Range Behavior of Correlation Effects

The formation of the correlation energy in AO-MP2 consists of the transformation (Eq. [178]) and the contraction step (Eq. [179]). We start our discussion by considering the distance dependence of correlation contributions. 

Some discussion in this section is similar to considerations of Ayala and Scuseria.However, we present here a different argument for deriving rigorous and tight upper bounds for estimating transformed integral products following the work in Ref. 24. 

Due to the orthogonality properties of P and P (similar to the standard one-particle density matrix P of SCF theory and its complement (1 - P) in an orthogonal basis), the transformation of the overlap leads to 

For the sake of notational simplicity, we have not made a distinction between distances of centers of untransformed or transformed charge distributions, because it is clear from the context. Considering in more detail the summation over the pv-part (the terms are omitted) of the first term of order 

If these criteria are fulfilled in the restricted space of indices defined by the multipole expansion within a threshold, then is zero and with that the 

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The MBIE screening formulas are crucial for the correct estimation of the long-range behavior of correlations effects ... 

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