Normal Vibration Analysis

A more detailed and complete understanding of the structure and potential function of a molecule requires that we be able to predict correctly the numerical values of its normal vibrational frequencies. These arc determined by the kind of theoretical analysis outlined in Section I. 1., which we have seen invokes knowledge of the mass distribution and force field of the molecule. If we can show that the experimentally observed bands correspond not only qualitatively but also quantitatively to those predicted, we have a secure basis for claiming detailed knowledge about a molecule. 

Such normal vibration analyses have been applied to the spectra of macromolecules to only a limited extent. In the first place, the only structure which has been analyzed in detail is that of the planar zig-zag chain of CHg groups, i.e., polyethylene. Neither substituted planar zig-zag chains nor the helical chain structures characteristic of many polymers [Bunn and Holmes (28)] have been submitted to such a theoretical analysis. In the second place, even for the case of polyethylene the answers are not in all instances unambiguous. Different assumptions as to the nature of the force field, and lack of knowledge of some of the force constants, has led to varying predictions of band positions in the observed spectrum. For the identification of certain modes, viz., those which retain the characteristics of separable group frequencies, such an analysis is not of primary importance, but for knowledge of skeletal frequencies and of interactions 

In these equations, m is the mass of an element (a CHa group), kc is the force constant for stretching of a C—C bond, ka is the force constant for bending of the CCC angle, and p is the phase difference between the motions of adjacent elements of the chain. Since we are interested only in the factor group modes, i.e those in which the vibrations of corresponding elements in neighboring unit cells are in phase, we require that 

In cases where q = 4, e. g., when the positions of substituents on the chain increase the true identity period to four elements, we can show that two other skeletal modes can become potentially active. This of course assumes that the substituents can be neglected and the previous treatment used, an unlikely assumption. Nevertheless the results may serve as crude guides to the location of other possible skeletal frequencies for such polymers. Application of equations (34)—(37) shows that these modes are expected at 

In addition, an out-of-plane skeletal mode can be active whose frequency is given by [Pitzer (774)] 

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The repeat distance along the chain axis (0.468 nm) is significantly less than that calculated for a planar zigzag stmcture. Therefore, the polymer must be in some other conformation (65—67). Based on k and Raman studies of PVDC single crystals and normal vibration analysis, the best conformation appears to be where the skeletal angle, is 120°, and the torsional of opposite sign) is 32.5°. This conformation is in... 

Beware that this type of coupling often may go undetected in a normal vibration analysis. Since the ghost frequencies are relatively high compared to the expected real frequencies, they are often outside the monitored frequency range used for data acquisition and analysis. 

Summarizing, we may say that the normal vibration analysis permits a good understanding of the skeletal modes (and hydrogen modes, as we shall see later) of the polyethylene chain. Extension of such calculations to other polymers is needed. 

F= 17.4 MHz (fixed to a value derived from a normal vibration analysis)... 

The theoretical predictions based on the normal vibration analysis also give valuable help in assignments of bands in the FIR spectra of high-crystalline POE [22]. FIR spectrum of POE is shown in Fig. 3d. There is the 215 cm (prediction 211 cm ) band that should be assigned to skeletal deformation vibrations. The band at 165 cm (prediction 162 cm is related to torsional vibrations around the C-C bond, whereas tte band at 106 cm (prediction 108 cm ) - mostly to those around the C-O bond. 

Other attempts to ign absorption bantk in the FIR spectra of amorphous polymers using the theoretical normal vibration analysis have revealed its inadequacy in the case of low-frequency skeletal vibrations [5, 7, 8, 21]. 

As noted in Sect. 2, normal vibration analysis predicts absorptitm l nds due to the low-frequency skeletal modes of a chain of the carbon-carbon badtbone in the range 100-200 cm Apparently, the suppression of the segmental motion leads to a decrease of the absorption by wliich the short range order in the backbone chain is reflected and only the modes allowed by the selection rules are kept. 

Torres, F. J., Civalleri, B., Pisani, C, Musto, P, Albunia, A. R., Guerra, G. Normal vibrational analysis of a trans-planar syndiotactic polystyrene chain. /. Phys. Chem. B., Ill, 6327-6335 (2007). 

The method of vibrational analysis presented here ean work for any polyatomie moleeule. One knows the mass-weighted Hessian and then eomputes the non-zero eigenvalues whieh then provide the squares of the normal mode vibrational frequeneies. Point group symmetry ean be used to bloek diagonalize this Hessian and to label the vibrational modes aeeording to symmetry. 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

To perform a vibrational analysis, choose Vibrationson the Compute menu to invoke a vibrational analysis calculation, and then choose Vibrational Dectrum to visualize the results. The Vibrational Spectrum dialog box displays all vibrational frequencies and a simulated infrared spectrum. You can zoom and pan in the spectrum and pick normal modes for display, using vectors (using the Rendering dialog box from Display/Rendering menu item) and/or an im ation. 

Vibrations in proteins can be conveniently examined using normal mode analysis of isolated molecules. The results of such analyses indicate the presence of a variety of vibra-... 

Normal mode analysis provides a good example of information which is obtainable only through a theoretical calculation, since spectroscopic data does not directly indicate the specific type of nuclear motion producing each peak. Note that it is also possible to animate vibrational modes in some graphics packages. 

A continuing periodic change in a displacement with respect to a fixed reference. The motion will repeat after a certain interval of time. Vibration analysis monitors the noise or vibrations generated by plant machinery or systems to determine their actual operating condition. The normal monitoring range for vibration analysis is from less than 1 to 20,000 Hertz. 

The second method uses the slip frequency to monitor for loose rotor bars. The passing frequency created by this failure mode energizes modulations associated with slip. This method is preferred since these frequency components are within the normal bandwidth used for vibration analysis. 

In the Wilson matrix analysis, the normal vibrational modes in methyl radical were assumed to be the same as those in both methyl ions. This assumption is rather crude however, it is believed to influence the results very little. The following values for the normal vibrational modes were obtained (in cm" ) 3100 3100 2915 1620 1620 1030. 

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