Pauli-Principle Restriction on the Two Matrix of Bopp for Atomic Ground States

A lower bound to the ground-state energy of an atom is formulated, using the pair density matrix Γ(2). The major problem in the development of such methods is that of finding necessary conditions on Γ(2) not satisfied by the current optimal class of density matrices. It is shown that the Bopp two-matrix ansatz suffers mainly from the Pauli principle not being satisfied. With the aid of an extensive set of system-dependent identities for Γ(2), numerical computations indicate how energy improvement will follow from insistence on the Pauli principle. A new necessary condition is conjectured which combines the Pauli condition on Γ(1) with a maximum eigenvalue condition on Γ(2).