5.6 Time-Dependent Exterior Complex Scaling
In Sect. 3.5 we introduced exterior complex scaling as an extension of complex
scaling where the transformation is only applied outside a certain region. It was
noted that it shares an important feature with the global transformation: it naturally
imposes outgoing boundary conditions on the Schr€ odinger equation. We discuss
here to what extent this property applies to the time-dependent case.
Let us consider a scaling transformation similar to those illustrated in Fig. 3. We
further select a path on the real axis deep into region A that departs for the complex
plane at some point close to the boundary and eventually reaches the asymptotic
form r ! re
iθ . Following this scaling transformation, the time-dependent
Schr€ odinger equation can be formally cast into a set of equations:
i
∂ψ θ r; t
ð Þ
∂t
¼ ^
H
ECS
θ
t
ð Þψ θ r; t
ð Þ
ð69Þ
Ài
∂ψ θ r; t
ð Þ
∂t
¼ ^
H
ECS
θ
t
ð Þψ θ r; t
ð Þ:
ð70Þ
for left ψ θ r; t
ð Þ and right states ψ θ (r,t), where H ˆECS
θ (t) represents the scaled
Hamiltonian. Extrapolating from the discussion in Sect. 4.4 we can interpret (69)
as imposing purely outgoing boundary conditions and (70) as the incoming
counterpart.
In the theory of complex scaling, the calculation of the expectation value of an
observable O ˆ θ on the scaled path as of (10) involves left and right states on an
equal footing. This extends to the time-dependent case with the requirement of
having both left and right states at the same time to calculate O ˆ θ . Hence, we need,
in principle, to solve (69) and (70) simultaneously.
The fact that the scaling path lies exactly on the real axis in a certain region
simplifies the equations. In fact, on the real axis, left and right states are complex
conjugates: ψ θ r; t
ð Þ ψ Àθ r; t
ð Þ
½
* ¼ ψ θ r; t
ð Þ
* for r in the interior region. This is
particularly true when the system contains only a local potential and the propagation is initialized with a state localized in the unscaled region at t ¼ 0 and
propagating outward. If we restrict ourselves to observables in the unscaled region
and we want to describe a purely outgoing process, we resolve to use the right state
ψ θ (r,t) only. This state can be obtained by propagating (69) which involves only
right states [38]. Following this, most applications of exterior complex scaling are
limited to a use with the decaying right states and observables evaluated in the
unscaled region.
Equation (69) perfectly describes problems where imposing purely outgoing
boundary conditions represents an exact condition similar to that, for example, in
ionization processes. In those cases it can be regarded as equivalent to a transparent
boundary condition described with a Green function. Here, because we are dealing
with purely outgoing conditions, we should note that the title of perfect absorber is
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
253
In Sect. 3.5 we introduced exterior complex scaling as an extension of complex
scaling where the transformation is only applied outside a certain region. It was
noted that it shares an important feature with the global transformation: it naturally
imposes outgoing boundary conditions on the Schr€ odinger equation. We discuss
here to what extent this property applies to the time-dependent case.
Let us consider a scaling transformation similar to those illustrated in Fig. 3. We
further select a path on the real axis deep into region A that departs for the complex
plane at some point close to the boundary and eventually reaches the asymptotic
form r ! re
iθ . Following this scaling transformation, the time-dependent
Schr€ odinger equation can be formally cast into a set of equations:
i
∂ψ θ r; t
ð Þ
∂t
¼ ^
H
ECS
θ
t
ð Þψ θ r; t
ð Þ
ð69Þ
Ài
∂ψ θ r; t
ð Þ
∂t
¼ ^
H
ECS
θ
t
ð Þψ θ r; t
ð Þ:
ð70Þ
for left ψ θ r; t
ð Þ and right states ψ θ (r,t), where H ˆECS
θ (t) represents the scaled
Hamiltonian. Extrapolating from the discussion in Sect. 4.4 we can interpret (69)
as imposing purely outgoing boundary conditions and (70) as the incoming
counterpart.
In the theory of complex scaling, the calculation of the expectation value of an
observable O ˆ θ on the scaled path as of (10) involves left and right states on an
equal footing. This extends to the time-dependent case with the requirement of
having both left and right states at the same time to calculate O ˆ θ . Hence, we need,
in principle, to solve (69) and (70) simultaneously.
The fact that the scaling path lies exactly on the real axis in a certain region
simplifies the equations. In fact, on the real axis, left and right states are complex
conjugates: ψ θ r; t
ð Þ ψ Àθ r; t
ð Þ
½
* ¼ ψ θ r; t
ð Þ
* for r in the interior region. This is
particularly true when the system contains only a local potential and the propagation is initialized with a state localized in the unscaled region at t ¼ 0 and
propagating outward. If we restrict ourselves to observables in the unscaled region
and we want to describe a purely outgoing process, we resolve to use the right state
ψ θ (r,t) only. This state can be obtained by propagating (69) which involves only
right states [38]. Following this, most applications of exterior complex scaling are
limited to a use with the decaying right states and observables evaluated in the
unscaled region.
Equation (69) perfectly describes problems where imposing purely outgoing
boundary conditions represents an exact condition similar to that, for example, in
ionization processes. In those cases it can be regarded as equivalent to a transparent
boundary condition described with a Green function. Here, because we are dealing
with purely outgoing conditions, we should note that the title of perfect absorber is
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
253