  In this paper we analyse the importance of a detailed description of the
electronic transitions in ultra-small nanoparticles through the optical
response to very small changes of size in systems, whose dimensions are in the
subnanometric scale. We present a carefully calculation of the optical response
of these systems by using the exact eigen-energies and wave functions for
nanospheres with diameter smaller than 10nm, to obtain the dielectric function
under different conditions of confinement. The aim is to use the so obtained
dielectric function to present the absorption spectra of one electron confined
in a sphere in two cases: 1) infinite confinement and 2) finite confinement, in
which, the value of the wells depth is carefully calculated after adjusting the
number of atoms that composed each sphere, so that the energies and dipole
matrix elements give a more accurate information of the optical response.
Moreover, we extend the calculation of this dielectric function for obtaining
the optical constants needed to find the plasmon frequency, throught a
numerical method of finite elements that solves the Maxwells equations, in
order to obtain the enhancement of the near electric field. We show an
interesting behavior for particles sizes less than 10 nm, finding that the
variation induced in the eigen-energies, through slight size changes in the
particle, provides significant variations in the optical response of these
nanoparticles. This effect, can be observed in the optical absorption spectra
and in the localized surface plasmon energies as confirmed in recent
observation of plasmonic phenomena at the subnanometer to atomic scale.
