126
8 Self-Energy and the Dyson Equation
where e n = E nn are the eigenvalues of A and x n is the vector of the p/ h-components,
x
(n)
p = X pn , of the full eigenvectors X n . This is an energy-dependent (pseudo-) eigenvalue equation for a one-particle problem, in which the self-energy matrix (ω)
can be seen as orbital representation of an energy-dependent, non-local one-particle
operator ˆ
(x, x
; ω). The enormous reduction of the dimensionality with respect to
Eq. (8.51) comes at a price, as the solutions of the pseudo-eigenvalue problem have
to be sought using iterative techniques, which are prone to computational problems.
It should be noted that the usual orthonormalization features do not apply to the
pseudo-eigenvectors x n (see Appendix A.1).
3. Dyson Orbitals
As a tool for visualizing many-body effects in ionization (or electron attachment)
the so-called Dyson orbitals have proven useful. Consider a final ionic state, say of
N − 1 electrons, |
N −1
n
. A Dyson orbital n (ξ) can be assigned to this state as
follows:
n (ξ) ==
N −1
n
| ˆ
ψ(ξ)| 0
=
r
N −1
n
|c r | 0 ψ r (ξ) =
r
x
(n)
r ψ r (ξ)
(8.57)
As in Eq. (1.3), ξ ≡ xσ combines the spatial and spin variables; ψ r (ξ) denote the
spin orbitals associated with the fermion operators, c
†
r , c r , and
ˆ
ψ(ξ) =
r
c r ψ r (ξ)
(8.58)
is the field operator according to Eq. (2.45). Alternatively, the Dyson orbital can be
obtained directly from the ground- and ionic-state wave functions,
n (ξ) =
√
N !
N −1
n
(ξ 2 , . . . , ξ N )
∗
0 (ξ, ξ 2 , . . . , ξ N ) dξ 2 . . . dξ N
(8.59)
where as in Eq. (1.5) the integration over ξ i comprises the summation over spin
variables. To show that both definitions are equivalent (see Exercise 8.4) one may
insert in Eq. (8.57) the resolution of the identity in terms of the coordinate eigenstates |ξ 1 . . . ξ N (respecting here Eq. 1.24) and evaluate ˆ
ψ(ξ)|ξ 1 . . . ξ N according
to Eq. (2.9).
Like spin orbitals, the Dyson orbitals n (ξ) can be written as products of a spatial
orbital and a spin function:
nγ (x, σ) = n (x)χ γ (σ)
(8.60)
Here, γ = α, β specifies the z-component of the spin in the final state, n ≡ nγ, and
γ denotes the spin quantum number complementary to γ, e.g., α = β. The spatial
8 Self-Energy and the Dyson Equation
where e n = E nn are the eigenvalues of A and x n is the vector of the p/ h-components,
x
(n)
p = X pn , of the full eigenvectors X n . This is an energy-dependent (pseudo-) eigenvalue equation for a one-particle problem, in which the self-energy matrix (ω)
can be seen as orbital representation of an energy-dependent, non-local one-particle
operator ˆ
(x, x
; ω). The enormous reduction of the dimensionality with respect to
Eq. (8.51) comes at a price, as the solutions of the pseudo-eigenvalue problem have
to be sought using iterative techniques, which are prone to computational problems.
It should be noted that the usual orthonormalization features do not apply to the
pseudo-eigenvectors x n (see Appendix A.1).
3. Dyson Orbitals
As a tool for visualizing many-body effects in ionization (or electron attachment)
the so-called Dyson orbitals have proven useful. Consider a final ionic state, say of
N − 1 electrons, |
N −1
n
. A Dyson orbital n (ξ) can be assigned to this state as
follows:
n (ξ) ==
N −1
n
| ˆ
ψ(ξ)| 0
=
r
N −1
n
|c r | 0 ψ r (ξ) =
r
x
(n)
r ψ r (ξ)
(8.57)
As in Eq. (1.3), ξ ≡ xσ combines the spatial and spin variables; ψ r (ξ) denote the
spin orbitals associated with the fermion operators, c
†
r , c r , and
ˆ
ψ(ξ) =
r
c r ψ r (ξ)
(8.58)
is the field operator according to Eq. (2.45). Alternatively, the Dyson orbital can be
obtained directly from the ground- and ionic-state wave functions,
n (ξ) =
√
N !
N −1
n
(ξ 2 , . . . , ξ N )
∗
0 (ξ, ξ 2 , . . . , ξ N ) dξ 2 . . . dξ N
(8.59)
where as in Eq. (1.5) the integration over ξ i comprises the summation over spin
variables. To show that both definitions are equivalent (see Exercise 8.4) one may
insert in Eq. (8.57) the resolution of the identity in terms of the coordinate eigenstates |ξ 1 . . . ξ N (respecting here Eq. 1.24) and evaluate ˆ
ψ(ξ)|ξ 1 . . . ξ N according
to Eq. (2.9).
Like spin orbitals, the Dyson orbitals n (ξ) can be written as products of a spatial
orbital and a spin function:
nγ (x, σ) = n (x)χ γ (σ)
(8.60)
Here, γ = α, β specifies the z-component of the spin in the final state, n ≡ nγ, and
γ denotes the spin quantum number complementary to γ, e.g., α = β. The spatial