8.3 Solving the Dyson Equation
133
region within the ionization/electron-affinity gap, sufficiently far from the two outermost poles, pp (ω) is a smooth function of ω, and the third-order expansion (8.82)
may represent that region quite reasonably, irrespective of its inadequacy in the pole
regions. Accordingly, for outer valence main (1h) states, where the graphical solutions of the respective diagonal Dyson equations lie in the gap region, a third-order
self-energy expansion may afford a good approximation. An approach based on the
third-order expansion of the self-energy is the widely used and utmost successful
outer valence Green’s function (OVGF) method [4–6]. A related scheme is the P3
method basing on a partial third-order self-energy expression [7].
The two main ingredients of the OVGF method are:
1. Diagonal approximation for the electron propagator: G pq (ω) ≈ δ pq G pp (ω)
2. A modified third-order expansion of the diagonal self-energy matrix elements:
pp (ω) ≈
(2)
pp (ω) +
1
1 − A p
(3)
pp (ω)
(8.83)
where A p < 1 is a predefined correction coefficient.
In the OVGF expression for the self-energy, a correction factor is attached to the
strict third-order contribution
(3)
pp (ω), the purpose of which is to extrapolate the
expansion to higher orders. The usual recipe is
A p =
5
i=2
T 1(i)| ω= p + T 2(i)| ω= p
M
(2)
pp ( p )
(8.84)
Here, T 1 and T 2 are the two ω-dependent third-order diagrams (see Fig. 8.9),
and i = 2, . . . , 5 label the time-orderings that have only one ω-denominator. Obviously, A p somehow measures the ratio of the third- and second-order self-energy
contributions (for the pth diagonal element), and, thus, the factor (1 − A p )
−1
=
1 + A p + A
2
p + . . . in (8.83) can be understood as sort of an extrapolation to higherorder contributions. It should be noted that there are two more extrapolation schemes
to be applied in specific cases (see Refs. [4, 6] for details).
Exercises
8.1 Consider a general one-particle system with a hamiltonian of the form ˆ
h = ˆ
h 0 +
ˆ
v, where ˆ
h 0 is a “free” hamiltonian and ˆ
v a perturbation. Accordingly, there are
resolvent operators ˆ
g(ω) = (ω − ˆ
h)
−1 and ˆ
g 0 (ω) = (ω − ˆ
h 0 )
−1 corresponding
to the full and free hamiltonian, respectively. Derive a Dyson-like equation for
ˆ
g(ω) and another equation analogous to Eq. (8.2). What are the analogues to the
improper and proper self-energy parts?
8.2 (a) Evaluate the three time-orderings contributing to
−
(ω) (t
> t) of the diagram T 1 in Fig. 8.9.
(b) Draw and evaluate the six time-orderings of diagram T 3 in Fig. 8.9.
133
region within the ionization/electron-affinity gap, sufficiently far from the two outermost poles, pp (ω) is a smooth function of ω, and the third-order expansion (8.82)
may represent that region quite reasonably, irrespective of its inadequacy in the pole
regions. Accordingly, for outer valence main (1h) states, where the graphical solutions of the respective diagonal Dyson equations lie in the gap region, a third-order
self-energy expansion may afford a good approximation. An approach based on the
third-order expansion of the self-energy is the widely used and utmost successful
outer valence Green’s function (OVGF) method [4–6]. A related scheme is the P3
method basing on a partial third-order self-energy expression [7].
The two main ingredients of the OVGF method are:
1. Diagonal approximation for the electron propagator: G pq (ω) ≈ δ pq G pp (ω)
2. A modified third-order expansion of the diagonal self-energy matrix elements:
pp (ω) ≈
(2)
pp (ω) +
1
1 − A p
(3)
pp (ω)
(8.83)
where A p < 1 is a predefined correction coefficient.
In the OVGF expression for the self-energy, a correction factor is attached to the
strict third-order contribution
(3)
pp (ω), the purpose of which is to extrapolate the
expansion to higher orders. The usual recipe is
A p =
5
i=2
T 1(i)| ω= p + T 2(i)| ω= p
M
(2)
pp ( p )
(8.84)
Here, T 1 and T 2 are the two ω-dependent third-order diagrams (see Fig. 8.9),
and i = 2, . . . , 5 label the time-orderings that have only one ω-denominator. Obviously, A p somehow measures the ratio of the third- and second-order self-energy
contributions (for the pth diagonal element), and, thus, the factor (1 − A p )
−1
=
1 + A p + A
2
p + . . . in (8.83) can be understood as sort of an extrapolation to higherorder contributions. It should be noted that there are two more extrapolation schemes
to be applied in specific cases (see Refs. [4, 6] for details).
Exercises
8.1 Consider a general one-particle system with a hamiltonian of the form ˆ
h = ˆ
h 0 +
ˆ
v, where ˆ
h 0 is a “free” hamiltonian and ˆ
v a perturbation. Accordingly, there are
resolvent operators ˆ
g(ω) = (ω − ˆ
h)
−1 and ˆ
g 0 (ω) = (ω − ˆ
h 0 )
−1 corresponding
to the full and free hamiltonian, respectively. Derive a Dyson-like equation for
ˆ
g(ω) and another equation analogous to Eq. (8.2). What are the analogues to the
improper and proper self-energy parts?
8.2 (a) Evaluate the three time-orderings contributing to
−
(ω) (t
> t) of the diagram T 1 in Fig. 8.9.
(b) Draw and evaluate the six time-orderings of diagram T 3 in Fig. 8.9.