Deviations from harmonicity

The results of such measurements would be unsurprising to the molecular spectroscopist where the system is, like ZrH2, reasonably harmonic but some systems show significant deviation from harmonicity, like NbHo.3, see Table 6.10 [64]. A spectral characteristic of these systems is the unusually weak transitions at the higher energies. It appears that the high energy wavefimctions are more extended than the... 

These deviations from harmonicity (i.e., from the proportionality of force and displacement) are related to the smaller and smaller force needed to elongate the bond and the larger and larger force needed to shorten the bond. 

The thermodynamic functions of Freon-20 in the ideal gas state were calculated in several works. As a rule, calculations are carried out by approximating them to the model of a rigid rotary-harmonic oscillator (RRHO). The deviation from harmonic oscillations was taken into account only in the work of Barho [0.37], the correction being found by semiempirical method. 

Deviations from harmonic behaviour are also found above about 200 K, however, only for the amorphous samples. These high temperature anharmonicities occur often far below Tg, which is typically around 300-350 K. They are supposed to be caused by residual solvents in the polymer matrix. We have also studied / T) for some polymers with a relatively low Tg of 250-300 K. In f T) decreases rapidly following a V c - T dependence as predicted by mode coupling theory (MCT). This is interpreted as the onset of local processes. Tc represents the transition from non-ergodic to ergodic behaviour, which occurs typically 30-150 K above the macroscopic glass transition temperature Tg. In Fig. 15.10 we show f T) for PROPS. The MCT fit is indicated by the broken line yielding Tc = 306 12 K whereas Tg k, 240 K. Simultaneously with the onset of anharmonic behaviour of/the Mofl-bauer resonance lines broaden and quasi-elastic lines appear close to Tc. 

Overtone spectra are due to deviations from harmonic vibrational motion and, compared to the fundamental ones, give valuable information on the harmonic frequencies coe, the anharmonicities Xe, the shape of the potential curve and hence on the dissociation energy, which is related to the former by... 

Three-body and higher terms are sometimes incorporated into solid-state potentials. The Axilrod-Teller term is the most obvious way to achieve this. For systems such as the alkali halides this makes a small contribution to the total energy. Other approaches involve the use of terms equivalent to the harmonic angle-bending terms in valence force fields these have the advantage of simplicity but, as we have already discussed, are only really appropriate for small deviations from the equilibrium bond angle. Nevertheless, it can make a significant difference to the quality of the results in some cases. 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

The harmonic frequencies calculated with different DFT functionals as a function of basis set are shown in Tables 11.17-11.19. The convergence as a function of basis set is similar to that observed for the HF method. The B3PW91 functional again shows the best performance. With the cc-pV5Z basis set the deviations from the experimental harmonic frequencies are only 0cm", 9 cm" and 17 cm", substantially better that the... 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

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. 

Table 6-1. Harmonic frequencies and experimental fundamentals for ozone [cm 11. Deviations from the experimental result [%] are given in parentheses.

The effects of deviations from the Born-Oppenheimer approximation (BOA) due to the interaction of the electron in the sub-barrier region with the local vibrations of the donor or the acceptor were considered for electron transfer processes in Ref. 68. It was shown that these effects are of importance for long-distance electron transfer since in this case the time when the electron is in the sub-barrier region may be long as compared to the period of the local vibration.68 A similar approach has been used in Ref. 65 to treat non-adiabatic effects in the sub-barrier region in atom transfer processes. However, nonadiabatic effects in the classically attainable region may also be of importance in atom transfer processes. In the harmonic approximation, when these effects are taken into account exactly, they manifest themselves in the noncoincidence of the... 

Studies on the statistical deviation from an ideal bond order support the relatively high aromaticity of 1,2,5-thiadiazole (Table 7). The harmonic oscillator model of aromaticity (HOMA) value for 1,2,5-thiadiazole has not yet been reported. 

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