The impossibility of expanding the square root of the electron density as a linear combination of elements of a complete set of basis functions

TL;DR

This paper proves that in orbital-free density functional theory, the square root of the electron density cannot generally be expanded as a linear combination of basis functions, except for one or two-electron systems.

Contribution

It establishes a fundamental limitation in representing the square root of electron density in OFDFT using complete basis sets, highlighting a key difference from bosonic systems.

Findings

The expansion is impossible for most electron systems.

The exception is for one or two-electron systems.

This result impacts the development of basis set methods in OFDFT.

Abstract

In orbital-free density functional theory (OFDFT), an equation exists for $\psi = \sqrt n$, the square root of the ground state electron density $n$. We show that $\psi$ cannot be expanded as a linear combination of elements of a complete set of basis functions except in the case of one or two electron systems. This is unlike the case for the ground state of a system of identical bosons in which the square root of the ground state bosonic density can have an expansion as a linear combination of elements of a complete set of basis functions.