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Existence and uniqueness of solutions to the time-dependent Kohn-Sham equations coupled with classical nuclear dynamics

Björn Baumeier, Onur Çaylak, Carlo Mercuri, Mark Peletier, Georg Prokert, Wouter Scharpach

Abstract

We prove existence and uniqueness of solutions to the initial-value problem associated with a class of time-dependent Kohn-Sham equations coupled with Newtonian nuclear dynamics. We consider a pure power exchange term within a generalisation of the Local Density Approximation (LDA), identifying a range of exponents for the existence and uniqueness of $H^2$ solutions to the Kohn-Sham equations.

Existence and uniqueness of solutions to the time-dependent Kohn-Sham equations coupled with classical nuclear dynamics

Abstract

We prove existence and uniqueness of solutions to the initial-value problem associated with a class of time-dependent Kohn-Sham equations coupled with Newtonian nuclear dynamics. We consider a pure power exchange term within a generalisation of the Local Density Approximation (LDA), identifying a range of exponents for the existence and uniqueness of solutions to the Kohn-Sham equations.

Paper Structure

This paper contains 15 sections, 15 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Physical motivation
  4. Paper outline
  5. Related questions
  6. Preliminaries
  7. Definition of the electronic feasible region Bel
  8. Definition of the nuclear feasible region Bnuc
  9. Lipschitz estimates
  10. Existence and uniqueness of nuclear configurations
  11. Existence and uniqueness of electronic configurations
  12. Proof of Theorem \ref{['thm:shorttimeexistence']}
  13. Acknowledgements
  14. CRediT author statement
  15. Notation

Key Result

Theorem 1.1

Let $q\geq 7/2$ and $\lambda\in\mathbb{R}$. Further, let $\psi^0\in H^2 (\mathbb{R}^3;\mathbb{C}^{N_{\mathrm{el}}} )$, $V^0\in\mathbb{R}^{3{N_{\mathrm{nuc}}}}$ and $X^0\in\mathbb{R}^{3{N_{\mathrm{nuc}}}}$ be given, with $X^0_{K}\neq X^0_{L}$ for $K\neq L$. Then, there exists $\tau>0$ such that the i

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • ...and 21 more