  When the number of strongly correlated electrons becomes larger, the
single-reference coupled-cluster (CC) CCSD, CCSDT, etc. hierarchy displays an
erratic behavior, while traditional multi-reference approaches may no longer be
applicable due to enormous dimensionalities of the underlying model spaces.
These difficulties can be alleviated by the approximate coupled-pair (ACP)
theories, in which selected $(T_2)^2$ diagrams in the CCSD amplitude equations
are removed, but there is no generally accepted and robust way of incorporating
connected triply excited ($T_3$) clusters within the ACP framework. It is also
not clear if the specific combinations of $(T_2)^2$ diagrams that work well for
strongly correlated minimum-basis-set model systems are optimum when larger
basis sets are employed. This study explores these topics by considering a few
novel ACP schemes with the active-space and full treatments of $T_3$
correlations and schemes that scale selected $(T_2)^2$ diagrams by factors
depending on the numbers of occupied and unoccupied orbitals. The performance
of the proposed ACP approaches is illustrated by examining the symmetric
dissociations of the $\text{H}_6$ and $\text{H}_{10}$ rings using basis sets of
the triple- and double-$\zeta$ quality and the $\text{H}_{50}$ linear chain
treated with a minimum basis, for which the conventional CCSD and CCSDT methods
fail.
