A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators^†

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed-form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two- and four-point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011