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Natural occupation numbers: when do they vanish?

K J H Giesbertz¹, R van Leeuwen

¹Theoretical Chemistry, Faculty of Exact Sciences, VU University, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands.

The Journal of Chemical Physics

|September 21, 2013

Summary

Natural orbital occupations in many-body systems are crucial for density matrix functional theory and Koopmans

Area of Science:

Quantum Chemistry
Many-Body Physics
Computational Chemistry

Background:

Natural orbital (NO) occupation numbers are key properties of the one-particle density matrix in many-body systems.
Their behavior impacts reduced density matrix functional theory and the extended Koopmans' theorem.

Purpose of the Study:

To investigate the relationship between the differentiability of the ground state wavefunction and the decay rate of NO occupations.
To determine conditions under which NO occupations do not vanish, particularly for two-particle systems.

Main Methods:

Application of Weyl's theorem to connect wavefunction differentiability with NO occupation decay.
Analysis of explicit two-particle systems, including those with Coulomb cusps and separable structures.

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Derivation of a general criterion for non-vanishing NO occupations in two-particle wavefunctions.

Main Results:

Wavefunctions with Coulomb cusps typically exhibit a power-law decay in NO occupations.
Infinitely differentiable wavefunctions generally show exponential decay of NO occupations.
Non-analytic wavefunctions, like those with Coulomb cusps, lead to non-vanishing NO occupations.
A derived criterion confirms non-vanishing NO occupations for Hookium (harmonically confined electrons with Coulomb interaction).

Conclusions:

The differentiability of the ground state wavefunction dictates the decay rate of natural orbital occupations.
Non-analyticity in wavefunctions, specifically Coulomb cusps, ensures non-vanishing natural orbital occupations.
This finding has significant implications for the theoretical frameworks of reduced density matrix functional theory and Koopmans' theorem.