Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the <i>v</i> -representability problem

TLDR

Universal variational functionals for electron densities, first‑order density matrices, and natural spin‑orbitals enable ground‑state calculations in atoms, molecules, and solids without requiring the trial function to correspond to a ground state of any external potential. The authors construct universal functionals that search constrained minima over all antisymmetric wavefunctions yielding a given density or density matrix, and they present implementations in Hartree–Fock, other restricted wavefunction theories, and natural spin‑orbital functional theory. The study identifies Percus’s functional as the universal Hohenberg–Kohn functional for kinetic plus electron–electron repulsion energies, establishes a universal functional for the electron–electron repulsion energy of any N‑representable first‑order density matrix, and thereby solves the long‑standing v‑representability problem. The Other section contains only the symbol ρ.

Abstract

Universal variational functionals of densities, first-order density matrices, and natural spin-orbitals are explicitly displayed for variational calculations of ground states of interacting electrons in atoms, molecules, and solids. In all cases, the functionals search for constrained minima. In particular, following Percus [Formula: see text] is identified as the universal functional of Hohenberg and Kohn for the sum of the kinetic and electron—electron repulsion energies of an N -representable trial electron density ρ. Q [ρ] searches all antisymmetric wavefunctions Ψ ρ which yield the fixed. ρ. Q [ρ] then delivers that expectation value which is a minimum. Similarly, [Formula: see text] is shown to be the universal functional for the electron—electron repulsion energy of an N -representable trial first-order density matrix γ, where the actual external potential may be nonlocal as well as local. These universal functions do not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus, the v -representability problem, which is especially severe for trial first-order density matrices, has been solved. Universal variational functionals in Hartree—Fock and other restricted wavefunction theories are also presented. Finally, natural spin-orbital functional theory is compared with traditional orbital formulations in density functional theory.