Symmetrized Darwin Interaction Energy

The gauge transformed potential energy Viz then reads 

We note that we may express this interaction energy equally well in terms of the momenta pj = mjfi. 

Finally, we emphasize that the last equation results from the specific choice of gauge function for the total time derivative added to the Lagrangian. Hence, the choice of the somewhat arbitrary gauge function, which solely fulfills the boundary condition to finally eliminate all terms that are not symmetric in the particle indices, determines the final expression for the interaction energy. In 

Harriman [59] gives a derivation which starts from the retarded potentials, Eqs. (2.142) and (2.143), for which he assumes Taylor expansions of the charge and current density of the integrand in terms of the retardation time r — r lc, 

This wonderful presentation of Einstein s general theory of relativity and its application to cosmology gives an almost complete introduction to special relativity at the beginning of the book. The reader will also find an interesting outline of the historical development of this epxjchal theory. 

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This is the correction to the Hartree-Fock energy. Obviously, only the terms that are diagonal in the spin and are spatially totally symmetric will contribute. These terms are the scalar relativistic corrections the mass-velocity term, the one- and two-electron Darwin terms, and the orbit-orbit term. Other terms such as the spin-spin term and the z component of the spin-orbit interaction contribute for open-shell systems, where the spin is nonzero. 

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