S-Diagnostic─An a Posteriori Error Assessment for Single-Reference Coupled-Cluster Methods

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 [CuCl₄]^2-, [Cu(NH₃)₄]²⁺, and [Cu(H₂O)₄]²⁺. Throughout the numerical investigations, the S-diagnostic is compared to other SRCC diagnostic procedures, that is, the T₁, D₁, max T₂, and D₂ diagnostics as well as different indices of multideterminantal and multireference character in coupled-cluster theory. Our numerical investigations show that the S-diagnostic outperforms the T₁, D₁, max T₂ and D₂ diagnostics and is comparable to the indices of multideterminantal and multireference 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 multireference regimes. The S-diagnostic, moreover, correctly detects the successful SRCC computations for [CuCl₄]^2-, [Cu(NH₃)₄]²⁺, and [Cu(H₂O)₄]²⁺, which have been known to be misdiagnosed by T₁ and D₁ diagnostics in the past. This shows that the S-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations.