In recent years, generalized Kohn–Sham schemes [37, 44, 45] have become
quite popular for calculating band gaps of solids. Generalized Kohn–Sham theory
means, in essence, using exchange-correlation (xc) functionals which contain an
admixture of Hartree–Fock (HF) nonlocal exchange. These functionals (for
instance B3LYP [46] or PBE0 [47]) have been crucial for the enormous success
of DFT in theoretical chemistry. Hybrid xc functionals for solids are not
unproblematic: HF exchange is difficult to implement in periodic systems, and
B3LYP fails for metals [48]. However, for insulators, hybrid functionals generally
produce excellent results, comparable to what can be achieved by more sophisticated many-body calculations [49–59]. Another promising approach is meta-GGA
functionals such as the Tran–Blaha exchange potential [60–64], which gives good
band structures and band gaps and is much less costly than hybrid functionals.
However, it has the drawback of not being derivable from an energy functional.
It should be emphasized that current implementations of hybrid xc functionals
do not actually calculate the Kohn–Sham gap (this would require a self-consistent
calculation with a local potential); instead, hybrid functionals yield an approximation to the quasiparticle gap. Local xc potentials can, in principle, be constructed
from hybrid functionals using the OEP (optimized effective potential) method,
which was successfully done for the case of exact exchange (see Betzinger
et al. [65] and references therein).
While the band gap can be measured using techniques in which electrons are
added or removed from the system (such as photoemission spectroscopy), the
optical gap refers to the lowest neutral excitation. The difference between quasiparticle band gap and optical gap is the lowest exciton binding energy, E
ex
0 . In the
previous section we have seen that excitons can be viewed as bound electron–hole
pairs, whose bound states form a Rydberg series, analogous to the hydrogen atom.
The band gap is given by the asymptotic limit of the excitonic Rydberg series [66]
(at least for direct-gap insulators and semiconductors).
4 Linear Response and Optical Properties in Periodic
Solids
4.1 Microscopic and Macroscopic Dielectric Functions
The interactions of electromagnetic fields and matter are governed by Maxwell’s
equations,
∇ Á D ¼ n f ;
ð6Þ
∇ Â E ¼ À
∂B
∂t
;
ð7Þ
Excitons in Time-Dependent Density-Functional Theory
191
quite popular for calculating band gaps of solids. Generalized Kohn–Sham theory
means, in essence, using exchange-correlation (xc) functionals which contain an
admixture of Hartree–Fock (HF) nonlocal exchange. These functionals (for
instance B3LYP [46] or PBE0 [47]) have been crucial for the enormous success
of DFT in theoretical chemistry. Hybrid xc functionals for solids are not
unproblematic: HF exchange is difficult to implement in periodic systems, and
B3LYP fails for metals [48]. However, for insulators, hybrid functionals generally
produce excellent results, comparable to what can be achieved by more sophisticated many-body calculations [49–59]. Another promising approach is meta-GGA
functionals such as the Tran–Blaha exchange potential [60–64], which gives good
band structures and band gaps and is much less costly than hybrid functionals.
However, it has the drawback of not being derivable from an energy functional.
It should be emphasized that current implementations of hybrid xc functionals
do not actually calculate the Kohn–Sham gap (this would require a self-consistent
calculation with a local potential); instead, hybrid functionals yield an approximation to the quasiparticle gap. Local xc potentials can, in principle, be constructed
from hybrid functionals using the OEP (optimized effective potential) method,
which was successfully done for the case of exact exchange (see Betzinger
et al. [65] and references therein).
While the band gap can be measured using techniques in which electrons are
added or removed from the system (such as photoemission spectroscopy), the
optical gap refers to the lowest neutral excitation. The difference between quasiparticle band gap and optical gap is the lowest exciton binding energy, E
ex
0 . In the
previous section we have seen that excitons can be viewed as bound electron–hole
pairs, whose bound states form a Rydberg series, analogous to the hydrogen atom.
The band gap is given by the asymptotic limit of the excitonic Rydberg series [66]
(at least for direct-gap insulators and semiconductors).
4 Linear Response and Optical Properties in Periodic
Solids
4.1 Microscopic and Macroscopic Dielectric Functions
The interactions of electromagnetic fields and matter are governed by Maxwell’s
equations,
∇ Á D ¼ n f ;
ð6Þ
∇ Â E ¼ À
∂B
∂t
;
ð7Þ
Excitons in Time-Dependent Density-Functional Theory
191