B868 P. HOHENBERG AND W. KOHN the Thomas-Fermi equation 2. The Gradient Expansion It is well k of the Thomas-Fermi equation is that n(r)must be a slowly varying function of r. This suggests study of th functional G[n], where n has the form (r=p(rro with It is obvious that this is quite a different class of systems than that considered in Part II (n=no+h, i/no<<1) III. THE GAS OF SLOWLY VARYING DENSITY since now we shall allow to have substantial varia- 1. The Thomas-Fermi equa aton tions. On the other hand, whereas in Part Il, i could contain arbitrarily short wavelengths, these are here For a first orientation we shall derive, from our general ruled out as ro becomes large variational principle, the elementary Thomas-Fermi We now make the basic assumption that for large ro, equation. For this purpose, we use the functional (18) the partial energy density g [n] may be expanded in and in(16)we neglect exchange and correlation effects, the form thus setting C2=0. We approximate the kinetic-energy term by its form for a free-electron gas, i.e. gr[n]=go(n(r))+2 g(n(r).Vn(r) g-[n]=To[kr(n)Pn (50) where the Fermi momentum k is given by +2 Lgi. d, 1)(n(r). vn(r) n(r) (51) +g,2)(n(r)Vvm(r)]+…(61) This results Here successive terms correspond to successive negative n(r)n(r) powers of the scale parameter ro. Quant E[n]= s(r)(r)dr+- - drdr go(n(r)), gi(n(r))etc, are functions (not functionals r-r, of n(r). No general proof of the existence of such an expansion is known to us, although it can be formally +io(3r)ia/[o(r)]8dr. (52) verified in special cases, e g-, when G[n(r)] can be ex- To determine n(r)we now set we know that, for a finite ro, the series does not strictly converge(see the discussion at the end of Sec. I1.3) may expect it to be useful (in the sense 6{En[n]-4/n(r)lr}=0 (53)totic convergence)for sufficiently large values of ro Now a good deal of progress can be made, using only where u is a Lagrange parameter. This results in the the fact that gi[n] is a universal functional of n independent of v (r). This requires gr[n] to be invariant under rotations about r. The coefficients g v(raf mdr+1(3T)/Ln(r)/3-H=O.(54)under rotations. Hence one finds by elementary being functions of the scalar n, are of course invariant siderations that gr[n] must have the form If we now introduce the "internal"potential gr[n]=80(n)+[g2)(x)V2n+g2()(n)(Vnwn)] +terms of order VA.(62) r (55) A further simplification results from the fact that we may eliminate from gr[n] an arbitrary divergence (54)is equivalent to the pair of equations 2iVhr'ln(see the end of Sec. 1. 3). It is then elemer n(r)=(1/3m)(2[u-o(r)-0(r)J)/,(56)tary to show that grLn]may be replaced by vvi (r)=-4rn(r (57)grLn]=go(0 )+g2 2)(n)Vn.Vn +{g42)()(V2n)(V2n)+g4)(n)(Vn)(VnV rom(56)and(57)we can eliminate n(r)and arrive at +g4(n)(Vn·Vn)2}+O(V).(63)P. HOHEN BERG AN D W. KOHN the Thomas-Fermi equation V'v;(r) = (—2 &'/37r)f p —v(r) —v, (r)5'& (58) 10- FxG. 2. Behavior of the kernel E'(q), as a function of q (electronic density =4 &10"cm 3). 2. The Gradient Expansion It is well known that one condition for the validity of the Thomas-Fermi equation is that ri(r) must be a slowly varying function of r. This suggests study of the functional Gfej, where e has the form 0 0 with N(r) = y(r/rp), ro~~ . (59) (60) III. THE GAS OF SLOWLY VARYING DENSITY 1. The Thomas-Fermi Equation For a erst orientation we shall derive, from our general variational principle, the elementary Thomas-Fermi equation. For this purpose, we use the functional (18) and in (16) we neglect exchange and correlation effects, thus setting C2=0. We approximate the kinetic-energy term by its form for a free-electron gas, i.e. , It is obvious that this is quite a di6erent class of systems than that considered in Part 11 (N=ep+n, 8/Np«1), since now we shall allow q to have substantial varia￾tions. On the other hand, whereas in Part II, rI, could contain arbitrarily short wavelengths, these are here ruled out as r0 becomes large. We now make the basic assumption that for large r0, the partial energy density g,fnj may be expanded in the form g,f&i]=gp(N(r))+g g,(n(r)) Vps(r) g f&5= i'oft~—(~)]'~, where the Fermi momentum kl: is given by k p(n) = (37r'e)'~'. (50) (51) +Z Lg ""(~(r)) V'~(r)V~(r) +g;,&'&(n(r)) V,V,&i(r)]+ . . (61) This results in 1 &i(r)&p(r') Z„fe]= v(r)e(r)dr+- drdr' 2 fr—r'f +r'p (3~')'" f~(r)]""«(52) To determine e(r) we now set 8 E„fe]—&i e(r)dr =0, (53) where p, is a Lagrange parameter. This results in the equation m(r') v(r)+ dr'+-', (3 r')7'"fm(r)]' ' &Ii=0 —(54). /r —r'f If we now introduce the "internal" potential n (r') v, (r)—= dr', (55) (56) (57) (54) is equivalent to the pair of equations N(r) = (1/3m ){2'—v(r) —v, (r)5)'&' VPv;(r) =—4v 0(r) . From (56) and (57) we can eliminate m(r) and arrive at Here successive terms correspond to successive negative powers of the scale parameter rp. Quantities like gp(e(r)), g;(n(r)) etc. , are functions (not functionals) of N(r). No general proof of the existence of such an expansion is known to us, although it can be formally verified in special cases, e.g., when Gfe(r) 5 can be ex￾panded in powers of fe(r)—Npj. At. the same time, we know that, for a finite r0, the series does not strictly converge (see the discussion at the end of Sec. II.3), but we may expect it to be useful (in the sense of asymp￾totic convergence) for suKciently large values of rp. Now a good deal of progress can be made, using only the fact that g,flj is a universal functional of n, independent of v(r). This requires g,fej to be invariant under rotations about r. The coeKcients g, ;, (n(r)), being functions of the scalar e, are of course invariant under rotations. Hence one 6nds by elementary con￾siderations that g,fnj must have the form g,fnj= gp(n)+ fgp&'(e) V'm+gpt &(n)(VN Vn)5 +terms of order V~4. (62) A further simpli6cation results from the fact that we may eliminate from g,fej an arbitrary divergence Q,V,h, 'fN] (see the end of Sec. I.3).It is then elemen￾tary to show that g,fnj may be replaced by g,f&r j=gp(e)+gp "&(N)Ve V&p +{g"&(e)(V'e)(V'I)+g i'&(n)(V'&i)(VN VN) +g4'4&(e) (VN Ve)')+O(V P). (63)