Meaning and magnitude of the reduced density matrix cumulants

Within the framework of a generalized normal ordering (GNO), invented by Mukherjee [1], the reduced density matrix cumulants of the (multiconfigurational) reference wave function play a central role, as they arise directly from the contraction rules. The extended Wick theorem allows contractions of an arbitrary number of active annihilators and creators through a cumulant of corresponding rank. Because the cumulant rank truncates naturally only at the number of active spin orbitals, practical applications of the GNO concept seem to rely on a fast convergence of the cumulant series, allowing one to neglect cumulants with high rank. By computing cumulant norms for selected systems (and up to rank 16), we demonstrate that the cumulants decay approximately exponentially with increasing rank for single reference cases, while the convergence is generally slower for multireference cases. When strong left-right correlation is present as in the singlet state of a dissociated N-2 molecule, even the cumulant with maximum rank, lambda(12) approximate to -1.5 for a CAS(6,6), is not negligible per se. Besides reporting numerical results, the authors reformulate the theory of reduced density matrices and their cumulants using a notation that is particularly easy to handle, highlighting the close connection to conventional statistics. From this statistical approach, a simple interpretation of reduced density matrices and cumulants follows, according to which an n-body cumulant is a measure for the correlation between the occupation numbers of n spin orbitals. This interpretation is also valid for cumulants with ranks exceeding the number of electrons in the system. (C) 2011 Elsevier B. V. All rights reserved.