decays as governed by the damping force, and the rate of decay is proportional to
the width of the resonance peak. Second, if we consider the phase of the oscillation
of the system with respect to that of the external force, we see that it shifts quickly
by up to π as the energy passes that of the resonance. The rate with which it shifts is
inversely proportional to the decay rate.
We mention here a few commonly studied types of resonance in atomic,
molecular, and condensed-matter physics:
• Plasmon resonances where the whole electron charge density in a material
resonates with incoming light. Surface plasmon resonances are central to the
field of plasmonics.
• Scattering resonances where incident electrons interact with an atom or molecule. Near a resonance energy, the electrons couple strongly and the scattering
cross-section shows a peak. The process may be understood as the incoming
electron becoming temporarily trapped in a metastable state before escaping.
• Asymmetric Fano resonances [10]. These occur when two coupled excitation
pathways interfere with each other.
• Autoionizing resonances, wherein a system such as an atom or molecule is
unstable with respect to the ejection of one or more electrons. These are similar
to those that would be observed in time-resolved spectroscopies and electron
scattering experiments as mentioned above.
• Electron transport processes with molecular junctions, where a bias voltage
causes electrons to jump from one metallic lead across a metastable state at a
molecule, then escapes through another lead. Such processes have, for example,
been studied using DFT plus non-equilibrium Green functions represented with
atomic basis sets [11–14].
• Adsorption of an atom onto a surface where the continuum states of the surface
couple with the discrete atomic states which then become unstable, broadening
into resonances. The Newns–Anderson model [15] describes this process for a
one-electron adsorbate.
There are many further classes of resonance which we do not mention here.
Below we consider only a small class of resonances, namely scattering or
autoionizing ones. In this context, a resonance is a metastable quantum mechanical
state that the system possesses, and which can be associated with a wavefunction.
Below we describe some mathematical properties of such resonances, with the
objective of eventually calculating them from static or time-dependent DFT.
2.1 Definition and Properties of Resonant States
Let us consider a typical scattering experiment where an incoming electron is
captured by atom and is temporarily trapped before it escapes again. Whereas
scattering processes are clearly time-dependent, resonances can nevertheless be
captured from time-independent methods as static properties of the system. A
conceptually simple method is to study the phase δ of the wavefunction in the
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
223
the width of the resonance peak. Second, if we consider the phase of the oscillation
of the system with respect to that of the external force, we see that it shifts quickly
by up to π as the energy passes that of the resonance. The rate with which it shifts is
inversely proportional to the decay rate.
We mention here a few commonly studied types of resonance in atomic,
molecular, and condensed-matter physics:
• Plasmon resonances where the whole electron charge density in a material
resonates with incoming light. Surface plasmon resonances are central to the
field of plasmonics.
• Scattering resonances where incident electrons interact with an atom or molecule. Near a resonance energy, the electrons couple strongly and the scattering
cross-section shows a peak. The process may be understood as the incoming
electron becoming temporarily trapped in a metastable state before escaping.
• Asymmetric Fano resonances [10]. These occur when two coupled excitation
pathways interfere with each other.
• Autoionizing resonances, wherein a system such as an atom or molecule is
unstable with respect to the ejection of one or more electrons. These are similar
to those that would be observed in time-resolved spectroscopies and electron
scattering experiments as mentioned above.
• Electron transport processes with molecular junctions, where a bias voltage
causes electrons to jump from one metallic lead across a metastable state at a
molecule, then escapes through another lead. Such processes have, for example,
been studied using DFT plus non-equilibrium Green functions represented with
atomic basis sets [11–14].
• Adsorption of an atom onto a surface where the continuum states of the surface
couple with the discrete atomic states which then become unstable, broadening
into resonances. The Newns–Anderson model [15] describes this process for a
one-electron adsorbate.
There are many further classes of resonance which we do not mention here.
Below we consider only a small class of resonances, namely scattering or
autoionizing ones. In this context, a resonance is a metastable quantum mechanical
state that the system possesses, and which can be associated with a wavefunction.
Below we describe some mathematical properties of such resonances, with the
objective of eventually calculating them from static or time-dependent DFT.
2.1 Definition and Properties of Resonant States
Let us consider a typical scattering experiment where an incoming electron is
captured by atom and is temporarily trapped before it escapes again. Whereas
scattering processes are clearly time-dependent, resonances can nevertheless be
captured from time-independent methods as static properties of the system. A
conceptually simple method is to study the phase δ of the wavefunction in the
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
223