On the intraatomic Coulomb and exchange energies in degenerate narrow bands

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On the intraatomic Coulomb and exchange energies in

degenerate narrow bands

G. Allan, J. Friedel, C.M. Sayers

To cite this version:

G. Allan, J. Friedel, C.M. Sayers.

On the intraatomic Coulomb and exchange energies in

de-generate narrow bands.

Journal de Physique Lettres, Edp sciences, 1980, 41 (12), pp.287-289.

�10.1051/jphyslet:019800041012028700�. �jpa-00231781�

L-287

On the intraatomic Coulomb and

_exchange

_energies

in

_degenerate

narrow

bands

G. Allan

Laboratoire de Physique des Solides _{(*), I.S.E.N.,}

3, rue _{François-Baës,} 59 Lille, France

J. Friedel

Laboratoire de Physique des Solides (*), Université Paris-Sud, 91405 Orsay, France

and C. M.

_Sayers

AERE, Harwell, Oxfordshire, England

(Re~u le 25 fevrier 1980, accepte le 20 _{avril 1980)}

Résumé. 2014 Dans les bandes étroites

dégénérées étudiées en liaisons fortes, les fonctions de Wannier des bandes

partielles

ne _{peuvent être utilisées pour calculer les interactions de Coulomb intraatomiques, parce que chacune}

d’elles met en

_{jeu plusieurs}

fonctions

atomiques,

sur des sites différents. Dans

_{l’approximation}

_{(de Hartree-Fock)}

qui utilise un déterminant fait de fonctions de Bloch, le changement d’énergie de Coulomb intraatomique entre les

atomes libres et le métal

_{paramagnétique}

varie ainsi de façon

parabolique

avec le nombre d’états _atomiques

occupés, et non avec le nombre d’états occupés dans chaque bande

_partielle.

_{Ceci justifie} des traitements _précédents de ce _problème et _corrige un travail récent de Thalmeier et Falicov. Le _{changement d’énergie d’échange}

intra-atomique est _également

_explicité.

Abstract. 2014 In

degenerate narrow bands studied in the _{tight binding limit,} Wannier functions of

_partial

bands

cannot be used for

_computing

intraatomic Coulomb interactions, because each of them involves several atomic states, on different sites. In the _{(Hartree-Fock)}

_{approximation using}

one determinant made of Bloch functions,

the _{change in intraatomic Coulomb energy} from free atoms to _paramagnetic metal thus varies

_{parabolically}

with the numbers of

_occupied

atomic states, and not with the number of

_occupied

states in each

_partial

band. This

justifies earlier treatments of the _problem and corrects a recent work by Thalmeier and Falicov. The _change in intraatomic exchange energy is also

explicitly

given.

J. Physique - LETTRES 41 (1980) L-287 - L-289 15 JUIN 1980,

Classification

Physics Abstracts 71.10

1. Wannier versus atomic states in

_degenerate

bands. - In a recent work

_[1],

Thalmeier and Falicov have

computed

the intraatomic Coulomb _{energy in} a

degenerate

narrow band.

_They

have

_equated

the Coulomb _energy of two

_electrons,

of

_{opposite spins,}

in the same Wannier state of the

metal,

to the Coulomb energy U of two electrons in the same atomic state in the free atom. With this

_{approximation,}

these authors have

_computed

the contribution to lowest order

_(one

determinant mode of Bloch

_functions)

of this _energy

to cohesion.

_They

have shown that this contribution is sensitive to the

_crystal

structure and also differs

(*) L.A. au C.N.R.S.

much from an earlier estimate made

_by

one of us

_{[2, 3]}

and used in more recent studies of second order

terms

_[4,

5,

6].

Practical

_applications

are to

_degenerate

d bands of transitional metals studied in the

_simplest

tight binding

limit.

We wish to

_point

out that the Coulomb _energy of

two electrons in the same Wannier state is

_usually

different from

_U,

and indeed varies with the subband considered as well as with the _way Wannier states are .

defined

_[7].

The reason is

_simply

that,

in a

_degenerate

band studied in

_{tight binding,}

a Wannier state of one

subband extends over more than one atomic site

_[8].

Thalmeier and Falicov’s treatment if furthermore inconsistent in that

_they

do not use Wannier functions

to _compute the total Coulomb _energy in the metallic

L-288 JOURNAL DE _{PHYSIQUE -} LETTRES

state, but

only

that _part which is in excess of the Coulomb _{energy in} free atoms.

2. Intraatomic _{Coulomb energy}

_using

atomic states. -A more correct way of

computing

the Coulomb energy is to introduce

_directly

the _energy U of electrons

sitting

in atomic states on the same

_site,

and to

eva-luate the

_change

due to U, in the

_simplest

Hartree-Fock

_scheme,

when one _goes from free atoms to the

paramagnetic

metal. This is what was done in our

earlier _treatment, which

_explicitly

used the random uncorrelated distribution of the electrons over Bloch

states. We _repeat here the _argument,

_using

a more

explicit

formulation.

We write the total

_function

_{I tp >}

of the metal as a

determinant of Bloch functions with

spatial

part

The summation R is over lattice sites. The summation i

is over the n

_degenerate

atomic states

_(n

= 5 for d

bands).

A

_{straightforward}

and classical

_computation

gives,

for the total Coulomb _energy between the d electrons in the metal :

1, 2,

I,

m are indices

_denoting

electrons.

_kcc

and

_k~c

denote states k and k’ that are

_occupied,

with

_energies

below the Fermi level of the metal. The factors 2 and 1

come from the

_spin

_degeneracy;

the first terms are

the classical Coulomb terms and the second ones are

the

_exchange

terms.

We

denote,

for _any

_{i, j}

on _any site R :

and

_neglect

all the other Coulomb terms. If N is the total number of

sites,

A then reduces to

Here

is the number of electrons with one

_spin

direction in the atomic orbital i on site R. It is such

_that,

if there are

p electrons per atom in the

_paramagnetic

_metal,

Equation (1)

can also be written

It has a

simple physical

meaning,

which was used

directly previously

[4,

5,

6] :

- The first term is the intraatomic Coulomb

interaction of electrons in different atomic

_{orbitals i,}

j

when distributed at random over N sites with _average numbers

_Z¡,

_Zj

_per site.

- The third term is the

corresponding

intraatomic

exchange

energy.

- The second term is the

special

intraatomic Coulomb _energy of electrons in the same atomic

orbital,

when correction has been made of the

cor-responding

exchange

term, or

_equivalently

when the Pauli

_principle

has been taken into account.

The

_{corresponding}

terms in N free

_(magnetic)

atoms can be written in the same _way

(1) :

Finally,

the contribution to the cohesive _{energy per} atom of the Coulomb and

_{exchange (Hartree-Fock)}

terms is

-where

L-289 INTRAATOMIC COULOMB AND EXCHANGE ENERGIES IN NARROW BAND

Thus _{the Coulomb part}

_of

_AE,,

varies

_{parabolically}

with the average

population Zi

of

each atomic orbital i,

and not with the

_population

of each

subband,

as stated

_by

Thalmeier and Falicov

_[1].

It is also

_easily

deduced from these formulae that

_Me

= 0 for _a

completely

_empty

band

(all Zi’s

_{= p} =

0)

as well as for a

_completely

full band

_{(all Zi’s}

= 1

and p

= 2

n).

The Coulomb

_part

in

A~c

is

always

negative,

with a broad minimum for a

_nearly

half filled band

_(p

=

n) ;

the

exchange

_part in

A~c

has an acute minimum for a half filled band

_(p

=

n).

In alternant structures

_[8]

where the band structure is

_symmetrical

in energy, one has

Zi(p)

=1- Z~(2

n - p),

and thus

_Zi(n)

=

2.

The minimum

of Ee

is then

exactly

for a half filled band

(p

=

n)

and is

equal

_to

Even in non _{alternant structures,}

_Me

does not deviate much from these characteristics.

Indeed numerical

_{computations [9]}

have shown

that,

in the

_closepacked

structures, the various d orbitals have

_nearly

_{equal populations}

for most

_filling

of the d band. In our

_{previous analysis [4],}

we have used therefore the

_simple

approximation :

Then

_AEe(~)

is

_symmetrical

with _respect to _{p = n,} and can be written

We should

_point

out that a numerical error occurred in the value of the coefficient of the

_exchange

term J in references

_[4]

to

_[6]

which however does not alter the

_general

variation of the

_exchange

term nor its minimum

(negative)

value. Also

_{approximation}

₍₂₎

is less valid for

_nearly

full or

_nearly

_empty

bands.

References

[1] TALMEIER, P. and FALICOV, L. M., Phys. Rev. B 20 (1979) 4637.

[2] FRIEDEL, J., J. Physique Radium 16 (1958) 829.

[3] FRIEDEL, J., in the _{Physics of} Metals. I. _Electrons, edited by

J. M. Ziman _{(Cambridge University} Press, Cambridge) 1969, p. 340.

[4] FRIEDEL, J., Ann. Phys. 1 (1976) 257.

[5] FRIEDEL, J. and SAYERS, C. M., J. _Physique 38 ₍₁₉₇₇₎ 697 ;

J. _Physique Lett. 38 _{(1977) L-263 ;} 39 (1978) L-59.

[6] FRIEDEL, J., J. _Magnet. Mat., under press.

[7] BLOUNT, E. I., Solid State Phys. 13 (1962) 305.

[8] FRIEDEL, J., J. _Physique 39 _{(1978) 651,} 671.

[9] ASDENTE, M. and DELITALA, M. D., Phys. Rev. 163 (1967) 497; DUCASTELLE, F. and CYROT-LACKMANN, F., J. _Phys. Chem.

Solids 31 (1970) 1295;

DESJONQUÈRES, M. C. and CYROT-LACKMANN, F., J. Chem. _Phys.

64 (1976) 3707 ;