SELF-CONSISTENT EQUATIONS A1135 where the symbols r and r are understood to include an atomic nucleus, and (2)it does not lead to quantum lectron spin coordinates and integration is understood density oscillations, such as the density fluctuations to include summation over spin coordinates. One next due to atomic shell structures. By not making the re- sumes that the wave functions can be approximated placement(2. 20), we avoid both of these shortcomings by plane waves which results in Let us now qualitatively discuss the appropriateness kp(rr k(r)-k2. k+kp(r) of our procedure for various classes of electronic 2kkp(r) Ik-kp(),(2. 17) In atoms and molecules one can distinguish three regions: (1)A region near the atomic nucleus, where There kF(r)=(3nn(r))a. Finally, one averages %xk the electronic density is high and therefore, in view of over the occupied state k, which results in case(b)above, we expect our procedure to be satis- (3/2 m)(3m n(r)) 1a.(2.18) factory (2)The main body of the charge distribution In our procedure (neglecting correlation)we obtain, in varying, so that our approximation (2. 3)for exe is ex- place of Slaters vx smaller by a factor of 3. From the discussion in Appen- validity and therefore we expect this region to be the dix I, it follows that while A= gives the exchange correc- main source of error. We do not expect an accurate de tion of the density correct to order v1, inclusive, Ux scription of chemical binding In large atoms, of course as indeed any other function of n(r)] leads to errors of this"surface?"region becomes of less importance. (The order v|3. The same comment applies to any extension surface is more satisfactorily handled in the nonlocal consistent potential For metals, alloys, and small-gap insulators we have We may note that our result is equivalent to taking, of course, no surface problem and we expect our ap k=kp();i.e, the effective exchange potential for a change and correlation effects. In large-gap insulators, state at the top of the Fermi distributions. This is however, the actual correlation energy will be con physically understandable since density adjustments siderably reduced compared to that of a homogeneous come about by redistribution of the electrons near the electron gas of the same densit Fermi level (b)High density. This regime is characterized by the B. Nonlocal Effective Potential condition r/ao<<1, where do is the Bohr radius. In this Instead of the Hartree-type procedure discussed in case, the entire exchange and prelatic smaller than the kinetic energy by a factor of order Sec. IIA it is also possible to obtain a scheme which in- (/ao)and hence our inaccuracy in representing these cludes exchange effects exactly. We write in place of portions becomes negligible. (2.3) The reader will have noticed that while in Eg.(2.3) Ex[n]=2[n]+|n(re(n()dr(221) re approximate the exchange and correlation energy by made no approximation for the kinetic-energy func- where Ex[] is the exchange energy of a Hartree-Fock tional T[n]of Eq.(2.2). This procedure is responsible system of density n(r)and e(n)is the correlation energy for the exactness of the high-density limit, the density is rapidly varying, such as in the vicinity tionary property of (2. 1)leads to the following system of an atomic nucleus proximation. If in Eq.(2.2), we had ap TLn] by its form appropriate to a system of -2v2+g(r)+H(r);(r varying density, T[n] (3rn)2n dr (2.20) ∫/-,02 would have been led to the general μ=d(ne)/, omas-Fermi method suggested by method shares with the Thomas-Fermi n(r)-=∑yr (2.24) shortcomings: (1)It leads to an infinite density near d p(r), n(r) are define Eqs.(2.6)and(2.9)SELF—CONSISTENT EQUATIONS A 1135 where the symbols r and r' are understood to include electron spin coordinates and integration is understood to include summation over spin coordinates. One next assumes that the wave functions can be approximated by plane waves which results in () (2.17) k—kr(r) kr(r) — krs(r) k—s k+kr r 1+ ln 2kkr (r) w„i,(r)=— where kr(r)—={3srt(r)}Us. Finally, one averages v & over the occupied state k, which results in n„(r)=—(3/2') {3s'I(r)}'" (2.18) In our procedure (neglecting correlation) we obtain, in place of Slater's v„ t *()=—( /~){ ~'I()}'" (2 ) smaller by a factor of —', . From the discussion in Appen￾dix I, it follows that while p gives the exchange correc￾tion of the density correct to order ~ V~', inclusive, s t as indeed any other function of e(r)] leads to errors of order ~ V ~'. The same comment applies to any extension of Slater's exchange to include correlation in the self￾consistent potential. We may note that our result is equivalent to taking, not the average of (2.17), but rather its value at k=kr(r); i.e. , the effective exchange potential for a state at the top of the Fermi distributions. This is physically understandable since density adjustments come about by redistribution of the electrons near the Fermi level. (b) High dertsity. This regime is characterized by the condition r,/a,((1, where as is the Bohr radius. In this case, the entire exchange and correlation energy is smaller than the kinetic energy by a factor of order (r,/as) and hence our inaccuracy in representing these portions becomes negligible. The reader will have noticed that while in Eq. (2.3) we approximate the exchange and. correlation energy by the expression valid for a slowly varying density, we made no approximation for the kinetic-energy func￾tional T,Lnj of Eq. (2.2). This procedure is responsible for the exactness of the high-density limit, even when the density is rapidly varying, such as in the vicinity of an atomic nucleus. We now Inake a few further remarks about our ap￾proxirnation. If in Eq. (2.2), we had approximated T,LN) by its form appropriate to a system of slowly varying density, B. Nonlocal Effective Potential Instead of the Hartree-type procedure discussed in Sec. IIA it is also possible to obtain a scheme which in￾cludes exchange effects exactly. We write in place of Eq. (2.3) F,PN)=E Lrtj+ rt(r)e, (e(r)) dr (2.21) where Z LNj is the exchange energy of a Hartree-Fock system of density I(r) and e,(m) is the correlation energy per particle of a homogeneous electron gas. Applying this ansatz in conjunction with Eq. (2.2) and the sta￾tionary property of (2.1) leads to the following system of equations: Ni(r, r') , 4'(r') d~= e'4'(r), (2 22) lr—r'I (2.20) where T,Lm|~ —,', (3x'I)'t'rt dr, an atomic nucleus, and (2) it does not lead to quantum density oscillations, 4 such as the density fluctuations due to atomic shell structures. By not making the re￾placement (2.20), we avoid both of these shortcomings. Let us now qualitatively discuss the appropriateness of our procedure for various classes of electronic systems. In atoms and molecules one can distinguish three regions: (1) A region near the atomic nucleus, where the electronic density is high and therefore, in view of case (h) above, we expect our procedure to be satis￾factory. (2) The main "body" of the charge distribution where the electronic density n(r) is relatively sjowly varying, so that our approximation (2.3) for e„, is ex￾pected to be satisfactory as discussed in case (a) above. (3) The "surface" of atoms and the overlap regions in mole cules. Here our approximation (2.3) has no validity and therefore we expect this region to be the main source of error. We do not expect an accurate de￾scription of chemical binding. In large atoms, of course, this "surface" region becomes of less importance. (The surface is more satisfactorily handled in the nonlocal method described under 8 below. ) For metals, alloys, and small-gap insulators we have, of course, no surface problem and we expect our ap￾proximation (2.3) to give a good representation of ex￾change and correlation effects. In large-gap insulators, however, the actual correlation energy will be con￾siderably reduced compared to that of a homogeneous electron gas of the same density. we would have been led to the generalization of the Thomas-Fermi method suggested by Lewis. This method shares with the Thomas-Fermi method two shortcomings: (1) It leads to an in6nite density near ' H. W. Lewis, Phys. Rev. 111, 1554 (1958). p.=d (rte.)/dN, I I (r r )=Z It't(r)A*(r ) j~1 (2.23) (2.24) and p(r), rt (r) are defined as before, Eqs. (2.6) and (2.9)