A1134 W. KOHN AND L.J. SHAM where exo(n)is the exchange and correlation energy per Seitz radius and ro is a typical length over which there electron of a uniform electron gas of density n Our sole is an appreciable change in density. In this case, as approximation consists of assuming that (2.3)consti- shown in HK, we can expand the true exchange and tutes an adequate representation of exchange and corre- correlation energy as follows lation effects in the systems under consideration. We shall regard Exe as known from theories of the homo- Ex[n]= Exe(on)n dr geneous electron gas. 6 From the stationary property of Eg. (2. 1)we now obtain, subject to the condition +/c(n)vndr+…,(21) on(rdr=0 (24) the second term in the energy expansion in powers of the equation the gradient operator. In this regime we may similarly [n] a nI on(r)e(r)+ +an(()}ar=0;(2.5) g(=t(y+/ +|)va2ar+…(2.12) and chemical potential of a uniform gas or tribution.(2.7) From HK, expecially Sec. III 2, we have the following kx(n)=d(n∈o(x)/lnt is the exchange and correlation ce to the n(r)n(r) E[n]= +(r)n(r)dr dr dr Equations (2.4)and (2.5) are precisely the same as one obtains from the theory of Ref 3 when applied to a system of noninteracting electrons, moving in the given potential p(r)+xe(n(r). Therefore, for given p and +/0+/()计…,(1 u,one obtains the n(r)which satisfies these equations where simply by solving the one-particle Schrodinger equation go(n)={10(3r2)2+ex(n)}n -号V2+[φ(r)+kx2((r)》(r) and g2)(m)={ex(2)(n)+t(2)(m)}n,(2.15) Since in our approximation (2.3), the V 2 term of n(r)=∑|(r)|2, ( 2.9) Eq.(2. 11)is neglected, it is clear that for a gas of slowly varying density our expression(2.10) for the energy has errors of the order VI, or equivalently, of the order It is physically very satisfactory that Hxe appears in o Eq.(2.8)as an additional effective potential so that Surprisingly, our procedure determines the density gradients of wze lead to forces on the electron fluid in a with greater accuracy, the errors being of order v 4 nanner familiar from thermodynamics. This is shown in Appendix I Equations (2.6)-(2.9)have to be solved sel At this point a comparison of our procedure and th ently One begins with an assumed n(r), co pp(r) from(2.6)and uxe from(2.7), and finds a ts original work does not include correlation effects.? But from(.8)and(2.9). The energy is given by ( r) even the exchange correction is different from ours. To obtain Slater's exchange correction, one may begin by n(r)n(r writing the Hartree-Fock exchange operator in the form =∑E- dr dr of an equivalent potential acting on the hth wave function n(r)lex(n(r))-uxen(r)]dr.(2.10) 4(2/(*(m( The results of our procedure are exact in two limiting cases ψk*(r)y(r),(2.16) ent to the original paper by Slater, there have been (a) Slowly varying density. This regime is character ized by the condition ra/ro<1, where r, is the wigner ry Excitations in Solids (1964);S. Lundqvis min, Inc, New York, 1963) Ufford, Phys. Rev. 139AN SH~M d. ro» a yp th over which sit . In t»s ical lengt ' se as it.z, ra ius l hange in dens ty hange and n appreciab e c nd the true exc is an can expan ene gy a F,fej= a .(e)e. dr c ('&(e))Vn('dr+ ~ ~ ~, (2.11) A &&34 prrelation ene e istheexcha g gy Pel f densltyn. O n e and corre ur sole where &x.( ) . iectron gas 2 3) consti￾f a uniform e e that electron o ts pf assum~~g nd corre￾l matipn con pf exchang apprpxima resentatipn o . tion. e e uaterep consl era tutes ana q . s stems un e the homo￾cts in the sy theories lation e I,npwn from 2.1) we now eiectron g ' rty of Eq' gene From th ' statio ary prop obtain, subject to th,e ,o„dition the equation be(r) dr=0, (2.4) e and correlation p ortion o the n rg xpansi ansion in powers o expand T, e in be(r) q(r)+ t(„. n ()) d 0 ( ) r =, 2. here e(r') ~(r) =~(r)+ dr', (2 6) T,Ln) = —,' (3x'e)'"e dr t('&(n)iVei'dr+ . (2.12) 2 From HK expecially Sec.III 2,, we have the following expression for the energy. and ) + correlation c ' is the exc to th o e obtain om t e e)}e, Therefore or g 3m'e)'"+e .e po p~ si a + go(e) dr+ g22&'&(e) [Ve)'dr+ (2.13) (2.14) ir n r e,(n(r)) n r —t(.(e(r))g r dr. (2.10) .*(r)A '(r')4' (r)A(r' [r—r'[ N w.,(r)=- Z The results of ou r p roce du re are exactt in twow limiting ensi . ime is c aracter￾cases. ' ' ized by n r, ro~(1, where r, is the i P', elementary x ' ' 'n 6F r a review see D. Pines, r Solzds in Inc. , (W. A. Benjamin, ew A*(r)4'(r) (2.16) ter there have been m ts to add corre a R Schrieffer P y s 9 ' on Phys. Rev. 136, S and C W. UGord, Phys. R 139 h&ch sat&sfies these e uat 6„,('& (n)+t(" (n) )e. of the order nd setting 'g 'y e(r)=Z '(r) I', I ' i=1 l ou re determines t e e of o d ivi4. iv is the num er o d and that dd't' l ff pp E. (2.8) as an a ii g familiar from . ns (26—)—(2.9) h to ed mr, g begin by Equatio . — . o the form ne begins wi and fin s ane ter's exc ange o erator in (i 2.6) an pxc i' o (2.7, is iven y -F k h f . 2.9 . The energy is g' writing w the ' ' eHartree- oc f (2.8) and (2.9 . is f equivalent po en function Z=P c,—— 1 —r