Excitation function vibrational

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

The interactions of photons with molecules are described by molecular cross-sections. For IR spectroscopy the cross-section is some two orders of magnitude smaller with respect to UV or fluorescence spectroscopy but about 10 orders of magnitude bigger than for Raman scattering. The peaks in IR spectra represent the excitation of vibrational modes of the molecules in the sample and thus are associated with the various chemical bonds and functional groups present in the molecules. The frequencies of the characteristic absorption bands lie within a relatively narrow range, almost independent of the composition of the rest of the molecule. The relative constancy of these group frequencies allows determination of the characteristic... 

Indeed, several identifiable resonance fingerprints in experimental observables were found.26-31 Concurrent theoretical simulations and analyses not only confirmed the experimental conjectures, but also provided deeper insights into the nature of this resonance state. For the integral cross-sections, a distinct step for Ec < 1 kcal/mol was observed in the reactive excitation function (i.e. the translation energy dependence of the reactive cross-section) for the HF+D product channel, whereas it is totally absent for the other DF+H product channel.26 Anomalous collision energy dependence of the HF vibration branching was also observed.28 For Ec < 1 kcal/mol more than 90% of the HF products are populated in the v = 2 state. However, as the energetic threshold for the formation of HF( / = 3) from... 

F(2P3/2) + HD, which occurs at Ec = 1.16kcal/mol, is reached, a sudden drop (growth) of the v = 2 (v = 3) branching was observed. In fact, the vibration state-specific excitation functions displayed two distinctive features a steplike feature span from 0.2 to 1 kcal/mol was detected for... 

For typical values of p, re and V, encountered in molecules, Eq. (l.ll) is an excellent approximation to the exact solution (better than l part in 109). The Morse potential is the simplest member of a family of potentials that give rise to a vibrational spectrum of the functional form E(v) = coc(v +1/2) -a>exe(v +1/2)2. This is quite realistic at lower levels of excitation. The vibrational spectrum does not however suffice, by itself, to specify the potential uniquely. The dependence of the eigenvalues on the rotational state is therefore important. For / 0 (as well as for the / = 0) the energy eigenvalues are given by... 

Excitation Eunctions of O2 and 02-Doped Ar Eilms. Resonances can be best identified by the structures they produce in excitation functions of a particular energy-loss process (i.e., the incident-electron energy dependence of the loss). Fig. 7 is reproduced from a recent study [118] of the electron-induced vibrational and electronic excitation of multilayer films of O2 condensed on the Pt(lll) surface and shows the incident electron energy dependence of major losses at the indicated film thickness and scattering angles. Also shown in this figure is the scattered electron intensity of the inelastic background... 

Notice that as one moves to higher vf values, the energy spacing between the states (Evf -Evf-i) decreases this, of course, reflects the anharmonicity in the excited state vibrational potential. For the above example, the transition to the vf = 2 state has the largest Franck-Condon factor. This means that the overlap of the initial state s vibrational wavefunction Xvi is largest for the final state s %vf function with vf = 2. 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

R. W. Field I must apologize for not being sufficiently clear about the excitation scheme we use for our acetylene experiments. Although the initial and final states are both on the acetylene X1 g surface, the final state we prepare is the result of two electronic transitions (A X followed by A —X) rather than one vibrational-rotational infrared or Raman transition. There is a profound difference between the knowledge of the excitation function needed to describe electronic versus vibrational processes. 

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

In chemiluminescence experiments such as those described previously in the experimental section, emission spectra characteristic of the excited products of ion-neutral collisions are obtained, that is, intensities of the emitted radiation as a function of wavelength. This permits identification of the electronically excited states produced in the reaction as well as determination of the relative populations of these states. In addition if the luminescence measurements are made using beam techniques, excitation functions (intensity of a given transition as a function of the translational energy of the reactants) can be measured for certain transitions. As is discussed later, some of the observed transitions exhibit translational-energy thresholds. In the emission spectra from diatomic or polyatomic product molecules, band systems are sometimes observed from which the relative importance of vibrational and rotational excitation accompanying electronic excitation may be assessed. 

When potential surfaces are available, quasiclassical trajectory calculations (first introduced by Karplus, et al.496) become possible. Such calculations are the theorist s analogue of experiments and have been quite successful in simulating molecular reactive collisions.497 Opacity functions, excitation functions, and thermally averaged rate coefficients may be computed using such treatments. Since initial conditions may be varied in these calculations, state-to-state cross sections can be obtained, and problems such as vibrational specificity in the energy release of an exoergic reaction and vibrational selectivity in the energy requirement of an endo-... 

Excitation functions measured for different ionizing electron energies importance of vibrational energy decreases with increasing ion translational energy 93... 

J(ro) is the classical counterpart of the rotational quantum number j of the fragment molecule. (Note that J(to) is defined as a dimensionless quantity.) It represents the final angular momentum of the fragment molecule as a function of all initial variables to- In the same way, we define the vibrational excitation function (Miller 1974, 1975, 1985)... 

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

The free BC oscillator is assumed to be harmonic with force constant k and equilibrium separation r the parameter e controls the coupling between the dissociation coordinate R and the vibrational coordinate r. For e = 0 (elastic limit) the equations of motion for (R, P) and (r, p) decouple and energy cannot flow from one degree of freedom to the other. As a consequence, the vibrational energy of the oscillator remains constant throughout the dissociation and the corresponding vibrational excitation function, which for zero initial momentum po is given by... 

The final vibrational state distribution essentially reflects the initial distribution of the vibrational coordinate mediated by the excitation function N(ro). 

Because of the lack of quantum mechanical interference effects classical mechanics is well suited for the treatment of direct dissociation. Very few trajectories actually suffice to construct the rotational and the vibrational excitation functions which establish the unique relation between (ro,7o) and (n,j). /(70) and N(ro) are the links between the multi-dimensional PES on one hand and the final state distributions on the other. 

The vibrational reflection principle outlined in Section 6.4 provides a simple, but yet quantitative explanation of final vibrational state distributions and their variation with the coupling strength and the total energy. The central quantity is the vibrational excitation function N(ro). It comprehensively manifests the dynamical details of the fragmentation process in the upper electronic state. Usually, one needs only very few trajectories to construct N(ro) which makes the simple classical theory outlined in Section 6.4 very efficient for calculating and understanding final state distributions. This is particularly beneficial for fitting experimental data. 

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