Generalization of Heisenberg Hamiltonians to non half-filled bands : a magneto-angular effective Hamiltonian for boron clusters

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Generalization of Heisenberg Hamiltonians to non

half-filled bands : a magneto-angular effective

Hamiltonian for boron clusters

A. Pellegatti, F. Marinelli, M. Roche, Daniel Maynau, Jean-Paul Malrieu

To cite this version:

A. Pellegatti, F. Marinelli, M. Roche, Daniel Maynau, Jean-Paul Malrieu. Generalization of

Heisen-berg Hamiltonians to non half-filled bands : a magneto-angular effective Hamiltonian for boron

clus-ters. Journal de Physique, 1987, 48 (1), pp.29-43. �10.1051/jphys:0198700480102900�. �jpa-00210421�

Generalization of

_Heisenberg

Hamiltonians

to

non

half-filled bands :

a

magneto-angular

effective Hamiltonian for boron

clusters

A.

_Pellegatti,

F.

_Marinelli,

M. Roche

_(*),

D.

_Maynau

₍₊₎

and J. P. Malrieu

₍₊₎

Centre de

_{Thermodynamique}

et de

_{Microcalorimétrie,}

_(U.P.

7461 du

C.N.R.S.),

26, rue du 141e _R.I.A.,

13003 Marseille, France

(+)

Laboratoire de

_Physique

_Quantique,

_(U.A.

505 du

_C.N.R.S.),

Vniversité Paul Sabatier,

118, route de Narbonne, 31062 Toulouse Cedex, France

(Requ

le 26 mars 1986,

accepté

le 16

_septembre

₁₉₈₆₎

Résumé. 2014

On propose un hamiltonien effectif pour l’étude des états neutres

_{d’agrégats}

de Bore, en

_{généralisant}

les hamiltoniens de

_Heisenberg

au cas d’une bande

_remplie

au 1/6e.

_L’espace

modèle est

_porté

_{par les}

_produits

des fonctions fondamentales

_2P(s2p)

des atomes. L’hamiltonien effectif

_implique,

outre des termes

_d’échange

de

_spin

de _type

_Heisenberg,

des

_{opérateurs d’échange}

du moment

_angulaire

et des

opérateurs changeant

à la fois le

_spin

et le

moment

_angulaire

des atomes. La

_logique

de cet hamiltonien effectif est _{obtenue par}

_{développement}

au

3e ordre de la théorie des

_{perturbations}

_{quasi dégénérées.}

En

_pratique

l’hamiltonien est obtenu _{directement par} les

équations

de Bloch à

_partir

de calculs

_précis

des fonctions d’onde de

_B2.

La transférabilité de cet hamiltonien effectif

a été testée sur le

système

B3

linéaire ; la

_partie

basse du spectre est correctement

_reproduite.

Ce modèle sera utilisé

pour l’étude du

ferromagnétisme

possible

de chaînes linéaires de Bore. L’article discute le rôle de

_{l’hybridation}

et

souligne

la

_portée

de cette

_{approche, qui}

_{étend la rationalisation par Anderson des hamiltoniens de}

_Heisenberg

à des

systèmes

autres _{que les bandes 1/2}

_remplies.

Abstract. 2014 An effective Hamiltonian has been

proposed

for the

_study

of the neutral states of boron clusters, i.e. for a 1/6th filled band. This

_{generalization}

of

_Heisenberg

Hamiltonians rests _{upon the definition of} a model space

spanned by

products

of

_ground

state

_2P(s2p)

atomic wave-functions. The

_resulting

effective Hamiltonian involves, besides

_{Heisenberg-type}

effective

_{spin exchange}

terms,

angular-momentum

exchange

operators and

spin-and-angular-momentum exchange

operators. The

_logic

of this effective Hamiltonian has been derived

_through

3d order

quasi-degenerate perturbation

theory.

In

_practice

the effective interactions are obtained from accurate MO-CI

_large

basis set calculations on

_{B2 through}

the use of a

_spectral

definition of

Heff.

The

_{transferability}

has been tested on the linear

_B3

_system,

_{by comparing}

the

Heff

_predictions

with accurate MO-CI results. The

_{transferability}

_proves to be

satisfactory

for the lower _part of the spectrum. The model will be used for the treatment of the

_magnetic

character of

(possibly high spin) Bn

linear chains. The role of

_{hybridization}

_{processes has been discussed and the}

_possible

extension of this

_approach,

derived from Anderson’s rationalization of

_{Heisenberg Hamiltonians,}

has been outlined. Classification

Physics

Abstracts 31.10 -

31.20D - 31.20R - 75.10

1. Introduction.

The theoretical foundations of

_Heisenberg

Hamil-tonians

_[1]

are known to be based on the

_theory

of effective Hamiltonians and on the use of

quasi-degen-erate

_{perturbation theory}

_[2].

These effective Hamil-tonians are _easy to derive for

_systems

_involving

one

electron in one Atomic Orbital

(AO)

on each centre

[2],

and

_(although

less

_evidently

_[3])

for

_systems

(*)

Permanent address :

Ddpartement

de Chimie, Univer-site de Provence, Place Victor

_Hugo,

13331 Marseille Cedex 3, France.

involving

p electrons in p AO’s per centre. In both

cases the

_systems

involve a half-filled-band

_(the

other bands

_being

full or

empty).

Besides their

typical

applications

for Mott insulators and

_magnetic

_problems

in coordination

_chemistry,

the

_Heisenberg

Hamil-tonians have

_proved

to be

_{surprinsingly}

efficient tools for

_problems

where the electronics delocalization is far from

_being

_small,

_namely

in

_{organic chemistry}

for the

study

of

_conjugated

molecules

_[4]

and even for alkali

atom clusters or solids

_[5].

The basic idea of the

_present

work is to

_attempt

an

extension of this

_approach

to non-half filled

bands,

for

instance to 1/6-filled

_bands,

as will occur in

_s2p

atoms

(B

or Al

clusters).

The motivation and the

_strategy

are

developed

in section

_{2 ;}

the choice of _{the model space,}

spanned by

the Valence Bond

_(VB)

determinants built from

_ground

state

2p

_{( s2p )}

atomic

_{wave-functions,}

is

explained

and discussed. The

_{ferromagnetic}

character of the

_B2

_ground

state

_suggests

a

_possible

_{ferromagnetic}

ordering

of the linear

_Bn

chains.

Section 3 derives the formal structure of the effective

Hamiltonian from the

_application

of the

quasi-degener-ate

_perturbation

_theory

_(to

third

_order)

to the

_B2

problem.

It is shown

that,

besides

_{angular dependant}

repulsion

terms and other first order terms, the effective Hamiltonian

_presents

a

_large

_variety

of

_operators,

of second-and third-order

_origin.

These

_operators

act on

local

_{spins only, angular}

momenta

_only

or on both

_spin

and

_angular

momenta, and

_imply

2 AO’s

_only

_(of

either the same or

_orthogonal

_subsystem)

or 4 AO’s. The effective Hamiltonian for this

_problem

is thus a

mag-neto-angular

effective Hamiltonian since it acts on both

the local

_{(i.e. atomic) spin}

and

_angular

momenta. Its formal structure is

_developed

in

_{second-quantization}

formalism. From accurate molecular orbital

configura-tion interaction

_(MO-CI)

and

_orthogonal

V.B. calcula-tions on

_B2

numerical effective Hamiltonians are

pro-posed, refering

to a direct

_{(non-perturbative)}

definition

of the effective

_Hamiltonian,

_analogous

to those of Bloch

_[6]

and Des Cloiseaux

_[7].

In a fourth section the

_{transferability}

of these

effec-tive Hamiltonians to heavier

_systems

is tested on the

linear

_B3

_problem.

For this molecule accurate MO-CI

calculations have been

_{performed assessing}

the exact state

_ordering.

_{It appears that in this}

_system

all the lowest neutral states

_essentially

_keep

the

_s2p

character of the atoms. The effective

Hamiltonian,

extracted

from the relevant lowest

_eigenstates

of

_{B2, gives}

surprisingly

accurate

_energies

for all the lowest states of

B3

of various

_{spin multiplicities}

and

_symmetries.

This

implicit

treatment of the

_{hybridization problem}

is

discussed and a further

_{generalization}

of the

approach

to the simultaneous treatment of different bands is

suggested. Applications

to

_longer

linear chains is

plan-ned for the near future.

2. Situation of the

_problem.

2.1 HALF-FILLED BANDS : THE HEISENBERG HAMIL-TONIANS. - For

a

_system

involving n

electrons and n

active

_AO’s,

the

_procedure

to obtain

_Heisenberg

Hamiltonians consists first in the definition of a model space ; this model space is

spanned

by

the Valence Bond

_single

_{determinants which occupy each of the n} active AO’s of the

_system

once and

_only

once. Then all the determinants of the _{model space have the} same

space

part ;

they only

differ

_by

the

_spin

distribution,

and _{the model space is}

_composed

of

_{C :/2}

determinants.

These determinants are all neutral in the sense of V.B.

theory,

and

_they

are the

determinants

_{of lowest energy}

in the V.B. matrix

_(at

least in

_non-polar

well-defined

problems).

The second

_step

of the

_procedure

consists in

applying

the

QDPT

[8]

to _{this model space, i.e.} to construct

_{perturbatively}

the effective

_energies

and effective

_coupling

of the neutral

determinants

under the influence of the ionic determinants. For a

two-electron,

two-AO

_(a

and

_b),

two-centre

_system,

Ander-son shows that the effective

_exchange

between

the two

neutral

_determinants

_I

_ab

_{I and}

_lab

_l

is

where

_Kab

is the direct bicentric

_{exchange, Fab}

the

hopping

integral

between the two centres and AE is the

« transition »

energy from the neutral to the ionic

I aal [ or

lbbl

I

determinants. The

_Heisenberg

Hamil-tonian treats the electronic delocalization

_(i.e.

the mixture between neutral and ionic

_{determinants)}

as

effective

_spin

_operators

between the neutral deter-minants.

This

_straight

forward

_approach

is not

_directly

generalizable

to the

_general

half-filled band

_problems

(p

electrons in p AO’s per

centre),

if one uses the

well-known Bloch

_[6]

or Des Cloizeaux

[7]

definitions of the

effective

Hamiltonian,

as shown

_{by Maynau et}

al.

_[3]

recently ;

new definitions of the effective Hamiltonian

are to be

_proposed

to obtain a

_good

_{transferability}

of

the effective

_couplings

from the two-centre

_problem

to

larger

systems.

Nevertheless the

_general

foundations of the

_Heisenberg

Hamiltonians for half-filled band

prob-lems are clear.

The

_Heisenberg

Hamiltonians have

_recently

received

new

_{applications,}

far from the

_typical

field of Mott

insulators.

_They

have been

_applied

with

_surprising

success to the

_7r-systems

of

_conjugated

molecules in

organic chemistry [4].

A

_precise

MO-CI calculation on

ethylene

makes

_possible

the accurate extraction of two

operators

(a spin-dependant

o

potential

+ 7r

_repulsion

and an effective

_exchange)

which are functions of the

interatomic distances. This

_Heisenberg

Hamiltonian

predicts

very

accurately

the

_geometries,

isomerization

energies,

rotational

barriers,

lowest excited states

ener-gies

and conformations of

_conjugated

molecules

involv-ing

up to 10 carbon atoms

_[4].

A similar

_attempt

has been

_performed

on an

isoelec-tronic

_{problem, namely}

the alkali atom clusters. A

Heisenberg

Hamiltonian has been extracted from

accu-rate MO-CI

_potential

curves of

_{Na2 ;}

it proves to

_give

realistic

_energies

of small clusters and the cohesive

energy of the bulk

[5].

The

_application

of ab initio extracted efficient

Heisenberg-type

Hamiltonian for the

_study

of metal clusters may appear as a

_{tempting challenge}

in view of

the

_difficulty

of usual ab initio MO-CI

_techniques

for the treatment of such versatile

_systems,

_presenting

certainly

several

_{nearly degenerate optimal}

conforma-tions with

_possibly

different

_{spin multiplicities.}

2.2. NON HALF-FILLED BAND : TOWARDS NEW TYPES

OF EFFECTIVE HAMILTONIANS. - The _{basic purpose}

of the

_present

_{paper is} to establish the form of an

effective

_{Hamiltonian, inspired by}

the

strategy

leading

to the

_Heisenberg

Hamiltonians but

_dealing

with non

half-filled bands. The fundamental idea is

_that,

as in

Anderson’s

_{legitimization}

of the

_Heisenberg

Hamil-tonian,

one _may:

i)

choose as _{model space the} set of neutral

determin-ants with maximum

_spread

of the electrons in the concerned

band,

ii)

apply

the

_quasi

_degenerate

_perturbation

_theory,

or an alternative variational formalism

_[9],

to build the effective Hamiltonian

_spanned

_by

this _{model space,} to treat the effect of ionic

_{configurations}

on the neutral

subspace.

As a model

_problem

we consider a one-sixth

(1/6)-filled

_band,

i. e. a cluster of B

_{(or Al)}

atoms in their

s2p (2P)

ground

state.The s band is

_supposed

to be filled up, and the

hybridization

will be considered as a

perturbation only.

We choose the model space

So

as

built of all the determinants in which each atom A bears one

_(and

_only

_one)

_p-electron

in a _PXI

_PY

or _p,

AO. The

_{projector Po}

on the model space is written as :

where

Then it is clear that the model space determinants do not have the same

_{space-part ;}

for each atom one has

two

_degrees

of freedom

_namely

the

_{orientation x, y}

or z

of the

_occupied

AO and its

_spin.

The size of the

effective matrix for a 2 n-atom

_problem

is

32n

_C2n,

larger

than for a

_typical

_Heisenberg

_Hamiltonian,

but

tremendously

smaller than the full V.B.-matrix size. The

_logic

of the effective Hamiltonian is of course

different from that of a

_Heisenberg

Hamiltonian since

it deals with both the local

_spin

and

_angular

momentum.

One may

expect

that the effective

_operators

will be

both bicentric

_{spin exchange}

_operators

and

_I-exchange

operators,

as will be confirmed in the next section. The

logical

structure of the effective Hamiltonian has to be established from

_{analytic perturbative developments}

using

the

_{quasi degenerate perturbation theory}

Qn a

two centre

_problem

in the

_spirit

of the Anderson-Brandow

_approach

to

_Heisenberg

Hamiltonians.

Two

_types

of

_problems

_{may arise}

_along

this

_{path :}

-

non-convergence of the

quasi

degenerate

pertur-bation

_{expression, especially}

at short

_{distances ;}

- mixing

with

_hybridized

structures in which the

sp2

excited atomic

_{configuration}

becomes dominant. These

_{configurations}

_{may appear} as intruder states in

JOURNAL DE PHYSIQUE.-T. 48, N° 1, JANVIER 1987

the

QDPT

_expansion,

_questionning

the

_validity

of the choise of the model space. This

problem

will be

_briefly

discussed in the

_present

_{paper but the}

_perturbative

difficulties will be avoided

_by

a

_spectral

definition of

the effective Hamiltonian. The

_{generalization}

of this

approach

to the simultaneous treatment of several

bands is out _{of the scope of the}

_present

work. 2.3 INTEREST IN B CLUSTERS. - At least at

long

distances,

the reduction of the model space to the

determinants built from the

_{s 2p (2p)}

_ground

state of the atoms is

_{valid ;}

the lowest

_{Sp2 (2p)}

atomic state is 3.5 eV above the

_ground

state, and the

_{corresponding}

Sp2 (2 D )

is 6 eV above the

_ground

state. The

_ground

state of

_B2

is of

_3-v;

_symmetry,

is

_essentially

built from

A ( S2px) x B ( S2py )

atomic

_{configurations,}

and is

ferromagnetic :

This

_point

is

_interesting

since it

_suggest

that :

i)

the

_s2p

_{configuration}

_may dominate the

_ground

state of at least linear B clusters.

ii)

the linear B chain may be a

_high

_{spin compound}

since it may be viewed as dominated

_by

a

_spin-angular

arrangement

i.e.

_presenting

a molecular

_magnet.

Be-sides the formal

_problem

treated

_here,

the

_present

work may then be of

practical

interest.

3. Structure of the

_B2

effective Hamiltonian.

3.1 LOWER PART OF THE SPECTRUM FROM A MO-CI

APPROACH. - The _{number of papers} related to be

B2

molecule is rather limited.

_Among

those

_concerning

the order of the

_low-lying

states, let us note first the calculations of

_Padgett

and

_Griffing

_[10]

at the SCF

level and of Bender and Davidson

_[11]

at the CI level. In the

last,

the CI was not

_sufficiently

extended so that

the

_{5E u}

is

_assigned

to the

_ground

state in both _papers,

instead of the true

_{3 V9}

state

_[12].

At the same

time,

Bader,

Henneker and Cade

_[13]

focused their attention

on the

_charge

distributions and the chemical

binding.

More

_{recently Dupuis}

and Liu

_{[14] performed}

a more

extended CI calculation and found a

_good

_agreement

between their results and

_Douglas

and

_Herzberg’s

experimental interpretation

of the

_symmetry

of the

ground

state

_{( 3 Ii)}

and of the first

303A3 :> 3 Ii

transition. The

_{51 u}

state was found 0.006 a.u.

_higher

than the

_ground

state. Becke

_{[15] using}

a numerical

Hartree-Fock-Slater method studied

_only

the

_ground

state

_303A3g.

He found a dissociation _{energy of 3.8}

_eV,

slightly

larger

than the

_experimental

value

Fig.

1. - MO-CI

[xy] -symmetry

potential

curves of

_B2.

Fig.

2. - MO-CI

[xx,

yy,

_zz]

-symmetry

potential

curves of

B2. L1

states,

degenerate

with those of the

_{[xy ]}

_-symmetry have been removed from the

_figure.

( 3.1

± 0.2 eV

_{[16] ) .}

All these calculations found an

interatomic distance close to the

_experimental

value 3.005 a.u.

_[17].

The most accurate results are those of

Dupuis

and Liu but

_{unfortunately}

_{they only}

concern

the X

_{3 Xi ’ a 5X;;}

and the two first

_303A3u;

states. As will be seen in next

_sections,

a series of

_potential

curves of

_B2

are

_required

for the construction of the

desired effective Hamiltonians.

_By

_{the way, it is also}

interesting

to test the

_validity

of our MO-CI calculations

for this

_simple

case and in

_{particular concerning}

the small difference

_separating

the

_{3 Xi}

and

_{5 X;;}

states. At the SCF level we used the Pshondo program

_[18],

an

extansion of Hondo

_[19].

From the basis evaluated for the B atom

_[20]

we extracted a

_triple-ts

_{(contracted [3,}

1, 1]),

a

double- 03B6p (contracted [3, 1])

and one

_{single- 03B6d}

basis set. The

_d-exponent

_parameter

was chosen

_equal

to 0.5. Correlation effects have been introducted

_by

means of the CIPSI

_algorithm

[21].

In this

method,

a

multiconfigurational

zero-order wave function is first

obtained from a variational

_{configuration}

interaction

(CI)

calculation in a restricted

_subspace

_{built up from} the most

_important

determinants for the states

_being

studied ;

then this wave function is

_perturbed

_up to second order in energy.

The

₃₁₉

state is

_clearly

found to be the

_ground

state. The

_{spectroscopic}

characteristics

_(re

= 3.05 _a.u.,

D,

= 3.10

eV)

_are in

good

_agreement

with

experiment

(re

= 3.005 _a.u.,

D,

= 3.10

eV).

As also

predicted by

Dupuis

and

Liu,

the

_{5 X;;}

state is obtained at 0.01 a.u.

above the

_ground

state, with a

_slightly

smaller

equilib-Fig.

3. - MO-CI

rium distance

_{( re}

= 3.00

_a.u..

One must

_immediately

notice that this

_503A3u;

state is

_necessarily

spanned

by

B

Sp2 (2 D)

x B

_s2p

_{(2p )V.B.}

structures, and it is

neces-sarily

outside of our _{model space}

(cf.

_Eq.

(1)).

The relevant

_potential

curves of

_B2

are

pictured

in

figures

1-3. Curves

_{corresponding}

to various irreduct-ible

_{representations}

of the

_{D 00 h point}

_{group have} been

gathered

into three reducible

_{representations,}

which may be reached from the use of _{px p,} and _{pZ AO’s}

(z

being

the internuclear

_{axis) ;}

this

_presentation

is

espe-cially

convenient for the

_forthcoming

construction of the effective Hamiltonians. The

_{figures only}

_report

the

potential

curves of the states

_having

_significant

compo-nents _{in the model space}

_(with

the

_exception

of the above mentioned

_503A3u-

state, which appears as a dotted

line).

The concerned states are

_always

the lower ones of their

_symmetry,

with three

_{exceptions only :}

the

_highest

in each

_symmetry.

Other states, of redundant

_(spin

or

space)

symmetry

and/or of irrelevant V.B. character

were found between the two

_reported

_{303A3u ,}

30394u

and

1 _Vu,

_{3 llg}

and

_{1 llg}

states. In

_particular,

one notices

irregular shapes

of the

_3IIg

and

_{1 llg}

_potential

curves,

which result from an avoided

_crossing

_{with upper}

configurations.

Since the

_{corresponding eigenstates}

kept,

at short interatomic

_distances,

a

_{non-negligible}

(although non-dominant)

component

in the model V.B. space, we decided to use these states in the

definition of the

_{« target »}

stable

_subspace

_defining

Heff.

_{This may be called} an adiabatic definition of the

target

space S

(see

below for a more detailed

dis-cussion).

3.2 DEFINITION OF THE EFFECTIVE HAMILTONIAN. 3.2.1 Model _{space, size} and

_symmetry

_{partitioning.}

-According

to the basic choice of our treatment, the model space for

_B2

reduces to the neutral

A ( S2p )x B ( S2p )V.B.

determinants

_(cf.

_Eq.

_(1)).

where

_PA,i

is a

_spin

orbital

Px,

y or z on the atom A. The model space

splits

into

_{Sz = ± 1, SZ = 0}

indepen-dent

_subspaces,

the last one

_being

the

_largest

one and

containing

all the relevant information.

_{Equations (1)}

and

_{(2) implicity}

assume the use of

_orthogonal

valence

AO’s. This may be achieved

by

a

_preliminary

S- 1/2

transformation,

but,

as noticed elsewhere

_[4,

_5],

the

precise

specification

of this basis is not _{necessary. This}

orthogonality problem,

connected with

_{transferability}

to

_larger

_systems,

will be discussed in the

_subsequent

section. This model space may be

split

into three

different

_symmetries

[xa Yb, y a xb

and

_xa

_{Zb’ Za}

_Xb]

_(and

its

_degenerate

[ya

Zbl Za

Yb

counterpart)

where the atoms are

de-noted a and b for convenience. The size of the effective matrices will be

_6,

4 and 4

_{respectively for Sz}

=

0,

and

3, 2, 2,

for

_Sz

= ± 1. The u, g

symmetry

will be

exploited

later on.

Once _{the model space is defined the} effective

Hamil-tonian may be obtained

_by

two different routes :

- the

perturbative approach, using quasidegenerate

perturbation theory

in one of its

_{versions ;}

- the direct use of the Bloch or Des Cloiseaux

equations,

since in that

_{precise problem,}

almost exact

energies

and wave functions are available from the

MO-CI

_preliminary

calculations. The latter solution is

equivalent

to an infinite order summation of the

perturbative

expansion,

and is

_by

far more

_precise,

but

the low-order

_perturbative

_{approach gives}

the main structure of the effective Hamiltonian and the

_physical

origin

of the effective interactions.

3.2.2 Structure

_of

the

_effective

Hamiltonian

_from

third order

_{perturbative expansion.}

- Some details about the

general

structure were

_given

in

_previous

_papers

_[3-5].

Let us recall that the

_perturbation

_operator

is defined

by :

with

where

_So

_{is the model space. In} an

_{Epstein-Nesbet}

definition

_[22]

of

_Ho,

the V

_operator

is

_purely

ex-tradiagonal. Up

to second order the effective Hamil-tonian matrix elements take the form:

and to third order

where

a, (3

may be ionic or

_hybridized

neutral deter-minants.

Since the ionic determinants are known to

_play

the dominant role in the

_{interpretation}

of the effective

antiferromagnetic coupling

in half-filled

bands,

we shall concentrate our attention on their effect in our

determin-ants to those which

_{keep s2}

electron

_pairs

on each

_atom,

the V.B. matrix may be

schematically pictured

as in

figure

4,

for the

_[xx,

_yy,

_zz]

_symmetry,

which is the

largest

and most

_complex

one. The zeroth order

energies

of the neutral determinants are

_repulsive,

but to a

_larger

extent for _{za zb}

_spatial

_parts

_{(Rzz:> Rxx ) ;}

the same differences _{appear among ionic} states. The

largest

extradiagonal

elements are

hopping integrals

F

coupling

the

_{neutral xa xb (resp. ya yb}

or _za

_Tb)

and ionic

xa xa (resp. Ya 37a

or

_{za ia)}

determinants. One should also notice the occurrence of

_{non-negligible}

monocen-tric

_{exchange integrals K}

between ionic determinants.

The

_hxy

_(and

_hxz)

_integrals

_represent

electrostatic in-teractions between the atomic xy

(or xz) quadrupoles ;

the other

_integrals

involve bicentric

_overlap

distribu-tions and are much smaller.

To first

_order,

the different

_symmetries

involve 6

parameters

The

_identity

of

_{the dxy}

and

_{dxz_y2}

solutions at first order leads to the

_following

relation

Starting

from the neutral _xa

_xb

determinant one _may

go to an ionic _xa

_{xa or xb xb}

determinant. From it one

may either move back to

_{the xa xb}

_original

determinant

(and

obtain a

_diagonal

second order

_correction)

or

reach the

_{x b , 7}

determinant ;

the effective

_coupling

is then an interatomic

_exchange,

shown in the

_following

scheme where * _{and 0 stand for a and}

p

spins

respectively.

This back-and-forth _{second order process in} the

’IT x

subsystem

repeats

the well-known Anderson’s mechanism of

_{antiferromagnetic}

_coupling.

_{One may}

write

_{diagonal - gxx IXa Xb) (xa xb l}

and

_off-diagonal

- gxxlxaXb) (xbxal

operators,

with

_{g = gxx}

=

2

_F2/AExx,

which

_give

the

_amplitude

of the

_negative

effective

_{exchange integral}

in the _{’IT x}

_system.

_Analogous

contributions occur in the

’IT y (gyy = gl)

and

u (gzz = gí)

systems.

A second

_type

of

_coupling

occurs at third

order,

pictured

as follows

which

_{couples xa xb and ya yb}

_through

(the

factor 2 reflects the

_possible

travel

_through

A+ A- and A- B+

structures).

This

_operator

_changes

the

_angular

_part

of the

_{wave-function,}

and therefore cannot enter a

_Heisenberg

Hamiltonian.

_Analogous

operators

occur between x and z _{or y and} z

spatial

parts,

with a different

_amplitude

_exz =

Eyz.

Table I

summarizes the

_expected

structure of the effective

Fig.

4. - Structure of the V.B. matrix

spanned

by

the non

_{hybridized, Sz}

=

0, determinants

in the

[xx,

yy,

zz]

-symmetry. Bicentric

_{exchange integrals}

are not taken into account. Identities between matrix

_elements,

_resulting

from the

_degeneracy

of 7r

Hamiltonian

in the

_[xx,

yy,

zz]

_symmetry

_(Sz

= 0

and 1).

In the

_{[xy ]}

_symmetry,

_analogous

_{processes take}

place

leading

to second order

_diagonal

and

_off-diagonal

corrections,

with for

_example

the

_{corresponding off-diagonal}

operator

where one _{may notice that}

_{AExy =}

_{AExx -}

_{2 Kxy,}

since one does not _go

_through

an ionic determinant with 2

electrons in the same

_AO,

as occurred in the Anderson

typical

problem

and in the

_{[xx, yy, zz]}

_symmetry.

A _{third order contribution also appears}

_through

processes such as

leading

to

_{non-Heisenberg}

_operators

Table I

_gives

the structure of the third order effective Hamiltonian in the

_[xy]

_symmetry

for

_Sz

= 0 and

Sz

= 1. The

_identity

of

_{the SZ}

= 0

and Sz

= 1

triplet

eigenvalues requires

that

This relation is

_easily

understood from the difference in

the energy denominators

so that

Similar derivations are

_posible

for the

_[xz]

(and

[yz] )

symmetry

and are also shown in table I. However one

should notice that :

i)

the

_diagonal

second order corrections are

while the

off-diagonal

second-order correction obtained

from the

_operator

is of different

_amplitude

than the

_diagonal

correction

and

ii)

the

_{Fzz hopping}

_integral

is

_positive,

_resulting

in

apparent

changes

in

_sign,

also active in

_{the £3}

third

order terms.

The

_identity

of

the 3

_Sz

=

0)

and 3

_{Sz = 1)}

solu-tions for the

_{[ xz ]}

_symmetry

leads to the

_following

relations

analogous

to

_equation

_(3).

3.2.3

_{Physical typology of}

the

_effective

_operators.

-One may summarize the various

types

of

_operators

according

to several criteria :

-

diagonal/non-diagonal

character ;

- order of

perturbation

concerned ;

- number of AO’s

involved ;

- action

on the local

_spin

and

_angular

momenta. On the

_diagonal

one first has

_purely

_{( la, lb)}

depen-dent terms R which are

_independent

of the

_spin.

One also

_{has - g}

stabilization second order contributions.

The terms

only

involve two AO’s of the same

_subsystem,

and

appear when these two AO’s bear electrons of different

spins.

The -

_{g2/xa Y7b) 1xa 3Tbi I - g31xa Zb) xa zb l}

terms

imply

two AO’s

_belonging

to two

_orthogonal

subsys-tems.

The

operators

appear in

the Sz

= 1

(or -1)

subspaces. They

involve two AO’s of two

_orthogonal

_{subsystems bearing}

parallel spins.

Off the

_diagonal

one finds

_typical

second order

spin

exchange integrals involving

only

two AO’s of the same

subsystem,

as in all

_Heisenberg

Hamiltonians

and

_analogous

_operators

for

_{yy ( - gl)}

and

_{zz ( - g1 )}

determinants.

Other second order

_operators

of the

_type

Table II. -

Characters

_of

the various

_{off-diagonal effective}

_operators.

momentum. These

_operators

involve four AO’s

belong-ing

to two

_{orthogonal subsystems.}

The second order

_off-diagonal

operators

change

the

_angular

momenta of the atoms without

_changing

their

_spins.

One should notice that due to the

_identity

of the _{’IT x} and

_{’IT y systems ii2}

_{= 92}

and g

=

g2.

The third order

_operators

change

the

_spins

of two

_{spin-orbitals}

of different

_spins

belonging

to two

_orthogonal

_subsystems,

without

_acting

on _{the space}

_part.

The third-order

_operators

change

the

_angular

momentum on the atom without

changing

the local

_{spin ; they}

involve four atomic orbitals

_belonging

to two

_orthogonal

_subsystems.

Finally

the third-order

_operators

change

both the

_angular

and

_spin

momenta on the

atom,

involving

therefore four AO’s on two

_orthogonal

subsystems.

This

_typology

is

_clearly

_pictured

in

table

II. 3.3 AB INITIO DERIVATION OF THE EFFECTIVE HAMILTONIANS FROM THE BLOCH EQUATION. -Since the effective Hamiltonian satisfies the Bloch

equation

for n roots

with

a

_spectral

definition of

Heft

is

given

by

where

_{Po 03C81}

is the

_biorthogonal

vector associated with the

_{projection Po}

_03C8i

of the

_eigenstate

_03C8i

of H into

_So.

If the exact

_spectrum

of H and its

_eigenvectors

are

known,

a selection of n

_eigenvalues

and

eigenvectors

03C8i

defines

the stable

_subspace,

which we call «

_target

_{space »,} and

Heff

is

_entirely

defined

_by

_{P 0’ P s}

and the exact

eigenenergies,

without any reference to a

_possibly

divergent

perturbation

expansion.

If one uses the wave

operator

n,

then :

Since for the

_B2

_molecule,

accurate

_energies

and

wave functions are available from the MO-CI

cal-culations,

it is

_tempting

to use

_{equation (5)}

to obtain

directly

the

Heff

matrix,

in a

_purely

numerical form : in

this section the

_target

_{space is} defined as

spanned by

the

_eigenvectors

of the correct

_symmetry,

lowest _energy and

_{having significant}

_components

on the model _space.

Another definition of the

_target

_space

_might

_pick

_up the

eigenvectors having

the

_largest

_components

onto the

model space. From

symmetry

considerations,

the

fol-lowing correspondence

between model

_subspaces

and

The A states obtained from

_{[xx, yy, zz ]}

and

_{[xy ]}

model

_subspaces

are of course

_degenerate.

All the vectors which are the

_only

ones of their

symmetry

in the concerned model

_subspace

have their

components

in _{the model space}

_entirely

defined

_by

symmetry.

This means that the

_components

of

_{P 0 f/J}

_i are

_immediately

obtained. For instance for

_3-v9

one

necessarily

has +--+

_components

on the

_{xa Yb ,}

I

yb xa ,

IYa

xb

_{y and}

xb

Ya

model

determinants. The

only

exception

concerns the

_pairs

of

I!;

and

_303A3u+

states

belonging

to the

_[xx,

_yy,

_{zz ]}

_symmetry.

The

_degree

of

freedom concerns of course the ratio of

Xa 3Fb I ( =

IYa Yb I)

over

za

_zb

I

_components.

In

_principle

the

_projected

_eigenvectors

of the same

symmetry

are not

_orthogonal

This results in a non-Hermitian structure of the Bloch effective Hamiltonian. For the sake of

_simplicity,

we

preferred

to use Hermitian structure. One

_possible

solution refers to an

S-ll2

transformation of the

projec-tions,

and is known as the des Cloizeaux

_[7]

effective

Hamiltonian definition. This solution

_{symmetrically}

distorts

_{projections ;}

_previous

work

_[3]

on

_Heisenberg

Hamiltonians for a half-filled band with 2 e-

_/2

AO _per

centre has shown that this distortion _{may result in}

important

deviations from the

_expected

structure and

difficulties in the transfer of effective

_operators

to

larger

systems.

A better solution consists of a

Schmidt-type

orthogonalization keeping

the lowest

_projected

eigenvector unchanged,

and

_{orthogonalizing}

the second

one

_(of

_higher

_energy)

to the first one. This solution has been

_applied

for

_lz+

and

_303A3

_symmetries.

The ratios of

_{I xa}

_3Eb

_{l (}

=

ya

j7b

over

za

_zb

I

pro-jected

components

are

_given

in table

III,

other

pro-jected eigenvectors

on the model space

_{being trivially}

obtained

_by

_symmetry.

In this

table,

the first state of each

_symmetry

_corresponds

to the lowest _{in energy.}

The

_changes

of the order of

_magnitude

_(for

_103A3g;

_states)

and of the

_signs

_(for

_303A3u:

_states),

between the two states

of each

_symmetry,

_occurring

between 3.3 and 4.5 a.u.,

both

_correspond

to avoided

_crossings.

For 3.3 a.u., the

two ratios smaller than 1.0 found for

_103A3g+

states do not

mean that the states are both «

z2 dominant

» _states,

since the

_{[xx] -component}

has to be counted twice

xx

I and

[ yy I

determinants).

The first

_’,V+

state is a

xx

,

I YY I ] -dominant

state. The

corresponding

ef-fective Hamiltonians are

_given

in table

_IV,

where the

energy of the two

_separated

atoms

_{(5.18 a.u.)}

has been taken as zero of _energy.

3.4 COMPARISON BETWEEN THE PERTURBATIVELY

DERIVED AND NUMERICALLY OBTAINED EFFECTIVE

HAMILTONIANS. - For the

_diagonal

R «

_repulsive

»

matrix elements one should notice

_(cf.

_Sz

=

1,

Exx, yy . zz ]

symmetry)

that

_{Rzz ( = R[)}

is the most

repulsive,

as

_expected.

The

_large

_off-diagonal

matrix

elements

_correspond

to the effective

_exchanges

_g. Since

the

_overlap

_(and

_hopping

_integrals)

are of

_larger

amplitude

in the

_u-system

than in the 1T _ones, one

should have

The numerical values

_actually

show the

_large

_{9 uu value}

(0.115 a.u.)

at r = 3 _a.u., but the three other

integrals

are of the same order of

_magnitude

_(-

0.03 to 0.02

_a.u.).

One should not

_forget

that our third-order

perturbative

development

did not involve

_hybridized

configurations,

and deviations are

_logically

_expected.

Nevertheless the numerical effective Hamiltonian

_keeps

the

_predicted

structure.

Table- III. - Ratio

of the projected

components

of

xa

Xb

[ over

za

ib

I

for

the states in the

_[xx,

_yy,

_{zz]- symmetry,}

Table IV. - « Exact» numerical

Heff

matrix elements at various B-B distances. Deviations

from

the

theoretical

third order structure are

_clearly

outlined

_by

a direct

comparison

to the.second

column.

The effective Hamiltonians have been calculated for five interatomic distances. All matrix elements

R,

g, E decrease

_{exponentially}

with the interatomic

distance,

since

_they

involve interatomic

_overlap

distributions. The

_{quadrupole-quadrupole}

interactions h decrease as

r-

8.

_Analytic

_fittings

of the matrix elements are _easy

and _{may be used for}

_geometry

_{optimizations}

in

_larger

clusters.

4.

_{Transferability}

to the linear

_B3

_system.

Transferability

of the effective Hamiltonian has been tested on the

_{simplest larger problem, namely}

the

B3

linear

_system

_{keeping D 00 h}

_symmetry.

4.1

QUALITATIVE

CONSIDERATIONS. - Close to the

equilibrium

geometry,

the lowest states of

_B2

are, in

increasing

order :

_{303A3g and 1 L1g}

_{( [xy ]-symmetry}

_{L-, )}

and

_3IIu

_{([ xz ] -symmetry: L-).}

_Starting

from two B

interactions to form the linear

_B3

_molecule,

we _may

predict

to

_find,

as lowest states for

_B3,

a structure

resulting

from the

_junction

of two such

_{[xy ]-structures}

or

one

_{[xy ] -}

and one

_{[xz ] -structure : , ,}

Moreover,

as also outlined in the

introduction,

the

distribution

_{corresponding}

to all

_{spins parallel,}

i. e. a

quartet

symmetry

in that case, should be low in energy.

4.2 AB INITIO MO-CI CALCULATIONS. - Accurate MO-CI calculations have been

_performed

on

_B3.

The

symmetry

of the

_{B3 ground}

state was

_investigated

first.

Keeping

in mind that d-functions were of fundamental

importance

in the determination of the

_right

_symmetry

for the

_ground

state of

_B2,

we used the same

_triple-Cs,

double-( p

and

_single-Cd

valence basis set as for

_B2.

According

to the CIPSI

_algorithm,

a

_great

_part

of the

valence correlation energy has been treated

variational-ly

and the

_remaining

part

perturbatively.

Figure

5 shows the three lowest states close to the

_equilibrium

internuclear distance which is found of the same order as that of

_B2.

The

_21Iu

state is the

_ground

state. The

quartets

_4IIg

and

_{4 _Vu-}

_suggested

_by

our

_qualitative

Fig.

5. -

MO-CI

_potential

curves for the three lowest states

of B3.

ground

state, the

_4IIg

state

_being

_slightly

more stable

than the

_{4_V u -state.}

In a second

_step,

in order to

_lighten

the

_{calculations,}

we eliminated the d-functions from the basis set.

Figures

6a and 7a show the lowest states for the three

subsymmetries.

These three

_{subsymmetries}

are denoted

here:

_[xxx]

_(which

also

_includes

_{I XYY l and}

_{I xzz}

_l

determinants and the

_degenerate

_ones),

_[zzz]

_(with

also

[ xxz

I

and yyz

determinants)

and

_[xyz].

The u

and g characters are then

_assigned

_appropriatly

to each MO in MOCI calculations while _a, b and c

_subscripts

characterize the atoms to each A.O. in the V.B.

formalism. The three lowest states are still found to be :

2IIu,

4 IIg

and

_4,X.-

in the same order.

4.3 COMPARISON BETWEEN AB INITIO AND

EFFEC-TIVE HAMILTONIAN RESULTS FOR

_B3.

- Malrieu et

al.

_[5]

discussed in detail the

_problem

of the

orthogonal-ity

between A.O.’s. The

_{orthogonalized}

A.O.’s. for a

two-atom

_system

and those for a three-atom

_system

are

not the same. We must

_keep

in mind that this could affect the

_{transferability}

to some extent. Their calcula-tions on alkali metals also have shown

that,

as

_expected,

three-body

increments

_{5 gab( c )}

are

small,

at least in linear chains.

The

_B3

effective Hamiltonian

_splits

into three blocks of dimensions

18,

21 and 21 for the

_{[xyz], [xxx]}

and

[zzz]

pseudo-symmetries respectively.

The

long-dis-tance

_{(A-C)-interactions}

have been checked to have

negligible

effects. The

_ground

state is

_correctly

pre-dicted to be of

_{2H .}

_symmetry

_by

our

_model,

and the

Fig.

6a. - MO-CI

[xxx] -symmetry

lowest

_potential

curves

of B3.

Fig.

6b. -

Heff

[xxx]

-symmetry lowest

_potential

curves of

Fig.

7a. - MO-CI

[xyz] -

and

_{[zzz]-symmetry}

lowest

potential

curves of

_B3.

Fig.

7b. -

Heff

[xyz ] -

and

_{[zzz ]}

_-symmetry lowest

_potential

curves of

_B3.

equilibrium

distance is correct

_(3.08

a.u. to be

com-pared

to 3.01 in the ab initio

_{calculations).}

From a V.B.

point

of view it may be

analysed

as a resonance

between

The two _upper states are _{very close in energy ;}

_they

are

quartets

of

_4ng

and

_403A3u-

_symmetries.

The best MO-CI

calculation

_predicts

them at

_Te

= 0.14 eV and 0.19 eV

above the

_ground

state

_minimum,

while the effective

Hamiltonian

_predicts

0.22 and 0.03 eV

_{respectively:}

this inversion

_corresponds

to an error of = 0.2

_eV,

which is not too bad.

_Deleting

the d A.O.’s in the MO-CI calculation leads to

_Te

= 0.11 and 0.24 eV

respect-ively

for

_4IIg

and

_4!u.

The

_important

feature is the differential effect on the interatomic distances

(cf.

Figs.

6-7 and Table

_V)

which are

_correctly

_reproduced

by

the effective Hamiltonian. The V.B. content of these wave functions may be

_pictured

as

A direct

_comparison

between the

_potential

curves of

upper states _{may be obtained}

_{by comparing figures}

6a and 6b for the states of II and 0

_symmetries,

and

figures

7a and

7b

for the states of

_{-V-, d}

and

11

symmetries.

The results are also summarized in table V. There is

a

_significant

_gap

_{(= 0.5 eV)}

between the three lowest

states and a _{group of five} states which lie in a narrow

energy range of 0.4 eV. These states are of

2 llg,

2_Vu,

4 _V g

2 Llu

and

_2.I;

_character,

in order of

increasing

energy in the MO-CI calculation. The effec-tive Hamiltonian

_only

_permutes

the

_position

of the

ZIIg

and

_{2 Llu}

states in this _{group. The}

_largest

_equilibrium

distances concern the

2.I¡;

and

2 Llu

states in both

approaches.

The two next states are

6Hu

and

2.I:

in the MO-CI

g

calculation. The former cannot be reached from a

three-open-shell

model _space and the second one is

essentially

hybridized ;

from that

_{region important}

discrepancies

begin

to _appear between the MO-CI and

effective Hamiltonian

_spectra.

5. Conclusion.

The

_{major conceptual}

interest of the

_present

derivation

concerns the

_{logical possibility}

and cost of an extension

of the

_Heisenberg

Hamiltonian to a non half-filled band

problem.

The basic

_philosophy

is

_again

the choice of a

Table V. - Lowest

spectroscopic

states

_of

_B3.

The values in

_parentheses

are relative to the

_largest

basis set.

and the use of

_quasi

_{degenerate perturbation theory}

or

spectral

definition of the effective Hamiltonian. The

use of the

_{perturbative expansion}

has the

_advantage

of

putting

in

_place

the

_logical

structure of

_Heff,

_showing

which are the

_leading

terms and their

_physical

_origin.

The use of the

_spectral

definition of

Heff permits

avoidance of

_{divergence problems}

of the

_perturbative

expansion

and introduces

_{high-quality (infinite order)}

information.

The effective

_operators

_involve,

beside the classical

Heisenberg

type

spin exchange

operators

between A.O.’s of the same

_{subsystem :}

-

spin-exchange

operators

between A.O.’s of two

orthogonal

subsystems ;

-

angular momenta-exchange

operators ;

-

spin-

and

_{angular momenta-exchange}

_{operators ;}

the two last

_types

_implying

_necessarily

_four

A.O.’s

belonging

to two

_orthogonal

_subsystems.

The

_{generalization}

of the

_Heisenberg

effective Hamiltonian

_approach

to

_(p/n)-filled

bands is thus in

principle possible although

some

_{supplementary}

dif-ficulties _may arise from the existence _{of p electrons per}

atom. This

_approach

_{may be useful for the}

_study

of the

dn systems

of clusters of transition metals.

Concerning

the

_precise

case studied

here,

the main

proble,m

concerned of course the restriction of our

model to a

_single

band

_through

the choice of a model

space

spanned by

products

of atomic

_S2p

_{configurations.}

The

_{hybridization}

_(i.e.

the involment of

_sp2

atomic

configurations)

should

_{play a}

_very

_important

role,

and this is clear from the existence of a low

_lying

_quintet

state

503A3u-

of

_B2,

which cannot enter our model. The fact

that we used the exact

_{eigenenergies}

of

_B2

for the

spectral

definition of

Heft

has taken

_implicitly

into account most of the

_{hybridization mixing.}

One

_might

have defined the

_target

_space S as

_spanned

_by

the

eigenvectors

of H

_having

the

_largest

_components

on the

model space

So,

some of them

_being

_higher

in _energy.

Attempts along

this line failed to

_give

reliable results

except

for a few almost

_{non-hybridized}

states

_(the

ground

state was

_{correctly predicted}

to be

_2IIu).

The

success of our model for

_B3

is

_certainly

linked to the

fact that the

_strongly

_hybridized

states _go to

_higher

energies

on linear chains at

_least,

since the lowest

essentially hybrid

state

6IIu

_appears

_only

in

9th

_position

in the MO-CI calculation.

Such success would be

unlikely

for bi- and

tridimen-sional clusters. We intend to

_apply

our effective

Hamil-tonians to the

_prediction

of the

_magnetic

character of linear

_Bn

chains.

It is clear that the main further

_step

in the construc-tion of effective Hamiltonians

_{spanned by}

all neutral

V.B. structures would consist in

_{explicitly treating}

the

hybridization,

i.e. several

_bands,

_spanning

the model space

by

all the

possible products

of neutral

ground

and excited valence atomic

_{configurations.}

This

_strategy,

now under

_study,

faces three

_types

of

_{problem ;}

- the

large

size of the model _{space makes} the ab

initio extraction

_{difficult ;}

the use of the

recently

proposed

intermediate Hamiltonians

_[23]

_may solve

some intruder state

_{problems, necessarily occurring}

in

such

situations ;

- effective

integrals

will be numerous, and their

handling

in

_larger

_problems

_{may become tedious. Some}

analytic fittings

of effective

_integrals

_(in

terms of

overlap

for

_instance)

_may become _{necessary ;}

- the number of neutral V.B. determinants for

a

moderate-size cluster or molecule will be

_large

and the

diagonalization

of the

_{generalized Heisenberg}

Hamil-tonian matrix may become rather

expensive.

Sanchez et al.

_[24]

_recently

_proposed

a

_procedure

to truncate