Force constant calculations

One of the best available texts describing the principles of Raman scattering from crystals. Includes factor group calculations, polarization measurements, force constant calculations, and many other aspects of crystal physics. 

Force constant calculations are normally done in Cartesian coordinates. Suppose we have N atoms whose position vectors are Ri, R2,. .., Ra - Each of the atoms vibrates about its equilibrium position Ri g, Ri.e, , R v,e-The first step in our treatment is to define mass-weighted displacement coordinates... 

The molecule S12, like Se, is of Dsd symmetry but in the soHd state it occupies sites of the much lower C211 symmetry [163]. Due to the low solubihty and the thermal decomposition on melting only solid state vibrational spectra have been recorded [2,79]. However, from carbon disulfide the compound Si2-CS2 crystallizes in which the S12 molecules occupy sites of the high Sg symmetry which is close to 03a [163]. The spectroscopic investigation of this adduct has resulted in a revision [79] of the earher vibrational assignment [2] and therefore also of the earlier force constants calculation [164]. In Fig. 24 the low-temperature Raman spectra of S12 and Si2-CS2 are shown. 

The spectra of S12 and Si2-CS2 clearly reflect the differing site symmetries of the S12 molecule The two Eg vibrations at 188 and 245 cm degenerate in Si2 CS2, split into their components in pure S12 with its non-degenerate site symmetry of C2h- The valence force constants calculated for S12 reproduce the observed wavenumbers within 14 cm [79]. The two most interesting constants are/r = 2.41 and/ r = 0.60 N cm ... 

By means of force constant calculations based on a UBFF it has turned out that the force constant responsible for the interaction of neighboring bonds/rr is relatively high (fir/fr 0.25) [80]. The variation of bond length Ar of one bond with length r causes an opposite variation of the adjacent bond(s) with length / by —A/ Eq. (4) ... 

Since vibrational spectra of S2O2 have not yet been observed, the force constants calculated by ab initio MO methods were used to predict the harmonic vibrational wavenumbers of ds-S202 (C2v) and trans-S202 (C2I1) see Table 3 [34, 57]. 

Fredin, L., B. Nelander, and G. Ribbegard. 1977. Infrared spectrum of the water dimer in solid nitrogen. I. Assignment and force constant calculations. J. Chem. Phys. 66,4065. 

Liu and Zhou29 have computed the quadratic force field of cis-hexatriene by a systematic scaling of ah initio force constants calculated at the planar C2V structure. Their results reproduce satisfactorily the observed spectral features of this molecule. 

Kofraneck and coworkers24 have used the geometries and harmonic force constants calculated for tram- and gauche-butadiene and for traws-hexatriene, using the ACPF (Average Coupled Pair Functional) method to include electron correlation, to compute scaled force fields and vibrational frequencies for trans-polyenes up to 18 carbon atoms and for the infinite chain. 

The values of the force constant calculated from classical lADO and quantum chemical lNDO/2 methods, along with corresponding experimental values, are given in Table 1. Close agreement with experiments indicates that these two simple methods can be easily used to determine the force constant from ions in solution for which experimental values are not available. 

Although IR frequencies provide a useful measure of the extent of v bonding in carbonyl complexes, a better quantitative picture can be obtained from C—O force constants. These values are commonly derived from IR data by means of the Cotton-Kraihanzel force-field technique.33 This procedure makes certain simplifying assumptions in order to provide a practical solution to a problem that would be extremely difficult to solve rigorously Among the important assumptions are that the C—O vibrations are not coupled to any other vibrational modes of the molecule and that the observed frequencies can be used without correction for enharmonic effects. The results of force constant calculations of this type provide a means of setting up a 77-acceptor series 34... 

Force constant calculations for the in-plane vibrations of PtyX /t-X) - show that the terminal stretching force constants are larger than the bridging stretching force constants but that the difference diminishes for the heavier halides.1920... 

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 number of force constants calculated in the general potential function is very large. For molecules having no symmetry, such as carbohydrates, this number is equal to 1 + 2+3 + - - + (3 N-7) + (3 N-6) = (l/2)(3 N—6)(3 N — 5), so that it can be determined for small molecules only. For such large and complex molecules as sugars and their derivatives, additional information may be obtained from studies employing isotopes and model molecules. 

Force constant calculations have been especially valuable in the important field of carbonyl complexes M tL>(CO)z, where M is one of the d block elements and L represents some other ligand(s). It has already been noted that the C-O stretches around 2000 cm-1 are very useful in characterising such compounds. The C-O stretching force constants, if obtainable, should provide additional information. The nature of the bond between M and CO is discussed in more detail in Chapter 8 for the moment, we may say that the stronger the M-C bond, the weaker will be the C-O bond, if the conventional description is valid. Furthermore, the theory predicts the relative strengths of C-O bonds in carbonyls where the CO ligands are not all equivalent. For example, in complexes such as these ... 

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