In CCSD the ground–state energy is (as for CC2) evaluated as

E_{CC} | = ⟨HF|H|CC⟩ = ⟨HF|H exp(T)|HF⟩ , | (11.1) |

T_{1} | = ∑
_{ai}t_{ai}τ_{ai} , | (11.2) |

T_{2} | = ∑
_{aibj}t_{aibj}τ_{aibj} . | (11.3) |

Ω_{μ1} | = ⟨μ_{1}| + [,T_{2}]|HF⟩ = 0 , | (11.4) |

Ω_{μ2} | = ⟨μ_{2}| + [,T_{2}] + [[Ĥ,T_{2}],T_{2}]|HF⟩ = 0 , | (11.5) |

E_{MP3,tot} = | E_{HF} + E_{MP2} + E_{MP3} | (11.6) |

= | ⟨HF|Ĥ + [Ĥ,T_{2}^{(1)}]|HF⟩ + ∑
_{μ2}t_{μ2}^{(1)}⟨μ_{
2}|[Ŵ,T_{2}^{(1)}]|HF⟩ | (11.7) |

⟨μ_{1}|[,T_{1}^{(2)}] + [Ŵ,T_{
2}^{(1)}]|HF⟩ = | 0 | (11.8) |

⟨μ_{2}|[,T_{2}^{(2)}] + [Ŵ,T_{
2}^{(1)}]|HF⟩ = | 0 | (11.9) |

⟨μ_{3}|[,T_{3}^{(2)}] + [Ŵ,T_{
2}^{(1)}]|HF⟩ = | 0 | (11.10) |

E_{MP4} = | ∑
_{μ2}t_{μ2}^{(1)}⟨μ_{
2}|[Ŵ,T_{1}^{(2)} + T_{
2}^{(2)} + T_{
3}^{(2)}] + [[Ŵ,T_{
2}^{(1)}],T_{
2}^{(1)}]|HF⟩ . | (11.11) |

Explicitly-correlated CCSD(F12) methods:
In explicitly-correlated CCSD calculations the double excitations into products of virtual orbitals,
described by T_{2} = ∑
_{aibj}t_{aibj}τ_{aibj}, are augmented with double excitations into the
explicitly-correlated pairfunctions (geminals) which are described in Sec. 9.5:

T | = T_{1} + T_{2} + T_{2′} | (11.12) |

T_{2′} | = ∑
_{ijkl}c_{ij}^{kl}τ_{
kilj} | (11.13) |

Ω_{μ1} | = ⟨μ_{1}| + [,T_{2} + T_{2′}]|HF⟩ = 0 , | (11.14) |

Ω_{μ2} | = ⟨μ_{2}| + [,T_{2} + T_{2′}] + [[Ĥ,T_{2} + 2T_{2′}],T_{2}]|HF⟩ = 0 , | (11.15) |

Ω_{μ2′} | = ⟨μ_{2′}|[,T_{2′}] + + [,T_{2}]|HF⟩ = 0 . | (11.16) |

E_{CCSD(F12)-SP} | = L_{CCSD(F12)} = ⟨HF|H|CC⟩ + ∑
_{μ2′}c_{μ2′}Ω_{μ2′} | (11.17) |

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 [18], which gives essentially identical
energies as CCSD(F12). Also available are the CCSD[F12] (Ref. [18]), CCSD-F12a
(Ref. [146]) and CCSD-F12b (Ref. [147]) approximations as well as the perturbative
corrections CCSD(2)_{F12} and CCSD(2*)_{F12} (see Refs. [18,148,149]). 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 c_{ij}^{kl}. 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 c_{ij}^{kl}.
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 + 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 - 3N). If
the geminal coefficients are determined by solving Eq. (11.16) instead of using fixed
amplitudes, the costs per CCSD(F12) iteration increase to ≈ (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 ≈ (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.

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

CC calculations with restricted open-shell (ROHF) references: The MP2 and all CC calculations for ROHF reference wavefunctions are done by first transforming to a semi-canonical orbital basis which are defined by the eigenvectors of the occupied/occupied and virtual/virtual blocks of the Fock matrices of alpha and beta spin. No spin restrictions are applied in the cluster equations. This approach is sometimes also denoted as ROHF-UCCSD.

Note that if a frozen-core approximation is used, the semicanonical orbitals depend on whether the block-diagonalization of the Fock matrices is done in space of all orbitals or only in the space of the correlated valence orbitals. The two approaches lead thus to slightly different energies, but none of two is more valid or more accurate than the other. The ccsdf12, pnoccsd and ricc2 programs uses the former scheme with the block-diagonalization done in the space of all molecular orbitals. The same scheme is used e.g. in the CFOUR program suite, but other codes as e.g. the implementation in MOLPRO use a block-diagonalization restricted to the active valence space.

Perturbative triples corrections: To achieve ground state energies a high accuracy which systematically surpasses the acccuracy MP2 and DFT calculations for reaction and binding energies, the CCSD model should be combined with a perturbative correction for connected triples. The recommended approach for the correction is the CCSD(T) model

| (11.18) |

which includes the following two terms:

Integral-direct implementation and resolution-of-the-identity approximation: The computationally most demanding (in terms floating point operations) steps of a CCSD calculation are related to two kinds of terms. One of the most costly steps is the contraction

| (11.22) |

where a, b, c, and d are virtual orbitals. For small molecules with large basis sets or basis sets with
diffuse functions, where integral screening is not effective, it is time-determing step and can most
efficiently be evaluated with a minimal operation count of O^{2}V ^{2} (where O and V are number of,
respectively occupied and virtual orbitals), if the 4-index integrals (ac|bd) in the MO are
precalculated and stored on file before the iterative solution of the coupled-cluster equation, 11.4
and 11.5. For larger systems, however, the storage and I/O of the integrals (ac|bd) leads to
bottlenecks. An alternatively, this contribution can be evaluated in an integral-direct was
as

| (11.23) |

which, depending on the implementation and system, has formally a 2–3 times larger operation
count, but allows to avoid the storage and I/O bottlenecks by processing the 4-index integrals
on-the-fly without storing them. Furthermore, integral screening techniques can be
applied to reduce the operation count for large systems to asymptotic scaling with
(^{4}).

In TURBOMOLE only the latter algorithm is presently implemented. (For small systems other codes will therefore be faster.)

The other class of expensive contributions are so-called ring terms (in some publications denoted
as C and D terms) which involve contractions of the doubles amplitudes t_{aibj} with several 4-index
MO integrals with two occupied and two virtual indeces, partially evaluated with T_{1}-dependent
MO coefficients. For these terms the implementation in TURBOMOLE employs the
resolution-of-the-identity (or density-fitting) approximation (with the cbas auxiliary basis set)
to reduce the overhead from integral transformation steps. Due this approximation
CCSD energies obtained with TURBOMOLe will deviate from those obtained with
other coupled-cluster programs by a small RI error. This error is usually in the same
order or smaller the RI error for a RI-MP2 calculation for the same system and basis
sets.

The RI approximation is also used to evaluate the 4-index integrals in the MO basis needed for the perturbative triples corrections.

Disc space requirements:
In difference to CC2 and MP2, the CCSD model does no longer allow to avoid the
storage of double excitation amplitudes (t_{aibj}) and intermediates of with a similar size.
Thus, also the disc space requirements for the CCSD calculation are larger than for
RI-MP2 and RI-CC2 calculation for the same system. For a (closed-shell) CCSD ground
state energy calculations the amount of disc space needed can be estimated roughly
as

| (11.24) |

where N is the number of basis functions, O the number of occupied orbitals and m_{DIIS}
the number of vectors used in the DIIS procedure (by default 10, see Sec. 21.2.17 for
details).

For (closed-shell) CCSD(T) calculations the required disc space is with

| (11.25) |

somewhat larger.

For calculations with an open-shell (UHF or ROHF) reference wavefunctions the above estimates should be multiplied by factor of 4.

Memory requirements:
The CCSD and CCSD(T) implementation in Turbomole uses multi-pass algorithms to avoid
strictly the need to store any arrays with a size of N^{3} or O^{2}N^{2} or larger as complete array in main
memory. Therefore, the minimum memory requirements are relatively low—although is difficult to
give accurate estimate for them.

On should, however, be aware that, if the amount of memory provided to the program in the data
group $maxcor becomes too small compared to O^{2}N^{2}∕(128 × 1024) MBytes, loops will be broken
in many small batches at the cost of increased I/O operations and a decrease in performance. As
mentioned above, it is recommended to set $maxcor to 66–77% of the physical core memory
available for the calculation.

Important options: The options to define the orbital and the auxiliary basis sets, the maximum amount allocatable core memory ($maxcor), and the frozen-core approximation ($maxcor) have been mentioned above and described in the previous chapters on MP2 and CC2 calculations. Apart from this, CCSD and CCSD(T) calculations require very little additional input.

Relevant are in particular some options in the $ricc2 data group:

$ricc2

ccsd

ccsd(t)

conv=7

oconv=6

mxdiis=10

maxiter=25

ccsd

ccsd(t)

conv=7

oconv=6

mxdiis=10

maxiter=25

The options ccsd and ccsd(t) request, respectively, CCSD and CCSD(T) calculations. Since CCSD(T) requires the cluster amplitudes from a converged CCSD calculation, the option ccsd(t) is implies the ccsd option.

The number given for mxdiis defines the maximum number of vectors included in the DIIS procedure for the solution of the cluster equations. As mentioned above, it has some impact on the amount of disc space used by a CCSD calculation. Unless disc space becomes a bottleneck, it is not recommended to change the default value.

With maxiter one defines the maximum number of iterations for the solution of the cluster equations. If convergence is not reached within this limit, the calculation is stopped. Usually 25 iterations should be sufficient for convergence. Only in difficult cases with strong correlation effects more iterations are needed. It is recommended to increase this limit only if the reason for the strong correlation effects is known. (Since one reason could also be an input error as e.g. unreasonable geometries or orbital occupations as a wrong basis set assignment.)

The two parameters conv and oconv define the convergence thresholds for the iterative
solution of the cluster equations. Convergence is assumed if the change in the energy
(with respect to the previous iteration) has is smaller than 10^{-conv} and the euclidian
norm of the residual (the so-called vector function) is smaller than 10^{-oconv}. If conv is
not given in the data group $ricc2 the threshold for changes in the energy is set to
value given in $denconv (by default 10^{-7}). If oconv is not given in the data group
$ricc2 the threshold for the residual norm is by default set to 10 times the threshold
changes in the energy. With the default settings for these thresholds, the energy will
thus be converged until changes drop below 10^{-7} Hartree, which typically ensures an
accuracy of about 1 μH. These setting are thus rather tight and conservative even
for the calculation of highly accurate reaction energies. If for your application larger
uncertainites for the energy are tolerable, it is recommended to use less tight thresholds,
e.g. conv=6 or conv=5 for an accuracy of, respectively, at least 0.01 mH (0.03 kJ/mol) or 0.1
mH (0.3 kJ/mol). The settings for conv (and oconv) have not only an impact on the
number of iterations for the solution of the cluster equations, but as they determine the
thresholds for integral screening also to some extend on the costs for the individual
iterations.

CCSD(T) energy with a second-order correction from the interference-corrected MP2-F12: The error introduced from a CCSD(T) calculation with a finite basis set can be corrected from second-order corrections of the the interference-corrected MP2-F12 (INT-MP2-F12) (Ref. [150]). The approximate CCSD(T)-INT-F12 at the basis set limit is given from

| (11.26) |

From define, in the submenu $ricc2 select the ccsd(t) method and add the keyword intcorr

$ricc2

ccsd(t)

intcorr

ccsd(t)

intcorr

Then, switch on the f12 method (approximation A or B, inv or fixed). The corrected
CCSD(T)-INT-F12 energy will be printed in the end of the calculation. It is highly recommended
to start the CCSD(T)-INT-F12 calculation from a converged SCF calculation with symmetry,
which is transfromed to C_{1}. It is furthermore recommended to use Boys localized orbitals in the
$rir12 submenu. A table with the corrected second-order pair-electron energies and the
corresponding interference factors can also be printed in the output by using the keyword
intcorr all instead of intcorr.

Excitation energies with CCSD: Since release V6.5 electronic excitation energies can also be computed at the (conventional) CCSD level. For this the data group $excitations has be added (the same keyword as for CC2 apply). The implementation is currently restricted to vertical excitation energies (no transition moments or properties available) and in the closed–shell case to singlett excited states.

Note that for single-excitation dominated transitions CCSD is as CC2 correct through second-order in H and does not neccessarily more accurate than CC2. It is, however, for double excitations still correct through first-order, while CC2 describes double excitations only in a zero-order approximation. Therefore, CCSD results are more robust with respect to double excitation contributions to transitions and are thus usefull to check if CC2 is suitable for a certain problem.