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Theory

In the following, only the most important ideas are presented and discussed with a focus on the PERI-CC2 model. The essential concept is the introduction of an environment coupling operator $ \hat{G}$(DCC)


$\displaystyle \hat{G}$(DCC) = $\displaystyle \hat{G}^{\text}_{}$es + $\displaystyle \hat{G}^{\text}_{}$pol(DCC) (9.24)

with the electrostatic contribution


$\displaystyle \hat{G}^{\text}_{}$es = $\displaystyle \sum_{{m = 1}}^{M}$$\displaystyle \sum_{{k = 0}}^{K}$$\displaystyle \sum_{{pq}}^{}$Θ(k)m, pqQm(k)$\displaystyle \hat{{E}}_{{pq}}^{}$ (9.25)

and the polarization contribution


$\displaystyle \hat{G}^{\text}_{}$pol(DCC) = $\displaystyle \sum_{{u = 1}}^{U}$$\displaystyle \sum_{{pq}}^{}$Θ(1)u, pqμuind(DCC)$\displaystyle \hat{{E}}_{{pq}}^{}$ . (9.26)

Here, Θ(k)m, pq are multipole interaction integrals of order k and μuind are the induced dipoles which can be obtained from the electric field Fu and the polarizability α u at a site u :

μuind = Fu$\displaystyle \mbox{\boldmath$\alpha$}$u (9.27)

Because the induced dipoles depend on the electron density and vice versa, their computations enter the self-consistent part of the HF cycle. Introducing $ \hat{G}$(DCC) into standard equations for the HF reference state and the CC2 equations leads to a general PE-CC2 formulation. To maintain efficiency, a further approximation has been introduced which makes the operator only dependent on a CCS-like density term. These general ideas define the PERI-CC2 model and allow to formulate the corresponding Lagrangian expression



LPERI-CC2(t,$\displaystyle \bar{{\mathbf{t}}}$) = EPE-HF + 〈HF|$\displaystyle \hat{{W}}$($\displaystyle \hat{{T}}_{1}^{}$ + $\displaystyle \hat{{T}}_{2}^{}$ + $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \hat{{T}}_{1}^{2}$)| HF〉 +  
    $\displaystyle \sum_{\mu_1} \bar{t}_{\mu_1} \ensuremath{\langle \mu_1\vert} \til...
...{PE}, \hat{T}_1 ] + [\tilde{W}, \hat{T}_2] \ensuremath{\vert\text{HF}\rangle} +$  
    $\displaystyle \sum_{\mu_2} \bar{t}_{\mu_2} \ensuremath{\langle \mu_2\vert} \tilde{W} + [ \hat{F}^\text{PE}, \hat{T}_2 ] \ensuremath{\vert\text{HF}\rangle}$  
    $\displaystyle - \frac{1}{2} \sum_{uv} F_u^{\text{elec}}(\mathbf{D}^{\Delta\prime}) R_{uv} F_v^{\text{elec}}(\mathbf{D}^{\Delta\prime})$ (9.28)

from which all PERI-CC2 equations including the linear response terms may be derived. Note that the dependency on the density couples the CC amplitude and multiplier equations for the ground state solution vector.


next up previous contents index
Next: Computational details: SCF calculations Up: Polarizable embedding calculations Previous: Polarizable embedding calculations   Contents   Index
TURBOMOLE M