Hamiltonian modes anharmonic coupling

Now, recall that for weak hydrogen bonds the high-frequency mode is much faster than the slow mode because 0 m 20 00. As a consequence, the quantum adiabatic approximation may be assumed to be verified when the anharmonic coupling parameter aG is not too strong. Thus, neglecting the diabatic part of the Hamiltonian (22) and using Eqs. (18) to (20), one obtains... 

Several studies of Fermi resonances in the absence of H bond have been made [76-80]. We shall account for this situation by simply ignoring the anharmonic coupling between the fast and slow modes (a = 0). The theory then describes the coupling between the fast mode and a bending mode through the potential Htf, with both of these modes being damped in the same way. Because aG = 0, the slow mode does not play any role, so that the total Hamiltonian does not refer to it ... 

In the Heitler-London approximation, with allowance made only for biquadratic anharmonic coupling between collectivized high-frequency and low-frequency modes of a lattice of adsorbed molecules (admolecular lattice), the total Hamiltonian (4.3.1) can be written as a sum of harmonic and anharmonic contributions ... 

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. 

As a consequence of the above equations, the full Hamiltonian describing the fast mode coupled to the H-bond bridge (via the strong anharmonic coupling theory) and to the bending mode (via the Fermi resonance process) may be written within the tensorial basis (222) according to [24] ... 

Once Eq. (3.15) has been reached, it is generally possible to separate the small-amplitude (harmonic) modes from the large-amplitude modes and treat any remaining anharmonic coupling terms by perturbation methods. The resulting small-amplitude vibrational Hamiltonian is given by ... 

State calculations. With the extensions provided, the method can be applied to the full Watson Hamiltonian [51] for the vibrational problem. The efficiency of the method depends greatly on the nature of the anharmonic potential that represents couphng between different vibrational modes. In favorable cases, the latter can be represented as a low-order polynomial in the normal-mode displacements. When this is not the case, the computational effort increases rapidly. The Cl-VSCF is expected to scale as or worse with the number N of vibrational modes. The most favorable situation is obtained when only pairs of normal modes are coupled in the terms of the polynomial representation of the potential. The VSCF-Cl method was implemented in MULTIMODE [47,52], a code for anharmonic vibrational spectra that has been used extensively. MULTIMODE has been successfully applied to relatively large molecules such as benzene [53]. Applications to much larger systems could be difficult in view of the unfavorable scalability trend mentioned above. 

We have seen that as a consequence of the harmonic Hamiltonian that has been set up thus far, our oscillators decouple and in cases which attempt to capture the transport of energy via heating, there is no mechanism whereby energy may be communicated from one mode to the other. This shortcoming of the model may be amended by including coupling between the modes, which as we will show below, arises naturally if we go beyond the harmonic approximation. The simplest route to visualizing the physics of this problem is to assume that our harmonic model is supplemented by anharmonic terms, such as... 

As an alternative that solves the kinetic coupling problem. Miller and co-work-ers suggested an all-Cartesian reaction surface Hamiltonian [27, 28]. Originally this approach partitioned the DOF into atomic coordinates of the reactive particle, such as the H-atom, and orthogonal anharmonic modes of what was called the substrate. If there are N atoms and we have selected reactive coordinates there will he Nyi = 3N - G - N-g harmonic oscillator coordinates and the reaction surface Hamiltonian reads... 

Thus, the Morse potential is the most appropriate for estimating bond energy in a molecule [4-9], The C-X and C-Y bond stretching modes in YCX3 molecules are treated as anharmonic Morse-like diatomic, non-linear coupled oscillators. The well-known [26-37] effective Hamiltonian of the local and normal mode model is used. 

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