Rotational thermodynamic parameters for symmetric-top, linear-top and spherical-top molecules: classical versus quantum approach and New analytical partition functions

Abstract

We investigated classical and quantum rotational partition functions, rotational energies and rotational entropies for linear-, spherical- and symmetric-top molecules in a canonical system within a wide range of temperatures under the rigid rotor approximation. We then derived the temperature, $$\mathbf {T_{limit}}$$ , below which classical approach is not valid and the complex quantum approach is strongly recommended. This temperature certainly increases with the inertia moment of the molecule and follows an exact combination of three decreasing exponential functions for linear- and spherical-top molecules when rotational partition functions are considered. However, another good news is that contrary to what is being commonly known, quantum approach is not needed at all for energetic rotational parameters, whatever the temperature. Therefore, there is no real frontier between the classical and quantum approaches when energetic rotational parameters are concerned, though the frontier is naturally a bounded decreasing sequence converging to zero when rotational partition functions are rather concerned. Particularly, classical approach (as implemented in several quantum chemistry computational codes) fails describing rotational partition functions for light molecules or molecules at low temperatures. As a matter of fact, we noted that as far as the partition function is concerned, $$\mathbf {T_{limit}}=\frac{100}{3}\theta$$ for linear-top molecules and $$\mathbf {T_{limit}}=25\,\theta$$ for spherical-top molecules, with $$\theta$$ the characteristic rotational temperature. Moreover, all the closed-forms expressions proposed in the whole literature fail drastically approaching quantum and exact results for light molecules or molecules at low temperatures. In regard of the importance of partition functions in the evaluation of rate constants for chemical reactions and accurate intensities lines in spectroscopy, we proposed two new accurate and simple analytic expressions of the rotational partition functions of linear-top and spherical-top molecules, valid for light molecules or at low temperatures. These new closed-forms reproduce excellently the exact values and are exceptionally better than previous proposed closed-forms in the literature, and they are therefore strongly recommended for light molecules or at low temperatures.