PHYSICAL REVIEW VOLUME 136 NUMBER 3B 9 NOVEMEBR 1964 Inhomogeneous Electron Gas" P. HoHENBERGt Ecole Normale Superieure, Paris, france Ecole Normale Superieure, Paris, France and Faculte des Sciences, Orsay, france University of California at San Diego, La Jolla, California This paper deals with the ground state of n external potential u(r). It is proved that there exists a universal functional of the density, FLr(r)], independent of o(r), such that the ex- (2)n(r)=p(r/ro)with e arbitrary an elation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approac also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented heoretical considerations is a description of this D URING the last decade there has been considerable functional. Once known, it is relatively easy to deter- progress in understanding the properties of a mine the ground-state energy in a given external homogeneous interacting electron gas. I The point of potential view has been general, to regard the electrons as In Part II, we obtain an expression for F[n] when n ction of noninteracting particles deviates only slightly from uniformity, i.e, n(r)=no additional concept of collective +n(r), with A/no-0. In this case FLn] is entirely excitations ole in terms of On the other hand. there has been in existence since and the exact electronic polarizability a(q) of a uniform the 1920,s a different approach, represented by the electron gas. This procedure will describe correctl Thomas-Fermi method? and its refinements, in which the long-range Friedel charge oscillations set up by the electronic density n(r)plays a central role and in a localized perturbation. All previous refinements of the which the system of electrons is pictured more like a Thomas-Fermi method have failed to include these classical liquid. This approach has been useful, up In Part III we consider the case of a slowly val now, for simple though crude descriptions of inhomo- but not necessarily almost constant density, n(r) geneous systems like atoms and impurities in metals. =p(r/ro), fo-c0 For this case we derive an expansion Lately there have been also some important advances of FLn] in successive orders of ro or, equivalently of along this second line of approach, such as the work of the gradient operator v acting on n(r). The expansion and Borowitz, Baraf, 7and Du Bois and Kivelson. s The ground-state energy and the exact linear, quadratic present paper represents a contribution in the same area. etc, electric response functions of a uniform electron In Part I, we develop an exact formal variational gas to an external potential o(r). In this way we recover, Sitv ciple for the ground-state energy, in which the den- quite simply, all previously developed refinements of enters a universal functional F[n(r)], which applies to somewhat further. Comparison of this case with the all electronic systems in their ground state no matter nearly uniform one, discussed in Part II, also reveals what the external potential is. The main objective of why the gradient expansion is intrinsically incapable Supported in part by the U. S. Office of Naval Research. of properly describing the Friedel oscilations or adial oscillations of the electronic density in an atom tfor a review see, iof oin Info New s s Elementary Excitations summation of the gradient expansion can be carried which reflect the electronic shell structure. A partial Phy. 6, 1 rev95) of work up to 1956, see N. H. March, Advan. out(Sec. I4 ) but its usefulness has not yet been neets and E. S. Pavlovskii, Zh, Eksperir 31, 427(1956)[English transl. Soviet Phys.-JETP I EXACT GENERAL FORMULATION 4 D. A. Kirzhnits, Zh. Eksperim i. Teor 13 32,115(1957) The Density as Basic Variable H. W. Lewis. P We shall be consider number of electrons, enclosed in a large box and movin 8 D. F. Du Bois and M. G. Kivelson, Phys. Rev. 127, 1182 J. Friedel, Phil. Mag. 43, 153(1952). B864PHYSICAL REVIEW VOLUM E 136, NUM B ER 3 8 9 NOVEMEBR 1964 InhOmOgeIIeouS EleCtrOn Gaa* P. HOHENBERGt Ecole Xornzale Superzeure, I'aris, France AND W. KonNt Ecole Xonnale Superieure, I'aris, Prance and I'aculte des Sciences, Orsay, France and University of Calzfo&nia at San Diego, La Jolla, Calzfornia (Received 18 June 1964) This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, FtI(r) g, independent of v(r), such that the ex￾pression E—=fs(r)n (r)dr+Ft I(r)j has as its minimum value the correct ground-state energy associated with s(r). The functional FLn(r)j is then discussed for two situations: (1) n(r) @san(r), 8/ao((1, and (2) a(r) =q (r/ra) with p arbitrary and 1'p ~~.In both cases Fcan be expressed entirely in terms of the cor￾relation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented. INTRODUCTION ' ' &~IJRING the last decade there has been considerable progress in understanding the properties of a homogeneous interacting electron gas. ' The point of view has been, in general, to regard the electrons as similar to a collection of noninteracting particles with the important additional concept of collective excitations. On the other hand, there has been in existence since the 7920's a different approach, represented by the Thomas-Fermi method' and its re6nements, in which the electronic density n(r) plays a central role and in which the system of electrons is pictured more like a classical liquid. This approach has been useful, up to now, for simple though crude descriptions of inhomo￾geneous systems like atoms and impurities in nietals. Lately there have been also some important advances along this second line of approach, such as the work of Kompaneets and Pavlovskii, ' Kirzhnits, ' Lewis, ' Baraff and Borowitz, ' Bara6, ' and DuBois and Kivelson. ' The present paper represents a contribution in the same area. In Part I, we develop an exact formal variational principle for the ground-state energy, in which the den￾sity tz(r) is the variable function. Into this principle enters a universal functional PLtr(r)), which applies to all electronic systems in their ground state no matter what the external potential is. The main objective of *Supported in part by the U. S. Once of Naval Research. f NATO Post Doctoral Fellow. f Guggenheim Fellow. ' For a review see, for example, D. Pines, Elementary E'.'xci tati ons in Solids (W. A. Benjamin Inc., New York, 1963). ' For a review of work up to 1956, see N. H. March, Advan. Phys. 6, 1 (1957). A. S. Kompaneets and E. S. Pavlovskii, Zh. Eksperim. i. Teor. Fiz. 51, 427 (1956) [English transl. : Soviet Phys.—JETP 4, 328 (1957)j. D. A. Kirzhnits, Zh. Eksperim. i. Teor. Fiz. 32, 115 (1957) I English transl. : Soviet Phys.—JETP 5, 64 (1957)j. ' H. W. Lewis, Phys. Rev. 111, 1554 (1958). ' G. A. 13araff and S. Borowitz, Phys. Rev. 121, 1704 (1961). 7 G. A. BaraG, Phys. Rev. 123, 2087 (1961). 'D. F. Du Bois and M. G. Kivelson, Phys. Rev. 127, 1182 (1962). theoretical considerations is a description of this functional. Once known, it is relatively easy to deter￾mine the ground-state energy in a given external potential. In Part II, we obtain an expression for FLnj when tr deviates only slightly from uniformity, i.e., n(r)=1'cp +ts(r), with ts/tss —& 0; In this case FLej is entirely expressible in terms of the exact ground-state energy and the exact electronic polarizability n(g) of a uniform electron gas. This procedure will describe correctly the long-range Friedel charge oscillations' set up by a localized perturbation. All previous refinements of the Thomas-Fermi method have failed to include these. In Part III we consider the case of a slowly varying, but +of necessarily almost constant density, tr (r) = p(r/rs), rs —&oo. For this case we derive an expansion of F)trj in successive orders of rs ' or, equivalently of the gradient operator V acting on e(r). The expansion coeKcients are again expressible in terms of the exact ground-state energy and the exact linear, quadratic, etc. , electric response functions of a uniform electron gas to an external potential w(r). In this way we recover, quite simply, all previously developed refinements of the Thomas-Fermi method and are able to carry them somewhat further. Comparison of this case with the nearly uniform one, discussed in Part II, ,also reveals why the gradient expansion is intrinsically incapable of properly describing the Friedel oscillations or the radial oscillations of the electronic density in an atom which reQect the electronic shell structure. A partial summation of the gradient expansion can be carried out (Sec. III.4), but its usefulness has not yet been tested. I. EXACT GENERAL FORMULATION I. The Density as Basic Variable Ke shall be considering a collection of an arbitrary number of electrons, enclosed in a large box and moving ' J. Friedel, Phil. Nag. 45, 155 (1952)