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Orbital-unrelaxed first-order properties

The unrelaxed first-order properties are calculated from the variational excited states Lagrangian [114], which for the calculation of unrelaxed properties is composed of the unrelaxed ground state Lagrangian, Eq. (9.12), and the expression for the excitation energy:
Lur 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 (9.18)
    + $\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}}$ + [F0 + β$\displaystyle \hat{{V}}$, T2]| HF〉  

where it is assumed that the left and right eigenvectors are normalized such that $ \sum_{{\mu\nu}}^{}$$ \bar{{E}}_{{\mu}}^{}$μ| τνEν = 1 and H = H0 + βV . The first-order properties are calculated as first derivatives of Lur CC2, ex(E,$ \bar{{E}}$, t,$ \bar{{t}}^{{(ex)}}_{}$, β) with respect to the field strength β and are evaluated via a density formalism:
Vur, ex = $\displaystyle \cal {R}$$\displaystyle \left(\vphantom{\frac{\partial L^{ur,ex}(E,\bar{E},t,\bar{t}^{(ex)},\beta)
}{\partial \beta} }\right.$$\displaystyle {\frac{{\partial L^{ur,ex}(E,\bar{E},t,\bar{t}^{(ex)},\beta)
}}{{\partial \beta}}}$$\displaystyle \left.\vphantom{\frac{\partial L^{ur,ex}(E,\bar{E},t,\bar{t}^{(ex)},\beta)
}{\partial \beta} }\right)_{0}^{}$ = $\displaystyle \sum_{{pq}}^{}$Dur, expqVpq  , (9.19)

(Again ℜ indicates that the real part is taken.) The unrelaxed excited-state properties obtained thereby are related in the same way to the total energy of the excited states as the unrelaxed ground-state properties to the energy of the ground state and the differences between excited- and ground-state unrelaxed properties are identical to those identified from the second residues of the quadratic response function. For a detailed description of the theory see refs. [114,113]; the algorithms for the RI-CC2 implementation are described in refs. [111,12]. ref. [111] also contains a discussion of the basis set effects and the errors introduced by the RI approximation.

The calculation of excited-state first-order properties thus requires the calculation of both the right (Eμ ) and left ( $ \bar{{E}}_{\mu}^{}$ ) eigenvectors and of the excited state Lagrangian multipliers $ \bar{{t}}^{{(ex)}}_{\mu}$ . The disk space and CPU requirements for solving the equations for $ \bar{{E}}_{\mu}^{}$ and $ \bar{{t}}^{{(ex)}}_{\mu}$ are about the same as those for the calculation of the excitation energies. For the construction of the density matrices in addition some files with $ \cal {O}$(nrootN2) size are written, where nroot is the number of excited states.

The single-substitution parts of the excited-states Lagrangian multipliers $ \bar{{t}}^{{(ex)}}_{\mu}$ are saved in files named CCNL0-s--m-xxx.

For the calculation of first-order properties for excited states, the keyword exprop must be added with appropriate options to the data group $excitations; else the input is same as for the calculation of excitation energies:

$ricc2
  cc2
$response
  fop unrelaxed_only operators=diplen,qudlen
$excitations
  irrep=a1 nexc=2
  exprop states=all operators=diplen,qudlen


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