  Obtaining sub-chemical accuracy (1 kJ mol${}^{-1}$) for reaction energies of
medium-sized gas-phase molecules is a longstanding challenge in the field of
thermochemical modeling. The perturbative triples correction to CCSD, CCSD(T),
constitutes an important component of all high-accuracy composite model
chemistries that obtain this accuracy, but can be a roadblock in the
calculation of medium to large systems due to its $\mathcal{O}(N^7)$ scaling,
particularly in HEAT-like model chemistries that eschew separation of core and
valance correlation. This study extends the work of Lesiuk [J. Chem. Phys. 156,
064103 (2022)] with new approximate methods and assesses the accuracy of five
different approximations of (T) in the context of a subset of molecules
selected from the W4-17 dataset. It is demonstrated that all of these
approximate methods can achieve sub-0.1 kJ mol${}^{-1}$ accuracy with respect
to canonical, density-fitted (T) contributions with a modest number of
projectors. The approximation labeled $\tilde{Z}T$ appears to offer the best
trade-off between cost and accuracy and shows significant promise in an
order-of-magnitude reduction in the computational cost of the CCSD(T) component
of high-accuracy model chemistries.
