The variation of the energy functional with respect to the density matrix of first order

Abstract

The Hohenberg-Kohn theorem may be extended to prove the existence of an universal energy functional based upon the reduced density matrix of first order. We use a set of parameters representing these density matrices uniquely: the eigenvalues (occupation numbers), the projection operators on the corresponding eigenspaces and their dimensions. Without solving the corresponding variational equations explicitly, it is possible to show that, in the stationary case, the eigenfunctions of an "effective Hamiltonian" may serve as natural spin orbitals. Thus the connection to the familiar orbital representation is established. Because interacting many particle systems may lead to partial occupation numbers, the number of involved natural orbitals may exceed the number of particles in this scheme. The unique parametrization of the density matrix, the consideration of all necessary constraints and the possibility for partial occupation numbers consequently lead to more general and complicated variational equations. From these we derive an "average energy theorem" generalizing results of other authors.