The name NTOs was first suggested by Martin in 2003, but this form of analysis was actually introduced much earlier by Luzanov and co-workers. It has since be rediscovered (and expressed in the notation used above) by others. The terminology for these orbitals reflects the sense in which "natural" is used in quantum chemistry to mean eigenfunctions of a density matrix. Just as natural orbitals are eigenfunctions of P (even in the case of a correlated wave function), with eigenvalues that constitute natural occupation numbers, the NTOs diagonalize the TDM. Within a single-excitation model, this is equivalent to diagonalizing the difference density matrix ∆P, although that equivalence is lost in the case of a correlated wave function. (In that case, one must distinguish between NTOs that diagonalize the TDM and natural difference orbitals that diagonalize the difference density matrix. ) Similarly, natural ionization orbitals diagonalize the difference density obtained upon electron removal. None of these quantities should be confused with natural bond orbitals or any of the other "natural" concepts introduced by Weinhold and co-workers.