a.u.

respectively)

and the numbers _represent the deviations

from

the

Heisenberg

structure

₍₀

_symbolizes

a zero matrix element in the

Heisenberg

formalism).

The second number on the

_diagonals

concern the

1-2,

3-4

or 5-6 matrix element.

iii)

the

_diagonal

_energy of the determinants

I XaYa XbYb

I

which

_satisfy

the Hund’s rule on both

atoms and take benefit of the

antiferromagnetic

interatomic

coupling

are almost

_degenerate

with

those of the

_determinants

_{I xa Ya Xb Yb}

_I

which lose the benefit of Hund’s rule.

Table IIIb

_gives

the des Cloizeaux effective

Hamilto-nian,

according

to the same

conventions;

one _may

notice that

The deviation from a

_Heisenberg

structure is _very

large

and the

_preceding

comments

_ii)

and

_iii)

remain valid. The near

_degeneracy

between the

_diagonal

terms

concerning

the determinants xa

ya xb Yb

and Xa

Ya Xb

Yb reflects the fact that the

S112

transformed vector

§?

has

_{equal weights}

on these two

_types

of determinants. Our

_proposal

of an Hierarchic Hermitization leads

to an effective Hamiltonian which

_weakly

deviates from a

_Heisenberg

structure. The

_largest

deviation

concerns the

four-body

_operators

One may notice that the

determinants

I Xa Ya Xb Yb

[

are now

_{significantly}

lower in _energy than the

X.

Ya

xb

Yb

I ones,

as one would

_expect.

This

_analysis

demonstrates

_clearly

that for

_systems

having

more than one active electron _{per atom,}

Heisenberg

Hamiltonians cannot be considered as

effective Hamiltonians derived from the full Hamil-tonian

_by

a natural choice of the model space

(as

the neutral determinants with one electron _per n

_AO)

and one of the canonic definition of effective Hamil-tonians.

_Heisenberg

Hamiltonians can

_only

be

obtain-ed

_through

new definitions of effective

Hamiltonians,

one of them

_being

the above defined

_{hierarchically}

Hermitized

definition;

this

_point

will be

_briefly

discussed in the conclusion. This

problem

is related

to the _convergence

_problem,

since the

_large

effect of the

_pseudo-ionic

neutral determinants is

_responsible

for both the

divergence

of

_QDPT

and for the

_large

overlaps

between the

_{1 E:}

_{projected eigenvectors,}

resulting

in _strong

_{non-hermiticity.}

But the

_divergence

of the

_QDPT

_expansion

is a minor

_problem

in front

of the structural defects of the Bloch and des Cloizeaux effective Hamiltonians.

4. Test of

_{transferability}

to

_larger

_systems.

The three effective Hamiltonians obtained in the

previous

section _may be used to treat

_larger

_systems,

such as

_polyines.

The smallest

_larger

molecule is the

linear three-centre

_analog

of

_acetylene

_C3H2.

Our

present purpose is

essentially methodological,

and we

simply

want to see the

_ability

of the effective

Hamil-tonians to

_predict

the

_ground

state nature and the

excited state

_ordering

of this model

_molecule,

for which accurate MO-CI calculations have been per-formed

_by

us,

using

the same basis set as for

_acetylene.

These MO-CI calculations are used as benchmarks.

The fact that the Bloch and des Cloizeaux effective

operators

strongly

deviate from a

Heisenberg

structure

does not invalidate

them;

they might

be as efficient as the

_{Hierarchically}

Hermitized effective operator,

which

_keeps

a

_Heisenberg

structure and has smaller

four-body

operators.

The matrix for the three-centre

_problem

involves 20

vectors. The _operators assume a

tight-binding

approxi-mation,

i.e. all determinants

_{presenting spin exchanges}

between atoms 1 and 3 are

_supposed

to have a zero

interaction.

Since the most favorable electronic distribution

is

_{ferromagnetic}

on each atom and

that the lowest _energy determinant in

_C3H2

are

which the

_Heisenberg

model

_predict

to lie at -

_{4 g}

below the

_{septuplet; they}

_belong

to the

_Sz

= _± 1

subspace.

One _may thus

_expect

that the

_ground

state

is

_triplet

in nature, and its

symmetry

would then be

3 Eg-.

This is

_actually

verified

_by

MO-CI calculation.

The Sz

=

0 ’A.

state lies 1.2 eV above the

3 Eu-.

Its

com-ponents

necessarily

violate Hund’s rule on the three

atoms if one wants to maintain the

_{antiferromagnetic}

(energy

3 K -

_{4 g}

in the

Heisenberg model)

ordering

in all bonds. If one introduces a

ferro-magnetic

coupling

on two atoms, such as in

one

_destroys

one

_{antiferromagnetic}

bond stabilization

at

_least,

the _energy

_becoming

K -

_{3 g}

in the

Heisenberg

model. As noticed

_previously,

the

_{diagonal four-body}

terms in the Bloch or des Cloizeaux effective

Hamil-tonians

_destroy

the stabilization of the structures

having parallel spins

on the same atom with

_respect

to those

_{having antiparallel spins.}

When transferred in the

_{C3H2 problem,}

the Bloch

and

des Cloizeaux Hamiltonians result in a

_major

failure (Fig. 2) :

they

predict

a 1 L1u

ground

state, the

_{3 Ig-}

state

_being

0.65 eV

above;

the above remarks

_explain

this failure as a

consequence of the underestimation of the

ferro-Fig.

2. -

Lower part of the _spectrum of

_C3H2

obtained from the _{fragment operators} of

_C2H2

in the three cases: B: Bloch

operator; dC : des Cloizeaux _{Hermitization;} HH : Hierar-chical _{Hermitization, compared} with an MO-CI calcula-tion. Note the

_similarity

of B and dC results.

magnetic

intraatomic

_coupling

in these effective

operators.

On the _contrary the

_{Hierarchically}

Hermitized effective Hamiltonian

_gives

a correct

_prediction

of the

_ground

state

_symmetry

and of

_{the 1 Au/3 L g-}

_energy

splitting (Fig.

2).

We also calculated _upper states of this molecule. One may notice that the Bloch and des Cloizeaux Hamiltonians also reverse the

u /3,E.

ordering

of the third and fourth excited states and a

5 L g- /3 L g-

ordering

in an _{upper part} of the

_spectrum.

The

_Heisenberg

type Hamiltonian does not

_present

such

_{misorderings.}

It is true that the transition

energies

to these excited states are not

_{quantitatively}

correct

_(deviations

of about 0.5

_eV),

but one should

remember that some low

_lying

neutral excited states

have their

_largest

_components on Valence Bond

structures with

_{doubly occupied}

_AO’s;

in

_C3H2

two such states

_{(1 Lu}

and

_{3 Au)}

_appear in

_figure

2. Their

_specific

interactions with the

_low-lying

states

spanned by

the

_{singly occupied}

AO structures _may

result in an _energy

_lowering

of the latter states. Notice that the

_{5 La-}

does not interact with states

_involving

doubly occupied

AO’s

_{(since they necessarily}

involve

two closed shell atoms and are therefore at most of

triplet spin

symmetry)

and is

correctly positionned

by

the

_Heisenberg

Hamiltonian. For

_larger

molecules the

_part

of the _spectrum

_correctly

treated

_by

the

Heisenberg

Hamiltonian should involve more and more states.

5. Conclusion.

The

_present

_paper shows that the theoretical derivation of a

Heisenberg

Hamiltonian from the exact Hamil-tonian is not

_{straightforward}

for half-filled band

systems

where some atoms

_{bring p( > 1)}

electrons in

p AO.

The derivation

_{proposed by}

Anderson

_[7]

of effective

_exchange

_operators

_reflecting

the

_coupling

through

ionic determinants in a

_Quasi

_Degenerate

Perturbation

_Theory

works

_perfectly

well for one

electron/one

AO

_systems.

_{For p e-/p}

AO per atom

systems,

the second order

_{approximation}

leads to a

Heisenberg

formalism,

but its third order

_development

breaks the

_hermiticity

and the fourth order

pertur-bation would

_diverge.

The

_{Quasi Degenerate}

Per-turbation

_Theory

is unable to

_produce

Heisenberg-type

Hamiltonians for

_systems

like

_acetylene

since

the s2

_px

determinant of the carbon atom is not much

higher

in energy than the

s2 px py

determinant. The lack of convergence of the

Quasi

Degenerate

Perturbation

_Theory

is not a

major problem

since

the effective Hamiltonians that it should prove are

well

_defined,

either in the Bloch or in the des Cloizeaux

versions;

the

_knowledge

of accurate variational wave

functions and

_energies

is sufficient to define the

effective Hamiltonian

_{spanned by}

a

_given

model _space.

The _{present work} has shown that the deviation from

hermiticity

of the Bloch effective Hamiltonian is very

large ;

its structure

_largely

deviates from that of

215

Hamiltonian is of course Hermitian but it also deviates

strongly

from the structure of a

Heisenberg

Hamil-tonian. These

_deviations,

which are

large four-body

terms,

destroy

the benefit of the

_{expected ferromagnetic}

intraatomic

_coupling.

_{The consequence is} a

complete

failure of the

_simplest

_attempt to transfer these

Hamiltonians,

namely

in the

_prediction

of the

_ground

state of the

_C3H2

linear molecule.

These difficulties are linked to the

non-orthogo-nality

of the

_projections

in the model _space of the first and second

_{i Eg}

states of

_acetylene.

This is due

to the _strong

_mixing

of the second

_{1 Eg}

with a third

1 Eg

state, which is

_essentially

built on closed shell

atoms

_pxA2

_p:2,

and which acts as an intruder state in the

_{perturbation expansion.}

The

_symmetrical

treat-ment of the

_overlap

between the

_projected

11:,+

vectors, as

_{managed by}

the des Cloizeaux

S - il2

transform,

results in a _strong distortion of the lowest

I-r+

projected

vector, and this lack of information

on the lowest

eigenvector

destroys

the

_{transferability.}

It

_might

seem

_astonishing

that the Bloch effective

Hamiltonian

suffers the same

defect,

since its

eigen-vectors

I ïji 1ft >

are not

_{orthogonalized.}

But one should remember that in the

_spectral

definition of the Bloch effective Hamiltonian

the (

bra

I vectors

are

biorthogonal

(i.e.

S -1

trans-forms of

_the

_{I t/lm »}

and

_strongly

modified

_by

the S

overlap.

It seems reasonable to _say that when the

projections

of the

_eigenvectors

in the model _space

are far from

_{orthogonality,}

the

_{transferability}

of the Bloch and des Cloizeaux effective Hamiltonians to

larger

systems

is

_{questionable.}

A solution to this dilemna has been

_proposed.

It consists in a

_{non-symmetrical}

treatment of the

pro-jected eigenvector

(or

of the model

_space).

Two classes of

_{projected eigenvectors}

are

_defined,

the lowest ones are

kept unchanged,

while the second class is

projected

in the

_{complementary subspace (and eventually}

sym-metrically orthogonalized

in a later

_step).

If the

lowest vectors are

orthogonal by

_symmetry

the final

eigenvectors

are

_orthogonal

and this new

hierarchi-cally

hermitized effective Hamiltonian has

only

des-troyed

some information

concerning

the

eigenvectors

of the second

class,

i.e. excited

_{eigenvectors;}

when

applied

to

_acetylene

this effective Hamiltonian turns out to

_give

a

_{satisfactory Heisenberg}

structure, with

very small

four-body

terms. It

_{reproduces nicely}

the lowest

_part

of the C-H _spectrum,

_presenting

a

_good

transferability.

This

_partition

of the

_eigenvectors

in two classes

(or

more than

two)

_may be shown to

_correspond

to a

partition

of the model space into a main

subspace

and intermediate model

_subspace.

The above des-cribed hierarchic Hermitization does not lead to an

analytic perturbative expansion

as did the Bloch and

des Cloizeaux definitions. However an

_analytic

expan-sion has been

_{recently proposed by}

one of us and coworkers

[11],

and it will be

_applied

to the same

problem.

If one had

_only

one vector in each

class,

the hierarchic Hermitization would

_simply

consist in

a Schmidt

orthogonalization

of the

projected

eigen-vectors

_(instead

of an

S-1/2

orthogonalization).

In our

precise

problem

the two

_procedures

are identical.

Our hierarchic Hermitization is

certainly

one

member of a

_large

class of

_generalized

effective Hamiltonians which _accept to sacrifice some

infor-mation on the

_{highest eigenvectors}

and

_eventually

on the

_{highest eigenenergies.}

The

_{presently proposed Heisenberg}

Hamiltonian for

_triple

bonds will be extracted for various C=C distances. It will allow the treatment of

_polyines

in their

_ground

and lowest excited states.

_By

_combining

it with the

_{previously proposed Heisenberg}

Hamil-tonians for

_conjugated

_molecules,

it will be

_possible

to treat cumulenes and mixed

_systems

with double

(or aromatic)

and

_triple

bonds. As-C=N- double

bonds have been treated in the same formalism

_[12],

-C N

_triple

bonds

_might

be studied in a similar

fashion. Of course the

_Heisenberg

Hamiltonians increase

_rapidly

in size when the number of centres

increases but some size-consistent

_approximate

treat-ments have been

_{recently proposed}

and tested with

success

_[13],

opening

the _way to the treatment of rather

_large

molecules.

It is

_quite

clear that this work must also be consi-dered as a

_step

on the _way to a

magnetic

treatment of transition metals. The

_ground

state of Cr atom for instance is

sd’,

i.e. with s and d half filled

bands,

and

one

_might

be

_tempted

to define a

Heisenberg

Hamil-tonian for Cr clusters.

_Heisenberg

Hamiltonians are

also _very

_important

for the

study

of

_magnetic

pro-perties

of transition metal

_complexes.

In both cases

the neutral states where some d orbitals are

_doubly

occupied

will act as intruder states with

_respect

to a

model space where each active orbital is

_{only occupied}

once.

References

[1] For a review, see _{HERRING, C., Magnetism} Vol. IIB,

G. T. Rado and H. Suhl, Eds.

_(Academic

Press,

New _York) 1966.

[2]

KLEIN, D. _J., Int. J. _Quantum Chem. 12 ₍₁₉₇₇₎ 255 ;

KLEIN, D. J. and GARCIA-BACH, M. A.,

Phys.

Rev. B 19 ₍₁₉₇₉₎ 877.

[3] OVCHINIKOV, A. _O., Theor. Chim. Acta 47 ₍₁₉₇₈₎ 297 ;

BULAEWSKII, L. _N., Zh.

_Eksp.

Teor. Fis. 51 _{(1966) 230 ;}

Engl.

Ed., 24 (1967) 154.

[4] MALRIEU, J. P. and _{MAYNAU, D.,} J. Am. Chem. Soc. 104

₍₁₉₈₂₎

_3021;

[5] SAID, M., MAYNAU, D., MALRIEU, J. P. and

GARCIA-BACH, M. A., J. Am. Chem. Soc. 106

₍₁₉₈₄₎

_{571 ;}

SAID, M., MAYNAU, D. and MALRIEU, J. _P., ibid. 106

(1984) 580.

[6] SHAVITT, I. and REDMON, L. _T., J. Chem. _Phys. 73

(1980) 5711.

[7]

ANDERSON, P. _{W., Phys.} Rev. 115 ₍₁₉₅₉₎ 1 ; Solid State

Phys.

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99.

[8] BRANDOW, B. H., Rev. Mod.

_Phys.

39 _{(1967) 771 ;}

Adv. Quantum Chem. 10 ₍₁₉₇₇₎ 187 ; Int. J.

Quan-tum Chem. 15 (1979) 207.

[9] BLOCH, C., Nucl.

_Phys.

6 ₍₁₉₅₈₎ 329.

[10] DES _{CLOIZEAUX, J.,} Nucl. _Phys. 20 (1960) 321.

[11] MALRIEU, J. P., DURAND, Ph., DAUDEY, J. P., J. _{Phys. A} 18 (1985) 809.

[12] SANCHEZ-MARIN, J. and MALRIEU, J. _P., J.

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