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 alpha < 2/pi, where alpha denotes the fine structure constant. We are interested in the ground state energy in the simultaneous limit Z -> infinity, alpha -> 0 such that kappa = Z alpha is fixed. The leading term in the energy asymptotics is independent of kappa, 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(alpha Z)Z(2). The current paper extends the result of Solovej [Commun. Pure Appl. Math. LXIII, 39-118 (2010)] 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 by Solovej [Commun. Pure Appl. Math. LXIII, 39-118 (2010)], is unchanged by including a magnetic field. We also prove new Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic fields. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3697417]