A review on the derivation of the spin-Restricted Hartree—Fock (RHF) Self-Consistent Field (SCF) equations for open-shell systems. Description of different methods to handle the off-diagonal Lagrangian multipliers coupling closed and open shells

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A detailed review is presented on the spin-Restricted Hartree—Fock (RHF) Self-Consistent Field (SCF) method (i) for high-spin ( M s = S) open-shell systems and (ii) for singlet systems with two occupied open-shell orbitals, either of two different irreducible representations (Irreps) or of the same Irrep. Basic knowledge, such as the Born—Oppenheimer separation and approximation, the model of independent particles, Slater determinants and important rules of Group Theory, is shortly summarized. The RHF equations are derived by a minimization of the energy expectation value for the lowest state of a given Irrep and spin multiplicity, described by a spin-adapted wavefunction composed of a single Slater determinant in the high-spin case and a linear combination of two determinants in the open-shell singlet case. Molecular symmetry is considered throughout the paper. The Roothaan method as well as two other widely used methods to handle the orthogonality constraints for the spin orbitals of the respective Slater determinant(s) are described. Detailed algorithms for performing an RHF—SCF procedure with each of these methods are presented.