Vibrational anharmonicity rotational levels

Figure 1.4 The very anharmonic energy levels of the C6H6 - Ar stretch motion (cf. Figure 0.3) (adapted from Neusser, Sussman, Smith, Riedle, and Weber, 1992). The computed values of the rotational constant Bv [the coefficient of 1(1+1) in the expression for the energy eigenvalues] are given in the figure, as are the vibrational spacings.

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

If there was no interaction between vibration and rotation, the energy levels would be given by the simple sum of the expression giving the vibrational levels for the anharmonic oscillator, equation (6.188), and that describing the rotational levels of the rigid rotor, equation (6.162). There is an interaction, however during a vibration the moment of inertia of the molecule changes, and therefore so also does the rotational constant. We may therefore use a mean value of Bv for the rotational constant of the vibrational level considered, i.e. 

Vibrationally excited diatomic molecules will only emit if they are polar, and most of the available results are for reactions which produce diatomic hydrides. Because of their unusually small reduced mass, these molecules have high frequency and very anharmonic vibrations, and their rotational levels are widely spaced. Consequently, their spectra can be resolved more easily than those of nonhydrides, where there are many more individual lines in the vibration-rotation spectrum. Furthermore, the molecular dynamics of these reactions are particularly interesting because of the special kinematic features that arise when an H atom is involved in a reactive collision and because these... 

The vibrational and rotational constants of the respective electronic levels were taken from Rosen (2 ). The thermodynamic functions are calculated using first-order anharmonic corrections to and 0 in the partition function Q = Q,j.EQ Q gj exp(-... 

Figure 10.5 Rotation/vibration levels of carbon monoxide. V and J are the quantum numbers of vibration and rotation. The fundamental vibration corresponds to V = +l and 7 = +1. (a) A rotation-vibration band corresponds to all of the allowed quantum transitions. If the scale of the diagram is in cm , the arrows correspond to the wavenumbers of the absorptions (b) branch R corresponds to A7 = +1 and the band P to A7 = — 1. They are situated either side of band Q, absent from the spectrum (here it can be supposed that A7 = 0 corresponding to a forbidden transition) (c) below, vibration-rotation absorption band of carbon monoxide (pressure of 1000 Pa). The various lines illustrate the principle of the selection rules. The difference (wavenumbers) between successive rotational peaks are not constant due to anharmonicity factors.

To calculate n E-E, the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The former approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Harmonic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by performing an appropriate normal mode analysis as a function of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to determine anharmonic energy levels for the transitional modes [27]. 

The preceding normal-mode/rigid-rotor sampling assumes the vibrational-rotational levels for the polyatomic reactant are well described by separable normal modes and separability between rotation and vibration. However, if anharmonicities and mode-mode and vibration-rotation couplings are important, it may become necessary to go beyond this approximation and use the Einstein-Brillouin-Keller (EBK) semiclassical quantization conditions [32]... 

The RRKM rate constant as given by equation (6.73) in the previous chapter is expressed as a ratio of the sum of states in the transition state and the density of states in the reactant molecule. An accurate calculation of this rate constant requires that all vibrational anharmonicity and vibrational/rotational coupling be included in calculating the sum and density. The vibrational energy levels in units of wavenumbers can be represented by a power series ... 

In the visible region of the spectrum water vapour is transparent and all further absorptions of interest occur in the infrared or at even longer wavelengths. These are associated with transitions between vibrational levels of the molecule, the fundamental modes for which are shown in fig. 1.4, and have a fine structure dependent upon the rotational levels involved. Since each of the three normal modes has a direct effect upon the dipole moment of the molecule, they aU lead to absorption bands. Because the interatomic potentials have appreciable anharmonic components from terms of cubic or higher order in the displacements, the relation between... 

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