Abstract A new formula is developed for nuclear dynamics on a Jahn—Teller split potential energy surface. There are two single-valued electronic wave functions for the potential energy surface. We construct a two-component wave function for the nuclear dynamics. One component is a nuclear wave function for one of the two electronic wave functions and the other component is a nuclear wave function for the other electronic wave function. As is evident from the two components of the wave function, every vibronic level is at least doubly degenerate. The nuclear Hamiltonian does not have the permutation symmetry of identical nuclei because of the gauge potential arising from singularities of the Born—Oppenheimer approximation. But it is shown that for a scattering problem on the ground electronic surface for the X 3 E⊗e system, the two-component wave function is unitary-transformed to wave functions belonging to irreducible representations for the permutation symmetry group of three in asymptotic regions.

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