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Numerical Analysis of Finite Dimensional Approximations in Finite Temperature DFT

Ge Xu, Huajie Chen, Xingyu Gao

TL;DR

The paper develops a density-matrix variational framework for finite-temperature DFT and analyzes the ground-state problem within a finite-dimensional Galerkin setting. It proves coercivity and weak lower semicontinuity of the Helmholtz free energy, establishes existence of minimizers, and derives first- and second-order optimality conditions linked to the Mermin–Kohn–Sham equations. The authors prove convergence of Galerkin approximations and provide an optimal a priori error estimate via the inverse function theorem, supported by numerical experiments using plane-wave discretizations that exhibit exponential convergence with energy cutoff. This work delivers a rigorous convergence theory for finite-temperature DFT discretizations and lays groundwork for robust numerical methods applicable to metallic and insulating systems alike.

Abstract

In this paper, we study numerical approximations of the ground states in finite temperature density functional theory. We formulate the problem with respect to the density matrices and justify the convergence of the finite dimensional approximations. Moreover, we provide an optimal a priori error estimate under some mild assumptions and present some numerical experiments to support the theory.

Numerical Analysis of Finite Dimensional Approximations in Finite Temperature DFT

TL;DR

The paper develops a density-matrix variational framework for finite-temperature DFT and analyzes the ground-state problem within a finite-dimensional Galerkin setting. It proves coercivity and weak lower semicontinuity of the Helmholtz free energy, establishes existence of minimizers, and derives first- and second-order optimality conditions linked to the Mermin–Kohn–Sham equations. The authors prove convergence of Galerkin approximations and provide an optimal a priori error estimate via the inverse function theorem, supported by numerical experiments using plane-wave discretizations that exhibit exponential convergence with energy cutoff. This work delivers a rigorous convergence theory for finite-temperature DFT discretizations and lays groundwork for robust numerical methods applicable to metallic and insulating systems alike.

Abstract

In this paper, we study numerical approximations of the ground states in finite temperature density functional theory. We formulate the problem with respect to the density matrices and justify the convergence of the finite dimensional approximations. Moreover, we provide an optimal a priori error estimate under some mild assumptions and present some numerical experiments to support the theory.
Paper Structure (11 sections, 7 theorems, 90 equations, 2 figures)

This paper contains 11 sections, 7 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free energy in density matrix formulation
  3. Ground state solutions and optimality conditions
  4. Euler-Lagrange equations
  5. Second-order optimality condition
  6. Numerical analysis of the Galerkin approximations
  7. Finite dimensional approximations
  8. Convergence analysis
  9. A priori error estimate
  10. Numerical experiments
  11. Concluding remarks

Key Result

Lemma 2.1

If the assumptions A1- A2 are satisfied, then there exist positive constants $a$ and $b$ such that

Figures (2)

  • Figure 5.1: (Example 1. Silicon) Left: diamond structure in a unit cell. Middle: energy error with respect to the energy cutoffs. Right: $L^2$-error of the electron density with respect to the energy cutoffs.
  • Figure 5.2: (Example 2. Aluminium) Left: FCC lattice in a unit cell. Middle: energy error with respect to the energy cutoffs. Right: $L^2$-error of the electron density with respect to the energy cutoffs.

Theorems & Definitions (20)

  • Lemma 2.1: Coercivity
  • proof
  • Lemma 2.2: Weak lower semicontinuity
  • proof
  • Remark 2.1: Orbital-based formulation: the free energy
  • Remark 3.1: Orbital-based formulation: variational form
  • Remark 3.2: Orbital-based formulation: MKS equations
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 10 more