Interaction modified Gaunt

We start the derivation of the modified Gaunt interaction by examining the effect of sandwiching a between two projection operators ... 

While the Coulomb interaction is diagonal and appears only on the main diagonal of the Hamiltonian (457) in terms of the ll, Is etc. components, the Gaunt, as well as the Breit interaction is non-diagonal and their matrix representations, in terms of the modified metric are... 

The modified operators for the Gaunt and Breit interactions can be derived in an analogous manner. The derivation is somewhat more involved than for the Coulomb interaction due to the presence of the a matrices. Here we derive the Gaunt terms, but the gauge term for the Breit interaction is considerably more complicated (and the derivation may be found in appendix F). 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

We now turn to the Gaunt interaction, and use the terms from the modified Dirac representation in (15.54) to derive the Breit-Pauli operators. These terms need no renormalization, because they are all of order 1/c. The three classes of operators defined in (15.54) are considered in turn. 

Reference values for the various 2-component relativistic Hamiltonians are provided by the 4-component Dirac-Coulomb Hamiltonian, but we have also included orbital energies obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian. As already mentioned, the Gaunt term brings in spin-other-orbit (SOO) interaction. Since spin-orbit interaction induced by other electrons will oppose the one induced by nuclei we see from Table 3.3 that the spin-orbit splitting of orbital levels is overall reduced. However, one should note that the Gaunt term also modifies /2 levels. 

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