The procedure above could be followed for any difference density matrix, including the relaxed one from a TD-DFT calculation or one that is obtained from a correlated wave function. In the special case that ∆P is the unrelaxed difference density matrix from a TD-DFT calculation [Eq. (2.10)], it follows that ∆P attach ≡ ∆P elec and ∆P detach ≡ ∆P hole . Although this equivalence has been noted before, it does not seem to be widely appreciated. It arises from a unique feature of single-excitation theories, namely, a direct correspondence between CI coefficients and matrix elements of the TDM. For example, x ia = Ψ exc |â † a âi |Ψ 0 in the CIS case. Considering the specific case of ∆P in Eq. (2.10), qualitative insight into the nature of an excited state can often be gleaned by analyzing its particle and hole components ∆P elec and ∆P hole , but it must be borne in mind that electron/hole separation does not survive the contribution from orbital relaxation, i.e., from Z in Eq. (2.9). Nonzero matrix elements Z ia = Z ai introduce occupied-virtual coupling, in contrast to the occupied-occupied and virtual-virtual terms that define the unrelaxed difference density [Eq. (2.12)]. However, one may construct the relaxed difference density,