  Optical nonlinearities of matter are often associated with the response of
individual atoms. Here, using a toy oscillator model, we show that in the
confined geometry of a two-dimensional dielectric nanoparticle a collective
nonlinear response of the atomic array can arise from the Coulomb interactions
of the bound optical electrons, even if the individual atoms exhibit no
nonlinearity. We determine the multipole contributions to the nonlinear
response of nanoparticles and demonstrate that the odd order and even order
nonlinear electric dipole moments scale with the area and perimeter of the
nanoparticle, respectively.
