The Shin-Metiu model 39 is a one-dimensional molecular system that describes a proton-coupledelectron transfer reaction between a donor and an acceptor ion. The model consists of a transferring proton, an electron, and two fixed ions (a donor ion and an acceptor ion). The total Hamiltonian of the molecular system is ĤM = TR + Ĥel , where TR = P 2 R /2m p is the kinetic energy of the transferring proton, with the proton mass m p . In addition, Ĥel is the electronic Hamiltonian