In general, the problem of finding the Green function for an arbitrary system is hard
to solve. However, including in A most of the atomic and molecular structure leaves
us in B with a problem which, in many cases, can be easily solved.
The simplest case consists of choosing B to represent the empty space, and the
method lends itself to the description of scattering or ionization [62, 63]. On a more
advanced level, one may choose B to represent a bulk system and, in conjunction
with a time-dependent potential, create a base model for electron transport [64,
65]. Alternatively, by mixing both bulk and empty space Green functions, the
frameworks can adapt to the description of ionization from surfaces [66, 67].
The approach is adaptable to a large variety of situations. This versatility has,
however, to face the fact that discretizing the otherwise exact equations often leads
to computationally demanding implementations with limited application. On the
practical level, either one introduces an approximation which affects the quality of
the results, or one just uses a simple time propagation of a full-dimensional system,
which represents a challenging task [68].
In spite of the technical limitations, the approach provides a fundamental and
illustrative description of the open boundary problem. Below we discuss two of the
most notable derivations present in the literature.
5.2 Time-Dependent Embedding
The original Green function embedding was developed in the context of surface and
solid state physics for the static Schr€ odinger equation by Inglesfield [69]. It was
subsequently extended to the time-dependent case in Inglesfield [67, 70] by the
same author, but similar derivations have been proposed earlier in different fields,
for instance, to describe the interaction of a strong laser with atoms in Boucke
et al. [62] and Ermolaev et al. [63], and for electron transport in Hellums and
Frensley [64].
Below we introduce the theory following an approach similar to the one used to
describe molecular transport with TDDFT by Kurth [65].
4 We first restrict ourselves to the single-electron case and then discuss the extension to the manyelectron one with TDDFT.
Let us consider the case of a system in contact with a reservoir as shown in
Fig. 9. We want to find a closed set of conditions that have to be imposed on the
equations for a wavefunction in A such that it correctly matches its outer part in
B for all times. Following the division in the figure, we can write the timedependent Schr€ odinger equation for the system A coupled with a reservoir/environment B using a block matrix representation:
4 An analogous approach was first presented by Hellums [64] in a single-particle picture.
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
245
to solve. However, including in A most of the atomic and molecular structure leaves
us in B with a problem which, in many cases, can be easily solved.
The simplest case consists of choosing B to represent the empty space, and the
method lends itself to the description of scattering or ionization [62, 63]. On a more
advanced level, one may choose B to represent a bulk system and, in conjunction
with a time-dependent potential, create a base model for electron transport [64,
65]. Alternatively, by mixing both bulk and empty space Green functions, the
frameworks can adapt to the description of ionization from surfaces [66, 67].
The approach is adaptable to a large variety of situations. This versatility has,
however, to face the fact that discretizing the otherwise exact equations often leads
to computationally demanding implementations with limited application. On the
practical level, either one introduces an approximation which affects the quality of
the results, or one just uses a simple time propagation of a full-dimensional system,
which represents a challenging task [68].
In spite of the technical limitations, the approach provides a fundamental and
illustrative description of the open boundary problem. Below we discuss two of the
most notable derivations present in the literature.
5.2 Time-Dependent Embedding
The original Green function embedding was developed in the context of surface and
solid state physics for the static Schr€ odinger equation by Inglesfield [69]. It was
subsequently extended to the time-dependent case in Inglesfield [67, 70] by the
same author, but similar derivations have been proposed earlier in different fields,
for instance, to describe the interaction of a strong laser with atoms in Boucke
et al. [62] and Ermolaev et al. [63], and for electron transport in Hellums and
Frensley [64].
Below we introduce the theory following an approach similar to the one used to
describe molecular transport with TDDFT by Kurth [65].
4 We first restrict ourselves to the single-electron case and then discuss the extension to the manyelectron one with TDDFT.
Let us consider the case of a system in contact with a reservoir as shown in
Fig. 9. We want to find a closed set of conditions that have to be imposed on the
equations for a wavefunction in A such that it correctly matches its outer part in
B for all times. Following the division in the figure, we can write the timedependent Schr€ odinger equation for the system A coupled with a reservoir/environment B using a block matrix representation:
4 An analogous approach was first presented by Hellums [64] in a single-particle picture.
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
245