is the exchange-correlation kernel computed as a functional of the density of the core-ionized state. All of the above integrals use molecular orbitals that were optimized for the coreionized state. Within generalized Kohn-Sham theory, these expressions reduce to electron-affinity time-dependent Hartree-Fock (EA-TDHF) in the limit of the HF functional and represent an exact first-order correction to particle-hole interaction errors encountered in TDDFT calculations that employ the adiabatic approximation. While we could have written Eq. 1 without invoking the orbital energy ε i , we chose this form to illustrate that EA-TDDFT comes from the sum of two linear responses, the first of which simply modifies the reference energy by ε i . This form is also useful because it is simple to see that after factoring Ẽn δ ab out of the A matrix, that the matrix A -B is positive definite, allowing for the non-Hermitian eigenvalue equation of EA-TDDFT to be solved by transformation into a Hermitian one, 43