Hamiltonian modes approximations

In the classical vibrational spectroscopy, the subject of investigation is the vibrational-rotational motion of polyatomic molecules near the very bottom of their potential-energy surface in the electronic ground state. In this case, the normal-mode approximation proves quite applicable. Indeed, one can expand as a Taylor series the potential energy of the molecule near the equilibrium position (the potential-energy minimum) and write down the molecular Hamiltonian in the form... 

The analogous coupling between the antisyimnetric stretch and bend is forbidden in the H2O Hamiltonian because of syimnetry.) The 2 1 resonance is known as a Femii resonance after its introduction [ ] in molecular spectroscopy. The 2 1 resonance is often very prominent in spectra, especially between stretch and bend modes, which often have approximate 2 1 frequency ratios. The 2 1 couplmg leaves unchanged as a poly ad number the sum ... 

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

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... 

Applying the adiabatic approximation, we restrict the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes an undamped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

We shall give here a brief summary of our previous work [71,72] that was concerned with the introduction of the relaxation phenomenon within the adiabatic treatment of the Hamiltonian (77), as was done in the undamped case by Witkowski and Wojcik [74]. Following these authors, we applied the adiabatic approximation and then we restricted the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes a damped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

Applying the exchange approximation and neglecting the zero-point energy terms, we may safely limit the representation of the Hamiltonian Hsf ex within the following reduced base which accounts for the ground states of each mode and for the first (second) overtone of the fast (bending) mode ... 

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

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 ... 

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