Pavlyukh Y., Berakdar J., Köksal K.

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(N³) scaling in commonly used brute-force implementations.

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