Top Curr Chem (2016) 368: 185–218
DOI: 10.1007/128_2014_610
# Springer International Publishing Switzerland 2014
Published online: 25 March 2015
Excitons in Time-Dependent DensityFunctional Theory
Carsten A. Ullrich and Zeng-hui Yang
Abstract This chapter gives an overview of the description of the optical and
dielectric properties of bulk insulators and semiconductors in time-dependent density-functional theory (TDDFT), with an emphasis on excitons. We review the linearresponse formalism for periodic solids, discuss excitonic exchange-correlation kernels, calculate exciton binding energies for various materials, and compare the
treatment of excitons with TDDFT and with the Bethe–Salpeter equation.
Keywords Bethe–Salpeter equation Á Dielectric function Á Exchange-correlation
kernel Á Excitons Á Time-dependent density-functional theory
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2 What Is an Exciton? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3 A Tale of Three Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4 Linear Response and Optical Properties in Periodic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.1 Microscopic and Macroscopic Dielectric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.2 Linear-Response Theory and TDDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5 TDDFT for Excitons in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.1 Why Are Excitons a Difficult Problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.2 Formalism: Direct Calculation of Exciton Binding Energies . . . . . . . . . . . . . . . . . . . . . . . 197
5.3 Why Does ALDA Fail? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.4 Excitonic xc Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.A. Ullrich (*)
Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA
e-mail: ullrichc@missouri.edu
Z.-h. Yang
Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA
Department of Physics, Temple University, Philadelphia, PA 19122, USA
DOI: 10.1007/128_2014_610
# Springer International Publishing Switzerland 2014
Published online: 25 March 2015
Excitons in Time-Dependent DensityFunctional Theory
Carsten A. Ullrich and Zeng-hui Yang
Abstract This chapter gives an overview of the description of the optical and
dielectric properties of bulk insulators and semiconductors in time-dependent density-functional theory (TDDFT), with an emphasis on excitons. We review the linearresponse formalism for periodic solids, discuss excitonic exchange-correlation kernels, calculate exciton binding energies for various materials, and compare the
treatment of excitons with TDDFT and with the Bethe–Salpeter equation.
Keywords Bethe–Salpeter equation Á Dielectric function Á Exchange-correlation
kernel Á Excitons Á Time-dependent density-functional theory
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2 What Is an Exciton? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3 A Tale of Three Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4 Linear Response and Optical Properties in Periodic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.1 Microscopic and Macroscopic Dielectric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.2 Linear-Response Theory and TDDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5 TDDFT for Excitons in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.1 Why Are Excitons a Difficult Problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.2 Formalism: Direct Calculation of Exciton Binding Energies . . . . . . . . . . . . . . . . . . . . . . . 197
5.3 Why Does ALDA Fail? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.4 Excitonic xc Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.A. Ullrich (*)
Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA
e-mail: ullrichc@missouri.edu
Z.-h. Yang
Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA
Department of Physics, Temple University, Philadelphia, PA 19122, USA