A novel interpretation of reduced density matrix and cumulant for electronic structure theories

We propose a novel interpretation of the reduced density matrix (RDM) and its cumulant that combines linear and exponential parametrizations of the wavefunction. Any n-particle RDM can be written as a weighted average of "configuration interaction" amplitudes. The corresponding n-particle cumulant is represented in terms of two types of contributions: "connected" (statistical averages of substitution amplitudes) and "disconnected" (cross-correlations of substitution amplitudes). A diagonal element of n-RDM represents the average occupation number of the orbital n-tuple. The diagonal elements of 2- and 3-cumulants take particularly elegant forms in the natural spin-orbital basis: they represent the covariances (correlated fluctuations) of the occupation numbers of the orbital pair and triples, respectively. Thus, the diagonal elements of the cumulants quantify the correlation between the orbital occupation numbers. Our interpretation is used to examine the weak to strong correlation transition in the "two electrons in two orbitals" problem.