Theory of disordered Coulomb system: The COPW method for localized and delocalized states in liquid metals

Solid State Comunications, Printed in Great Britain.

Vo1.56,No.l,

pp.29-34,

1985.

0038-1098/85 $3.00 + .OO Pergamon Press Ltd.

THEO= OF DISORDEREDCOULOMB SYS!CEM: THE COPW MEZHOD FOR LOCALIZEDAHD DELOCALIZED STATES IIVLIQUID M.Bl!ALZ V.B.Bobrov and S.A.Trigger Institute of High Temperatures, Academy of Sciences of the USSR Izhorskaya lj/lB, Moecow 127412, USSR ( Received 27 June 1985 by G.S.Zhdanov ) Effective self-consistent Hamiltonian for the electron subsystem in random Coulomb field8 of nuclei is constructed by means of the completely orthogonalized plane-wave basis. Localized electron states are described in the one-centre approximation with due account of the medium influence. The concept of the electron-ion pseudopotential is employed for description of the delocalized states. An explicit expression is obtained for the effective ion-ion interaction.

Considerable advances have been made in theory of simple met;l; using the pseudopotential approach ’ . However, the elaborated pseudopotential method is faced with difficulties when dealing with electronic properties of non-simple metals. Employing the COPW basis, we show in the present work that pseudopotentials of electron-ion and electronelectron interaction8 arise in considering of delocalized electron states. The self-consistent scheme proposed here for determination of delocalized electron state8 makes it possible to resolve an uncertainty in constructing the COPW basis. The problem is stated as follows. There is a system of N,,,, nuclei of a charge Zcore, and N, electrons, where Z

coreNcore

electron subsystem by means of the second-quantization formalism, we use a set of one-electron states which are divided in two groups: localized (locstates) which are similar to atomic bound states and specified by a set of quantum numbers n and a spin number 6, and delocalized (free-statea) where an electron can move throughout the whole system. The delocalized states are specified with a set of quantum numbers % and the spin number6(the meaning of the quantum numbers nand % will be explained below 1. To construct an appropriate complete set of functions we use a formalism3 proposed to construct the COPW basis for investigation of electron states in solids. The COPW basis is composed as follows. A number of states \cn> which is equal to the number of localized states ie removed from the plane-wave basis I:> . The remaining plane waves are denoted The COPW basis contains by lx>

= N,

The subsystem of nuclei will be considered in the framework of classical statistical mechanics. Consequently, the electron subsystem is in a field of disordered nuclei. Analysing the

l

I n>

1% 29

=tQ

- localized states,

- gcnk(ln>

-Ii;,,),(2)

30

THEORY OF DISORDERED COULOMB SYSTEM.

- the delocalized COPW states. The COPW states must be orthogonal to the localized states, so the condition

= 0,

Vn,

it ;

Z#Zn,

results in an equation determining the coefficients cnk, yM,=&llll,_(n~n,> One can prove that the COPW tasis is complete and orthonormalizedJ. We will assume that the COPW basis and the plane-wave basis are related by

Vol. 56, No. I

problem considered. As is known, the best one-particle approximation is the Hartree - Pock method'. Therefore we adopt the Hartree - Fock approximation in the analysis of interactions between electrons in localized states and in localized and COPW states. The interaction between electrons in the COPW states will be described exactly at the present stage. Under the above mentioned aesumptions, the electron subsystem Hamiltonian, up to an additive constant, is

Geff e

+ I 28

Z W,,(iTl(g6a))a$ q,tkaJ I61a;22 6 '833 6 "I?6 44 + '

a unitary traneformation 2 , S>

= 21:)

; In) = 21gn> .

(3)

seen from the definition of the COPW basis in (2), the choice of statevectors lZn> is not unique. An apparently reasonable assumption is \cn>>>kF, 60 physically relevant states are not removed from the analysis. Besides, As

2+(

; +

Gel)

; =

G+ Gei

(4)

are the electron where a& and 46 creation operators in the states \n6> respectively, W,i is a nonand Ike>, local pseudopotential describing interaction of COPW-state electrons with ions (i.e. nuclei with electrons localized near them), Wee is a COPW-state electron interaction pseudopotential. Explicitly, (5)

L thie assumption enables one to neglect a difference between the set of plane waves Is> and the plane-wave basis 16 when evaluating real functiona. The construction of the COPW basis is applicable to an arbitrary set of orthonormal localized states In) , its choice depends on the physical

Here $ is the kinetic energy operator, veC is the Coulomb electron-nucleus interaction, vee(q) = 4TLe2/q2 is the Fourier transformation of the Coulomb electron-electron interaction potential,

Vol. 56, No. I

31

THEORY OF DISORDERED COULOMB SYSTEM

localized electron deneity, averaged over the electron subsyetem ( iy 1, f(6 1 - the Fermi - Dirao energy distribution. The electron energy in a localized state is determined by the equation

AB seen in equalities (5)-(71, the pseudopotentials W,i and Wee are noalocal, 80 a further analysis is too complioated. Therefore it is reasonable model local pseudopotentials to use which ha8 been applied SUCCeSSfUlly in

where jsfree(?) is the electron denaity in the COPW states, averaged over the electron subey~tem. Equation (8) may have solution6 describing delocalized states; such aolutionlsare discarded. Note that in equations (4)-(g)

the theory of simple metals5r6.

is the

The

definition of the model local pseudopotentials ia a substitution,

+ wei(q)

w,,ct;{~a) 1-c

9

W,,( 9)

we omitted the Fock term in the electron-electron interaction in order to make the exposition free of cumbersome calculations. Let LIBconsider equalities (5)-(7) in some detail. Here we have eleotronion and electron-electron pSeUdOpotentials for delocalized electrons, Wei

and Wee, derived within the scheme adopted.

Though the evaluation of the

paeUdOpOtential8

for a given Bet of lo-

calized states io a difficult oalculaMona1

problem when treated consistently

and accurately,

the significance of the

This is not a rigorous Operation, 80 the behaviour of W,i(q) and Wee(q) cannot be unambigoue, but stems from reasonable physical assumptions with introduction

of fitting parameters.

The standard aasumption is Wee(q) =

v,,(q) ( possible q-dependence of W,,(q) has not been previously diecussed in the literature, though the existence of the pseudopotential wa8 mentioned7). Thus the effective Hemiltonian (4) is reduced to

GEff =C Ena&pna+ & q&q6 ++ g6 n6 +I 2v

): q,W)

‘ee(

Wee

Wei(3)a&p~+~ , 6

+c (IO) q)a~~Ia~62a~-~,62a~I+~,61 Let us atudy equation (8) for a

pseudopotential W,i i8 mainly in that it is a Justification of the almost-free

self-consistent

electron model in the description of the

calized etates. It ie impossible to

delocalized

states, where W,i appear8 as

a perturbation

potential.

+

determination

of lo-

find exact electron states in a random field of many scatterers, 80 one

32

THEORY OF DISORDERED COULOMB SYSTEM

needs to simplify the problem with account of the specific physical aim. In the study of liquid metals, as well as for some other systemof particles with the Coulomb interaction, one can discard molecular (and some other multi-center > localized electron states. In this case it is auffi-

Vol. 56, No. 1

w En0 are localized at different points of the system. In this respect, there is a degeneracy in B,, so this quantity can be treated as an additional "quantum number" 8 . Now we turn to determination of the effective one-center potential. In the basic approximation one has

where

&oc!k) =s

cient to consider the electron states with discrete energy levels in an effective potential (to be determined appropriately) of a separate nucleus;

1; + I

~ C V

(II’)

$dgexp(-is)(
j=I k

( -

Perturbation theory in the pseudopotential Wei is employed to evaluate F free(')* The resulting equation for the localized states is

~Zi(k) +Wei(k>Vee(k)free(k))

x exdik(r-Rj))) specified with a set of quantum numbers no, corresponding to a bound state in a potential with the central , symmetry, the spin quantum number and a parameter which is the radiusvector of the localization centre, l

X

EneIn6)

(12)

where Q(k) = Zcore-~loc(k), and Zi(0) = Nfree/Ncore can be considered as the charge of a l'point-likell ion; is the density-density linear free(k) 3 response function for homogeneous electron gas in the compensating positive background. To determine states localized at the 1-th nucleus we use the following transformation:

I

,$Teexp(iSj) - exp(iG7-I)+<$loreexp(i~~)>core =s

core(k)exp(i~l)+ <2W3f’core&(i!)

=

(13)

where S

gcore(r) cwe(k) = 1 +BcoreSdrfexp(-im(

( j is the nucleus number ), since the electron states with a given ener-

I )

(13’)

is the statical structure factor, gcoretr) is the pair distribution for

THEORY OF DISORDERED COULOMB SYSTEM

Vol. 56, No. 1

the subsystem of nuclei, 'Q core = B ,_ore/V,(...> cOre stands for averaging over positions of nuclei. With the account of (12),(13) one has

d3k 3

33

An investigation beyond the onecentre approximation is possible by means of the perturbation theory in the fluctuating field which is proportional to the difference

exp(ii;(~-~l))Score(k)vloc(k)} =

(14)

=&&j&6) where vloc(k)

= - =$Zi(k)

+ weiWvee(JdJ$ree(k)

Equation (14) is a way to aelf-consietent determination of localized electron states in the one-centre approximation with due account of the medium influence. It is easily seen that in the extreme rarefication limit and with the Pock term added to the Hamiltonian equation (14) is reduced to the Hartree Fock equation for a single atom. The effects of the medium, a screening due to localized electrons and a shortrange order in the system, are responsible for a short-range behaviour of the effective nuclear potential, so the number of states localized at a single

eff u core(EI - 82) = -$g

V

eff core(k) = -+;(k)-

(14’?

&e exp(iGj)

-(&oreexp(ij;iij)>core. j#l

As seen from (14), to find the localized states one needs an information on the structure factor Score(k) and the eleotron chemical potential_Ele. Determination of the free energy of the electron subsystem with the Hamiltonian Giff, in the one-centre approximation for the localized states and using the second-order perturbation theory In the pseudopotential Wei, enables one to get the structure part of the effective inter-nuclear interaction potential:

v~~~,(k)exp(ii;(a',~I - $>),

2vloc(k)3jloc(k)+ Iwei(k$Xfree(k)

(15)

(15’)

I

nucleus is finite. Note also that a change in the system thermodynamical parameters, i.e. its temperature T=l/ B and the density of nuclei 9 core' induces variations in the energy levels of the localized states. Probably, equation (14) provides a fairly effective description of the Yott transition.

It is evident from (15) that U$ce is a short-range potential, so the methods of theory of simple liquids are adequate for the determination of Score(k). The chemical potentialye is obtained in the one-centre approximation for the localized states and neglecting the pseudopotential W,i. The result is

34

N,

THEORY OF DISORDERED COULOMB SYSTEM

1 + exP(j3 ( a2m

=

Lpe)))-I * N,,, &(I

Vol. 56, No. 1

+ exp(p(&nO-jje))F1

e

(16) is a contribution to the whereR 2' thermodynamical potential of homogeneous electron gas from the exchangecorrelation effectls'. in summary, equations (14)-(16) are a cloeed eyatem of eelf-consistent relations which determine the localized state8 in the one-centre approximation with a model pseudopotential W,i. Thus

we resolve an uncertainty in the choice of localized states in oonstruoting the COPW baeis (Z), and the effective Hamiltonian ieff in (10) is determined compe letely. The pertuzbation theory in the interaction a? = R, - ;tzffcan be constructed to find concrete properties of systems under investigation.

REFERENCES t. 2. 3. 4. 5. 6. 7.

a. 9.

"Pseudopotentials in the Theory of Metals" HBRRISON, W.A., (W.A.Ben$amin, New York - Amsterdam, 1966). HEIWE, V., COHEN, M.L. & WEAIRE, D., "The Pseudopotential Concept" Solid State Physics, vol.24 (Academic Press* New York - London, 1970). GIRARDEAU, ti.D., J.Math. Phys. 12, 165 (1971). Methods for Many-Particle Systems" KIRZHBITS, D.A., *'Field-Theoretical (in Russian) (Gosatomizdat, MOBCOW, 1963). WAWG, S. & SO, C.B., J. Phys. F: kietalPhys. 1, 1439 (1979). ASHCROFT, N.M. i%STROUD, D., Solid dt. Phys. 2, 1 (1978). BASSANI, F., ROBIBSOW, J., GOODMAN, B. & SCHRIEFFER, J.R., Phys. Rev. 127, 1969 (1962). BONCH-BRUEVICH, V.L., "Electron Theory of Disordered Semiconductors" (in Ruesian) (Nauka, MOBCOW, 1981). GUPTA, U. & RAJAGOPAL, A.K., Phys. Rev. g, 2792 (1980).