The term ∂F iaσ /∂P jbσ in A, along with the entirety of B, has the form of a coupling matrix that modifies the zeroth-order (independent-particle) excitation energies, ε aσ -ε iσ . The solution of Eq. ( ), for the Ith excited state, consists of an excitation energy (eigenvalue) ω I = E I -E 0 , along with vectors x I and y I consisting of excitation amplitudes and de-excitation amplitudes, respectively. We will use indices i, j, . . . to denote occupied MOs and a, b, . . . for virtual MOs, and p, q, . . . for arbitrary MOs, as in the p that appears in Eq. ( ). The solution vector (x, y) to Eq. ( ) parameterizes the transition density ρ 0I (r, r ) of the excited state in question: