CLOSED-FORM EXPRESSIONS FOR LOW-TEMPERATURE ROTATIONAL PARTITION FUNCTIONS

Author Institution: University of California, Los Alamos National LaboratoryModern non-equilibrium spectroscopic cooling techniques, as well as possible astrophysical applications, make it necessary to evaluate rotational partition functions, $Q_{r}$, at temperatures low enough for the usual closed-form expressions to become inaccurate, but still high enough for direct summation to be cumbersome. We will demonstrate that analytic expressions for $Q_{r}$ can be derived that provide much greater accuracy at all temperatures, and extend the range of validity to significantly lower temperatures. For spherical-top $XY_{n}$ molecules, such a formula is $$Q_{r}=\sigma^{\ast}\Pi^{1/2}e^{\alpha/4}\alpha^{-3/2}(1+\delta)(1+ 15D/4\alpha B)$$ where $\alpha = hcB/kT$ and $igma^{\ast} = (2I_{Y^{\ast}}1)n/\sigma (1_{Y}$ = nuclear spin of the Y nuclei, $\sigma =$ classical symmetry number). Here the first four factors are the usual high-temperature quantum-mechanical partition function, accurate to within 1\% for $\alpha 38 K$ for $CH_{4}$). The factor in 6 corrects for low-temperature quantum effects, and the last factor accounts for centrifugal distortion. For $CH_{4}, \delta = ^{1}\zeta \exp(2.31 - 1.10/\alpha)$, and this formula is accurate to within 0.002\% for $58 < T \leqslant 500 K$, and to within 0.1\% down to 18 K. For octahedral hexafluorides, $\delta = -\exp(-0.35 - 0.60/\alpha)$, and $Q_{r}$ is accurate to within 0.002\% down to 1.2 K for $SF_{6}$