# Coupled cluster

Coupled cluster method is a technique used for description of the many-body systems. The method was initially developed by Fritz Coester and Hermann Kümmel in 1950's for studying nuclear physics phenomena but it became more frequently used after Jiři Čížek and Josef Paldus reformulated the method for studying electronic correlation in atoms and molecules in 1960's. It is now one of the most prevalent methods in quantum chemistry that include electronic correlation. The method is based on exponential Ansatz:

[itex] \vert{\Psi}\rangle = e^\hat{T} \vert{\Phi_0}\rangle [itex],

where [itex]\vert{\Psi}\rangle[itex] is the wave function, [itex]\vert{\Phi_0}\rangle[itex] is the reference function (e.g. Hartree-Fock function), and [itex]\hat{T}[itex] is the cluster operator:

[itex] \hat{T}=\hat{T}_1 + \hat{T}_2 + \cdots [itex],

where in the formalism of second quantization:

[itex] \hat{T}_1=\sum_{i}\sum_{a} t_{i}^{a} \hat{a}_{i}\hat{a}^{\dagger}_{a}, [itex]

[itex] \hat{T}_2=\frac{1}{4}\sum_{i,j}\sum_{a,b} t_{ij}^{ab} \hat{a}_{i}\hat{a}_j\hat{a}^{\dagger}_{a}\hat{a}^{\dagger}_{b}. [itex]

In the above formulae [itex]\hat{a}[itex] and [itex]\hat{a}^{\dagger}[itex] denote the creation and annihilation operators and i,j stand for occupied and a,b for unoccupied orbitals. The coupled cluster equations are usually derived using diagrammatic technique and result in nonlinear equations which can be solved in an iterative way.

In the simplest version one considers only [itex]\hat{T}_2[itex] operator (double excitations). This method is called coupled cluster with doubles (CCD in short).

The method gives exact non-relativistic solution of the Schrödinger equation of the n-body problem if one includes up to the [itex]\hat{T}_n[itex] cluster operator. However, the computational effort of solving the equations grows steeply with the order of the cluster operator and in practical applications the method is limited to the first few orders.

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