6104
J. Phys. Chem. 1996, 100, 6104-6106
Reference-State Density Functional Theory R. K. Nesbet IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099 ReceiVed: September 15, 1995; In Final Form: NoVember 21, 1995X
Many theoretical methods for N-electron systems use a reference state whose electronic density in general differs from that of the associated eigenstate. In contrast, the density functional theory of Kohn and Sham postulates a model reference system for which these two densities are identical. An alternative postulate, requiring the reference state to have maximum projection on the true wave function (the Brueckner-Brenig construction), is shown here to provide a theory in which the ground-state energy is a functional of the reference-state density. This revised formalism makes it possible to identify elements of the theory with specific formulas and concepts of formal many-body theory. In particular, the one-electron wave functions are in one-to-one correspondence with physical quasiparticles of a generalized Fermi-liquid theory, and the one-electron energies are Landau quasiparticle energies.
I. Introduction The density-functional theory (DFT) of Hohenberg, Kohn, and Sham1,2 postulates a model state described by a singledeterminant wave function whose electronic density function is identical to that of the correlated N-electron ground state. Although it has been shown that Kohn-Sham one-electron energies are derivatives with respect to occupation numbers of the DFT total energy,3 the orbital wave functions of the model state have no clear relationship to the quasiparticles defined in many-body theory. Quasiparticle energies correspond to true energy differences for the addition or removal of one electron. A reformulation of density-functional theory is presented here in which the Kohn-Sham model state is replaced by a reference state determined so as to have maximum projection on the ground-state eigenfunction.4,5 This construction defines a Landau energy functional of quasiparticle occupation numbers,6 as postulated in Fermi-liquid theory.7,8 One-electron energies are occupation-number derivatives of this functional. The electronic density of this reference state is not constrained to be identical to the full correlated density. There is no systematic theory of corrections to standard DFT functionals or theory that justifies use of DFT one-electron energy differences as approximate excitation energies. An important goal of the present theory is to validate use of DFT as an initial step in applications of many-body theory that describe excitations or perturbations of the ground state. Individual terms in the present density functional and the resulting one-electron energies have well-defined counterparts in many-body theory. II. Formalism Any N-electron wave function Ψ determines a reference-state Slater determinant Φ by the condition that the projection of Φ on Ψ is maximized. A theorem of Brenig5 shows that this maximal property implies that Φ has no one-electron matrix elements with the orthogonal remainder of Ψ. The set of orthonormal occupied one-electron orbital wave functions of Φ is determined by this condition up to a unitary transformation.4,9 If Φ is not unique in some special case, it will be assumed here that one such function is selected by some criterion. The normalization convention to be assumed here, X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6104$12.00/0
(Φ|Ψ) ) (Φ|Φ) ) 1, implies, by a second basic theorem10,11 that the unsymmetric formula E ) (Φ|H|Ψ) is exact if Ψ is an eigenfunction of H with eigenvalue E. The full N-electron Hamiltonian is H ) T + V + U, where T and V are the kinetic energy and external potential operators, and U is the two-electron Coulomb interaction. By the Brenig theorem, (Φ|H1|Ψ) ) (Φ|H1|Φ) for any one-electron Hamiltonian operator H1. Because T and V are one-electron operators, the unsymmetric energy formula implies that their contributions to N-electron energy eigenvalues can be evaluated as mean values in the reference state. Hence the external potential function can be taken to interact with the electronic density F of the reference state Ψ rather than with the electronic density Fˆ of Ψ. These two density functions are in general not identical. An alternative density functional theory based on the referencestate density is developed here. In the standard Hohenberg-Kohn theory, a universal functional Fˆ [Fˆ] is defined such that the ground-state energy for given external potential V is equal to the energy functional
Eˆ [Fˆ] ) Fˆ [Fˆ] +∫VFˆ d3r
(1)
Spin indices are suppressed in the notation here. The Hohenberg-Kohn theorems imply12,13 that the density Fˆ is determined by
δ Fˆ /δFˆ + V ) µ
(2)
where µ is the chemical potential. If Fˆ is specified, this equation determines V ) VF. Since the constrained-search definition by Levy14 of Fˆ [Fˆ] determines a wave function Ψ as a functional of Fˆ, the Brueckner-Brenig construction determines a referencestate Φ and a reference-state density F as functionals of the specified Fˆ. The unsymmetric energy formula for the groundstate energy EF takes the form
E[F] ) F[F] + ∫ VFF d3r
(3)
where F[F] is a functional whose value is (Φ|T + U|Ψ) for any ground-state wave function Ψ. Explicitly,
F[F] ) Fˆ [Fˆ] + ∫ VF(Fˆ - F) d3r
(4)
for any reference-state density F that corresponds to some ground-state density Fˆ. © 1996 American Chemical Society
Reference-State Density Functional Theory
J. Phys. Chem., Vol. 100, No. 15, 1996 6105
F[F] can be defined directly as a universal functional, independent of the external potential, following the constrainedsearch logic of Levy:
F[F] ) min (Φ|T+U|Ψ)
(5)
ΨfΦfF
As indicated by the notation Ψ f Φ f F, given a spin-density F(r), a search is implied over all N-electron wave functions Ψ whose reference-state density is that specified. This constrainedsearch procedure can be expressed in a formalism in which the potential function VF is a Lagrange multiplier field determined by the assigned density function F. The detailed mathematical argument is given elsewhere.15 Because the unsymmetrical energy formula (Φ|T+V+U|Ψ) is not itself a variational functional, this search procedure may require a restriction on the form of trial wave functions Ψ to avoid the possibility of including energy values below the ground-state energy E[V]. The restricted search obtains the ground-state wave function ΨF and the corresponding reference function ΦF as functionals of F. The kinetic energy functional T[F] is defined by (ΦF|T|ΦF) in the present formalism. A universal exchange-correlation energy functional is obtained by subtraction. Denoting the classical Coulomb energy of the reference density by U[F],
Exc[F] ) F[F] - T[F] - U[F]
(6)
As in standard theory, the present universal functional F is not in general the eigenvalue of an N-electron Hamiltonian. Given reference spin-density F, the spin-indexed potential function VF and the corresponding operator VF are determined by the constrained search such that ΨF is the ground-state eigenfunction of the Hamiltonian H ) T + VF + U with energy eigenvalue EF ) E[F], using the unsymmetric energy formula. Here VF and EF are determined only up to a common additive constant. The only assumption required by this logic is that the local potential VF(r) exists for any physically meaningful F. It will be shown below that E[F] can be used to determine ground-state F if V is given. Equation 3 defines E[F] for V ) VF. Equations 1, 3, and 4 imply that E[F] can be expressed in terms of the HohenbergKohn energy functional:
E[F] ) Eˆ [Fˆ] - ∫ (V - VF)Fˆ d3r
(7)
This equation can be used to define variations of E[F] if V * VF, or equivalently, if F varies for fixed V. For variations about a ground state, with fixed external potential V ) VF but variable number of electrons N, these equations imply
∫ d3r [δF/δF + V - µ]δF ) ∫ d3r [δFˆ /δFˆ + V - µ]δFˆ
(8)
where µ ) ∂E/∂N and ∫ δF ) ∫ δFˆ. Equations 2 and 8 imply that the reference-state density and the external potential are related by
δF/δF + V ) µ
(9)
Equations 3 and 9 imply that E[F] is stationary for fixed V ) VF and fixed N. The minimal value must correspond to the ground state. If eq 9 has multiple solutions F for given V, it must be assumed that the ground-state reference density F is selected. An effective Schro¨dinger equation for the orbital wave functions of the reference state can be derived from functional derivatives of E[F], as in the original theory of Kohn and Sham.2 The present theory is operationally equivalent to Kohn-Sham theory, except that the computed electronic density is no longer
interpreted as the true correlated electronic density. Equation 4 implies that density functionals derived from the interacting electron gas, widely used in the local density approximation (LDA), are equally valid in the present formalism, since the electronic densities of the reference state and the N-electron ground state are identical for the uniform electron gas. The general success of LDA using these functionals also supports the present theory. However, if the density functional is deduced from accurate calculations on finite systems, or if densitygradient corrections are included, as in many recent refinements of LDA,16-18 the theory presented here implies quantitative changes and modified methodology. III. Relation to Landau Fermi-Liquid Theory In the Landau theory of interacting Fermi systems,7,8,19 not restricted to ground states, the energy functional is parametrized as a function of orbital occupation numbers, denoted here by {n}, indexed to correspond to the orbital functions {φ}. To validate the postulate of continuously variable occupation numbers, it will be assumed here that atoms or molecules are represented as noninteracting equivalent cells of an infinite space lattice. Occupation numbers can then take arbitrary rotational values, as they do in energy-band theory. The alternative of considering a statistical ensemble of atoms or molecules in different charge states conflicts with the definition of an optimal reference function here and with use of the unsymmetric energy formula. Given the external potential function, an exact N-electron wave function is specified by the occupation numbers of a reference state, by the orbital basis set, and by a set of coefficients {c} that are elements of an eigenvector of the Hamiltonian matrix in the CI (configuration-interaction) representation. It has been shown6 that a Landau functional is implied if the Brueckner-Brenig construction is used to determine an optimal reference state and the implied orbital basis set for an exact wave function. An energy eigenvalue is stationary with respect to the CI coefficients {c} by solution of the matrix eigenvalue equation and stationary with respect to the set of orbital basis functions {φ} because of the Brenig theorem. To lowest order in infinitesimal variations, the only residual energy variation for fixed external potential is due to variations of the occupation numbers {n}, which become the free parameters in a resulting Landau functional. The unsymmetrical energy formula is valid for all energy eigenstates and is used as an interpolation formula for continuously variable occupation numbers. In the case of finite variations, the CI and Brueckner-Brenig calculations must be repeated over a range of occupation numbers. A finite energy difference is obtained by integrating over this range. The universal ground state exchange-correlation functional defined above implies an equivalent Landau functional in which the explicit dependence of the variational energy on the CI coefficients is replaced by dependence on the reference-state density function F. A universal functional is only defined for ground states, giving derivatives that are valid for infinitesimal variations about a ground state. Finite excitations, with finite changes of occupation numbers, require augmentation of the density functional theory by an appropriate form of many-body theory. Because the coefficients {c} for a ground state are properties of the eigenfunction, they are determined by the constrainedsearch procedure for F[F] as universal functionals of F. The orbital basis set {φ}, defined by the Brueckner-Brenig construction, is also a ground-state functional of F. It follows that for infinitesimal variations about the ground state, for fixed
6106 J. Phys. Chem., Vol. 100, No. 15, 1996
Nesbet
external potential,
E[F] ) (ΦF|T+V|ΦF) + U[F] + Exc[F]
(10)
acts as a Landau functional. The explicit dependence on occupation numbers is given by expanding F ) ∑nφφ*. Then
E[F] ) ∑ ni(i|t+V|i) + 1/2 ∑ ninj(ij|u|ij) + Exc[F] i
(11)
i,j
where t is the one-electron kinetic energy operator, and u denotes the Coulomb interaction. Differentiation of this expression with respect to an occupation number ni gives the Landau one-particle energy,
i ) ∂E/∂ni ) (i|t+V+Vcl+µxc|i)
(12)
in analogy to the theorem of Janak.3,20 Here Vcl is the classical Coulomb potential due to F and µxc is the functional derivative δ Exc/δF. The implied computational procedure is to solve the eigenvalue problem for the effective one-electron Hamiltonian defined by eq 12 and to iterate the reference state spin density to self-consistency. The chemical potential (Fermi level) and occupation numbers are defined as in Kohn-Sham theory. In the present formalism, for atoms or molecules, the Landau oneelectron energies, derivatives with respect to occupation numbers, are distinct from the Dyson one-electron energies associated with poles of the one-particle Green function, which correspond to removal or addition of one electron. Changes of total energy corresponding to finite changes of occupation numbers in general require integrating these energy derivatives over a continuous range of occupation numbers. IV. Discussion The present formalism provides a methodology for applying density-functional theory to strongly correlated states. Unlike the model state postulated in Kohn-Sham theory, the reference state used here is well-defined in detailed CI or many-body theory. The ground-state functional Exc[F] has an explicit representation in CI theory,6 using the Brueckner-Brenig basis {φ} and the CI coefficients cijab for two-particle virtual excitations of the reference state. Introducing occupation numbers for the ground state, this is
Exc ) -1/2 ∑ ninj(ij|u|ji) + i,j
ninj ∑(1 - na)(1 - nb)(ij|uj|ab)cijab ∑ i
