7.8 Coupled-Cluster Excited-State and Open-Shell Methods 7.8.22 Job Control for EOM-CC(2,3)7.8.24 Potential Energy Surface Crossing Minimization

7.8.23 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 $N^{7}$ 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}}

(7.60)

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:

	$\displaystyle\tilde{\sigma}_{ijk}^{abc}(\mu)$	$\displaystyle=$	$\displaystyle\langle\Phi_{0}\|({L_{1}}_{\mu}+{L_{2}}_{\mu})(He^{(T_{1}+T_{2})})% _{c}\|\Phi_{ijk}^{abc}\rangle$		(7.61)
	$\displaystyle\sigma_{ijk}^{abc}(\mu)$	$\displaystyle=$	$\displaystyle\langle\Phi_{ijk}^{abc}\|[He^{(T_{1}+T_{2})}({R_{0}}_{\mu}+{R_{1}}% _{\mu}+{R_{2}}_{\mu})]_{c}\|\Phi_{0}\rangle$

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.⁷³⁰

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. 602.

Similar corrections are available for EOM-IP-CCSD,⁶⁰³ 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 .

7.8.23.1 Job Control for Non-Iterative Triples Corrections

Triples corrections are requested by using METHOD or EOM_CORR:

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
       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.

7.8.23.2 Examples

Example 7.80 EOM-EE-CCSD(fT) calculation of CH ${}^{+}$ .

$molecule
   1 1
   C
   H  C CH

   CH  = 2.137130
$end

$rem
   INPUT_BOHR             true
   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

Example 7.81 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
$end

Example 7.82 EOM-IP-CCSD(dT) calculations of Mg.

$molecule
   0 1
   Mg    0.000000     0.000000    0.000000
$end

$rem
   JOBTYPE      sp
   METHOD       eom-ccsd(dt)
   BASIS        6-31g
   IP_STATES    [1,0,0,0,0,1,1,1]
$end

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

7.8.23.1 Job Control for Non-Iterative Triples Corrections

7.8.23.2 Examples