Two-Electron Terms and the Douglas-Kroll-Hess Approximation

Two-Electron Terms and the Douglas-Kroll-Hess Approximation 

So far we have made no reference to the two-electron terms of the Hamiltonian. Performing the free-particle Foldy-Wouthuysen transformation took the potential from an even operator to a combination of even and odd operators. The same will be true of the two-electron terms, except that a transformation must be performed on both electron coordinates. For the Coulomb interaction the resulting operator is 

To be entirely consistent with the one-particle terms, a second transformation should be applied to this operator. However, it has been found that this first transformation of the two-electron operators is about as important in scalar relativistic calculations as the transformations of the one-electron operators up to fifth order (Wolf et al. 2002). The second transformation of the two-electron operators is therefore unlikely to be of great significance. The reason is the strength of the nuclear potential, which is a factor of Z larger than the electron-electron interaction and is attractive rather than repulsive. 

The transformed two-electron operator bears a striking resemblance to the operator from the modified Dirac equation given in (15.43). We need only define [f = and t = [c /( p- -mc )],4 [f andtheidentityiscomplete. Theanalysisoftheterms of the modified Dirac equation into scalar and spin-orbit terms in section 15.4 can then be transferred directly to the above equation. The kinematic factors are reintroduced at the end to obtain the final expressions. 

Even though the number of transformed integrals is no larger than in a nonrelativistic calculation, the cost of the integral evaluation remains. What we would like is an approximation that is no more severe than the truncation of the transformed one-electron operator and that reduces the integral evaluation work. 

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