Various approximations such as A, B, C, HY1, HY2 exist for the matrix elements of the first-order Hamiltonian (see I). They differ in the way the RI approximations are made. In the limit of a complete RI basis, approximations B and C are identical and yield the exact result for a given wave function ansatz. We generally recommend approximation C, which is simpler and more efficient than approximation B. Normally, the union of the AO and RI basis sets is used to approximate the resolution of the identity (CABS approach). In the hybrid approximations (HY1, HY2, HX) only the AO basis is used in some less important terms. Together with the recommended approximation C, HY1 or HY2 can be used; HY2 is more accurate, HY1 more efficient. In most cases, approximation 3C(HY1) provides an excellent compromise between accuracy and efficiency. In approximation A, all terms involving exchange operators are neglected. This approximation is used along with local approximations in our low-order scaling LMP2-F12/3*A(loc) method that can be applied to large molecules (cf. section 35.11)
If the extended Brillouin condition (EBC, see I) is assumed, the explicitly correlated and conventional amplitude equations decouple and can be solved independently. These approximations are denoted by a star, e.g. 3*C.