Calculation of the Rovibrational Partition Function Using Classical Methods with Quantum Corrections Frederico V. Prudente, Antonio Riganelli, and Anto ´ nio J. C. Varandas* Departamento de Quı ´mica, UniVersidade de Coimbra, P-3049 Coimbra Codex, Portugal ReceiVed: December 6, 2000; In Final Form: February 28, 2001 The rovibrational partition functions of diatomic molecules are calculated using a classical framework plus quantum, semiclassical, and semiempirical corrections. The most popular methods to calculate such corrections are briefly reviewed and applied to the benchmark H 2 molecule. A novel hybrid scheme is proposed and applied to H 2 , HCl, and ArO. Each method is analyzed with a view to find an economical way to calculate such corrections for polyatomic systems. 1. Introduction Accurate values of the rovibrational partition function (q vr ) are frequently needed in chemistry and physics. They are required to calculate the equilibrium properties of molecular systems, 1 and the rates of chemical reactions using transition state theory. 2,3 In principle, q vr can be evaluated exactly in quantum statistical mechanics by carrying out the following explicit summation of Boltzmann factors where  ) 1/(k B T), k B is the Boltzmann constant, T is the temperature, ǫ i is the rovibrational energy associated with state i, and g i is the corresponding degeneracy factor. However, the calculation of highly excited states with spectroscopic accuracy is currently feasible only for systems with a few degrees of freedom, 4-6 which limits the evaluation of the rovibrational partition function using eq 1 to small molecules and low rotational states. 7-10 In turn, q vr assumes in classical statistical mechanics the form 1,11 with H CM (q, p) being the classical Hamiltonian, h the Planck constant, n the number of degrees of freedom, q the generalized coordinate vector, and p its conjugate momenta. In turn, the subscript B implies that the hypervolume of integration is restricted to phase space regions corresponding to a bound state situation, i.e.,0 e H CM (q, p) e D e , where D e is the dissociation energy of the molecule 12 (throughout this work we assume as reference energy the minimum of the potential energy surface). A major advantage of the classical approach is the appreciably smaller computational cost in comparison to the quantum one, which allows a treatment of molecular systems with a large number of degrees of freedom. Yet, it is well-known that q vr CM overestimates q vr QM at low temperatures, while converging to the latter at the high-temperature limit; 1 for a recent discussion on the range of applicability of classical statistical mechanics to calculate the vibrational partition function for triatomic systems, see ref 13. For molecular simulations it is essential to have an economical recipe to introduce corrections into the partition functions evaluated within the classical framework. The idea for introduc- ing such corrections comes from the early days of quantum mechanics. Such corrections can be based on “strictly” quantum, semiclassical and semiempirical formulations. A well established strictly quantum approach comes from the seminal work of Wigner 14 and Kirkwood. 15 They have shown that the quantum probability function (and hence the quantum partition function) in phase space can be obtained through an expansion in powers of p ) h/2π. The Wigner-Kirkwood (WK) expansion has been extensively employed in molecular simulations of liquids, 16-19 and to calculate the partition functions of hindered rotors 20 and diatomic molecules. 21,22 Another procedure to correct q vr CM from quantum mechanics is due to Green 23 and Oppenheim and Ross 24 (GOR). Similar to the WK method, the GOR approach consists of writing the quantum Hamiltonian within the phase space formalism as an expansion in powers of p, which is then used in eq 2; for applications of the GOR method in molecular simulations, see ref 25. Other approaches to correct the classical partition function come from path integral formulations, 26,27 and are of semiclas- sical nature. One such a proposal due to Feynman and Hibbs 27 uses the Feynman path integral formulation of quantum mechanics, and consists of approximating the integration of the energy functional over all paths by using an effective potential. Since their proposal, several forms of the effective potential have been suggested and applied to various physical prob- lems, 19,28-34 including the calculation of the molecular rovibra- tional partition function. 35 The second proposal comes from Miller and co-workers 36-39 and is based on an approximation of the Feynman path integrals by their classical counterpart which, are in turn approximated by using a “semiclassical” potential. Such an approach has been utilized in the context of molecular vibration-rotation dynamics. 40 A simple semiempirical procedure to correct the classical partition function was suggested by Pitzer and Gwinn. 41 In the Pitzer-Gwinn method, the quantum partition function is ap- proximated as the classical partition function scaled by the ratio of the quantum and classical partition functions for a reference q vr QM (T) ) ∑ i g i exp(-ǫ i ) (1) q vr CM (T) ) 1 h n ∫∫ B exp {- H CM (q,p)} dq dp (2) 5272 J. Phys. Chem. A 2001, 105, 5272-5279 10.1021/jp0043928 CCC: $20.00 © 2001 American Chemical Society Published on Web 05/04/2001