Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. I. The equations satisfied by the density cumulants

Abstract

Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BC_k and the k-particle contracted Schrodinger equations, CSE_k. These conditions express the k-particle density matrices γ_k in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γ_k in terms of their cumulants λ_k, but this is not sufficient to make the BC_k or CSE_k separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBC_k and ICSE_k of the BC_k and CSE_k can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants λ_k, which are additively separable quantities and which scale linearly with the system size. The IBC_k or the ICSE_k are necessary conditions for ν and the λ_k to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate λ₂ to ν and γ₂. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of ν and the λ_k are determined by the IBC_k or the ICSE_k, the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants λ_k can be truncated at any particle rank, which would not be possible for the density matrices γ_k. For closed-shell states the one-particle approximation agrees with Hartree-Fock.