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We propose a novel a posteriori error assessment for the single-referencecoupled-cluster (SRCC) method called the $S$-diagnostic. We provide aderivation of the $S$-diagnostic that is rooted in the mathematical analysis ofdifferent SRCC variants. We numerically scrutinized the $S$-diagnostic, testingits performance for (1) geometry optimizations, (2) electronic correlationsimulations of systems with varying numerical difficulty, and (3) thesquare-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 ofmulti-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 ofmulti-determinantal and multi-reference character in coupled-cluster theory intheir individual fields of applicability. The experiments investigating theperformance of the $S$-diagnostic for geometry optimizations using SRCC revealthat the $S$-diagnostic correlates well with different error measures at a highlevel of statistical relevance. The experiments investigating the performanceof 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)$_4$]$^{2+}$, whichhave 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 posterioridiagnostic for SRCC calculations.

The $S$-diagnostic -- an a posteriori error assessment for single-reference coupled-cluster methods