Frequency dependent polarizability of small metallic grains.

We study the dynamic electronic polarizability of a single nano-scale spherical metallic grain using a quantum mechanical approach. We introduce the model for interacting electrons bound in the grain allowing us numerically to calculate the frequency dependence of the polarizability of grains of different sizes. We show that within this model the main resonance peak corresponding to the surface plasmon mode is blue-shifted and some minor secondary resonances above and below the main peak exist. We study the behavior of blue-shift as a function of grain size and compare our findings with the classical polarizability and with other results in the literature.