Combining active-space coupled-cluster approaches with moment energy corrections via the CC(P;Q) methodology: connected quadruple excitations

We have recently proposed the CC(P;Q) methodology that provides a systematic approach to correcting the energies obtained in active-space coupled-cluster (CC) calculations for the remaining, mostly dynamical, correlations. We report the development of the CC(t,q;3) and CC(t,q;3,4) methods, which use the CC(P;Q) formalism to correct the energies obtained with the CC approach with singles, doubles and active-space triples and quadruples, termed CCSDtq, for the remaining triples [CC(t,q;3)] or the remaining triples and quadruples [CC(t,q;3,4)] missing in CCSDtq. By examining the double dissociation of water, the Be + H-2 -> HBeH insertion, and the singlet-triplet gap in the (HFH)(-) system, we demonstrate that the CC(t,q;3) and CC(t,q;3,4) methods, particularly the latter one, offer significant improvements in the CCSDtq results, reproducing the total and relative energies obtained with the CC approach with singles, doubles, triples and quadruples (CCSDTQ) to within fractions of a millihartree, at the tiny fraction of the computational effort involved in the CCSDTQ calculations, even when electronic quasi-degeneracies become substantial. We also show that the previously examined CC(t;3) approach, which corrects the active-space CCSDt energies for the triples missing in CCSDt, accurately reproduces the CCSDT energetics.