First-order response derivatives for the variational Lagrangian

In the variational Lagrangian theory, since the wave function parameters \(\require{physics} \vb{p}^\star(\boldsymbol{\eta})\) and the associated Lagrange multipliers \(\boldsymbol{\lambda}(\boldsymbol{\eta})\) have been variationally determined, we can recover a Hellmann-Feynman theorem for the calculation of first-order properties. Instead of having to calculate the partial perturbational derivative of the energy function, we now have to calculate the first-order perturbational derivative of the Lagrangian: \[\begin{equation} \eval{ \dv{ \mathcal{E}(\boldsymbol{\eta}) }{\eta_m} }_{\boldsymbol{\eta}_0} = \eval{ \dv{ \mathscr{L} ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0), \boldsymbol{\lambda}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} = \eval{ \pdv{ \mathscr{L} ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0), \boldsymbol{\lambda}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} \thinspace , \end{equation}\] in which we can switch from normal derivatives to partial derivatives precisely because the variational conditions are met on the Lagrangian. Furthermore, after filling in the definition of the Lagrangian, we obtain a formulate to calculate first-order properties: \[\begin{align} \eval{ \dv{ \mathcal{E}(\boldsymbol{\eta}) }{\eta_m} }_{\boldsymbol{\eta}_0} &= \eval{ \pdv{ \mathscr{L} ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0), \boldsymbol{\lambda}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} \\ &= \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0)) }{\eta_m} }_{\boldsymbol{\eta}_0} + \sum_a^y \qty( \eval{ \pdv{ \lambda_a^\star(\boldsymbol{\eta}) }{\eta_m} }_{\boldsymbol{\eta}_0} ) f_a ( \boldsymbol{\eta}_0, \vb{p}^\star(\boldsymbol{\eta}_0) ) \notag \\ & \hspace{12pt} + \sum_a^y \lambda_a^{\star}(\boldsymbol{\eta}_0) \qty( \eval{ \pdv{ f_a ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} ) \label{eq:non-var_dE_deta_eq2} \\ &= \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0)) }{\eta_m} }_{\boldsymbol{\eta}_0} + \sum_a^y \lambda_a^{\star}(\boldsymbol{\eta}_0) \qty( \eval{ \pdv{ f_a ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} ) \thinspace , \end{align}\] where the second term vanishes because the constraining equations are fulfilled. This means that we don’t require any first-order responses, not from the wave function parameters, nor from the Lagrangian multipliers, in order to calculate first-order molecular properties. In the end, this means that we will be able calculate properties after putting in the extra effort to determine \(\boldsymbol{\lambda}^\star(\boldsymbol{\eta}_0)\) (whose cost is described above) and the matrix of the partial derivatives of the constraining equations: \[\begin{equation} \eval{ \pdv{ f_a ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} \thinspace . \end{equation}\]