# Reconstruction of the On-Top Two-Electron Density from Natural Orbitals and Their Occupation Numbers

## TL;DR

This paper shows how to reconstruct the on-top two-electron density using natural orbitals and their occupation numbers, offering a new way to validate electronic structure theories.

## Contribution

A novel method to reconstruct the on-top two-electron density from natural orbitals and their occupation numbers is introduced.

## Key findings

- Spatial derivatives of natural orbitals at nodal surfaces encode information about the on-top two-electron density.
- The reconstruction becomes exact with an infinite number of nodal surfaces and a multiplicative constant derived from NOs and their occupation numbers.
- This method provides a new consistency check for electronic structure formalisms using natural orbitals.

## Abstract

Spatial derivatives of the natural orbitals (NOs) at
their nodal
surfaces are shown to encode information about the on-top two-electron
density Φ2(r⃗) in an approximate
manner. This encoding, which becomes exact at the limit of an infinite
number of nodal surfaces, allows the reconstruction of Φ2(r⃗) up to a multiplicative constant
that can be retrieved from an identity involving the NO in question
and its occupation number. This reconstruction provides a new consistency
check for electronic structure formalisms, such as the one-electron
reduced density matrix theory, that employ NOs as primary quantities.