Scott correction for large atoms and molecules in a self-generated magnetic field

TL;DR

This paper analyzes the ground state energy of large molecules with self-generated magnetic fields, showing that magnetic effects influence only the Scott correction term in the energy expansion as the molecule size grows.

Contribution

It extends the Scott correction analysis to include self-generated magnetic fields, revealing their impact on the energy expansion of large atoms and molecules.

Findings

Ground state energy has a two-term expansion with magnetic effects only in the second term.

The leading term aligns with non-magnetic Thomas-Fermi theory.

Magnetic field influences the Scott correction term in the energy expansion.

Abstract

We consider a large neutral molecule with total nuclear charge $Z$ in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that $Z\al^2\le \kappa_0$ for a sufficiently small $\kappa_0$, where $\al$ denotes the fine structure constant. We show that, in the simultaneous limit $Z\to\infty$, $\al\to 0$ such that $\kappa =Z\al^2$ is fixed, the ground state energy of the system is given by a two term expansion $c_1Z^{7/3} + c_2(\kappa) Z^2 + o(Z^2)$. The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion.