Necessary conditions for the N-representability of pair distribution functions

A necessary condition for the N-representability of the electron pair density proposed by one of the authors (E. R. D.) is generalized. This shows a link between this necessary condition and other, more widely known, N-representability conditions for the second-order density matrix. The extension to spin-resolved electron pair densities is considered, as is the extension to higher-order distribution functions. Although quantum mechanical systems are our primary focus, the results are also applicable to classical systems, where they reduce to an inequality originally derived by Garrod and Percus. As a simple application, bounds to the average angle between an electron pair are derived. It is shown that computational methods based on variational minimization of the energy with respect to the electron pair density can give extremely poor results unless robust N-representability constraints are considered. For reference, constraints for the N-representability of the pair density are summarized.