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The frequency dependent Sterheimer equation in TDDFT

Authors: M.A.L. Marques

Ref.: Time-Dependent Density Functional Theory - Gordon Research Conference, C. Ullrich and K. Burke (chairs), Colby College, Maine, 15-20/7/2007 (2007)

Abstract: Often we are interested in the response of an electronic system to a weak perturbing field. These underlie many different spectroscopy tools, and are therefore a window to the quantum mechanical world. It is then of little surprise that a multitude of methods appeared over the years to calculate response properties. In this talk, we look at a very old method: the solution of the Sternheimer equation. It is well known that this is the method of choice when calculating static response, like static polarizabilities, phonon frequencies, etc. Although a perturbative technique, it avoids the use of empty states, has a quite good scaling (N2) with the number of atoms, and a relatively small prefactor.

The Sternheimer method can be trivially extended to frequency dependent perturbations, giving us access to a variety of dynamic responses. The simplest of these is perhaps the dynamic polarizability alpha. With basically the same effort we can access the first hyperpolarizability beta, that is responsible for the processes of second-harmonic generation, optical rectification and Pockles effect. Van der Waals C6 coefficients are obtained by changing the frequency of the perturbing field from real to imaginary. Finally, it is possible to use the solution of the Sterheimer equation to define the linear-response of the electron localization function (lr-ELF) -- a quantity that can be used to help understanding electronic excitations in complex systems. All these phenomena are illustrated with benchmark calculations for molecules and clusters.

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