through each orbital and the full many-body Green function G[n]. The total
embedding operator can thus be interpreted as the coupling functional ν
B [n] with
the environment. Obviously, this connection involving the full many-body Green
function is of little use in practical situations, but it provides a clear starting point
for further approximations.
5.3 Absorbing Boundaries
Describing charge transfer between a system and its environment implies a modification of the isolated Hamiltonian. In the previous section we showed how the exact
condition requires the addition of an embedding operator ^
ℰ ψ A
½ t
ð Þ that turns the
Hamiltonian non-Hermitian. The evaluation of such an operator can, however, be
very demanding and one needs to resort to simpler strategies.
Absorbing boundaries (ABs) or boundary absorbers are cheaper options. They
can be defined as any approximation of the form
^
H AB ψ A t
ð Þ
½
t
ð Þ % ^
ℰ ψ A
½ t
ð Þ
ð63Þ
to an embedding operator such as the one given by (60) or (61). This approximation
is specific to the case where B represents the empty space and we only have to
absorb outgoing electrons. We know that ^
ℰ ψ A
½ t
ð Þ can be spatially localized on the
boundary surface. The absorbing boundary operator is instead generally allowed to
act on the wavefunctions over a larger region close to the boundaries, as illustrated
in Fig. 10. In the large majority of approximations, this operator is taken to be a
local potential:
^
H AB ψ A t
ð Þ
½
t
ð Þ¼ ^
V AB t
ð Þψ A t
ð Þ:
ð64Þ
Its purpose is to absorb completely any outgoing wave packet entering the region
(striped in the figure) of its support. The main goal here is to apply the absorber that
best simulates the exact embedding operator with the minimum computational cost.
From a TDDFT perspective, when we apply H ˆ
AB to each Kohn–Sham orbital,
on top of all the approximations which might be involved in the description of the
embedding operator, we are also approximating the interaction between the system
and the environment by setting it to zero.
The absorbing properties of a boundary depend strongly on the numerical
implementation. We do not enter any specific implementation here but just point
out the fact that none of the absorbers presented in the literature are completely free
from reflections. We refer to De Giovannini et al. [74] for a recent review on the
reflection properties of members of each boundary family.
We discuss below two of the most popular families of absorbing boundaries: the
complex absorbing potentials (CAPs) and the mask function absorbers (MFAs).
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
249
embedding operator can thus be interpreted as the coupling functional ν
B [n] with
the environment. Obviously, this connection involving the full many-body Green
function is of little use in practical situations, but it provides a clear starting point
for further approximations.
5.3 Absorbing Boundaries
Describing charge transfer between a system and its environment implies a modification of the isolated Hamiltonian. In the previous section we showed how the exact
condition requires the addition of an embedding operator ^
ℰ ψ A
½ t
ð Þ that turns the
Hamiltonian non-Hermitian. The evaluation of such an operator can, however, be
very demanding and one needs to resort to simpler strategies.
Absorbing boundaries (ABs) or boundary absorbers are cheaper options. They
can be defined as any approximation of the form
^
H AB ψ A t
ð Þ
½
t
ð Þ % ^
ℰ ψ A
½ t
ð Þ
ð63Þ
to an embedding operator such as the one given by (60) or (61). This approximation
is specific to the case where B represents the empty space and we only have to
absorb outgoing electrons. We know that ^
ℰ ψ A
½ t
ð Þ can be spatially localized on the
boundary surface. The absorbing boundary operator is instead generally allowed to
act on the wavefunctions over a larger region close to the boundaries, as illustrated
in Fig. 10. In the large majority of approximations, this operator is taken to be a
local potential:
^
H AB ψ A t
ð Þ
½
t
ð Þ¼ ^
V AB t
ð Þψ A t
ð Þ:
ð64Þ
Its purpose is to absorb completely any outgoing wave packet entering the region
(striped in the figure) of its support. The main goal here is to apply the absorber that
best simulates the exact embedding operator with the minimum computational cost.
From a TDDFT perspective, when we apply H ˆ
AB to each Kohn–Sham orbital,
on top of all the approximations which might be involved in the description of the
embedding operator, we are also approximating the interaction between the system
and the environment by setting it to zero.
The absorbing properties of a boundary depend strongly on the numerical
implementation. We do not enter any specific implementation here but just point
out the fact that none of the absorbers presented in the literature are completely free
from reflections. We refer to De Giovannini et al. [74] for a recent review on the
reflection properties of members of each boundary family.
We discuss below two of the most popular families of absorbing boundaries: the
complex absorbing potentials (CAPs) and the mask function absorbers (MFAs).
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
249