Second-order distortions

It was shown by several workers that in this case the first-order Jahn-Teller distortion is due to an ej vibration, and that the second-order distortion vanishes. Therefore, in terms of simple Jahn-Teller theoi, the moat around the symmetric point should be a Mexican hat type, without secondary minima. This expectation was borne out by high-level quantum chemical calculations, which showed that the energy difference between the two expected C2v structures ( A2 and Bi) were indeed very small [73]. 

According to Eq. (11), the force constant for the normal vibration Q, can be identified with the term in braces and can be negative if the second term, which is positive, exceeds the first term. If the force constant is negative, the energy should be lowered by the nuclear deformation Qi, and the second-order distortion from the symmetrical nuclear arrangement would occur spontaneously. 

It is worth to be noted that these definitions of first- and second-order distortions according to Warren-Averbach are model-free. From a linear or a quadratic increase of peak breadths it can neither be concluded in reverse that strain broadening, nor that paracrystalline disorder were detected. 

Madison, D.H. and Winters, K.H. (1983). A second-order distorted-wave model for the excitation of the 21P state of helium by electron and positron impact. J. Phys. B At. Mol. Phys. 16 4437-4450. 

In the expressions (184) and (184b) the second, temperature-dependent term defines the Born effect due to superposition of the two non-linear processes of second-order distortion and reorientation of permanent dipole moments in the electric field. Buckingham et al. determined nonlinear polarizabflities If and c for numerous molecules by Kerr effect measurements in gases as a function of temperature and pressure. It is here convenient to use the virial expansion of the molar Kerr constant, when the first and second virial coefficients Ak and Bk result immediately from equations (177), (178), and (184). Meeten et al. determined nonlinear molecular polarizabilities by measuring K in liquids as a function of temperature. 

Term III is a second-order distortion effect in which both electrons are in 2p states on cither nucleus. The distortion effects in the ionic... 

In an intensity modulated link the amplitude of the intermodulation terms depend primarily on the linearity of the modulation device, in this case the diode laser. The IM-free DR is defined as the maximum difference between the noise floor and the fundamental output, which produces distortion terms of equal amplitude to the noise floor. The noise floor in turn depends on the link bandwidth (BW), which varies by application. Consequently to make the IM-free DR measurements have general applicabihty, the results are often given in terms of a 1-Hz bandwidth. To use such results in a specific apphcation simply requires scaling the 1 -Hz data to the application bandwidth. The bandwidth scaling exponent depends on the order of the dominant distortion for second-order distortion, the IM-free DR scales as (BW) or the square root of bandwidth, for third-order distortion, the IM-free DR scales as (BW). ... 

Unlike the diode laser, the transfer function of many modulators is easily expressible in a simple analytical form. For example, the Mach-Zehnder transfer function is a simple raised cosine, 1 + cosrp, as shown by the solid curve in Fig. 9.55. Consequently, the linearity is a predictable function of the chosen bias point. For example, operating a Mach-Zehnder at /2 forces aU of the even-order distortion terms to zero. Using this bias point in a broad-band application means that the IM-free DR is determined by the odd-order distortion, which is dominated by the third-order term. Recall that for narrow-band systems the second-order distortion terms fall outside the pass band and consequently can be filtered out. Thus, for such applications the bias point can be moved away from /2 with no system consequence. One reason... 

A typical plot of third-order IM-free DR for a Mach-Zehnder biased at V /2 is shown in Fig. 9.60. Although the data in this plot were taken at 60 MHz, they should be valid at any frequency. This is because for a modulator, unlike the diode laser, the shape of the electrical-to-optical transfer function is independent of RF frequency. This has been experimentally demonstrated up to 20 GHz (Betts, Cox and Ray, 1990). A directional coupler modulator, when biased where the second-order distortion is zero, has almost identical third-order IM-free DR. 

Figure 2.26. Second-order distortions decrease on more powerful instruments. The spectrum corresponds to that shown in Figure 2.25 but was recorded on a 400-MHz instrument. Note that the distance between the inner two peaks is now greater (40.3 Hz) and the distortion between the sizes of the outer and inner peaks is smaller.

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

Remembering that the polarizabilities are expressed in units of it follows that the largest positive eigenvalue corresponds to the energetically most favorable second-order bond distortion, and the type... 

The second-order bond distortions predicted on the basis of the Huckel values are not in good agreement with those predicted using... 

The symmetries of the lowest excited states listed in Table 1 are nothing but the symmetries to which the most soft second-order bond distortions belong. It is seen that the types of symmetry reduction predicted using the symmetry rule are in complete agreement with those obtained on the basis of the dynamic theory. 

Table 1. The second-order bond distortion in the ground states of nonalternant hydrocarbons...

Using the same method as described in II.B, Binsch and Heil-bronner have examined the second-order bond distortion in the lowest excited states of nonalternant hydrocarbons (I, IV—VII, X, XI, XIII — XV and XVII), and have shown that, of the molecules examined, only VI and XVII suffer a molecular-symmetry reduction in the lowest... 

The type of the most favorable second-order bond distortion is given by the eigenvector corresponding to the largest eigenvalue. 

The relation between diamond and zinc blende shown above is a formal view. The substitution of carbon atoms by zinc and sulfur atoms cannot be performed in reality. The distortion of the NiAs structure according to Fig. 18.4, however, can actually be performed. This happens during phase transitions (Section 18.4). For example, MnAs exhibits this kind of phase transition at 125 °C (NiAs type above 125 °C, second-order phase transition another transition takes place at 45 °C, cf. p. 238). 

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