Here, i, j, • • • represent the valence occupied orbitals, I, J, • • • represent the core occupied orbitals, and a, b, • • • represent the unoccupied orbitals. The state-index k in |R k refers to either the ground state g or the final state f . Note that we have projected out the CCSD state in the RHS of Eq. ( ) to account for its omission in the sum over valence "uS" states in Eqs. (19) and (20).