, where the spin label indicates the α or β spin-orbitals in Eq. . It is easy to see that γ ∆Ms=0 triplet matrix is constructed through a † pα a qαa † pβ a qβ excitations between the bra and ket states. Although it is a triplet excitation operator, for the consistency with other triplet excitation operators, the γ ∆Ms=0 triplet matrix should be further divided by √ 2, as per definition of T 1,0 in Eq. . If the spin projections of bra and ket differ by ±1, the corresponding excitation operators are spin-flip triplet operators, and no decomposition is needed. However, for spintensor form they should be multiplied by the appropriate phase factors from Eqs. ( ) and . We note that for these spin-flipping excitations right-to-left and left-to-right density matrices gain opposite phase factors. Once the density matrix between the states with the same spin projection is computed, one can easily calculate u from Eq. ( ) as