The variational Lagrangian
Suppose we have a wave function model \(\require{physics} \ket{\Psi(\vb{p})}\) whose the optimal parameters \(\vb{p}^\star(\boldsymbol{\eta})\) don’t solve any stationary equations, but rather are the solutions of a (not necessarily linear) system of \(y\) equations: \[\begin{equation} \forall \boldsymbol{\eta}: \qquad f_a ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}) ) = 0 \thinspace , \end{equation}\] like the equations for the cluster amplitudes for coupled-cluster types of calculations, or any general projected Schrödinger equations. These types of methods have a separate energy function \[\begin{equation*} E(\vb{p}, \boldsymbol{\eta}) \thinspace , \end{equation*}\] so solving the equations doesn’t necessarily lead to a vanishing gradient of this energy function. This, in turn, means that we cannot ignore the first-order contribution due to the wave function response in the Hellmann-Feynman principle. The corresponding wave functions, energies and parameters \(\vb{p}^\star(\boldsymbol{\eta})\) are therefore said to be non-variational, short for non-variationally determined.
This non-variationality, however, is short-lived. According to Helgaker (T. Helgaker, Jørgensen, and Olsen 2000) (Saue 2002) (Wilson and Diercksen 1992), we can propose the Lagrangian \[\begin{equation} \mathscr{L}(\boldsymbol{\eta}, \vb{p}, \boldsymbol{\lambda}) = E(\boldsymbol{\eta}, \vb{p}) + \sum_a^l \lambda_a f_a(\boldsymbol{\eta}, \vb{p}) \end{equation}\] whose goal is to minimize the energy function \(E(\boldsymbol{\eta}, \vb{p})\), subject to the constraint that the optimality equations should be fulfilled. The vector \(\boldsymbol{\lambda}\) thus represents \(y\) Lagrangian multipliers, one for every (non-)linear equation. The variational conditions are now expressed with respect to the Lagrangian \(\mathscr{L}\): \[\begin{align} & \forall \boldsymbol{\eta}: \qquad \eval{ \pdv{ \mathscr{L} ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}), \boldsymbol{\lambda}) }{\lambda_a} }_{ \boldsymbol{\lambda}^\star(\boldsymbol{\eta}) } = f_a ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}) ) = 0 \\ % & \forall \boldsymbol{\eta}: \qquad \eval{ \pdv{ \mathscr{L} ( \boldsymbol{\eta}, \vb{p}, \boldsymbol{\lambda}^\star(\boldsymbol{\eta}) ) }{p_i} }_{ \vb{p}^\star(\boldsymbol{\eta}) } = \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{ \vb{p}^\star(\boldsymbol{\eta}) } + \sum_a^y \lambda_a^\star(\boldsymbol{\eta}) \qty( \eval{ \pdv{ f_a(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{ \vb{p}^\star(\boldsymbol{\eta}) } ) = 0 \thinspace , \end{align}\] in which now both sets of optimized parameters \(\vb{p}^\star\) and \(\boldsymbol{\lambda}^\star\) have implicit dependences on the external perturbation \(\boldsymbol{\eta}\): \[\begin{equation} \vb{p}^\star(\boldsymbol{\eta}) \qqtext{and} \boldsymbol{\lambda}^\star(\boldsymbol{\eta}) \thinspace . \end{equation}\] We should note that the first set of stationary equations recovers the original equation constraints because the Lagrangian parameters \(\boldsymbol{\lambda}\) enter the Lagrangian linearly. The second set of the stationary equations then constitutes a linear system of equations for the Lagrange multipliers, which requires the extra cost of calculating the partial derivatives \[\begin{equation} \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{ \vb{p}^\star(\boldsymbol{\eta}) } % \qquad \qqtext{and} \qquad % \eval{ \pdv{ f_a(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{ \vb{p}^\star(\boldsymbol{\eta}) } \thinspace . \end{equation}\]
After the solutions \(\vb{p}^\star(\boldsymbol{\eta})\) and \(\boldsymbol{\lambda}^\star(\boldsymbol{\eta})\) have been obtained, the energy is straightforwardly calculated as an evaluation of the Lagrangian function: \[\begin{equation} \mathcal{E}(\boldsymbol{\eta}) = E ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}) ) = \mathscr{L} ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}), \boldsymbol{\lambda}^\star(\boldsymbol{\eta}) ) \thinspace , \end{equation}\] because the variational conditions on the Lagrangian hold at the optimal point \((\vb{p}^\star, \boldsymbol{\lambda}^\star)\).
Through the use of the variational Lagrangian, we have succeeded in making a non-variational theory, variational, similarly to regular variationally determined wave functions:
- For first-order response properties, we can recover a Hellmann-Feynman theorem;
- For second-order response properties, we require the first-order parameter and multiplier responses.