Density Matrices and Density Functionals

Abstract

The Schrödinger equation satisfied by the square root of the electron density is derived without approximation from the theory of marginal and conditional amplitudes. The equation arises from a factorization of the total N—electron wavefunction defined by the normalisation appropriate to a conditional amplitude. The effective potential is, in contrast to that in the one—electron Dyson equation, a scalar, local potential that only depends upon one stationary state of the N—electron system. The equation is easily transformed into an exact differential equation for the electron density itself; the transformation leaves the potential energy and the energy eigenvalue unchanged. In this exact equation the kinetic energy is identical with the kinetic energy in the Thomas-Fermi-Weizsäcker differential equation. The exact equation applies to excited states as well as to the ground state, thus extending the Hohenberg-Kohn density-functional theorem to excited states. The exact equation provides a basis for self-consistent-density calculations. This Schrödinger equation is an exact dynamical model for computing effective one-electron potentials from known N—electron wavefunctions, or from experimental electron densities. Two forms of the theory are presented: 1) the static nuclei model pertinent to theoretical calculations within the Born-Oppenheimer separation, and 2) the non-Born-Oppenheimer (vibrationally averaged) model appropriate to computation of the effective one-electron potential from experimental electron densities.