Long-range interactions and the sign of natural amplitudes in two-electron systems

TL;DR

This paper investigates how long-range Coulomb interactions influence the sign of natural amplitudes in two-electron systems, revealing their connection to wave function structure and occupation number behavior.

Contribution

It establishes a link between long-range interactions and natural amplitude signs, and introduces an avoided crossing mechanism explaining non-zero natural occupation numbers.

Findings

Long-range Coulomb interactions lead to positive natural amplitudes.

Natural amplitudes exhibit avoided crossing behavior as a function of interaction parameters.

Natural occupation numbers never vanish except in special interaction cases.

Abstract

In singlet two-electron systems the natural occupation numbers of the one-particle reduced density matrix are given as squares of the natural amplitudes which are defined as the expansion coefficients of the two-electron wave function in a natural orbital basis. In this work we relate the sign of the natural amplitudes to the nature of the two-body interaction. We show that long-range Coulomb-type interactions are responsible for the appearance of positive amplitudes and give both analytical and numerical examples that illustrate how the long-distance structure of the wave function affects these amplitudes. We further demonstrate that the amplitudes show an avoided crossing behavior as function of a parameter in the Hamiltonian and use this feature to show that these amplitudes never become zero, except for special interactions in which infinitely many of them can become zero simultaneously when changing the interaction strength. This mechanism of avoided crossings provides an alternative argument for the non-vanishing of the natural occupation numbers in Coulomb systems.