The strictly-correlated electron functional for spherically symmetric systems revisited

TL;DR

This paper investigates the strong-interaction limit of the Hohenberg-Kohn functional for spherically symmetric systems, analyzing the validity of conjectured co-motion functions and their impact on the interaction energy and functional derivatives.

Contribution

It revisits the conjectured co-motion functions for radially symmetric densities, demonstrating they are nearly optimal and retain correct functional derivatives even when not truly optimal.

Findings

Conjectured maps are not always optimal but yield nearly minimal interaction energy.

The functional built from these maps has the correct functional derivative.

Numerical results show close approximation to true minimum energy.

Abstract

The strong-interaction limit of the Hohenberg-Kohn functional defines a multimarginal optimal transport problem with Coulomb cost. From physical arguments, the solution of this limit is expected to yield strictly-correlated particle positions, related to each other by co-motion functions (or optimal maps), but the existence of such a deterministic solution in the general three-dimensional case is still an open question. A conjecture for the co-motion functions for radially symmetric densities was presented in Phys.~Rev.~A {\bf 75}, 042511 (2007), and later used to build approximate exchange-correlation functionals for electrons confined in low-density quantum dots. Colombo and Stra [Math.~Models Methods Appl.~Sci., {\bf 26} 1025 (2016)] have recently shown that these conjectured maps are not always optimal. Here we revisit the whole issue both from the formal and numerical point of view, finding that even if the conjectured maps are not always optimal, they still yield an interaction energy (cost) that is numerically very close to the true minimum. We also prove that the functional built from the conjectured maps has the expected functional derivative also when they are not optimal.