The values n r = 1-λ 2 r represent the hole that is created and n r = λ 2 r correspond to the excited electron, which demonstrates why ∆P elec and ∆P hole have the same eigenvalues up to a sign. Redundant orbitals (n r = 0) have been eliminated by the SVD in Eq. (3.4). Although the direct connection between the excitation amplitudes, transition density, and one-electron density matrix for the excited state is a unique feature of the single-excitation ansatz, the concept of attachment and detachment densities as eigenfunctions of ∆P, separable based on the sign of the eigenvalues in Eqs. (2.16) and (2.17), is generalizable to wave functions of arbitrary complexity. The individual eigenfunctions of ∆P, which are the NDOs, then generalize the concept of NTOs for many-body theories, without the need to introduce "correlated NTOs". Notice also that the TDM is diagonal in the NTO basis: