32
3 One-Particle Green’s Function
For notational brevity here and in the following, atomic units will be supposed where
= 1. In the chosen representation, the electron propagator or one-particle Green’s
function (GF) G(t, t
) is a matrix of time-dependent functions (components) defined
according to
G pq (t, t
) = −iθ(t − t
) 0 |c p [t]c
†
q [t
]| 0 + iθ(t
− t) 0 |c
†
q [t
]c p [t]| 0
(3.3)
Here, θ(t) denotes the step function
θ(t) =
1, t > 0
0, t < 0
(3.4)
The notation can be shortened by using the time-ordering operator ˆ
T T T , also referred
to as Wick’s operator. Acting on a product of time-dependent fermion operators, ˆ
T T T
reorders the factors in such a way that operators with larger times are to the left of
those with smaller times; ˆ
T T T also introduces a sign (−1)
P , where P is the permutation
transforming the original order into the final one. The result is also referred to as
time-ordered product. In the case of two operators, the time-ordered product is
simply given by
ˆ
T T T
c p [t]c
†
q [t
]
=
c p [t]c
†
q [t
], t > t
−c
†
q [t
]c p [t], t < t
(3.5)
This allows us to write the electron propagator components in the more compact
form
G pq (t, t
) = −i 0 | ˆ
T T T
c p [t]c
†
q [t
]
| 0
(3.6)
As seen from the definition (3.3), the electron propagator consists of two parts,
G(t, t
) = G
+
(t, t
) + G
−
(t, t
)
As will be shown below, the two parts contain spectral information related to electron
attachment (G
+ ) and electron removal or ionization (G
− ). Accordingly, the two parts
are also referred to as (N +1)- and (N −1)-electron parts, respectively.
Let us have a closer look at the physical content of the (N +1)-electron part. As a
first step, one may insert the explicit definition of the time-dependent operators (3.2)
in Eq. (3.3):
G
+
pq (t, t
) = −iθ(t − t
) 0 |e
i ˆ
Ht c p e
−i ˆ
Ht e
i ˆ
Ht
c
†
q e
−i ˆ
Ht
| 0
= −iθ(t − t
)e
i E 0 (t−t
)
0 |c p e
−i ˆ
H (t−t
) c
†
q | 0
(3.7)
This shows that the GF components depend only on the difference t − t
of the
two time arguments. To proceed, we insert the resolution of the identity in terms of
3 One-Particle Green’s Function
For notational brevity here and in the following, atomic units will be supposed where
= 1. In the chosen representation, the electron propagator or one-particle Green’s
function (GF) G(t, t
) is a matrix of time-dependent functions (components) defined
according to
G pq (t, t
) = −iθ(t − t
) 0 |c p [t]c
†
q [t
]| 0 + iθ(t
− t) 0 |c
†
q [t
]c p [t]| 0
(3.3)
Here, θ(t) denotes the step function
θ(t) =
1, t > 0
0, t < 0
(3.4)
The notation can be shortened by using the time-ordering operator ˆ
T T T , also referred
to as Wick’s operator. Acting on a product of time-dependent fermion operators, ˆ
T T T
reorders the factors in such a way that operators with larger times are to the left of
those with smaller times; ˆ
T T T also introduces a sign (−1)
P , where P is the permutation
transforming the original order into the final one. The result is also referred to as
time-ordered product. In the case of two operators, the time-ordered product is
simply given by
ˆ
T T T
c p [t]c
†
q [t
]
=
c p [t]c
†
q [t
], t > t
−c
†
q [t
]c p [t], t < t
(3.5)
This allows us to write the electron propagator components in the more compact
form
G pq (t, t
) = −i 0 | ˆ
T T T
c p [t]c
†
q [t
]
| 0
(3.6)
As seen from the definition (3.3), the electron propagator consists of two parts,
G(t, t
) = G
+
(t, t
) + G
−
(t, t
)
As will be shown below, the two parts contain spectral information related to electron
attachment (G
+ ) and electron removal or ionization (G
− ). Accordingly, the two parts
are also referred to as (N +1)- and (N −1)-electron parts, respectively.
Let us have a closer look at the physical content of the (N +1)-electron part. As a
first step, one may insert the explicit definition of the time-dependent operators (3.2)
in Eq. (3.3):
G
+
pq (t, t
) = −iθ(t − t
) 0 |e
i ˆ
Ht c p e
−i ˆ
Ht e
i ˆ
Ht
c
†
q e
−i ˆ
Ht
| 0
= −iθ(t − t
)e
i E 0 (t−t
)
0 |c p e
−i ˆ
H (t−t
) c
†
q | 0
(3.7)
This shows that the GF components depend only on the difference t − t
of the
two time arguments. To proceed, we insert the resolution of the identity in terms of