A1136 KOHN AND L SHAM The energy is now n(r)n(r) drdr′ fxe(n)=f(n)-fo(n) (3.5) where f and fo are the free energies per electron of an 1frn1〔r)n1(r',r) interacting and noninteracting gas, respectively. dr dr 0=(r)+(GLn]/6n(r)+kx(t(r))-μ,(3.6) where p(r) is given, as before, by Eq.(2.6)and +/n()(s(()-(()a.(2) uxo(n=d(xe(n))/dn (37 This procedure may be regarded as a Hartree-Fock Equation (3.6) is identical to the corresponding equa method corrected for correlation effects. It is no more tion for a system of noninteracting electrons in the complicated than the uncorrected Hartree-Fock method effective potential o+xe. Its solution is therefore de but, because of the nonlocal operator appearing in Eg. termined by the following system of equations (2.22), very much more complicated than the method described in Sec. IIA. Since at least exchange effects 一2V2+g(r)+x(n();=;,(3.8) are now treated exactly we must expect, in general, and more accurate results than from the method of sec. IIA n()=∑|(x)|2/{e()hr+1}.(3. In particular, near the surface of an atom the effective potential now is correctly(1/r) whereas in Sec. IIA it approaches zero much faster. Even here, however, u is determined as usual by the total number of particles correlation effects are not correctly described near the from Eq.(3.9). This value also represents our approxi- mation for the chemical potential of the interacting III FREE ENERGY: SPECIFIC HEAT Of special interest for metals and alloys is the low- temperature heat capacity. This may be obtained by We can generalize the consideration of the ground making an expansion, in powers of T, of the above state to finite temperature ensembles by using the finite system of equations. An equivalent, but more con temperature generalization of Eq.(2.1)given by venient, method is as follows: From thermodynamics Mermin. He has shown that the grand canonical po- and Eq.( 3. 1)we have tential can be written in the form 1n〔)n(r S[n]=--(+N)v pr+ =|v()n()dr+ dr dr (r) +Gn]-/n()ar,(1) (3.10) where GLn] is a unique functional of the density at a The integral vanishes because of the stationary property given temperature and u is the chemical potential. For of &, so that the correct n this quantity is a minimum S[n]=-(aG[]/ar)n(r).v (3.11) In analogy with(2.2)we now write The same argument, applied to a system of noninter GLn]=Ga[nI+Fxon]; (3.2)acting electrons of density n(r),gives here G[n]=T[n]-7S[] SLn]=-(0G[n/07) where T[n] and s[n] are, respectively, the kinetic Combining Eqs.(3.11),(3.12),(3.2),and(3.4),we energy and entropy of noninteracting electrons with density n(r)at a temperature T; and Fx[ tion, the exchange and correlation contribution to the free energy. For the latter quantity, we make the For small r it is well known that S, is given by fxeLn]= n(r)x(o(r)dr (34) S[n]=N3TkTge(uy rhere g, is the single-pan of states in the where fxe(n)is the exchange and correlation contribu- effective potential p tion to the free energy per electron of a uniform electron (afro(n)/ar)n(r), y=3m" -[g(uA(n)-go(uo(n)), 9 N. D. Mermin, Phys. Rev. 137, A1441 (1965).K AND L' I gHAM The energy is now 1V' E=P p,—— 1 e(r)e(r') ei(r,r')ei (r',r (3.7) (3.9) III. FREE ENERGY; SPECIFIC HEAT (3.10) &&36 as of de»ity 0 n (3.5) are the free energie Per elect.ron of an 1 . ~here f and fp are t gas respective y. ~ nd noninteracting g ( )+(bG [e7/be(r))+I)~p(e 1 is given, as before, b Fq. (2.6) and )=q( f ())/d". dr. (2' ) "-" ndln eq ' p qg p (e(r))—pp(e( )) ]. to the corresPon e 1' &o 3 6) is identlca ' electr()ns in the k Equation ~ f noninteracting a beregar e as ion a Hartree-Foc t' for a, system o +~-' 1"""1 tion is therefo This procedure may Beets. It is no mo effective potent W . tern of equat o eth()d ( orrected o t d Hartree-Foc&»e ter»ined by t { ~, (3.8) plicated than the nn 1 tor appearing» l i~2+ &(r)+pxo(e(r)))~' N i 2/{ (e(—p)/pe+ 1) described in 1 we must expect, n&= j,=1 n.ow from the m ctive r of a,rticles more e surface o an accurate resu ts a f atom the effec iv d as usua by th t talnumbero P roxi￾rticular, near the p is determined ' teracting proaches zero muc hf str . E n here, howeve, tl described near t e i p allo s is the low￾correlation e ec s surface. p n ex ansion, in po ut more con￾tion o gr g h 6 A 1 ad namics re ensembles by using t e ni temperature generalization o E . (2.1) i m b q that the gran can 8G ' He has shown t a 5 e7= (0+—1)c—'(T)—v=- ~()+ be(r) tential can be wri en BT ()e(r) (BG[e7 +G[e7—) e(r) dr,r 3.1) ni ue functiona 1 of thee density at a h h ' gi " perature g and p is t e c e 1 rrect n this quan i y In analogy with (2 2) e now write G[e7=G,[e7+F,[e7; G,[e7=—T,[e7—rS,[e7, (3.3) Ln ~ are respectiveely, the kinetic 1 to th where, n energy a (1 t, py of noninteracting e R tion, ge an corre 1a tion o tib to free energy. For te approximation (3.2) here ~-[ 7= ()f*.( ())d, (3 4) on to the h s. Rev. 137, A1441 (19651. 9 N. D. Mermin, Phys. Rev. where, n an e and correlation contribu￾ti free energy perr electron e o auniUo 1 The integral i vanishishes because of thee stationary property 3.11 of 0, so that ~~ n(r), V ~ 5[e7=—(()G[e7/ ) ) ment, applied to a system of noni oninter￾o ' ' Coinbining Eqs.s. (3.11, 3.12), (3.2), and (3.4), we obtai n 5 e7= 5 [e7+ e(r) (Bf (nn 8787' ~(,( ),vdr. (3.13) ell known that 5, ' g' For small r it is we is iven by 5,[e7=Ã-', x'k'~g, (p), (3.14) le- article densityit of states in the (bf..()/~ ).„.=-.-'ir'k'r[g - . (d(o,(e))—gp(((p e (3.15)