Participation of two-dimensional hindered internal rotations in activated complexes

Abstract

It is postulated that the bending motions in an activated complex, of the form AB2, may be treated as a two-dimensional internal rotation hindered by a sinusoidal potential function. The Shrödinger equation for these degrees of freedom takes the form of the oblate spheroidal equation. For various values of the barrier to internal rotation, this equation has been solved to find the lowest 231 energy levels. A series expansion has been found for the energies of the bound states. The contributions of these degrees of freedom to the heat capacity, the enthalpy function, and the free energy function have been calculated. Approximations to the latter quantities are also deduced and are shown to be valid in certain temperature ranges. This type of motion has been incorporated into activated complex theory. Replacement of the usual harmonic bending potential by a sinusoidal one has the following effects: (i) the concept of reaction path degeneracy is replaced by nondegenerate states of opposite symmetry, (ii) the zero point energy of the complex is decreased, (iii) at low temperatures, partition functions, activation energies, and Arrhenius plot curvature increase more rapidly with increasing temperature, (iv) at high temperatures, partition functions and activation energies increase less rapidly and curvature declines with increasing temperature. At high temperatures, the expression for the rate constant has the same form as the expression from simple collision theory. Expressions for the collision theory steric factor and activation energy are deduced. As an example, calculations are performed for the reaction of D with H2.