An efficient way to include connected quadruple contributions into the coupled cluster method

The general inclusion of the $T_4$ operator into the coupled cluster equations requires an $n^{10}$ computational procedure, and even $n^9$ in the lowest order, as in the CCSDTQ-1 (coupled cluster singles, doubles, triples and lowest-order quadruples) method. That level of $n$ -dependence makes it difficult to apply the method to larger systems. In this paper we circumvent this difficulty by a factorization approximation that requires only an $n^7$ procedure, but that provides results nearly identical to those obtained with the CCSDTQ-1 method. This observation offers a practical and accurate method to go beyond the CCSDT (coupled cluster singles, doubles and triples) approach. We also consider noniterative ${\mathrm{CCSDT(Q}}_{\mathrm{f}}\mathrm{)}$ (coupled cluster singles, doubles, triples and noniterative quadruples) and ${\mathrm{CCSD(TQ}}_{\mathrm{f}}\mathrm{)}$ (coupled cluster singles and doubles with noniterative triples and quadruples) methods.