5.4.3 Exact Exchange
The frequency-dependent xc kernel can be formally constructed from many-body
perturbation theory, using a diagrammatic expansion [93, 94]. The first-order term
of this expansion is the exact exchange kernel f x (r, r
0 , ω), which can be represented
as the sum of five diagrams (see Fig. 4a). Translated into formulas using the
standard diagrammatic rules, one obtains [95, 96]
ð
d
3 r 1
ð
d
3 r 2 χ s r; r 1 ; ω
ð
Þf x r 1 ; r 2 ; ω
ð
Þ χ s ðr 2 , r
0 , ωÞ
¼ R V ðr, r
0 , ωÞ þ R Σ ðr, r
0 , ωÞ:
ð50Þ
R v is the first-order vertex diagram (third on the right-hand side in Fig. 4a),
R V r, r
0 , ω
ð
Þ¼À2
X
ijkl
φ i r
ð Þφ
*
j r
ð Þφ
*
k r
0
ð Þφ l r
0
ð Þ il
h jw jk
j i
ð f i À f j Þ f k À f l
ð
Þ
z À ω i j
À
Á
z À ω lk
ð
Þ
; ð51Þ
where z ¼ ω + i0
+
, and the f j are the usual occupation factors. R Σ denotes the sum of
all the remaining four diagrams (the self-energy diagrams):
R Σ r,r
0 ,ω
ð
Þ¼4
X
i jk
φ i r
ð Þφ
*
i r
0
ð Þφ j r
ð Þφ
*
k r
0
ð Þ j
h jΔ k
j i
ω k j
f k À f i
ð
Þω ik
z 2 À ω 2
ik
À
ð f j À f i Þω i j
z 2 À ω 2
i j
(
)
;
ð52Þ
where Δ(r 1 , r 2 )¼Σ x (r 1 , r 2 )Àν x (r 1 )δ(r 1 Àr 2 ). Here, Σ x is the exchange part of the
self-energy, and ν x is the exact exchange potential of DFT, defined as an orbital
functional via the optimized effective potential (OEP) method [45, 97]. It is also
x
v
x
v
x
f
ex
xc
f
a
b
Fig. 4 Diagrammatic representations of (a) the exact exchange kernel f x (50) and (b) the excitonic
xc kernel f
ex
xc (61), the so-called nanoquanta kernel. Full lines represent noninteracting Kohn–Sham
Green’s functions and dashed lines represent quasiparticle Green’s functions. Thin wavy lines are
bare Coulomb interactions and the thick wavy line is a screened interaction
202
C.A. Ullrich and Z.-h. Yang
The frequency-dependent xc kernel can be formally constructed from many-body
perturbation theory, using a diagrammatic expansion [93, 94]. The first-order term
of this expansion is the exact exchange kernel f x (r, r
0 , ω), which can be represented
as the sum of five diagrams (see Fig. 4a). Translated into formulas using the
standard diagrammatic rules, one obtains [95, 96]
ð
d
3 r 1
ð
d
3 r 2 χ s r; r 1 ; ω
ð
Þf x r 1 ; r 2 ; ω
ð
Þ χ s ðr 2 , r
0 , ωÞ
¼ R V ðr, r
0 , ωÞ þ R Σ ðr, r
0 , ωÞ:
ð50Þ
R v is the first-order vertex diagram (third on the right-hand side in Fig. 4a),
R V r, r
0 , ω
ð
Þ¼À2
X
ijkl
φ i r
ð Þφ
*
j r
ð Þφ
*
k r
0
ð Þφ l r
0
ð Þ il
h jw jk
j i
ð f i À f j Þ f k À f l
ð
Þ
z À ω i j
À
Á
z À ω lk
ð
Þ
; ð51Þ
where z ¼ ω + i0
+
, and the f j are the usual occupation factors. R Σ denotes the sum of
all the remaining four diagrams (the self-energy diagrams):
R Σ r,r
0 ,ω
ð
Þ¼4
X
i jk
φ i r
ð Þφ
*
i r
0
ð Þφ j r
ð Þφ
*
k r
0
ð Þ j
h jΔ k
j i
ω k j
f k À f i
ð
Þω ik
z 2 À ω 2
ik
À
ð f j À f i Þω i j
z 2 À ω 2
i j
(
)
;
ð52Þ
where Δ(r 1 , r 2 )¼Σ x (r 1 , r 2 )Àν x (r 1 )δ(r 1 Àr 2 ). Here, Σ x is the exchange part of the
self-energy, and ν x is the exact exchange potential of DFT, defined as an orbital
functional via the optimized effective potential (OEP) method [45, 97]. It is also
x
v
x
v
x
f
ex
xc
f
a
b
Fig. 4 Diagrammatic representations of (a) the exact exchange kernel f x (50) and (b) the excitonic
xc kernel f
ex
xc (61), the so-called nanoquanta kernel. Full lines represent noninteracting Kohn–Sham
Green’s functions and dashed lines represent quasiparticle Green’s functions. Thin wavy lines are
bare Coulomb interactions and the thick wavy line is a screened interaction
202
C.A. Ullrich and Z.-h. Yang