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Density of States for Interacting Electrons*

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Density of States for Interacting Electrons* JOU RNAL OF RESEARCH 01 the Nationa l Bureau of Standards - A. Physics and Chemistry Vol. 74A, No.3, May-June 1970 Beyond the One-Electron Approximation: Density of States for Interacting Electrons* L. Hedin, B. I. Lundqvist, and S. Lundqvist Chalmers University of Technology, Goteborg, Sweden (October 10, 1969) The concept " dens it y of states" can be given many different meanings wh e n we go beyond the o ne­ e lectron approximation. In this s urvey we concentrate on the de fin iti o n ti e d to excitati on processes, where one electron is added or re moved from the solid. We discuss the one-parti cle s pectral (uncti on for conduction a nd core e lectrons in me tals, how it c an be approx imately ca lc ulate d, and ho w it can be re­ late d to diffe re nt types of expe ri me nts like x-ray photoemi ssion, x-ray e mi ssion and absorpti on , photoemission and optica l absorpti on in th e ultraviolet, and the CO)nplOn effect. We a lso di scuss the form of the exc hange-corre lation pote ntia l for use in band structure calcu lations_ Key words: De nsity of states: interacting e lectrons; one- partic le Green function; oscillator stre ngths; quas i particle d ensity of states; x-ray e mission and absorption_ 1. Introduction functions. The quantity which is most easy to define is the true density of states of the fully inte racting In the one-electron approximation the concept "elec- systems tronic density of states" is unique and simple and is defined by the formula N(w) = L o(w - EII)' (3) n I (1) where Ell are the energy levels of the syste m. Although l being a key quantity in thermodynamics it is of no use for the description for e.g. optic al, x-ray or e nergy loss where Ei denotes the single-particle energies_ Similarly spectra. We rather need the proper extensions of eqs the optical density of states is gi ven by the joint density (1) and (2) to describe one- and two-electron properties of state for electrons and holes, thus of the system. A particularly simple case is that of parti­ cles having a Lorentzian energy distribution (2) (4) These functions only describe the cruder aspects of experiments such as soft x-ray emission or optical ab­ The density of states is then given by sorption_ They must be augmented with the proper oscillator strengths, whic h provide important selection N J(w) = LAk(W). (5) '-" rules and as well give rise to deviations from the de ns ity k of state curves. I In cases where the line width r k is small compared with I When one goes beyond the one-electron approxima­ the width of the band, the formulas (5) and (1) obviously ~ tion and considers interactions not included in a self- will give similar results. consiste nt field theory the situation becomes non-trivial The simple case just mentioned does not really carry and one enco un ters a wide varie ty of density of states us beyond the one-electron theory. We have in th e general case to abandon a concept based on single-par­ * An invit ed paper presented at the 3d Material s Researc h Symposium , El.ectronic Density ojSt.ates, Nuve mber 3·6, 1969, Gai t h e r s Lur~. Md . ticle levels and instead use distribution fun ctions or 417 spectral densities of which the function A", (w) given dynamic interaction between the electrons and above is a simple and almost trivial case. Such distribu­ between the electrons and phonons. (' tion functions include the proper oscillator stre ngths, In section 4 we take up the core electron spectrum ;1 and they refer to formally exact many-electron states, and discuss calculations to first order in the dynamic properly weighted according to the physical process interaction with the valence electrons. We also survey considered. They are closely related to correlation the recent work by Langreth who obtained the exact functions and Green functions. An example is the solution to an important model problem. In section 5 we density-density correlation function (p(rt)p(r't')), give a qualitative disc ussion of some experimental \' which correlates density fluctuations at different times results for photon absorption and emission, photoemis- and different positions. This function is of basic im­ sion and Compton scattering. In section 6 finally we portance for describing optical experiments and energy give some concluding remarks. loss spectra. Unfortunately it is very difficult to analyze its structure a nd we will here instead concentrate on a 2. The One-Electron Density of States in an. / simpler entity, the one-particle Green function and its Interacting System and its Connection with X- -~ assoc iated s pectral function [1] . Ray Photoemission and with X-Ray Emission l The one-particle spectral function is a generalization and Absorption of the usual density of states N1(w). It gives an asymp­ In this section we introduce some theoretical techni· totically exac t description of x-ray photoemission, is calities, which however seem unavoidable in order to connected with x-ray emission and absorption and also establish a firm connection between theory and experi­ has some relevance for uv photoemission and absorp­ ment. tion. The one-electron spectral weight function is defined It is, however, not capable to describe edge effe cts in as general, and does not at all account for final state or A (x, x' ; w) = 2: i s (x)ls* (x')a (w - Es). (6) particle-hole interactions. Such effects, which in their simplest form are described by N2(w) or generalizations In the independent-particle approximation the quan­ therefrom provide intricate problems upon which we tities Is and Es are the one-electron wave fun ctions and shall only briefly touch here. These questions will be energies. In an interac ting system the oscillator ~ take n up in de tail in the lecture by Mahan. strength functionls is defined as It has recently been noticed that the one-electron spectral func tion should exhibit some marked and (7) strong structure. This structure is primarily associated with the interaction between the electron and plasmon where t{J(x) is the electron field operator. It gives the r excitations. It should be quite important in x-ray probability amplitude for reaching an excited state "'­ photoemission, influencing the line shape of the core IN - 1, s) , when an electron is suddenly removed from electron peak and giving low energy satellites both to the ground state IN). The quantity Es is the excitation the core electron and conduction band structures. It is energy important in x-ray emission and absorption spectra Es=E(N) - E(N -l,s). (8) showing up at the threshold, and giving rise to satellite structures. Its effect may also be noticeable in uv These definitions apply for w in eq (6) smaller than photoemission and optical absorption spectra. the chemical potential fl. For w larger than fl, states We should also mention the structure in the one-elec­ with N + 1 particles are involved. tron spectrum caused by coupling to phonons. This In applications of the theory it is often practical to structure is limited to a region of the order of the De bye represent the spectral func tion as a matrix in a space energy around the Fermi level, but in that region it is spanned by some suitable complete orthonormal set of quite pronounced, causing e.g. the well-known single·particle states flk(x), thus enhancement of the quasi particle density of states. In the next section we will disc uss the connections A kk , (w) = Jui (x)A (x, x'; w)uk,(x')dxdx' between the one-electron spectrum and x-ray photoemission, and with x-ray emission and absorption. =2: (Nl atIN - 1, s) (N s In section 3 we discuss the spectrum of the conduction electrons from calculations to lowest order in the 418 · The spectral weig ht function is the general di stribution taking eK=E. Using an average momentum matrix ele­ fun c ti on describing one-electron properties, and is a ment and neglecting nondiago nal terms in A we obtain :( generalization of the one-electron density matrix, which the simple result is obtained after an integration over the frequencies. Di stributions with respect to a single variable are in the (15) usual way obtained by summation over all other varia­ The neglect of nondiagonal terms s hould be a good bles. Thus, the one-electron density of states is defined approximation if the one-electron functions are care­ >, as fully chosen Bloch functions and core-electron func­ N(w) =Tr A (w) tions. We note that if the electron is ejected from a core level and if we neglect excitations of the core electron = jA(x, x; w)dx=LAH(w). (10) (13) k system, the operator air in eq must destroy a core electron (air = ad, giving As indicated, the definition is independent of the y' choice of matrix basis. W ~ L I(N:, sINv)p"" 128(w-E,,+Es). (16) The s pectral weight function is a key quantity in the K , S Green function formulation of the many-electron Here INvl denotes the ground state of the valence elec­ problem. A more detailed account is give n In many tron system, and IN,:, Sl are excited states in the texts, e.g.
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