Unrestricted Projected Hartree-Fock Solutions for Two-Electron Systems

The singlet and triplet projections of a two-electron unrestricted Slater determinant (USD) are obtained. Then the S, Σg⁺, etc. projections are considered and a correspondence is found between these functions and those that arise from a special choice of superposition of configurations. In the atomic case (S states), if the variation method is applied to a specific multiconfiguration wave function, one can readily obtain the corresponding USD which is the Roothaan-type solution to the unrestricted, projected Hartree-Fock equations for that case, in a basis {Ril|m|(r)Yₗₘ(θ, φ)ξ_mₛ(σₛ), i=1, ⋯4 (i=1, ⋯8 when m≠0); l=0, 1, ⋯∞; m=-l, ⋯l; mₛ=±1/2}. Similar results are found for even-P states (Lz=0), Σg⁺ and Σg^- states. This applies to the ground state as well as to the excited states of the same symmetry type. In this way a value better than E=-2.9030 atomic units may be obtained for the nonrelativistic ground-state energy of the He atom, which means that more than 98% of the correlation energy can be accounted for by a single unrestricted projected Slater determinant. The quantitative effect of the use of spin-mixed spin orbitals is discussed, as well as some problems arising in the theory of unrestricted projected Hartree-Fock equations as applied to excitation processes. The use of a multiconfigurational form of the wave function constitutes only an artifice, and the unrestricted projected Hartree-Fock scheme at the present moment does not represent a reasonable approach to the two- and many-electron problems. Nevertheless, the present results indicate that it may become an alternative worthy of investigation.