Inapplicability of exact constraints and a minimal two-parameter generalization to the DFT+$U$ based correction of self-interaction error

TL;DR

This paper demonstrates that exact constraints are inapplicable for correcting self-interaction errors in DFT using a minimal two-parameter generalization of DFT+U, which effectively recovers the exact energy in a model system.

Contribution

It introduces a generalized DFT+U functional with separate linear and quadratic correction terms, enabling improved self-interaction error correction and simultaneous property corrections.

Findings

Generalized DFT+U recovers exact energy in H2+ model

Pure constraints fail to preserve symmetry

Proposed scheme estimates Hubbard parameters from first principles

Abstract

In approximate density functional theory (DFT), the self-interaction error is an electron delocalization anomaly associated with underestimated insulating gaps. It exhibits a predominantly quadratic energy-density curve that is amenable to correction using efficient, constraint-resembling methods such as DFT + Hubbard $U$ (DFT+$U$). Constrained DFT (cDFT) enforces conditions on DFT exactly, by means of self-consistently optimized Lagrange multipliers, and while its use to automate error corrections is a compelling possibility, we show that it is limited by a fundamental incompatibility with constraints beyond linear order. We circumvent this problem by utilizing separate linear and quadratic correction terms, which may be interpreted either as distinct constraints, each with its own Hubbard $U$ type Lagrange multiplier, or as the components of a generalized DFT+$U$ functional. The latter approach prevails in our tests on a model one-electron system, $H_2^+$, in that it readily recovers the exact total-energy while symmetry-preserving pure constraints fail to do so. The generalized DFT+$U$ functional moreover enables the simultaneous correction of the total-energy and ionization potential or the correction of either together with the enforcement of Koopmans condition. For the latter case, we outline a practical, approximate scheme by which the required pair of Hubbard parameters, denoted as U1 and U2, may be calculated from first-principles.