Calculation of corrections to Fresnel optics from density response

Abstract

We develop a computational scheme for the Feibelman d parameters, which characterize the surface corrections to Fresnel formulas at a smooth jellium surface. The d parameters are determined from the nonretarded electronic density response to a long-wavelength field. We find this density response via the time-dependent Hartree approximation, which in turn requires the solution of a one-dimensional integral equation. The integral equation is analyzed in Fourier space, which allows us to isolate explicitly the nonanalytic structure in the kernel and to avoid the difficulties of long-ranged Friedel oscillations in the real-space kernel. The detailed formulas and procedures necessary to produce an efficient yet accurate computer code are described. As an initial illustration of the method, we calculate the linear dispersion coefficient of surface plasmons in a single, finite-step barrier model for the electrons. The results are compared to earlier calculations and to infinite barrier values. The evolution of the dispersion coefficient with barrier height shows interesting structure below the threshold for photoemission.