312
Appendix
A.7 Linear Response Theory and the Polarization
Propagator
The polarization propagator discussed in Chap. 13 is closely related to polarizabilities
or, more general, to the linear response of properties, as, e.g., dipole moments, to a
time-dependent “external” perturbation [5]. In this Appendix, we briefly outline that
relation and show how the ADC approach can be used to treat linear response properties. For a more comprehensive presentation, general references, and a discussion
of the computational performance, the reader is referred to Ref. [16].
Let ˆ
H x (t) denote the hamiltonian for an external perturbation,
ˆ
H x (t) = ˆ
B f (t)
(A.7.1)
where
ˆ
B =
B rs c
†
r c s
(A.7.2)
is a (constant) one-particle operator and f (t) a time-dependent function describing
the time-dependence of the perturbation. The perturbation is assumed to be turned
on at t 0 ; that is, f (t) = 0 for t ≤ t 0 . The full hamiltonian is given by
ˆ
H
(t) = ˆ
H + ˆ
H x (t)
(A.7.3)
where ˆ
H is the hamiltonian of the unperturbed system. Initially, that is, at and prior
to t 0 , the system is assumed to be in the ground state | 0 of ˆ
H .
To construct the solution of the TDSE, we may resort to the procedure presented
in Sect. 4.1, where now ˆ
H (rather than ˆ
H 0 ) is the time-independent part of the full
hamiltonian. As in Eq. (4.13), the time-dependent wave function can be written in
the form
|(t) = e
−i ˆ
Ht
| x (t)
(A.7.4)
where | x (t) denotes the wave function in the interaction picture with ˆ
H acting as
the unperturbed part of the hamiltonian. Analogous to Eq. (4.19), we may write
| x (t) = ˆ
U (t, t 0 )| 0
(A.7.5)
where the time-evolution operator in the interaction picture, ˆ
U (t, t 0 ), is subject to
an integral equation of the type featured in Eq. (4.26). Solving this integral equation
in an iterative way generates a perturbation expansion (see Eq. 4.27). In the present
case, where the time-dependent part of the hamiltonian is given by Eq. (A.7.1), the
expansion through first order reads
ˆ
U (t, t 0 ) = ˆ
1 − i
t
t 0
ˆ
B I (t
) f (t
) dt
+ . . .
(A.7.6)
Appendix
A.7 Linear Response Theory and the Polarization
Propagator
The polarization propagator discussed in Chap. 13 is closely related to polarizabilities
or, more general, to the linear response of properties, as, e.g., dipole moments, to a
time-dependent “external” perturbation [5]. In this Appendix, we briefly outline that
relation and show how the ADC approach can be used to treat linear response properties. For a more comprehensive presentation, general references, and a discussion
of the computational performance, the reader is referred to Ref. [16].
Let ˆ
H x (t) denote the hamiltonian for an external perturbation,
ˆ
H x (t) = ˆ
B f (t)
(A.7.1)
where
ˆ
B =
B rs c
†
r c s
(A.7.2)
is a (constant) one-particle operator and f (t) a time-dependent function describing
the time-dependence of the perturbation. The perturbation is assumed to be turned
on at t 0 ; that is, f (t) = 0 for t ≤ t 0 . The full hamiltonian is given by
ˆ
H
(t) = ˆ
H + ˆ
H x (t)
(A.7.3)
where ˆ
H is the hamiltonian of the unperturbed system. Initially, that is, at and prior
to t 0 , the system is assumed to be in the ground state | 0 of ˆ
H .
To construct the solution of the TDSE, we may resort to the procedure presented
in Sect. 4.1, where now ˆ
H (rather than ˆ
H 0 ) is the time-independent part of the full
hamiltonian. As in Eq. (4.13), the time-dependent wave function can be written in
the form
|(t) = e
−i ˆ
Ht
| x (t)
(A.7.4)
where | x (t) denotes the wave function in the interaction picture with ˆ
H acting as
the unperturbed part of the hamiltonian. Analogous to Eq. (4.19), we may write
| x (t) = ˆ
U (t, t 0 )| 0
(A.7.5)
where the time-evolution operator in the interaction picture, ˆ
U (t, t 0 ), is subject to
an integral equation of the type featured in Eq. (4.26). Solving this integral equation
in an iterative way generates a perturbation expansion (see Eq. 4.27). In the present
case, where the time-dependent part of the hamiltonian is given by Eq. (A.7.1), the
expansion through first order reads
ˆ
U (t, t 0 ) = ˆ
1 − i
t
t 0
ˆ
B I (t
) f (t
) dt
+ . . .
(A.7.6)