124
8 Self-Energy and the Dyson Equation
8.3 Solving the Dyson Equation
1. The Dyson Secular Matrix
Using the general form (8.18) for the self-energy and the matrix notation G
0
(ω)
−1
=
ω1 − for the inverse of the free electron propagator, the formal solution (8.11) of
the Dyson equation can be written more explicitly as
G(ω) =
G
0
(ω)
−1
− (ω)
−1 = (ω1 − − (∞) − M(ω))
−1
(8.44)
Moreover, one may adopt the following matrix notation for the dynamical self-energy
parts:
M(ω) = M
+
(ω) + M
−
(ω), M
±
(ω) = m
± †
ω1 −
±
−1 m
±
(8.45)
Here,
± denote the diagonal matrices of the pole positions in the spectral representations (8.19) of the two M
±
pq (ω) parts,
+
νν = ω ν , ν ∈ {N + 1}, ,
−
μμ = ω μ , μ ∈ {N − 1}
(8.46)
and m
± are the corresponding matrices of the amplitudes,
m
+
ν p = m
(ν)
p
∗ , ν ∈ {N + 1}, m
−
μ p = m
(μ)
p
∗ , μ ∈ {N − 1}
(8.47)
As a result, the Dyson equation takes on the form
G(ω) =
ω1 − − (∞) − m
− † (ω1 −
−
)
−1 m
−
− m
+ † (ω1 −
+
)
−1 m
+
−1
(8.48)
which has some semblance of the partitioning formulas reviewed in Appendix A.1.
In fact, one may introduce the following Dyson secular matrix
A =
⎛
⎝
+ (∞) m
− † m
+ †
m
−
−
0
m
+
0
+
⎞
⎠
(8.49)
Using the partitioning formulas (A.1.26) and (A.1.27) for the upper left diagonal
block of A, that is, A 11 = + (∞), the right-hand side of Eq. (8.48) can readily
be identified with the upper left diagonal block of the inverse of (ω1 − A):
G(ω) = (ω1 − A)
−1
11
(8.50)
Let us note that in the upper left diagonal block, denoted by | 11 , the matrix indices
are the HF one-particle indices, comprising both occupied (h) and unoccupied ( p)
orbitals.