Bound-state-type resonance

The bound-state type of approach that was followed by Hylleraas—albeit fraught with a couple of erroneous conclusions—eventually evolved in the 1960s into methods aiming at the calculation and identification of resonance states from the behavior of roots of diagonalized energy matrices constructed from discrete basis sets, as a function of a scaling parameter, or size of discrete basis sets, or size of the artificial box inside which the continuous spectrum is artificially discretized. 

The wavepacket is propagated until a time where it is all scattered and is away from the interaction region. This time is short (typically 10-100 fs) for a direct reaction. Flowever, for some types of systems, e.g. for reactions with wells, the system can be trapped in resonances which are quasi-bound states (see section B3.4.7). There are eflScient ways to handle time-dependent scattering even with resonances, by propagating for a short time and then extracting the resonances and adding their contribution [69]. 

Feshbach-type resonances [51], also known as Fano resonances [52] and Floquet resonances [22] depending on the system studied, are formed in a different manner. We encounter this type of metastable states whenever a bound system is coupled to an external continuum. In the same spirit as before, one can define a reference Hamiltonian in which the closed channel containing the bound states is uncoupled from the open channel through which the asymptote can be reached. When the coupling is introduced, the previously bound state decays into the continuum of the open channel. The distinction from shape-type resonances, described above, is that the resonance state decays into a different channel of the reference Hamiltonian. 

Partitioning technique refers to the division of data into isolated sections and it was put into successful practice in connection with matrix operations. Lowdin, in his pioneering studies, [21, 22] developed standard finite dimensional formulas into general operator transformations, including treatments appropriate for both the bound state and the continuous part of the spectrum, see also details in later appendices. Complementary generalizations to resonance-type problems were initiated in Ref. [23], and simple variational formulations were demonstrated in Refs. [24,25]. Note that analogous forms were derived for the Liouville equation [26, 27] and further developed in connection with a retarded-advanced subdynamics formulation [28]. 

The present volume of the Advances in Quantum Chemistry is the sequel of the first volume, mentioned above, i.e., Unstable States in the Continuous Spectra, Part II Interpretation, Theory and Applications. It contains six chapters with contents varying from a pedagogical introduction to the notion of unstable states to the presence and role of resonances in chemical reactions, from discussions on the foundations of the theory to its relevance and precise limitations in various fields, from electronic and positronic quasi-bound states and their role in certain types of reactions to applications in the field of electronic decay in multiply charged molecules and clusters, as well. 

Resonances of the type illustrated in Figure 12.2 are called Feshbach resonances (Child 1974 ch.4 Fano and Rao 1986 ch.8 see also Figure 12.5). The quasi-bound states trapped by the Vn(.R) potential can only decay via coupling to the lower vibrational state because asymptotically the n = 1 channel is closed and therefore cannot be populated. This is different from the dissociation of CH30N0(Si), for example, [see Figure 7.10(a)] where the resonances can either decay via tunneling or alternatively by nonadiabatic coupling to the lower states. 

The second type of predissociation observed for diatomic molecules is known as electronic predissociation the principles are illustrated in figure 6.28. A vibrational level v of a bound state E lies below the dissociation asymptote of that state, but above the dissociation asymptote of a second state E2. This second state, E2, is a repulsive state which crosses the bound state E as shown. The two states are mixed, and the level v can predissociate via the unbound state. It is not, in fact, necessary for the potential curves of the two states to actually cross. It is, however, necessary that they be mixed and there are a number of different interaction terms which can be responsible for the mixing. We do not go into the details here because electronic predissociation, though an important phenomenon in electronic spectroscopy, seldom plays a role in rotational spectroscopy. Since it involves excited electronic states it could certainly be involved in some double resonance cases. 

As discussed above, continuum resonance occurs when the excitation laser energy is higher than the dissociation limit of an excited, bound electronic state or directly with purely repulsive states. Continuum resonance Raman spectra of gaseous molecules are very sensitive to the position and shape of the potential functions involved in this type of light scattering as well as to the electronic transition moments between ground and excited states. Since it is possible to calculate the relevant spectra using both the KHD... 

Figure 13 Calculated absorption-type spectrum for DCO. Energy normalization is such that E = 0 corresponds to D-I-CO with CO at equilibrium. The dotted line marks the quantum mechanical threshold and the numbers indicate the pure CO stretching states (0, U2,0). There are 29 bound states. Because of the logarithmic scale, the Lorentzian resonance profiles have an unusual shape. Reproduced, with permission of the American Institute of Physics, from Ref. 15.

The case of decay by energy transfer akin to Feshbach-type resonance(s) in collision theory, presents a probe of greater sensitivity to bound-state dynamics. This is clearly exemplified by the doorway channel model.69 In this model, the bound manifold is coupled to a dissociative manifold via a single ( doorway ) channel. This forces the system, if excited to some arbitrary state, to diffuse to the subspace spanned by the doorway channel in order to dissociate. 

It follows from the above discussion that considering excitons and intramolecular phonons as independent particles is approximate. It is clear, in particular, that a sufficiently strong exciton-phonon interaction can create propagating bound states, when electronic and vibronic excitations are centered on the same molecule. These states correspond to the previously discussed weak resonant interaction case. But the existence of such states in the vibronic spectrum does not exclude the existence of free excitons and intramolecular phonons states. Both types of states can usually coexist in the vibronic spectrum, in analogy to the case of two interacting particles, where continuum states, corresponding to free particles, coexist with bound states. 

In the preceding section it was shown that the formation of bound states of phonons leads to the appearance of a new type of resonance of the dielectric tensor ij(co). It is clear, of course (23), that the nonlinear polarizabilities should have analogous resonances, and this also concerns, besides biphonons, other types of bound states of quasiparticles, such as biexcitons, electron-exciton complexes, etc. 

An investigation of the contribution of the bound states of quasiparticles to the nonlinear polarizabilities is of interest for many reasons. The main ones are the new opportunities for studying the properties of bound states, as well as the gigantic values of the nonlinear polarizabilities that can be reached, precisely as a result of the new type of resonances. 

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