DKH Transformation of Two-Electron Terms

The DKH transformation of the two-electron terms requires more effort, which is the reason why this is neglected in standard DKH calculations that sacrifice this transformation for the sake of efficiency. For this transformation it is vital to appreciate the double-even structure of the Coulomb operator and the double-odd structure of the Breit operator as highlighted in Eq. (8.89). Then, following the protocol for the transformation of the one-electron operators in the preceding section, the innermost unitary transformation of the two- 

This expression can formally be evaluated without assuming any further simplification for the electron-electron interaction potential energy operator i.e., g i,j) is still defined by Eq. (8.69) as the sum of the Coulomb and frequency-independent Breit interactions. As in the case of the one-electron DKH transformation, g i,j) acts as an integral operator in momentum space and hence does not commute with any expression containing momentum operators such as Aj or R , for example. With the abbreviation gij = g i,j) the total two-electron interaction operator reads 

The DKH transformations (m = 0,1.) are designed to decouple, i.e., block-diagonalize, the total one-electron terms step by step. In practice, the decoupling can be done most efficiently in an approximate way if one restricts all transformations to the nuclear potential Vnuc- Then, the untransformed electron-electron interaction operators can be considered in some sort of perturbation theory by employing the same unitary transformations to obtain transformed operators of which the upper left block is then chosen for the 

For the Coulomb-only electron-electron interaction only the four terms of the first line of Eq. (12.70) yield even expressions for both electrons under consideration. All other terms of Eq. (12.70) have at least one odd component. The result may be written as 

This free-particle Foldy-Wouthuysen-transformed two-electron part has been investigated by various authors [225,623,627,643,653,654]. Still, these are expressions of the four-component framework. Similarly to the procedure for one-electron operators the restriction to the electronic, i.e., upper-left part of 

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