Anharmonicity bond lengths

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

The HF vibrational frequencies are too high by about 7% relative to the experimental harmonic values, and by 10-13% relative to the anharmonic values. This overestimation is due to the incorrect dissociation and the corresponding bond lengths being too short (Table 11.1), and is consequently quite general. Vibrational frequencies at the HF level are therefore often scaled by 0.9 to partly compensate for these systematic errors. 

Wu and Farges (1999) have made use of eqn (9.17) relating bond valence to the coefficient of thermal expansion to confirm that it is possible to resolve the different thermal expansions of the long and short Th-O bonds in thorite (o -ThSi04) from XAFS spectra measured between room temperature and 1700K. They also use this relation to estimate the anharmonic corrections needed for the bond lengths determined from XAFS (Brown et al. 1995, pp. 358-9). 

Intriguing evidence for an inductive effect comes from computations that treat nuclei quantum mechanically.11 This takes account of anharmonicity and leads to bond lengths and atomic charges that vary with isotopic substitution. Whether those charge variations are large enough to account for the IEs on acidity, independently of changes of vibrational frequencies, is not yet clear. 

The repulsive frequency shift, Av0, is expressed explicitly in terms of the first and second derivatives of the excess chemical potential (equation 2) along with the vapor phase vibrational transition frequency, vvib, equilibrium bond length, re, and harmonic and anharmonic vibrational force constants, f and g (232528). 

Complex CO bond length Exr> Calc MC bond length Exr> Calc VcoUp [4] (anharmonic) vcoCalc (harmonic) vco1 / VC0Calc... 

The experimental MC and CO bond lengths are well reproduced by the calculations. For the CO stretching mode, the ratio of the experimental anharmonic (2143 cm-1) to the harmonic frequency (2170 cm-1) is equal to 0.988. The calculated harmonic vcoreproduces well the harmonic CO frequency. The vco 2x11 / Vco Cal° ratios obtained fit this value with an accuracy equal to or better than 1%, showing that the method used for the frequency calculation is accurate enough. Hence, the chosen method appears to be adequate for our study,... 

Near the equilibrium bond length qe the potential energy/bond length curve for a macroscopic balls-and-spring model or a real molecule is described fairly well by a quadratic equation, that of the simple harmonic oscillator (E = ( /2)K (q — qe)2, where k is the force constant of the spring). However, the potential energy deviates from the quadratic (q ) curve as we move away from qc (Fig. 2.2). The deviations from molecular reality represented by this anharmonicity are not important to our discussion. 

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