Variationally-determined wave functions
For energies and wave function parameters to be called variationally determined (or, fully variational (T. Helgaker 1998)), the optimal parameters \(\require{physics} \vb{p}^\star(\boldsymbol{\eta})\) have to satisfy the \(x\) (one equation for every parameter) variational conditions: \[\begin{equation} \forall \boldsymbol{\eta}: \qquad \eval{ \pdv{ E(\boldsymbol{\eta}, \vb{p}) }{p_i} }_{\vb{p}^\star(\boldsymbol{\eta})} = 0 \thinspace . \end{equation}\] These equations are also called the stationary conditions and they mean that the energy function (at a given perturbation \(\boldsymbol{\eta}\)) is optimized with respect to the wave function parameters \(\vb{p}\). Here, optimized is used in the mathematical sense that the gradient of the energy (with respect to the wave function parameters) should vanish. More loosely, this means that the wave function parameters should adapt to accomodate the changes in the external perturbation as much as possible.