Exact Retarded Electromagnetic Interaction Energy

After having considered the approximate retarded interaction, i.e., the Breit interaction, from the point of view of classical electrodynamics, we now follow Rosenfeld [207] and derive the exact retarded interaction energy of two 

It is clear that all retardation effects are now coded in the exponential function under the integral. The unretarded potentials are obtained once the exponential vanishes, which is the case if tOmn is set equal to zero as we will investigate below. Then, the total interaction energy expectation value can be generalized from Eq. (2.150), i.e., expressed as the 4-current of electron 1 in the electromagnetic 4-potential generated by electron 2, 

Equally well, we could have considered electron 2 in the 4-potential created by electron 1, which would have created a similar expression but with cu/fc. If we assume that electron 2 changes its state from tpm to ipn upon interaction, we may associate a change in energy by that must be compensated by the other electron because the total energy must remain unchanged [178]. Hence, we choose 

The electromagnetic interaction energy will play a significant role in current-density functional theory developed in section 8.8. Then, we will also consider (Pauli) exchange effects which so far have been completely neglected — an approximation in accord with the early work by Moller [210,211]. For cu m — 0 and cou — 0 the interaction energy expectation value reduces to 

According to our derivation, the scaled energy difference in the integral stems from electron 2, which we may explicitly write as which in what follows would create an asymmetry in the particle labels [210]. Therefore, since the two interacting electrons are indistinguishable we eventually obtain a symmetric expression if we rewrite the square of the scaled energy difference with Eq. (8.34) as 

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