7 Summary
We have discussed a selection of methods which in different ways allow the
calculation of properties of open quantum systems, with the objective of describing
electron emission processes.
In scattering experiments and spectroscopy, the concept of resonances is of
particular importance, and we have taken care to describe in detail the complex
scaling method which, by a transformation of the real-space coordinates, causes the
exponentially divergent resonant states to localize and become representable as
square integrable states which emerge as eigenstates of the transformed,
non-Hermitian Hamiltonian. This, to a large extent, makes them accessible using
standard bound-state methods. The extension of ordinary ground-state DFT with
complex scaling allows for a computationally tractable means of extracting resonant states and properties such as energy and lifetimes in many-body systems.
Although resonances can be captured from static calculations reminiscent of
ground-state DFT, truly dynamic processes require explicit time propagation
approaches. We have subsequently examined several methods used to describe
dynamics leading to electron emission. We have studied these methods from the
perspective of a complex system A which is in contact with a different system B that
acts as a reservoir. Representing the wavefunction in B by means of Green functions
provides a flexible way of accessing the full open-boundary problem, allowing
transfer of particles into and out of a system. Using Green function embedding, one
can calculate the wavefunctions in A that automatically satisfy the boundary
conditions emulating their contact with B. However such embedding techniques
suffer the disadvantage of being computationally demanding when employed to
solve fully three-dimensional problems with first-principles methods.
Absorbing boundaries provide more computationally practical ways of
accessing ionization processes in which charge leaves the system. We have considered absorbing boundaries and mask functions which are simple methods to
absorb outgoing waves in time-dependent simulations. The boundary absorbers
are meant to absorb waves that leave the system, so that an outgoing wave
disappears rather than reflects on the simulation box. The complex scaling method
provides a particularly elegant way to absorb outgoing waves, allowing one,
in principle, to impose perfectly absorbing boundaries.
Having discussed the problem of describing total ionization with the appropriate
choice of boundary conditions, we turned to the problem of describing electron
photoemission probabilities with TDDFT. We examined three approaches suitable
for the task. The sampling point method, where the energy-resolved probability is
calculated by Fourier transforming the time evolution of each Khon–Sham orbital
in the energy domain; the surface flux method, where the photoelectron probability
is generated by recording the electron flux through a closed surface surrounding the
system; and finally we discussed the mask method where, by means of a mask
function, it is possible to generate a split real/momentum-space propagation scheme
where electrons, moving from a bounded volume into the empty space, seamlessly
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
265
We have discussed a selection of methods which in different ways allow the
calculation of properties of open quantum systems, with the objective of describing
electron emission processes.
In scattering experiments and spectroscopy, the concept of resonances is of
particular importance, and we have taken care to describe in detail the complex
scaling method which, by a transformation of the real-space coordinates, causes the
exponentially divergent resonant states to localize and become representable as
square integrable states which emerge as eigenstates of the transformed,
non-Hermitian Hamiltonian. This, to a large extent, makes them accessible using
standard bound-state methods. The extension of ordinary ground-state DFT with
complex scaling allows for a computationally tractable means of extracting resonant states and properties such as energy and lifetimes in many-body systems.
Although resonances can be captured from static calculations reminiscent of
ground-state DFT, truly dynamic processes require explicit time propagation
approaches. We have subsequently examined several methods used to describe
dynamics leading to electron emission. We have studied these methods from the
perspective of a complex system A which is in contact with a different system B that
acts as a reservoir. Representing the wavefunction in B by means of Green functions
provides a flexible way of accessing the full open-boundary problem, allowing
transfer of particles into and out of a system. Using Green function embedding, one
can calculate the wavefunctions in A that automatically satisfy the boundary
conditions emulating their contact with B. However such embedding techniques
suffer the disadvantage of being computationally demanding when employed to
solve fully three-dimensional problems with first-principles methods.
Absorbing boundaries provide more computationally practical ways of
accessing ionization processes in which charge leaves the system. We have considered absorbing boundaries and mask functions which are simple methods to
absorb outgoing waves in time-dependent simulations. The boundary absorbers
are meant to absorb waves that leave the system, so that an outgoing wave
disappears rather than reflects on the simulation box. The complex scaling method
provides a particularly elegant way to absorb outgoing waves, allowing one,
in principle, to impose perfectly absorbing boundaries.
Having discussed the problem of describing total ionization with the appropriate
choice of boundary conditions, we turned to the problem of describing electron
photoemission probabilities with TDDFT. We examined three approaches suitable
for the task. The sampling point method, where the energy-resolved probability is
calculated by Fourier transforming the time evolution of each Khon–Sham orbital
in the energy domain; the surface flux method, where the photoelectron probability
is generated by recording the electron flux through a closed surface surrounding the
system; and finally we discussed the mask method where, by means of a mask
function, it is possible to generate a split real/momentum-space propagation scheme
where electrons, moving from a bounded volume into the empty space, seamlessly
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
265