Here, L 0 | and |R 0 correspond to the CCSD state; note that often, for X2PA, the CCSD state is the initial state g. Ẽn and Ln | and | Rn are the n th eigenenergy and left and right eigenvectors, respectively, that diagonalize the ĤuS block, which is defined as the Ĥval block with singles-doubles, doubles-singles, and doublesdoubles blocks zeroed out and the CCSD state projected out. These sums in Eqs. (19) and (20) are separately recast into closed-form expressions using response wave functions that span separate determinant spaces as follows: