χ qp f
ex
xc e χ % χ qp f
ex
xc χ qp . Second, find a diagrammatic representation of e χ À χ qp ; the
details of this representation are quite technical and are not given here (in essence, it
involves two-particle Green’s functions and four-point vertex functions [1, 14, 15,
125]). The key point is that this diagrammatic representation can be very easily
approximated, and one ends up with the following expression:
ð
d3
ð
d4χ qp 1; 3
ð Þf
ex
xc 3; 4
ð Þχ qp 4; 2
ð Þ ¼
ð
d3
ð
d4G qp 1; 3
ð ÞG qp 4; 1
ð ÞW 3; 4
ð ÞG qp 3; 2
ð ÞG qp 2; 4
ð Þ:
ð61Þ
This is the xc kernel of Reining et al. [89] and many others [also known as the
“nanoquanta” kernel (http://www.cmt.york.ac.uk/nanoquanta/)]. Figure 4b shows
its diagrammatic representation. The numbers in (61) represent space-time arguments, e.g., 1 ¼ (r 1 , t 1 ). G qp is a quasiparticle Green’s function, and W is a screened
interaction, formally defined as
W 1; 2
ð Þ ¼ w 1; 2
ð Þþ
ð
d3
ð
d4w 1; 3
ð Þe χ 3; 4
ð ÞW 4; 2
ð Þ;
ð62Þ
where w(1,2) is the bare Coulomb interaction, and e χ is approximated by χ qp .
The excitonic xc kernel f xc
ex of (61) has been widely applied in a variety of
systems [14, 89, 90, 102, 126–131]. Its performance is, in general, found to be
excellent, at par with results obtained from solving the full BSE. This provided an
important proof of concept that TDDFT is very well capable of capturing excitonic
properties. The price to be paid, however, is that the many-body xc kernel (61) is
not simple to implement and is computationally costly (it is, essentially, as expensive as the BSE when it comes to calculating optical spectra, but somewhat more
favorable if the full dielectric matrix is needed).
5.4.7 The “Bootstrap” Kernel
Compared to the exact exchange and nanoquanta kernels, the simplicity of the LRC
kernel is desirable for practical use. The adjustable parameter A LRC requires prior
knowledge to the system, however, and therefore the LRC kernel cannot be used as
a black-box method. The bootstrap kernel proposed by Sharma et al. [132, 133] can
be seen as an attempt to determine the A LRC parameter (which now depends on q,
G, and G
0 ) self-consistently while retaining the simplicity of the LRC kernel. The
original definition [132] is written in terms of symmetrized quantities to avoid
singularities: for example, f
sym
xc ðq, G, G
0
Þ ¼ v
À1=2
G
q
ð Þf xc ðq, G, G
0
Þv
À1=2
G 0
q
ð Þ and
χ
sym
ðq, G, G
0
Þ ¼ v
1=2
G q
ð Þχðq, G, G
0
Þv
1=2
G 0 q
ð Þ. In terms of regular, non-symmetrized
quantities, the kernel is defined as
206
C.A. Ullrich and Z.-h. Yang
ex
xc e χ % χ qp f
ex
xc χ qp . Second, find a diagrammatic representation of e χ À χ qp ; the
details of this representation are quite technical and are not given here (in essence, it
involves two-particle Green’s functions and four-point vertex functions [1, 14, 15,
125]). The key point is that this diagrammatic representation can be very easily
approximated, and one ends up with the following expression:
ð
d3
ð
d4χ qp 1; 3
ð Þf
ex
xc 3; 4
ð Þχ qp 4; 2
ð Þ ¼
ð
d3
ð
d4G qp 1; 3
ð ÞG qp 4; 1
ð ÞW 3; 4
ð ÞG qp 3; 2
ð ÞG qp 2; 4
ð Þ:
ð61Þ
This is the xc kernel of Reining et al. [89] and many others [also known as the
“nanoquanta” kernel (http://www.cmt.york.ac.uk/nanoquanta/)]. Figure 4b shows
its diagrammatic representation. The numbers in (61) represent space-time arguments, e.g., 1 ¼ (r 1 , t 1 ). G qp is a quasiparticle Green’s function, and W is a screened
interaction, formally defined as
W 1; 2
ð Þ ¼ w 1; 2
ð Þþ
ð
d3
ð
d4w 1; 3
ð Þe χ 3; 4
ð ÞW 4; 2
ð Þ;
ð62Þ
where w(1,2) is the bare Coulomb interaction, and e χ is approximated by χ qp .
The excitonic xc kernel f xc
ex of (61) has been widely applied in a variety of
systems [14, 89, 90, 102, 126–131]. Its performance is, in general, found to be
excellent, at par with results obtained from solving the full BSE. This provided an
important proof of concept that TDDFT is very well capable of capturing excitonic
properties. The price to be paid, however, is that the many-body xc kernel (61) is
not simple to implement and is computationally costly (it is, essentially, as expensive as the BSE when it comes to calculating optical spectra, but somewhat more
favorable if the full dielectric matrix is needed).
5.4.7 The “Bootstrap” Kernel
Compared to the exact exchange and nanoquanta kernels, the simplicity of the LRC
kernel is desirable for practical use. The adjustable parameter A LRC requires prior
knowledge to the system, however, and therefore the LRC kernel cannot be used as
a black-box method. The bootstrap kernel proposed by Sharma et al. [132, 133] can
be seen as an attempt to determine the A LRC parameter (which now depends on q,
G, and G
0 ) self-consistently while retaining the simplicity of the LRC kernel. The
original definition [132] is written in terms of symmetrized quantities to avoid
singularities: for example, f
sym
xc ðq, G, G
0
Þ ¼ v
À1=2
G
q
ð Þf xc ðq, G, G
0
Þv
À1=2
G 0
q
ð Þ and
χ
sym
ðq, G, G
0
Þ ¼ v
1=2
G q
ð Þχðq, G, G
0
Þv
1=2
G 0 q
ð Þ. In terms of regular, non-symmetrized
quantities, the kernel is defined as
206
C.A. Ullrich and Z.-h. Yang