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

E_CCSD(T) = E_CCSD + E⁽⁴⁾_DT + E⁽⁵⁾_ST (10.12)

which includes the following two terms:

$$\sum_{{\mu_2}}^{}$$ E⁽⁴⁾_DT = (10.13)

$$\sum_{{\mu_1}}^{}$$ E⁽⁵⁾_ST = (10.14)

where the approximate triples amplitudes evaluated as:

$${\frac{{\langle {{ijk}\atop{abc}}\vert[\hat{H},T_2]\vert\mathrm{H... ... }}{{ \epsilon_a-\epsilon_i + \epsilon_b-\epsilon_j + \epsilon_c-\epsilon_k }}}$$ (10.15)

In the literature one also finds sometimes the approximate triples model CCSD[T] (also denoted as CCSD+T(CCSD)), which is obtained by adding only E⁽⁴⁾_DT to the CCSD energy. Usually CCSD(T) is slightly more accurate than CCSD[T], although for closed-shell or spin-unrestricted open-shell reference wavefunctions the energies of both models, CCSD(T) and CCSD[T] model, are correct through 4.th order perturbation theory. For a ROHF reference, however, E⁽⁵⁾_ST contributes already in 4.th order and only the CCSD(T) model is correct through 4.th order perturbation theory.