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Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals

Andrea Bordignon, Geneviève Dusson, Éric Cancès, Gaspard Kemlin, Rafael Antonio Lainez Reyes, Benjamin Stamm

TL;DR

The work provides fully guaranteed, computable a posteriori bounds for the energy in periodic Kohn–Sham DFT with convex density functionals, using a density-matrix and planewave discretization framework. By proving an abstract energy-difference bound under convexity and combining it with rigorous bounds for linear eigenvalue problems, the authors decompose the total error into discretization and SCF-iteration contributions and present practical strategies based on operator splitting and Neumann-series inverses. Numerical experiments on 1D toy models and 3D insulating Si/HF systems demonstrate that the zeroth-order bound, despite its simplicity, closely tracks the true energy error and enables adaptive refinement guidance; nonconvex xc functionals still yield informative bounds in practice. Overall, the results offer a certify-and-refine approach for reliable electronic structure computations in periodic systems with convex functionals and provide actionable insights for extensions to Brillouin-zone discretization and Neumann-series truncation errors.

Abstract

In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.

Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals

TL;DR

The work provides fully guaranteed, computable a posteriori bounds for the energy in periodic Kohn–Sham DFT with convex density functionals, using a density-matrix and planewave discretization framework. By proving an abstract energy-difference bound under convexity and combining it with rigorous bounds for linear eigenvalue problems, the authors decompose the total error into discretization and SCF-iteration contributions and present practical strategies based on operator splitting and Neumann-series inverses. Numerical experiments on 1D toy models and 3D insulating Si/HF systems demonstrate that the zeroth-order bound, despite its simplicity, closely tracks the true energy error and enables adaptive refinement guidance; nonconvex xc functionals still yield informative bounds in practice. Overall, the results offer a certify-and-refine approach for reliable electronic structure computations in periodic systems with convex functionals and provide actionable insights for extensions to Brillouin-zone discretization and Neumann-series truncation errors.

Abstract

In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.
Paper Structure (23 sections, 3 theorems, 97 equations, 7 figures, 4 tables)

This paper contains 23 sections, 3 theorems, 97 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem setting, discretization, and practical resolution
  3. Functional setting
  4. Traces of operators and density matrices
  5. Problem formulation
  6. Kohn--Sham DFT equations
  7. Discretization
  8. Self-consistent field iterations
  9. A posteriori analysis
  10. Abstract analysis
  11. Error bounds for linear eigenvalue problems
  12. Practical strategies for efficient computation of error bounds based on operator splitting
  13. Error bounds for the nonlinear problem
  14. Numerical results
  15. Numerical framework
  16. ...and 8 more sections

Key Result

Lemma 3.1

For any $\mu \in \mathbb{R},$ for any $\gamma_1,\gamma_2\in \mathcal{M},$ there holds

Figures (7)

  • Figure 1: Tracking of the error $E(\gamma_{N,m}) - E(\gamma_\star)$ for a 1D toy model. (Left) full-inversion bound and its zeroth- and first-order approximations. (Right) Zeroth-order bound and its splitting between SCF and discretization contributions, as in \ref{['eq: full bound']}.
  • Figure 2:
  • Figure 3: Tracking of the error $E(\gamma_{N,m}) - E(\gamma_\star)$ for a Si crystal (one $\bm k$-point). (Left) Full-inversion bound with zeroth- and first-order approximations. (Right) Zeroth-order bound and its splitting between SCF and discretization contributions, as in \ref{['eq: full bound']}.
  • Figure 4: Tracking of the error $E(\gamma_{N,m}) - E(\gamma_\star)$ for a Si crystal (one $\bm k$-point), with the zeroth-order bound (left) and first-order bound (right). We added the associated guaranteed bounds by estimating the remainders and their optimization with respect to the shift.
  • Figure 5: Tracking of the error $E(\gamma_{N,m}) - E(\gamma_\star)$ for a Si crystal (eight $\bm k$-points). (Left) Full-inversion bound with zeroth- and first-order approximations. (Right) Zeroth-order bound and its splitting between SCF and discretization contributions, as in \ref{['eq: full bound']}.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Remark 2.1: Aufbau principe
  • Remark 2.2: Reduced Hartree--Fock model
  • Remark 2.3: Spins
  • Remark 2.4: Brillouin zone discretization
  • Lemma 3.1
  • Proof 1
  • Corollary 3.3
  • Proof 2
  • Remark 3.4
  • Remark 3.5: Negative Sobolev norms
  • ...and 4 more