Here we analyze the use of the resolution of the identity (RI) in approximating many-electron matrix elements in linear R12 theories. A modified standard approximation is proposed that involves expansion in the orthogonal complement to the span of orbital basis set (OBS). The new formulation is labeled complementary auxiliary basis set (CABS) approach. CABS MP2-R12 method has a smaller RI error vis-A-vis the standard ABS approach. Both ABS and CABS approaches are most accurate if the auxiliary basis includes OBS explicitly. The CABS approach found to be more numerically robust than the ABS counterpart. (C) 2004 Elsevier B.V. All rights reserved.