  Surface-response functions are one of the most promising routes for bridging
the gap between fully quantum-mechanical calculations and phenomenological
models in quantum nanoplasmonics. Within all the currently available recipes
for obtaining such response functions, \emph{ab initio} calculations remain one
of the most predominant, wherein the surface-response function are retrieved
via the metal's non-equilibrium response to an external perturbation. Here, we
present a complementary approach where one of the most appealing
surface-response functions, namely the Feibelman $d$-parameters, yield a finite
contribution even in the case where they are calculated directly from the
equilibrium properties described under the local-response approximation (LRA),
but with a spatially varying equilibrium electron density. Using model
calculations that mimic both spill-in and spill-out of the equilibrium electron
density, we show that the obtained $d$-parameters are in qualitative agreement
with more elaborate, but also more computationally demanding, \emph{ab initio}
methods. The analytical work presented here illustrates how microscopic
surface-response functions can emerge out of entirely local electrodynamic
considerations.
