Analytical Mechanics of Chemical Reactions. IV. Classical Mechanics of Reactions in Two Dimensions

Abstract

The natural collision coordinates of Part III are used to treat the analytical mechanics of chemical reactions, AB + C→A + BC. Other than in Part II, the classical analytical mechanics of chemical reactions on a smooth potential surface have not been explored previously in the literature. A Hamilton–Jacobi formalism is used, apparently for the first time in calculating a reaction rate. The “vibrationally adiabatic” reaction serves as the zeroth-order approximation and nonadiabatic corrections are obtained. Theoretical expressions yield the rotational and vibrational energy distribution of reaction products, angular distribution, and reaction probability, as a function of impact parameter, initial translational velocity of relative motion, and initial rotational–vibrational state of reactants. The results are not intended to apply to reactions which show very large excursions from vibrational adiabaticity. In the zeroth approximation (vibrational adiabaticity), an adiabatic separation of variables is achieved. Here, the vibrational action is constant; however, the rotational–orbital action changes by a known increment from one constant value to another, on transition of that motion into a bending vibration. The resulting “adiabatic” correlation shows several interesting features. For reactions in which there are no large mass ratios, the state of vibration of AB, of rotation of AB, and of orbital motion of AB + C correlate with the state of vibration of BC, of rotation of BC, and of orbital motion of A + BC, respectively. For reactions with unusual mass ratios, such as H + Cl2→HCl + Cl, the correlation equations show instead the “adiabatic” transformation of Cl2 rotation into HCl + Cl orbital motion, thereby reflecting the expected result of angular momentum conservation. Had the rotational–orbital cross term in the kinetic energy been neglected, an incorrect correlation would have resulted in the latter case. Extension of the present work to three dimensions involves an added approximation, to be given in a subsequent paper. The expressions and method also permit comparison of one- and two-dimensional computer results on a more similar basis and, because of certain similarities in computer results for energy distributions in two and three dimensions, perhaps comparison with experimental results on energy distributions. In conjunction with the computer results information can be obtained on various approximations, such as near adiabaticity. The present theory can also be used to analyze and perhaps extend various statistical-type theories in the literature.