# Amplitudes

## 1.Brief description

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

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

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

## 2.Specification

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

$$| \Psi \rangle = e^{\hat T} | \Phi \rangle,$$

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.

## 3.Literature

Created: 2022-09-19 Mon 15:00