Density functionals in unresticted Hartree–Fock theory

Abstract

The Hohenberg–Kohn theorem is generalized to unrestricted Hartree–Fock (UHF) wave functions. It is postulated that each N-electron density can be expanded in many sets of orbitals, and these orbital sets are continuously deformable into one another. Subject to these postulates several conclusions follow: The UHF wave function functional ψ[ρ] exists for all such densities. It consists of those orbitals which minimize the sum of kinetic and exchange energies, and it can always be chosen to be a ground state wave function. The exact exchange energy, including self-repulsion, is also a functional of the electron density. It is implicitly defined as the expectation value of exchange energy for the wave function functional ψ[ρ]. This work also addresses some general properties of local exchange potentials. A local exchange potential is a multiplicative potential which replaces the exchange operator in the UHF equations. Every method which employs a local exchange potential is variationally equivalent to minimization of kinetic energy alone, subject to constant density constraint. Many ad hoc local exchange potentials are currently in use, but it is here shown that the functional derivative of exchange energy δK/δρ (r) is the optimal choice for a local exchange potential. This choice is best because the resultant orbitals reproduce the correct UHF ground state energy and electron density.