Electron affinities of the light atoms in self-interaction corrected LDA

HAL Id: jpa-00211090

https://hal.archives-ouvertes.fr/jpa-00211090

Submitted on 1 Jan 1989

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.

Electron affinities of the light atoms in self-interaction

corrected LDA

P. Cortona, G. Böbel, F.G. Fumi

To cite this version:

Electron

affinities of the

_light

atoms

in

self-interaction

corrected

LDA

P.

_Cortona,

G. Böbel and F. G. Fumi

Dipartimento

di _Fisica, Università di Genova, 1-16146 Genova,

Italy

GNSM/CNR-CISM/MPI, Unità di Genova,

Italy

(Reçu

le 28

_février

1989, révisé et

_accepté

le 28 avril

₁₉₈₉₎

Résumé. 2014 Nous

présentons

les valeurs des affinités

_{électroniques}

des atomes

_légers

(Z 36)

que nous avons calculées en utilisant

_{l’approximation}

locale

_(LDA)

à la théorie de la fonctionnelle de densité et deux

_façons

différentes de

_corriger

les effets de la self-interaction. Nous trouvons un bon accord entre nos valeurs et les meilleures données

_{expérimentales}

disponibles

dans le cas des ions

_négatifs

obtenus en

_ajoutant

un électron s ou _{p à} un atome neutre, tandis que l’accord est mauvais

_{lorsqu’il s’agit}

d’un électron d. Nous avons

_{également comparé}

les valeurs obtenues par la méthode de correction de la self-interaction

qui

donne les meilleurs résultats avec des données très récentes,

qui

ont été _{calculées par la methode Hartree-Fock} et les

plus

évoluées des

_{approximations}

non-locales à la fonctionnelle

_énergie

de corrélation

actuelle-ment

_disponibles.

Nous trouvons _{que dans de nombreux} cas les

_premiers

résultats sont

_{plus précis}

que les seconds.

Abstract. 2014 We

report values for the electron affinities of the

light

atoms

_{(Z 36 )}

calculated in the

_{local-density approximation}

(LDA)

to the

_{density-functional theory}

with two different forms of the self-interaction correction. We obtain

_good

agreement with the best available

_experimental

values for the

_negative

ions

_involving

the addition of an s or a _{p electron} to a _{neutral atom, while}

the agreement is not

_good

for the

_negative

ions

_involving

the addition of a d electron to a

transition metal atom. The values obtained with the form of the self-interaction correction

giving

better results are

_compared

with very recent values obtained

_using

the Hartree-Fock method and the best available

_{approximations}

to _{the correlation energy functionals.}

_Surprisingly

the former values turn out to be more accurate than the latter in a number of cases.

Classification

Physics

Abstracts 31.20J - 31.20D

Introduction.

The

_{density-functional theory (DFT) [1, 2]}

states that the

_ground

state

_{charge density}

and total energy of an electronic

_system

can be obtained

_exactly

from the solutions of the Kohn and Sham one-electron

_equation

where

_Ven

is the electron-nucleus

_potential

and

_Veff

is an effective

_potential

which describes the electron-electron interactions.

_{Generally Veff is}

divided in two _terms,

_Vc

and

V xc’

where the first one is the

_ordinary

coulombic

_potential

and the second one, the

_{exchange-correlation potential,}

contains all of the

_remaining

electron interactions and is

_{given by}

the functional derivative of the

_{exchange-correlation ground}

state

energy

Exc

with

_respect

to the electron

_density

p :

As the

_explicit

form of the functional

_{Exc [p ]}

is

unknown,

one needs some

_{approximation}

of it in order to use

equation

(1)

in actual calculations. The most common _one, the

_{local-density}

approximation (LDA),

consists in

_assuming

that the

_dependence

of

_E,,c

from p is the same as for an

_homogeneous

electronic gas.

The

_problem

of

_correcting

_{equation (1)}

for the electronic self-interaction effects arises because the coulombic

_{potential (Eq. (2))}

contains a

_non-physical

self-interaction term. The Kohn and Sham

_theory,

which is based on the use of an orbital

_independent

effective

potential,

states that to cancel out

_explicitly

this term is not _{necessary. This is}

_accomplished

implicitly by

the exact effective

_potential

which verifies the

_following

three conditions :

_i)

the

«

exchange-correlation

hole » satisfies the « sum rule »

[3], ii) Vxc

reduces to -

_Vc

for a

_system

containing only

one

_electron,

_iii)

at

_large

distance from a neutral finite

system

Ven

+

_{V eff}

decreases

_{proportionally}

to

_{1 /r}

_[4].

These

_properties,

which are _{very natural} on

_physical

ground,

are

_rigorously

true for the exact Kohn and Sham

_potential.

The first one, in

particular,

is considered of crucial

_importance

for the success of any

_{approximation}

to the

exact

_theory.

This is the case, for

_example,

of the LDA

_[3],

which, however,

does not

_verify

the other two

_properties.

This drawback has a number of consequences, such as the

_instability

of the

_negative

_ions,

or the fact that the

_highest

_energy

_eigenvalue

of an isolated finite

_system

is

_only

a bad

approximation

of the first ionization

_potential,

while in the exact

_theory

these

two

_quantities

should coincide

_[4].

A natural way of

taking

into account the self-interaction effects consists in

_separating

the

exchange

from the correlation and in

_treating

the first as in the Hartree-Fock

_{(HF) theory.}

Also in this way,

however,

if one uses the LDA for the correlation

_potential,

one has to do

something

like a self-interaction

correction,

because the

_expressions

derived from the

homogeneous

gas

theory

fail to vanish in the case of a

_system

_{containing only}

one electron. The main

_advantage

of

_proceeding

in this way is that one

_{requires only}

an

_approximate

description

of the « correlation hole », which has a

_vanishing

total

_charge.

The obvious drawback is that one has to

_{perform complicated}

HF-like calculations. For this latter reason one is

interested,

in

general,

in

approximations

of the

exchange

contribution as well. Some forms of the self-interaction correction.

Before

_examining

some

_possible

_{ways of}

_performing

the self-interaction correction in the case of

_{inhomogeneous}

_systems,

let us recall a result of the

_theory

of the

_homogeneous

electronic gas.

Let us consider

_Nu

electrons of

_{spin a}

in a box of volume V = 1 with

conditions. The interelectron

_exchange

_energy

_Eie

has been calculated

_by

Rae

_[5]

and is

_given

by :

where

_Exu

is the total

_exchange

_energy

_{(including self-interaction)}

of the

_homogeneous

_gas and

_{y (N u)}

is a function of

_N,

defined

_by

the two

_{equations :}

We note that if

_Nu

=

1, 8 =

2 and _y =

0 ;

and that if

N, --*

_oo,

/3

= 0 and _y = 1. So

equation (4)

takes the correct values in these

_limiting

cases.

Historically

the first self-interaction corrections were due to Hartree and to Fermi and Amaldi. As is well

_known,

in the Hartree

_theory

each electron interacts with an orbital

dependent potential

which is

_{given by}

the coulombic term

_{(Eq. (2))}

minus the contribution to

this

_potential

due to the electron itself. Fermi and Amaldi

_{proposed [6],}

in the context of the Thomas-Fermi

_{theory, subtracting}

to _{the coulombic energy}

_{E, [p ]}

an _average

_{self-interaction}

of the electrons

_{given by :}

where N is the number of electrons in the

_system.

Starting

from these two basic ideas many versions of the self-interaction correction were elaborated. In

_particular,

the Fermi-Amaldi correction can be

_{easily generalized}

in order to

derive an

_expression

for the interelectron

_exchange

and correlation.

_{Indicating by}

Ex,

[p , ] and

by

Ec.,, [p 1 ,

p 1

]

the total

exchange

and the total correlation of the

homogeneous

gas and with

Eie

and

Ec,,,,,

the

_{corresponding}

interelectron

_parts,

one obtains :

These two

_expressions

have been

_recently

used in self-consistent calculations of atomic

properties.

The first one

_by

Cedillo et al.

_[7],

who

_{performed exchange-only}

calculations

where the average coulombic self-interaction was cancelled out

_{by using equation}

_(7).

The second one

_by

Vosko and Wilk

[8]

and

_{by Lagowski}

and Vosko

[9],

who used

_equation

(9)

in order to include the correlation into Hartree-Fock-like calculations.

The use of the Rae

_equations

in the context of the DFT was first considered

_by

Bôbel and Cortona

_[10]

who,

assuming

that

_equation

₍₄₎

could be used to

_approximate

the interelectron

exchange

of an

_{inhomogeneous}

_system,

derived a self-interaction-free one-electron

_equation.

Also in this case, as in the Fermi-Amaldi-like

theories,

an

_{explicit dependence}

from the total number of electrons is introduced in the

_theory.

equations

(4-6)

by neglecting

the

_higher

_{powers of}

_/3

in

_equation

₍₅₎

and in

_equation

₍₆₎

to

obtain

and

and then

_replacing

this latter

_equation

with the

_following

in order to obtain an

_exchange

_{energy which vanishes when}

_{N cr}

= 1.

The

_{resulting expression}

is,

in

fact,

still

_{given by equation (8).}

_Lindgren

noted, however,

that this

_equation

coincides,

in the

_particular

case of the

_homogeneous

_{gas, with another} natural

_expression

for the interelectron

_exchange

_{energy :}

where i indicates the

_quantum

numbers of the

_occupied

orbitals,

and he

_{proposed using}

this

expression

in the

_{inhomogeneous}

case.

The

_{Lindgren approach}

was then

_{generalized by}

Perdew and

_Zunger

_[12],

who included the correlation

_{by assuming}

the

_{following equation}

as the base of their

_{theory (Exc}

=

exchange-correlation energy).

Perdew and

Zunger

made extensive tests of this method

_finding,

for a number of

_{properties, important improvements}

with

respect

to the

_{corresponding}

LDA results.

Further progress in this method was done

_by

Harrison

_[13]

who focused on _{the way of}

performing

the

_spherical

_{average in atomic calculations. A} well known characteristic of an orbital

_dependent

scheme

is,

in

fact,

the non-invariance under

_unitary

transformation of the orbitals. In the case of

_equation

(14), performing

the usual

_spherical

_{average of the orbital}

charge

densities one has no

_problem

because the results become

orbital-independent.

But,

using non-spherical

orbital

_charge

densities in

_{equation (14)}

and then

_performing

the

spherical

average, one

_gets

different results for different choices of the orbitals. It was

suggested

that the

_optimum

choice is

_{given by}

_{the orbitals which minimize the total energy of} the

_system

_[12].

Harrison showed that these orbitals are not those which

_give

the best results : in atomic

calculations,

spherical-harmonic

orbitals

_give

a _{total energy lower than cartesian}

orbitals,

but the latter

_{produce considerably}

better

results,

for

_example

for the

intercon-figurational energies

of the transition metal atoms

_[14].

In a recent _{paper, Cortona}

_[15]

has

_re-analyzed

the use of

_equation

(4)

in the

in-homogeneous

case. In

fact,

it is not obvious that

_N,

should be identified with the total number of electrons in the

_system.

For

_example,

in the case of a

_system

_composed

of two well

separated

groups of electrons

y (N,,) Ex, [pl, + y (N2,) Ex, IP2,1

is

certainly

a better

approximation

for

_E(§

than

_{-y (Nlo, + N2u) Exu[P1u + pzu].}

On the other

_hand,

the latter should be better than the first one when there is a

_{large overlap}

between the two

_charge

example,

in the case of atomic

calculations,

it was

_suggested

that the electronic shells are the natural units to do this

_partition.

Furthermore,

to

_neglect

_completely

the

_intergroups

exchange

is too

_rough

an

_{approximation :}

it seems better to treat it

_{by using}

the LDA and to

include in the interelectron

_exchange

energy a term similar to

_{equation (13),}

but with Piu

replaced by

the

density

of each group. This

gives

the

following expression :

where s indicates the different groups of electrons.

About the

_equation

above some remarks are in order. First of

all,

the

_{intergroups exchange}

energy does not contain self-interaction. So to use for it the usual LDA does not contrast with the use of

_equation

(4)

for the

intragroups exchange.

Second,

in the case of _{groups localized}

very far

apart,

the

_{intergroups exchange correctly}

vanishes,

leaving

a sum of terms of the

_type

of

_equation

_(4).

Third,

the

_intergroups

_exchange

is,

for its nature, nonlocal. A local

approximation

to it is

_necessarily

_very

_rough,

and one can

_expect

bad results for all the

properties

which are

_essentially

determined from it.

_{Finally, equation (13)}

is a

_particular

case of

_equation

_{(15) :}

the two

_expressions

coincide if all the electrons are localized in well

separated regions

of space. If this is not the case,

equation

(15)

takes into account more

completely

the results of the

_homogeneous

_gas

_theory,

_and,

for this reason, we believe that it is the natural local

_expression

for the interelectron

_exchange

of an

_{inhomogeneous}

_system.

In two recent _papers

_{[15, 16],}

Cortona has

_{applied equation}

₍₁₅₎

in self-consistent calculations of atomic

_properties.

_{The method gave}

_{general improvement}

of both the

local-spin-density

and the Perdew and

_{Zunger approximations}

for all the

_properties

that were studied. In

_{particular, remarkably good}

results were obtained for the total

_{energies (errors}

: 0.04

%)

and for the

interconfigurational

energies

of the transition metals

(errors

reduced

by

50 % or

_more).

*

In the

_following

section we will extends this

_{analysis by giving}

a

_complete

discussion of the electron affinities of the

_light

atoms

_{(Z «-- 36)}

wich are known to have stable

_negative

ions. Calculations of electron affinities of

_thèse

elements have

_previously

been

_{reported by}

Cole and Perdew

_[17],

who used the Perdew and

_Zunger

method

_{(Eq. (14)),}

and

_by

_Lagowski

and Vosko

_[9],

whose work will be

_briefly

discussed later on.

The électron affinities of the

_light

atoms.

Equation (15),

when

_applied

to the atomic _case, takes the form :

where

_V",,i,,

the orbital

_{dependent exchange potential,}

is

_{given by}

In these

expressions

p,,f, is the

density

of the shell of

quantum

numbers

ne (T

normalized to

1 and

_Nnf,

is the number of electrons in this shell. The coefficient

_N,’,?,

_{[1 -}

y

(N nier)]’

which enters in the effective

_potential,

characterizes the

_{approximation}

and

_{distinguishes}

it,

in

practice,

from the Perdew and

Zunger

one. This coefficient

is,

in

principle,

a function of the

N ne a-

can be

_replaced

_by

the

_degeneracy

_De

=

(2

f

₊

1 )

of the shell. The effective

interelectron

_{exchange potential}

takes then the form

where the coefficient

_{DÎ13 [1 - y (De )}

_]

is

_equal

to 1 for s

_electrons,

to 1.130153 for p electrons and to 1.167141 for d electrons. The method based on

_equations

₍₁₆₎

and

₍₁₈₎

was called D-SIC. We shall use this notation and we shall indicate the Perdew and

_Zunger

method

by

its usual name SIC.

All the results that we

_present

_{in this paper have been obtained}

_by

self-consistent

_spin

polarized

calculations. As in the

_preceding

_papers, we have included the correlation

contributions

_{by using}

the Perdew and

_Zunger

_{[12] parametrization}

of the

_Ceperley

and Alder

[18]

Monte Carlo data for the

_homogeneous

_{gas, that} we have corrected for the self-interaction effects

_using

the Perdew and

_Zunger

method.

_Furthermore,

we have taken into

account the

_{non-orthogonality}

of the orbitals

_by

_performing

a Schmidt

_{orthogonalization}

after each iteration.

In table 1 we show the differences between the calculated electron affinities

_(EA)

and the

experimental

ones

(1).

The theoretical values have been obtained as differences of total

energies (ASCF method),

while the

_experimental

data are the « recommended » values

_given

by Hotop

and

_{Lineberger [20].}

The

_precision

of these latter data is

_generally

_very

_good

(except

in a few cases, such as Ga and

Ge)

_and,

in any case,

_quite

sufficient for our _purposes. In

_compiling

the table we have

_{distinguished}

if the electron added to the atom is a _{s, p} or d electron. We have done this in order to

_emphasize

_{the different accuracy in the}

_description

of these electrons

_{given by}

the two theoretical methods. The s electrons are treated

_by

SIC and D-SIC

_exactly

in the same way. The small differences in the EA are due to the differences in the core

_charge

densities which can

_{only produce}

minor effects

_owing

to the

_{large spatial}

separation

between the core electrons and the electron which is concerned in the EA. As one can see in table

I,

the EA

_{predicted by}

SIC and D-SIC for s electrons are about the same and

they

are

_quite

_accurate, with errors of some hundredths of eV.

Larger

differences between the two methods are _{obtained for p electrons. D-SIC}

_gives

EA with an _average error of 0.09

_eV,

while the

_{corresponding}

error of SIC is about four times

greater.

In

_particular,

we note _{that the accuracy of D-SIC is about the} same as for the s electrons.

The case of the d electrons is more difficult. The intershells

interactions,

that the two

theoretical methods treat

_by

the local

_{approximation,}

have an

_important

role and

_they

can be

expected

to introduce

_large

errors in the calculated EA. That this is indeed the case can be seen in table

I,

even

_though,

in

_analyzing

these

data,

one should consider that the relativistic effects are not included in our calculations. These effects are

_{practically negligible}

in all the cases _{of interest in this paper}

_except

for the d states of the transition metal atoms

_{[9, 17].}

In this case, in

fact,

they

reduce the EA

roughly by

0.2 eV thus

reducing

also the differences between calculated and

_experimental

values

_(except

for the D-SIC EA of

_Fe).

_Finally,

we

note that for the d electrons as

_well,

D-SIC

_gives

EA

_considerably

better than SIC.

(1)

We have

_{performed corresponding}

calculations for Ca and Sc but these do not

_{give stability}

for the

pertinent negative

ions which have a «

peculiar »

electronic structure with a delocalized

_4p

electron.

Configuration

interaction calculations and Hartree-Fock calculations with the full LDA correlation

Table 1. -

Experimental

electron

_affinities

and the

_differences

between the theoretical and the

experimental

values

_{(in eV).}

The calculated electron

_afjînities

are obtained as

differences of

total

_energies.

An _{alternative way of}

_evaluating

the EA consists in

_taking

the

_negative

of the

_highest

one-electron energy of every

negative

ion. The results of this

_procedure

are

_reported

in table II for s and p electrons and in table III for transition metal atoms. The

_analysis

of table II is

straightforward

and

_quite

similar to that of table 1 : D-SIC and SIC

_give

similar results for s

electrons,

while D-SIC is better for p electrons. The accuracy of D-SIC is

_{quite good}

and the errors are of the same order of

_magnitude

as in the ASCF calculations

(except

for

_H,

0 and

F).

Table III

_requires

some more comments. In

_fact,

_{by comparing}

the theoretical EA with the

Table II. -

Experimental

electron

_affinities

and the

_differences

between the theoretical and the

experimental

values

_{(in eV).}

The calculated electron

_affinities

are obtained

by taking

the

negative of

the

_{highest eigenvalue of}

_every

_negative

ion.

Table III. -

Experimental

and theoretical electron

_affinities

_{(in eV).}

In the column labelled exp 2 the

experimental

electron

affinities

are

referred

to the

_{(3d)n - (4s )1 configuration of}

the neutral atom. The theoretical values are obtained

by taking

the

_negative

of

the

highest

number. The reason of this is that for all these ions the

_{highest eigenvalue corresponds}

to a 4s electron.

_Taking

off this electron leaves a neutral atom in the

_{configuration}

_{(3d )" -1 (4s )1}

which is not the

_{experimental ground}

state

_except

in the case of Cr and Cu. So the natural

quantities

to _{compare with the theoretical values} are not the

_experimental

EA but the sums of the latter and the

_{experimental [21]}

_{interconfigurational energies}

_{(3d )" -1 (4s )1 - (3d)n - 2}

(4s

)2 .

This is the

_meaning

of the data labelled as _exp

_2,

which agree better with the calculated values.

Recently, Lagowski

and Vosko

_{(LV) [9]}

have calculated a number of ionization

_potentials

(IP)

and of EA in order to test the nonlocal correlation functionals

_{proposed by}

Hu and

Langreth (HL) [22]

and

_by

Perdew

_{(P) [23].}

These functionals are based on a modification of the

_gradient

correction

_{suggested by}

the

_analysis

of this

_{approximation performed by}

Langreth

and Perdew

_{[24, 25],}

and

_they

have been used

_by

LV in order to calculate the correlation contribution to the IP and to the EA to be added to the

_{corresponding}

HF

Table IV. -

Experimental

electron

_affinities

and the

_differences

between the theoretical and the

experimental

values

_{(in eV).}

The calculated electron

_affinities

are obtained as

_{differences of}

total

_energies.

The D-SIC values

_{differ from}

those

_of

table 1

_for

the addition

_of

relativistic contributions. HL and P indicate results

_{of Lagowski}

and Vosko

_[9]

obtained

_including

in

Hartree-Fock-like calculations the nonlocal correlation

_functionals

_proposed

_by

Hu and

quantities.

We note that the HL and P functionals are the most

_{sophisticated}

correlation functionals at

_present

in use and that to combine them with the HF method

_gives

the most accurate

_{approximation}

that one can conceive in the context of the DFT for atomic calculations.

We

report

the LV results in table IV

_together

with our D-SIC EA. In order to

_directly

compare the three sets of data we have added to the D-SIC EA the relativistic

_{contributions,}

which were calculated

_by

a

_perturbative

method

_by

LV and which are included in their HL and P results. The

_comparison

is

_{quite surprising}

and reveals that D-SIC is

_generally

the more

accurate of the three methods. More in

_detail,

_{the accuracy of the D-SIC and P results is} about the same for most of the s electrons and p

_electrons,

while D-SIC is better for

_0,

_F,

and for all the transition metal atoms. On the other

_hand,

the HL

functional,

which is based on the random

_{phase approximation}

and which makes no use of the Monte Carlo results for the

homogeneous

gas, is often less accurate than the other two methods.

It is difficult to rationalize these results : the D-SIC

_exchange

is

_{approximated,}

while in HL and in P the

_exchange

is exact ; D-SIC is a local

_method,

while in HL and in P the nonlocal corrections are taken into account ; on the other hand the self-correlation is cancelled out in D-SIC but not in HL and in P.

_Perhaps

the more

_important

factor is the use of similar

approximations

for the

_exchange

and _{for the correlation functionals. In any} case we believe that table IV illustrates one more time the

_difficulty

of

_{going really beyond}

the

_{local-density}

approximation.

Conclusions.

We have used a

_{recently proposed}

method

(D-SIC)

of

_correcting

the LDA for the

self-interaction effects in order to calculate the EA of

_light

atoms. We have found that this method

reproduces

the

_experimental

values with a

_precision

of

_roughly

0.1 eV for the EA of

_negative

ions obtained

_{by adding}

an s or a _{p electron} to a neutral atom

_(except

the cases of

_H,

0 and F if the EA are

_{computed by taking}

the

_negative

of the

highest eigenvalue),

while the errors are

considerably

greater

in the case of the addition of a d electron to a transition metal atom.

We have

_compared

the results of D-SIC with the

_{analogous quantities}

calculated

_{by using}

the self-interaction correction

_{proposed by}

Perdew and

Zunger

(SIC).

We have found that D-SIC

_{gives clearly}

better results in all the cases where

_appreciable

differences between the two

methods could be

_expected.

This is

_analogous

to the

_findings

of

_preceding

_papers

_{[15, 16]}

for total and

_{exchange energies,}

ionization

_potentials

and

_{interconfigurational energies.}

We have also

_compared

the D-SIC EA with values calculated

_{[9] by using}

the best nonlocal

correlation functionals

_presently

available and the HF method. Also in this case D-SIC turns out to be

_generally

the more accurate method. This is

_quite

_surprising,

in

_particular

in the case of d electrons : these have

_important

nonlocal contributions to the EA and

_yet

the nonlocal methods discussed

_give

rather inaccurate results.

References

[1]

HOHENBERG P. and KOHN W.,

Phys.

Rev. 136

₍₁₉₆₄₎

B864.

[2]

KOHN W. and SHAM L. J.,

Phys.

Rev. 140

₍₁₉₆₅₎

A1133.

[3]

GUNNARSSON O. and LUNDQVIST B. I.,

Phys.

Rev. B 13

₍₁₉₇₆₎

4274.

[4]

ALMBLADH C. O. and VON BARTH U.,

Phys.

Rev. B 31

₍₁₉₈₅₎

3231.

[5]

RAE A. I. M., Mol.

Phys.

29

₍₁₉₇₅₎

467.

[7]

CEDILLO A., ORTIZ E., GAZQUEZ J. L. and ROBLES J., J. Chem.

Phys.

85

₍₁₉₈₆₎

7188.

[8]

VOSKO S. H. and WILK L., J.

_Phys.

B 16

₍₁₉₈₃₎

3687.

[9]

LAGOWSKI J. B. and VOSKO S. H., J.

_Phys.

B 21

₍₁₉₈₈₎

203.

[10]

BÖBEL G. and CORTONA P., J.

_Phys.

B 16

₍₁₉₈₃₎

349.

[11] LINDGREN

I., Int. J.

_Quantum

Chem. S 5

₍₁₉₇₁₎

411.

[12] PERDEW

J. P. and ZUNGER A.,

Phys.

Rev. B 23

₍₁₉₈₁₎

5048.

[13]

HARRISON J. G., J. Chem.

Phys.

78

₍₁₉₈₃₎

4562.

[14]

HARRISON J. G., J. Chem.

_Phys.

79

₍₁₉₈₃₎

2265.

[15]

CORTONA P.,

Phys.

Rev. A 34

₍₁₉₈₆₎

769.

[16]

CORTONA P.,

Phys.

Rev. A 38

₍₁₉₈₈₎

3850.

[17]

COLE L. A. and PERDEW J. P.,

Phys.

Rev. A 25

₍₁₉₈₂₎

1265.

[18]

CEPERLEY D. M. and ALDER B. _J.,

_Phys.

Rev. Lett. 45

₍₁₉₈₀₎

566.

[19]

FROESE-FISCHER C., LAGOWSKI J. B. and VOSKO S. H.,

Phys.

Rev. Lett. 59

₍₁₉₈₇₎

2263.

[20]

HOTOP H. and LINEBERGER W. C., J.

_Phys.

Chem.

_Ref.

Data 14

₍₁₉₈₅₎

731.

[21] MOORE

C. E., Atomic

Energy

Levels, Natl. Bur. Stand.

_(U.S.)

Circ. No. 467

_(U.S.

GPO,

Washington,

D.C.)

1949 and 1952, Vols. I and II.

[22]

HU C. D. and LANGRETH D. C.,

Phys.

Scr. 32

₍₁₉₈₅₎

391.

[23] PERDEW

J. P.,

Phys.

Rev. B 33

₍₁₉₈₆₎

8822.

[24]

LANGRETH D. C. and PERDEW J. P., Solid State Commun. 31

₍₁₉₇₉₎

567.