ia ) and de-excitation amplitudes y (n) = (x (n) ia ), for the nth excited state whose vertical excitation energy is ω n . Throughout this work, we use indices i, j, . . . to denote occupied MOs, a, b, . . . to indicate virtual (unoccupied) MOs, and r, s, . . . to denote arbitrary MOs. Spin indices are omitted here; see Ref. 1 for a version of these equations that includes them. The matrices A and B in Eq. (2.2) are Hessians with respect to orbital rotations. In the canonical MO basis that diagonalizes the Fock matrix F, their matrix elements are