Normal mode analysis of molecular vibrations

Molecules consist of atoms held together, bonded, by interatomic forces. These interatomic forces are electromagnetic interactions arising from the motion of electrons and determine the shape and size of molecules. At absolute zero, in the classical interpretation, the molecules have an equilibrium configuration that is determined by the energy minimum of these interatomic forces. We shall analyse how the motions of atoms in a molecule can be calculated and describe how the interactions between atoms in molecules are calculated. 

The atoms in a molecule undergo vibrations around their equilibrium configuration within the quantum mechanical picture, even at zero temperature. The application of elementary djmamical principles to these small amplitude vibrations leads to normal mode analysis. Crystalline solids can naively be thought of as big molecules but solving the equations becomes impossible imless the periodicity of the unit cell is included whereupon major simplifications of the algebra are introduced. 

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Sidebar 10.3 outlines the useful analogy to normal-mode analysis of molecular vibrations, where the null modes correspond to overall translations or rotations of the coordinate system that lead to spurious alterations of coordinate values, but no real internal changes of interatomic distances. For this reason, the internal metric M( of (10.29) is the starting point for analyzing intrinsic state-related (as opposed to size-related) aspects of a given physical system of interest. 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

To decompose the vibrational energy of a flexible molecule into single-mode contributions, it is useful to perform an instantaneous normal-mode analysis - of the vibrational dynamics. In this approach, we choose a structure at the instantaneous position r(t) and consider the normal mode vibrations around this reference structure. We expand the molecular potential energy up to second order... 

De Man and van Santen ° performed a normal mode analysis of both cluster and periodic models of zeolite lattices using the GVFF developed by Etchepare et al. In an attempt to find a relation between specific normal modes and the presence of particular substructures, de Man and van Santen compared spectra of zeolite lattices with those of lattice substructures, projected eigenvectors of a substructure in the framework onto the eigenvectors of the molecular model of the structure, and constructed the difference and sum spectra of frameworks with and without particular structural units. The study concluded that there is no general justification for correlating the presence of large structural elements with particular features in the vibrational spectra. 

The study of oriented polymers with polarized infrared radiation is an equally important tool in the detailed analysis of the vibrational spectrum, since it permits us, within certain restrictions, to determine the orientation (with respect to the molecular structure) of the transition moment for a given normal mode. This makes it possible, as we shall see, not only to classify bands in the spectrum but to establish their origin. Although polarizers are available with commercial spectrometers, their use has not yet become as general as would be desirable. Some comments... 

The amount of information contained in a measured vibrational spectrum is exploited to some, but not full extent. For example, vibrational spectra are never used to characterize all bonds of the molecule and to describe its electronic structure and charge distribution in detail. Of course, aspects of such investigations can be found off and on in the literature, however, both quantum chemists and spectroscopists fail to use vibrational spectra on a routine basis as a source of information on bond properties, bond-bond interactions, bond delocalization or other electronic features. Therefore, it is correct to say that the information contained in the vibrational spectra of a molecule is not fully utilized. This has to do with the fact that the analysis of vibrational spectra is always carried out in a way that is far from chemical thinking. The basic instrument in this respect is the normal mode analysis (NMA), which describes the displacements of the atomic nuclei during a molecular vibration in terms of delocalized normal modes [1-6]. 

To this point in our discussion of molecular vibrations, we have shown how to generate the symmetries of the vibrational modes by reduction of the representation and subtraction of translational and rotational modes. When pictures of the normal modes of vibration are available, we have also been able to demonstrate how to assign the different symmetries to the correct normal modes of vibration. However, we have not yet attempted to show the origins of the normal modes themselves. The reason for this is because a full-blown norma/ mode analysis (NAM) is a... 

Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates... 

Normal mode analysis is a versatile technique which is capable of providing a compact description of the vibrational dynamics of both small molecules and proteins and nucleic acids. For small molecules in particular, the technique is closely coupled to both the interpretation of vibrational spectroscopic data and the development of molecular mechanical force fields. When normal modes are determined using a force field model, vibrations of specific frequencies can be assigned to particular correlated atomic displacements. Force field parameters can be tested and refined by comparing... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

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