Natural occupation numbers: When do they vanish?
Abstract
The non-vanishing of the natural orbital (NO) occupation numbers of the one-particle density matrix of many-body systems has important consequences for the existence of a density matrix-potential mapping for nonlocal potentials in reduced density matrix functional theory and for the validity of the extended Koopmans' theorem. On the basis of Weyl's theorem we give a connection between the differentiability properties of the ground state wavefunction and the rate at which the natural occupations approach zero when ordered as a descending series. We show, in particular, that the presence of a Coulomb cusp in the wavefunction leads, in general, to a power law decay of the natural occupations, whereas infinitely differentiable wavefunctions typically have natural occupations that decay exponentially. We analyze for a number of explicit examples of two-particle systems that in case the wavefunction is non-analytic at its spatial diagonal (for instance, due to the presence of a Coulomb cusp) the natural orbital occupations are non-vanishing. We further derive a more general criterium for the non-vanishing of NO occupations for two-particle wavefunctions with a certain separability structure. On the basis of this criterium we show that for a two-particle system of harmonically confined electrons with a Coulombic interaction (the so-called Hookium) the natural orbital occupations never vanish.
- Publication:
-
Journal of Chemical Physics
- Pub Date:
- September 2013
- DOI:
- 10.1063/1.4820419
- arXiv:
- arXiv:1306.4142
- Bibcode:
- 2013JChPh.139j4109G
- Keywords:
-
- density functional theory;
- ground states;
- many-body problems;
- orbital calculations;
- wave functions;
- 31.15.ep;
- Variational particle-number approach;
- Physics - Atomic Physics;
- Condensed Matter - Strongly Correlated Electrons;
- Quantum Physics
- E-Print:
- 15 pages, 2 figures