Exact classical vibrational-rotational partition function for Lennard-Jones and Morse potentials

Abstract

Exact analytical expressions for the classical vibrational-rotational partition function of the Lennard-Jones (m, n) and Morse potentials are derived. The values obtained from such expressions are compared to accurate evaluations of the corresponding quantum-mechanical partition function for the LJ (10, 6) and Morse potentials. Both sets of values are found to be in close agreement over a wide range of temperatures.

On the Rovibrational Partition Function of Molecular Hydrogen at High Temperatures

“…They may then be important in computations of thermodynamical properties for larger systems where quantum calculations are out of reach and classical statistical mechanics and/or FPI methods are the only viable routes . They are also relevant in relation to floppy polyatomic systems such as van der Waals molecules, where classical and quantum results are in good agreement even at low temperatures. ,, Indeed, for these systems, the temperature at which the configurational integral based methods (classical and FPI) start to diverge is expected to be only a few hundred Kelvin or so. We should conclude by noting that it is unclear whether the correct prescription for calculating a partition function is always to preclude contributions from dissociative states.…”

“…Thus, the classical rovibrational partition function assumes the form where H ( q , p ) is the classical Hamiltonian, q is the vector of generalized coordinates, and p is the vector of conjugate momenta; we will assume as reference throughout this work the energy of the minimum of the potential energy surface (curve). Note that in some situations − the classical picture gives results accurate enough to make quantum calculations unnecessary, whereas in others (for which the sum over states is still computationally out of reach), one can think of coping with the breakdown of classical statistical mechanics (especially at low temperatures) by using quantum corrections. For recent work which discusses how to account for the deviation of the classical rovibrational partition function from the exact quantum result, the reader is referred to refs −19.…”

Monte Carlo Simulation Approach to Internal Partition Functions for van der Waals Molecules

“…For some diatomic potential models such as the generalized Lennard-Jones (m,n), Morse, and Woolley curves, it is even possible to get the analytical solution of eq 15. 18,19 To investigate the validity of the classical approach, we first compare the diatomic results for the internal partition function obtained from eq 15 with their accurate quantum analogs. For this purpose, we have calculated Q int in eq 15 for ArO using two different models: EHFACE2 20 (see also ref 21) and Lennard-Jones (6-12) potential functions (the latter is defined by 22 R min ) 6.65a 0 and ) -0.0002798E h ).…”

“…The integral in eq 15 is a one-dimensional integral and can be resolved easily by any conventional numerical method. For some diatomic potential models such as the generalized Lennard-Jones ( m , n ), Morse, and Woolley curves, it is even possible to get the analytical solution of eq 15. , …”

“…In turn, the exact quantum value of 𝒬 int has been obtained from eq. 10 with the ε vj calculated by solving numerically the 1 D nuclear Schrödinger equation using the standard Numerov−Cooley algorithm. , Classically, the integration of eq 15 for the EHFACE2 potential function has been carried out using the trapezoidal method, while for the Lennard-Jones (6−12) potential function, we have used the analytical solution proposed by Guérin. , From the calculated 𝒬 int for ArO, we have then determined the equilibrium constant ( K eq ≡ K 2 ) for reaction 4 by applying the standard formalism of statistical mechanics . Table shows the results obtained from both the CSM and QSM approaches using the above two potential energy curves over a wide range of temperatures.…”