The projected Schrödinger coupled-cluster equations
Given the cluster operator \[\begin{equation} \require{physics} \hat{T} = \sum_\mu t_\mu \hat{\tau}_\mu \thinspace , \end{equation}\] in which \(\hat{\tau}_\mu\) is a general excitation operator and \(\vb{t} = \qty{t_\mu}\) represents the coupled-cluster amplitudes, the coupled cluster method uses an exponential parametrization as its wave function model: \[\begin{equation} \ket{\Psi(\vb{t})} = \exp(\hat{T}) \ket{\Psi_0} \thinspace , \end{equation}\] i.e. it creates excitations on top of its reference \(\ket{\Psi_0}\). By projecting the electronic Schrödinger equation \(\eqref{eq:electronic_schrodinger_equation}\) onto the reference \(\ket{\Psi_0}\), we find the coupled cluster energy \[\begin{equation} E(\boldsymbol{\eta}, \vb{t}) = \frac{ \ev{ \exp(-\hat{T}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(\hat{T}) }{\Psi_0} }{ \braket{\Psi_0} } \end{equation}\] and by projecting on the excited determinants \[\begin{equation} \ket{\mu} = \hat{\tau}_\mu \ket{\Psi_0} \thinspace , \end{equation}\] we find the equations that determine the coupled cluster amplitudes \(\vb{t}\): \[\begin{equation} \matrixel{\mu}{ \exp(-\hat{T}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(\hat{T}) }{\Psi_0} = 0 \thinspace . \end{equation}\] Since the form of the cluster operator \(\hat{T}_P\) determines the excited determinants that are projected on, there are as many coupled cluster equations as there are amplitudes.
In order to make the connection with the general PSE framework, we define the PSE reference as \[\begin{equation} \bra{0} = \bra{\Psi_0} \exp(-\hat{T}) \end{equation}\] and the rest of the projection space as \[\begin{equation} \bra{i} = \bra{\mu} \exp(-\hat{T}) \end{equation}\] and fill in these expressions into the PSE energy \(\eqref{eq:pse_energy_function}\) and functions \(\eqref{eq:pse_function}\). It is very interesting to notice that in the PSE functions, the resulting term involving the energy, i.e. \[\begin{equation} E(\boldsymbol{\eta}, \vb{t}) \matrixel{\mu}{ \exp(-\hat{T}) \exp(\hat{T}) }{\Psi_0} \end{equation}\] vanishes because any excited determinant is orthogonal to the coupled cluster reference \(\ket{\Psi_0}\). We thus conclude that coupled-cluster theory fits perfectly into the general PSE framework we have established and that by including the coupled cluster amplitudes into the projection space (in a dual way), the resulting equations simplify considerably.
In the previously discussed form, coupled-cluster theory is non-variational, but by constructing the coupled-cluster Lagrangian Stein, Henderson, and Scuseria (2014) \[\begin{equation} \mathscr{L}_{\text{CC}}(\boldsymbol{\eta}, \vb{t}, \boldsymbol{\lambda}) = \ev{ \exp(-\hat{T}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(\hat{T}) }{\Psi_0} + \sum_\mu \lambda_{\mu} \matrixel{\mu}{ \exp(-\hat{T}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(\hat{T}) }{\Psi_0} \thinspace , \end{equation}\] we can again recover a fully variational theory. Often, this Lagrangian is written in a more compact form by introding the Lagrange multiplier-dependent deexcitation operator \[\begin{equation} \hat{\Lambda}_{\text{CC}} = \sum_\mu \lambda_\mu \hat{\tau}^\dagger_\mu \thinspace , \end{equation}\] leading to \[\begin{equation} \mathscr{L}_{\text{CC}}(\boldsymbol{\eta}, \vb{t}, \boldsymbol{\lambda}) = \ev{ (1 + \hat{\Lambda}_{\text{CC}}) \exp(-\hat{T}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(\hat{T}) }{\Psi_0} \thinspace . \end{equation}\]
Inspired by this approach, we can implicitly define a Lagrange multiplier-dependent PSE lambda operator as \[\begin{equation} \bra{0} \hat{\Lambda}_{\text{PSE}} = \sum_a^S \lambda_a \bra{a} \end{equation}\] such that we can rewrite the PSE Lagrangian as \[\begin{equation} \mathscr{L}_{\text{PSE}}( \boldsymbol{\eta}, \vb{p}, \boldsymbol{\lambda} ) = \matrixel{0}{ \qty( \frac{1}{ \braket{0}{\Psi(\vb{p})} } + \hat{\Lambda}_{\text{PSE}} ) \hat{\mathcal{H}}(\boldsymbol{\eta}) }{\Psi(\vb{p})} - E(\boldsymbol{\eta}, \vb{p}) \matrixel{0}{ \hat{\Lambda}_{\text{PSE}} }{\Psi(\vb{p})} \thinspace . \end{equation}\] As we have seen that the last term vanishes in coupled cluster theory because the projection space is such that the vanishing occurs, it could prove useful to tailor projection spaces to the wave function models in such a way that the right term vanishes or is simple to calculate.