Unfortunately, explicit calculations show that the performance of the PGG kernel is
disappointing for solids: it does not produce any bound excitons at all, despite
having a nonzero head contribution with the correct 1/q
2 behavior [19]. How can
this be reconciled with the fact that the PGG kernel seems to work well in finite
systems such as atoms and molecules? Periodic systems are dominated by the head
of the xc kernel in reciprocal space; however, the situation is very different in finite
systems, where the electron dynamics can be viewed as coming entirely from localfield effects. Thus, the strongly attractive nature of the PGG kernel in finite systems
would at most translate into a strong body of the xc matrix in periodic systems
(which, however, is irrelevant for excitons), but does not necessarily guarantee a
strong head. Indeed, if one fits the head of the PGG kernel to the LRC kernel (48),
one finds that the resulting constant A
PGG
LRC is orders of magnitude too weak [19].
The underlying reason for the failure of the PGG kernel for periodic insulators
can be inferred from its real-space definition (53), which can be written in the form
f
PGG
x
¼ À
ρðr, r
0
Þ
2 = 2
r À r
0
n r
ð Þn r
0
ð Þ
Â
Ã
, where ρ(r, r
0 ) is the Kohn–Sham density
matrix. For periodic solids, the long-range behavior of f
PGG
x
is determined by both
the Coulomb singularity and the density matrix. It is a well-known fact [120–124]
that the one-particle density matrix in insulators decays exponentially as ρ(r, r
0 )
~ exp(Àγ|r–r
0 |). This effectively cuts off the required long-range behavior and
explains why the head of the PGG kernel is so weak.
5.4.6 The “Nanoquanta” Kernel
Let us introduce the so-called proper response function e χ as
e χ ¼ χ s þ χ s f xc e χ
ð58Þ
(for simplicity, we drop all arguments and integrals). At the beginning of this
section we defined the quasiparticle and excitonic parts of the xc kernel; see (46).
It is then easy to write down the following relations for the two parts of f xc [91]:
χ qp ¼ χ s þ χ s f
qp
xc χ qp ;
ð59Þ
e χ ¼ χ qp þ χ qp f
ex
xc e χ:
ð60Þ
Here, χ qp is the quasiparticle response function, which uses quasiparticle states as
input. Hence, χ qp has the quasiparticle gap built in by default, and the roles of f
qp
xc
and f
ex
xc are clear from (59) and (60). Our focus here is on the excitons: all we need to
do, then, is start with a good approximation for χ qp . Usually, one obtains it from the
GW approach (see Sect. 3), but other approximations that yield good quasiparticle
gaps (such as hybrids or the scissors operator) can be used as well.
The exact proper response function e χ is, of course, unknown. To construct an
approximation for f
ex
xc from (60) one can proceed in two steps. First, replace
Excitons in Time-Dependent Density-Functional Theory
205
disappointing for solids: it does not produce any bound excitons at all, despite
having a nonzero head contribution with the correct 1/q
2 behavior [19]. How can
this be reconciled with the fact that the PGG kernel seems to work well in finite
systems such as atoms and molecules? Periodic systems are dominated by the head
of the xc kernel in reciprocal space; however, the situation is very different in finite
systems, where the electron dynamics can be viewed as coming entirely from localfield effects. Thus, the strongly attractive nature of the PGG kernel in finite systems
would at most translate into a strong body of the xc matrix in periodic systems
(which, however, is irrelevant for excitons), but does not necessarily guarantee a
strong head. Indeed, if one fits the head of the PGG kernel to the LRC kernel (48),
one finds that the resulting constant A
PGG
LRC is orders of magnitude too weak [19].
The underlying reason for the failure of the PGG kernel for periodic insulators
can be inferred from its real-space definition (53), which can be written in the form
f
PGG
x
¼ À
ρðr, r
0
Þ
2 = 2
r À r
0
n r
ð Þn r
0
ð Þ
Â
Ã
, where ρ(r, r
0 ) is the Kohn–Sham density
matrix. For periodic solids, the long-range behavior of f
PGG
x
is determined by both
the Coulomb singularity and the density matrix. It is a well-known fact [120–124]
that the one-particle density matrix in insulators decays exponentially as ρ(r, r
0 )
~ exp(Àγ|r–r
0 |). This effectively cuts off the required long-range behavior and
explains why the head of the PGG kernel is so weak.
5.4.6 The “Nanoquanta” Kernel
Let us introduce the so-called proper response function e χ as
e χ ¼ χ s þ χ s f xc e χ
ð58Þ
(for simplicity, we drop all arguments and integrals). At the beginning of this
section we defined the quasiparticle and excitonic parts of the xc kernel; see (46).
It is then easy to write down the following relations for the two parts of f xc [91]:
χ qp ¼ χ s þ χ s f
qp
xc χ qp ;
ð59Þ
e χ ¼ χ qp þ χ qp f
ex
xc e χ:
ð60Þ
Here, χ qp is the quasiparticle response function, which uses quasiparticle states as
input. Hence, χ qp has the quasiparticle gap built in by default, and the roles of f
qp
xc
and f
ex
xc are clear from (59) and (60). Our focus here is on the excitons: all we need to
do, then, is start with a good approximation for χ qp . Usually, one obtains it from the
GW approach (see Sect. 3), but other approximations that yield good quasiparticle
gaps (such as hybrids or the scissors operator) can be used as well.
The exact proper response function e χ is, of course, unknown. To construct an
approximation for f
ex
xc from (60) one can proceed in two steps. First, replace
Excitons in Time-Dependent Density-Functional Theory
205