Force constants transferability

The lower vibrational levels have been studied by infrared (IR) and Raman spectroscopies [16]. Normal coordinate analyses based on force constants transferred from other molecules (Urey-Bradley type) or from ab initio HF calculations have played a part in the construction of the vibrational assignments [17]. The observed fundamental frequencies are given in Table 2.3. 

The vibrational frequencies were obtained from force constants transferred from the trlhallde given by Herzberg (1.). The bond length Is taken between those of the mono and trlhallde and the angle Is estimated to be the same as that In the trlhalldei... 

Three vibrational frequencies for both B OFg and B OFg molecules (or ions) were deerrained by vibrational analysis of the 5800 - A bands of the emission spectrum by Mathews and Innes (2). These values are corrected to the average isotopic species and adopted. The last three frequencies are estimated from values calculated by the valence-force method, using force constants transferred from C0F2(g). 

The functions are similar to those calculated by Hisatsune (5) based on the analysis of Devlin and Hisatsune (6) of the Raman and infrared spectra in terms of Urey - Bradley force constants. Devlin and Hisatsune used observed isotopic shifts in the spectra along with force constants transferred from nitrogen oxides and oxyhalides in order to select the N-N bond distance and the 0-N-N bond angle. 

The observed frequencies of the nitroamminecobalt(lll) complex ions can be explained by the force constants transferred from [Co(NH and [Co(NOj)/ . 

The Stable carbonyl and thiocarbonyl halide molecules have been studied by IR as well as Raman spectroscopy. Normal coordinate analyses based on force constants transferred from other molecules (Urey-Bradley type), or from ab initio calculations, have aided in the vibrational assignments. Some of the unstable molecules which have been observed in the microwave have been characterized by infrared spectroscopy. The somewhat lower sensitivity of this method means that long path lengths of the gas may be needed. The identification of the various stable and unstable species in the microwave spectrum is simplified by the fact that the absorption lines are usually well resolved from each other. The widths of the bands in the infrared may make the transient species difficult to detect against the stronger absorptions of the stable side products. IR and Raman spectroscopies do have the advantage that they can be used on solid and liquid samples. Since the bands in a low temperature rare gas matrix have a narrower profile, the infrared spectrum is usually simplified over the room temperature gas phase spectrum. Moreover, the vibrational frequencies are only mildly perturbed by solid state effects. For example, CF Se has not been observed in the vapor phase, yet its vibrational dynamics are known from its matrix isolation spectrum. Table 9 gives the vibrational data for the carbonyl, thiocarbonyl, seleno-carbonyl and formyl halides. 

Very early force fields were used in an attempt to calculate structures, enthalpies of formation, and vibrational spectra, but it was soon found that accuracy suffered severely in either the structure-energy calculations or the vibrational spectra. Force constants were, on the whole, not transferable from one field to another. The result was that early force fields evolved so as to calculate either structure and energy or spectra, but not both. 

When the same energy terms are used in two force helds, it may be acceptable to transfer bond-stretching and angle-bending terms. These are fairly stiff motions that do not change excessively. The force constants for these terms vary between force helds, much more than the unstrained lengths and angles. 

Fig. 18. Rate constant calculated with the use of (2.80a) plotted against (m/mH). The hydrogen transfer rate is assumed to be 10 s the effective symmetric vibration mass 125mH. The ratio of force constants corresponding to the intra (Kq) and intermolecular (K,) vibrations is (Ki/Ko) = 2.5 x 10 , 5 x 10 and l.Ox 10 for curves 1-3, respectively.

The key study for our development of molecular mechanics was that by Scbachtschneider and Snyder (1969), who showed that transferable force constants can be obtained provided that a few off-diagonal terms are not neglected. These authors found that olf-diagonal terms are usually largest when neighbouring atoms are involved. A final point for consideration is that the C atom in OCS is obviously chemically different from a C atom in ethane and from a C atom in ethyne. It is necessary to take account of the chemical environment of a given atom. 

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. 

In particular, the TS trajectory remains bounded for all times, which satisfies the general definition. The constants c and c in Eq. (39) depend on the specific choice of the TS trajectory. Because the saddle point of the autonomous system becomes a fixed point for large positive and negative times, one might envision an ideal choice to be one that allows the TS trajectory to come to rest at the saddle point both in the distant future and in the remote past. However, this is impossible in general because the driving force will transfer energy into or out of the bath modes in such a way that... 

Terms representing these interactions essentially make up the difference between the traditional force fields of vibrational spectroscopy and those described here. They are therefore responsible for the fact that in many cases spectroscopic force constants cannot be transferred to the calculation of geometries and enthalpies (Section 2.3.). As an example, angle deformation potential constants derived for force fields which involve nonbonded interactions often deviate considerably from the respective spectroscopic constants (7, 7 9, 21, 22). Nonbonded interactions strongly influence molecular geometries, vibrational frequencies, and enthalpies. They are a decisive factor for the transferability of force fields between systems of different strain (Section 2.3.). 

The subscript e means that the second derivatives are taken for internal parameters of the known or assumed molecular geometry.) It then follows that, in principle, to every molecule a F-matrix is assigned with molecule-specific force constants Fd.Transferability can only be expected for molecules which contain qualitatively and quantitatively very similar internal coordinates For instance, a force field which reproduces very well frequencies of n-al-kanes (from which it was derived) gives significantly worse results for branched alkanes... 

For frequency calculations one usually starts out with a set of approximate existent force constants (e.g. taken over from similar, already treated molecules under the preliminary tentative assumption of transferability), and subsequently varies the force constants in a systematic way by means of a least-squares procedure until the calculated frequencies (square roots of the eigenvalues of Eq. (10)) agree satisfactorily with the experimental values. Clearly, if necessary, the analytical form of the force field is also to be modified in the course of this fitting process. 

F specific energy (= force constant), kj/kg FF fouling factor (heat transfer), J/(m2 s °C)... 

Equation 9.6 and Equation 9.7 require that any interface between two fluids in equilibrium must have the constant curvature H. Appearance of two points with different curvatures results in appearance of a corresponding AP, which forces substance transfer until equalization of H. This gives rise to a... 

It may be concluded that for reactions where the proton is less or more than one-half transferred in the transition state, i.e. the A—H and H—B force constants are unequal, the primary hydrogen-deuterium kinetic isotope effect will be less than the maximum of seven. The maximum isotope effect will be observed only when the proton is exactly half-way between A and B in the transition state. This relationship is also found for carbon kinetic isotope effects where the isotopically labelled carbon is transferred between two atoms in the reaction10,11. This makes interpreting carbon isotope effects difficult. 

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