TOMBO Ver.2 Manual

2.5 The GW approximation 18
The GW self-energy can be separated into two terms ( Σ GW = Σx + Σ c ). The exchange part
is given by
Σ x (r,r ′ ) = i ∫
2π
dωe iω0+ G(r,r ′ ;ω)ν(r − r ′ ) = −
occ
∑
nk
ψ nk (r)ψ ∗ nk (r)
|r − r ′ |
(2.30)
In equation (2.30), the symbol occ in the sum means that the summation is taken over the
occupied states only. The diagonal matrix elements of this exchange part of the self-energy
become
⟨
n,k
∣ ∣Σx (r,r ′ ) ∣ ∣ n,k
⟩
= −
occ
∑
∑
n ′ q
∑
G
4π
⟨
ΩG 2
∣
n,k∣e i(q+G)·r∣ ⟩⟨
∣ ∣n ′ ,k − q n ′ ∣
k − q∣e −i(q+G)·r′∣ ⟩
∣ ∣n,k
(2.31)
The correlation part, Σ c can be evaluated by using generalized plasmon-pole (GPP)
model [19]. Σ c can also be evaluated by using the full ω integration [20]:
Σ c (r,r ′ ;ω) = i ∫
2π
dω ′ e −iω0+ G(r,r ′ ;ω − ω ′ ) [ W(r,r ′ ;ω ′ ) − ν(r − r ′ ) ] (2.32)
In Equation (2.32), it is difficult to perform the ω ′ integral along the real axis, since W and
G have a strong structure on this axis. In order to avoid this difficulty, Godby et al.[21–24]
restricted the values of ω to small imaginary numbers and changed the contour of the ω ′
integral from real axis to imaginary axis [16]. Then, by analytic continuation, the resulting
Taylor series is used to estimate the matrix elements for real values of ω. Ishii and Ohno et
al. [20, 25] suggested that this intergration method employed by Godby et al. [16] can be
extended rather easily to real number of ω by slightly modifying the contour. The contour
along the real ω ′ axis from −∞ to +∞ for the integral Equation (2.32) can be replaced by
the contour C shown in Fig.2.2.
Here we further use the symmetry W(ω) = W(−ω) to reduce the contour to the positive
real and imaginary parts only. The diagonal matrix elements of the correlation part of the
self-energy become
⟨
n,k
∣ ∣Σc (r,r ′ ;ω) ∣ ∣ n,k
⟩
= ∑
∑
n ′ q
× i
2π
⟨
∑
G,G ′
∫
∣
n,k∣e i(q+G)·r∣ ⟩⟨
∣ ∣n ′ ,k − q n ′ ∣
k − q∣e −i(q+G′ )·r ′∣ ⟩
∣ ∣n,k
dω ′ [ W G,G ′(q,ω ′ ) − (4π/ΩG 2 ]
)δ G,G ′
1
× (
ω + ω ′ +
− ε k−q,n − iδ k−q,n
1
ω − ω ′ − ε k−q,n − iδ k−q,n
)
(2.33)