Proton transfer Hamiltonian equation

The beautiful point of this formulation is that the Marcus-Levich-Dogonadze result Equation (5) is the solution of the Hamiltonian Equation (7) in the deep tunneling limit. In addition, the solution of the Hamiltonian Equation (7) in the classical limit reproduces the TST result, corrected for recrossings of the barrier and for memory effects.12 These results mean that the Zwanzig Hamiltonian provides a unified description of proton transfer reactions in all the three parameter regions defined earlier in this section. 

The instanton theory of tunneling splittings in hydrogen-bonded systems and decay of metastable states in polyatomic molecules was studied by Nakamura et al. [182, 192, 195, 201-204, 216] They formulated a rigorous solution of the multidimensional Hamiltonian-Jacobi and transport equations, developed numerical methods to construct a multidimensional tunneling instanton path, and applied this method to HO [201], malonaldehyde [192, 195], vinyl radical [203], and formic acid dimer [202]. Coupled electron and proton transfer reactions were recently reviewed by Hammes-Schiffer and Stuchebrukhov [209]. 

Equation (106) shows that the interaction of the proton with the motion of the center of mass, described by the terms proportional to fx, is formally of the same form as the interaction with the medium atoms, and the first three terms in the Hamiltonian in Eq. (106) are equivalent to addition of one more degree of freedom to the vibrational subsystem. Thus, this problem does not differ from that for the process of tunnel transfer of the particles stimulated by the vibrations which were discussed in Section IV. So we may use directly the expressions obtained previously with substitution of the appropriate parameters. 

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