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Relativistic electrons coupled with Newtonian nuclear dynamics

Umberto Morellini

TL;DR

The paper develops a rigorous framework for the time dependent coupling of relativistic electrons, described by the Bogoliubov-Dirac-Fock dynamics, with classical nuclei moving under Newtonian mechanics. Using a finite ultraviolet cut-off and the $P^0$-trace to manage vacuum polarization, the authors prove local existence and uniqueness via a Schauder fixed-point approach and then establish global well-posedness by energy conservation for subcritical coupling $\alpha<4/\pi$. The analysis is carried out in the setting of a finite nuclear system with two nuclei treated in detail, and it extends to any finite number of nuclei; the results provide a mathematically rigorous foundation for relativistic molecular dynamics incorporating vacuum effects. This work lays groundwork for studying dynamical phenomena where both relativistic and quantum effects in the electronic structure are essential, while treating nuclei classically to maintain tractability.

Abstract

In the study of the electronic structure of heavy atoms, relativistic effects cannot be neglected and the Dirac operator naturally appears in place of the Schrödinger operator, raising a number of additional difficulties. The complexity of these systems has been addressed by various approximations. We consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei. The electrons are described by the Bogoliubov-Dirac-Fock (BDF) equation of quantum electrodynamics (QED). This system can be seen as a first step in the study of molecular dynamics phenomena, when both relativistic and quantum effects have to be taken into account for the electrons. We address the Cauchy problem for this model and prove global well-posedness.

Relativistic electrons coupled with Newtonian nuclear dynamics

TL;DR

The paper develops a rigorous framework for the time dependent coupling of relativistic electrons, described by the Bogoliubov-Dirac-Fock dynamics, with classical nuclei moving under Newtonian mechanics. Using a finite ultraviolet cut-off and the -trace to manage vacuum polarization, the authors prove local existence and uniqueness via a Schauder fixed-point approach and then establish global well-posedness by energy conservation for subcritical coupling . The analysis is carried out in the setting of a finite nuclear system with two nuclei treated in detail, and it extends to any finite number of nuclei; the results provide a mathematically rigorous foundation for relativistic molecular dynamics incorporating vacuum effects. This work lays groundwork for studying dynamical phenomena where both relativistic and quantum effects in the electronic structure are essential, while treating nuclei classically to maintain tractability.

Abstract

In the study of the electronic structure of heavy atoms, relativistic effects cannot be neglected and the Dirac operator naturally appears in place of the Schrödinger operator, raising a number of additional difficulties. The complexity of these systems has been addressed by various approximations. We consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei. The electrons are described by the Bogoliubov-Dirac-Fock (BDF) equation of quantum electrodynamics (QED). This system can be seen as a first step in the study of molecular dynamics phenomena, when both relativistic and quantum effects have to be taken into account for the electrons. We address the Cauchy problem for this model and prove global well-posedness.
Paper Structure (10 sections, 9 theorems, 83 equations)

This paper contains 10 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The Bogoliubov-Dirac-Fock model
  3. The classical dynamics for nuclei
  4. The general setting and main result
  5. Local existence
  6. Local existence for the BDF equation
  7. Local existence for the Newton equation
  8. Schauder fixed-point argument
  9. Uniqueness
  10. Global-in-time existence

Key Result

Theorem 2.1

Let $0\leq\alpha<4/\pi$ and $f_k\in L^1\left(\mathbb{R}^3\right)\cap L^2\left(\mathbb{R}^3\right)\cap\mathcal{C}$, $k=1,\ldots M$. Let $P_I$ be an orthogonal projector such that $Q_I=P_I-P^0\in\mathcal{H}_\mathrm{\Lambda}$ and $x^0_k,v^0_k\in\mathbb{R}^3$, $k=1,\ldots M$. Then there exists a unique of system eq:M-nuclei. Moreover, $Q\left(t\right)=P\left(t\right)-P^0$ is $P^0$-trace class and an

Theorems & Definitions (12)

  • Remark 1.1
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Lemma 3.6
  • Proposition 4.1
  • Remark 4.2
  • ...and 2 more