On the validity of power functionals for the homogeneous electron gas in reduced.density-matrix-functional theory

TL;DR

This paper investigates the physical validity and accuracy of power functionals in reduced density-matrix functional theory for the homogeneous electron gas, focusing on the parameter range that yields physically sound and accurate exchange-correlation energies.

Contribution

It refines the constraints on power functionals for the electron gas, identifying the parameter regimes that produce physically valid and accurate results, including the exact correlation energy.

Findings

Valid power functionals require pprox; 0.6 for physical soundness.

Accurate correlation energies are obtained at low densities (pprox; 10).

Narrow regimes of validity are identified for the power parameter pprox; 0.6 and low densities.

Abstract

Physically valid and numerically efficient approximations for the exchange and correlation energy are critical for reduced density-matrix functional theory to become a widely used method in electronic structure calculations. Here we examine the physical limits of power functionals of the form $f(n,n')=(n n')^\alpha$ for the scaling function in the exchange-correlation energy. To this end we obtain numerically the minimizing momentum distributions for the three- and two-dimensional homogeneous electron gas, respectively. In particular, we examine the limiting values for the power $\alpha$ to yield physically sound solutions that satisfy the Lieb-Oxford lower bound for the exchange-correlation energy and exclude pinned states with the condition $n({\mathbf k})<1$ for all wave vectors ${\mathbf k}$. The results refine the constraints previously obtained from trial momentum distributions. We also compute the values for $\alpha$ that yield the exact correlation energy and its kinetic part for both the three- and two-dimensional electron gas. In both systems, narrow regimes of validity and accuracy are found at $\alpha\gtrsim 0.6$ and at $r_s\gtrsim 10$ for the density parameter, corresponding to relatively low densities.