Explicitly correlated coupled-cluster theory using cusp conditions. I. Perturbation analysis of coupled-cluster singles and doubles (CCSD-F12).

Geminal functions based on Slater-type correlation factors and fixed expansion coefficients, determined by cusp conditions, have in recent years been forwarded as an efficient and numerically stable method for introducing explicit electron correlation into coupled-cluster theory. In this work, we analyze the equations of explicitly correlated coupled-cluster singles and doubles (CCSD-F12) theory and introduce an ordering scheme based on perturbation theory which can be used to characterize and understand the various approximations found in the literature. Numerical results for a test set of 29 molecules support our analysis and give additional insight. In particular, our results help rationalize the success of the CCSD(F12) approximation which is based on a very systematic cancellation of the neglected, otherwise individually large third-order geminal-geminal coupling terms. Further approximations to CCSD(F12) can be introduced without sacrificing the accuracy if the entire set of third-order coupling terms between the conventional doubles cluster amplitudes and the geminal doubles amplitudes is retained, leading to the recently proposed CCSD[F12] and CCSD(F12(∗)) models, which have negligible overhead compared to conventional CCSD calculations. Particularly, the importance of the ring-term type contribution is pointed out which may be used to improve on other existing approximations such as CCSD-F12b. For small basis sets, it might be advantageous to keep certain higher-order terms leading to CCSD-F12(∗), which, for the case of the SP ansatz, merely involves a noniterative correction to CCSD(F12(∗)).