First-order response derivatives for variationally determined wave functions

For variationally determined wave functions, we can calculate the first-order perturbational derivative by applying the chain rule: \[\begin{align} \require{physics} \eval{ \dv{ \mathcal{E}(\boldsymbol{\eta}) }{\eta_m} }_{ \boldsymbol{\eta}_0 } &= \eval{ \dv{ E (\boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0)) }{\eta_m} }_{ \boldsymbol{\eta}_0 } \\ &= \eval{ \pdv{ E (\boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0)) }{\eta_m} }_{\boldsymbol{\eta}_0} + \sum_i^x \qty( \eval{ \pdv{ E(\boldsymbol{\eta}_0, \vb{p}) }{p_i} }_{\vb{p}^\star(\boldsymbol{\eta}_0)} ) \qty( \eval{ \pdv{ p_i^\star(\boldsymbol{\eta}) }{\eta_m} }_{ \boldsymbol{\eta}_0 } ) \thinspace . \end{align}\] The first term can be interpreted as the explicit perturbation-dependence of the energy. The implicit perturbation-depence that is described by the second term is then given through a quantity \[\begin{equation} \eval{ \pdv{ p_i^\star(\boldsymbol{\eta}) }{\eta_m} }_{ \boldsymbol{\eta}_0 } \end{equation}\] that we call the first-order wave function response. In words, it contains information as to how the optimal wave function parameters \(\vb{p}^\star\) change if we turn on a perturbation \(\boldsymbol{\eta}\). As we do not know the explicit dependence \(\vb{p}^\star(\boldsymbol{\eta})\), this quantity is, in general, not straightforward to calculate.

For variationally determined wave functions, the variational (or: stationary) conditions are fulfilled. This means that the first factor in the second term vanishes, such that we obtain for the first-order total derivative: \[\begin{equation} \eval{ \dv{ \mathcal{E}(\boldsymbol{\eta}) }{\eta_m} }_{ \boldsymbol{\eta}_0 } = \eval{ \pdv{ E (\boldsymbol{\eta}, \vb{p}^\star(\boldsymbol{\eta}_0)) }{\eta_m} }_{\boldsymbol{\eta}_0} \thinspace . \end{equation}\] This very important result is what we will refer to as the Hellmann-Feynman theorem: for variationally determined wave functions, the first-order wave function response is not needed to calculate first-order molecular properties and so they are solely determined by the explicit first-order partial derivative of the energy function.