Converging High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions

We propose a new approach to the determination of accurate electronic energies that are equivalent to the results of high-level coupled-cluster (CC) calculations. The approach is based on merging the $\mathrm{CC}(P;Q)$ formalism, which corrects energies obtained with an arbitrary truncation in the cluster operator, with the stochastic configuration interaction and CC ideas. The advantages of the proposed methodology are illustrated by molecular examples, where the goal is to recover the energetics obtained in the CC calculations with a full treatment of singly, doubly, and triply excited clusters.

Figure 1

Convergence of the $\mathrm{CC}(P)$ (red filled circles and dashed lines) and $\mathrm{CC}(P;Q{)}_{\mathrm{EN}}$ (black open squares and solid lines) energies toward CCSDT for the ${\mathrm{F}}_2/\mathrm{cc}\text{−}\mathrm{pVDZ}$ molecule in which the F–F distance $R$ was set at (a) $R_e$, (b) $1.5R_e$, (c) $2R_e$, and (d) $5R_e$, where $R_e=2.66816$ bohr is the equilibrium geometry. The $P$ spaces consisted of all singles and doubles and subsets of triples identified during the $i\text{−}\mathrm{FCIQMC}$ propagations with $\delta \tau =0.0001\mathrm{a}.\mathrm{u}.$ (depicted by the green lines representing the corresponding projected energies). The $Q$ spaces consisted of the triples not captured by the $i\text{−}\mathrm{FCIQMC}$. All energies are errors relative to CCSDT in millihartree, and the insets show the percentages of triples captured during the $i\text{−}\mathrm{FCIQMC}$ propagations.

Figure 2

Same as Fig. 1 except that the subsets of triples included in the $\mathrm{CC}(P)$ calculations are now identified by the $i\text{−}\mathrm{CCSDT}\text{−}\mathrm{MC}$ simulations and the corresponding $Q$ spaces consist of the triples not captured by $i\text{−}\mathrm{CCSDT}\text{−}\mathrm{MC}$.

Figure 3

Same as Fig. 1 for the ${\mathrm{F}}_2$ molecule with $R$ set at $2R_e$ using the (a) cc-pVTZ and (b) aug-cc-pVTZ basis sets.