Atom–molecule reactive scattering and symmetrized cross section for the system containing identical nuclei

A formulation of atom-diatomic molecule (or molecular ion) collision on a single potential surface of ground electronic state of triatomic (or ionic) system containing identical nuclei is presented based on the formal multiarrangement channel collision theory. A properly symmetrized scattering matrix which is unitary over all arrangement channels is introduced. The matrix makes the formulation quite systematic. Furthermore, it enables us to extend the formulation efficiently to more general collision systems containing several species of the identical particles by partitioning the arrangement channels into collections of the indistinguishable ones. This is illustrated in the simple three particle atom–diatom system of the present work. Through the matrix, formulas of the optical theorem satisfied by the symmetrized cross sections are also obtained. The adiabatic electronic wave functions are found to be either symmetric or antisymmetric under interchange of nuclear coordinates. It is determined by the electronic states but is independent of the fermion or boson characteristic of the nuclear spin. Formulas of the properly symmetrized cross sections that arise from the nuclear spin statistics of the indistinguishable nuclei are obtained for the symmetric as well as the antisymmetric electronic wave function of the triatomic system. They are obtained in terms of either resultant nuclear spins of the molecules or individual nuclear spin quantum numbers. These cross sections are the only observable ones in the experiment. Depending on the interchange symmetries of the electronic wave functions with respect to nuclear coordinates, the observable cross sections of the reaction for indistinguishable nuclei result in quite different characteristics as manifested, for example, in the principle of detailed balance. When the scattering takes place through direct and exchange collisions coherently, either resultant nuclear spin conversion or rotational parity conversion between initial and final molecules is an indication that the reaction has taken place via exchange scattering for certain type of collisions. However, for other type of collisions, only the resultant spin conversion is possible in the reactive (exchange) scattering. When a collision proceeds through exchange scattering only, an unusual spin statistical factor appears in the symmetrized cross section. The factor is not only a function of the spin quantum number of the identical nuclei and rotational quantum number of homonuclear molecule, but is also dependent on the symmetry property of the electronic wave function mentioned above. Our discussions are mainly confined to the cases where either the electronic wave function is nondegenerate on the potential surface or no conical intersection is present when it is degenerate so that the reaction takes place through continuous single valued electronic wave function with respect to the nuclear coordinates. The cases where the conical intersection is present are only briefly discussed by applying the recent result to the present formulas.