  We further develop the theory of quantum finite-size effects in metallic
nanoparticles, which was originally formulated by Hache, Ricard and Flytzanis
[J. Opt. Soc. Am. B 3, 1647 (1986)] and (in a somewhat corrected form) by
Rautian [Sov. Phys. JETP 85, 451 (1997)]. These references consider a metal
nanoparticle as a degenerate Fermi gas of conduction electrons in an
infinitely-high spherical potential well. This model (referred to as the HRFR
model below) yields mathematical expressions for the linear and the third-order
nonlinear polarizabilities of a nanoparticle in terms of infinite nested
series. These series have not been evaluated numerically so far and, in the
case of nonlinear polarizability, they can not be evaluated with the use of
conventional computers due to the high computational complexity involved.
Rautian has derived a set of remarkable analytical approximations to the series
but direct numerical verification of Rautian's approximate formulas remained a
formidable challenge. In this work, we derive an expression for the third-order
nonlinear polarizability, which is exact within the HRFR model but amenable to
numerical implementation. We then evaluate the expressions obtained by us
numerically for both linear and nonlinear polarizabilities. We investigate the
limits of applicability of Rautian's approximations and find that they are
surprizingly accurate in a wide range of physical parameters. We also discuss
the limits of small frequencies (comparable or below the Drude relaxation
constant) and of large particle sizes (the bulk limit) and show that these
limits are problematic for the HRFR model, irrespectively of any additional
approximations used. Finally, we compare the HRFR model to the purely classical
theory of nonlinear polarization of metal nanoparticles developed by us earlier
[Phys. Rev. Lett. 100, 47402 (2008)].
