  We propose a novel a posteriori error assessment for the single-reference
coupled-cluster (SRCC) method called the $S$-diagnostic. We provide a
derivation of the $S$-diagnostic that is rooted in the mathematical analysis of
different SRCC variants. We numerically scrutinized the $S$-diagnostic, testing
its performance for (1) geometry optimizations, (2) electronic correlation
simulations of systems with varying numerical difficulty, and (3) the
square-planar copper complexes [CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$, and
[Cu(H$_2$O)$_4$]$^{2+}$. Throughout the numerical investigations, the
$S$-diagnostic is compared to other SRCC diagnostic procedures, that is, the
$T_1$, $D_1$, and $D_2$ diagnostics as well as different indices of
multi-determinantal and multi-reference character in coupled-cluster theory.
Our numerical investigations show that the $S$-diagnostic outperforms the
$T_1$, $D_1$, and $D_2$ diagnostics and is comparable to the indices of
multi-determinantal and multi-reference character in coupled-cluster theory in
their individual fields of applicability. The experiments investigating the
performance of the $S$-diagnostic for geometry optimizations using SRCC reveal
that the $S$-diagnostic correlates well with different error measures at a high
level of statistical relevance. The experiments investigating the performance
of the $S$-diagnostic for electronic correlation simulations show that the
$S$-diagnostic correctly predicts strong multi-reference regimes. The
$S$-diagnostic moreover correctly detects the successful SRCC computations for
[CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$, and [Cu(H$_2$O)$_4$]$^{2+}$, which
have been known to be misdiagnosed by $T_1$ and $D_1$ diagnostics in the past.
This shows that the $S$-diagnostic is a promising candidate for an a posteriori
diagnostic for SRCC calculations.
