The Hohenberg-Kohn Theorem for Relativistic -Particle Systems

The Hohenberg-Kohn Theorem for Relativistic N-Particle Systems 

Let us start with the usual A -electron Dirac Hamiltonian including the external fields. The energy contribution from the external field is given by 

With our previous expressions for the current density and charge density in (4.41) and (4.42) we may write this energy contribution as 

Starting from these considerations, we can repeat the reasoning of the nonrelativistic Hohenberg-Kohn theorem in a relativistic setting. We thus need to show the 1 1 

If we follow the lines of the nonrelativistic proof, we need to establish the inequality 

---