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Physical review. B, Condensed matter

Extrema of the density functional for the energy: Excited states from the ground-state theory

John P. Perdew, Mel Levy

Published May 15, 1985• 136 citations
Publisher

Abstract

Although every stationary-state density ${n}_{i}$(r\ensuremath{\rightarrow}) of a many-particle system is not an extremum of the ground-state density functional ${E}_{v}$[n], every extremum of ${E}_{v}$[n] [i.e., every solution of the Euler equation \ensuremath{\delta}${E}_{v}$/\ensuremath{\delta}n(r\ensuremath{\rightarrow})=\ensuremath{\lambda}] is a stationary-state density ${n}_{i}$(r\ensuremath{\rightarrow}). Always, ${E}_{v}$[${n}_{i}$]\ensuremath{\le}${E}_{i}$, where ${E}_{i}$ is the lowest stationary-state energy for density ${n}_{i}$(r\ensuremath{\rightarrow}); the equality holds if and only if ${n}_{i}$(r\ensuremath{\rightarrow}) is an extremum of ${E}_{v}$[n]. The extrema lying above the absolute minimum are excited-state densities which fail to be pure-state v-representable. Surprisingly, infinitesimal number-conserving density variations \ensuremath{\delta}n(r\ensuremath{\rightarrow}) about an extremum n(r\ensuremath{\rightarrow}) do not lead to energy variations \ensuremath{\delta}${E}_{v}$ of order (\ensuremath{\delta}n${)}^{2}$ when \ensuremath{\delta}n(r\ensuremath{\rightarrow})/${n}^{1/2}$(r\ensuremath{\rightarrow}) fails to be square-integrable; in fact, variations \ensuremath{\delta}${E}_{v}$ of order \ensuremath{\Vert}\ensuremath{\delta}n\ensuremath{\Vert} about the ground state are exemplified by the recently discovered ``derivative discontinuities of the energy.'' This unconventional behavior of ${E}_{v}$[n] may be traced in part to an asymptotic divergence of ${\ensuremath{\delta}}^{2}$${E}_{v}$/\ensuremath{\delta}n(r\ensuremath{\rightarrow})\ensuremath{\delta}n(r\ensuremath{\rightarrow}'). Conditions are presented under which a self-consistent solution of the Kohn-Sham single-particle problem represents an extremum of ${E}_{v}$[n]. The multiplets of the ground-state orbital configuration of the carbon atom are examined. The local-density and Langreth-Mehl approximations are found to yield a remarkably accurate account of the degeneracy of the various ground-state densities for this system, but no estimate of the multiplet splitting is obtained. Finally, aspects of v-representability are discussed, with emphasis on the iron atom.

DOI
10.1103/physrevb.31.6264

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We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00
The well-known and well-used Density Functional Theory (DFT) is a variational theory for the ground-state. There are no guarantees that it delivers good results for excited state properties such as quasiparticle energies or wave function. All that is guaranteed is that the total energy is exact, the chemical potential is correct, and that the eective electronic states (Kohn-Sham states) generate the right electron density at the variational minimum. Luttinger-Ward approaches to electronic structure are attractive as beyond-DFT schemes since, in principle, they oer a variational functional of the one-electron Green’s function that yields both the correct total energy and all one-particle properties at the variational extremum. However, compared to DFT, these methods face a number of additional challenges beyond the familiar problem of choosing an exchange-correlation functional: (a) they have a much greater computational cost compared to DFT, (b) one must choose some set of trial Green’s functions, and (c) one must locate an extremum (not necessarily a minimum). This talk aims both to be pedagogical and to provide a summary of our most recent work in this area. We focus mainly on the random-phase approximation (RPA) for the correlation functional which yields the GW self-energy for the Green’s function. We will describe ecient approaches for computing the RPA correlation energy; how various types of approximations to the correlation energy create a ladder of approximate self-energy operators and what type of physics is included in these approximations (exchange, screening, Hubbard U, etc.); and how the extremization is in fact highly problematic. We will end with some ideas for making practical progress with the extremization and their relation to the successful Quasiparticle Self-consistent GW (QSGW) scheme [1].
The Hohenberg–Kohn theorem implies the existence of an energy functional based solely on the first-order reduced density matrix of the ground state of an atomic or molecular system. Application of the variational principle for the functional EvM[γ] generates a set of coupled Euler equations for the representation coefficients and spin orbitals of a rank-M approximation to the exact ground-state density matrix. Defining the (assumed Hermitian) kernel FM[γ;x′,x]≡δEvM/δγ (x,x′), the equations in an arbitrary representation for the approximate density matrix, γ (x,x′) =ΣijMψ1(x) γijψj* (x′) are the following: ℱdξFM[γ;x′,ξ]ψi(ξ) =μψi(x′); i=1, 2,...,M; ℱdξFM[γ;x′,ξ] ΣMiψi(ξ) γ ij=ΣMiψi(x′) λij; j=1, 2, ...,M. The quantity μ is the chemical potential of the system of interest and the λij are a set of M2 Lagrange multipliers constraining the orthonormality of the spin orbital basis {ψ}. The coefficients γij must be chosen such that FM has the degenerate eigenvalue spectrum, FiiM=μ, i=1, 2,...,M, for all partially occupied orbitals. The spin orbitals, for a fixed set of coefficients, must be simultaneously determined so that λ is Hermitian. Provided all occupation numbers lie on the open interval (0,1), the stationary density matrix itself obeys the eigenvalue equation ℱdξFM[γ;x,ξ]γ (ξ,x′) =μγ (x,x′), and for any stationary density matrix the following commutation rules are valid: [FM[γ], γ]=0; [λ, γ]=0. The matrices γ and λ consequently can be brought simultaneously to diagonal form, and the canonical representation of the energy functional provides an eigenvalue equation determining the natural spin orbitals. The value of the chemical potential is μ=εi/ni, i=1, 2, ...,M; ε=diag (λ); from which follows the distribution function governing the occupation numbers of a stationary density matrix. Two limiting forms of the variational principle are examined, the exact and Hartree–Fock functionals, and related previous work by Gilbert is discussed. The physical content of the equations is illuminated by identification of the chemical potential as the negative of the electronegativity; Sanderson’s Principle of Electronegativity Equalization follows.