Paper detail

Relativistic Scott correction in self-generated magnetic fields

We consider a large neutral molecule with total nuclear charge $Z$ in a model with self-generated classical magnetic field and where the kinetic energy of the electrons is treated relativistically. To ensure stability, we assume that $Z α< 2/π$, where $α$ denotes the fine structure constant. We are interested in the ground state energy in the simultaneous limit $Z \rightarrow \infty$, $α\rightarrow 0$ such that $κ=Z α$ is fixed. The leading term in the energy asymptotics is independent of $κ$, it is given by the Thomas-Fermi energy of order $Z^{7/3}$ and it is unchanged by including the self-generated magnetic field. We prove the first correction term to this energy, the so-called Scott correction of the form $S(αZ) Z^2$. The current paper extends the result of \cite{SSS} on the Scott correction for relativistic molecules to include a self-generated magnetic field. Furthermore, we show that the corresponding Scott correction function $S$, first identified in \cite{SSS}, is unchanged by including a magnetic field. We also prove new Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic fields.