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.
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