Rotational-vibrational quantum states

The introduction of laser technology also permitted the determination of related data, such as the distribution of rotational-vibrational quantum states of the products of the unimolecular dissociation [75, 76]. It was found that in the threshold energy region of a unimolecular dissociation,... 

In general, though, Raman spectroscopy is concerned with vibrational transitions (in a manner akin to infrared spectroscopy), since shifts of these Raman bands can be related to molecular structure and geometry. Because the energies of Raman frequency shifts are associated with transitions between different rotational and vibrational quantum states, Raman frequencies are equivalent to infrared frequencies within the molecule causing the scattering. 

The excitation of a molecule may result in a change of its electron and rotational-vibrational quantum numbers. In the adiabatic approximation," the total wavefunction of a molecule can be presented as a product of the electron wave and the rovibrational wavefunction. In those cases where the former is weakly affected by the changes in the relative position of the nuclei (this is usually the case with lower vibrational levels), we can use the Condon approximation considering the electron wavefunction only at equilibrium configuration of the nuclei. In this case the oscillator strength factorizes into an electron oscillator strength and the so-called Frank-Condon factor, which is the overlap integral of the vibrational wavefunctions of the initial and the final states of the molecule.115,116... 

Fig. 4.1.13(a-c) shows partial cross-sections for reactions with the reactant molecules in vibrational quantum states n = 0,1,2 and rotational quantum state J = 0 and products in vibrational states n = 0,1,2, respectively, and any rotational quantum state. Note that the abscissa axis in this plot is the translational energy and not the total energy as in Fig. 4.1.12. The translational energy is found in the latter plot by subtracting the molecular energy En

Fig. 4.1.14 shows another example of the calculation of partial cross-sections for product molecules in specified rotational quantum states J and any vibrational quantum state. 

The applications of lasers in kinetic studies are essentially twofold. Firstly, they can be used to produce a particular species. This might be a vibration—rotationally defined quantum state of a molecule [21], or it could be an ion [22—24] or fragment [25—28] produced by photoionization or photodissociation [29, 30] of some parent. The combination of specific frequency, short pulse duration and high powers makes selective control of chemical reactions possible. Secondly, they can be used as detectors of specific species and quantum states [31, 32]. There are a number of different methods of using lasers to detect small concentrations of a species in a chemical reaction. Lin and McDonald [33] have broadly reviewed the generation and detection of reactive species in static systems with particular emphasis on the use of lasers for this purpose. 

Extending the theory to interpret or predict the rovibrational state distribution of the products of the unimolecular dissociation, requires some postulate about the nature of the motion after the unimolecularly dissociating system leaves the TS on its way to form products. For systems with no potential energy maximum in the exit channel, the higher frequency vibrations will tend to remain in the same vibrational quantum state after leaving the TS. That is, the reaction is expected to be vibrationally adiabatic for those coordinates in the exit channel (we return to vibrational adiabaticity in Section 1.2.9). The hindered rotations and the translation along the reaction coordinate were assumed to be in statistical equilibrium in the exit channel after leaving the TS until an outer TS, the PST TS , is reached. With these assumptions, the products quantum state distribution was calculated. (After the system leaves the PST TS, there can be no further dynamical interactions, by definition.)... 

A truly mechanistic (in the sense of classical mechanics) description of a molecule s reaction is in fact prohibited by Heisenberg s uncertainty relations (Equation 2.1). Some reaction mechanisms of small molecules in the gas phase have been elucidated in the utmost detail, that is, reaction rate constants have been determined for individual rotational and vibrational quantum states of the reactant. We take a more modest view a reaction mechanism is the step-by-step sequence of elementary processes and reaction intermediates by which overall chemical change occurs. 

Gas phase Raman measurements can, of course, resolve individual rotational transitions, so it is necessary to consider rotational-vibrational electronic states in the general transition polarizability (2.18). The classical isotropic averages derived in the previous section provide a useful background because, in accordance with van Vleck s principle of spectroscopic stability [14, a quantum-statistical average over all allowed transitions should yield the... 

The particularly useful feature of REMPl is that the resonance wavelength for the intermediate absorption step is normally different for transitions between the various rotational and vibrational quantum states. Therefore, using REMPl, each ro-vibrational level of a molecule can be ionized individually by varying the wavelength of the excitation laser. 

A rovibrational spectrum is a set of transitions in which both the rotational and vibrational quantum state may change. 

The extension to diatom-surface scattering is in principle straightforward, but the number of coupled equations increases since rotational and vibrational states of the diatom also need to be included. Thus the expansion of the wavefunction eq. (5.3) includes additional quantum numbers. For a diatomic molecule we have the additional quantum numbers v, j, mj for the vibration, rotational, and rotational projection quantum states, respectively. This rather pedestrian approach... 

When a substance is heated, the kinetic energy of atoms and molecules increases. E.g., if methane is heated, the kinetic energy of translation, vibration and rotation of methane molecules increases, as discussed in section 1.2. As heat is applied, higher vibrational states are increasingly populated. In higher vibrational quantum states, the average C-H bond distance increases until finally the C-H bond breaks. The result is the formation of a methyl radical and a hydrogen atom. 

Figure A3.9.9. Dissociation probability versus incident energy for D2 molecules incident on a Cu(l 11) surface for the initial quantum states indicated (u indicates the mitial vibrational state and J the initial rotational state) [100], For clarity, the saturation values have been scaled to the same value irrespective of the initial state, although in reality die saturation value is higher for the u = 1 state.

The site specificity of reaction can also be a state-dependent site specificity, that is, molecules incident in different quantum states react more readily at different sites. This has recently been demonstrated by Kroes and co-workers for the Fl2/Cu(100) system [66]. Additionally, we can find reactivity dominated by certain sites, while inelastic collisions leading to changes in the rotational or vibrational states of the scattering molecules occur primarily at other sites. This spatial separation of the active site according to the change of state occurring (dissociation, vibrational excitation etc) is a very surface specific phenomenon. 

Figure B2.3.10. Potential energy eiirves [42] of the ground X and exeited A eleetronie states of the hydroxyl radieal. Several vibrational levels are explieitly drawn in eaeh eleetronie state. One vibrational transition is explieitly indieated, and the upper and lower vibrational wavefiinetions are plotted. The upper and lower state vibrational quantum numbers are denoted V and v", respeetively. Also shown is one of the three repulsive potential energy eurves whieh eorrelate with the ground 0( P) + H dissoeiation asymptote. These eause predissoeiation of the higher rotational and vibrational levels of the A state.

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

Thus far, exaetly soluble model problems that represent one or more aspeets of an atom or moleeule s quantum-state strueture have been introdueed and solved. For example, eleetronie motion in polyenes was modeled by a partiele-in-a-box. The harmonie oseillator and rigid rotor were introdueed to model vibrational and rotational motion of a diatomie moleeule. 

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

The emission spectrum observed by high resolution spectroscopy for the A - X vibrational bands [4] has been very well reproduced theoretically for several low-lying vibrational quantum numbers and the spectrum for the A - A n vibrational bands has been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrationm and rovibrational transitions appear to be the largest for the X ground state followed by those... 

The vibrational and rotational motions of the chemically bound constituents of matter have frequencies in the IR region. Industrial IR spectroscopy is concerned primarily with molecular vibrations, as transitions between individual rotational states can be measured only in IR spectra of small molecules in the gas phase. Rotational - vibrational transitions are analysed by quantum mechanics. To a first approximation, the vibrational frequency of a bond in the mid-IR can be treated as a simple harmonic oscillator by the following equation ... 

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