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Recently the molecular electronic structure theories for efficiently treatingstatic (or strong) correlation in a black-box manner have attracted muchattention. In these theories, a spin projection operator is used to recover thespin symmetry of a broken-symmetry Slater determinant. Very recently, PonsViver proposed the practical and exact implementation of L\"owdin's spinprojection operator [Int. J. Quantum Chem. 119, e25770 (2019)]. In the presentstudy, we attempt to supply mathematical proofs to Pons Viver's proposals andshow a condition for establishing Pons Viver's implementation. Moreover, weexplicitly derive the (spin projected) extended Hartree-Fock (EHF) equations onthe basis of the model of common orbitals (i.e., closed-shell orbitals used inthe restricted open-shell Hartree-Fock (ROHF) method), which was combined byPons Viver with the EHF method.

A mathematical discussion of Pons Viver's implementation of L\"owdin's spin projection operator