Approximations to the Coulomb-Breit matrix elements

Since this chapter is meant to focus on 4-component type methods I will give some attention to the recent developments that reduce the time spent in evaluation of matrix elements over g. It not trivial to reduce the amount of work associated with the separate upper and lower component basis sets since the norm of the small component wave function may be rather large for heavy atoms. One possibility is to use the locality of the small components of the wave function to replace long-range interactions by a classical interaction [27]. If we distinguish between the upper (large) two and lower (small) two components of the basis 4-spinors 

The fourth contribution requires most of the work since this term is evaluated via an expansion in the basis for the lower components. Restriction of the exact evaluation of this term to one-center contributions only [28] is possible and gives significant speed-up with little loss of accuracy. Similar techniques can also be used for the second and third term leaving the computation of first term as the dominant one. 

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