The projected Schrödinger Lagrangian
The projected Schrödinger equation framework may be extended with a variational Lagrangian formulation in order to yield a stationary formalism. The main benefit for this is that response properties may be calculated analogously to regularly variationally determined wave functions.
We can use the PSE energy function and the PSEs (which are to be seen as the constraints on the optimal wave function parameters \(\require{physics} \vb{p}^\star\)) in a Lagrangian (T. Helgaker, Jørgensen, and Olsen 2000): \[\begin{equation} \mathscr{L}(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\lambda}) = E(\boldsymbol{\eta}, \vb{p}) + \sum_a^S \lambda_a f_a(\boldsymbol{\eta}, \vb{p}) \thinspace . \end{equation}\] An alternative form may be retrieved by plugging in the form of the projected Schrödinger equation \(f_a(\boldsymbol{\eta}, \vb{p})\): \[\begin{equation} \mathscr{L}(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\lambda}) = \qty( 1 - \sum_a^S \lambda_a \braket{a}{\Psi(\vb{p})} ) E(\boldsymbol{\eta}, \vb{p}) + \sum_a^S \lambda_a \matrixel{a}{\hat{\mathcal{H}}(\boldsymbol{\eta})}{\Psi(\vb{p})} \thinspace . \end{equation}\]
In order to minimize the PSE energy, subject to the PSEs being fulfilled, we require stationarity on the wave function parameters \(\vb{p}\) and Lagrange multipliers \(\boldsymbol{\lambda}\):
Requiring stationarity on the Lagrange multipliers \(\boldsymbol{\lambda}\) yields the original PSEs: \[\begin{equation} \eval{ \pdv{ \mathscr{L}(\boldsymbol{\eta}, \vb{p}^\star, \boldsymbol{\lambda}) }{\lambda_a} }_{\boldsymbol{\lambda}^\star} = f_a(\boldsymbol{\eta}, \vb{p}^\star) = 0 \thinspace . \end{equation}\]
Requiring stationarity on the wave function parameters \(\vb{p}\) yields linear equations that can be used to determine the values of the Lagrange multipliers: \[\begin{equation} \eval{ \pdv{ \mathscr{L}(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\lambda}^\star) }{p_i} }_{\vb{p}^\star} = \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{\vb{p}^\star} + \sum_a^S \lambda_a^\star \qty( \eval{ \pdv{ f_a(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{\vb{p}^\star} ) = 0 \thinspace . \end{equation}\]
Analogously to the optimal wave function parameters, the ‘optimal’ Lagrange multipliers \(\boldsymbol{\lambda}^\star(\boldsymbol{\eta})\) are also implicit functions of the external perturbation \(\boldsymbol{\eta}\). In order to solve these equations, we thus need the parameter derivatives of the energy function: \[\begin{equation} \pdv{E(\boldsymbol{\eta}, \vb{p})}{p_i} = \frac{ \matrixel{0}{ \hat{\mathcal{H}}(\boldsymbol{\eta}) }{ \pdv{ \Psi(\vb{p}) }{p_i} } }{ \braket{0}{\Psi(\vb{p})} } - \frac{ \matrixel{0}{ \hat{\mathcal{H}}(\boldsymbol{\eta}) }{\Psi(\vb{p})} }{ \braket{0}{\Psi(\vb{p})}^2 } \braket{0}{ \pdv{ \Psi(\vb{p}) }{p_i} } \thinspace , \end{equation}\] and of the PSEs: \[\begin{equation} \pdv{ f_a(\boldsymbol{\eta}, \vb{p}) }{p_i} = \matrixel{a}{ \hat{\mathcal{H}}(\boldsymbol{\eta}) - E(\boldsymbol{\eta}, \vb{p}) }{ \pdv{ \Psi(\vb{p}) }{p_i} } - \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{p_i} \braket{a}{\Psi(\vb{p})} \thinspace , \end{equation}\] which means that the parameter derivative of the wave function model \[\begin{equation} \ket{ \pdv{ \Psi(\vb{p}) }{p_i} } \equiv \pdv{p_i} \ket{\Psi(\vb{p})} \end{equation}\] should be known.
Using a variational Lagrangian, we have achieved a stationary formulation of the PSE framework. Therefore, we can use it to calculate first-order and second-order response properties. As a special case of using first-order response properties, we may also formulate orbital optimization.