B866 P. HOHENBERG AND W. KOHN FLn] the classical Coulomb energy and write and 1 n(r)(r) f(rdr=0 (23) FLn] drdr+GLnT (14) Here we clearly must have a formal expansion of the so that E[n becomes 1 n(r)n(r) E[n]=|(r)n()dr+ 2/r- drdr+GLn],(15)GLn]=GLno]+ K(r-r)n(r)(r)drdr here G[n] is a universal functional like FL. L(r,r, r")i(r)i(r)n(rdrdr'dr (24) Now from the definition of F[n], Eg.(9), and G[n1, g.(14), we see that In this equation there is no term linear in i(r)since 1/C2(r, r) by translational invariance the coefficient of i(r)would G[n]=- V,r1(r,r,)lrerdr+- drdr.(16) be independent of readi ding to zero, by(23). The kernel appearing in the quadratic term is a functional of r-r'l Here n(r, r)is the one-particle density matrix; and only and may therefore be written as (r, r) is the two-particle correlation function defined in terms of the one-and two-particle density matrices as K(r-r)=(1/9∑K(q)e,(rr).(25) C(r, r)=n2(r,r;r,r)-n(r, r)n(r, r).(17) The higher order terms will not be further discussed Of course n1(r, r)=n(r) One may also quite trivially introduce a density From(16)we see that we can define an energy-density function grLn]=2V,Vrn1(r,r) er gLn]=8(n0)+/K(r)(r+r)n(r-r)dr+…,(20) 1C2(r-r/2;r+r/2) dr'(18)where go(no) is the density function of a unifor (kinetic, exchange such that energy). GEn]=g[n]dr (19) 2. Expression of the Kernel K in Terms of the Electronic Polarizability The fact that gr n] is a functional of n follows of course We shall now see that the kernel K appearing from the fact that y and hence ni and ne are Eqs.(24)and(26)is completely and exactly expressible It should be remarked, that wl while G[n] is a unique in terms of the electronic polarizability a(q). The latter functional of n, gr[n] is of course not the only possible is defined as follows: Consider an electron gas of mean energy-density functional. Clearly the functional small additional positive external-charge density 10 Write the electronic density, to first order in A, where the h() are entirely arbitrary, give equivalen A2∑b1(q)e results when used in conjunction with(19) Then eal with GIn] and grLn] in Let us now define the operator II. THE GAS OF ALMOST CONSTANT DENSITY 1. Form of the Functionals G[n] and g[n] (21) operators. Then, by first-order perturbation heory, O We consider here a gas whose density has the form where ck", Gx are the usual creation and annihilate n(r)=#0+(r) with a(g)(0 pan)(n e-glo) b1(q)=-(8丌)∑ (31)P. HOHEN BERG AND W. KOHN F[n] the classical Coulomb energy and write and 1 F[n]=— 2 so that E,.[n] becomes iz (r)n (r') drdr'+ G[n], l r—r'l (14) R(r)dr=0 (23) Here we clearly must have a formal expansion of the following sort: 1 n (r)n (r') E,„[n]= p (r)n(r) dr+ — drdr'+G[n], (15) G[n]=G[np]+ E(r—r')R (r)R (r') drdr' l r—r'l Cz(r,r')=nz(r, r'; r,r') —nz(r, r)nz(r', r'). (17) Of course nz(r, r)=—n(r). From (16) we see that we can define an energy-density functional g p[n] = z V~V~ 1zz(r)r ) l g—~ where G[n] is a universal functional like F[n]. Now from the definition of F[n], Eq. (9), and G[n], Eq. (14), we see that 1 C,(r,r') G[n]=— V,V,.n~(r, r') l, , dr+ — drdr'. (16) l r—r'l Here n, (r,r') is the one-particle density matrix; and Cz(r,r') is the two-particle correlation function defined in terms of the one- and two-particle density matrices as + I(r,r',r") R(r) R(r') R(r"} dr dr' dr"+ . . (24) ln this equation there is no term linear in R(r) since by translational invariance the coefficient of R(r) would be independent of r leading to zero, by (23). The kernel appearing in the quadratic term is a functional of l r—r l only and may therefore be written as It(r—r')=(1/~l)Z It(q)e "' "' (25) g,[n]=gp(np)+ IC(r')R(r+-,'r')R(r ——, 'r')dr'+, (26) The higher order terms will not be further discussed here. One may also quite trivially introduce a density function such that 1 C,(r—r'/2; r+r'/2) dr' (18) where gp(np) is the density function of a uniform gas of electron density np (kinetic, exchange, and correlation energy). G[n]= g,[n]dr. The fact that g,[n] is a functional of n follows of course from the fact that 4' and hence e~ and e2 are. It should be remarked, that while G[n] is a unique functional of n, g,[n] is of course not the only possible energy-density functional. Clearly the functionals 2. Expression of the Kernel Xin Terms of the Electronic Polarizability We shall now see that the kernel IC appearing in Eqs. (24) and (26) is completely and exactly expressible in terms of the electronic polarizability n(q). The latter is defined as follows: Consider an electron gas of mean density eo in a background of uniform charge plus a small additional positive external-charge density i9 g,[n]=g,[n]yP ts, &'&[n], (20) n.„,(r) = (X/Q)g a(q) e-'q'. Write the electronic density, to first order in X, as (27) where the h~') are entirely arbitrary, give equivalent results when used in conjunction with (19). The following sections deal with G[n] and g,[n] in some simple cases. n(r) =np+ ()/Q)p b,(q)e-'q'. ~(V)—=bz(a)/~(a). Let us now define the operator (28) (29) n(r) =np+R(r), (21) R(r)/n, «1 (22) II. THE GAS OF ALMOST CONSTANT DENSITY 1. Form of the Functionals G[n] and g„[n] We consider here a gas whose density has the form Pq=g Ck q Ck, k (3o) ~(v) (oI pql n)(nip-. l o) b~(q) =—(8~) (31) where c~*, c~ are the usual creation and annihilation operators. Then, by first-order perturbation theory