Truncating the cluster operator \( \hat{T} \) at the \( n=2 \) level, defines +CCSD approximation to the Coupled Cluster wavefunction. The +coupled cluster wavefunction is now given by +
+ +$$ +\begin{equation*} + \vert \Psi_{CC}\rangle = e^{\hat{T}_1 + \hat{T}_2} \vert \Phi_0\rangle, +\end{equation*} +$$ + +where
+$$ +\begin{align*} + \hat{T}_1 &= + \sum_{ia} + t_{i}^{a} a_{a}^\dagger a_i \\ + \hat{T}_2 &= \frac{1}{4} + \sum_{ijab} + t_{ij}^{ab} a_{a}^\dagger a_{b}^\dagger a_{j} a_{i}. +\end{align*} +$$ + + +where
+$$ +\begin{align*} + f_q^p &= \langle p\vert \hat{t} \vert q\rangle + \sum_i \langle pi\vert \hat{v} \vert qi\rangle \\ + \langle pq \vert \vert rs \rangle &= \langle pq \vert \hat{v} \vert rs \rangle \\ + \mathrm{E}_0 &= \sum_i \langle i\vert \hat{t} \vert i\rangle + \frac{1}{2} \sum_{ij} \langle ij\vert \hat{v} \vert ij\rangle. +\end{align*} +$$ + + + + + + + + + +Truncating the cluster operator \( \hat{T} \) at the \( n=2 \) level, defines +CCSD approximation to the Coupled Cluster wavefunction. The +coupled cluster wavefunction is now given by +
+ +$$ +\begin{equation*} + \vert \Psi_{CC}\rangle = e^{\hat{T}_1 + \hat{T}_2} \vert \Phi_0\rangle, +\end{equation*} +$$ + +where
+$$ +\begin{align*} + \hat{T}_1 &= + \sum_{ia} + t_{i}^{a} a_{a}^\dagger a_i \\ + \hat{T}_2 &= \frac{1}{4} + \sum_{ijab} + t_{ij}^{ab} a_{a}^\dagger a_{b}^\dagger a_{j} a_{i}. +\end{align*} +$$ + + +where
+$$ +\begin{align*} + f_q^p &= \langle p\vert \hat{t} \vert q\rangle + \sum_i \langle pi\vert \hat{v} \vert qi\rangle \\ + \langle pq \vert \vert rs \rangle &= \langle pq \vert \hat{v} \vert rs \rangle \\ + \mathrm{E}_0 &= \sum_i \langle i\vert \hat{t} \vert i\rangle + \frac{1}{2} \sum_{ij} \langle ij\vert \hat{v} \vert ij\rangle. +\end{align*} +$$ + + + + + + + + + +