The one-electron picture in the Piris natural orbital functional 5 (PNOF5)

Abstract

The natural orbital functional theory provides two complementary representations of the one-electron picture in molecules, namely, the natural orbital (NO) representation and the canonical orbital (CO) representation. The former arises directly from the optimization process solving the corresponding Euler equations, whereas the latter is attained from the diagonalization of the matrix of Lagrange multipliers obtained in the NO representation. In general, the one-particle reduced-density matrix (1-RDM) and the Lagrangian cannot be simultaneously brought to the diagonal form, except for the special Hartree-Fock case. The 1-RDM is diagonal in the NO representation, but not the Lagrangian, which is only a Hermitian matrix. Conversely, in the CO representation, the Lagrangian is diagonal, but not the 1-RDM. Combining both representations we have the whole picture concerning the occupation numbers and the orbital energies. The Piris natural orbital functional 5 leads generally to the localization of the molecular orbitals in the NO representation. Accordingly, it provides an orbital picture that agrees closely with the empirical valence shell electron pair repulsion theory and the Bent’s rule, along with the theoretical valence bond method. On the other hand, the equivalent CO representation can afford delocalized molecular orbitals adapted to the symmetry of the molecule. We show by means of the extended Koopmans’ theorem that the one-particle energies associated with the COs can yield reasonable principal ionization potentials when the 1-RDM remains close to the diagonal form. The relationship between NOs and COs is illustrated by several examples, showing that both orbital representations complement each other.