Aspuru-Guzik [5–7]. This group also proposed a description of open quantum
systems in terms of a unitarily evolving closed Kohn–Sham system [7, 8]. The
theory of open quantum systems (OQS) mostly deals with the situation where the
environment exchanges energy and momentum with the system but particle number
is conserved ([9, Chap. 10]). What happens in the case when the environment
exchanges particles with the system is an equivalently important problem which
has been less developed. Here we intend to review methods developed to address
this kind of problem. We describe the theoretical frameworks and approximations
that can be used to describe particle exchange.
Solving the problem of describing a system which exchanges electrons with the
environment is only half the challenge. In fact, even in the ideal case where one is
able to calculate the correct time-dependent wavefunction, one is faced with the
additional problem that some observables may require the knowledge of the
complete wavefunction or of eigenstates in the continuum. This includes ionization
products such as photoelectron spectra and resonance lifetimes/widths, and is also
connected to the measurement process of an open system. These problems are even
more severe in the case of DFT and TDDFT, where the density is the only physical
object, and where finding the explicit density-functional linking to a physical
observable is a daunting task.
This review is structured as follows. We first introduce the general concept of
resonance in Sect. 2 and describe how it can be observed in many different physical
situations. Then in Sect. 3 we introduce the reader to the basic concepts of the
complex scaling theory which is one of the most important tools for studying shape
resonances in a static framework. In Sect. 4 we review the successful extension of
the complex scaling theory to the realm of DFT, including some recent work
adapting the method to the time-dependent realm. In Sect. 5 we review several
methods for the incorporation of boundary conditions with the TDDFT equations in
order to include the dynamic exchange of electrons with an environment/reservoir.
We discuss the strategies for describing specific observables in Sect. 6 where we
focus on the case of electron photoemission.
Unless otherwise specified, atomic units are used throughout
( h ¼ m e ¼ e ¼ 4πε 0 ¼ 1).
2 Resonances
Consider a system acted upon by an external oscillating force characterized by
some energy and corresponding frequency. If the system responds particularly
strongly close to a particular frequency, we call that a resonance process. The
typical textbook case is that of a classical damped harmonic oscillator acted upon
by an external sinusoidal force. For each frequency the system responds by oscillating with some amplitude, and the resonances appear as strong narrow peaks in
the amplitude.
This simple model has two important properties that are very general to any type
of resonance: First, if the oscillatory force is turned off, the resonant oscillation
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A.H. Larsen et al.
systems in terms of a unitarily evolving closed Kohn–Sham system [7, 8]. The
theory of open quantum systems (OQS) mostly deals with the situation where the
environment exchanges energy and momentum with the system but particle number
is conserved ([9, Chap. 10]). What happens in the case when the environment
exchanges particles with the system is an equivalently important problem which
has been less developed. Here we intend to review methods developed to address
this kind of problem. We describe the theoretical frameworks and approximations
that can be used to describe particle exchange.
Solving the problem of describing a system which exchanges electrons with the
environment is only half the challenge. In fact, even in the ideal case where one is
able to calculate the correct time-dependent wavefunction, one is faced with the
additional problem that some observables may require the knowledge of the
complete wavefunction or of eigenstates in the continuum. This includes ionization
products such as photoelectron spectra and resonance lifetimes/widths, and is also
connected to the measurement process of an open system. These problems are even
more severe in the case of DFT and TDDFT, where the density is the only physical
object, and where finding the explicit density-functional linking to a physical
observable is a daunting task.
This review is structured as follows. We first introduce the general concept of
resonance in Sect. 2 and describe how it can be observed in many different physical
situations. Then in Sect. 3 we introduce the reader to the basic concepts of the
complex scaling theory which is one of the most important tools for studying shape
resonances in a static framework. In Sect. 4 we review the successful extension of
the complex scaling theory to the realm of DFT, including some recent work
adapting the method to the time-dependent realm. In Sect. 5 we review several
methods for the incorporation of boundary conditions with the TDDFT equations in
order to include the dynamic exchange of electrons with an environment/reservoir.
We discuss the strategies for describing specific observables in Sect. 6 where we
focus on the case of electron photoemission.
Unless otherwise specified, atomic units are used throughout
( h ¼ m e ¼ e ¼ 4πε 0 ¼ 1).
2 Resonances
Consider a system acted upon by an external oscillating force characterized by
some energy and corresponding frequency. If the system responds particularly
strongly close to a particular frequency, we call that a resonance process. The
typical textbook case is that of a classical damped harmonic oscillator acted upon
by an external sinusoidal force. For each frequency the system responds by oscillating with some amplitude, and the resonances appear as strong narrow peaks in
the amplitude.
This simple model has two important properties that are very general to any type
of resonance: First, if the oscillatory force is turned off, the resonant oscillation
222
A.H. Larsen et al.