where M is chosen as the proton mass, ω b = 1000 cm -1 is the barrier frequency, and E b = 2120 cm -1 is the barrier height. For the matter Hamiltonian ĤM = T + V , the vibrational eigenstates |ν i ⟩ and eigenenergies E i are obtained by solving ĤM |ν i ⟩ = E i |ν i ⟩ numerically using the discrete variable representation (sinc-DVR) basis with 1001 grid points in the range of [-2.0, 2.0]. To facilitate the mechanism analysis, we diabatize the two lowest eigenstates and obtain two energetically degenerate diabatic states