7.10.26 Non-Iterative Triples Corrections to EOM-CCSD and CCSD

The effect of triple excitations to EOM-CCSD energies can be included via perturbation theory in an economical $𝒪(N^7)$ computational scheme. Using EOM-CCSD wave functions as zero-order wave functions, the second order triples correction to the $\mu$th EOM-EE or SF state is:

where $i,j$ and $k$ denote occupied orbitals, and $a,b$ and $c$ are virtual orbital indices. ${\omega }_{\mu }$ is the EOM-CCSD excitation energy of the $\mu$th state. The quantities $\tilde{\sigma }$ and $\sigma$ are:

where, the $L$ and $R$ are left and right eigen-vectors for $\mu$th state. Two different choices of the denominator, $D_{ijk}^{abc}$, define the (dT) and (fT) variants of the correction. In (fT), $D_{ijk}^{abc}$ is just Hartree-Fock orbital energy differences. A more accurate (but not fully orbital invariant) (dT) correction employs the complete three body diagonal of $\overline{H}$, $⟨{\mathrm{\Phi }}_{ijk}^{abc}|{(H{e}^{(T_1+T_2)})}_C|{\mathrm{\Phi }}_{ijk}^{abc}⟩$, $D_{ijk}^{abc}$as a denominator. For the reference (e.g., a ground-state CCSD wave function), the (fT) and (dT) corrections are identical to the CCSD(2)${}_T$ and CR-CCSD(T)${}_L$ corrections of Piecuch and coworkers. ¹⁰¹⁷ Piecuch P., Włoch M.
J. Chem. Phys.
(2005), 123, pp. 224105. Link

The EOM-SF-CCSD(dT) and EOM-SF-CCSD(fT) methods yield a systematic improvement over EOM-SF-CCSD bringing the errors below 1 kcal/mol. For theoretical background and detailed benchmarks, see Ref.  829 Manohar P. U., Krylov A. I.
J. Chem. Phys.
(2008), 129, pp. 194105. Link .

Similar corrections are available for EOM-IP-CCSD, ⁸³⁰ Manohar P. U., Stanton J. F., Krylov A. I.
J. Chem. Phys.
(2009), 131, pp. 114112. Link where triples correspond to $3h2p$ excitations and EOM-EA-CCSD, where triples correspond to $2h3p$ excitations.

Note:  Due to the orbital non-invariance problem, using (dT) correction is discouraged.

Note:  EOM-IP-CCSD(fT) correction is now available both in CCMAN and CCMAN2 .