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 $S$, ${{\Sigma }_g}^{+}$, 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 {$R_{\mathrm{il}\left|m\right|\mathrm{}}\mathrm{}\left(r\right)\mathrm{}{Y}_{\mathrm{lm}}(\theta , \phi ){\xi }_{m_s}\left({\sigma }_s\right)$, $i=1, \cdots 4$ ($i=1, \cdots 8$ when $m\ne 0$); $l=0, 1, \cdots \infty$; $m=-l, \cdots l$; $m_s=\pm \frac{1}{2}$}. Similar results are found for even-$P$ states ($L_z=0\mathrm{}$), ${{\Sigma }_g}^{+}$ and ${{\Sigma }_g}^{-}$ 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 $E=-2.9030\mathrm{}$ 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.