In explicitly-correlated CCSD calculations the double excitations into products of virtual orbitals, described by T2 = taibjτaibj, are augmented with double excitations into the explicitly-correlated pairfunctions (geminals) which are described in Sec. 8.5:
|T||= T1 + T2 + T2'||(10.6)|
|Ωμ1||= 〈μ1| + [, T2 + T2']| HF〉 = 0 ,||(10.8)|
|Ωμ2||= 〈μ2| + [, T2 + T2'] + [[, T2 +2T2'], T2]| HF〉 = 0 ,||(10.9)|
|Ωμ2'||= 〈μ2'|[, T2'] + + [, T2]| HF〉 = 0 .||(10.10)|
|ECCSD(F12)-SP||= LCCSD(F12) = 〈HF| H| CC〉 + cμ2'Ωμ2'||(10.11)|
exampoption in $rir12 (see Sec. 8.5 for further details on the options for F12 calculations; note that the
examp noinvoption 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 + NCABS/N), where N is the number of basis and NCABS the number of complementary auxiliary basis (CABS) functions (typically NCABS 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 (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 (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.