6 Comparison of TDDFT and the BSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
1 Introduction
Time-dependent density-functional theory (TDDFT) is a universal approach to the
dynamical many-body problem. A detailed, up-to-date coverage of TDDFT can be
found in two books [1, 2]. An easy and concise introduction is given in a recent
review article by Ullrich and Yang [3].
At present, the majority of applications of TDDFT take place in the field of
computational (bio)chemistry, to obtain excitation energies and excited-state properties of molecules. However, applications in condensed-matter physics and materials science are emerging at a rapid rate. In this chapter we give an introduction and
overview of TDDFT for extended periodic systems, focusing on the optical properties of semiconducting and insulating systems. In particular, we address the
question of how TDDFT can be used to calculate excitonic binding energies and
optical spectra with excitonic features. The present state-of-the-art approach in this
field is given by Green’s function-based many-body techniques, most notably, the
combination of GW [4, 5] and the Bethe–Salpeter equation (BSE) [6–13]. We
compare and contrast this approach with TDDFT and discuss their performance and
the various pros and cons for bulk semiconductors and insulators.
TDDFT for periodic solids was reviewed a few years ago by Onida et al. [14] and
Botti et al. [15]. Since then, many new developments have occurred, and in this
chapter we attempt to cover the more recent progress in this field, including our own
recent work [16–19]. We use atomic units (ℏ ¼ m ¼ e ¼ 4πε 0 ¼ 1) unless otherwise
indicated.
2 What Is an Exciton?
The optical properties of materials are determined by the way in which the electrons
and the ions respond to light. In this chapter we focus exclusively on the electronic
response and ignore the lattice dynamics or any effects related to the coupling of
electronic and lattice excitations (such as polarons; for details see, e.g., Yu and
Cardona [20]).
The response of a system of N electrons is often characterized as having either
“single-particle” or “collective” character. What do we mean by this? If a system
consists of noninteracting particles, the response is always purely single-particle, or
can be viewed as the sum of many individual, uncorrelated single-particle excitations. In the presence of interactions this simple picture is no longer valid, because
any change of state of one electron has an immediate influence on all other electrons
186
C.A. Ullrich and Z.-h. Yang
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
1 Introduction
Time-dependent density-functional theory (TDDFT) is a universal approach to the
dynamical many-body problem. A detailed, up-to-date coverage of TDDFT can be
found in two books [1, 2]. An easy and concise introduction is given in a recent
review article by Ullrich and Yang [3].
At present, the majority of applications of TDDFT take place in the field of
computational (bio)chemistry, to obtain excitation energies and excited-state properties of molecules. However, applications in condensed-matter physics and materials science are emerging at a rapid rate. In this chapter we give an introduction and
overview of TDDFT for extended periodic systems, focusing on the optical properties of semiconducting and insulating systems. In particular, we address the
question of how TDDFT can be used to calculate excitonic binding energies and
optical spectra with excitonic features. The present state-of-the-art approach in this
field is given by Green’s function-based many-body techniques, most notably, the
combination of GW [4, 5] and the Bethe–Salpeter equation (BSE) [6–13]. We
compare and contrast this approach with TDDFT and discuss their performance and
the various pros and cons for bulk semiconductors and insulators.
TDDFT for periodic solids was reviewed a few years ago by Onida et al. [14] and
Botti et al. [15]. Since then, many new developments have occurred, and in this
chapter we attempt to cover the more recent progress in this field, including our own
recent work [16–19]. We use atomic units (ℏ ¼ m ¼ e ¼ 4πε 0 ¼ 1) unless otherwise
indicated.
2 What Is an Exciton?
The optical properties of materials are determined by the way in which the electrons
and the ions respond to light. In this chapter we focus exclusively on the electronic
response and ignore the lattice dynamics or any effects related to the coupling of
electronic and lattice excitations (such as polarons; for details see, e.g., Yu and
Cardona [20]).
The response of a system of N electrons is often characterized as having either
“single-particle” or “collective” character. What do we mean by this? If a system
consists of noninteracting particles, the response is always purely single-particle, or
can be viewed as the sum of many individual, uncorrelated single-particle excitations. In the presence of interactions this simple picture is no longer valid, because
any change of state of one electron has an immediate influence on all other electrons
186
C.A. Ullrich and Z.-h. Yang