Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. III. Systems of noninteracting electrons

Abstract

We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBC_k) and the irreducible contracted Schrodinger equations (ICSE_k) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC₁ leads to a Liouville equation for the one-particle density matrix γ₁ = γ, consistent with our earlier analysis that the IBC₁ holds both for a pure and an ensemble state. The IBC₁ or the ICSE₁ must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for ν. For a closed-shell state ν is idempotent, i.e., all natural spin orbitals (NSO's) have occupation numbers 0 or 1, and all cumulants λ_k with k ≥ 2 vanish. For open-shell states there are NSO's with fractional occupation number, and at the same time nonvanishing elements of λ₂, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of ν. There are essentially three possibilities to construct the diagonal elements of ν: (i) to consider the perturbation expansion of the characteristic polynomial of ν, especially the idempotency for closed-shell states, (ii) to rely on the ICSE₁, which (at variance with the IBC₁) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction.