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

(December 20, 2021)

The effect of triple excitations to EOM-CCSD energies can be included via perturbation theory in an economical $N^{7}$ 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:

 $\Delta E^{(2)}_{\mu}=-\frac{1}{36}\sum_{i,j,k}\sum_{a,b,c}\frac{{\tilde{\sigma% }_{ijk}^{abc}}(\mu){\sigma_{ijk}^{abc}}(\mu)}{D_{ijk}^{abc}-\omega_{\mu}}$ (7.73)

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:

 $\displaystyle\tilde{\sigma}_{ijk}^{abc}(\mu)$ $\displaystyle=$ $\displaystyle\langle\Phi_{0}|({L_{1}}_{\mu}+{L_{2}}_{\mu})(He^{(T_{1}+T_{2})})% _{c}|\Phi_{ijk}^{abc}\rangle$ (7.74) $\displaystyle\sigma_{ijk}^{abc}(\mu)$ $\displaystyle=$ $\displaystyle\langle\Phi_{ijk}^{abc}|[He^{(T_{1}+T_{2})}({R_{0}}_{\mu}+{R_{1}}% _{\mu}+{R_{2}}_{\mu})]_{c}|\Phi_{0}\rangle$

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 $\bar{H}$, $\langle\Phi_{ijk}^{abc}|(He^{(T_{1}+T_{2})})_{C}|\Phi_{ijk}^{abc}\rangle$, $D_{ijk}^{abc}$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. 895 Piecuch P., Włoch M.
J. Chem. Phys.
(2005), 123, pp. 224105.

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.  730 Manohar P. U., Krylov A. I.
J. Chem. Phys.
(2008), 129, pp. 194105.
.

Similar corrections are available for EOM-IP-CCSD, 731 Manohar P. U., Stanton J. F., Krylov A. I.
J. Chem. Phys.
(2009), 131, pp. 114112.
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 .

## 7.10.26.1 Job Control for Non-Iterative Triples Corrections

Triples corrections are requested by using METHOD or EOM_CORR:

METHOD

METHOD
Specifies the calculation method.
TYPE:
STRING
DEFAULT:
No default value
OPTIONS:
EOM-CCSD(DT) EOM-CCSD(dT), available for EE, SF, and IP EOM-CCSD(FT) EOM-CCSD(fT), available for EE, SF, IP, and EA EOM-CCSD(ST) EOM-CCSD(sT), available for IP
RECOMMENDATION:
None

EOM_CORR

EOM_CORR
Specifies the correlation level.
TYPE:
STRING
DEFAULT:
None No correction will be computed
OPTIONS:
SD(DT) EOM-CCSD(dT), available for EE, SF, and IP SD(FT) EOM-CCSD(fT), available for EE, SF, IP, and EA SD(ST) EOM-CCSD(sT), available for IP
RECOMMENDATION:
None

Note:  In CCMAN2, EOM-IP-CCSD(fT) can be computed with or without USE_LIBPT = TRUE.

## 7.10.26.2 Examples

Example 7.102  EOM-EE-CCSD(fT) calculation of CH${}^{+}$.

$molecule 1 1 C H C 1.13092$end

$rem METHOD eom-ccsd(ft) BASIS general EE_STATES [1,0,1,1] EOM_DAVIDSON_MAX_ITER 60 increase number of Davidson iterations$end

$basis H 0 S 3 1.00 19.24060000 0.3282800000E-01 2.899200000 0.2312080000 0.6534000000 0.8172380000 S 1 1.00 0.1776000000 1.000000000 S 1 1.00 0.0250000000 1.000000000 P 1 1.00 1.00000000 1.00000000 **** C 0 S 6 1.00 4232.610000 0.2029000000E-02 634.8820000 0.1553500000E-01 146.0970000 0.7541100000E-01 42.49740000 0.2571210000 14.18920000 0.5965550000 1.966600000 0.2425170000 S 1 1.00 5.147700000 1.000000000 S 1 1.00 0.4962000000 1.000000000 S 1 1.00 0.1533000000 1.000000000 S 1 1.00 0.0150000000 1.000000000 P 4 1.00 18.15570000 0.1853400000E-01 3.986400000 0.1154420000 1.142900000 0.3862060000 0.3594000000 0.6400890000 P 1 1.00 0.1146000000 1.000000000 P 1 1.00 0.0110000000 1.000000000 D 1 1.00 0.750000000 1.00000000 ****$end


View output

Example 7.103  EOM-SF-CCSD(dT) calculations of methylene.

$molecule 0 3 C H 1 CH H 1 CH 2 HCH CH = 1.07 HCH = 111.0$end

$rem METHOD eom-ccsd(dt) BASIS 6-31G SF_STATES [2,0,0,2] N_FROZEN_CORE 1 N_FROZEN_VIRTUAL 1 CCMAN2 false !only works in ccman1$end


View output

Example 7.104  EOM-IP-CCSD(dT) calculations of Mg.

$comment This job segfaults on some platforms.$end

$molecule 0 1 Mg 0.000000 0.000000 0.000000$end

$rem METHOD eom-ccsd(dt) BASIS 6-31g IP_STATES [1,0,0,0,0,1,1,1] CCMAN2 false NYI in ccman2$end


View output