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Energy-Adaptive Riemannian Conjugate Gradient Method for Density Functional Theory

Daniel Peterseim, Jonas Püschel, Tatjana Stykel

TL;DR

This work addresses efficient Kohn–Sham energy minimization by formulating a constrained optimization on the infinite-dimensional Stiefel manifold and introducing an energy-adaptive Riemannian conjugate gradient (EARCG) method. The core approach combines a problem-aware shifted-Hamiltonian metric, a polar retraction with a differentiated vector transport, and a hybrid FR-PRP CG update, along with an adaptive step-size strategy. Key contributions include a rigorously defined energy-adaptive gradient via a Sylvester-equation-based gradient evaluation, a coercivity-guaranteeing shifting strategy, and comprehensive numerical validation showing EARCG's competitiveness with state-of-the-art SCF-based methods on plane-wave discretizations. The methods promise robust performance for non-metallic crystals and lay groundwork for extensions to other metrics, metallic systems, and integration with common SCF heuristics.

Abstract

This paper presents a novel Riemannian conjugate gradient method for the Kohn-Sham energy minimization problem in density functional theory (DFT), with a focus on non-metallic crystal systems. We introduce an energy-adaptive metric that preconditions the Kohn-Sham model, significantly enhancing optimization efficiency. Additionally, a carefully designed shift strategy and several algorithmic improvements make the implementation comparable in performance to highly optimized self-consistent field iterations. The energy-adaptive Riemannian conjugate gradient method has a sound mathematical foundation, including stability and convergence, offering a reliable and efficient alternative for DFT-based electronic structure calculations in computational chemistry.

Energy-Adaptive Riemannian Conjugate Gradient Method for Density Functional Theory

TL;DR

This work addresses efficient Kohn–Sham energy minimization by formulating a constrained optimization on the infinite-dimensional Stiefel manifold and introducing an energy-adaptive Riemannian conjugate gradient (EARCG) method. The core approach combines a problem-aware shifted-Hamiltonian metric, a polar retraction with a differentiated vector transport, and a hybrid FR-PRP CG update, along with an adaptive step-size strategy. Key contributions include a rigorously defined energy-adaptive gradient via a Sylvester-equation-based gradient evaluation, a coercivity-guaranteeing shifting strategy, and comprehensive numerical validation showing EARCG's competitiveness with state-of-the-art SCF-based methods on plane-wave discretizations. The methods promise robust performance for non-metallic crystals and lay groundwork for extensions to other metrics, metallic systems, and integration with common SCF heuristics.

Abstract

This paper presents a novel Riemannian conjugate gradient method for the Kohn-Sham energy minimization problem in density functional theory (DFT), with a focus on non-metallic crystal systems. We introduce an energy-adaptive metric that preconditions the Kohn-Sham model, significantly enhancing optimization efficiency. Additionally, a carefully designed shift strategy and several algorithmic improvements make the implementation comparable in performance to highly optimized self-consistent field iterations. The energy-adaptive Riemannian conjugate gradient method has a sound mathematical foundation, including stability and convergence, offering a reliable and efficient alternative for DFT-based electronic structure calculations in computational chemistry.

Paper Structure

This paper contains 19 sections, 5 theorems, 72 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Kohn-Sham energy minimization problem
  3. Problem setting
  4. The infinite-dimensional Stiefel manifold
  5. Riemannian conjugate gradient method
  6. Retraction
  7. Vector transport
  8. Step size strategy
  9. Calculation of CG parameters
  10. Energy-adaptive optimization
  11. Energy-adaptive metric
  12. Energy-adaptive gradient
  13. Shifting strategy
  14. Numerical experiments
  15. Plane-wave discretization
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $\bm{H}_\star$ be the set of all $p$-frames from $\bm{H}$ with linearly independent components. For ${\bm \psi} \in \bm{H}_\star$, consider the polar decomposition eq:polar. Then $\mathrm{pf} : \bm{H}_\star \to \mathrm{St}(p,\bm{H})$ and $\sf: \bm{H}_\star \to \mathrm{Herm}(p)$ are both real Fré where the derivative $\mathrm{D}\,\! \sf({\bm \psi})[\bm{v}] \in \mathrm{Herm}(p)$ is the unique so

Figures (1)

  • Figure 1: Performance comparison of the different methods in terms of iterations, Hamiltonian operator applications and CPU time for three different models.

Theorems & Definitions (13)

  • Proposition 3.1
  • proof
  • Remark 3.2: Convergence analysis
  • Remark 4.1
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof
  • Corollary 4.5
  • proof
  • ...and 3 more