Symmetry preserving and improved energy states derived from asymmetric Hartree-Fock

Symmetry preserving Hartree-Fock (HF) states, give higher energies than those of the unrestricted HF. However, in analyzing an unrestricted HF state (UHF) in terms of states transforming according to the Irreducible Representations of the exact Hamiltonian of a many electron system, we prove that some of these states have lower energy than the initial UHF state. The case of rotational and spin symmetries are discussed.