Mechanical anharmonicity

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

A molecule may show both electrical and mechanical anharmonicity, but the latter is generally much more important and it is usual to define a harmonic oscillator as one which is harmonic in the mechanical sense. It is possible, therefore, that a harmonic oscillator may show electrical anharmonicity. 

One effect of mechanical anharmonicity is to modify the Au = t infrared and Raman selection rule to Au = 1, 2, 3,. .., but the overtone transitions with Au = 2, 3,... are usually weak compared with those with Au = t. Since electrical anharmonicity also has this effect both types of anharmonicity may contribute to overtone intensities. 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

In a diatomic molecule one of the main effects of mechanical anharmonicity, the only type that concerns us in detail, is to cause the vibrational energy levels to close up smoothly with increasing v, as shown in Figure 6.4. The separation of the levels becomes zero at the limit of dissociation. 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

B. The Main Mechanism Strong Coupling Theory of Anharmonicity... 

On the other hand, one has to take into account the influence of the surrounding which must induce an irreversible evolution of the H-bond system when its fast mode is excited the fast mode may be directly damped by the medium that is the direct relaxation mechanism. It may be also damped through the slow mode to which it is anharmonically coupled, that is the indirect relaxation mechanism. A schematical illustration of these two damping mechanism is given in Fig. 2. Of course, the role played by damping must be more important for H bonds in condensed phase. 

The cornerstone of the strong anharmonic coupling theory relies on the assumption of a modulation of the fast mode frequency by the intermonomer distance. This behavior is correlated by many experimental observations, and it is undoubtly one of the main mechanisms that take place in a hydrogen bond. Because the intermonomer distance is, in the quantum model, represented by the dimensionless position coordinate Q of the slow mode, the effective angular frequency of the fast mode may be written [52,53]... 

There are two kinds of damping that are considered within the strong anharmonic coupling theory the direct and the indirect. In the direct mechanism the excited state of the high-frequency mode relaxes directly toward the medium, whereas in the indirect mechanism it relaxes toward the slow mode to which it is anharmonically coupled, which relaxes in turn toward the medium. 

It may be shown [8] that both semiclassical [83,84], and full quantum mechanical approaches [7,32,33,58,87] of anharmonic coupling have in common the assumption that the angular frequency of the fast mode depends linearly on the slow mode coordinate and thus may be written... 

The concept of discrete or quantized energy levels can be superimposed on this diagram by representing them as a series of horizontal lines, the spacing of which becomes closer with increasing energy due to the anharmonic nature of the vibration. In quantum mechanical terms, these levels are labelled V =... 

T. Di Paolo, C. Bourderon, and C. Sandorfy, Model calculations on the influence of mechanical and electrical anharmonicity on infrared intensities Relation to hydrogen bonding. Can. 

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