Asymptotics of the ground state energy of heavy molecules in self-generated magnetic field

TL;DR

This paper analyzes the asymptotic behavior of the ground state energy of heavy molecules in a self-generated magnetic field, including corrections like Scott, Schwinger, and Dirac, and explores related charge and ionization properties.

Contribution

It provides the first detailed asymptotic analysis of heavy molecules' ground state energy considering self-generated magnetic fields and includes various quantum corrections.

Findings

Derived asymptotics including Scott, Schwinger, and Dirac corrections.

Established bounds and properties for excessive charge and ionization energy.

Analyzed the effect of self-generated magnetic fields on molecular stability.

Abstract

We consider asymptotics of the ground state energy of heavy atoms and molecules in the self-generatedl magnetic field. Namely, we consider $$ H=((D-A)\cdot\boldsymbol{\sigma})^2-V $$ with $$V=\sum_{1\le m\le M} \frac{Z_m}{|x-y_m|}$$ and a corresponding Multiparticle Quantum Hamiltonian $$ \mathsf{H}=\sum_{1\le n\le N} H_{x_n} +\sum_{1\le n < n'\le N}|x_n-x_{n'}|^{-1} $$ on the Fock space $\wedge _{1\le n\le N} L^2(\mathbb{R}^3, \mathbb{C}^2)$. Here $A$ is a self-generated magnetic fiels. Then the ground state energy is given by $$ \mathsf{E}(A)=\inf \operatorname{Spec}(\mathsf{H})+\frac{1}{\alpha}\int |\nabla \times A|^2\,dx $$ where the last term is the energy of magnetic field. Under assumption $\alpha Z\le \kappa^*$ (with a small constant $\kappa^*$) we study the ground State Energy $$ \mathsf{E}^*=\inf _{A}\mathsf{E}(A). $$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections. We also consider related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.