Vibrational amplitude effects

Abstract Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. 

Vibrational amplitude effects for diatomic molecules can be straightforwardly calculated from spectroscopic data. Mean amplitudes and mean square amplitudes for diatomics are expressed in Equations 12.8a and 12.8b (see, e.g. Levine 1975)... 

Whereas there is little that one can do to overcome the effects of static disorder, the effects of thermal vibration can be significantly decreased by performing experiments at low temperatures, and, in fact, many solid samples are typically run at liquid nitrogen temperatures just to minimize such effects. An example of the effect of thermal vibration can be ascertained in Fig. 8 A, where the EXAFS amplitude decreases precipitously due to the large vibrational amplitude of the Cu—O bond. In general, failure to consider the effects of thermal vibration and static disorder can result in large... 

Eventually the average vibration amplitudes become large enough to saturate the effect as a result of increased entropy. Then the normal decline of flow stress with temperature begins. 

Fig. 12.1 (continued) (c) Isotope effects on mean square amplitudes (upper curve) and root mean square amplitudes (lower curve) as a function of temperature for hypothetical nondissociating molecules. At low temperatures the molecules are in the ground state and the amplitude is nearly independent of temperature. At higher temperature the vibrational amplitudes increase due to excitation into upper levels (Fig. 12.1) but the ratios drop smoothly to the classical value of unity at very high temperature (Fig. 12.1)... 

Here, AT is a constant, f is the incoming intensity, R is the distance of the scattered wave from the molecule (in practical terms, it is the distance between the scattering center and the point of observation), i and j are the labels of atoms in the jV-atomic molecule, g contains the electron scattering amplitudes and phases of atoms, 5 is a simple function of the scattering angle and the electron wavelength, I is the mean vibrational amplitude of a pair of nuclei, r is the intemuclear distance r is the equilibrium intemuclear distance and is an effective intemuclear distance), and k is an asymmetry parameter related to anharmonicity of the vibration of a pair of nuclei. 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

Actually, however, at least for an isolated XH Y system, the change in length of the H-bonds occurs rhythmically with the frequency of the v(XH Y) vibration. In effect the vKK vibration is frequency modulated by the v(XB. Y) vibration. From a consideration of this more precise classical picture Batuev [29, 30] has shown that the broad band should actually consist of a series of sub-bands of frequencies rXH (XH Y). This is the frequency modulation theory of the origin of the broad vXH bands. This explanation also implies that the band should become narrow at low temperatures when the amplitude of the H-bond stretching vibration is small. 

In the gas phase, the 180=C=160 band would not be doubled in this way. The only effects of isotopic modification would be to make the symmetric stretch (vj) weakly allowed, and to have slightly more stretching of C=160 in v3, and slightly less in vt. In an asymmetric environment, intermolecular forces on the two ends of the C02 molecule must balance, but the intermolecular force constants can differ. Since in v3, the vibrational amplitude of the C=160 bond is slightly greater than of the C=180 bond, the mode is more sensitive to the intermolecular force constant at the 160 terminus. When 160 is on the end with a higher force constant, the v3 frequency is shifted to higher frequency, while the opposite is true when the lsO is on the other end. 

This increase in A with increasing n is simply due to shortening of the tunneling distance with increasing vibration amplitude 8ln, and it is equivalent to the effect of increasing temperature for the incoherent tunneling rate [Benderskii et al., 1992b],... 

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