Vibrating-rotator energy levels

Each and every electronic energy state, labelled k, has a set, labelled L, of vibration/rotation energy levels k,L and wavefiinctions... 

The homonuclear rare gas pairs are of special interest as models for intennolecular forces, but they are quite difficult to study spectroscopically. They have no microwave or infrared spectmm. However, their vibration-rotation energy levels can be detennined from their electronic absorjDtion spectra, which he in the vacuum ultraviolet (VUV) region of the spectmm. In the most recent work, Hennan et al [24] have measured vibrational and rotational frequencies to great precision. In the case of Ar-Ar, the results have been incoriDorated into a multiproperty analysis by Aziz [25] to develop a highly accurate pair potential. 

Far-infrared and mid-infrared spectroscopy usually provide the most detailed picture of the vibration-rotation energy levels in the ground electronic state. However, they are not always possible and other spectroscopic methods are also important. 

In using symmetry to help simplify moleeular orbital or vibration/rotation energy level ealeulations, the following strategy is followed ... 

Figure 2. Vibrational and vibrational-rotational energy levels for a two-atomic molecule.

The use of tunable lasers as sources in electronic absorption and emission spectroscopy has made possible a very considerable increase in resolution and precision. Electronic spectra are often difficult to analyze because of the many transitions involved. However, with a tunable laser source, one can tune the laser frequency to a specific absorption frequency of the molecule under study and thus populate a single excited electronic vibration-rotation energy level the resulting fluorescence emission spectrum is then simple, and easy to analyze. 

For H2, accurate theoretical calculations3 of the vibration-rotation energy levels have been done by solving the radial differential equation (4.11) using numerical integration. The potential-energy function used is that found from a 100-term variational electronic wave function. 

Shapiro, M. and Balint-Kurti, G.G. (1979). A new method for the exact calculation of vibrational-rotational energy levels of triatomic molecules, J. Chem. Phys. 71, 1461-1469. 

For HCN the situation is somewhat better, because the data on DCN are much more effectively independent of the HCN data. This molecule has also been the subject of much high-resolution spectroscopic study, so that the vibration-rotation energy levels are particularly well known and its vibrational spectrum is free of accidental resonances. Table 8 compares the results of three quite different calculations. The calculation by Strey and Mills is the most recent, and was based on the latest spectroscopic data the refinement was made to a and x values rather than to the vibrational levels and rotational constants as used by both the earlier workers. Strey and Mills also constrained 3 of the quartic interaction constants to zero, and refined to cubic and quartic force constants in a separate calculation to the quadratic refinement. The level of agreement between the calculations leads to conclusions rather similar to those made above for C02 in particular, standard errors should be multiplied by at... 

These corrections are of two types. The first arises from quantum effects in the calculation of vibration-rotation energy levels and was originally discussed by Dunham himself [23], He used semiclassical methods, discussed in section 6.13.2, to derive his results. Use of the first-order quantisation condition... 

Rotational Levels and Transitions. The vibrational-rotational energy levels for a linear molecule are similar to those for a diatomic molecule and to a good approximation are given in cm units by the sum G u V2...) + where ... 

The vibration-rotation energy levels of a diatomic molecule are given in traditional notation (Herzberg, 1950) by the expression... 

We have already mentioned the interpretation of photodetachment spectra of FH2 and XHX (X = Cl, Br, I), in terms of quantized transition state resonances. Similar experiments have been carried out for IDF, OHF , OHC1, OHOH , and HOHOH (174-177), and these experiments have been interpreted in terms of resonances and other types of vibration-rotation energy level structure associated with the transition state species of the neutral product (10,11,17-19,162,163,174-178). The FH2 and FD " photodetachment experiments provide a particularly striking example of the observation of quantized transition states in experimental spectra (133-135). In theoretical work carried out to analyze recent experimental work on photodetachment, in particular for OHC1 (176), the calculated cumulative reaction probability for the O + HC1 reaction showed steps at quantized hindered rotor energies (as well as sharper resonances due to trapped states), but the steps had transmission coefficients considerably smaller than unity. 

The preceding presentation describes how the collision impact parameter and the relative translational energy are sampled to calculate reaction cross sections and rate constants. In the following, Monte Carlo sampling of the reactant s Cartesian coordinates and momenta is described for atom + diatom collisions and polyatomic + polyatomic collisions. Initial energies are chosen for the reactants, which corresponds to quantum mechanical vibrational-rotational energy levels. This is the quasi-classical model [2-4]. 

There is a unique topologically independent path for each mode, and each vibrational-rotational energy level is specified by a set of quantum numbers kt. 

Vibrational, rotational, and vibrational/rotational energy levels are found by first transforming the classical Hamiltonians described in the previous section to the appropriate quantum mechanical operator H. The eigenvalue equation... 

Table 2.6. Variational Caiculations of Vibrational and Vibrational-Rotational Energy Levels.

Vibration/rotation energy levels can always be determined with the use of variational procedures. In this approach the wave function for a given vibration-rotation level is written as a linear combination of basis functions AfiRjn... 

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