Self-consistent description of finite multielectron systems: new approach

(1)

HAL Id: jpa-00247846

https://hal.archives-ouvertes.fr/jpa-00247846

Submitted on 1 Jan 1993

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.

Self-consistent description of finite multielectron systems: new approach

M. Amusia, V. Shaginyan

To cite this version:

M. Amusia, V. Shaginyan. Self-consistent description of finite multielectron systems: new approach.

Journal de Physique II, EDP Sciences, 1993, 3 (4), pp.449-463. �10.1051/jp2:1993143�. �jpa-00247846�

(2)

Classification

Physics Abstracts

31.10 71.10

Self-consistent description of finite mhltielectron systems:

new

approach

M. Ya. Amusia (~' _*) ^and ^V. ^R.

Shaginyan (~)

(~) ^Institlit fir Theoretische Physik, Universit£t Frankfurt, 6000 Frankfurt _am Main, 11

Germany

(~) St-Petersburg ^Nuclear Physics Institute, Gatchina, St-Petersburg 188350, Russia

(Received

29 July 1992, accepted in final form 6

Jauuary1993)

Abstract. An analytical expression for the effective interaction in _a finite multielectron system is derived. The interaction is of finite radius and is density dependent. An analytical expression is also derived for the effective single particle potential and _an equation is presented

for the linear _response function of the considered system. ^Thus ^it ^is demonstrated that the spectrum, photoeffect and electron scattering _cross section for such _a system _may be calculated by solving _a complete system of

integral

and differential equations.

1. Introduction.

llecently

_an essential _progress has been achieved in

describing

finite multielectron systems, which is connected

mainly

with the construction of

Density

Functional

Theory (DFT),

based

on [1, 2] and with

development

of the Random Phase

Approximation

with

Exchange (RPAE)

[3]. ^This DFT in

principle permits

to calculate

precisely

the characteristics of the

ground

state of the system, ^such _as ^the

ground

state energy and selfconsistent

single particle

effective

potential.

However out of its _scope _are those

dynamic properties

of the system which _are

closely

related to the behaviour of the system in

time-dependent

external fields. This

shortcoming

of DFT _was eliminated in papers [4,

5],

where the

Density

Functional method _was

generalized

to

time-dependent

densities. After

this,

DFT

acquires

the

ability

to consider

formally

static and

dynamic

characteristics of _a finite multielectron system. ^However ^DFT is unable to derive the

density

functional itself.

So,

for its

application,

_some

approximations

of this functional _must be constructed [6]. ^Most

frequently

used is the Local

Density Approximation (LDA),

in which

one has [6, 7]

Exciii

₌

f 6xc(P(r,f))P(r,f)

dr.

(I)

(°) On leave of absence from Physical-Technical Institute, St-Petersburg K-21, Russia.

(3)

Here

E~c

is the

exchange-correlation

part ofthe

density functional, e~c(p(r,t))

is the

exchange-

correlation _energy of the

homogeneous

electron gas of the

density p(r,t),

related to a

single particle.

The

exchange-correlation potential

is of the form [2, 6]

Vxc(r)

₌

((j

lp=po,

(2)

with _po

being

the

ground

state

one-particle density.

The

exchange-correlation

effective inter-

action,

_or sc-called

exchange-correlation

kernel [5, 6], ^is

given by Rxc(r,

f;

r', f')

~2

=

6(f f')6(r r') p it(r) 6xc(P(r))I

_lp=po

(3)

However, in this _case the

potential V~c(r) (2)

^has a wrong

asymptotic

behaviour at

large

distances _r from _a system [6, 7], ^while ^the ^effective interaction

R(r,tj r',t')

is of _zero range.

These limitations _are

quite

essential

i

because for instance the correct

asymptotic V~c(r)

at

long

distances

r is of decisive

importance

for the correct

description

of autoionization resonances,

photoionization,

_van der Waals forces etc. [3,

6].

The finite range

dependence

of the effective interaction is also essential in

considering

_processes related to

transferring

considerable _mc-

mentum such _as the electron

scattering

on atoms and the

photoeffect

_{[3, 7].} We remark that in

the frame of _a selfconsistent

description

the effective interaction and the effective

potential

_are

closely

linked to each

other,

that is _an incorrect effective interaction

gives

rise to the incorrect

effective

potential. Partly

this

problem

is solved in

RPAEI

where _an effective

potential

with the correct

asymptotics

at

large

r is constructed and the effective interaction is of finite range [3].

While

including

the

inhomogeneity

of the system

(mainly

it _was

applied

to

atoms),

RPAE _uses the effective interaction which

completely neglects polarization phenomena,

I-e- is

independent

of the electron

density.

This

dependence

is included in _some of the

generalizations

of RPAE, but

semiphenomenologically.

To include this effect

precisely, analysing

and

accounting

for _a

number of _sequences of the

many-body theory diagrams

is

required

which makes the

problem

rather

complicated.

The aim of this work is the construction of _a _new

approach

to the

description

of finite multielectron systems. In the frame of this

approach,

_a _new

equation

will be

derived,

which determines the effective interaction R and the effective _one electron

potential V,

which allow to calculate static and

dynamic

characteristics of _a finite multielectron system.

This _paper is

organized

in the

following

_way. In part 2 the

equation

for the effective interaction is derived and the _means to find its solution _are considered. In part 3 the effective

one-electron

potential

is considered. Part 4 concentrates _on the derivation of

analytical

ex-

pressions

for the effective interaction and

potential.

Part 5 summarizes the main results of the paper.

2. Functional

equation

for the effective interaction.

The functional

equation

for the effective interaction R has been derived in [8, 9]. ^Here we

present the main

points

of this derivation in order to

clarify

the

presentation

of the account, and for

completeness.

Let _us consider _a system of N

interacting particles

in its

ground

state.

According

to the

prescriptions

of the

Density

Functional Method instead of this _a system of N fictitious

particles

_may be considered which _are

moving

in _an effective field

V(r),

determined

by equation

_[1, _2]

~

~~~~

_~

@ ^~~ ^~0~'

^~~~

(4)

Here To ^{is the} ^kinetic energy of

noninteracting

fictitious

particles,

while E is the total _energy of the system. We remark that To and E _are functionals of the

density

_p. In what follows _we will not

emphasize

the difference between these

particles

and the fictitious _ones if this will not lead _to confusion.

The

equation (4)

_may be

presented

_as

V(r)

₌

f _gll(r _r')P(r')

_dr'

+

)

_Exc.

₍₅₎

Here

gv(r r')

is the

interparticle

interaction in _vacuum, g is the

coupling

constanti while E~c is the

exchange-correlation

_energy. The functional derivative in

(5)

is calculated at p =

po(r),

where

po(r)

is the

ground

state one-electron

density.

This condition is valid also for other functional derivatives calculated below. Instead of E~c the

exchange-correlation

part of the functional of action A~c _can be used [4, 5],

leading

to the

equation

V(r, ii

₌

f

g

v(r r'jp(r', ii ^dr'+

~

~~

A~c.

(6)

If the

ground

state is considered

(the

functional derivative is calculated at p =

po(r)),

the effective

potential

must be

independent

of

time,

V(r)

₌

f _gll(r _r')P(r')

_dr'

+

~~)

~~

Axc

(7)

The effective interaction R is determined

by

the

equation

~~'''~ 6p(~,t2) ^~~~~~'

Of _course, here

R(ri,

iii _r2,

t2) depends only

_upon the time difference (11

t2)

₌ t:

Note, ^that ^the ^second term in the

right

part ^of

equation (8)

is

equal

to

exchange-correlation

kernel

f~c

introduced in [5].

Equations (5)

^and

(8)

_are in fact

only

formal definitions. In order to _use them _one needs to express E~c via such

functions,

which

permit explicit

calculation of the variational derivative _over

density.

For this _reason it is convenient to _use the _response function apparatus. ^The energy E~c _may be

expressed

via the linear _response function

and,

on the other

hand,

there _are rules _on how _to take variational derivatives of it _over the

density.

Let _us _use the known

expression

for E~c

[10]:

Erc _"

f _l'fl~X(~l,

~2i

")

+

~P(~l)6(")6(~l

_~2)1

x

g'v(ri r2) ~§'

^~~

dri dr2 (9)

g ~

Here

x(ri,r21w)

is the linear _response function and the

integration

over the

frequency

_w is

performed

from 0 to +cxJ, while

integration

_over the

coupling

constant goes from 0 to _a real value _g. Let _us _now

complement equations (5, 8, 9) ^by

an

equation connecting

the function

x(ri,r21w)

and the effective interaction

R(ri,r21w)

which is the Fourier

image

^of ^the ^time-

dependent

function [5,

8,

9]:

R(ri,r21w)

=

xp~(ri,r2;w)-x~~(ri,r21w). ₍₁₀₎

(5)

More convenient is another form:

x(ri,

_r2

w)

₌

x(ri,

_r2

w)

+

xo(ri, r[ w) R(r[, r[ w)x(r[1r2 w)dr[dr[. II)

Here

xo(ri, r21w)

^is the linear _response of fictitious Kohn-Sham

particles, moving

in _an effective

single particle

field

V(r).

We note that

exchange-correlation

_energy E~c reduces to the

exchange

energy E~ if

x(ri,

_r2

iw)

is

replaced by xo(ri,

_r2i

w)

in

equation (9).

In this _case the

integration

over the

coupling

constant is trivial. The function

xo(ri, r21w)

is determined

by

the

expression ill,

_12]

X0(rl,

_r2i

")

_"

~ _n;(§2) (rl)i°I(r2)G(rl,

_{r2i 6i} ₊ _W) +

~Di(rl)i'l(r2)G*(rl,

_{r2i 6i} _W)j.

₍₁₂₎

I

Here _n; is _an

occupation number, G(ri, _r21w)

is the

single particle

Green's function determined

by

the

equation

'' ~ ^~~~~~~~~~

' ~~~~

where

~a;(r)

and _e; _are the

single particle

Kohn-Sham _wave functions and

energies, respectively,

q is an

infinitesimally

small

positive

^number. These functions and

energies

_are determined

by

the

equation(~)

~~j

q2 ₊

v(~l) ^i°I(~l

^"

^~Si'S(~l' ^(i~l

Thus

equations (5, 8,

9, 11) ^form ^the system of

equations

which

permits

to calculate the functions

R(ri

,

r21WI, V(r), x(r1

, r2

w)

^and

exchange-correlation

_energy E~c ^of a finite _many

particle

system,

interacting

with each other

by

the

pair potential gv(ri-r2)

For _a multielectron system one has

gv(ri r2)

"

1/[ri

_-r2 _(. ^This system is

considerably simplified

in the _case of _an

infinite

homogeneous medium;

its solution _we will _use _as _a basis to construct

approximations

for the effective interaction and the

one-particle

effective

potential

of _a finite system. In this

case the effective

potential V(r)

is _a constant, and

equation (11)

_may be solved

analytically

after

performing

the Fourier transformation _over the _space variables:

x(q,

_W)

=

~(j~]j)(

~~

~~ (15)

Substituting equation (9)

into

equation (5)

and

using equation (8),

_we get

~~''

_~~ _~~~ _~~'~~~ _~~~ $2

2

6p(ri,ti)6p(r2,t2)

~

[~fllX(,,W)

~

~P(~~)~( ~)~(")j

x

g'v(r[ r[) ~l'

^~"

dr[ dr[. ₍₁₆₎

g ^~

The function

x(q,w),

which is determined

by equation (15),

is

conveniently presented

in the

following

form:

~~~'"~

_~°_~~'

"~

~ ~

l

~~~)~()i)

^~~

^~

(~ The atomic system ôf ûnits îs ûsed ⁱⁿ ^this paper.

(6)

Let _us _pass in

equation (16)

to the variables _q and _w

by

the Fourier transformation _over _space and time variables.

Notei

that

R(ri,

r2i11

t2)

for _a

homogeneous

system

depends only

_on

the coordinate difference

[rI r2(. Performing

the Fourier transformation of

equation (16)

and

using equation (17),

the

following equation

is obtained

R(q,W)

~2

=

RHF(q,W) j

~~~~ ~~~~~~

~~ (18)

~

i

_~~

^l

~i~~)j~)~i')j

^~'~~~~

(~$3 ^~~' ~''

Here

RHF(q,W)

₌

gll(q) ~~~~((~,~~ _(lo)

x

f'mlto(k,W')1gll(k) ) ^j

Choosing

in _a

special

_way the contour of

integration

_over the

frequency

_w, one can show that

[10,

13] ^the

following

relation is valid

i~ 'fI~X(")

"

f~ X(I") ~) ⁽²⁰⁾

-co

With the

help

of

equation (20),

_one obtains [8, _9]

R(q,W)

₌

RHF(q,W)

~ ~~

~((

~~

~~ (21)

~

j ^xi (k, iW')R(k, i~') ,~~~j

dk dW'

q

i

R(k,

iw'

)xo(k, iwi) (2x)4 g'

^'

where

i 52

~~~ _~'" _" _~~ _~~

i 6P(q,W)6P(q,W)

x

f _xo(k,

SW')g

1l(k) (())' (22)

Equation (21)

determines the effective interaction from which _one calculates the linear _re-

sponse function

x(q,w)

of the system and the

exchange-correlation

_energy E~c.

Unfortunately,

equations (16)

and

(21)

are the variational

equations, being

in fact _a compact

expression

of

an infinite system ^of

coupled equations

lo]. ^In ^order to demonstrate

this,

let _us

clarify

how to calculate the functional derivative

6x/6p.

Such _a

technique

will be useful later. In order to calculate this functional

derivative,

let _us _use _an

equation connecting

the

density

fluctuation

6p(r,w),

which

appeared

in the system ^under the action of _a weak external field

lvext(r,w)

[121 13]

6P(r,

_W) ₌

f f

( ~~(("~

^x~~~

^(r,

ri,

, rk wi,

,

wk)

k=I

x lvext

(ri,

_wi

lvext(rk,

_wk

)2x6(w

_WI _wk

(23)

(7)

Here

x(k)

is the causal _response function ofthe order k. The

equation (23) pernfits

to determine the linear _response function

x(~)

_%

x(r, ri;w) using

the

following equation

$p(T,

_UJ

~

(~

_~ _~ ~~

$(~

_wi

), (~~)

$)y~~~(Ti,UJi)

j=Q

~~

Similarly

the _response functions of

higher

orders may be defined. Now let _us calculate the functional derivative [9]:

~X(~l, ~2i") f I'ext('~'~) ~~P(~li") l~~i

^~~i

(~~)

6p(r3,w2) 6p(r3,w2) 61vext(r[,w[)61vext(r2,wi)

^~ ^~'

>=0

Taking

into account

equation (24),

_we get

)$~j~

~

i

~

~~~'~ ~'~'~~~~~~~''~ ~'~'~'~'~~

x2x

6(w

_wi _w2 ^dr

[, (26)

where

x(~)

is the second order response function and x~

(r, r', w)

is determined

by

the

following equation

x ^~

(r, r',

_w _x

(r',

_ri

,

w dr ^'

=

6(r

_ri

(27)

The calculation of the next functional derivatives is

performed sinfilarly.

Thus the function

R(ri,

_r2

iw)

^and ^the linear _response function

x(ri, r21w) (Eqs. (ll,

21,

22)

_are deternfined

by

the response functions of the second and third orders.

Taking

the variational derivatives of both sides of

equation (21),

_an

equation

is derived which _connects the _response function of the second order with the response functions of the third and fourth orders etc. This

procedure

leads to _an infinite chain of

equations.

This infinite chain _may be transformed into _a closed system of

equations

if _an

approximate

^relation _can be

established,

which connects the n-th and

(n+I)th

_response functions _or _an

approximate expression

for the variational derivative

can be found. This _can be done

using

_a local

approximation

which works

successfully

for the infinite medium. In this

approximation

the variational derivative of the functional

F(T,

_{~p]) is}

substituted

by

the

following approximation:

6F(r, lpi)

~+

d

F(r, iii)

~ ~~ ~ ~~~~ ~~~~

6P(

_ri

,

t) dP

p= p~

Note that the

approximation (28)

is transformed into _an

equalityi

when

integrating

both sides of

equation (22)

_over _rI and t

j 6F(r, iii)dridf

~

d

F(r, iii) 6p(ri,t) dp

~~g~

Therefore, equation (28)

_preserves the _sum rule for the

compressibility

ofthe system, ^which ^is a

feature of

precise

solution. Let _us

apply

the

approximation (28)

to transform

equation (16)(or Eq. (21))

into _an

ordinary integrc-differential equation.

As _a

result,

for the

homogeneous

electron fluid

(or gas)

the

following equation

is obtained [8, 9]

R(q,

_g,

p)

₌

RHF(q,

_g,

P)

li~ I

I

~i~)iw)

[/[2 ^~ ~''

_~~~~

(8)

Here

R~~

(q

_g,

p)

₌

~(~

+ R~

(q,

_g,

p), (31)

where

~~~~'~'~~ ~i l'~[

^~~

^

^~~

^~)

+

~

j

~~~~

The electron

density

_p is connected to the Fermi momentum

by

^the usual relation _p

=

pi /3x~.

Having

in hand the effective interaction

R(qi

_g,

p),

_one _can calculate the linear _response func-

tion, given by equation (15),

as well _as the correlation _energy Ec.

Substituting equation (15)

into

equation (9)

and

separating

the

exchange

_energy

E~,

_we obtain

~~ _~~~

~2 I

I

~i~,~iw)

~~

'"

_~~~~

Here _cc is the correlation _energy _per electron of _an electron _gas of the

density

_p.

Based _on

equation (30),

_we will construct in the

following

parts of the _paper

approximate

solutions for

equations (5)

and

(8).

3.

Asymptotic

behaviour of the effective

potential V(r).

In this part we will consider the

asymptotic

potential V(r)

at r - cxJ.

To do

this,

it b convenient to separate

V(r)

into two terms: the

exchange V~(r)

^and ^the

correlation

l~(r) [14]. Substituting equation (9)

into

equation (5)

and

separating

the

exchange

term, we get

vr(r)

=

-1 ^p~l

w~

f _{Jm xo(ri}

r2 ~J) ⁺

~P(ri

_6(~J)

6(ri _r2)1

x

gv(ri

_r2 dri dr2 ^~~

x

It is easy ^to demonstrate

using equations (25)

and

(26),

that the variational derivative 6

/6p(r,

_w

is transformed into _a

frequency independent 6/6p(r), just

_as _one should expect:

~~ Sir) I ~°~''"~

_~ _~~~~~~

_~~"~

_~~~~ _~~~~

x

gv(ri r2) dri dr2

^~~

(34)

x

The correlation

potential

is of the form

~~~~ Sir) I

~~

~~~''~'~ ~°~~~~~'~'~~

x

g'v(ri r2) dri dr2

^~~

~l' (35)

~ g

Let _us calculate the

potential

and its

asymptotics

at r - cxJ

using equation (26).

^As a result

one has

v~(r)

=

f12

^im

^{xi ~(r,}

^r

^')xi~~(r',

^ri, ^{r~; w,}

^o) ⁽³⁶⁾

+

~6(ri r2)6(r ri)6(w)] gv(ri r2)

dri dr2

dr'~~'

~

JOURNAL DE PHYSIQUE II T 3, N' 4, APRIL ,993

(9)

Here

xi~(r',ri,r2iw,0)

_is _the _second _order

response function for the Kohn-Sham

particles

moving

ⁱⁿ ^the ^effective

potential V(r).

The function

xi~

_is known to be

analytical

_[12] and is determined

by

Xi~(~',~l, ~2i"1,"2)

_"

£ ni(i'l(~') i'i(~2)G(r',

_rl16i ₊ _"1 ₊

"2)

I

x

G(ri,

_r21e; +

w2)

+

~7;(r')~o)(r2)G* (r'i

_rip _e; _wi

_w2)

x

G*(rii

_{r2i e;}

w2)

+

i';(ri)i'l(r2)G(r'i

_ripe; +

wi)

x

G*(r',r~;e; w2)

+

bermutations

of

(ri,wit r2,w2)]). (37)

The function

xo(r, r')

is

given by

Xo(~,~')

_"

Xo(r,r'i")(W=o,

and

xp~(r,r')

is

given by

)xi _(r, _{r') xo(r',}

ri dr'

=

6(r

_ri

(38)

After

integration

over

frequency

_w and _some

algebra, equation (36)

is transformed into [15]

Vr(r)

₌

f _xi~(r, _r') _L n;ni1i2i(r2)i2i(ri)i2;(r')i2l (ri) XG(~21~'16i)

+

§~1(~2)§~I(~l)§~1(~')§~i(~l)

xG*(r2, r'; e;)]

_gv

(ri r2)dri dr2

dr'.

(39)

Equation (39)

determines the exact

exchange potential

of

Density

Functional

Theory.

Note that the

potential V~(r)

for

gv(ri r2)

₌

1/[ri r2(

coincides with the known Talman- Sha~dwick

potential [16],

^which was derived in the Hartree-Fock

approximation

for the

ground

state energy. So _we _resume that

equation (39)

is the

generalization

of this

potential.

In order to calculate

V~(r)

at _r _- _cxJ let _us _use the

equation (39)

and _assume, that if _r _- _cxJ then _r2 _- _cxJ also.

Using

the

fact,

that _r2 _> _ri and

performing

the

integration

_over _ri, _we

obtain

~~~~~~~" x(I(r,~i)~~_

~

-

~

~~)

h~;

(r ')

G*

I)11([) j[j)~(([l'

_{~' ~'~}

~°

~~'~'~ _{~°~~'~ ~2)} 9»(r2) dr~

dr,

Note,

that

gv(ri r2)

_"

gv(r2)

_as _r2 _- _cxJ.

Using equation (38),

_we get

Vx(r)

=

f6(r-r2)gll(r2)dr2

=

-gll(r). (41)

In the _case, when

gv(r)

₌

I/r,

_one has the known result:

V(r)lr-m

₌

-)

(10)

Note that the

equation (41)

confirms _our

assumption

that _r2 _- _cxJ when

r - cxJ. Unfor-

tunately,

^{it is}

impossible

to obtain _an

analytical expression

for

I~(r),

because this

requires knowledge

of the

precise

linear _response function _x, _or effective interaction R. However the

asymptotic expression

at

large

_r is known 11?]

1(~)ir-cc

_"

$, ₍₄₂₎

where _a is the static

polarizability.

4. The system of

equations

for

description

of _a finite multielectron system.

In order to describe _a finite multielectron system, such _as _an atom _or _a metallic

cluster,

the

equations (9,

II,

16, 35, 39)

must be solved. As _a result _one _can obtain the

ground

state energy, the effective

potential,

the effective interaction and the linear _response function.

Having

the linear _response

function,

it is

possible

to calculate the

probabilities

and _cross sections of different processes

taking place

with the system ⁱⁿ an external

field, just

_as the

characterbtics of the system.

Thus, having

the _response

function,

_one _can calculate the inelastic electron

scattering

cross section _upon _an atom. In what follows

by

"atom~' _we _mean _any finite

multielectron system, such _as, e-g-, a metallic cluster _or _an atom itself.

The inelastic

scattering

_cross section in the Distorted Wave Born

Approximation

is

given by

_[18]

~lll)~

= Im

I

_e'~~~~ _-~~

~x

(r1

, r2 W) an

dr2 ~'()

^~~

⁽⁴³⁾

Here _q and _w _are transferred momentum and energy

loss,

while

da~'/dq

is the elastic electron- electron

scattering

_cross section.

Similarly

^the interaction of _an atom with

radiationi

I-e- with monochromatic

spatially

constant electric field in z-direction with _a

frequency

_w, _can be

considered [3,

7].

For instance, the

photoabsorption

_cross section is

given by

_[7]

a(w

= -4xw Im

zz'x(r, r';

_w dr dr'.

(44)

The linear _response function

x(ri, r2iw)

has the

following spectral representation [10]:

~j ,~~

~

~ (°l#(ri)Inllnl#(r2)1°1 °1#lr2)In)Inl>in)1°)j

' ' ~~~~

~~~ ^w

(E~ Eo)

+

in

_w +

(E~ Eo)

+

in

Here @(r) ^is ^the ^electron

density

_operator, _[0 _> and

in

> are the

many-particle system's

exact

ground

^and excited states

respectively.

Eo ^and En _are

energies

of these states. It is _seen from

(45)

that the

poles

of

x(ri,r2iw)

as a function of _w determine the spectrum of the system, while the residues determine the

probabilities

of the transitions from _< 0[ ^to

in

> states.

So,

we see that the _response function includes vast information _upon the described system.

Let _us _now construct the response function _x. This function _may be obtained

by solving

the

equations (9,

11,

16, 35, 39).

This is _a system of functional

equations,

the solution of which is connected with

big

difficulties of both mathematical and

computing

character.

Therefore,

let _us _use

equation (30)

in order to calculate

approximations

for the effective interaction

R(ri,r2iw),

the effective

potential

V(r)

and the _response function

x(ri, r2iw). Equation (30)

determines

frequency independent

effective interaction in _an infinite

homogeneous

multielectron system

(jellium model). Theni

having

at hand function

R(q,p),

_one _can in the local

density approximation

construct the

(11)

Table I. Correlation _energy _per electron in eV of _an electron _gas of

density

_rs. The Monte Carlo results [19]

et

are

compared

with the calculations of [8]. cc denotes the results of _exact

calculations,

and

e)

denotes the results of the calculations when the eiTective interaction R _was

approximated by

RHF

~S ~C

5 -o.77

-o.56

effective interaction

R(ri,r2)

in

a finite system. The local

density approximation

_proves to be rather successful _even in atomic systems, where the

density

varies rather fast and therefore

one can expect, ^that in _more

homogeneous

systems, ^like ^metallic

clusters,

this

approximation

will work _even better. On the other

hand,

it appears that the

dependence

of the effective interaction _upon the

frequency

is inessential while

describing

such _processes _as

photoionization

or

photoabsorption,

where its contribution is estimated _as

being

less than _{I To} [5].

Thus,

_one

may expect that the function

R(q, p)

^is

quite

suitable for

constructing

the effective interaction in _a finite system.

Then, knowing

the effective interaction

R(ri,r2),

_one _can next _use the local

density approximation

and construct the effective

single particle potential V(r),

which

permits

to solve the

single particle Schr6dinger equation (14)

in _a self-consistent _way. As _a

result,

after

calculating

the Kohn-Sham orbitals _~o; and

energies

_ei, _one _can construct the _response function

xo(ri,r2iw) (Eq. (12))

and the linear _response function

x(ri,r2iw) (Eq. (ll))

which _may be used to consider different _processes discussed above.

So,

in accordance with the described

program, let _us start with the construction of

R(ri,r2).

As _can be _seen from table

I,

R(q, p), being

the solution of

equation (30), permits

to describe the electron _gas correlation _energy _cc

(see Eq.(33))

in _a rather broad

interval of

density

variation.

Note,

that at rs = 50 the mistake is _no _more than 5 $l

as

compared

to Monte Carlo

calculations,

at rs = 30 the mistake is I

Sl,

while at rs - 0 the

results, quite naturally

_are in

agreement

with the Random Phase

Approximation.

Here _r~

=

(3/4~p)~/~

So,

one can use the numerical solution of

equation (30)

to construct the function

R(ri,r2).

However, it is _more convenient to

employ

the

analytical approximation RHF(q,p) given by equation (31).

Note, that the one-electron

density

of _a metallic cluster is

nearly equal

to that of _a metal. It is _seen from table I that RHF

gives

_a _very

good description

of the correlation

energy ^at metallic densities. The result becomes _exact when _r~ _- 0 [8]. ^So we conclude that _one _can _use the function

RHF

_as the effective interaction between electrons in _an infinite system. None the less

we can correct this

approximation

in _a rather

simple

_way. The solution

of

equation (30)

_may be written _as

R(q, P)

_"

(~

⁺

^~F(q, ^P)

⁺

^Rc(q, ^P) ⁽⁴⁶⁾

(12)

It follows

straightforwardly

from

equations (31), (32)

that

RF(q, p)lq=o

= d2

~ (6r p), (47)

with _e~

being

the

exchange

_energy _per

electroni

while

Rc(qiP)lq=o

₌

pl6cP),

~2

⁽⁴⁸⁾

as it follows from

equations (30), (33)

and

(46),

where _cc is correlation _energy _per electron. The function

Rc(q, p)

is calculated

numerically,

and _as _a result the ratio

A(q)

₌

Rc(q, p)/RF(q, PI

may be estimated [9]

(see Eqs. (46, 47)) A(ql

_lq=o ₌

$

(6cP) / $ _(6rP)

m

~~[~

~~

i

(49)

where

ec(rs)

is the correlation _energy _per electron of the electron gas with

density

_rs, measured in atomic units.

Using

^table

I,

it is _easy to demonstrate that for _rs

= I _one has A _cS

0.05,

for r~ = 10 it is A _cS 0.15 and _even for _r~

= 50 the

quantity

A becomes

equal

to A cS 0.25.

So,

for reasonable electron densities the correction

Rc(qi pi

is

comparatively small,

and

Rc(q, p)

_may

be accounted for

using

the

simple approximation R(q,P)

₌

)

+

RF(q,P) II

+

A(q)lq=oi (5°)

Equation (50)

_preserves

equation (49)

and

reproduces

^the

Rc(q, p) dependence

_upon _q _reason-

ably

well for the electron densities of interest

[21].

^Of course, one may use a more

sophisticated

approximation

_{[9] for}

llc(q, p),

but _we should bear in nfind that this function is _a

comparatively

small correction.

Now let _us _use

equation (50)

to construct the effective interaction for _a finite

inhomogeneous

system in the local

density approximation.

It _means, that _we shall _assume that the effective interaction in _a finite system ^at a

point

_ri with the one-electron

density p(ri)

is the _same _as in _an infinite

homogeneous

^medium ^with ^the

density

_p ₌

p(ri).

In coordinate

representation

R(ri,r2)

_may be derived

using

the Fourier transformation

R(ri, r2)

=

f _R(qi

P)

^e'~~~~^~~~~

~j(~ ⁽⁵¹⁾

After

straightforward

calculations _one obtains

l~(~l,~2)

_"

I'(~l ~r2) (1

⁺

) ^~ ^~~

~~ ^()

~

d~

jri

⁺ ^r2

(ri

⁺ ^r2 ^sin ^2z

^sin~ _zj

~ dP~ ^~ ² ^~~ ^~ ²

~~~~~

Z

~ Z~

Here

v(ri r2)

_#

1/[ri r2(,

and

Z "

(rlPF(rl) r2PF(~2)(. (~~)

The function

pF(r)

is related to the

single-electron density pF(r)

=

13~~P(r)l~/~ (54)

(13)

We remark that the effective interaction

(52) depends symmetrically

_on the

spatial

coordinates

R(rl,r2)

_"

R(r2,rl). (55)

Using equation (11)

and

equation (55)

_one _can obtain

X

(~l,

_r2

")

_" _X

(r2,

_rl _LJ),

(56)

as it should be.

To calculate the effective

single-particle potential V(r)

let _us _use

equation (8). Bearing

in mind that the effective interaction

R(ri,r2)

does not

depend

_on the

frequency

_wi we obtain [5,

8122]

R(~l

_~2

"

~ ^v

(~l (57)

P r2

To solve

equation (57)

let _us _use the local

density approximation

and act

just

_as when calcu-

lating

R(ri,r2)

of _a finite system. ^The contribution _to

V(r)

related with the correlation

Rc(q, p) (see Eq. (46))

_we shall take into account in the

simplest

_way

Vc(r)

₌

) 16c(P(r))P(r)1. (58)

Such

procedure

^enables us to derive

V(r)

in _an

analytical

form. After

performing straightfor-

ward calculations _one has:

3 sin 2z Slll~ _Z

V(ri)

_-

j~~~~

^~~~~~~~~

dr2

~

"

_~

~

(~g)

+

~

j6c(P(rl))P(~l)1

~~

l sin 2zi Slll~ _~l dr2 +3

fv(ri

r2)P°~~~~4

2 _~~

Here _z is

given by (53),

_xi ₌

(ri -r2( pF(ri ),

and

po(r)

is the

single-electron density,

normalized

so that

J po(r)dr

₌ I. The third term in

equation (59)

vanishes in the _case of _an infinite medium and

provides

the correct

asymptotic

potential (see Eq. (40))

Vlr)lr-m

₌

160)

Taking

the results of section 3 _as _a basis and

using equations (26, 34, 35),

_one _can _express

V(r)

in terms of the _response functions of the first and the second

order,

_as _was done when

calculating

the

exchange potential V~(r)

~'~~~

i~~~~

^~

~~'~'~ ~~''''~'~

+~6(ri

_r2

)6(r ri)6(w)] dri

dr2 ^~~

(61)

[ri r2(

x

Equation (61)

_ensures the correct

asymptotics

of

V(r)

at

large

_ri but from _a numerical

point

of view it is _very difficult to _use

eqution (61)

to calculate

V(r).

(14)

Now let _us

briefly

consider _a

pair

correlation function

g(ri,r2). g(ri,r2)

is related to the linear _response function [10]

P(rl)P(r2)(g(rl, r2) II

_#

f _ImX(rl,

r2i WI

P(~l)~(~l r2). (~~)

As follows from

equations (62)

and

(56), 9(r1,r2)

"

9(r21rl). (~~)

With

help

of

equation (62)

_we _may conclude that the well-known _sum rule is satisfied

f P(ri) lglri, r2) II

dri ₌ I

164)

The relation

(64) immediately

follows from

equation (62)

^if we make _use of the relation [12]

/x(ri, r21WI

dr2 _" 0.

(65)

The

exchange-correlation

_energy

E~c,

_as follows from

equations (9)

and

(62)1

is

given by

E~c ₌

f~g(riir2) 1]p(ri)p(r2)g'v(ri r2) ^~~ dri dr2. (66)

2 g"

So _we _may conclude that in _our

approach

the function

g(ri, r2)

is defined in _a selfconsistent manner, while the well known relations

(63, 64, 66)

_are satisfied.

In this section _we have derived the

equations

for

description

of _a finite electron system.

Having

in mind the

description

of _a

many-electron

finite system we have to

simplify

_our _equa-

tions _as much _as

possible.

This

problem

is resolved.

Indeedi

the effective

potential

is

given by equation (59). Using

the

single-particle equation (14),

we calculate the

one-particle

Kohn-Sham

orbitals

~a;(r)i

the

energies

_e; and the

single

electron

density P(r)

₌

L

n; i2l

(r) i2;(r).

The linear Kohn-Sham _response function

xo(ri,r2iw)

is

given by equation (12),

while the effective interaction

R(ri1r2)

is defined

by equation (52). Having

in hand the functions _xo and R and

using equation (11)

_one can calculate the linear _response function

x(ri,r2iw). Using

x(ri,r2iw),

_as _was shown

above,

the characteristics of different _processes _may be calculated.

5.

Sununary

and discussion.

Let us now discuss the main results. We have

suggested

_a _new

approach

to calculate the effective interaction

R(ri, r2),

the effective

single particle potential V(r)

and the linear _response function

x(ri,

_r2i

w)

^of a finite

inhomogeneous many-electron

_system. Our

approach

^is free from any

adjustable

parameters, the function

R(ri,r2)

has _a finite _range coordinate

dependencei correspondingly

the function

V(r)

has the

asymptotic

^form

V(r)[r_m

₌

-I/r. Having

in hand the function

x(ri~r2iw)

_one _can calculate the

spectrum

^of ^the

many-electron

system, the characteristics of its interaction with

electromagnetic

radiation and the electron

scattering

upon the system.

(15)

Now _we shall

briefly

_compare the

equations

obtained here with

equations

of the

approach grounded

_on Time

Dependent

Local

Density Approximation (TDLDA)

_16,_7~

22]

ând with the well-known RPAE [3]. În ^TDLDA ^the êffective interaction has _a

b-type

coordinate

dependence,

while the effective

potential

falls off

exponentially

at

largest

_r. The finite _range

dependence

of the effective interaction and the

asymptotic

potential

_are of crucial

importance

in

studying negative

^ions ^and autoionization resonances, as well _as _a

photoabsorp-

tion and

a

photoeffect

_cross section in the

vicinity

of its threshold. Also these features _are essential in

considering

_processes connected with

transferring

considerable momentum~ ^{for in-} stance, in electron

scattering

on atoms and in the

photoelfect

_{[3, 7,}

22].

The

equations

derived

by

_us _may be reduced _to those of TDLDA if _we _use _a

b-type

coordinate

dependent

effective interaction instead of the finite _range effective interaction.

In

RPAE,

which is based _on the Hartree-Fock

approximation

for the

ground

state, the effective interaction is taken _as _a _sum of the bare Coulomb interaction and the

exchange

one, so the correlation contribution to the effective interaction and the effective

potential

is almost

completely neglected

[3]. ^Note ^that ^this contribution is of great

importance

at _a

comparatively

low

density

_as it is, for

instance,

in _a metallic cluster. The RPAE

equations

may be obtained from the derived

equations

if the

exchange-correlation

part of the effective interaction is substituted

by

the

exchange

Coulomb interaction.

Acknowledgments.

One of the authors

(M.A.)

is

grateful

to the Alexander _von

Humboldt-Stiftung,

whose

help

makes

possible

his stay ^and

performing

research in

Germany.

We thank R.M. Dreizler and E.K.U. Gross for valuable discussions.

References

ill Hohenberg

P. and Kolln W., Pllys. Rev. 136

(1964)

8864.

[2] ^Kohn ^W. ^and ^Sham L., Pllys. Rev. 140

(1965)

A1133.

[3] ^Amusia M.Ya., Atomic Photoeffect

(Plenum

Press, New York and London,

1990).

[4] Runge E. and Gross E.K.U., Pllys. Rev. Lent. 52

(1984)

997.

[5] ^Gross ^E.K.U. ^and ^Kohn W., Density Functional Theory of Many-Fermion Systems, S.B. Trickey Ed. Adv.Quaut.Diem. 21

(1990)

255.

[6] ^Williams ^A.R. ^and vou Barth U., Theory of the Inhomogeneous Electron gas, S. Lundqvist and N.H. March Eds.

(Plenum

Press, New York and London,

1983).

[7] Zangwill A. and Soven P., Pllys. Rev. A 21

(1980)

1561.

[8] Shaginyan V-R-, Preprint LNPI-973, Leningrad, 1984; Solid State Commun. 55

(1985)

9.

[9] ^Khodel ^V,A.and Shaginyan V-R-, Yad. Fiz. 46

(1987)

1638;

Khodel V.A. and

Shaginyan

V-R-, Nucl. Pllys. A 500

(1989)

242.

[10] ^Pines ^D. ^and ^Nozieres Ph., The Theory of Quantum Liquids, vol.i

(Benjamin

Press, N-Y.,

1966).

[11] ^Khodel ^V-A- ^and Saperstein E-E-, Phys. Rep. 92

(1982)

183.

[12] Senatory G. and Subbaswamy K-R-, Phys. Rev. A 35

(1987)

2440.

[13] Landau L-D- and Lifshitz E-M-, Statistical Physics

(Nauka,

Moscow,

1976).

[14] Levy M. and Perdew J-P-, Phys. Rev. A 32

(1985)

2010.

[15]

Shaginyan

V-R-, Preprint PNPI-1781,

St.Petersburg, (1992);

Phys. Rev. A, in _press.

[16] ^Talman ^J-D- ^and ^Shadwick W.F., Phys. Rev. A14

(1976)

36.

(16)

[17] ^Almbladll ^C.-O. ^and von Barth U., Phys. Rev. B 31

(1985)

3231.

[18] Czyz ^W. ^and ^Gotthied K., Auu. Phys. 21

(1963)

47.

[19] Ceperley D. and Alder B., Phys. Rev. Lent. 45

(1980)

566.

[20] ^Amusia ^M.Ya. ^and

Shaginyan

V-R-, UFTP Preprint-307

(1992);

J. Phys. B 25

(1992)

L345.

[21] Shaginyan V.R., unpublished.

[22] ^Dreizler ^R-M- ^and ^Gross E-K-U-, Density Functional Theory