Semiclassical study on multidimensional effects in tunneling chemical reactions: Tunneling paths and tunneling tubes

Abstract

The effects of multidimensionality in the quantum mechanical tunneling of chemical reactions are investigated. The aim of the present report is twofold. In the first place, we construct a new semiclassical theory to describe the tunneling by incorporating nonclassical solutions of the time-dependent Hamilton–Jacobi equation into the Feynman kernel. A systematic class of complex-valued (nonclassical) solutions for the time-independent Hamilton–Jacobi equation has been found that are generated along non-Newtonian paths in real-valued configuration space [K. Takatsuka and H. Ushiyama, Phys. Rev. A 51, 4353 (1995)]. In the present paper, the straightforward extension is applied to the time-dependent Hamilton–Jacobi equation, the solutions of which describe the tunneling in chemical reactions. It is shown that no damping factor due to the tunneling arises from the preexponential factor in the thus obtained nonclassical kernel, since it is still real valued, aside from the complex phase due to the Maslov index, and moreover its functional form is essentially the same as in the nontunneling case. Thus only the imaginary part of the action integral is responsible for the damping. A quasiclassical treatment of the semiclassical mechanics is developed to characterize the real-valued tunneling paths. In the second-half of this paper, some typical tunneling reactions in collinear three atomic systems on the LEPS (London–Eyring–Polanyi–Sato) potential surface are investigated in terms of our semiclassical theory. The effect of the initial energy distribution among the vibrational and translational modes is investigated asking which is preferable for tunneling and what is the resultant distribution of the energy in the product molecules. The following two factors to control the tunneling reactions are mainly examined as our first case study: (a) the mass effects featuring heavy–light–heavy and light–heavy–light patterns and (b) the anisotropy of the potential surface, namely, the early or late barrier. Tunneling paths of the types of Marcus–Coltrin and Miller–George are both generated spontaneously. A path of Marcus–Coltrin type takes a major role when the translational energy dominates in tunneling, while that of Miller–George type is dominant in a case where the vibrational excitation is important. As a distinguished feature of the multidimensionality in tunneling, we have identified what we call a tunneling tube, in which a bunch of the tunneling paths are involved emanating from the so-called caustic line. It turns out that the width of the tunneling tube determines in part the final energy distribution among the product vibrational modes.