Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. II. Spin-free formulation

Abstract

Recently [W. Kutzelnigg and D. Mukherjee, Chem. Phys. Lett. 317, 567 (2000); D. Mukherjee and W. Kutzelnigg, J. Chem. Phys 114, 2047 (2001)] the irreducible k-particle Brillouin conditions IBCk and the irreducible k-particle contracted Schrödinger equations ICSEk were derived. These permit the definition of a hierarchy of k-particle approximations for the direct calculation of the cumulants λk of the k-particle density matrices. Now, the spin-free form of these conditions, appropriate for a spin-free Hamiltonian, is given. This is particularly useful for open-shell states. The definition of the cumulants of the reduced densities has to be generalized for these anyway, making use of irreducible tensor operators with respect to SU2. There are two alternative definitions of spin-free cumulants, of which the one in terms of spin-free reduced density matrices appears to be preferable. Alternatively to the straight spin-free formulation, we also present a theory in terms of spin-free operators adapted to the symmetric group. Partial trace relations that relate the elements of the cumulants of different particle rank are derived. There are partial trace relations for “exchange elements,” which are determined by the total spin quantum number S. From these relations the individual exchange elements of Λ2 can be obtained in special cases. This allows a simple formulation of the stationarity conditions for open-shell states of any spin multiplicity.