The paper considers a free electron gas model based on the equations of Breit generalized to the case of many interacting electrons. An equation is obtained for calculating the relativistic corrections to deep-lying electron energy levels in atoms, of the order of α2Z4 and α2Z3. A comparison with quantum-mechanical calculations for a number of atoms shows qualitative agreement. The possible causes of the quantitative discrepancies are discussed. An expression is derived for the local exchange potential in the relativistic description of an atom.