for excitation energies ω and amplitudes {x, y}. Matrices A and B are orbital Hessians, as discussed elsewhere. Transition amplitudes x ia and y ia indicate excitation and de-excitation, respectively, between an occupied MO ψ i and an unoccupied (virtual) MO ψ a . Importantly, TD-DFT excitation energies and excitedstate properties are invariant to unitary transformations of either the occupied or the virtual MOs, so at this point it does not need to be stipulated which MOs (canonical or otherwise) we mean by ψ i and ψ a . This allows the freedom to use localized MOs or other transformations that are favorable for visualization or interpretative purposes, 2 without affecting observables.