Internal valence force field

The second problem is due to the fact that the experimental infrared and Raman spectra of zeohtes are characterized by a relatively small number of broad and strongly overlapping bands. Hence, the number of force constants extremely exceeds the number of observable absorptions, and it is impossible to derive the complete force field from experimentally observed vibrational frequencies. In internal coordinates, assuming that the bonds in the lattice are largely covalent, the complete internal valence force field (IVFF) is given by... 

This type of representation of the potential energy in terms of the internal (valence) degrees of freedom is called a Valence Force Field. Valence force fields have long been used in vibrational spectroscopy in order to carry out normal mode analysis[j ]. Basically what the terms in equation (2) express are the energies required to deform each internal coordinate from some unperturbed... 

Sokel carried out the variational calculations necessary to calculate C44 for a few materials. In a similar way Chadi and Martin used the special points method to calculate C44 with the artificial restriction that all nearest-neighbor distances remain fixed. They then did a valence force field calculation, such as will be discussed in the next section, to correct for internal displacements. The results of these calculations are listed in Table 8-3. In spite of the extra complications, the theory seems to be about as accurate as it was for the calculation of c, — c,2. 

We saw in Section 8-C that the strain 64 introduced four complications in the problem that were not present in the strain ei- The valence force field bypasses three of these, but leaves us with internal displacements. These are of interest in their own right and must be included if we wish to predict C44 in terms of the valence force field and parameters obtained from c, and c,2. That will be an interesting prediction since it gives some measure of the validity of the valence force field model, so let us proceed with it. 

Force constant calculations are facilitated by applying symmetry concepts. Group theory is used to find the appropriate linear combination of internal coordinates to symmetry-adapted coordinates (symmetry coordinates). Based on these coordinates, the G matrix and the F matrix are factorized, which makes it possible to carry out separate calculations for each irreducible representation (c.f. Secs. 2.133 and 5.2). The main problem in calculating force constants is the choice of the potential function. Up until now, it has not been possible to apply a potential function in which the number of force constants corresponds to the number of frequencies. The number of remaining constants is only identical with the number of internal coordinates (simple valence force field SVFF) if the interaction force constants are neglected. If this force field is applied to symmetric molecules, there are often more frequencies than force constants. However, the values are not the same in different irreducible representations, a fact which demonstrates the deficiencies of this force field (Becher, 1968). 

Draeger, J.A. (1985) The methylbenzenes. 1. Vapor-phase vibrational fundamentals, internal rotations and a modified valence force-field, Specwochim. Acta., 41A, 607-627. Bemshtein, V. and Oref, I. (2001) Dependence of collisions lifetimes on translational energy, J. Phys. Chem. A. 105, 3454-3457. 

If one wishes to determine the vibrational force constants for a polyatomic molecule, isotopic substitutions are essential. Die general valence force field for a non-linear polyatomic molecule, expressed in internal coordinates, contains a total, of (3N-6)(5N-5) force constants. Some of these may equal others by the symmetry of the molecule, but a maximum of only (3N-6) vibration frequencies may be observed. 

Draeger, J.A. (1985). The methylbenzenes—I. Vapor-phase vibrational fundamentals, internal rotations and a modified valence force field. Spectrvchim. Acta A 41, 607. 

The general valence force field of pyramidal XY3-type molecules contains six independent constants. In internal coordinates, these are the stretching and bending force constants f and f , and four interaction constants f r for two adjacent bends, fr and fr . for a stretch and an adjacent and a nonadjacent bend, and fo two bends. Besides the four fundamental frequencies, additional spectroscopic data such as isotopic frequency shifts, centrifugal distortion constants, and Coriolis coupling constants are required to fix a unique set of force constants. Various authors have performed such calculations, and the sets so obtained [13, 14, 25, 26, 35 to 38, 45] agree reasonably well. As a representative example, the symmetry and valence force constants, all in mdyn/A, of Sawodny et al. [25] are presented below. 

The force constants frr = 6.503 mdyn/A, i = 0.70Q mdyn-A, frr =-0.068 mdyn/A, and fra = 0.706 mdyn were taken from an ab initio-computed general valence force field of 4th order in internal coordinates of NH2 in the X B state [1]. Force constants were also calculated by ab initio [2 to 5] and semiempirical [6, 7] procedures for the X Bi and A Ai states. 

Table 6.5 Valence Force Fields of cis- and trans-Polyacetylene in Internal Coordinates...

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