for which ô and v are projectors onto the occupied and virtual spaces respectively. Using this relation the UW12 energy for a given geminal may be evaluated either using the virtual space or entirely within the occupied space. In the virtual space, the total UW12 energy may be evaluated for a fixed set of orbitals using equation (1). In the occupied space, the UW12 energy may be written solely in terms of the one-particle reduced density matrix allowing the orbitals to be optimized self-consistently within a generalized Kohn-Sham scheme.