ROTATIONAL PARTITION FUNCTIONS FOR SYMMETRIC TOPS
Robin S. McDowell · The Knowledge Bank (The Ohio State University) · 1990
A simple, accurate closed-form expression for the rotational partition functions of symmetric - top molecules is derived, which includes the effects of nuclear-spin statistics (significant at very low temperatures), quartic and sextic centrifugal distortion terms (moderate and high temperatures), and inversion (all temperatures):$Q_{2} - \\sigma^{\\ast}(nm)^{1/2} e^{\\beta(4-n)/12}\\beta^{-3/2}[1 + \\beta^{2}(1-m)^{2}/90] (1 + \\delta) (1 + p_{2}\\beta^{-1}+p_{2}\\beta^{-2} + p_{2}\\beta^{-3}) (1 - \\beta \\Delta G/2B)$ where $\\beta$ - hcB/kT. m = B/A for prolate tops or B/C for oblate tops, and $\\sigma^{\\ast} - \\Pi_{b}(2I_{i}+1)/\\sigma$ where $1_{i}$ is the spin of nucleus 1 and 0 is the classical symmetry number. The low-temperature nuclear-spin correction is $\\delta = 2(21_{y}+1)^{-2}\\exp[\\pi^{2}m(\\beta m-6)/54\\beta)$ for $C_{3v} XY_{3}$ and WXY3 molecules; similar expressions hold for other molecular models. The $p^{\\prime}s$ are functions of the centrifugal distortion constants $D_{J}. D_{JK}, D_{K}, H_{J}$, etc., and $\\Delta G$ is the inversion splitting in the ground vibrational state. Applications to the partition and thermodynamic functions of $NH_{3}, CH_{3}D, CHD_{3}$, allene $(C_{3}H_{4})$, and ethane, including isotopic variants, will be discussed