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

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), $\mathrm{\Delta }{I}_{ND}[\text{(T)}]=\overline{I_{ND}}[\text{CCSD(T)}]-\overline{I_{ND}}[\text{CCSD}]$. 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., $r_I[(T)]=\mathrm{\Delta }{I}_{ND}[\text{(T)}]\mathrm{/}\mathrm{\Delta }{I}_T[\text{(T)}]$. The first diagnostic can be extended to higher-order improvements in the wave function, e.g., $\mathrm{\Delta }{I}_{ND}[\text{(Q)}]=\overline{I_{ND}}[\text{CCSDT(Q)}]-\overline{I_{ND}}[\text{CCSDT}]$. In general, a small $\mathrm{\Delta }{I}_{ND}$[$\text{level}{}_1$] 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 $\mathrm{\Delta }{I}_{ND}$[$\text{level}{}_2$] 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.