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.
Highlights
AND MOTIVATIONIn this work, we rigorously generalize the latter result, building upon the foundation provided by Wu and Van Voorhis (W-VV)’s Constrained density functional theory (cDFT) stationary point classification, first showing that the analysis becomes inconclusive when electronic screening effects are considered
Constrained density functional theory [1] is a generalization of density functional theory (DFT) [2,3] in which external constraints are applied in order to simulate excitation processes, to calculate response properties, or to impose a physical condition that is not met by the unconstrained approximate exchange-correlation functional
Constraining potentials that are nonlocal or orbital dependent may be introduced, moving beyond formal DFT [4]. cDFT enables individual excited states to be studied within the well-established framework of ground-state DFT [4,5,6,7,17,18,19], those excited states which may be represented as the ground state for some potential
Summary
Constrained density functional theory (cDFT) [1] is a generalization of density functional theory (DFT) [2,3] in which external constraints are applied in order to simulate excitation processes, to calculate response properties, or to impose a physical condition that is not met by the unconstrained approximate exchange-correlation functional. Wu and Van Voorhis (W-VV) [5] carried out the pioneering and enabling work in this area, analyzing the relevant derivatives, and their principal results have been subsequently synopsized in numerous works [1,4,18,24,52] It was concluded by W-VV [5] on the basis of nondegenerate perturbation theory that a nontrivial stationary point, for an arbitrary constraint on the electron density, arises only at a maximum of the total energy with respect to a cDFT Lagrange multiplier, and that this solution is unique. This is a central result in cDFT, suggesting the feasibility of its routine automated optimization, which has been extended to multivariate cases in Refs. [1,4]
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