Explicitly-correlated CCSD(F12) methods:

In explicitly-correlated CCSD calculations the double excitations into products of virtual orbitals, described by T₂ = ${\frac{{1}}{{2}}}$$\sum_{{aibj}}^{}$t_aibjτ_aibj, are augmented with double excitations into the explicitly-correlated pairfunctions (geminals) which are described in Sec. 8.5:

T = T₁ + T₂ + T_2' (10.6)

$${\frac{{1}}{{2}}} \sum_{{ijkl}}^{}$$ T_2' (10.7)

where τ_kilj| ij〉 = $\hat{{Q}}_{{12}}^{}$f₁₂| kl〉 (for the defintion $\hat{{Q}}_{{12}}^{}$ and f₁₂ see Sec. 8.5). This enhances dramatically the basis set convergence of CCSD calculations ([113]). Without any further approximations than those needed for evaluating the neccessary matrix elements, this extension of the cluster operator T leads to the CCSD-F12 method. CCSD(F12) is an approximation ([114,113]) to CCSD-F12 which neglects certain computationally demanding higher-order contributions of $\hat{{T}}_{{2'}}^{}$. This reduces the computational costs dramatically, while the accuracy of CCSD(F12) is essentially identical to that of CCSD-F12 [115,116]. In the CCSD(F12) approximation the amplitudes are determined from the equations:

$$\hat{{\tilde{H}}} \hat{{\tilde{H}}}$$ Ω_μ1 (10.8)

$$\hat{{\tilde{H}}} \hat{{\tilde{H}}} \hat{{H}}$$ Ω_μ2 (10.9)

$$\hat{{F}} \hat{{\tilde{H}}} \hat{{\tilde{H}}}$$ Ω_μ2' (10.10)

Similar as for MP2-F12, also for CCSD(F12) the coefficients for the doubles excitations into the geminals, c^kl_ij can be determined from the electronic cups conditions using the rational generator (also known as SP or fixed amplitude) approach. In this case Eq. (10.10) is not solved. To account for this, the energy is then computed from a Lagrange function as:

$$\sum_{{\mu_{2'}}}^{}$$ E_CCSD(F12)-SP (10.11)

This is the recommended approach which is used by default if not any other approch has been chosen with the examp option in $rir12 (see Sec. 8.5 for further details on the options for F12 calculations; note that the examp noinv option should not be combined with CCSD calculations). CCSD(F12)-SP calculations are computationally somewhat less expensive that CCSD(F12) calculations which solve Eq. (10.10), while the boths approaches are approximately similar accurate for energy differences.

The CPU time for a CCSD(F12) calculation is approximately the sum of the CPU time for an MP2-F12 calculation with the same basis sets plus that of a conventional CCSD calculation multiplied by (1 + N_CABS/N), where N is the number of basis and N_CABS the number of complementary auxiliary basis (CABS) functions (typically N_CABS $\approx$ 2 - 3N). If the geminal coefficients are determined by solving Eq. (10.10) instead of using fixed amplitudes, the costs per CCSD(F12) iteration increase to $\approx$ (1 + 2N_CABS/N) the costs for conventional CCSD iteration. Irrespective how the geminal coefficients are determined, the disc space for CCSD(F12) calculations are approximated a factor of $\approx$ (1 + 2N_CABS/N) larger than the disc space required for a conventional CCSD calculation. Note that this increase in the computational costs is by far outweighted by the enhanced basis set convergence.