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Physical Review A

Variational principle for the ground-state energy as a functional of the one-particle density matrix: Beyond Hartree-Fock theory

Abraham Klein, Reiner M. Dreizler

Published Apr 1, 1998• 11 citations
Publisher

Abstract

In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.

DOI
10.1103/physreva.57.2485

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In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.
We discuss the one-particle density-matrix functional theory for the N-particle ground-state problem. Using the variational principle we obtain a set of self-consistent equations which could be useful for practical calculations. The derived relation between the eigenvalues of the density matrix and the chemical potential agrees with that of Gilbert. The present theory is compared with the Kohn–Sham and the Hartree–Fock approach to the ground state problem. It is found that both approaches can be obtained from the present theory under certain assumptions.
The analysis in this paper shows that the Hohenberg-Kohn theorem is the constellation of two statements: (i) the mathematically rigorous Hohenberg-Kohn lemma, which demonstrates that the same ground-state density cannot correspond to two different potentials of an external field, and (ii) the hypothesis of the existence of the universal density functional. Based on the obtained explicit expression for the nonrel-ativistic particle energy in a local external field, we prove that the energy of the system of more than two non-interacting electrons cannot be a functional of the inhomogeneous density. This result is generalized to the system of interacting electrons. It means that the Hohenberg-Kohn lemma cannot provide justification of the universal density functional for fermions. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect. In the framework of the density matrix functional theory, the hypothesis of the existence of the universal density matrix functional corresponds to the cases of noninteracting particles and to interaction in the Hartree-Fock approximation.
The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of 4He^{4}\mathrm{He}4He atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.
A density-matrix formalism is developed based on the one-particle density-matrix of a single-determinantal reference state. Unlike traditional density-functional-theory approaches, the vvv-representable problem does not appear in the proposed method, nor the need to introduce functionals defined by a constrained search. The correlation-energy functionals are nonuniversal, in the sense that they depend on the external potential. Nevertheless, model systems can still be used to derive universal energy functionals. Variational and nonvariational energy functionals are introduced that yield the target-state energy when the reference state---or its corresponding one-particle density matrix---is constructed from Brueckner orbitals. Nonvariational energy functionals yield generalized Hartree-Fock equations involving a nonlocal correlation potential and the Hartree-Fock exchange operator; these equations are obtained by imposing the Brillouin-Brueckner condition. The same equations---for the most part---are obtained from variational energy functionals using functional minimization, yielding the (kernel of the) correlation potential as the functional derivative of correlation-energy functionals. Approximations for the correlation-energy functions are introduced, including a one-particle-density-matrix variant of the local-density approximation, a variant of the Colle-Salvetti functional, and a linear combination of the two that is a variant of the correlation-energy functional within the hybrid, three-parameter, Becke-Lee-Yang-Parr density functional (B3LYP).
A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.
The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for NNN-representable density kernels, that any finite non-negative differentiable function is an NNN-representable particle density.