Explicitly correlated coupled-cluster singles and doubles method based on complete diagrammatic equations.

The explicitly correlated coupled-cluster singles and doubles (CCSD-R12) and related methods-its linearized approximation CCSD(R12) and explicitly correlated second-order Moller-Plesset perturbation method-have been implemented into efficient computer codes that take into account point-group symmetry. The implementation has been largely automated by the computerized symbolic algebra SMITH that can handle complex index permutation symmetry of intermediate tensors that occur in the explicitly correlated methods. Unlike prior implementations that invoke the standard approximation or the generalized or extended Brillouin condition, our CCSD-R12 implementation is based on the nontruncated formalisms [T. Shiozaki et al., Phys. Chem. Chem. Phys. 10, 3358 (2008)] in which every diagrammatic term that arises from the modified Ansatz 2 is evaluated either analytically or by the resolution-of-the-identity insertion with the complementary auxiliary basis set. The CCSD-R12 correlation energies presented here for selected systems using the Slater-type correlation function can, therefore, serve as benchmarks for rigorous assessment of other approximate CC-R12 methods. Two recently introduced methods, CCSD(R12) and CCSD(2)(R12), are shown to be remarkably accurate approximations to CCSD-R12.