Calculations of vibrational frequencies

Assuming that a reasonable force field is known, the solution of the above equations to obtain the vibrational frequencies of water is not difficult However, in more complicated molecules it becomes very rapidly a formidable one. If there are N atoms in the molecule, there are 3N total degrees of freedom and 3N-6 for the vibrational frequencies. The molecular symmetry can often aid in simplifying the calculations, although in large molecules there may be no true symmetry. In some cases the notion of local symmetry can be introduced to simplify the calculation of vibrational frequencies and the corresponding forms of the normal modes of vibration. 

Regardless of the force field chosen, the calculation of vibrational frequencies by the method outlined above is based on the harmonic approximation. Tabulated values of force constants can be used to calculate vibrational frequencies, for example, of molecules whose vibrational spectra have not been observed. However, as anharmonicities have been neglected in the above analysis, the resulting frequency values are often no better than 5% with respect to those observed. 

We shall not discuss all the numerous energy minimisation procedures which have been worked out and described in the literature but choose only the two most important techniques for detailed discussion the steepest descent process and the Newton-Raphson procedure. A combination of these two techniques gives satisfactory results in almost all cases of practical interest. Other procedures are described elsewhere (1, 2). For energy minimisation the use of Cartesian atomic coordinates is more favourable than that of internal coordinates, since for an arbitrary molecule it is much more convenient to derive all independent and dependent internal coordinates (on which the potential energy depends) from an easily obtainable set of independent Cartesian coordinates, than to evaluate the dependent internal coordinates from a set of independent ones. Furthermore for our purposes the use of Cartesian coordinates is also advantageous for the calculation of vibrational frequencies (Section 3.3.). The disadvantage, that the potential energy is related to Cartesian coordinates in a more complex fashion than to internals, is less serious. 

Calculations of vibrational frequencies in a three-center bond as a function of Si—Si separation were performed by Zacher et al. (1986), using linear-combination-of-atomic-orbital/self-consistent field calculations on defect molecules (H3Si—H—SiH3). The value of Van de Walle et al. for H+ at a bond center in crystalline Si agrees well with the value predicted by Zacher et al. for a Si—H distance of 1.59 A. 

The most isotope sensitive motions in molecules are the vibrations, and many thermodynamic and kinetic isotope effects are determined by isotope effects on vibrational frequencies. For that reason it is essential that we have a thorough understanding of the vibrational properties of molecules and their isotope dependence. To that purpose Sections 3.1.1, 3.1.2 and 3.2 present the essentials required for calculations of vibrational frequencies, isotope effects on vibrational frequencies (and by implication calculation of isotope effects on thermodynamic and kinetic properties). Sections 3.3 and 3.4, and Appendices 3.A1 and 3.A2 treat the polyatomic vibrational problem in more detail. Students interested primarily in the results of vibrational calculations, and not in the details by which those results have been obtained, are advised to give these sections the once-over lightly . 

There are situations where exact equilibrium structures must be used. The most conspicuous is for the calculation of vibrational frequencies, as well as thermodynamic properties such as entropies obtained from calculated frequencies. As already discussed in Chapter 7, this is because the frequencies derive from the second derivative term, E", in a Taylor series expansion of the total energy. 

III. Computational Calculation of Vibrational Frequencies, and Band Assignments... 

With the development of analytical energy derivative methods135 l67, the calculation of vibrational frequencies (second derivatives of the energy with regard to atomic coordinates) and infrared absorption intensities (derivatives of the energy with regard to components of electronic field and atomic coordinates, i.e. dipole moment derivatives) both at the HF and correlation corrected levels has become routine168. There are six (two a " + four e)... 

In summary, calculations of vibrational frequencies confirm a crucial role for zero-point energies in secondary IEs on acidity. They are especially important when the C H bond or C-D bond is antiperiplanar or (to a lesser extent) synperiplanar to a lone pair. 

Calculations of vibrational frequencies are never accurate enough to verify that the secondary IE arises entirely from zero-point energies. Therefore although they do confirm a role for zero-point energies, which was never at issue, they cannot exclude the possibility of an additional inductive effect arising from changes of the average electron distribution in an anharmonic potential. The question then is whether it is necessary to invoke anharmonicity to account for a part of these secondary IEs. 

Let us now turn our attention to the calculated vibrational frequencies of H20,02F2, and B2H6. First of all, it should be mentioned that the calculation of these frequencies is a computationally expensive task. As a result, high-level calculations of vibrational frequencies are performed only for relatively small systems. When the calculated frequencies are examined and compared with experimental data, it is found that the former are often larger than the latter. Indeed, after an extensive comparison between calculation and experiment, researchers have arrived at a scaling factor of 0.8929 for the HF/6-31G(d) frequencies. In other words, vibrational frequencies calculated at this level are... 

The true minimum of the total energy is calculated in the 6-31G basis for the asymmetrically twisted pyramidal form 118 as 0.16 kcal mol -1 further stabilization with respect to 117. The additional calculation of vibrational frequencies indicates only positive force constants, which as second derivatives of the energy must all be positive for a minimum on the potential energy curve. 

For the points denoted by numbers in Figure 5 we performed additionally a calculation of vibration frequencies to determine the kind of extremal values. For 120, which corresponds to a local maximum of the energy curve or, more precisely, to a saddle point of the energy hypersurface, all but one force constants (as second derivatives of the total energy) are positive, which is the mathematical criterion for a saddle point181. 

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