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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}}^{}$taibjτaibj , are augmented with double excitations into the explicitly-correlated pairfunctions (geminals) which are described in Sec. 8.5:

T = T1 + T2 + T2' (10.12)
T2' = $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \sum_{{ijkl}}^{}$cklijτkilj (10.13)

where τkilj| 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 ([123]). 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 ([124,123]) 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 [125,126]. In the CCSD(F12) approximation the amplitudes are determined from the equations:

Ωμ1 = 〈μ1|$\displaystyle \hat{{\tilde{H}}}$ + [$\displaystyle \hat{{\tilde{H}}}$, T2 + T2']| HF〉 = 0  , (10.14)
Ωμ2 = 〈μ2|$\displaystyle \hat{{\tilde{H}}}$ + [$\displaystyle \hat{{\tilde{H}}}$, T2 + T2'] + [[$\displaystyle \hat{{H}}$, T2 +2T2'], T2]| HF〉 = 0  , (10.15)
Ωμ2' = 〈μ2'|[$\displaystyle \hat{{F}}$, T2'] + $\displaystyle \hat{{\tilde{H}}}$ + [$\displaystyle \hat{{\tilde{H}}}$, T2]| HF〉 = 0  . (10.16)

Similar as for MP2-F12, also for CCSD(F12) the coefficients for the doubles excitations into the geminals, cklij can be determined from the electronic cusp conditions using the rational generator (also known as SP or fixed amplitude) approach. In this case Eq. (10.16) is not solved. To account for this, the energy is then computed from a Lagrange function as:

ECCSD(F12)-SP = LCCSD(F12) = 〈HF| H| CC〉 + $\displaystyle \sum_{{\mu_{2'}}}^{}$cμ2'Ωμ2' (10.17)

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. (10.16), while both approaches are approximately similar accurate for energy differences.

The SP approach becomes in particular very efficient if combined with the neglect of certain higher-order explicitly-correlated contributions which have a negligible effect on the energies but increase the costs during the CC iterations. The most accurate and recommeded variant is the CCSD(F12*) approximation [127], which gives essentially identical energies as CCSD(F12). Also available are the CCSD[F12] (Ref. [127]), CCSD-F12a (Ref. [128]) and CCSD-F12b (Ref. [129]) approximations as well as the perturbative corrections CCSD(2) $\scriptstyle \overline{{\mathrm{F12}}}$ and CCSD(2) $\scriptstyle \overline{{\mathrm{F12*}}}$ (see Refs. [130,131,127]). Note that these approximations should only be used with ansatz 2 and the SP approach (i.e. fixed geminal amplitudes).

For MP3 the approximations (F12*), and (F12) to a full F12 implementation become identical: they include all contributions linear in the coefficients cklij . The explicitly-correlated MP4 method MP4(F12*) is defined as fourth-order approximation to CCSD(F12*)(T). Note that MP4(F12*) has to be used with the SP or fixed amplitude approache for the geminal coefficients cklij . MP3(F12*) and MP4(F12*) are currently only available for closed-shell or unrestricted Hartree-Fock reference wavefunctions.

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 + NCABS/N) , where N is the number of basis and NCABS the number of complementary auxiliary basis (CABS) functions (typically NCABS $ \approx$ 2 - 3N ). If the geminal coefficients are determined by solving Eq. (10.16) instead of using fixed amplitudes, the costs per CCSD(F12) iteration increase to $ \approx$ (1 + 2NCABS/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 + 2NCABS/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.

In combination with the CCSD(F12*) approximation (and also CCSD[F12], CCSD-F12a, CCSD-F12b, CCSD(2) $\scriptstyle \overline{{\mathrm{F12}}}$ and CCSD(2) $\scriptstyle \overline{{\mathrm{F12*}}}$ ) the CPU time for the SP approach is only about 20% or less longer than for a conventional CCSD calculation within the same basis set.

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