Irreducible Brillouin conditions and contracted Schrödinger 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 BCk and the k-particle contracted Schrödinger equations, CSEk. These conditions express the k-particle density matrices γk in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γk in terms of their cumulants λk, but this is not sufficient to make the BCk or CSEk 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 IBCk and ICSEk of the BCk and CSEk 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 λk, which are additively separable quantities and which scale linearly with the system size. The IBCk or the ICSEk are necessary conditions for γ and the λk 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 γ and γ2. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of γ and the λk are determined by the IBCk or the ICSEk, 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 λk can be truncated at any particle rank, which would not be possible for the density matrices γk. For closed-shell states the one-particle approximation agrees with Hartree–Fock.