  In this paper we extend the rank-reduced coupled-cluster formalism to the
calculation of non-iterative energy corrections due to quadruple excitations.
There are two major components of the proposed formalism. The first is an
approximate compression of the quadruple excitation amplitudes using the Tucker
format. The second is a modified functional used for evaluation of the
corrections which gives exactly the same results for the exact amplitudes, but
is less susceptible to errors resulting from the aforementioned compression. We
show, both theoretically and numerically, that the computational cost of the
proposed method scales as the seventh power of the system size. Using reference
results for a set of small molecules, the method is calibrated to deliver
relative accuracy of a few percent in energy corrections. To illustrate the
potential of the theory we calculate the isomerization energy of
\emph{ortho}/\emph{meta} benzyne (C$_6$H$_4$) and the barrier height for the
Cope rearrangement in bullvalene (C$_{10}$H$_{10}$). The method retains a
near-black-box nature of the conventional coupled-cluster formalism and depends
on only one additional parameter that controls the accuracy.
