PHYSICAL REVIEW VOLUME 140, NUMBER 4A I 5 NOVEMBER 1965 Self-Consistent Equations Including Exchange and Correlation Effects"* W. KOHN AND L. J. SHAM University of California, San Diego, La Jolla, California (Received 21 June 1965) heory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system ing electrons are developed. These methods are exact for systems of slowly varying or high der respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion Electronic systems at finite temperatures and in fields are also treated by similar methods. An appendix deals with a further correction for with short-wavelength density oscillations I INTRODUCTION In Secs. III and iv, we describe the n recent years a great deal of attention has been cations to deal with the finite-temper given to the problem of a homogeneous gas of inter- and with the spin paramagnetism of an ith a considerable degree of confidence over a wide Of course, the simple methods which are here pro- range of densities. Of course, such a homogeneous gas represents only a mathematical model, since in all real origins: a too rapid variation of density and, for finite ystems(atoms, molecules, solids, etc. )the electronic systems, boundary effects. Refinements aimed at re- is nonuniform ducing the first type of error are briefly discussed in then a matter of interest to see how properties Appendix I of the homogeneous gas can be utilized in theoretical studies of inhomogeneous systems. The well-known II. THE GROUND STATE nethods of Thomas- Fermil and the Slater 2 exchange A. Local Effective Potential hole are in this spirit. In the present paper we use the It has been shown that the ground-state energy of an formalism of Hohenberg and Kohn to carry thi pproach further and we obtain a set of self-consistent interacting inhomogeneous electron gas in a static po- quations which include, in an approximate way, ex tential v(r) can be written in the form change and correlation effects. They require only a n(rn( knowledge of the true chemical potential, HA(), of a E- o(r)n(r)dr+a dr dr+G[n] geneous interacting elect r-rI 3(21) We derive two alternative sets of equations where n(r) is the density and GLn] is a universal func- LEgs.(2.8)and(2.22)] which are analogous tional of the density. This expression, furthermore, is a tively, to the conventional Hartree and hartre minimum for the correct density function n(r). In this equations, and, although they also include co section we propose first an approximation for GLn] effects, they difficult to solve which leads to a scheme analogous to Hartree s method e& The local effective potentials in these equations are but contains the major part of the effects of exchange ique in a sense which is described in Sec. II. In par- and correlation ticular, we find that the Slater exchange-hole potential We first write besides its omission of correlation effects, is too large by a factor of是 GLn]=TLn+Exon] Apart from work on the correlation energy of the where TaLn] is the kinetic energy of a system of non- omogeneous electron gas, most theoretical many-body interacting electrons with density n(r)and Excln] is studies have been concerned with elementary excita- by our definition, the exchange and correlation energ. tions and as a result there has been little recent progress of an interacting system with density n(r). For an arbi- in the theory of cohesive energies, elastic constants, trary n(r), of course, one can give no simple exact ex etc,of real (i. e,, inhomogeneous) metals and alloys. pression for Exo[n]. However, if n(r) is sufficiently The methods proposed here offer the hope of new slowly varying, one can showd that progress in this latter area Supported in part by the U. S Office of Naval Research Ex[n]=n(r)exe(n(r))dr (23) dge Phil. Soc. 23, 542( 1927) 4W.Kohn and L. J. Sham, Phys. Rev. 137, A Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964);. For such a system it follow referred to hereafter as HK A1133
PHYSICAL REVIEW VOLUM E 140, NUM B ER 4A 15 NOVEM B ER 1965 Self-Consistent Equations Including Exchange and Correlation Effects* W. KOHN AND L. J. SHAM Unieersity of Ca/Bfornia, San Diego, la Jolta, California (Received 21 June 1965l From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of -';.) Electronic systems at finite temperatures and in magnetic lelds are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations. I. INTRODUCTION 'N recent years a great deal of attention has been - - given to the problem of a homogeneous gas of interacting electrons and its properties have been established with a considerable degree of confidence over a wide range of densities. Of course, such a homogeneous gas represents only a mathematical model, since in all real systeins (atoms, inolecules, solids, etc.) the electronic density is nonuniform. It is then a matter of interest to see how properties of the homogeneous gas can be utilized in theoretical studies of inhomogeneous systems. The well-known methods of Thomas-Fermi' and the Slater' exchange hole are in this spirit. In the present paper we use the formalism of Hohenberg and Kohn' to carry this approach further and we obtain a set of self-consistent equations which include, in an approximate way, exchange and correlation effects. They' require only a knowledge of the true chemical potential, tie(e), of a homogeneous interacting electron gas as a function of the density n. We derive two alternative sets of equations [Eqs. (2.8) and (2.22)) which are analogous, respectively, to the conventional Hartree and Hartree-Fock. equations, and, although they also include correlation effects, they are no more difficult to solve. The local effective potentials in these equations are unique in a sense which is described in Sec. II. In particular, we And that the Slater exchange-hole potential, besides its omission of correlation effects, is too large by a factor of —, '. Apart from work. on the correlation energy of the homogeneous electron gas, most theoretical many-body studies have been concerned with elementary excitations and as a result there has been little recent progress in the theory of cohesive energies, elastic constants, etc., of real (i.e. , inhomogeneous) metals and alloys. The methods proposed here offer the hope of new progress in this latter area. ~ Supported in part by the U. S. Ofhce of Naval Research. 'L. H. Thomas, Proc. Cambridge Phil. Soc. 23, 542 (1927); E. Fermi, Z. Physik 48, 73 (1928). ' J. C. Slater, Phys. Rev. 81, 385 (1951). ' P. Hohenberg and W. Kohn, Phys. Rev. 136, 3864 (1964l; referred to hereafter as HK. In Secs. III and IV, we describe the necessary Inodidcations to deal with the finite-temperature properties and with the spin paramagnetism of an inhomogeneous electron gas. Of course, the simple methods which are here proposed in general involve errors. These are of two general origins4: a too rapid variation of density and, for 6nite systems, boundary effects. Refinements aimed at reducing the 6rst type of error are brieQy discussed in Appendix II. II. THE GROUND STATE A. Local Effective Potential It has been shown' that the ground-state energy of an interacting inhomogeneous electron gas in a static potential n(r) can be written in the form 1 e(r)e(r') Z= tt(r)e(r) dr+ — dr dr'+G[e), r r'[— i:,, (2.1) where e(r) is the density and G[e) is a universal functional of the density. This expression, furthermore, is a minimum for the correct density function e(r). In this section we propose first an approximation for G[e), which leads to a scheme analogous to Hartree's method but contains the major part of the effects of exchange and correlation. We first write G[e)=T.[e)yZ,[e), (2.2) where T,[e) is the kinetic energy of a system of noninteracting electrons with density e(r) and F,[e) is, by our definition, the exchange and. correlation energy of an interacting system with density e(r). For an arbitrary e(r), of course, one can give no simple exact expression for E,[e). However, if e(r) is sufliciently slowly varying, one can show' that F,[e)= e(r)e,(e(r)) dr, (2.3) 4 W. Kohn and L.J. Sham, Phys. Rev. 137, A1697 (1965). ~ For such a system it follows from HK that the kinetic energy is in fact a unique functional of the density. 1138
A1134 W. KOHN AND L.J. SHAM where exo(n)is the exchange and correlation energy per Seitz radius and ro is a typical length over which there electron of a uniform electron gas of density n Our sole is an appreciable change in density. In this case, as approximation consists of assuming that (2.3)consti- shown in HK, we can expand the true exchange and tutes an adequate representation of exchange and corre- correlation energy as follows lation effects in the systems under consideration. We shall regard Exe as known from theories of the homo- Ex[n]= Exe(on)n dr geneous electron gas. 6 From the stationary property of Eg. (2. 1)we now obtain, subject to the condition +/c(n)vndr+…,(21) on(rdr=0 (24) the second term in the energy expansion in powers of the equation the gradient operator. In this regime we may similarly [n] a nI on(r)e(r)+ +an(()}ar=0;(2.5) g(=t(y+/ +|)va2ar+…(2.12) and chemical potential of a uniform gas or tribution.(2.7) From HK, expecially Sec. III 2, we have the following kx(n)=d(n∈o(x)/lnt is the exchange and correlation ce to the n(r)n(r) E[n]= +(r)n(r)dr dr dr Equations (2.4)and (2.5) are precisely the same as one obtains from the theory of Ref 3 when applied to a system of noninteracting electrons, moving in the given potential p(r)+xe(n(r). Therefore, for given p and +/0+/()计…,(1 u,one obtains the n(r)which satisfies these equations where simply by solving the one-particle Schrodinger equation go(n)={10(3r2)2+ex(n)}n -号V2+[φ(r)+kx2((r)》(r) and g2)(m)={ex(2)(n)+t(2)(m)}n,(2.15) Since in our approximation (2.3), the V 2 term of n(r)=∑|(r)|2, ( 2.9) Eq.(2. 11)is neglected, it is clear that for a gas of slowly varying density our expression(2.10) for the energy has errors of the order VI, or equivalently, of the order It is physically very satisfactory that Hxe appears in o Eq.(2.8)as an additional effective potential so that Surprisingly, our procedure determines the density gradients of wze lead to forces on the electron fluid in a with greater accuracy, the errors being of order v 4 nanner familiar from thermodynamics. This is shown in Appendix I Equations (2.6)-(2.9)have to be solved sel At this point a comparison of our procedure and th ently One begins with an assumed n(r), co pp(r) from(2.6)and uxe from(2.7), and finds a ts original work does not include correlation effects.? But from(.8)and(2.9). The energy is given by ( r) even the exchange correction is different from ours. To obtain Slater's exchange correction, one may begin by n(r)n(r writing the Hartree-Fock exchange operator in the form =∑E- dr dr of an equivalent potential acting on the hth wave function n(r)lex(n(r))-uxen(r)]dr.(2.10) 4(2/(*(m( The results of our procedure are exact in two limiting cases ψk*(r)y(r),(2.16) ent to the original paper by Slater, there have been (a) Slowly varying density. This regime is character ized by the condition ra/ro<1, where r, is the wigner ry Excitations in Solids (1964);S. Lundqvis min, Inc, New York, 1963) Ufford, Phys. Rev. 139
AN SH~M d. ro» a yp th over which sit . In t»s ical lengt ' se as it.z, ra ius l hange in dens ty hange and n appreciab e c nd the true exc is an can expan ene gy a F,fej= a .(e)e. dr c ('&(e))Vn('dr+ ~ ~ ~, (2.11) A &&34 prrelation ene e istheexcha g gy Pel f densltyn. O n e and corre ur sole where &x.( ) . iectron gas 2 3) constif a uniform e e that electron o ts pf assum~~g nd correl matipn con pf exchang apprpxima resentatipn o . tion. e e uaterep consl era tutes ana q . s stems un e the homocts in the sy theories lation e I,npwn from 2.1) we now eiectron g ' rty of Eq' gene From th ' statio ary prop obtain, subject to th,e ,o„dition the equation be(r) dr=0, (2.4) e and correlation p ortion o the n rg xpansi ansion in powers o expand T, e in be(r) q(r)+ t(„. n ()) d 0 ( ) r =, 2. here e(r') ~(r) =~(r)+ dr', (2 6) T,Ln) = —,' (3x'e)'"e dr t('&(n)iVei'dr+ . (2.12) 2 From HK expecially Sec.III 2,, we have the following expression for the energy. and ) + correlation c ' is the exc to th o e obtain om t e e)}e, Therefore or g 3m'e)'"+e .e po p~ si a + go(e) dr+ g22&'&(e) [Ve)'dr+ (2.13) (2.14) ir n r e,(n(r)) n r —t(.(e(r))g r dr. (2.10) .*(r)A '(r')4' (r)A(r' [r—r'[ N w.,(r)=- Z The results of ou r p roce du re are exactt in twow limiting ensi . ime is c aractercases. ' ' ized by n r, ro~(1, where r, is the i P', elementary x ' ' 'n 6F r a review see D. Pines, r Solzds in Inc. , (W. A. Benjamin, ew A*(r)4'(r) (2.16) ter there have been m ts to add corre a R Schrieffer P y s 9 ' on Phys. Rev. 136, S and C W. UGord, Phys. R 139 h&ch sat&sfies these e uat 6„,('& (n)+t(" (n) )e. of the order nd setting 'g 'y e(r)=Z '(r) I', I ' i=1 l ou re determines t e e of o d ivi4. iv is the num er o d and that dd't' l ff pp E. (2.8) as an a ii g familiar from . ns (26—)—(2.9) h to ed mr, g begin by Equatio . — . o the form ne begins wi and fin s ane ter's exc ange o erator in (i 2.6) an pxc i' o (2.7, is iven y -F k h f . 2.9 . The energy is g' writing w the ' ' eHartree- oc f (2.8) and (2.9 . is f equivalent po en function Z=P c,—— 1 —r
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 Appendix I, it follows that while p gives the exchange correction 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 selfconsistent 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 functional 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 approxirnation. 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 includes 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 stationary 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 replacement (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 satisfactory. (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 expected 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 description 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 approximation (2.3) to give a good representation of exchange and correlation effects. In large-gap insulators, however, the actual correlation energy will be considerably 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)
A1136 KOHN AND L SHAM The energy is now n(r)n(r) drdr′ fxe(n)=f(n)-fo(n) (3.5) where f and fo are the free energies per electron of an 1frn1〔r)n1(r',r) interacting and noninteracting gas, respectively. dr dr 0=(r)+(GLn]/6n(r)+kx(t(r))-μ,(3.6) where p(r) is given, as before, by Eq.(2.6)and +/n()(s(()-(()a.(2) uxo(n=d(xe(n))/dn (37 This procedure may be regarded as a Hartree-Fock Equation (3.6) is identical to the corresponding equa method corrected for correlation effects. It is no more tion for a system of noninteracting electrons in the complicated than the uncorrected Hartree-Fock method effective potential o+xe. Its solution is therefore de but, because of the nonlocal operator appearing in Eg. termined by the following system of equations (2.22), very much more complicated than the method described in Sec. IIA. Since at least exchange effects 一2V2+g(r)+x(n();=;,(3.8) are now treated exactly we must expect, in general, and more accurate results than from the method of sec. IIA n()=∑|(x)|2/{e()hr+1}.(3. In particular, near the surface of an atom the effective potential now is correctly(1/r) whereas in Sec. IIA it approaches zero much faster. Even here, however, u is determined as usual by the total number of particles correlation effects are not correctly described near the from Eq.(3.9). This value also represents our approxi- mation for the chemical potential of the interacting III FREE ENERGY: SPECIFIC HEAT Of special interest for metals and alloys is the low- temperature heat capacity. This may be obtained by We can generalize the consideration of the ground making an expansion, in powers of T, of the above state to finite temperature ensembles by using the finite system of equations. An equivalent, but more con temperature generalization of Eq.(2.1)given by venient, method is as follows: From thermodynamics Mermin. He has shown that the grand canonical po- and Eq.( 3. 1)we have tential can be written in the form 1n〔)n(r S[n]=--(+N)v pr+ =|v()n()dr+ dr dr (r) +Gn]-/n()ar,(1) (3.10) where GLn] is a unique functional of the density at a The integral vanishes because of the stationary property given temperature and u is the chemical potential. For of &, so that the correct n this quantity is a minimum S[n]=-(aG[]/ar)n(r).v (3.11) In analogy with(2.2)we now write The same argument, applied to a system of noninter GLn]=Ga[nI+Fxon]; (3.2)acting electrons of density n(r),gives here G[n]=T[n]-7S[] SLn]=-(0G[n/07) where T[n] and s[n] are, respectively, the kinetic Combining Eqs.(3.11),(3.12),(3.2),and(3.4),we energy and entropy of noninteracting electrons with density n(r)at a temperature T; and Fx[ tion, the exchange and correlation contribution to the free energy. For the latter quantity, we make the For small r it is well known that S, is given by fxeLn]= n(r)x(o(r)dr (34) S[n]=N3TkTge(uy rhere g, is the single-pan of states in the where fxe(n)is the exchange and correlation contribu- effective potential p tion to the free energy per electron of a uniform electron (afro(n)/ar)n(r), y=3m" -[g(uA(n)-go(uo(n)), 9 N. D. Mermin, Phys. Rev. 137, A1441 (1965)
K AND L' I gHAM The energy is now 1V' E=P p,—— 1 e(r)e(r') ei(r,r')ei (r',r (3.7) (3.9) III. FREE ENERGY; SPECIFIC HEAT (3.10) &&36 as of de»ity 0 n (3.5) are the free energie Per elect.ron of an 1 . ~here f and fp are t gas respective y. ~ nd noninteracting g ( )+(bG [e7/be(r))+I)~p(e 1 is given, as before, b Fq. (2.6) and )=q( f ())/d". dr. (2' ) "-" ndln eq ' p qg p (e(r))—pp(e( )) ]. to the corresPon e 1' &o 3 6) is identlca ' electr()ns in the k Equation ~ f noninteracting a beregar e as ion a Hartree-Foc t' for a, system o +~-' 1"""1 tion is therefo This procedure may Beets. It is no mo effective potent W . tern of equat o eth()d ( orrected o t d Hartree-Foc&»e ter»ined by t { ~, (3.8) plicated than the nn 1 tor appearing» l i~2+ &(r)+pxo(e(r)))~' N i 2/{ (e(—p)/pe+ 1) described in 1 we must expect, n&= j,=1 n.ow from the m ctive r of a,rticles more e surface o an accurate resu ts a f atom the effec iv d as usua by th t talnumbero P roxirticular, near the p is determined ' teracting proaches zero muc hf str . E n here, howeve, tl described near t e i p allo s is the lowcorrelation e ec s surface. p n ex ansion, in po ut more contion o gr g h 6 A 1 ad namics re ensembles by using t e ni temperature generalization o E . (2.1) i m b q that the gran can 8G ' He has shown t a 5 e7= (0+—1)c—'(T)—v=- ~()+ be(r) tential can be wri en BT ()e(r) (BG[e7 +G[e7—) e(r) dr,r 3.1) ni ue functiona 1 of thee density at a h h ' gi " perature g and p is t e c e 1 rrect n this quan i y In analogy with (2 2) e now write G[e7=G,[e7+F,[e7; G,[e7=—T,[e7—rS,[e7, (3.3) Ln ~ are respectiveely, the kinetic 1 to th where, n energy a (1 t, py of noninteracting e R tion, ge an corre 1a tion o tib to free energy. For te approximation (3.2) here ~-[ 7= ()f*.( ())d, (3 4) on to the h s. Rev. 137, A1441 (19651. 9 N. D. Mermin, Phys. Rev. where, n an e and correlation contributi free energy perr electron e o auniUo 1 The integral i vanishishes because of thee stationary property 3.11 of 0, so that ~~ n(r), V ~ 5[e7=—(()G[e7/ ) ) ment, applied to a system of noni onintero ' ' Coinbining Eqs.s. (3.11, 3.12), (3.2), and (3.4), we obtai n 5 e7= 5 [e7+ e(r) (Bf (nn 8787' ~(,( ),vdr. (3.13) ell known that 5, ' g' For small r it is we is iven by 5,[e7=Ã-', x'k'~g, (p), (3.14) le- article densityit of states in the (bf..()/~ ).„.=-.-'ir'k'r[g - . (d(o,(e))—gp(((p e (3.15)
SELF-CONSISTENT EQUATIONS A1137 where uA(n) and uo(n)are, respectively, the chemical So far everything is formal and exact. We now write potentials of an interacting and a noninteracting homo- in the spirit of the previous sections geneous gas of density n, and g and go are the respective G-1(r,,,, [nD=G-1(r, r; [n]+Gxo-1(r,r; [n].(4.7) densities of states. 10 It follows immediately that the low-temperature heat The second term we approximate as for a slowl capacity is given by Cr=YT s, which giv Y=3+k Ng, (u)+ n(r)(g(un(n))-go(o(n))dr x[n]=(1/20)2(/V)×g(), (4.9) We shall not present a treatment, analogous to and x(n), xo(n)are, respectively, the susceptibilities for ec IIB, in which exchange effects are included exactly. uniform systems with and without interactions he development is straightforward but leads to nown divergence in the low-temperature specific heat. APPENDIX I: GRADIENT EXPANSION OF THE DENSITY IV. SPIN SUSCEPTIBILITY In this Appendix we show that for a system of slowly To obtain a theory of the spin susceptibility of an varying density our procedure gives the density electron gas, we first extend the theory of HK to include correct to order V 2 inclusive. When dealing with such the effects of spin interaction with an external magnetic a system we may proceed in two entirely equivalent direction and write the magnetic-moment density as Eqs.(2. and can solve the self-consistent equations, field. The result is that if we take the field in the s ways: (1) (2.9), for n(r), and(2)we can go bac m(r)=-(1/20)0y+*(r)1(t)-(x)4()10),(4.1)totheunderlyingvaiationalprinciple(25,make gradient expansion and determine n(r) directly. We the ground-state energy can be written in the form shall here follow the second route to estimate the errors (r) E, m =(o(r)n(r)-H(rm(r)dr From(2.5)and the expansion (2.12)of T.[n], we obtain where G is a universal functional of n and m, and the where u is the chemical potential [cf. HK, Eq.(68)] correct m(r), n(r)make a minimum Note however that because of our approximation of For small m we expand G in the form keeping only the first term in(2.11), some other contri outions of order V 2 are missing in(A1.1) G=GD]+ G(r,r; D ])m(r)m()dr dr+.;(4.3), To solve(A1. 1), let us write the external charge ity as the linear term vanishes for a paramagnetic system in ns()≡f(r/ro), which m=0 when H=0. From the stationary property where fo-co(slow spatial variation), and try the order and that n(r)=m(r)+n1(r), (A1.3) where ()+/Gr;[n])m(ar=0,(44) no(r)=fo(r/ro) 14) exactly neutralizes the external charge and n is assumed where n is the zero-field density. We now formally to approach zero as ro=o0. Neglecting, for the moment, invert this equation, which gives the terms of order V2 in(A1. 1)and substituting (A1.3)into(Al. 1), we obtain n(r)=/G-I(r,r; [n]H(r)dr' dr'tun(no)+ni(r)uA'(no)+o(n1? For a uniform field this gives for the susceptibility Now define y aH m(r)dr=G(r,r En]dr dr.(4.6) r and write 10 J. M. Luttinger, Phys. Rev. 119, 1153(1960) n1(r)≡f1(R), (A1.7)
SELF —Co N S1STENT EQ UATIONS So fal every thmg» fo~al » We now write ~ s sectjolls, G —i rr';[n cond pproxim ate va ryi gg which gives x[n]=x,[n]+— [x(n(r)) x n —xp(n(r))] r dr, (4.8) C,,=yv-, (3.16) where respective y the chemical „( ) r and Po(n) ' ' teract»g h where phi teracting and a n he respective tentials of an s of density ~ d and go are tIt follows immediately t a low Perature heat capacity is given by =-'ee Xg Q)+ «)(g6 n(. ~(n))—ap(t p(n))) « We shall no p treatment, analogous to b 1 d t Sec. known divergenence in the 1ow- e where = (1/2 )'(~/~)&&g, ( ), (4.9) res ective ly, the susceptibilities for unl orm sy stems wiith and wit hou t interactions. APPENDIX I:. GRADIEEINT EXPANSIONN OF IV. SPIN SUSCEPTIBILITY s in susceptibility of an as, we 6rstexten d the h fHK 1 d *()o ()-s * ~ theh groun-und-state energy can be written ln t e ().()-~() ()&d G G[n]+— 2 t e anisl e o pa rama agn etic system in th h h W lC eL=— 0f (4.2) we fin, or order and tha t I d I —H(r)+ G(r,r', [n])m(r')dr =0, p (4.4) n is the zero-field density. ensit . We now formally inver t this equation, wh h ' m r = G—'(r,r', [n])H(r')dr'. (4.5) ~ ~ e ' ' For a uniform Geld this gives for the susceptibihty ' ' or 8 x[n]=— G '(r,r'; m(r) dr= G [n]) dr dr . 4.6) — r,r; h p. Rpv. 119, 1153 (1960). 'o J. M. Luttinger, phys, Rcp, @Vi d'+GL () ()], ( 2 r—r'i universal functional of e an m, orrect m(r), ( 42) For small m we expan [n])m(r)m(r') dr dr'; + (4.3) 1 's endix we showw that or a 's en ' w f system of slow y ives the density In thi pp y o p corre e the self-consistent q d (2.9), for n(r), an 2) ob er i ' ' al rinciple and determine e r)di 1. W th dm t too estimate the error From (2.5) an d thee e pansion (2.12 o, n, otential [cf. HK, Eq.. (68)]. is the chemical po e ( k pngoonl y the first term in .11),, o t er co utlon t' ns of order 1 let us wri e To solvee A1. ), 'te the externa c (A1.2) density as d t the ' where ro~~— (slow spatia l variation), v an ry + ansatz where n(r) =np(r)+ni(r), (A1.3) Now de6ne and write R=r/rp, ni(r) =fi(R) . (A1.6) (A1.'/) no(r) =fp(r/«) (A1.4) exactly e y neu tralizes the external er charge annd e& is assumed e lecting, for the momen, terms of order V~' in (A1.1 (A1.3) A1.1), bt i in'to )+ .().'(.)+o( '). fr—r'/ (A1.5)
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 effective 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 correction to (2.3) to include such effects. In HK, the gradient expression for the energy functional 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 approximation, which are not available at present. The addition of (A2.3) to the effective potential obviously makes the solution of the self-consistent equations 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 evaluating 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 approximation (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