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Unrestricted Projected Hartree-Fock Solutions for Two-Electron Systems

1967; American Institute of Physics; Volume: 154; Issue: 1 Linguagem: Inglês

10.1103/physrev.154.70

1536-6065

Carlos F. Bunge,

Spectroscopy and Quantum Chemical Studies

The singlet and triplet projections of a two-electron unrestricted Slater determinant (USD) are obtained. Then the $S$, ${{\ensuremath{\Sigma}}_{g}}^{+}$, 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 {${R}_{\mathrm{il}|m|}(r){Y}_{\mathrm{lm}}(\ensuremath{\theta}, \ensuremath{\varphi}){\ensuremath{\xi}}_{{m}_{s}}({\ensuremath{\sigma}}_{s})$, $i=1, \ensuremath{\cdots}4$ ($i=1, \ensuremath{\cdots}8$ when $m\ensuremath{\ne}0$); $l=0, 1, \ensuremath{\cdots}\ensuremath{\infty}$; $m=\ensuremath{-}l, \ensuremath{\cdots}l$; ${m}_{s}=\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$}. Similar results are found for even-$P$ states (${L}_{z}=0$), ${{\ensuremath{\Sigma}}_{g}}^{+}$ and ${{\ensuremath{\Sigma}}_{g}}^{\ensuremath{-}}$ 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 $E=\ensuremath{-}2.9030$ 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.