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Time-Dependent Density-Functional TheoryConcepts and Applications$
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Carsten A. Ullrich

Print publication date: 2011

Print ISBN-13: 9780199563029

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199563029.001.0001

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The formal framework of linear-response TDDFT

The formal framework of linear-response TDDFT

Chapter:
(p.123) 7 The formal framework of linear-response TDDFT
Source:
Time-Dependent Density-Functional Theory
Author(s):

Carsten A. Ullrich

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199563029.003.0007

This chapter discusses the linear-response framework of time-dependent density-functional theory (TDDFT). It begins with a thorough and self-contained exposition of general linear-response theory for many-body systems. The time-dependent as well as the frequency-dependent response are discussed. Several exact representations and properties of the density-density response function are reviewed, such as the Lehmann representation and fluctuation-dissipation theorem, and the Kramers-Kronig relations. It is shown how several important spectroscopic observables are obtained from linear-response theory. Then, the TDDFT formulation of linear-response theory is derived. The key quantity in this formalism is the so-called exchange-correlation kernel. The remainder of the chapter demonstrates how this linear-response formalism can be used to calculate excitation energies in principle exactly with TDDFT. The Casida equation is derived, as well as several approximate schemes. A comparison is made with time-dependent Hartree-Fock theory.

Keywords:   linear-response theory, Lehmann representation, Kramers-Kronig relations, fluctuation-dissipation theorem, spectroscopic observables, Casida equation, excitation energies, time-dependent Hartree-Fock theory

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