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

Abstract

The Liouville-Lanczos approach to linear-response time-dependent density-functional theory is generalized so as to encompass electron energy-loss and inelastic X-ray scattering spectroscopies in periodic solids. The computation of virtual orbitals and the manipulation of large matrices are avoided by adopting a representation of response orbitals borrowed from (time-independent) densityfunctional perturbation theory and a suitable Lanczos recursion scheme. The latter allows the bulk of the numerical work to be performed at any given transferred momentum only once, for a whole extended frequency range. The numerical complexity of the method is thus greatly reduced, making the computation of the loss function over a wide frequency range at any given transferred momentum only slightly more expensive than a single standard ground-state calculation, and opening the way to computations for systems of unprecedented size and complexity. Our method is validated on the paradigmatic examples of bulk silicon and aluminum, for which both experimental and theoretical results already exist in the literature.

A Structure Preserving Lanczos Algorithm for Computing the Optical Absorption Spectrum

“…However, the projected symmetric tridiagonal matrixT k does not necessarily have a real spectrum that is symmetric with respect to the origin. Thus (34) is not structured preserving in general. In a subsequent paper [13], it was proposed that a structured starting vectorq 1 ∈ U 0 should be used in (34).…”

“…Thus (34) is not structured preserving in general. In a subsequent paper [13], it was proposed that a structured starting vectorq 1 ∈ U 0 should be used in (34). With such a structured starting vector, it can be shown thatT k is a real tridiagonal matrix whose diagonal entries are zeros.…”

“…As the transformation (35) is invertible, the two Lanczos procedures are mathematically equivalent. The Lanczos procedure (34) can be used to approximate the absorption spectrum as follows:…”

“…In the view of Gauss quadrature for estimating the absorption spectrum, such a zero eigenvalue is not a very useful Gauss quadrature node because σ (0) = 0 is known trivially. Therefore, an even number of Lanczos steps should be performed when computing (34). Similarly, the zero eigenvalue of T 2k is not very helpful.…”

A structure preserving Lanczos algorithm for computing the optical absorption spectrum

“…Thus (34) is not structured preserving in general. In a subsequent paper [13], it was proposed that a structured starting vectorq 1 ∈ U 0 should be used in (34). With such a structured starting vector, it can be shown thatT k is a real tridiagonal matrix whose diagonal entries are zeros.…”