The dynamic image plane in the long-wavelength limit has been calculated within the time-dependent density-functional approach for several simple-metal surfaces. Using the dynamical force sum rule it is proven that the image plane is given by the first moment of the induced density in the region outside the positive background. The imaginary part of the centroid d(ω) of the screening charge, which determines the probability for electron-hole pair excitations at the surface, is linear in ω up to about one-half of the work function and then rises steeply near the vacuum threshold. Close to 0.8 of the bulk plasma frequency Imd(ω) shows a peak which becomes very sharp for low bulk densities. If the response to the applied field is treated within the random-phase approximation, the spectral weight of Imd(ω) is shifted almost uniformly to higher frequencies. It is also found that the linear coefficient of Imd(ω) at low frequencies is in excellent agreement with the quasistatic predictions based on the golden-rule formula.

• Received 22 June 1987

DOI:https://doi.org/10.1103/PhysRevB.36.7378