B870 P, HOHENBERG AND W, KOHN Both of these conditions are necessary. For while( 81) where r, is the radius of igner-Seitz sphere defined would admit the case of a nearly uniform gas with a by small but short-wavelength nonuniformity similar cases are excluded by(82), as they must be. This expression is believed to be reasonably accurate 4. Partial Summation of Gradient Expansion only for ras1. At lower densities, such as occur in metals (25ras5), various approximate expressions In the preceding section we have expressed the coef- have been proposed. One is due to Wigner ficient gu(2)in terms of the expansion coefficient G, of the polarizability a(a), Eq(76). However, we may apply 2.210.9160.88 the expression(63)to the special case of the gas of almost constant density, discussed in Part II. This shows that the leading term go(n) and the subsequent sub- Other approximations are dr series involving coefficients gu/(n)may be summed to and Pines, 7 and Gaskell ls ue to Hubbard, 16Nozieres 点[n]=B()+/kn)Dm+)-m b.g2)(n) These coefficients are all determined in terms of the [n(r-r)-n(r)]dr+…(83) polarizability, a(a). For this latter quantity vailable, at present, a random- phase e ly from terms of the form of a di (41), which gives or of order in the superscript v of gn(). Here Kn(n(r)=-∑ d (2) The form(83)of g, has the merit of being exact in botl (89) limiting cases where either the density has everywhere 4丌24kr2k2 nearly the same value(see Part II)or is slowly varying 11 Its quantitative value for calculating the electronic structure of actual atomic, molecular, or solid-state 4180kx2k systems is at present uncertain but is being examined. Inclusion of the first of these in the energy expression it will, unlike the simple Thomas-Fermi theory, yield: agrees with a correction to the Thomas-Fermi energy (1)a finite density at the nucleus, and (2)oscillations An expression for a(g), allowing in an approximate in the charge density corresponding to shell structure manner for exchange effects has been proposed by Hubbard. 6 It is 5. Approximate Expressions for the Coefficients q2\.q2 In the previous section we have expressed the coef ficients gu) appearing in the gra udient expansion(63) in terms of properties of the uniform electron gas. We where S(a) is defined in Eq.(43). This form yields now collect some results of existing calculations refer- ring to the uniform electron gas which are useful for 6 ur present purposes. For typical metallic densities this has the opposite sign from the random-phase approximation expression(88 This is the sum of the kinetic +exchange+ce Thus we see that the lowest nonvanishing gradient ce nergy density of a uniform gas of density n ection to the Thomas-Fermi theory depends quite has available the high-density expansion of ge sensitively on refinements in the theory of the electronic polarizability, a(g) o(n)= 2.210916 +002m-006+0()}n,(5) 16 E P. Wigner, Phys. Rev. 40, 1002(19 ondon)A243,336(1957) 1: P. Nozieres and D. Pines, Phys. Rev. lll, 442(1958) 18 T. Gaskell, Proc. Phys. Soc. (London)77, 1182(1961);80, Gell-Mann and K. Brueckner, Phys. Rev. 106, 364(1957). 1091(1962P. HOHEN BERG AND AV. KOHN Both of these conditions are necessary. For while (81) would admit the case of a nearly uniform gas with a small but short-wavelength nonuniformity, this and similar cases are excluded by (82), as they must be. 4. Partial Summation of Gradient Expansion In the preceding section we have expressed the coef￾ficient g„(2& in terms of the expansion coefficient c, of the polarizabilityn(q), Eq. (76).However, we may apply the expression (63) to the special case of the gas of almost constant density, discussed in Part II.This shows that the leading term gp(n) and the subsequent sub￾series involving coeAicients g„("(n) may be summed to yield g,[n-t=gp(n(r))+ A. „(,i(r')[n(r+ —',r') —n(r) j where r, is the radius of the Wigncr-Seitz sphere defined by s4vrr,s= 1/n. (86) Other approximations are due to Hubbard, " Nozieres and Pines " and Gaskell " b. g„"'(n) This expression is believed to be reasonably accurate only for r,(1. At lower densities, such as occur in metals (2&r,&5), various approximate expressions have been proposed. One is due to Wigner" 2.21 0.916 0.88 gp(n)- — —— — n. r.' r, r,+7 8. These coefficients are all determined in terms of the electronic polarizability, n(q). For this latter quantity there is available, at present, a random-phase expres￾sion, Eq. (41), which gives y [n(r—-',r') —n(r)]dr'+ . (83) apart possibly from terms of the form of a divergence or of higher order in the superscript v of g ~'~. Here 27( n(q) = 1+ $(q) kg.' kp 2 (88) 1 2' 1 &.(.) (r') =—2 —, . ~—7q ~ r' en(r) (1) (84) and g ('& 1 1 7 4a 24 kc'kI;2 (89) g4(2) 1 1 (90) 4' 18O k„'kI, ' Inclusion of the erst of these in the energy expression agrees with a correction to the Thomas-Fermi energy functional derived by Kompaneets and Pavlovskii. ' An expression for n(q), allowing in an approximate manner for exchange effects has been proposed by Hubbard. " It is The form (83) of g„has the merit of being exact in both limiting cases where either the density has everywhere nearly the same value (see Part II) or is slowly varying. Its quantitative value for calculating the electronic structure of actual atomic, molecular, or solid-state systems is at present uncertain but. is being exaniined. However, it is already clear that if applied to an atom it will, unlike the simple Thomas-Fermi theory, yield: (1) a finite density at the nucleus, and (2) oscillations in the charge density corresponding to shell structure. S. Approximate Expressions for the CoefBcients of the Gradient Expansion In the previous section we have expressed the coef￾ficients g„("i appearing in the gradient expansion (63) in terms of properties of the uniform electron gas. We now collect some results of existing calculations refer￾ring to the uniform electron gas which are useful for our present purposes. 2 q2 1 n(q) = 1+— + 5(q) 2 q'+kg' kz' where 5(q) is defined in Eq. (43). This form yields 4x 24 kg 2kp2 k p' (91) (92) gp(n) = 2.21 O.91.6 +0.062 lnr, —0.096+0(r,) n, (85) rs '4 M. Gell-Mann and K. Brueckner, Phys. Rev. 106, 364 (1957). a., gp(n) This is the sum of the kinetic+exchange+correlation energy density of a uniform gas of density m. Here one has available the high-density expansion of Gell-Mann and Brueckner'4; For typical metallic densities this has the opposite sign from the random-phase approximation expression (88). Thus we see that the lowest nonvanishing gradient cor￾rection to the Thomas-Fermi theory depends quite sensitively on refinements in the theory of the electronic polarizability, u(q). "E.P. Wigner, Phys Rev. 40, 10.02 (1934). "J.Hubbard, Proc. Roy. Soc. (London) A243, 336 (1957). z7 P. Nozieres and D. Pines, Phys. Rev. 111,442 (1958). "T.Gaskell, Proc. Phys. Soc. (London) 77, 1182 (1961); 80, 1091 (1962)