The normalization condition for TD-DFT is 1 ia (x 2 ia -y 2 ia ) = 1 (7) so that the denominator in eq. 4 generally differs from unity unless the TDA is invoked, although it does normalize Λ such that 0 ≤ Λ ≤ 1. Larger values indicate a localized excitation whereas Λ = 0 indicates complete spatial separation of the excited electron and the hole. Several puzzling failures in early TD-DFT calculations were explained by using Λ to measure "hidden" CT character, although certain types of charge separation elude this particular metric. The Λ metric has undeniable intuitive appeal as an amplitude-weighted sum of donor-acceptor overlaps O ia , yet it is an incoherent average that does not correspond to the expectation value of any measurable (or even potentially measurable) physical quantity. A proper expectation value is expressible as the trace of an operator with a density matrix, as shown in Appendix A. From another point of view, the definition of Λ involves squaring the amplitudes, then constructing an incoherent superposition of the quantities O ia associated with ψ i → ψ a excitation, rather than evaluating an average using coherent superposition (i.e., an expectation value). A consequence is that Λ is not invariant to rotations of the occupied and virtual MOs, even within the TDA. As such, its numerical value depends on the choice of representation.