Complementary auxiliary basis set

In the original formulation of the R12 method [23], which is often referred to as "the standard approximation" or SA, a large uncontracted OBS was needed as the RI basis, strongly limiting the applicability of the R12 method to relatively small systems. Klopper and Samson [26] later pioneered the use of a separate RI basis set and considerably improved the applicability and accuracy of MP2-R12. Valeev developed an alternative formulation of the RI procedure [27] known as the complementary auxiliary basis set (CABS) approach, which had smaller RI errors. Ten-no, on the other hand, proposed the use of a numerical quadrature for the same purpose [28]. Kedzuch et al. later used the CABS approach to develop a particularly elegant and practical formalism for the MP2-R12 energies [30]. 

The advantage of this approach is its simplicity, but one has to use relatively large orbital basis sets to ensure that the approximated unity operators in Eq. (68) are well represented [22]. This limitation quickly becomes a bottleneck and apparently such approach gives reliable results only for relatively small systems (treated with large orbital basis sets). A natural idea is to introduce an additional basis set (different than the orbital one) that is used to represent the resolution-of-the-identity operators. There are two (so far) closely related models, the auxiliary basis set approximation (ABS) [47] and the complementary auxiliary basis set approximation (CABS) [20]. In the implementation in Turbomole only the latter one was considered. Within this approach the whole basis (singly primed), used for the representation of the unity operators, is the union of the orbital and some auxiliary basis (doubly primed)... 

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