Amplitudes

Amplitudes are a set of tensors, usually containing the closed-shell singles \(t_i^a\) and the doubles \(t_{ij}^{ab}\) amplitudes, that solve the employed coupled-cluster amplitude equation. The canonical closed-shell coupled-cluster correlation energy is given by

\begin{equation} E_\mathrm{cc} = \frac12 \sum_{abij} \left(t^{ab}_{ij} + t^a_i t^b_j\right) \left(2V^{ij}_{ab} - V^{ji}_{ab}\right). \end{equation}

Note, that the closed-shell doubles amplitudes are symmetric with respect to interchanging left and right inidices \(t^{ab}_{ij}=t^{ba}_{ji}\).

The Amplitudes parametrize the correlated wave function \(|\Psi\rangle\) using the coupled-cluster Ansatz

\begin{equation} | \Psi \rangle = e^{\hat T} | \Phi \rangle, \end{equation}

where \(|\Phi\rangle\) denotes the single Hartree–Fock slater determinant. The cluster operator \(\hat T = \hat T_1 + \hat T_2 + \ldots\) is expanded in increasing number of excitation levels. The single and double exciation parts of the cluster operator are given by

\begin{eqnarray} \hat T_1 = \sum_{ai} t^a_i \hat\tau^a_i, \\ \hat T_2 = \sum_{abij} t^{ab}_{ij} \hat\tau^{ab}_{ij}, \end{eqnarray}

where \(\hat \tau^{a\ldots}_{i\ldots} = \hat c^\dagger_a\ldots \hat c_i\ldots\) denotes the exciation operator. The coefficients \(t^a_i\), \(t^{ab}_{ij}\), \(\ldots\) are called coupled-cluster Amplitudes. The Amplitudes are generated by the CoupledCluster algorithm by solving the amplitude equation of the employed coupled-cluster method, described therein.