Rotational partition functions for symmetric-top molecules

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Abstract
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): Qr≂σ*(πm)1/2 exp[β(4−m)/ 12]β−3/21+β2(1−m)2/90+⋅⋅⋅](1+δ) (1+ρ1β−1+ρ2β−2+ρ3β−3), where β≡hcB/kT, m≡B/A for prolate tops or B/C for oblate tops (A>B>C are the rotational constants), and σ*≡Πi(2Ii+1)/σ, where Ii is the spin of nucleus i and σ is the classical symmetry number. For C3vXY3 and WXY3 molecules, the low-temperature nuclear-spin correction is δ≊2(2IY+1)−2 exp[π2m(βm−6)/54β]; similar expressions are given for other molecular models. The ρ’s are functions of the centrifugal distortion constants DJ, DJK, DK, HJ, etc. If inversion doubling occurs, with a splitting ΔG in the ground vibrational state, this expression for Qr is to be multiplied by (1−βΔG/2B+⋅⋅⋅). The molecules NH3, CH3D, CHD3, allene (C3H4), and ethane (C2H6), including isotopic variants, are discussed and illustrative calculations presented for T=10 to 2000 K.

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