Projections

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According to equations (16), (17), (19), (20) and (21) we finally summerize and define our wavefunction as

$$\Psi \rangle \sum_{\mu}^{} \mu \hat{T} \mu \mu \rangle$$ (21)

and the cluster operator as

$$\hat{T} \mu \varphi \mu \sum_{{\hat \tau_i(\mu) \in \mathbb{T}_\mu}}^{} \hat{\tau}_{i} \mu \mu \rangle \hat{\tau}_{i}^{} \mu$$ (22)

Inserting equation (22) into the Schrödinger equation we get

$$\hat{H} \sum_{\mu}^{} \mu \hat{T} \mu \mu \rangle \sum_{\mu}^{} \mu \hat{T} \mu \mu \rangle$$ (23)

Projecting equation (24) from the left onto $\langle$$\rho$| $\in$ $\mathbb {P}$^$\dagger$ $\cup$ $\mathbb {Q}$^$\dagger$ we obtain a system of equations linear in the reference coefficients c_$\mu$ and non linear in the amplitudes t_{| i$\rangle$}

$$\sum_{\mu}^{} \mu \langle \rho \hat{H} \hat{T} \mu \mu \rangle$$ (24)

with the additional equation

$$\sum_{\mu}^{} \mu$$ (25)

which collapses to the usual intermediate normalization of single reference coupled cluster theory in case of a one-determinantal reference space. It is possible to prove size consistency for this ansatz [15] although the cluster operator $\hat{T}$ is not connected. In actual applications [15,16] the ansatz performed very well.