Optimization of constrained density functional theory

TL;DR

This paper advances constrained density functional theory (cDFT) by developing a rigorous framework for energy derivatives, response functions, and stability analysis, enabling more efficient and reliable application of multiple constraints in electronic structure calculations.

Contribution

It provides a comprehensive theoretical foundation for automated Lagrange multiplier optimization in cDFT, including proofs of stationary point properties and analysis of solution multiplicity and stability.

Findings

Stable stationary points occur only at energy maxima with respect to Lagrange multipliers.

Multiple solutions, hysteresis, and energy discontinuities can occur in cDFT.

Derived expressions for dielectric function and condition number related to multi-constraint cDFT.

Abstract

Constrained density functional theory (cDFT) is a versatile electronic structure method that enables ground-state calculations to be performed subject to physical constraints. It thereby broadens their applicability and utility. Automated Lagrange multiplier optimisation is necessary for multiple constraints to be applied efficiently in cDFT, for it to be used in tandem with geometry optimization, or with molecular dynamics. In order to facilitate this, we comprehensively develop the connection between cDFT energy derivatives and response functions, providing a rigorous assessment of the uniqueness and character of cDFT stationary points while accounting for electronic interactions and screening. In particular, we provide a new, non-perturbative proof that stable stationary points of linear density constraints occur only at energy maxima with respect to their Lagrange multipliers. We show that multiple solutions, hysteresis, and energy discontinuities may occur in cDFT. Expressions are derived, in terms of convenient by-products of cDFT optimization, for quantities such as the dielectric function and a condition number quantifying ill-definition in multi-constraint cDFT.