First-order projected Schrödinger equation response properties
Let us now continue to provide formulas that can be used to calculate first-order properties. Since the PSE framework has been made variational through the use of a Lagrangian, first-order response properties may be calculated follows: \[\begin{equation} \require{physics} \eval{ \dv{ \mathcal{E}(\boldsymbol{\eta}) }{\eta_m} }_{\boldsymbol{\eta}_0} = \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0)) }{\eta_m} }_{\boldsymbol{\eta}_0} + \sum_a^S \lambda_a^{\star}(\boldsymbol{\eta}_0) \qty( \eval{ \pdv{ f_a ( \boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0) ) }{\eta_m} }_{\boldsymbol{\eta}_0} ) \thinspace , \end{equation}\] which means that we need the perturbation partial derivatives of the energy: \[\begin{equation} \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{\eta_m} = \frac{ \matrixel**{0}{ \pdv{ \hat{\mathcal{H}}(\boldsymbol{\eta}) }{\eta_m} }{\Psi(\vb{p})} }{ \braket{0}{\Psi(\vb{p})} } \end{equation}\] and the of PSEs: \[\begin{equation} \pdv{ f_a(\boldsymbol{\eta}, \vb{p}) }{\eta_m} = \matrixel**{a}{ \pdv{ \hat{\mathcal{H}}(\boldsymbol{\eta}) }{\eta_m} - \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{\eta_m} }{\Psi(\vb{p})} \thinspace . \end{equation}\]
Using this variational Lagrangian formulation, we can apply these formulas to obtain expressions for the PSE response density matrices (response DMs). For the response-1-DM, we find: \[\begin{align} D^{\text{r}}_{PQ} &= \dv{ E(\vb{h}, \vb{g}, \vb{p}^\star) }{h_{PQ}} \\ &= \qty( 1 - \sum_a^S \lambda_a^\star \braket{a}{\Psi(\vb{p}^\star)} ) \frac{ \matrixel{0}{ \hat{E}_{PQ} }{\Psi(\vb{p}^\star)} }{ \braket{0}{\Psi(\vb{p}^\star)} } + \sum_a^S \lambda_a^\star \matrixel{a}{ \hat{E}_{PQ} }{ \Psi(\vb{p}^\star) } \end{align}\] and for the response 2-DM: \[\begin{align} d^{\text{r}}_{PQRS} &= 2 \dv{ % E(\vb{h}, \vb{g}, \vb{p}^\star) % }{g_{PQRS}} \\ &= \qty( 1 - \sum_a^S \lambda_a^\star \braket{a}{\Psi(\vb{p}^\star)} ) \frac{ \matrixel{0}{ \hat{e}_{PQRS} }{ \Psi(\vb{p}^\star) } }{ \braket{0}{\Psi(\vb{p}^\star)} } + \sum_a^S \lambda_a^\star \matrixel{a}{ \hat{e}_{PQRS} }{ \Psi(\vb{p}^\star) } \thinspace . \end{align}\] Since the one- and two-electron integrals appear linearly in the Hamiltonian, we find that tracing over the response 1- and 2-DMs yields the energy: \[\begin{equation} \sum_{PQ}^M h_{PQ} D^{\text{r}}_{PQ} % + \frac{1}{2} \sum_{PQRS}^K g_{PQRS} d^{\text{r}}_{PQRS} % = E(\vb{p}^\star) \thinspace , \end{equation}\] in which the extra term that would arise vanishes because the PSEs are fulfilled at \(\vb{p}^\star\).
Using the first-order perturbation derivative of the Hamiltonian (T. Helgaker 2016), we find that a first-order property (in the real case) can be calculated using the response DMs: \[\begin{equation} \begin{split} \dv{E(\boldsymbol{\eta})}{\eta_m} % = % & \eval{ % \pdv{ % h_{\text{nuc}}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % + \sum_{pq}^K % \eval{ % \pdv{% h_{pq}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % D^{\text{r}}_{pq} \\ &+ \frac{1}{2} \sum_{pqrs}^K % \eval{ % \pdv{ % g_{pqrs}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % d^{\text{r}}_{pqrs} \\ &- % \sum_{pq}^K \eval{ % \pdv{ % S_{pq}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % \hat{\mathscr{F}}^{\text{r}}_{pq} \qty( % \hat{\mathcal{H}}(\boldsymbol{\eta}_0) % ) % \thinspace , \end{split} \label{eq:first_order_property_AO_metric} \end{equation}\] in which \(\hat{\mathscr{F}}^{\text{r}}_{pq} \qty(\hat{\mathcal{H}}(\boldsymbol{\eta}_0))\) is a quantity that we call the response Fockian matrix (cfr. section \(\ref{sec:orbital_optimization}\)). If the perturbation does not affect the AO metric, this reduces to: \[\begin{align} \dv{E(\boldsymbol{\eta})}{\eta_m} % = & \eval{ % \pdv{ % h_{\text{nuc}}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % + \sum_{pq}^K % \eval{ % \pdv{% h_{pq}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % D^{\text{r}}_{pq} \\ & + \frac{1}{2} \sum_{pqrs}^K % \eval{ % \pdv{ % g_{pqrs}(\boldsymbol{\eta}) % }{\eta_m} % }_{\boldsymbol{\eta}_0} % d^{\text{r}}_{pqrs} % \thinspace . \label{eq:first_order_property} \end{align}\] Note that due to the adequate definition of these response DMs and Fockians, the same functional form is recovered as in the Rayleigh-Ritz fully-variational case.