Invariance of the cumulant expansion under 1-particle unitary transformations in reduced density matrix theory

In the iterative solution of the contracted Schrِdinger equation (CSE) the 3- and 4-particle reduced density matrices (RDMs) are reconstructed from the 2-RDM via cumulant expansions. Under 1-particle unitary transformations, we establish that the connected (or cumulant) part of an RDM maps onto the connected part of the RDM in the transformed basis set. Consequently, neglecting the connected RDM in the CSE produces an error which is invariant under unitary transformations of the one-particle basis set. We illustrate this result with calculations on beryllium. The present results are applicable to unitary localization in linear-scaling RDM calculations for large molecules.