On the estimation of density of collectivized electrons in plasmonic spherical metal nanoparticles: quantum static versus classical dynamic approach

The density function of a gas of collectivized electrons emerges in different theoretical models of plasmonics describing nanostructured metals. This relates to either the most simple theories based on mechanics of classic particles (e.g., the Drude theory) or more complicated continuum and quantum mechanical based ones. The knowledge of the density function parameters allows predicting new physical phenomena in plasmonic systems, making mathematical simulation and automating the design of plasmonic devices in future. In this work we propose several approaches based on the density functional theory, single particle quantum mechanics and the continuum mechanics as well for definition of the mean electron density value in spherical metal nanoparticles. By using these tools we derive the quantitative estimates of electron density, the Fermi energy and velocity for large nanoparticles and compare them against the similar data known from the theory of bulk crystals. A significant difference between them has been shown. The attention is stressed on the universality of the results obtained together with the techniques of their derivation in context of application to different theories.