Because of the formal similarity between BSE and TDDFT in the transition
space representation, we can also define a BSE effective electron–hole interaction
potential V
BSE
eÀh in analogy to (72), using the BSE kernel K
W . Figure 7 shows a
comparison of V
xc
eÀh and V
BSE
eÀh for our one-dimensional model insulator. It can clearly
be seen that the nonlocal potentials in both cases are dominated by the diagonal
part; however, under further examination it turns out that V
xc
eÀh is shallower than
V
BSE
eÀh , and hence is only able to sustain a single bound exciton.
7 Conclusions
There exist two alternative, complementary first-principles methods to calculate
optical spectra and excitonic effects in extended periodic solids. The more traditional approach is based on many-body Green’s function techniques, exemplified
by the GW-BSE method; the other approach is based on TDDFT. The Green’s
function approach is formally straightforward, in the sense that excitonic particle–
hole interactions are built in by construction; however, it is computationally costly.
TDDFT, on the other hand, is computationally cheaper (provided that the approximate xc functionals are simple enough), but the price one has to pay is that
intrinsically nonlocal electron–hole interaction effects have to be described via
linearized local xc potentials; this is a somewhat unnatural way of dealing with
excitonic interactions, which makes it non-straightforward to construct good
approximations.
In this chapter we have focused on the TDDFT approach for excitons, and have
tried to bring across the following points:
• Excitons in TDDFT are a difficult problem, because they require an xc kernel
which has the long-range property f xc,00 (q, ω) ~ q
À2 for q ! 0. The popular local
and semilocal approximations such as ALDA and standard GGAs do not have
this property: although they work well for finite systems such as atoms and
molecules, they do not produce excitons in extended systems. New approximations are therefore required.
• There exist several approximate “excitonic” xc kernels, with various degrees of
sophistication. Some kernels involve adjustable parameters, others don’t. A
typical behavior of the simpler kernels is that they can reproduce some part of
the optical spectrum reasonably well (e.g., a bound exciton, or the continuum
part), but not all of it at the same time. The best excitonic xc kernel, the
“nanoquanta” kernel, is computationally not much simpler than the BSE.
• Adiabatic (i.e., frequency-independent) xc kernels cannot produce an excitonic
Rydberg series. At best, they can generate a single bound exciton (if only the
head of the xc coupling matrix is used). The TDDFT analog of the excitonic
Wannier equation features a nonlocal potential, which is too shallow to produce
more than one bound level.
Excitons in Time-Dependent Density-Functional Theory
213
space representation, we can also define a BSE effective electron–hole interaction
potential V
BSE
eÀh in analogy to (72), using the BSE kernel K
W . Figure 7 shows a
comparison of V
xc
eÀh and V
BSE
eÀh for our one-dimensional model insulator. It can clearly
be seen that the nonlocal potentials in both cases are dominated by the diagonal
part; however, under further examination it turns out that V
xc
eÀh is shallower than
V
BSE
eÀh , and hence is only able to sustain a single bound exciton.
7 Conclusions
There exist two alternative, complementary first-principles methods to calculate
optical spectra and excitonic effects in extended periodic solids. The more traditional approach is based on many-body Green’s function techniques, exemplified
by the GW-BSE method; the other approach is based on TDDFT. The Green’s
function approach is formally straightforward, in the sense that excitonic particle–
hole interactions are built in by construction; however, it is computationally costly.
TDDFT, on the other hand, is computationally cheaper (provided that the approximate xc functionals are simple enough), but the price one has to pay is that
intrinsically nonlocal electron–hole interaction effects have to be described via
linearized local xc potentials; this is a somewhat unnatural way of dealing with
excitonic interactions, which makes it non-straightforward to construct good
approximations.
In this chapter we have focused on the TDDFT approach for excitons, and have
tried to bring across the following points:
• Excitons in TDDFT are a difficult problem, because they require an xc kernel
which has the long-range property f xc,00 (q, ω) ~ q
À2 for q ! 0. The popular local
and semilocal approximations such as ALDA and standard GGAs do not have
this property: although they work well for finite systems such as atoms and
molecules, they do not produce excitons in extended systems. New approximations are therefore required.
• There exist several approximate “excitonic” xc kernels, with various degrees of
sophistication. Some kernels involve adjustable parameters, others don’t. A
typical behavior of the simpler kernels is that they can reproduce some part of
the optical spectrum reasonably well (e.g., a bound exciton, or the continuum
part), but not all of it at the same time. The best excitonic xc kernel, the
“nanoquanta” kernel, is computationally not much simpler than the BSE.
• Adiabatic (i.e., frequency-independent) xc kernels cannot produce an excitonic
Rydberg series. At best, they can generate a single bound exciton (if only the
head of the xc coupling matrix is used). The TDDFT analog of the excitonic
Wannier equation features a nonlocal potential, which is too shallow to produce
more than one bound level.
Excitons in Time-Dependent Density-Functional Theory
213