  We examine SERS from two perspectives: as a phenomenon described by the
Laplace Equation (the electrostatic or Rayleigh limit) and by the Helmholtz
Equation (electrodynamic or Mie limit). We formulate the problem in terms of
the scalar potential, which simplifies calculations without introducing
approximations. Because scattering is not usually calculated this way, we
provide the necessary theoretical justification showing that the
scalar-potential description is complete. Additional simplifications result
from treating the scatterer as a point charge q instead of a dipole. This
allows us to determine the consequences of including the longitudinal (Coulomb)
interaction between q and a passive resonator. This interaction suppresses the
mathematical singularities that lead to the unphysical resonant infinities in
first and second enhancements. It also modifies the effective restoring-force
constant of a resonant denominator, which permits us to explore the possibility
of dual resonance through a molecular pathway. We apply the formalism to
spherical inclusions of radius a for q located at polar and equatorial
positions. For small a, of the order of 1 nm or less, the low-l multipole terms
are important. For the more relevant case of radii of the order of 10 nm and
larger, the q-sphere interaction can be approximated by a model where q
interacts with its image charge for a dielectric plane, and the singularity
shifts in a discrete manner from \epsilon(\omega)=-2 to \epsilon(\omega)=-1.
These results are supported by more accurate calculations taking retardation
into account, although the use of only one spherical-harmonic term does not
fully represent the difference between forward- and backscattering.
