where the bra and ket vectors denote single configurations obtained by single excitations p → q and r → s from M S = +1 and -1 references, respectively. However, the nonorthogonality of alpha (or beta) MOs of bra and ket can arise some difficulties. Although one can compute the coupling terms by using a singular value decomposition of the overlap integral of the nonorthogonal molecular orbitals, we simply used the following spin-pairing coupling schemes in this work. We first define the two types of couplings can be defined by