The formula in Eq. ( ) has a straightforward interpretation. The leading term, which is given by Eq. ( ), is difference between eigenvalues E k and E 0 of the state-specific Schrödinger equation, where E k is correct through first order in perturbation theory. This eigenvalue difference becomes a free energy upon subtracting W f k(0) -W f 0 , which is the difference between the work required to polarize the fast charge on the excited state relative to that required on the ground state. Finally, W 0,k(0) is the aforementioned charge-separation penalty.