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# 6.10.3 Coupled Cluster Singles, Doubles and Triples (CCSDT)

(July 10, 2023)

The coupled-cluster method with single, double and triple substitutions, abbreviated as CCSDT 893 Noga J., Bartlett R. J.
J. Chem. Phys.
(1987), 86, pp. 7041.
includes single, double and triple excitation operators in the exponential ansatz. The theory of the method is very similar to that of CCSD – with triple excitations included fully. We only present the basic equations. These can be compared with the CCSD equations presented in the previous section, so as to understand the similarities and differences between CCSD and CCSDT. The CCSDT wave-function defined by

 $\left|{\Psi_{\mathrm{CCSD}}}\right\rangle=\exp\left({\hat{T}_{1}+\hat{T}_{2}+% \hat{T}_{3}}\right)\left|{\Phi_{0}}\right\rangle$ (6.38)

where, the operators, $\hat{T}_{1}$ and $\hat{T}_{2}$ are defined using Eqs. 6.33 and 6.34. The operator, $\hat{T}_{3}$ is defined by

 $\hat{T}_{3}\left|{\Phi_{0}}\right\rangle=\frac{1}{36}\sum_{ijk}^{\mathrm{occ}}% {\sum_{abc}^{\mathrm{virt}}{t_{ijk}^{abc}}}\left|{\Phi_{ijk}^{abc}}\right\rangle$ (6.39)

The CCSDT equations are coupled non-linear simultaneous equations of the tensors, $\hat{T}_{1}$, $\hat{T}_{2}$ and $\hat{T}_{3}$. However, the correlation energy depends only on $\hat{T}_{1}$ and $\hat{T}_{2}$ amplitudes (The energy equations is same as Eq 6.35). Thus the effect of triples is only in the way they affect the singles and doubles.

Currently, the CCSDT functionality is available for computation of correlation energy only.