Whitten-Rabinovitch

The value of the parameter a, which lies between 0 and 1, must be determined for each system by comparison with more accurate methods, but once this has been done the Whitten-Rabinovitch method gives an accurate parametric representation of the required density of states. 

Fe is a correction for the fact that the density of vibrational states is an increasing function of energy above the threshold, whereas the approximate formula has assumed it to be constant. Approximating the density of states instead by the Whitten-Rabinovitch formula, Eq. (86), gives... 

The other parameter was again modified to take account of the effective number of oscillators and the zero point energy, using the Whitten-Rabinovitch formula... 

Some of these assumptions can be softened at the expense of added complexity. For instance, the Whitten-Rabinovitch approximation (Forst, 1971 Whitten and Ra-binovitch, 1963, 1964) for AfiS ) ... 

Figure 9. Unimolecular decomposition rate constants ky for Mn(CO)x as a function of ion internal energy above threshold, Ef, calculated using RRKM theory employing Whitten-Rabinovitch state counting and bond energies listed in Table II, with log A = 15.

The Whitten Rabinovitch formula [15] for the vibrational density of states function is... 

Fig. 2. Densities of states as function of the freely distributable energy for ethoxy radicals calculated with the Beyer Swinehart (B S) algorithm or the Whitten-Rabinovitch (W-R) equation. Calculations were done with DenSum from the MultiWell distribution.

G.Z. Whitten and B.S. Rabinovitch. Accurate and Facile Approximation for Vibrational Energy-Level Sums. J. Chem. Phys., 38 2466-2473,1963. 

The w parameter is conveniently tabulated as a function of t. It may be noted that the correction to the semi-classical expression is largest when the vibrational frequencies are spread over a wide range. The method of Whitten and Rabinovitch [8] has been widely used and has provided a valuable service to the kinetics community. An expression for the density of states is obtained upon differentiation of eqn. (12). Their method also encompasses the case where there is an internal rotation. For the transition complex at low energies, W( e + ) is conveniently found by a straight count. 

This corrected approximation is a great improvement on the uncorrected classical method, but is an overcorrection, leading to a systematic overestimate of the density of states. Whitten and Rabinovitch [42] corrected this overestimate by introducing a parameter a to weight the zero-point energy in the correction formula. 

Although these integrals might look complicated, they can easily be evaluated if the density of states function is given in an analytic way as introduced by Whitten and Rabinovitch [15]. Essentially Pvib E) is expressed as a polynomial in E and the above integrals have analytic solutions (see, e.g.. Ref. [16]). 

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