Independent normal modes

Our findings lead us in a number of useful directions. One of these directions is a generalization of our basic instantaneous approach to dynamics. Our original linear INM formalism assumed the potential energy was instantaneously harmonic, but that the coupling was instantaneously linear [Equation (15)]. We still need to retain the harmonic character of the potential to justify the existence of independent normal modes (at least inside the band), but we are free to represent the coupling by any instantaneously nonlinear function we wish. A rather accurate choice for vibrational relaxation, for example, is the instantaneous exponential form ... 

Equation (3.51) expresses the general solution for A(x, t) as a sum over independent normal modes. qi(,t), obtained from Eq. (3.52), determines the time evolution of a mode, while Z>/(x), the solution to Eq. (3.53), detennines its spatial structure in much the same way as the time-independent Schrodinger equation determine the intrinsic eigenfunctions of a given system. In fact, Eq. (3.53) has the same structure as the time-independent Schrodinger equation for a free particle, Eq. (2.80). It admits similar solutions that depend on the imposed boundary conditions. If we use periodic boundary conditions with period L we find, in analogy to (2.82),... 

This follows from expressions such as (16.60), and more generally from the hannonic medium representation of dielectric response (Section 16.9) in which Er is made of additive contributions of independent normal modes including intramolecular modes. 

As yet, we have illustrated that the mechanics of harmonic vibrations can be reduced to the consideration of a series of independent normal modes. This normal mode decomposition is an especially powerful insight in that it points the way to the statistical mechanical treatment of the problem of the thermal vibrations of a crystal. The Hamiltonian for the problem of harmonic vibrations can be written as... 

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

The vibrations of a polyatomic molecule can be considered as a system of coupled anharmonic oscillators. If there are N atomic nuclei in the molecule, there will be a total of 3N degrees of freedom of motion for all the nuclear masses in the molecule. Subtracting the pure translations and rotations of the entire molecule leaves (3N-6) vibrational degrees of freedom for a non-linear molecule and (3N-5) vibrational degrees of freedom for a linear molecule. These internal degrees of freedom correspond to the number of independent normal modes of vibration. Note that in each normal mode of vibration all the atoms of the molecule vibrate with the same frequency and pass through their equilibrium positions simultaneously. 

The experimental data are not compatible with a regular Oh molecule vibrating in independent normal modes. See also [1, 2] for internal motion and mean geometiy deduced by electron diffraction. 

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... 

No matter how complicated the molecule, the motions of the atoms with respect to each other can be treated solely as if those atoms were moving as shown by the normal modes. This allows us to consider only the normal modes. More importantly, because of symmetry, some of the normal modes of large molecules are exactly the same as others. Consider again the C-H stretches of benzene, CgHg. Because of the sixfold symmetry of benzene, we might expect that they can be described equivalently, and to a certain extent this is true. As such, these normal modes have the same vibrational frequency. The total number of unique normal modes therefore depends on two things the number of atoms in the molecule (as indicated by the 3N — 5 or 3N — 6 number of normal modes) and the symmetry of the molecule. The higher the symmetry, the fewer the number of independent normal modes. 

It can be shown (Chapter 14) that the 3N — 6 internal degrees of freedom of motion of a nonlinear molecule correspond to 37V — 6 independent normal modes of vibration. In each normal mode of vibration all the atoms in the molecule vibrate with the same frequency and all atoms pass through their equilibrium positions simultaneously. The relative vibrational amplitudes of the individual atoms may be different in magnitude and direction but the center of gravity does not move and the molecule does not rotate. If the forces holding the molecule together are linear functions of the displacement of the nuclei from their equilibrium configurations, then the molecular vibrations will be harmonic. In this case each cartesian coordinate of each atom plotted as a function of time will be a sine or cosine wave when the molecule performs one normal mode of vibration (see Fig. 1.1). 

To evaluate the turning points we make the independent normal mode approximation, where the potential V ,(s, Q ) in mode m at s along the reaction coordinate is given by Eq. [118]. The turning point for vibrational state in this mode is obtained by solving the equation ... 

We assume that the set / can be converted into a set S of 3 mutually independent normal modes by a linear transformation. For this set, the fluctuation configuration function in a phantom network is... 

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