Natural occupation numbers: When do they vanish?
Giesbertz, K., & van Leeuwen, R. (2013). Natural occupation numbers: When do they vanish?. Journal of Chemical Physics, 139(10), Article 104109. https://doi.org/10.1063/1.4820419
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© 2013 AIP Publishing LLC. Published in this repository with the kind permission of the publisher.
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.
...


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