Harmonic oscillator anharmonic coupling

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

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. 

In the perfectly harmonic crystal the total vibrational energy equals the sum of the energies of the simple harmonic oscillators or normal modes which have infinitesimal amplitude and ignore each other. The introduction of anharmonic coupling between oscillators leads initially to small shifts in... 

The 03 dependence of the force constant k(Q) = rnoP iQ) creates an anharmonic coupling between both modes q and Q. It is at the origin of the exceptional width of bands of H-bonds, and of their shifts towards lower wavenumbers when compared to bands of the same X H molecules, when they do not establish H-bonds. These most intense bands are 0 1 transitions, or transitions between the ground vibrational state of q and its first excited state. The other possible transitions, 0 n with n > 1, have intensities equal to 0 in the case of an harmonic oscillator in q (no terms in q, (f, etc. in V q, Q)). Overtones of v, particularly those corresponding to 0 2 transitions in the 6000-6500 cm region, however often... 

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... 

Recently, the above approach has been applied to systems with a collection of harmonic oscillators coupled with rigid rotors and systems with a collection of anharmonic oscillators by Tou and Lin. For a system with a collection of w harmonic oscillators. . . , g, . . . g ) coupled... 

In the context considered here, a resonance is a near match of frequency between two coupled oscillations. Such a resonance will produce energy transfer from one of the oscillators to the other. A nonlinear resonance is a resonance arising from the nonlinearity of the restoring force in one or both of the oscillators, or in other words, due to the anharmonicity of one or both of the oscillators. For a harmonic oscillator, of course, the frequency of oscillation is independent of the energy or amplitude of the oscillation. Molecular vibrational modes, however, are both anharmonic, particularly at energies sufficient for unimolecular reaction, and the energy dependence of the oscillator frequency is critical to mode-mode energy transfer. 

The two baths are usually represented by sets of independent harmonic oscillators. For the molecular unit, we are mainly interested in 1-dimensional periodic chains made of N units with anharmonic force fields. System-bath coupling is either harmonic or nonlinear and is typically taken to be weak. We will assume that only the end atoms of the chain, 1 and Af, couple to the surfaces, L and R, respectively, and neglect direct interactions between the reservoirs. 

The vibrational spectra S co) after the second pulse in the cases of t = 134 and 201 fs are shown in Fig. 7.8, which clearly indicate that the amplitude of the hg(l) mode is enhanced for t = 134 fs and the predominant mode is switched to the ag(l) mode for t = 201 fs. In short, a Raman active mode is strongly excited if r is chosen to equal an integer multiple of its vibrational period TVib, and the energy of the mode takes the minimum if x is equal to a half-integer multiple of Tvib. This is known to be valid for the harmonic oscillator model. We proved that this is also the case for the potential surface of highly excited Ceo which includes anharmonic mode couplings by nature. 

Smoothed densities of levels p( ) for Morse oscillators coupled with harmonic oscillators etc. are given in refs. 6, 108. From these expressions the anharmonicity contributions to p( o) can be obtained easily. A further important expression which allows us to estimate the maximum contribution from rotation is derived as follows. If one assumes... 

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