Comparison of two expressions to compute the electronic energy of a crystal

HAL Id: jpa-00212507

https://hal.archives-ouvertes.fr/jpa-00212507

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Comparison of two expressions to compute the

electronic energy of a crystal

S. Pick

To cite this version:

2011

Short Communication

Comparison

of

two

_expressions

to

_compute

_{the electronic energy}

of

a

crystal

S.

Pick

J.

_Heyrovský

Institute of

_Physical

_Chemistry

and

_{Electrochemistry,}

Czechoslovak

_Academy

of

Sciences,

Dolej0161kova

3,

CS-182 23

_Prague

_8,

Czechoslovakia

(Received

on

May

28, l’990,

accepted

on

July

6,1990)

Abstract. 2014 Two

expressions

for the electronic energy, the second one of which is not

selfconsistent,

are

analyzed.

It _appears that the two formulas are

equivalent

_up to the first order in selfconsistent corrections.

J

_Phys.

France 51

₍₁₉₉₀₎

2011-2013 15 SEPTEMBRE

_1990,

Classification

Physics Abstracts

71.10

Severalyears

ago,

the

_legitimacy

of two formulas for the electronic

_(band)

_energy

_change

has been

discussed

_by

Masuda-Jindo et al.

_{[1] .}

1fie two formulas read

and

respectively.

Above,

ApA

is the

_change

of the local

_{partial density}

of electronic states associated

with the

_{spin-orbital a residing}

on the site

_i(A

=

(a,

i)),

NA

=

f EF

pAdE

is the

occupation

num-ber and

_OA

is the Coulomb

_{integral (diagonal}

matrix element of the

_{Hamiltonian). Expressions}

analogous

to

_equation

₍₁₎

are

commonly

used in the solid state

_computations

_[1-3].

1fie form we

adopt

here is somewhat more

general

than that considered in

_[1]

and it is identical to

_{equation (11)}

of reference

_[3]

_apart

from the second order contributions which are of the form

AOA ANA /2.

In the

_important

case of d-band

metals,

the latter terms vanish

_rigorously

when the local

_charge

neutrality

is

_{postulated (1, 2] .}

_However,

the local

_{charge neutrality}

is not obvious

_[3]

for

_validity

of

_equation

_(1).

Tb derive

_{equation (2),}

two similar

_{homogeneous crystals}

with

_slightly

différent Fermi level

_positions

are considered

[4].

The inclusion of the

_EF-term

reduces the

_inaccuracy

inherent to

_{equation (2)}

to a small term of order

_AEFAN.

In the

_present

_note, we consider two similar

_{nonhomogeneous}

_systems

for which Fermi levels

coincide

_(e.g.

various

_crystal

surface

_arrangements

or bulk

defects).

We

_suppose

that the

expres-sion

_{(2) corresponds}

to a non-selfconsistent treatment

_(i.e.

the

_{global charge neutrality}

condition

2012

EA

dNA

= 0 _can be

violated).

As for the

equation (1),

_{we assume} that the

charge selfconsistency

is

_imposed

which is controlled

_by

the

_changes

_AOA.

The main

_goal

of our discussion is to

_prove

that under these

_assumptions,

the

_{expressions (1)}

and

₍₂₎

are

equivalent

_up

to the first order in the

_perturbation

introduced

_by

the

_{selfconsistency.}

It is convenient to

_distinguish

the

_geometrical

_(d)

and selfconsistent

_(â)

corrections in

_equation

(1). Namely,

we introduce new notation

dp --+

Ap

+ 6p

and

similarly

for the other

quantities.

For

the "atomic levels"

_{jJ A}

_only

_60A

does not vanish. We treat the

_6-component

of

_{equation (1)}

as a

perturbation.

By

introducing

the Green function

_{G(z) (z}

= E +

iO),

_we transform the

equation

(1)

to the form

16 the first

_order,

(Here

and

_below,

we use the

identity

G2

=

-dG/dz.)

Inserting equation (4)

into

equation

(3)

and

_{integrating by}

_parts

we find

Further

_{manipulations yield}

The

_{selfconsistency}

_{grants, however,}

the

_{global charge}

_neutrality:

By

virtue of the latter

_identity,

which

_proves

the

_equivalence

of the

_expressions

_{(1), (2)}

in the sense claimed above.

The result arrived at

_gives

some

justification

to the

_attempts

to

_investigate

in a

highly

simplified

manner low

_{symmetry systems}

for which a

rigorous

treatment becomes cumbersome. Another

point

worth of attention is the fact that whatever the

_{selfconsistency}

_procedure

be

_(local

_charge

neutrality,

inclusion of Hubbard-like terms,

_{special electronegativity adjustment),}

the

_resulting

energy

change

is the same

_up

to the first order in the

_{selfconsistency}

corrections.

_Indeed,

the

_only

2013

References

[1]

MASUDA-JINDO

K.,

HAMADA N. and TERAKURA

K., J.

Phys.

C. 17

₍₁₉₈₄₎

L 271.

[2]

TRÉGLIA

_G.,

DESJONQUÈRES M.C. and SPANJAARD

_{D., J.}

_Phys.

C. 16

₍₁₉₈₃₎

2407.

[3]

VARMA C.M. and WILSON

A.J.,

Phys.

Rev. B 22

₍₁₉₈₀₎

3795.

[4]

HEINE

V.,

Solid State

_Phys.

35

₍₁₉₈₀₎

1.