Linearly dependent subspaces and the eigenvalue spectrum of the one-particle reduced density matrix

The structure of the one-particle reduced density matrix when expressed in a Cartesian Gaussian basis set is investigated. A set of exact linear depen dency conditions between products of basis functions, which result from the angular behaviour of the basis functions, is discovered. Some of these exa ct linear dependencies hold simultaneously in both position and momentum sp aces making it possible to alter the one matrix while keeping both the posi tion and momentum densities fixed. The magnitude of this space is easily pr edicted for the Pople and Dunning-Hay basis sets commonly used in quantum c hemical calculations, and we give simple rules for their enumeration. It is further shown that alteration of the one-matrix component in this space al ters the eigenvalue structure of the one-matrix and therefore has consequen ces for N-representability. Using the one-matrix corresponding to a wavefun ction as a starting point, the eigenvalue change is always in the same dire ction, that is small eigenvalues get more negative while large ones become more positive. For independent particle model wavefunctions, which are alre ady extreme in their eigenvalues, no change is possible without breaking th e N-representability conditions. (C) 2000 Elsevier Science B.V. All rights reserved.