A1138 W. KOHN AND L.J. SHAM With this notation,(A1.5)become such as a metal or y, the second term on the fI(R) R20 dr'+uA o (R)) effect of rapid density change on exchange and correlation f(RuA((R))+O(1).(A1.8) This Ex[n] again leads to a set of Hartree-type equations like Eq.(2.8), with an addition to the effec We may now write tive potential given by f(R)=(1/)1(R)+(1/(1)+…,(A19/ok2(;n()n() H=)+(1/r)m(2)+ (A110)×{n(+r)-m(r一r)2dr The first term of Eq.A1.9)is correctly determined by Eq.(A1.8)and not affected either by the inclusion of 2/Kx(r-r;n()(n()-n(r)a.(A2.3) terms of order V2 in(A1.5)or by the terms of order f12 in(A1.8). Hence, in spite of the errors of order v2 in Note that in the random-phase approximation K, 1), the density given by our procedure is correct to vanishes. Hence, in a calculation order 1/r or VI2, inclusive. Equation(A1. 8)shows that this curious result stems from the infinite range of effective potential (A2.3), we need estimates of the Coulomb interaction tion, which are not available at present, approxima- The addition of (A2. 3)to the effective potential ob APPENDIX II: EFFECT OF RAPID DENSITY viously makes the solution of the self-consistent equa OSCILLATION ON EXCHANGE AND tions much more difficult. However, assuming that the modification of n(r) produced by this term is small, one In Eg.(2.3), we approximated ExoN] by the first then, because of the stationary property, Eq.(2. 5) term in the gradient expansion. In actu systems, there are quantum density oscillatio one can obtain the correction to the energy by evaluat ffects on exchange and correlation are not 品油 ing the second term in (A2. 2)with the unmodified the approximation( 2.3). Now we put forward a correc- density. tion to(2.3) to include such effects Nole added in proof. point out tha In HK, the gradient expression for the energy func- it is possible, formally, to replac ce the many-electron tional is partially summed such that it is also correct problem by an exactly equivalent set of self-consistent a system of almost constant density even when the one-electron equations. This is accomplished quite density fluctuations are of short wavelength simply by using the expression(2. 2) [without the ap proximation(2. 3)] in the energy variational principle GLn]= go(n(r)dt K(r-r’;n() This leads to a set of equations, analogous to Eqs (2. 4)-(2.9), but with uxe(n) replaced by an effective one-particle potential vxe, defined formally as ×{n(r)-n(r)2drdr’,(A2.1) (r)=sExo[n]on(r where K(r-r; n)is determined by the polarizability of a homogeneous electron gas at density n, and Of course, an explicit form of vxo can be obtained only r=f(r+r). To the same approximation if the functional Exe[n], which includes all many-body effects, is known. This effective potential will reproduce Eac[n]= n(r)ex((r))dr-/Kx(r-r; n(r)) the exact density and the exact total energy is then (r)n(r) E where Kxe is the difference between K of the interacting Ir-r r dr+Ex[n] and that of the nor the same density. We believe that for an infinite system Txo(r)n(r)dr The second term of HK, Eg(83)is in error; it should be 2 K( n()(n(+r-n(r-ir)) Of course, if we make the approximation(2. 3)for Exo the above exact formulation reverts to the approximate The kernel K has the same meaning as in HK. heory of Sec. Il.A 1138 With this notation, (A1.5) becomes W. KOHN AND L. J. SHAM such as a metal or an alloy, the second term on the right-hand side of (A2.2) accounts adequately for the eGect of rapid density change on exchange and correlation. This F,fng again leads to a set of Hartree-type equations like Eq. (2.8), with an addition to the effec￾tive potential given by fi(R') dR'+I s(fs(R) ) [R—R'[ +fr(R)s.'(fs(R))+o(fr') (A1 8) We may now write fi(R)=(1/ro')fi"'(R)+(1/re')fi"&(R)+, (A1.9) (&If y. 1 2 p =p"'+ (1/ro') p"'+ (A1.10) X{e(r+sr') —N(r—rsr') }sdr' The 6rst term of Eq. (A1.9) is correctly determined by Eq. (A1.8) and. not affected either by the inclusion of terms of order V' in (A1.5) or by the terms of order fir in (A1.8). Hence, in spite of the errors of order 7' in (A1.1), the density given by our procedure is correct to order 1/rs' or ~V'~', inclusive. Equation (A1.8) shows that this curious result stems from the infinite range of the Coulomb interaction. APPENDIX II: EFFECT OF RAPID DENSITY OSCILLATION ON EXCHANGE AND CORRELATION In Eq. (2.3), we approximated E„,Lej by the erst term in the gradient expansion. In actual physical systems, there are quantum density oscillations' whose effects on exchange and correlation are not included in the approximation (2.3). Now we put forward a correc￾tion to (2.3) to include such effects. In HK, the gradient expression for the energy func￾tional is partially summed such that it is also correct for a system of almost constant density' even when the density Quctuations are of short wavelength": l GLriJ= gs(N(r)) dr E—(r——r'; n(r)) 2 X(rs(r)—e(r'))'dr dr', (A2.1) where &(r—r', m) is determined by the polarizability of a homogeneous electron gas at density n, and r= s r(r+r'). To the same aPProximation, p,Lrig= m(r)e„,(N(r)) dr—— Z„,(r—r', rs(r)) 2 —2 E,(r—r', rr(r) )(e(r)—rr(r')) dr'. (A2.3) Note that in the random-phase approximation E„, vanishes. Hence, in a calculation which includes the eRective potential (A2.3), we need reliable estimates of E„„calculated beyond the random-phase approxima￾tion, which are not available at present. The addition of (A2.3) to the effective potential ob￾viously makes the solution of the self-consistent equa￾tions much more di%cult. However, assuming that the modification of m(r) produced by this term is small, one may calculate n (r) and Z first without including it, and then, because of the stationary property, Eq. (2.5), one can obtain the correction to the energy by evaluat￾ing the second term in (A2.2) with the unmodified density. cVo&e added irr Proof We shou. ld like to point out that it is possible, formally, to replace the many-electron problem by an exactly equivalent set of self-consistent one-electron equations. This is accomplished quite simply by using the expression (2.2) Lwithout the ap￾proximation (2.3)$ in the energy variational principle. This leads to a set of equations, analogous to Eqs. (2.4)—(2.9), but with p, (ri) replaced by an effective one-particle potential v „dined formally as v„,(r)—=5E„,Leg/Bn (r) . Of course, an explicit form of v, can be obtained only if the functional E„,LN j, which includes all many-body effects, is known. This effective potential will reproduce the exact density and the exact total energy is then given by X(e(r)—e(r')) s dr dr' (A2.2) r—— i where K„,is the di6'erence between K of the interacting homogeneous gas and that of the noninteracting gas at the same density. We believe that for an infinite system, N(r) e(r') dr dr'+E, tier j t„(r) N(r) dr. "The second term of HK, Eq. (83) is in error; it should be E(r'; e r))(e r+ x') —e(r—~~r'))'dx'. @he kernel E has the same meaning as in HK. Of course, if we make the approximation (2.3) for E„, the above exact formulation reverts to the approximate theory of Sec. II