Vibrational state counting

The most accurate procedure for determining the density of harmonic vibrational states is by the direct count method. A particularly clever scheme for doing this was proposed by Beyer and Swinehart (1973). As demonstrated by Gilbert and Smith (1990), this approach is based on the convolution of state densities. Suppose that the system consists of s harmonic oscillators with vibrational frequencies, co, = v,/c (cm" )- Each will have a series of equally spaced states located at , = nco, (n = 0, 1,. . . ). We choose the zero of energy at the molecule s zero point energy, and divide the energy into bins. The vibrational frequencies must be expressed as integral numbers of bin sizes, for example, as multiples of 10 cm for a 10 bin size. A convenient bin size is 1 cm so that the s frequencies can be simply rounded off to the nearest wave-number. 

A comparison of the smoothed number of states ( ) with directly counted values for discrete states is shown in Fig. 1.21. One can see that the step function valid for discrete states approaches the smoothed curve at high energies. The assumption of a quasicontinuum of vibrational states in molecules with several oscillators thus is an excellent approximation at high energies. 

COUNT The vibrational state degeneracy g (e) is computed from 8=0 to s by direct count. 

The identification of an eigenstate as a vibrational state with a certain number of nodes can be accomplished by counting the zeros of the eigenvalues of the log derivative during an integration at the energy corresponding to a bound state. However, it appears to be a better... 

RRKM theory has been used widely to interpret measurements of unimolecular rate constants. However, harmonic state counting procedures are usually used in the RRKM calculations. This is not because enharmonic effects are thought to be unimportant, but because they are difficult to account for. The only comprehensive attempt to include the effect of anharmonicity has involved treating the vibrational degrees of freedom as separable Morse oscillators. However, since this correction is an obvious oversimplification it has not been widely used. The importance of anharmonicity is illustrated by comparing the trajectory unimolecular rate constant for C2H5 H + C2Hi dissociation at 100 kcal/mol (Fig. 4b), which is about 4.7 X 10 with that predicted by harmonic classical RRKM... 

The importance of enharmonic state counting in computing unimolecular rate constants must also be determined. In prior work it has been neglected and gross errors may be present in previous interpretations of unimolecular rate constants. If anharmonicity is important, statistical models could still be used to fit unimolecular rate constants by decreasing the number of effective vibrational modes in... 

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