In explicitly-correlated CCSD calculations the double excitations
into products of virtual orbitals, described by
*T*_{2} = *t*_{aibj}*τ*_{aibj},
are augmented with double excitations into the explicitly-correlated
pairfunctions (geminals) which are described in Sec. 8.5:

T |
= T_{1} + T_{2} + T_{2'} |
(10.6) |

T_{2'} |
= c^{kl}_{ij}τ_{kilj} |
(10.7) |

where

Ω_{μ1} |
= 〈μ_{1}| + [, T_{2} + T_{2'}]| HF〉 = 0 , |
(10.8) |

Ω_{μ2} |
= 〈μ_{2}| + [, T_{2} + T_{2'}] + [[, T_{2} +2T_{2'}], T_{2}]| HF〉 = 0 , |
(10.9) |

Ω_{μ2'} |
= 〈μ_{2'}|[, T_{2'}] + + [, T_{2}]| HF〉 = 0 . |
(10.10) |

Similar as for MP2-F12, also for CCSD(F12) the coefficients for the doubles excitations into the geminals,

E_{CCSD(F12)-SP} |
= L_{CCSD(F12)} = 〈HF| H| CC〉 + c_{μ2'}Ω_{μ2'} |
(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 `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 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**
[123], which gives essentially identical energies as CCSD(F12).
Also available are the CCSD[F12] (Ref. [123]), CCSD-F12a (Ref. [124])
and CCSD-F12b (Ref. [125]) approximations
as well as the perturbative corrections
CCSD(2)
_{} and CCSD(2)
_{}
(see Refs. [126,127,123]).
Note that these approximations should only be used with
ansatz 2 and the SP approach (i.e. fixed geminal amplitudes).

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} 2 - 3*N*).
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 + 2*N*_{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
(1 + 2*N*_{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.

In combination with the CCSD(F12*) approximation
(and also CCSD[F12], CCSD-F12a, CCSD-F12b,
CCSD(2)
_{} and CCSD(2)
_{})
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.