Open-shell Hartree-Fock theory RHF

The UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130], 

To illustrate the modifications of UHF formalism, it is convenient to consider pure spin symmetry for a single Slater determinant with Nc doubly occupied spatial orbitals Xi and N0 singly occupied orbitals y . The corresponding UHF state has Na mj = occupied spin orbitals / and Np rns = — J, occupied spin orbitals / f. The number of open-shell and closed-shell orbitals are, respectively Na = Na — Np 0 and Nc = Np. Occupation numbers for the spatial orbitals are nc = 2, n ° = 1. If all orbital functions are normalized, a canonical form of the RHF reference state is defined by orthogonalizing the closed- and open-shell sets separately. 

Nonzero Lagrange multipliers are required for orthogonalization of orbitals x° to Xc. The energy functional (4 // 4 ) is 

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