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