Vibrations. Force Constants

Vibrational anharmonicity constant Vibrational coordinates Internal coordinates Normal coordinates, dimensionless Mass adjusted Vibrational force constants "eAe A,s get Ri, r 0J, etc. Qr m-i ... 

Parr, R. G., and Crawford, B. L., Jr., J. Chem. Phys. 16, 526, Molecular orbital calculations of vibrational force constants. I. Ethylene."... 

The parameters D and a can be obtained by fitting 1E to the actual ground state of the given molecule (D is determined by the observed bond dissociation energy and a is determined by the vibrational force constant). This allows one to express J and K in terms of available experimental information. That is, from eqs. (1.47), (1.49), and (1.52) we obtain... 

The usefulness of quantum-chemical methods varies considerably depending on what sort of force field parameter is to be calculated (for a detailed discussion, see [46]). There are relatively few molecular properties which quantum chemistry can provide in such a way that they can be used directly and profitably in the construction of a force field. Quantum chemistry does very well for molecular bond lengths and bond angles. Even semiempirical methods can do a good job for standard organic molecules. However, in many cases, these are known with sufficient accuracy a C-C single bond is 1.53 A except under exotic circumstances. Similarly, vibrational force constants can often be transferred from similar molecules and need not be recalculated. 

It is interesting to contrast the rate ratio for reactions 10.1 and 10.4 where either H or D atoms react with H2 with that for reactions 10.1 and 10.7 where common H atoms react with either H2 or D2 (compare Figs. 10.1a and b). In the first case, (kH,HH/kD,HH), there is a ZPE difference in the transition state but not the ground state consequently the high temperature KIE is inverse. In the second (kH,HH/kH,DD), however, there are zero point energy differences in both the transition and ground states. We expect the vibrational force constants to be smaller in the more loosely bound transition as compared to the ground state. The isotope effects scale with the force constant differences. Consequently RT[ln(kH,HH/kH,DD)] =... 

In Equation 12.8 Be is the rotational constant, Be = h/(8jt2I), (I is the moment of inertia), coe is the vibrational frequency, 27T(oe = (k/ix)1, (k the vibrational force constant and x the reduced mass), re the equilibrium bond length (isotope independent to reasonable approximation), and ae is the vibration-rotation interaction constant... 

Stemheimer electric field shielding tensors (26), electronic reorganization terms in vibrational force constants (28, 29 softness kernels (30, 31 ... 

In order to support further the analogy between the electronic structures of alkaline earth oxides and the isoelectronic alkali fluorides, several of their properties are compared in Table 8 in the following way. The ratios of the experimental values of r, fi, p, and the vibrational force constant k are given for the two fluorides, the two oxides, and each of the two isoelectronic pairs. The single values given in Table 7 for the intemuclear distance fcr in the crystalline state apply to the NaCl structure those of k are derived from oo of the following origin. 

Table 7. Intemuclear distances fcr (A) and vibrational force constants kg (md/A) > in molecules...

Detailed measurements have been made of the low-frequency Raman spectra of [Zn(py)2X2] (X = C1 or Br) and of the far-IR spectra of the complex where X = Q at liquid nitrogen temperature. It is found that skeletal molecular vibrations couple with lattice vibrations in the crystal, except for the Zn—X stretching vibrations. Force constant calculations indicate the Zn—N bond to be stronger in the bromide, while the Zn—Cl bond is stronger than the Zn—Br bond.477... 

The magnitude of the Si—Si vibrational force constants in cyclopolysilane rings also depends on the ring size and on the substituents attached to silicon. Compared with other silanes like disilanes, the force constants are generally weaker. Detailed results are summarized elsewhere5d. 

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). 

The vibrational force constants are obtained from measured vibrational frequencies and bond lengths along with extended Barger s rule correlations (36). 

The set of functions, together with the collection of terms that parameterize them (kb, r0, etc.), is referred to as the force field. In some cases force field parameters can be related to experimentally determinable values. For example, the bond stretching force constant kb is approximately equivalent to the vibrational force constant derived from an infrared spectrum. However, in general die force field terms are derived empirically with the target of reproducing experimental structures and energy distributions. 

Reduced v(CH) IR or Raman vibrational frequency as a consequence of lowered vibrational force constant. 

Keywords molecular vibrations, force-constants, scaled force field, vibrational spectra... 

From the separations of the progression members the vibrational constants coe, cOgXe, (oeye,... (Equation 6.16) for II) can be obtained. Comparison of coe = 2322 cm-1 for the X2Zg state of II) with coe = 3115 cm 1 for the XlZg state of H2 shows that the vibrational force constant (see Equation 6.2) is much smaller in the ground state of H)" than in H2. This, in turn, implies a weaker bond in H)" consistent with the removal of a bonding electron. 

It also emerges that parameter schemes based on thermodynamic (heats of formation, isomer distributions), spectroscopic (vibrational force constants) or structural experimental data are different and primarily useful for modeling the corresponding properties. The MOMEC force field is largely based on structural data. 

A second type of model, derives the potential barrier from vibrational force constants and molecular geometrical parameters alone, assuming a certain barrier shape. Such a procedure is much less accurate (especially for high barriers) than the previous one, but it may be used for estimating barrier heights in systems for which no level splittings have been observed. It may thus be of appreciable practical usefulness. 

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