  We report a general macroscopic theory for the electrodynamic response of
semi-infinite metals (SIMs). The theory includes the hitherto overlooked
capacitive effects due to the finite spatial extension of a surface. The basic
structure of this theory is independent of the particulars of electron
dynamics. Analytical expressions have been obtained of the charge
density-density response function, which is naturally parsed into two parts.
One of them represents a bulk property while the other a pure surface property.
We apply the theory to study the responses according to several electronic
dynamics models and provide a unified view of their validity and limitations.
The models studied include the local dielectric model (DM), the dispersive
hydrodynamic model (HDM) and specular reflection model (SRM), as well as the
less common semi-classical model (SCM) based on Boltzmann's transport equation.
We show that, in terms of their basic equations, the SRM is an extension of the
HDM, just as the HDM is an extension of the DM. The SCM improves over the SRM
critically through the inclusion of translation symmetry breaking and surface
roughness effects. We then employ the response function to evaluate the
so-called dynamical structure factor, which plays an important role in particle
scattering. As expected, this factor reveals a peak due to the excitation of
surface plasma waves (SPWs). Surprisingly, however, the peak is shown to be
considerably sharper in the SCM than in other models, indicating an incipient
instability of the system according to this model. We also study the
distribution of charges induced by a charged particle grazing over a SIM
surface at constant speed. This distribution is shown to contain model-specific
features that are of immediate experimental interest.
