The Ground State Energy of Heavy Atoms According to Brown and Ravenhall:   Absence of Relativistic Effects in Leading Order
  It is shown that the ground state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas-Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order $\alpha^2$Ry. 
Roch Cassanas and Heinz Siedentop