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On the relativistic effect in the Dirac--Fock theory

Long Meng

TL;DR

This work delivers a rigorous mathematical derivation of the leading relativistic effect in nonlinear Dirac ground-state energies by comparing Dirac--Fock and Hartree--Fock theories. It proves that the DF ground-state energy satisfies $E_{c,q}=E_q^{\rm HF}+\mathcal O(c^{-2})$ and, under regular nuclear potentials, identifies a leading $\mathcal O(c^{-2})$ correction $E^{(2)}$ that splits into mass-velocity, Darwin, and spin-orbit contributions. The analysis hinges on precise control of projections onto the positive-energy subspace and a renormalization strategy that connects HF states to DF states, including a novel DF functional built from a retraction $\theta_c$. The results provide the first rigorous derivation of the leading relativistic correction for nonlinear Dirac ground-state energies and offer a framework to study relativistic effects in general nonlinear Dirac problems with potential implications for relativistic quantum chemistry and beyond.

Abstract

In this paper, we study the error bound of the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy. This error bound is called the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order $\cO(c^{-2})$ with $c$ being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the leading order relativistic correction, which comprises the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. The proof is based on a delicate study of projections onto the positive eigenspace of some Dirac operators. To our knowledge, it is the first mathematical derivation of the leading order relativistic correction for nonlinear Dirac ground-state energies. Our method paves the way to study the relativistic effects in general nonlinear Dirac problems.

On the relativistic effect in the Dirac--Fock theory

TL;DR

This work delivers a rigorous mathematical derivation of the leading relativistic effect in nonlinear Dirac ground-state energies by comparing Dirac--Fock and Hartree--Fock theories. It proves that the DF ground-state energy satisfies and, under regular nuclear potentials, identifies a leading correction that splits into mass-velocity, Darwin, and spin-orbit contributions. The analysis hinges on precise control of projections onto the positive-energy subspace and a renormalization strategy that connects HF states to DF states, including a novel DF functional built from a retraction . The results provide the first rigorous derivation of the leading relativistic correction for nonlinear Dirac ground-state energies and offer a framework to study relativistic effects in general nonlinear Dirac problems with potential implications for relativistic quantum chemistry and beyond.

Abstract

In this paper, we study the error bound of the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy. This error bound is called the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order with being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the leading order relativistic correction, which comprises the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. The proof is based on a delicate study of projections onto the positive eigenspace of some Dirac operators. To our knowledge, it is the first mathematical derivation of the leading order relativistic correction for nonlinear Dirac ground-state energies. Our method paves the way to study the relativistic effects in general nonlinear Dirac problems.

Paper Structure

This paper contains 50 sections, 40 theorems, 395 equations.

Table of Contents

  1. Introduction
  2. Linear Dirac eigenvalue problem
  3. Nonlinear Dirac ground-state problem
  4. Relativistic effect in nonlinear Dirac ground-state energy
  5. Dirac and Schrödinger operator, functional space and some notations
  6. Free Dirac and Schrödinger operator
  7. Functional spaces and density matrices
  8. Some notations
  9. Models and main results
  10. The DF and HF models for atoms and molecules
  11. The DF and HF operators and functionals
  12. Dirac--Fock ground state energy
  13. Hartree--Fock ground state energy
  14. Main result, Part I
  15. Main result, Part II
  16. ...and 35 more sections

Key Result

Theorem 3.2

Let $q\in {\mathbb R}^+$ and $z\in {\mathbb R}^+$ be fixed such that $q\leq z$. Then under Assumption ass:1 on $c$, the minimum problem eq:min-DF admits a minimizer $\gamma_*^c \in \Gamma_q^+$. In addition, ${\rm Tr}[\gamma_*^c]=q$, and any such minimizer can be written as with $0< \delta_c\leq {\mathds{1}}_{\{\nu_c\}}(\mathcal{D}^c_{\gamma_*^c})$ for some $\nu_c\in (0,c^2]$. When $q<z$, $\nu_c\i

Theorems & Definitions (70)

  • Theorem 3.2: Existence of minimizers in the DF theory sere2023new
  • Remark 3.3
  • Theorem 3.4: Existence of minimizers in the HF theory lieb1977hartreebach1992errorbachunfill
  • Remark 3.5
  • Remark 3.7
  • Theorem 3.8: Relativistic effect in the DF theory
  • Theorem 3.9: No unfilled-shell property
  • Remark 3.10
  • Remark 3.12
  • Theorem 3.13
  • ...and 60 more