Orbital optimization in the projected Schrödinger equation framework

In the variational PSE framework, if we want to search for the optimal set of orbitals that complement the wave function model \(\require{physics} \ket{\Psi(\vb{p})}\), we want to minimize the PSE energy of the rotated state, \[\begin{equation} E(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}) = \frac{ \matrixel{0}{ \exp(\hat{\kappa}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(-\hat{\kappa}) }{\Psi(\vb{p})} }{ \braket{0}{\Psi(\vb{p})} } \end{equation}\] subject to the condition that the PSEs are fulfilled for that rotated state: \[\begin{equation} f_a(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}) = \matrixel{a}{ \exp(\hat{\kappa}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(-\hat{\kappa}) }{\Psi(\vb{p})} - E(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}) \braket{a}{\Psi(\vb{p})} = 0 \thinspace . \end{equation}\]

We then write the orbital optimization Lagrangian as \[\begin{equation} \mathscr{L} ( \boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}, \boldsymbol{\lambda} ) = E(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}) + \sum_a^S \lambda_a f_a(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}) \thinspace , \end{equation}\] or equivalently as \[\begin{equation} \mathscr{L} ( \boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}, \boldsymbol{\lambda} ) = \qty( 1 - \sum_a^S \lambda_a \braket{a}{\Psi(\vb{p})} ) E(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\kappa}) + \sum_a^S \lambda_a \matrixel{a}{ \exp(\hat{\kappa}) \hat{\mathcal{H}}(\boldsymbol{\eta}) \exp(-\hat{\kappa}) }{\Psi(\vb{p})} \thinspace , \end{equation}\] and subsequently require stationarity on all the parameters at their optimal values \((\vb{p}^\star, \boldsymbol{\kappa}_0, \boldsymbol{\lambda}^\star)\).

Depending on the form of the orbital rotation generator \(\hat{\kappa}\), we can formulate different types of orbital optimization.