  We study the blueshift of the surface plasmon (SP) resonance energy of
isolated Ag nanoparticles with decreasing particle diameter, which we recently
measured using electron energy loss spectroscopy (EELS). As the particle
diameter decreases from 26 down to 3.5 nm, a large blueshift of 0.5 eV of the
SP resonance energy is observed. In this paper, we base our theoretical
interpretation of our experimental findings on the nonlocal hydrodynamic model,
and compare the effect of the substrate on the SP resonance energy to the
approach of an effective homogeneous background permittivity. We derive the
nonlocal polarizability of a small metal sphere embedded in a homogeneous
dielectric environment, leading to the nonlocal generalization of the classical
Clausius-Mossotti factor. We also present an exact formalism based on multipole
expansions and scattering matrices to determine the optical response of a metal
sphere on a dielectric substrate of finite thickness, taking into account
retardation and nonlocal effects. We find that the substrate-based calculations
show a similar-sized blueshift as calculations based on a sphere in a
homogeneous environment, and that they both agree qualitatively with the EELS
measurements.
