Quantum numbers vibrational predissociation

In closing the section on non-Hermitean approaches to continuum processes in atomic and molecular physics, we will also mention accurate examinations on resonance parameters in molecular predissociation displaying unexpected resonance overlapping [46,89]. The phenomenon of predissociation by rotation in HgH was analyzed via an isotopically combined potential due to Stwalley [120]. The potential, i.e., a relatively shallow energy curve with a nonzero /-value giving rise to a rotational barrier, supported novel metastable states above the dissociation limit. The Weyl s method was able to resolve the closely lying vibrational states v = 3 and v = 4 for the rotational quantum number K = 9. 

Fig. 12.5. Zeroth-order potentials Veff(R-,j,Sl,J) defined in (12.7) for fl = 0 and several total angular momentum quantum numbers J. The excited rotational states can decay either by tunneling (shape resonances) or by rotational predissociation ( Feshbach resonances) as indicated by the horizontal arrows. The excitation through the IR photon originates from the vibrational ground state n = 0 which is not shown in the figure.

Both electronic and vibrational shape resonances arise from a direct process and can be explained by a single potential (McKoy, et al., 1984). Shape resonances (single Vi(r) or Vj(R)) differ from autoionization resonances and predissociation (with the exception of predissociation by rotation), which involve two potentials or two states with different quantum numbers. 

Figure B2.3.10. Potential energy curves [42] of the ground X 11 and excited A electronic states of the hydroxyl radical. Several vibrational levels are explicitly drawn in each electronic state. One vibrational transition is explicitly indicated, and the upper and lower vibrational wavefimctions are plotted. The upper and lower state vibrational quantum numbers are denoted v and v", respectively. Also shown is one of the three repulsive potential energy curves which correlate with the ground 0( P) + H dissociation as miptote. These cause predissociation of the higher rotational and vibrational levels of the A E state. 

The collision-assisted predissociation in iodine B O + state merits a detailed discussion. It is well known that B state is weakly coupled to the dissociative A 1m state by rotational and hyperfine-structure terms in the molecular Hamiltonian. The natural predissociation rate strongly depends on the vibrational quantum number (pronounced maxima for o=5 and u = 25, a minimum for u= 15), this dependence being due to a variation of the Franck-Condon factor. " The predissociation rate is enhanced by collisions. In absence of a detailed theoretical treatment of the colhsion-assisted 12 predissociation, one can suppose that the asymmetric perturbation (breakdown of the orbital symmetry) in the collisional complex affects electronic and rotational wavefimctions but does not change the nuclear geometry. 

The vibrational bands are diffuse, the diffuseness increasing with vibrational quantum number, again indicating strong predissociation Rotational struc-... 

The widths for the predissociation of the state of HNO are much smaller than the widths shown for HCO in Fig. 18, typically smaller than 1 cm they depend markedly on the particular vibrational state and the K rotational quantum number (see Ref. 113). The widths depend, of course, also on the total rotational angular momentum quantum number J. However, because the term in the kinetic energy operator responsible for the... 

Not only do the experimental vibrational predissociation lifetimes require interpretation, so do the increasingly sophisticated theoretical calculations whose results often fall out of a web of coupled differential equations or the convoluted algebra of quantum mechanics. In order to offer a qualitative overview of dynamical processes in van der Waals molecules, we shall introduce a selection rule which can provide insight into possible relaxation channels of vibrationally excited molecules. This selection rule concerns the change in a quantum number, Anj., which is to remain small for efficient vibrational predissociation processes. It bears a close analogy to the selection rules of optical spectroscopy which require small changes in quantum numbers Au, AJ, AS, etc. for efficient transitions between molecular states. Let us review the origin of the vibrational predissociation selection rule which has been developed in more detail elsewhere. ... 

The quantum numbers for the fragments of vibrational predissociation are correlated through the same van der Waals molecule coordinates. For the radial coordinate r, the quantum number of the bound complex A-B C goes over into q, which is proportional to the translational quantum number of the fragments A-B + C. The free rotation of A-B relative to C about angle 6 is identified by the J quantum number. The quantum numbers of the vibrations involving chemical bonds have the same definitions in the fragments as in the complex. 

We are now prepared to set down the effective quantum numbers for use of the selection rule expression of eq. 5. Application of the analytical expression for vibrational predissociation rates of A-B C for a wide variety of van der Waals molecules bound by Morse intermolecular potential functions like those shown in Fig. 2 reveals the effective translational quantum number change... 

This quantum number change is essentially the difference between the effective number of nodes, q 2, of the translational wavefunction of the predissociation fragments, A-B + C, and the number of nodes, in the van der Waals stretching vibrational wavefunction of A-B ---C. The exponential dependence on these quantum numbers in eq. 5 is consistent with the poor Franck Condon type overlap expected for wavefunctions with widely differing numbers of nodes such as those shown for example by A-B C predissociation in Fig. 2. 

Figure 3. The total quantum number change, An, and lifetimes, r, for vibrational predissociation. The line is the selection rule expression of eq. 5. Experimental measurements are indicated by the open circles described in the text.

It has been shown that Fermi resonances between chemical bond vibrational levels and van der Waals modes can dramatically reduce vibrational predissociation lifetimes. The selection rule becomes altered because the definitions of the quantum numbers become blurred by the Fermi resonances. This is illustrated in the recent study of Tiller, Peet and Clary and shown in Fig. 5. Here 5 0,0,0> mixes with nearby 17 0,0,10> whose vibrational predissociation lifetime is calculated to be two orders of magnitude shorter than the prepared state. 

The predissociation lifetime of l2Hen is found to be dependent on the iodine vibrational level initially excited and for n = 1 ranges from 38 ps (v = 26) to 221 ps (v = 12) with a variation that is a nonlinear function of the iodine vibrational quantum"number v . The predissociation lifetimes for the neon hydrogen and deuterium van der Waals complexes with iodine are found to be somewhat shorter. As for the product iodine vibrational state distribution for l2Hen the A v = -n channel is dominant with a branching ratio of greater than 0.95 for v =... 

For example, let iicOi = 500 cm , a typical value. It seems plausible that the polyatom moiety s potential energy in this coordinate should change at least on the order of 0.1 cm i upon formation of a complex. Then ai(r) = (0.0004)1/2 = 0.02. Substitution of these values into Eq. (II.9) gives a matrix element of 7 cm i for Vi =1. Third, the vibrational predissociation rate constant for a one-quantum transition should vary linearly with the quantum number of the initial state. Since the energy lost to the van der Waals coordinate in a one-quantum change is independent of the initial state quantum number, the vibrational predissociation rate should vary with the square of the coupling function in Eq. (II.9). This equation indicates that the vibrational predissociation rate should then be proportional to the initial state vibrational quantum number Vi. 

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