Irreducible Brillouin conditions and contracted Schrodinger equations for n-elecrotron systems. II. Spin-free formulation

摘要

Recently [W. Kutzxelnigg and D. Mukherjee, Chem. Phys. Lett. 317, 567 (2000); D. Mukherjee and W. Kutzelnigg, J. Chem. Phys 114, 2047 (2001)] the irreducible k-particle Brilliuin conditions IBC_k and the irreducible k-particle contracted Schrodinger equations ICSE_k were derived. These permit the definition of a hierarchy of k-particle approximations for the direct calculation of th cumulants #lambda#_k of the k-particle density matrices. Now, the spin-free form of these conditions, appropriate for a spin-free hamiltonian, is given. This is particlarly useful for open-shell states. The dfinition of the cumulants of the reduced densities has to be generalized for these anyway, making use of irreducible tensor operators with respect to SU_2. there are two alternative definitions of spin-free cumulants, of which the one in temrs of spin-free reduced density matrices appears to be preferable. Alternatively to the straight spin-free formulation, we also present a theory in temrs of spin-free operators adapted to the symmetric groups. Partial trace relations that relate the elements of the cumulants of different particle rank are erived. There are partial trace relations for "exchange clements," which are determined by the total spin quantum number S. From these realtions the individual exchange elemtns of #LAMBDA#_2 canb e obtianed in special cases. this allows a simple formulation of the stationarity conditions for open-shell states of any spin multiplicity.