Mass Effects and Internal Space Geometryin Triatomic Reaction DynamicsTomohiro Yanao∗, Wang S. Koon†, and Jerrold E. Marsden‡Control and Dynamical Systems, MC 107-81,California Institute of Technology, Pasadena, CA 91125December 12, 2005AbstractThe effect of the distribution of mass in triatomic reaction dynamics is analyzed using thegeometry of the associated internal space. Atomic masses are appropriately incorporated intointernal coordinates as well as the associated non-Euclidean internal space metric tensor aftera separation of the rotational degrees of freedom. Because of the non-Euclidean nature of themetric in the internal space, terms such as connection coefficients arise in the internal equationsof motion, which act as velocity-dependent forces in a coordinate chart. By statistically averag-ing these terms, an effective force field is deduced, which accounts for the statistical tendencyof geodesics in the internal space. This force field is shown to play a crucial role in deter-mining mass-related branching ratios of isomerization and dissociation dynamics of a triatomicmolecule. The methodology presented can be useful for qualitatively predicting branching ratiosin general triatomic reactions, and may be applied to the study of isotope effects.∗E-mail address: [email protected]†E-mail address: [email protected]‡E-mail address: [email protected] INTRODUCTIONThe configuration of atomic masses in a molecule is an imp ortant factor in a determination ofbranching ratios and rates of reactions of the molecule. In particular, the roles of m ass configurationare of great interest in isotopic reactions. For example, branching ratios of the dissociation dynamicsof triatomic hydrogen ion H+3and its isotopomers, D2H+, H2D+, and D+3is the subject of extensivestudies, both experimentally and theoretically (see [1–4]). Another challenging problem is theanomalous isotope effect in the formation and exchange reactions of ozone O3( see [5–11]). Atomicmasses are also found to play a crucial role in intramolecular vibrational-energy redistribution (IVR)in N2O reactions (see [12]). Even in these fundamental reactions, there still exist many interestingand unknown issues on the role of atomic masses, and a general framework to understand the rolesof atomic masses should serve many purposes.There is a useful framework that allows one to characterize the m ass effect specifically forcollinear triatomic reactions such as A+BC → AB+C. This is the so-called skewed coordinatesystem framework [13, 14], in which the two inter-particle distances (A-B distance and B-C distance)are chosen as internal coordinates and are prop e rly embedded in a two-dimensional Euclidean space.As a result, the motion of the three atoms is expressed as a motion of a point mass (with constantmass) in this associated two-dimensional space. In this formalism, the mass effe ct is effectivelyincluded in the so-called skew angle between the two axes of the inter-particle distances. It iswell-known that the skew angle plays an important role in determining energy conversions [15] andtunneling probabilities [16].Mass effects are, without doubt, important in chemical reactions with higher degrees of freedom.However, it is not necessarily a trivial problem how to characterize the mass effect for reactiondynamics with more than two degrees of freedom. The reaction-path Hamiltonian formalism (see[17, 18]) may be useful for studying mass effects in high-dimensional reaction dynamics. In thisformalism, the curvature of the reaction path can be the key to quantify the mass effects. Onematter for concern is that this formalism em ploys the Eckart c onditions [19, 20] for a formalseparation of overall rotation and internal motion, which may disguise an imp ortant geometricproperties of molecular internal space by approximating the internal space as flat (Euclidean)space (see [21, 22]).For an appropriate desc ription of the mass effect in polyatomic molecules, correctly separatingrotational and internal motion is crucial already at the first stage. This is because rotational andinternal motion can couple dramatically, and a change in the mass distribution in a molecule playsan important role in the coupling process simply due to conservation of total angular momentum.In this respect, reduction theory from geometric mechanics [23, 24] and the associated gauge theory[21, 25–30] for n-body dynamics with rotational symmetry provides a quite universal basis for thestudy of the mass effect in m olecular reaction dynamics. According to these theories, atomic massesare incorporated into the metric tensor of internal space in a way that is independent of the choiceof body-fixed frame. Remarkably, the metric tensor is non-Euclidean for general three- and more-atom systems so that the internal space cannot be embedded into Euclidean space as in the caseof collinear triatomic reactions. Therefore, the characterization of the effect originating from thenon-Euclidean geometry of the internal space is quite important for understanding the mass effect.It should also be noted that the metric tensor of the internal space is independent of the potentialenergy surface, and is determined only by mass and shape of the system. Therefore kinematiceffects arising from the non-Euclidean nature of internal space should be of universal importance2in a variety of reactions.It is crucial to choose a good coordinate system for the study of the geometry of the internalspace. For triatomic systems, the so-called symmetrical coordinates [21, 30–32] and various hy-perspherical coordinates [34–39] are useful and have been extensively studied. An advantage ofthe symmetrical coordinates is that the me tric tensor of internal space becomes remarkably simple(conformally flat) [21, 30], and thereby a fairly uniform treatment of the three internal coordinatesis possible. For these reasons, we focus on these coordinates in this study. Its non-Euclidean natureindicates that trajectories in the internal space intrinsically possess “directionality” even withoutthe influence of the potential energy surface. This directionality will be shown to be important inthe branching processes in multi-channel reactions. A heuristic way to characterize this direction-ality is to investigate the behavior of a cloud of geodesics, which are the motion in the absenc e ofpotential energy surface. By comparing the directionality of multiple reaction