## ROTATIONAL PARTITION FUNCTIONS FOR SYMMETRIC-TOP MOLECULES

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Ohio State University##### Abstract:

It has recently been shown that improved closed-form expressions for the rotational partition functions of $linear^{1}$ and spherical-$top^{2}$ molecules allow one to obtain these with high accuracy over a wide temperature range without having to calculate and sum explicitly the rotational energy levels. This work has been extended to symmetric-top molecules, with particular attention to (1) a new treatment of the quantization correction which converges more rapidly at all temperatures; (2) corrections for the effect of nuclear-spin statistics at very low temperatures; and (3) corrections for centrifugal distortion of the rotating molecules at high temperatures. The derived form of the rotational partition function is $Q_{r}=\sigma^{\ast}(\pi m)^{1/2}\exp[\beta(4-m)/12]\beta^{-3/2}[1+\beta^{2}(1-m)^{2}/90\ldots](1+\delta)(1+\rho_{0}+ \rho_{1}\beta^{-1}+\ldots)$ where $\beta = hcB/kT; m= B/A$ for prolate tops or B/C for oblate tops; and for $XY_{<unclear>} molecules \sigma^{4} = (2I_{y}+1)^{n}/\sigma$, where $I_{Y}$ is the nuclear spin of the Y nuclei and $\sigma$ is the classical symmetry number. Here the first five factors are the high-temperature quantum-mechanical partition function, with an improved series development to account for quantization. The factor $1+\delta$ accounts for nuclear-spin statistics, and is given by $\delta<unclear>[2\exp(-\pi^{2}m/9\beta)\exp(\pi^{2}m^{2}/54)/(2I_{Y}+1)^{2}][1+\pi^{2}m^{2} [\pi^{2}m^{2}+72(1-m)\beta]/14580].$ The final factor gives the centrifugal distortion correction in the form derived by $Wilson,^{3}$ where in terms of the usual spectroscopic distortion constants $D_{J}, D_{JK}, D_{K}$, the parameters are $\begin{array}{lll}\rho_{0}<unclear>-[(8+2m-4m^{2}+3m^{3})D_{2}+m(2-2m+3m^{2}) D_{JK}+3m^{3}D_{K}]/12B,\\\rho_{2}<unclear>[(8+4m+3m^{2})D_{J}+m(2+3m)D_{JK}+3m^{2}D_{k}]/4B.\\ \end{array}$ The accuracy of this expression will be demonstrated with calculations on $NH_{3}, CH_{3}D$, and $CHD_{3}$.

##### Description:

$^{1}$ R. S. McDowell. J. Chem. Phys. 88. 356 (1988). $^{2}$ R. S. McDowell, J. Quant. Spectrosc. Radiat. Transfer 38, 337 (1987). $^{3}$ E. B. Wilson, Jr., J. Chem. Phys. 4, 526 (1936).

Author Institution: University of California, Los Alamos National Laboratory

Author Institution: University of California, Los Alamos National Laboratory

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