Theory of Molecular vibrations

For IR spectroscopy, the process of interest is absorption. Polarization and angle-dependent measurements are useful when using the transmission geometry. 

In general, when using unpolarized light, the vibrations both parallel and perpendicular to the interface are excited. However, when the light is polarized, selected dynamic dipoles are excited, from which the orientation of the dipoles can be inferred. 

Molecules possess discrete levels of rotational and vibrational energy, and transitions between vibrational levels occur by absorption of photons with frequencies in the mid-infrared range. There are four types of vibration  

Bending vibrations (8) changing bond angles but leaving bond lengths unaltered. 

Torsion vibrations (x) changing the angle between two planes through atoms. 

Molecules possess discrete levels of rotational and vibrational energy. Transitions between vibrational levels occur by absorption of photons with frequency v in the mid-infrared range (Table 8.1). The C-O stretch vibration, for example, is at 2143 cm 1. For small deviations of the constituent atoms from their equilibrium positions, the potential energy V(r) can be approximated by that of the harmonic oscillator  

V(r) is the interatomic potential r is the distance between the vibrating atoms req is the equilibrium distance between the atoms k is the force constant of the vibrating bond. 

vibrational frequencies increase with increasing bond strength and with decreasing mass of the vibrating atoms. 

The harmonic approximation is only valid for small deviations of the atoms from their equilibrium positions. The most obvious shortcoming of the harmonic potential is that the bond between two atoms cannot break. A physically more realistic potential is the Morse potential (Fig. 8.1)  

In this potential the energy levels are no longer equally spaced, and overtones, i.e. vibrational transitions with An 1, become allowed. The overtone of gaseous CO at 4260 cm-1 (slightly less than 2x2143 = 4286 cm ) is an example. For small deviations of r from equilibrium, however, the Morse potential is successfully approximated by a parabola, and for the interpretation of IR spectra the harmonic oscillator description is usually sufficient. 

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Wilson, E. B. Jr, Decius, J. C. and Cross, P. C. (1980) Molecular Vibrations, Dover, New York. Woodward, L. A. (1972) Introduction to the Theory of Molecular Vibrations and Vibrational Spectroscopy, Oxford University Press, Oxford. 

The book covers a variety of questions related to orientational mobility of polar and nonpolar molecules in condensed phases, including orientational states and phase transitions in low-dimensional lattice systems and the theory of molecular vibrations interacting both with each other and with a solid-state heat bath. Special attention is given to simple models which permit analytical solutions and provide a qualitative insight into physical phenomena. 

Abstract The theory of molecular vibrations of molecular systems, particularly in the harmonic approximation, is outlined. Application to the calculation of isotope effects on equilibrium and kinetics is discussed. 

L. Woodward, Introduction to the Theory of Molecular Vibrations and Vibrational Spectroscopy, Oxford University Press, Oxford, 1976. 

Each such null vector may be considered an invariant or symmetry of the thermodynamic system, because it corresponds to an operation (change of extensive variables Xt) that produces no response in any intensive state variable and thus leaves the thermodynamic state unaltered (Sidebar 7.2). As described in Sidebar 10.3, these invariants also play a role somewhat analogous to overall rotations and translations ( null eigenmodes of the Hessian matrix) in the theory of molecular vibrations. 

The elementary theory of molecular vibrations identifies the 3n - 5 and 3n - 6 rules for vibrating objects defined by sets of vertices forming linear and polyhedral shaped structures. 

At normal temperatures the lattice dynamics involves predominantly low amplitude atomic motions that are well described in a harmonic approximation. Therefore, potential models widely used in the theory of molecular vibration, such as a generalized valence force field (GVFF) model, may be of use for such studies. In a GVFF the potential energy of a system is described with a set... 

The treatment thus far has been for isolated molecules, either small or one-dimensional infinite helices. In some instances crystalline intermolecular interactions are important, and it is therefore necessary to be able to compute the normal modes of a molecular crystal. The general theory of crystal dynamics has been presented by Born and Huang (1954), and the theory of molecular vibrations in solids has been discussed by a number of authors (Fanconi, 1972a,b Zak, 1975 Neto et al., 1976 Decius and Hexter, 1977 Califano, 1977 Schrader, 1978). 

We prefer to support the output from standard packages that are continually developed. They are fast, accurate, widely supported, robust, simple to use and can be mastered quickly. They are widely exploited by optical spectroscopists and, especially important, these programs are familiar to the wider chemical community. Moreover, they automatically incorporate much of the theory of molecular vibrations [43]. 

From the theory of molecular vibrations, any internal coordinate can be Taylor-expanded about equilibrium in terms of its Cartesian coordinates ... 

There are many texts available which deal with the theory of molecular vibrations and infrared spectra. Complete treatments are given in the books by Herzberg (1945) and by Wilson et al. (1955). Other texts that can be highly recommended are referred to in Chapter 20, which deals with information sources. 

The theory of molecular vibrations predicts that an asymmetrical molecule which contains n atoms will have 3n - 6 modes of fundamental vibrations. This means that a molecule like CO should possess (3x3)- 6, i.e., 3 fundamental modes of vibrations. Likewise, methane should have 9 and ethane 18. 

It follows from the theory of molecular vibrations [188] that the value lies between the highest and the lowest values of the reactant molecule vibrational frequencies. Thus, A 10 —10 s At low temperatures (kT < hcoj, hcof), the pre-exponential factor is lower compared to this estimation. 

Csaszar P, Pulay P (1984) J Mol Struct 114 31 Malhiot RJ, Ferigle SM (1954) J Chem Phys 22 717 Woodward LA (1972) Introduction to the theory of molecular vibrations and vibrational spectroscopy. Oxford University Press, Oxford... 

R. M. Barrow, Molecular Spectroscopy, McGraw-Hill, New York (1962). This text provides an introduction to the theory of molecular vibrations. 

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