Molecular response properties
Suppose our system of interest is subject to an external perturbation. Introducing \(\boldsymbol{\eta}\) as set of external perturbations \(\eta_m\) on the system, we will then, in general, Taylor-expand the time-independent energy of the system up to second order as (T. Helgaker 1998) around the perturbation-free case \(\require{physics} \boldsymbol{\eta}_0 = \vb{0}\): \[\begin{equation} \mathcal{E}(\boldsymbol{\eta}) % = \mathcal{E}^{(0)} % + \sum_m % \eval{ % \dv{ % \mathcal{E}(\boldsymbol{\eta}) % }{\eta_m}% }_{\boldsymbol{\eta}_0} \eta_m % + \frac{1}{2!} \sum_{mn} % \eval{ % \frac{ % \dd{^2}{\mathcal{E}(\boldsymbol{\eta})} % }{ \dd{\eta_m} \dd{\eta_n} } % }_{\boldsymbol{\eta}_0} \eta_m \eta_n % + \order{\boldsymbol{\eta}^3} % \thinspace . \end{equation}\] \(\mathcal{E}^{(0)}\) is the energy in absence of an external perturbation: \[\begin{equation} \mathcal{E}^{(0)} % = \mathcal{E}(\boldsymbol{\eta}_0) % \thinspace . \end{equation}\] and the expansion coefficients \[\begin{equation} \mathcal{E}^{(1)}_m = % \eval{ % \dv{ % \mathcal{E}(\boldsymbol{\eta}) % }{\eta_m}% }_{\boldsymbol{\eta}_0} \end{equation}\] and \[\begin{equation} \mathcal{E}^{(2)}_{mn} = % \eval{ % \frac{ % \dd{^2}{\mathcal{E}(\boldsymbol{\eta})} % }{ \dd{\eta_m} \dd{\eta_n} } % }_{\boldsymbol{\eta}_0} \end{equation}\] describe the response of the system with respect to the external perturbations and are subsequently called molecular properties. Since we have expanded the energy about the perturbation-free case \(\boldsymbol{\eta}_0\), we calculate response properties as the derivatives evaluated at zero external perturbation, which loosely corresponds to a situation in which an external perturbation is switched on and immediately back off again. As a sidemark, we should note that some authors include the extra prefactor \(1/2!\) in the definition of the second-order property, but according to the theory on dipole moments (cfr. section \(\ref{sec:dipole_moments}\)), we can argue in favor of not doing so.
Most generally, a wave function model \(\ket{\Psi(\vb{p})}\), that depends on a set of \(x\) wave function parameters collected in the vector \(\vb{p}\), is determined by solving a set of equations: \[\begin{equation} \forall \boldsymbol{\eta}: \qquad % \vb{f} ( % \boldsymbol{\eta}, % \vb{p}^\star(\boldsymbol{\eta}) % ) = 0 % \thinspace , \end{equation}\] where \(\vb{p}^\star(\boldsymbol{\eta})\) is the optimal value for the wave function parameters, determined at the value of the perturbation \(\boldsymbol{\eta}\). energy of the system at that value of the perturbation \(\boldsymbol{\eta}\) can then be calculated by evaluating the \[\begin{equation*} E(\boldsymbol{\eta}, \vb{p}) \end{equation*}\] at the value of the perturbation \(\boldsymbol{\eta}\) and the corresponding optimal parameters \(\vb{p}^\star(\boldsymbol{\eta})\): \[\begin{equation} \mathcal{E}(\boldsymbol{\eta}) % = E (\boldsymbol{\eta}, % \vb{p}^\star(\boldsymbol{\eta}) % ) % \thinspace . \end{equation}\] We should already remark that it is important which kind of equations these optimal wave function parameters \(\vb{p}^\star(\boldsymbol{\eta})\) solve. In subsequent sections, we will see that we distinguish two cases.