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The Liouville-Lanczos approach to linear-response time-dependentdensity-functional theory is generalized so as to encompass electronenergy-loss and inelastic X-ray scattering spectroscopies in periodic solids.The computation of virtual orbitals and the manipulation of large matrices areavoided by adopting a representation of response orbitals borrowed from(time-independent) density-functional perturbation theory and a suitableLanczos recursion scheme. The latter allows the bulk of the numerical work tobe performed at any given transferred momentum only once, for a whole extendedfrequency range. The numerical complexity of the method is thus greatlyreduced, making the computation of the loss function over a wide frequencyrange at any given transferred momentum only slightly more expensive than asingle standard ground-state calculation, and opening the way to computationsfor systems of unprecedented size and complexity. Our method is validated onthe paradigmatic examples of bulk silicon and aluminum, for which bothexperimental and theoretical results already exist in the literature.

Electron energy-loss and inelastic X-ray scattering cross sections from time-dependent density-functional perturbation theory