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Physical Review A

Properties of the Goedecker-Umrigar functional for the many-electron problem and its generalization

A. Holas

Published May 1, 1999• 56 citations
Publisher

Abstract

The calculational scheme for an approximate ground-state energy of a many-electron system, recently proposed by Goedecker and Umrigar (GU) [Phys. Rev. Lett. 81, 866 (1998)], is analyzed in terms of the reduced-density-matrix approach. Its underlying object, the GU density-matrix functional, is generalized, possibly leading to improved energies. Properties of the solutions of the ground-state problem obtained from the original and the generalized GU functionals are established.

DOI
10.1103/physreva.59.3454

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The calculational scheme for an approximate ground-state energy of a many-electron system, recently proposed by Goedecker and Umrigar (GU) [Phys. Rev. Lett. 81, 866 (1998)], is analyzed in terms of the reduced-density-matrix approach. Its underlying object, the GU density-matrix functional, is generalized, possibly leading to improved energies. Properties of the solutions of the ground-state problem obtained from the original and the generalized GU functionals are established.
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The momentum distributions of a homogeneous electron gas, arising from a simple one-matrix functional for the electron-electron repulsion energy that generalizes the Hartree-Fock and Goedecker-Umrigar approximations, are analyzed in detail. Their properties are found to depend strongly on the exponent \ensuremath{\beta} that enters the functional. Smooth momentum distributions that, together with the corresponding energies per particle, exhibit simple scaling behavior with respect to the electron density \ensuremath{\rho}, are obtained only for $\frac{4}{5}<\ensuremath{\beta}<\frac{4}{3}$ and $\ensuremath{\rho}<~{\ensuremath{\rho}}_{\mathrm{crit}}(\ensuremath{\beta}).$
A density-matrix formulation of the ab initio methods of nonrelativistic quantum mechanics is presented. The variational equations for the standard calculational techniques of molecular quantum mechanics are derived from a single expression for the total energy of an N-particle system. A generalization of the Hohenberg-Kohn theorem is proved, showing that the ground-state energy of a many-body Hamiltonian is a functional of the single-particle density matrix and hence of the first-order reduced-density function (the diagonal of which is the particle density of the standard theorem). The theorem is used to develop a new method to calculate the total energy and single-particle properties of the many-body problem.
The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
The methods of quantum chemistry and solid state theory to solve the many-body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, NNN-representability), and their determining equations (contracted Schrödinger equations) and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasi-particle theory: from finite to extended systems (incremental method), from density to density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two-body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.