Application of a general classical variational treatment of the rates of atom–diatom reactions to the H+H2 and H+I2 reactions
Authors: Sverdlik Daniel I. , Koeppl Gerald W. 
Published: July 2002

The classical variational theory of chemical reaction rates gives the rate as the equilibrium flux of systems through a trial surface in the phase space of the reaction sysem. The surface divides the phase space into reactant and product regions and is varied to obtain a least upper bound for the rate of product formation. For atom–diatom reactions of the type A+BC→AB+C, we derived expressions which give the canonical rate coefficient and the microcanonical mean reaction cross section for the most general dividing surface defined by internal-configuration-space coordinates [J. Chem. Phys. 87, 5746 (1987)]. The dividing surface can be expressed as a power series in two of the internal coordinates and its flexibility can be systematically improved by introducing additional terms. We apply this variational formulation to the H+H2 and H+I2 reactions. Canonical rate coefficients are calculated using the downhill simplex algorithm to find the best values of three, six, and ten variational parameters in the first-, second-, and third-order expansions of the dividing surface. For the H+H2 reaction, canonical variational rate coefficients at 300 and 900 K show the expected improving trend for the first through third-order expansions of the dividing surface. The variational rate coefficient for the H+H2 reaction converges to the classical trajectory value at 300 K and exceeds the trajectory value at 900 K by a factor of 1.18±0.10. A reactivity map is devised to show the statistical importance of configurations on the dividing surface. For the quadratic dividing surface at 300 K, the most statistically important configuration on the dividing surface is nearly symmetric in terms of internuclear distances measured from the central H atom and has a ‘‘bond angle’’ for the arrangement H–H–H of 166 deg. The power series dividing surface for both the canonical and microcanonical formulations converges to a position which is close to the symmetric dividing surface of conventional transition state theory. Canonical variational rate coefficients
for the H+I2 reaction also show the expected improving trend with the expansion order of the dividing surface. However, the best variational rate coefficient for the H+I2 reaction exceeds the trajectory value by a factor of 1.767. The effective convergence of variational values of this ratio for the third-order expansion of the dividing surface shows that at this order, the dividing surface is nearly as good as it can be when its formulation is limited to configuration-space variables. For the quadratic dividing surface, the most statistically important configuration at 600 K has I–I and I–H internuclear separations of 5.10 and 4.65 a.u., respectively, and a bond angle for the arrangement I–I–H of 109 deg. The microcanonical formalism is applied to the H+I2 reaction and quadratic variational dividing surfaces are determined for seven values of the internal energy. The dividing surfaces show a weak dependence on the energy. The improvement obtained when the microcanonical results are used to evaluate the canonical rate coefficient at 600 K amounts to only 0.265%.