Toward an affordable density-based measure for the quality of a coupled cluster calculation

TL;DR

This work addresses the challenge of diagnosing static correlation effects in coupled cluster (CC) calculations by introducing three density-based diagnostics: $ΔI_{ND}[(T)]$, $ΔI_{ND}[(Q)]$, and the ratio $r_I[(T)]$. These diagnostics are defined from Matito's normalized nondynamical and total correlation measures and are evaluated across the W4-17/W4-11 benchmarks using the CFOUR suite, with cc-pVDZ to cc-pVQZ basis sets. The results show that $ΔI_{ND}[(T)]$ tracks density convergence between CCSD and CCSD(T), $ΔI_{ND}[(Q)]$ provides insight into higher-order corrections, and $r_I[(T)]$ correlates moderately well with the importance of post-CCSD(T) correlation effects (Pearson $R$ up to ~0.86 with %TAE$[(T)]$). Basis-set convergence is reasonably fast, with cc-pVTZ effectively converged, making these diagnostics practical for guiding higher-level CC treatments in systems with varying static correlation. The Be insertion into H$_2$ case demonstrates that $r_I[(T)]$ and its higher-order variant can outperform traditional density- or energy-based diagnostics in signaling when more advanced correlation is warranted.

Abstract

We propose two new diagnostics for the degree to which static correlation impacts the quality of a coupled cluster calculation. The first is the change in the Matito static correlation diagnostic $\overline{I_{ND}}$ between CCSD and CCSD(T), $ΔI_{ND}[\textrm{(T)}]=\overline{I_{ND}}[\textrm{CCSD(T)}]-\overline{I_{ND}}[\textrm{CCSD}]$. The second is the ratio of the same and of the corresponding change in the total correlation diagnostic $\overline{I_{T}}=\overline{I_{ND}}+\overline{I_{D}}$, i.e., $r_I[(T)]=ΔI_{ND}[\textrm{(T)}]/ΔI_{T}[\textrm{(T)}]$. The first diagnostic can be extended to higher-order improvements in the wave function, e.g., $ΔI_{ND}[\textrm{(Q)}]=\overline{I_{ND}}[\textrm{CCSDT(Q)}]-\overline{I_{ND}}[\textrm{CCSDT}]$. In general, a small $ΔI_{ND}$[\textrm{level$_1$}] value indicates that at this level$_1$ of theory, the density is converged and any further changes to the energy come from dynamical correlation, while larger $ΔI_{ND}$[\textrm{level$_2$}] indicates that the density is still not converged at level$_2$ and some static correlation remains. $r_I[(T)]$ is found to be a moderately good predictor for the importance of post-CCSD(T) correlation effects.

Figures (3)

• Figure 1: Heatmap of $R^2$ values with hierarchical clustering (9 clusters) between different diagnostics for static correlation, on the closed-shell subset of W4-11 with the cc-pVTZ basis set
• Figure 2: Heatmapped $R^2$ matrix between different diagnostics on the full W4-11 dataset (including open-shell species) with the cc-pVTZ basis set
• Figure 3: Evolution of diagnostics along the reaction profile of Be+H2.