Study of the correlation effects in a three-electron model system using the projected Hartree-Fock method and the natural spin orbital formalism

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Abstract
Various forms of the projected Hartree‐Fock (PHF) scheme are used to study the correlation effects in a simple three‐electron model system, represented by the PPP model of the allyl radical. The natural spin orbital formalism is used to calculate the correlation energy and spin densities. The unprojected independent particle trial function, which is used in the PHF scheme, always has lower symmetry than the Hamiltonian. The symmetries which are violated, either individually or simultaneously, are those of spin, space, and alternancy. The correlation energy and the spin density on the central atom obtained by the PHF method are compared with the corresponding values given by the exact solution of the model. Both quantities are examined for the whole range of the coupling constant. Due to the simplicity of the model, any two‐parameter symmetry breaking scheme gives results identical with the full CI exact solution. Thus, the same exact wavefunction is obtained in three different ways and has a different structure depending on the two‐parameter scheme used. Among the single symmetry breaking schemes, the spin symmetry violation is the most effective one. Its further improvement, however, is not straightforward and addition of the space symmetry violation leads to a complete reconstruction of the unprojected wavefunction. On the other hand, addition of alternancy symmetry violation to the space symmetry or spin symmetry violation results only in minor corrections to the unprojected wavefunction. This suggests that one should exercise extreme caution in selecting the trial wavefunction for the PHF scheme.