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CC2 Ground-State Energy Calculations

The CC2 ground-state energy is--similarly to other coupled-cluster energies--obtained from the expression

$\displaystyle E_{CC}$ $\displaystyle = \langle\mathrm{HF}\vert H \vert\mathrm{CC}\rangle = \langle\mathrm{HF}\vert H \exp(T) \vert\mathrm{HF}\rangle  ,$ (9.1)
  $\displaystyle = E_{SCF} + \sum_{iajb} \Big[t^{ab}_{ij} + t^a_i t^b_j \Big] \Big[ 2(ia\vert jb) - (ja\vert ib) \Big] ,$ (9.2)

where the cluster operator $ T$ is expanded as $ T= T_1 + T_2$ with

$\displaystyle T_1$ $\displaystyle = \sum_{ai} t_{ai} \tau_{ai}$ (9.3)
$\displaystyle T_2$ $\displaystyle = \frac{1}{2}\sum_{aibj} t_{aibj} \tau_{aibj}$ (9.4)

(for a closed-shell case; in an open-shell case an additional spin summation has to be included). The cluster amplitudes $ t_{ai}$ and $ t_{aibj}$ are obtained as solution of the CC2 cluster equations [82]:

$\displaystyle \Omega_{\mu_1}$ $\displaystyle = \langle\mu_{1} \vert \hat{H} + [\hat{H},T_2] \vert\mathrm{HF}\rangle = 0  ,$ (9.5)
$\displaystyle \Omega_{\mu_2}$ $\displaystyle = \langle\mu_{2} \vert \hat{H} + [F,T_2] \vert\mathrm{HF}\rangle = 0  ,$ (9.6)

with

$\displaystyle \hat{H} = \exp(-T_1) H \exp(T_1). $

The residual of the cluster equations $ \Omega(t_{ai},t_{aibj})$ is the so-called vector function. The recommended reference for the CC2 model is ref. [82], the implementation with the resolution-of-the-identity approximation, RI-CC2, was first described in ref. [10].



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