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Orbital-relaxed first-order properties and gradients

To obtain orbital-relaxed first-order properties or analytic derivatives (gradients) the Lagrange functional for the excited state in Eq. (9.18) is--analogously to the treatment of ground states--augmented by the equations for the SCF orbitals and the perturbations is also included in the Fock operator:
Lrel CC2, ex(E,$\displaystyle \bar{{E}}$, t,$\displaystyle \bar{{t}}^{{(ex)}}_{}$, β) = 〈HF| H| CC〉 + $\displaystyle \sum_{{\mu \nu}}^{}$$\displaystyle \bar{{E}}_{{\mu}}^{}$$\displaystyle \bf A_{{\mu\nu}}^{}$(t, β)Eν (9.20)
    + $\displaystyle \sum_{{\mu_1}}^{}$$\displaystyle \bar{{t}}^{{(ex)}}_{{\mu_1}}$μ1|$\displaystyle \hat{{H}}$ + [$\displaystyle \hat{{H}}$, T2]| HF〉  
    + $\displaystyle \sum_{{\mu_2}}^{}$$\displaystyle \bar{{t}}^{{(ex)}}_{{\mu_2}}$μ2|$\displaystyle \hat{{H}}$ + [F, T2]| HF〉 + $\displaystyle \sum_{{\mu_0}}^{}$$\displaystyle \bar{{\kappa}}^{{(ex)}}_{{\mu_0}}$Fμ0  .  

Compared to unrelaxed properties, the calculation of relaxed properties needs in addition for each excited state the solution of a CPHF equations for the Lagrangian multipliers $ \bar{{\kappa}}^{{(ex)}}_{{\mu_0}}$, for which the computational costs are similar to those of a Hartree-Fock calculation.

Orbital-relaxed properties are requested by adding the flag relaxed to the input line for the exprop option. The following is an example for a CC2 single point calculation for orbital-relaxed excited state properties:

$ricc2
  cc2
$excitations
  irrep=a1 nexc=2
  exprop states=all relaxed operators=diplen,qudlen
Note that during the calculation of orbital-relaxed excited-state properties the corresponding unrelaxed properties are also automatically evaluated at essentially no additional costs. Therefore, the calculation of unrelaxed properties can not be switched off when relaxed properties have been requested.

Again the construction of gradients requires the same variational densities as needed for relaxed one-electron properties and the solution of the same equations. The construction of the gradient contributions from one- and two-electron densities and derivative integrals takes approximately the same time as for ground states gradients (approx. 3-4 SCF iterations) and only minor extra disk space. The implementation of the excited state gradients for the RI-CC2 approach is described in detail in Ref. [14]. There one can also find some information about the performance of CC2 for structures and vibrational frequencies of excited states.

For the calculation of an excited state gradient with CC2 at a single point (without geometry optimization and if it is not a calculation with NumForce) one can use the input:

$ricc2
  cc2
$excitations
  irrep=a1 nexc=2
  xgrad states=(a1 2)
Note, that presently it is not possible to compute gradients for more than one excited state in one ricc2 calculation.

For geometry optimizations or a numerical calculation of the Hessian with NumForce the wavefunction model and the excited state for which the geometry should be optimized have to be specified in the data group $ricc2 with the keyword geoopt:

$ricc2
  geoopt model=cc2 state=(a1 2)
$excitations
  irrep=a1 nexc=2
If the geometry optimization should carried out for the lowest excited state (of those for which an excitation energy is requested in $excitation), one can use alternatively state=(s1).

Since the calculation of unrelaxed and relaxed first-order properties can be combined gradient calculations without significant extra costs, a request for excited state gradients will automatically enforce the calculation of the relaxed and unrelaxed dipole moments. If the keyword geoopt is used, the relaxed dipole moment for the specified excited state and wavefunction model will be written to the control file and used in calculations with NumForce for the evaluation of the IR intensities.


next up previous contents index
Next: Visualization of densities and Up: Excited State Properties, Gradients Previous: Orbital-unrelaxed first-order properties   Contents   Index
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