The matrix U o transforms the canonical occupied MOs into a set of "hole" orbitals that we will call {ψ hole i (r)}, while U v transforms the canonical virtual MOs into a corresponding set of "particle" (or "electron") orbitals {ψ elec i (r)} where i = 1, . . . , n occ in both cases, because even for the virtual orbital transformation in eq. (3.2) there are only n occ nonzero eigenvalues. These transformed orbitals are the NTOs for the hole and for the excited electron, respectively. These are useful analysis tools because they reduce the 2n occ n virt excitation amplitudes x ia and y ia into just n occ unique amplitudes, as will be discussed further in Section 3.2. For now, we simply note that the particle and hole densities are diagonal in the NTO representation, which in real space means