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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 (N^{2}) 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 C_{6} 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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