Anharmonic behavior

Due to a pronounced anharmonic behavior, the maximum position of the experimentally observed harmonic band 2vd is shifted to lower frequencies in comparison to the theoretically calculated value. Therefore, the calculated band of the harmonic vibration Id(2v) in Fig. 7.9a has been shifted 14 cm to lower frequencies to achieve a coincidence with the experimentally observed band. On the other hand, in the case of 2vq harmonic, theoretical band position matches well the experimentally observed maximum due much smaller anharmonicity of the G mode for the nanotubes. Therefore, G- and D-bands show very different anharmonic behavior as will be considered in more detail below. 

Before proceeding, it is necessary to specify more clearly what kind of anharmonic behavior will be dealt with in the framework of rovibrational modes. Consequently, a brief but important digression concerning the conventional treatment of interatomic potential functions in molecular studies is presented. We are interested in having at our disposal a potential energy function (either in analytical form or expressed as a series in powers of proper analytical terms) for reproducing the sequence... 

Note that the vibrational energy spacings on the Morse potential get smaller with increasing n, leading to anharmonic behavior. 

The most complicated anharmonic terms are found in the transverse [111] configuration, where both "forms" mentioned above appear at the same time forces are not parallel to the displacement, and th ir variation is not linear. Two calculations were performed with u = +0.004 (-2,1,1) a and the calculated forces were projected on the 3 perpendicular directions (-2,1,1), (111) and (0,-l,l). Whereas all projections on (0,-1,1) are zero, the "longitudinal" (111) component of the force at the first-neighbor site (<=+l) is, with the above ul, as much as 46 % of the "transverse" (-2,1,1) one at the sites <=-l, 2, the non-parallel components are still respectively 13 % and 20 % of the parallel ones. For obtaining the harmonic force constants, only the (-2,1,1) projection of the force was retained - and (in contrast to the transverse [100] case) a noticeable anharmonic behavior was still found k determined with displacements +u and -u were averaged, in order o eliminate the cubic contributions - because they still differed considerably for k j the cubic term lu in g(5.4.1) represents 7 Z of the ha onic one. It was verified" by a calculation with larger u that the influence of quartic anharmonicity is negligible. 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

Examination of Table 5 immediately reveals that the two anharmonic progressions diverge from the 3.3 p.m origin. One is shifted towards larger wavelengths (CH sym. stretch.) while the other is shifted towards smaller ones (CH asym. stretch.). This behavior can be understood from the shape of the potential as a function of the CH normal coordinates for the two vibrations (Figure 1). As illustrated, the two anharmonic surfaces deviate from the harmonic potentials in different ways. 

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]... 

Whereas the first microscopic theory of BaTiOs [1,2] was based on order-disorder behavior, later on BaTiOs was considered as a classical example of displacive soft-mode transitions [3,4] which can be described by anharmonic lattice dynamics [5] (Fig. 1). BaTiOs shows three transitions at around 408 K it undergoes a paraelectric to ferroelectric transition from the cubic Pm3m to the tetragonal P4mm structure at 278 K it becomes orthorhombic, C2mm and at 183 K a transition into the rhombohedral low-temperature Rm3 phase occurs. 

As mentioned earlier in this section, the convergence behavior of computed properties generally becomes less exponential as the quantities become less related to energies. By way of illustration, neither the application of the counterpoise procedure nor the addition of diffuse functions to the basis set improves the convergence behavior of the computed anharmonicities co x of the HF molecule (see Figure 10). Even in this case, however, both the uncorrected and corrected curves appear to be converging to the same limits. 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

In theory, the wave equations of quantum mechanics can be used to derive near-correct potential-energy curves for molecular vibrations. Unfortunately, the mathematical complexity of these equations precludes quantitative application to all but the very simplest of systems. Qualitatively, the curves must take the anharmonic form. Such curves depart from harmonic behavior by varying degrees, depending on the nature of the bond and the atom involved. However, the harmonic and anharmonic curves are almost identical at low potential energies, which accounts for the success of the approximate methods described. 

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