• DOI:10.1063/1.4820419
• Corpus ID: 10998567

Natural occupation numbers: when do they vanish?

@article{Giesbertz2013NaturalON,
  title={Natural occupation numbers: when do they vanish?},
  author={Klaas J. H. Giesbertz and Robert van Leeuwen},
  journal={The Journal of chemical physics},
  year={2013},
  volume={139 10},
  pages={
          104109
        }
}

• K. Giesbertz, R. van Leeuwen
• Published 18 June 2013
• Mathematics
• The Journal of chemical physics

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.