Fast computations of the dielectric response of systems with spherical or axial symmetry

When the size of a system does not permit a quantum-mechanical treatment it is still possible to obtain an accurate description of the collective electronic excitations by using a semiclassical approximation for the density-density response function. The dielectric function in this approach is a nonlocal quantity dependent on the electron density as a single parameter. The optical response is described by the Fredholm integral equation as demonstrated by Mukhopadhyay and Lundqvist [Nuovo Cimento B 27, 1 (1975)]. The optical response has to be evaluated numerically, except for few models amenable to analytic solutions ( typically with the density abruptly varying at the interface). We demonstrate that for the systems with a spherical or an axial symmetry this equation can be reduced to the Volterra integral equation, which furthermore can be solved as a system of differential equations. This observation leads to an efficient numerical scheme that scales as O(N), where N is the number of mesh points for the density. This is to be contrasted with O(N-3) scaling in commonly used brute-force implementations.