Renormalized Coupled-Cluster Methods: Theoretical Foundations and Application to Potential Function of Water

Conventional single-reference methods fail when bond breaking and other situations characterized by larger non-dynamical correlation effects are examined. In consequence, the adequate treatment of molecular potential energy surfaces involving significant bond rearrangements has been the domain of expert multi-reference methods. The question arises if one can develop practical single-reference procedures that could be applied to at least some of the most frequent multi-reference situations, such as single and double bond dissociations. This question is addressed in the present paper by examining the performance of the conventional and renormalized coupled-cluster (CC) methods in calculations of the potential enery surface of the water molecule. A comparison with the results of the highly accurate internally contracted multi-reference configuration interaction calculations including the quasi-degenerate Davidson correction (MRCI(Q)) and the spectroscopically accurate potential energy surface of water resulting from the use of the energy switching (ES) approach indicates that the relatively inexpensive completely renormalized (CR) CC methods with singles (S), doubles (D), and a non-iterative treatment of triples (T) or triples and quadruples (TQ), such as CR-CCSD(T), CR-CCSD(TQ), and the recently developed rigorously size extensive extension of the CR-CCSD(T), termed CR-CC(2,3), provide considerable improvements in the results of conventional CCSD(T) and CCSD(TQ) calculations at larger internuclear separations. It is shown that the CR-CC(2,3) results a posteriori corrected for the effect of quadruply excited clusters (the CR-CC(2,3)+Q approach) can compete with the highly accurate MRCI(Q) data. The excellent agreement between the CR-CC(2,3)+Q and MRCI(Q) results suggests ways of improving the global potential energy surface of water resulting from the use of the ES approach in the regions of intermediate bond stretches and intermediate and higher energies connecting the region of the global minimum with the asymptotic regions. In addition to the examination of the performance of the CR-CCSD(T), CR-CCSD(TQ), CR-CC(2,3), and CR-CC(2,3)+Q approaches, we provide a thorough review of the method of moments of CC equations (MMCC), as applied to ground electronic states, including the most recent biorthogonal formulation of MMCC theory employing the left eigenstates of the similarity-transformed Hamiltonian, and other mathematical and physical concepts that lie behind all renormalized CC approximations. In particular, we discuss the similarities and differences between the older CR-CCSD(T) and CR-CCSD(TQ) approximations and the recently formulated size extensive renormalized CC methods, such as CR-CC(2,3), and open questions that emerge in the process of designing higher-order schemes based on the biorthogonalMMCC formalism, such as CR-CC(2,4), which describe the combined effect of triples (already present in CR-CC(2,3) calculations) and quadruples in a proper manner.