Vibrational continuum

C22a+) state, followed by predissociation from vibrational levels, v > j 218-220 Another mechanism for collisional dissociation entails direct excitation of the ion to the vibrational continuum of the ground state. Both of these mechanisms may operate in competition in the same system.217... 

Since the velocities and angular distributions of products from collisional dissociations at low incident-ion energies have generally not been determined, the precise mechanism by which the products are formed is unknown. Thus in the collisional dissociation of H2+ with helium as the target gas, H+ may result from dissociation of H2+ that has been directly excited to the vibrational continuum... 

It is assumed that target states p are indexed for each value of q such that a smooth diabatic energy function Ep(q) is defined. This requires careful analysis of avoided crossings. The functions should be a complete set of vibrational functions for the target potential Vp = Ep, including functions that represent the vibrational continuum. All vibrational basis functions are truncated at q = qd, without restricting their boundary values. The radial functions fra should be complete for r < a. 

Bound electronic states exhibit a discrete spectrum of rovibrational eigenstates below the dissociation energy. The interaction between discrete levels of two bound electronic states may lead to perturbations in their rovibrational spectra and to nonradiative transitions between the two potentials. In the case of an intersystem crossing, this process is often followed by a radiative depletion. Above the dissociation energy and for unbound states, the energy is not quantized, that is, the spectrum is continuous. The coupling of a bound state to the vibrational continuum of another electronic state leads to predissociation. 

Figure 23 Predissociation of the v = 2 vibrational level of the bound electronic state by a vibrational continuum wave function of the dissociative electronic state after radiative excitation (arrows) from the electronic ground state Po- The circle around the potential curve crossing point indicates an area of large overlap between the vibrational wave functions.

Allowed excitation transitions are those between /V/1-states with different g, -symmetries, and most intense among them are those preserving the total spin (singlet-singlet and triplet-triplet transitions). The intermediary state in reaction (19.31) is a bound electronic state, but the transition takes place to the vibrational continuum of this state. The intermediary state in reaction (19.32) is a dissociative state that lies completely in the vibrational continuum. The reaction (19.33) is similar to reaction (19.26), except that the decay of (H ) resonance (auto-detachment) takes place in the vibrational continuum. 

The pre-dissociation results from a non-adiabatic coupling between electronic states (NA v) and (N A ec) at internuclear distances (usually smaller than the equilibrium distance) that correspond to the vibrational continuum of N A state, or when the N A state is a purely dissociative one. Autoionization and pre-dissociation probabilities for Npa(v) and Npir(v)(N > 4) states have been calculated in [68], while for the states with N < 3 they have been evaluated from experimental measurements [69,70]. Auto-ionization and pre-dissociation rates are very high ( 108s 1 for N = 3,4), and rapidly increase with increasing v. More experimental or theoretical information is needed for these processes. 

The correspondence between classical and quantum mechanics tells us that this trajectory corresponds to a resonance state with a localized wave function in the H-C-C vibration continuum. The quantum mechanical resonance state (discussed in detail in Section 15.2.4) will have nearly the same energy and be assignable with the semiclassical quantum numbers. It is also expected to have a very long lifetime as a result of the classical quasiperiodic motion. 

Interaction between a bound rotation-vibration level of one electronic state and the vibrational continuum of another electronic state (predissociation, Chapter 7). 

The state AB, often called the superexcited state, is an autoionized or resonance state. Autoionization is called preionization by Herzberg (1950). This can be justified by the analogy between preionization and predissociation. In predissociation, the interaction of a discrete state with the vibrational continuum of the nuclei allows this discrete state a finite probability of dissociation. In preionization, it is the mixing of a discrete state with the electronic continuum that provides a finite ionization probability. 

These /-levels, in addition to being weakly predissociated, are also very weakly autoionized (see also Fig. 8.23). Another example appears in the spectrum of Li2 (Chu and Wu, 1988). In many other cases, decay by predissociation can be very fast, particularly for electrostatic predissociations (see Tables 7.3 and 7.4). For example, in the spectrum of the NO molecule, most 2Il Rydberg states are predissociated by the vibrational continuum of the B2n valence state so rapidly that autoionization cannot compete (Giusti-Suzor and Jungen, 1984). 

In the limit of a vibrational continuum, the direct summation method fails, but there analytical methods become feasible. We shall show this for the simple case where the upper potential is linear (Mingardi and Siebrand,... 

In summary, the present study has shown that the measured thresholds for dissociation of NO and 02" correspond quite closely to known adiabatic dissociation energies of these ions if the initial internal energy state of the reacting ion is taken into account. The energy required for these dissociations can be derived both from the kinetic energy and from internal excitation of the ionic reactant. These processes can therefore be interpreted as involving adiabatic transitions to the vibrational continuum above the dissociation limit of the ground states or excited states of the... 

Figure 6 shows a higher resolution spectrum of the fluorescence from another excited state, l Ag (N-31, Fi) to the v=0 level of b IIu. Because these are both good Hund s case(b) states, due to small spin-orbit coupling, large rotational constants, and high J, the spectrum exhibits a P,Q,R pattern. The weaker features are due to collisional relaxation in the upper state, as in Na2 (Figure 1). The Q(N=31) line in this fluorescence scan is broader than the instrumental resolution of about 0.8 cm due to predissociation by the a Ey" vibrational continuum. The R(30) and P(32) lines are sharp. However, the collisionally relaxed Q(30) and Q(32) lines are sharp and the R and P lines adjacent to R(30) and P(32) are broad. The reason for this selectivity in the predissociation of the b II rotational levels is explained by examining Figure 7.

The initial state populates both types of resonance states of Interference between the vibrational discrete and the vibrational continuum autoionization resonances takes place although the resonance positions of the vibrationally-discrete autoionization-resonance (disc-res) states and the vibrationally-continuum autoionization-resonance states (cont-res) are very far from one another (as compared to their width). One may think that when E en(disc — res) there is only one dominant term in the series resonance expansion of tres(E) (see Eq. 33). This is however not the case. The numerators associated with the branch-cut resonances, an(cont — res), get complex values where both the real and the imaginary parts are larger than the corresponding ones of an(disc — res) by several orders of magnitude. In the numerical calculations the ampHtudes of the continuum t3q>e resonances... 

However, some details need to be explained in Fig. 4.5. First this weak vibrational continuum is superimposed on the high-frequency tail of the Boson peak, a band generally observed at relatively low frequencies between 10 cm and 100 cm in the Raman spectra of glasses. Secondly, the peaklike feature at around 108 cm is an artifact associated with the cutoff of filters used in the experiments therefore, this is not a real Raman peak. However, as evident in Fig. 4.5, the high-frequency tail of the Boson peak has a different slope for the different spectra and a clear feature can be found around 130 crar, indicating that this continuum indeed provides some structural information on the glasses. 

The idea of using B-spline basis sets for the representation of vibrational molecular wave functions emerged rapidly. For a Morse potential and a two-dimensional Henon-Heiles potential, we have assessed the efficiency of the B-splines over the conventional DVR (discrete variable representation) with a sine or a Laguerre basis sets [50]. In addition, the discretization of the vibrational continuum of energy when using the Galerkin method allows the calculation of photodissociation cross-sections in a time-independent approach. 

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