Here we use an extra prime ( ′ ) to denote quantities calculated at the second geometry, for example j ′ , b ′ , K ′ . Note that t j ′ b ′ = tj′b′ for a restricted calculation and Eq. ( ) requires evaluating the overlap of two singly-excited determinants, Φ a i Φ b ′ j ′ and Φ a i Φ b′ j′ . As a practical matter, all results below can be easily extended to triplet calculations, unrestricted calculation, or spin-flip calculations. See Appendix D and E.