Exact activation energies and phenomenological description of quantum tunneling for model potential energy surfaces. The F + H2 reaction at low temperature

Abstract

Activation energies Ea calculated as the negative of the logarithmic derivatives of rate constants with respect to the inverse of absolute temperature T, are presented for three potential energy surfaces previously introduced for the reaction F+H2→HF+H in the temperature range 10<T<350K. Exact benchmark rate constants from quantum mechanical calculations on each surface have been reported [V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, J.M. Lucas, Chem. Phys. 308 (2005) 237] and shown to deviate from Arrhenius behavior. The corresponding pronounced deviation from constancy of activation energy Ea represents a prototypical example of the role of quantum mechanical tunneling in propitiating a “sub Arrhenius” behavior. Four formulas are tested in order to provide a phenomenological description of the tunnel effect on reactivity, each introducing only one additional parameter with respect to the Arrhenius law. They correspond to: (i) the so-called Modified Arrhenius Equation, involving a linear dependence of EaversusT; (ii) the Curved Arrhenius Plot description, implying a linear dependence of Eaversus inverse T; (iii) the deformed Arrhenius law recently proposed and the corresponding inverse Ea – inverse T linear relationship recently derived from the deformed exponential distribution appearing in non – extensive statistical thermodynamics; (iv) a generalization of Mott’s law for electron and proton conduction in condensed matter, leading to an exponential relationship between Ea and T. Numerical investigation allows a discussion of merits of each formula from a temperature of 350K down to 50K. In all three cases, the deformed Arrhenius law provides a better description of the quantum mechanical trend.