possible to derive the exact exchange kernel directly as the functional derivative
f x (ω)¼δv x (ω)/δn(ω) [98, 99].
In periodic insulators, the exact-exchange kernel has the long-range behavior
necessary for the formation of excitons [100, 101]. However, the resulting
unscreened electron–hole interaction tends to lead to a dramatic overbinding of
the excitons; in extreme cases this causes a collapse of the optical spectra (i.e., the
exciton would be so strongly bound that it falls below the valence band edge). This
collapse can be prevented by a cutoff of the Coulomb singularity [100, 101]; this is
equivalent to an evaluation of the xc kernel with a screened interaction [15, 102].
5.4.4 Hybrid Functionals and Meta GGAs
Hybrid xc functionals [46–59] replace a portion of the semilocal exchange energy
with the exact exchange energy, so the long-range part of the corresponding xc
kernel resembles a screened exact exchange kernel; as we saw above, this can
produce bound excitons. In practice, however, hybrid functionals are used in a
different manner [103]: the exact exchange part is treated nonlocally, similar to the
time-dependent HF approach, instead of using the exact exchange kernel in (42).
The B3LYP hybrid functional has been used by Bernasconi et al. to calculate
optical spectra in several semiconductor materials [104–106]; they achieve a
generally good description of optical gaps, including excitonic features. Indeed,
we have obtained some preliminary results [107] which confirm that B3LYP can be
reasonably accurate for exciton binding energies in semiconductors, despite the fact
that the 0.2 mixing parameter of the exact exchange is optimized for finite
systems [46].
Range-separated hybrid functionals [50, 108, 109] are based on the idea of
separating the Coulomb interaction into different spatial ranges, which are then
treated differently, using either exact exchange or approximate semilocal exchange
functionals. Recent applications of range-separated hybrids to solids have produced
good quasiparticle gaps [51, 52, 54, 55, 57, 61]. In linear response, these functionals
are again closely related to the exact exchange kernel. However, if the rangeseparated functional uses semilocal exchange for the long-range part, it cannot
produce bound excitons for the same reason as in ALDA. Therefore, the popular
HSE06 functional [50, 51, 110] cannot yield bound excitons, although it may still
produce decent looking optical spectra of insulators [111].
The so-called meta-GGA functionals [112–115] depend not only on the density
and its gradients but also on the kinetic-energy density, which is expressed in terms
of Kohn–Sham orbitals and, hence, depends nonlocally on the density. This
nonlocality produces good quasiparticle gaps in solids [60–64], and opens up the
possibility of describing excitonic interactions with meta-GGAs. Nazarov and
Vignale [116] tested two types of meta-GGAs, TPSS [112] and VS98
[113, 114]. They found TPSS to be unsuited for describing dielectric properties
of solids; VS98, on the other hand, performed rather well. Further tests are needed;
Excitons in Time-Dependent Density-Functional Theory
203
f x (ω)¼δv x (ω)/δn(ω) [98, 99].
In periodic insulators, the exact-exchange kernel has the long-range behavior
necessary for the formation of excitons [100, 101]. However, the resulting
unscreened electron–hole interaction tends to lead to a dramatic overbinding of
the excitons; in extreme cases this causes a collapse of the optical spectra (i.e., the
exciton would be so strongly bound that it falls below the valence band edge). This
collapse can be prevented by a cutoff of the Coulomb singularity [100, 101]; this is
equivalent to an evaluation of the xc kernel with a screened interaction [15, 102].
5.4.4 Hybrid Functionals and Meta GGAs
Hybrid xc functionals [46–59] replace a portion of the semilocal exchange energy
with the exact exchange energy, so the long-range part of the corresponding xc
kernel resembles a screened exact exchange kernel; as we saw above, this can
produce bound excitons. In practice, however, hybrid functionals are used in a
different manner [103]: the exact exchange part is treated nonlocally, similar to the
time-dependent HF approach, instead of using the exact exchange kernel in (42).
The B3LYP hybrid functional has been used by Bernasconi et al. to calculate
optical spectra in several semiconductor materials [104–106]; they achieve a
generally good description of optical gaps, including excitonic features. Indeed,
we have obtained some preliminary results [107] which confirm that B3LYP can be
reasonably accurate for exciton binding energies in semiconductors, despite the fact
that the 0.2 mixing parameter of the exact exchange is optimized for finite
systems [46].
Range-separated hybrid functionals [50, 108, 109] are based on the idea of
separating the Coulomb interaction into different spatial ranges, which are then
treated differently, using either exact exchange or approximate semilocal exchange
functionals. Recent applications of range-separated hybrids to solids have produced
good quasiparticle gaps [51, 52, 54, 55, 57, 61]. In linear response, these functionals
are again closely related to the exact exchange kernel. However, if the rangeseparated functional uses semilocal exchange for the long-range part, it cannot
produce bound excitons for the same reason as in ALDA. Therefore, the popular
HSE06 functional [50, 51, 110] cannot yield bound excitons, although it may still
produce decent looking optical spectra of insulators [111].
The so-called meta-GGA functionals [112–115] depend not only on the density
and its gradients but also on the kinetic-energy density, which is expressed in terms
of Kohn–Sham orbitals and, hence, depends nonlocally on the density. This
nonlocality produces good quasiparticle gaps in solids [60–64], and opens up the
possibility of describing excitonic interactions with meta-GGAs. Nazarov and
Vignale [116] tested two types of meta-GGAs, TPSS [112] and VS98
[113, 114]. They found TPSS to be unsuited for describing dielectric properties
of solids; VS98, on the other hand, performed rather well. Further tests are needed;
Excitons in Time-Dependent Density-Functional Theory
203