Do Conventional and Explicitly Correlated Coupled Cluster Theory Converge to the Same Limits? Computational Evidence and Perturbation Theoretical Analysis

Coupled cluster theory with all singles and doubles (CCSD) accounts for the lion’s share of the electron correlation contribution to molecular heats of formation. Alas, like all wavefunction ab initio methods that account for dynamical correlation, it converges very slowly to the 1-particle basis set limit. Explicitly correlated (F12) methods, in which interelectronic distance terms are explicitly included in the wave function, greatly accelerate basis set convergence. Yet, since a full CCSD-F12 theory without approximations is computationally and algebraically unwieldy, a number of approximate approaches have instead been implemented, such as (in descending order of rigor) CCSD(F12*), CCSD[F12], CCSD-F12b, and CCSD-F12a.

For the W4-17 dataset, a diverse sample of nearly 200 first-and second-row small molecules, we have obtained CCSD basis set limits by extrapolation of AV6Z+d and AV7Z+d basis sets and compared to results using the aforementioned approximations to CCSD-F12. In addition, we have considered the latter approximations’ basis set convergence.

The difference between CCSD-F12a and CCSD-F12b is erratic and converges slowly with the basis set: CCSD-F12a cannot be recommended for any applications. The difference between CCSD-F12b and CCSD(F12*), on the other hand, is statistically well-correlated to several diagnostics for nondynamical correlation character: in a perturbation expansion, it can be traced to several fourth-order terms involving coupling between F12 and CCSD double excitation amplitudes. In molecules dominated by dynamical correlation, CCSD-F12b and CCSD(F12*) in sufficiently large basis sets both yield results almost identical to the conventional CCSD limits. In molecules with strong nondynamical correlation like ozone or BN, not only does CCSD-F12b yield unacceptable errors even with basis sets as large as aug-cc-pwCV5Z, but even CCSD(F12*) appears to suffer from some terms in higher order yet. Overall, CCSD(F12*) has the fastest basis set convergence of all the CCSD-F12 approximations. CCSD[F12], which retains all third-order but none of the fourth-order terns, is not practically useful, but does considerably help in the interpretation of the results.