Quantum harmonic oscillator thermal bath Hamiltonians

H° and HFree are, respectively, the Hamiltonians of the fast and slow modes viewed as quantum harmonic oscillators, whereas Hint is the anharmonic coupling between the two modes, which are given by Eqs. (15), (21), and (22). Besides, He is the Hamiltonian of the thermal bath, while Hint is the Hamiltonian of the interaction of the H-bond bridge with the thermal bath. 

Here, H is the Hamiltonian of the quantum harmonic oscillator coupled to the thermal bath, which is that given by Eq. (1.1). 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

Hamiltonians (P.26) are those of the undriven quantum harmonic oscillator describing the H-bond bridge moieties a and b. Hamiltonian (P.27) is that of the driven quantum harmonic oscillators describing the a H-bond bridge moiety. Finally, Hamiltonians (P.29) are dealing with the coupling of the H-bond bridge with the thermal bath, whereas Hamiltonian (P.28) is that of the thermal bath. 

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