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 A1133PHYSICAL 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 inter￾acting 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, ex￾change 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, respec￾tively, 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 par￾ticular, 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 excita￾tions 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 Inodid￾cations 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 pro￾posed 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 re￾ducing 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 po￾tential 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 func￾tional 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 non￾interacting 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 arbi￾trary e(r), of course, one can give no simple exact ex￾pression 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