The exact classical vibrational-rotational partition function for the Woolley potential: calculations of the equilibrium constants for the formation of Ar-Ar and Mg-Mg

Abstract

Exact analytical expressions are derived for the classical vibrational-rotational partition function and for the number of vibrational-rotational energy levels of the one-constant Woolley potential, which can be viewed as a deformed Lennard-Jones (12, 6) potential. These expressions are then used first to fit the Woolley potential to accurate analytic potentials of Ar2 and Mg2 by matching the total number of vibrational-rotational energy levels and, secondly, to calculate the corresponding classical equilibrium constants for the formation of these two molecules. The results are in excellent agreement with the exact classical and quantum-mechanical equilibrium constants calculated from the accurate analytic potentials by Dardi and Dahler (1990) with a maximum error of 4% for Ar2 and 2% for Mg2. In particular, they are in much better agreement than those calculated from various Lennard-Jones (12, 6) potentials.

Density functional theory of the structure of magnesium-doped helium nanodroplets

“…43, ⌽ i ͑r͒ is the wave function of the ith Mg atom and ⌿͑r͒ = ͱ ͑r͒ represents again the 4 He order parameter, where ͑r͒ is the 4 He atomic density. In this expression, E͑͒ is the 4 He potential-energy density.…”

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

“…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.…”