Double harmonic model

The zeroth-order approximation in the BK perturbation treatment of pure vibrational NLO is the double harmonic model. As far as electrical properties are concerned this approximation includes just the terms in the instantaneous property expression that are linear in the normal coordinates (there is no vibrational contribution from the constant term). To these are added the quadratic terms in the pure vibrational (or mechanical) potential which constitute the usual harmonic approximation. Then, in zeroth-order roughly half of the square brackets vanish leaving ... 

Fig. 4. Schematic representation of a 3-atom Double Harmonic Oscillator (DHO) with r, r2 interatomic distances and k, k2 force constants, and of TR-A, TR-B, TR-C temperature ranges corresponding to the DHO model.

Suitieri97 has also evaluated the anharmonic contributions to the nuclear relaxation y for some push-pull polyenes using analytical methods in a valence bond charge transfer model. Saal and Ouamerali98 have investigated the vibrational ft of N-fluorophemyl-2,5-dimethypyrrole in the double harmonic... 

RO, Fig. 3d) (2) higher-frequency, smaller amplitude, quasi-harmonic oscillations (QHO, Fig. 3a) and (3) double-frequency oscillations containing variable numbers of each of the two previous types. By far the most familiar feature of the BZ reaction, the relaxation oscillations of type 1 were explained by Field, Koros, and Noyes in their pioneering study of the detailed BZ reaction mechanism.15 Much less well known experimentally are the quasiharmonic oscillations of type 2,4,6 although they are more easily analyzed mathematically. The double frequency mode, first reported by Vavilin et al., 4 has been studied also by the present author and co-workers,6 who explained the phenomenon qualitatively on the basis of the Field-Noyes models of the BZ reaction. 

The two-dimensional PES shown in Figure 8.17 (as well as in Figures 8.3b and 8.7c) is typical of internal rotation coupled to inversion of the other part of the system. This situation is also realized in methylamine inversion, where the rotation barrier is modulated not by a harmonic oscillation but by motion in a double-well potential. The PES for these coupled motions can be modeled as follows ... 

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films. 

Figure 13 Energy level scheme for a system of two coupled oscillators. The isolated peptide states (left side) are coupled by some weak interaction, which mixes them to generate the excitonic states (right side). Anharmonicity, which is crucial for understanding the 2D pump probe spectra, is introduced into this model by lowering the energies of the double excited monomeric site states i2) and j2) by A from their harmonic energies 2eu- This anharmonicity mixes into all coupled states, giving rise to diagonal anharmonicity (Ae ) and off-diagonal anharmonicity (mixed-mode anharmonicity, Ae i) in the basis of the normal modes discussed in the text.

We can see from Eqs. (3.5) [see also Appendix in Ref. ] that the force constants F , F etc., are generally mass dependent quantities. To arrive at the iso-topically invariant potential function we must therefore express these quantities in terms involving mass independent valence force constants and to fit these to experimental spectra [cf. ]. For ammonia, this would represent a really formidable numerical problem. Taking into account the proposed limits of our model, the fact that we are mainly interested in the inversion—rotation structure of the spectra, we have overcome the above mentioned difficulties in the following way [see for details] (i) all the enharmonic force constants in Eq. (5.5) were neglected (ii) the p-dependent contributions to the harmonic force constants F [see Eq. (4.7)] were neglected (iii) the least squares fit of the double-minimum potential function parameters and the p-independent harmonic force constants were performed for light isotopes ( NHs, NHs) and heavy isotopes ( ND3, NTs) separately. 

This has the form of a double-well oscillator coupled to a transverse harmonic mode. The adiabatic approximation was discussed in great detail from a number of quantum-mechanical calculations, and it was shown how the two-dimensional problem could be reduced to a one-dimensional model with an effective potential where the barrier top is lowered and a third well is created at the center as more energy is pumped into the transverse mode. From this change in the reactive potential follows a marked increase in the reaction rate. Classical trajectory calculations were also performed to identify certain specifically quanta effects. For the higher energies, both classical and quantum calculations give parallel results. 

These correlation functions are double exponentials, and there is no single exponential relaxation so long as the motion is confined to one harmonic osdllator potential well, though, of course, the decay towards y(oo) will not deviate much from one exponential for t > P As a rule each molecule will have at least two distinct possible positions of stable equilibrium. In the long nm, thermally activated jumps between the potential wells centred on these equilibrium positions must give rise to exponential decay as in the initial two-site model of relaxation. 

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