Independent mode-displaced harmonic

A molecule with N atoms has a total of 37V degrees of freedom for its nuclear motions, since each nucleus can be independently displaced in three perpendicular directions. Three of these degrees of freedom correspond to translational motion of the center of mass. For a nonlinear molecule, three more degrees of freedom determine the orientation of the molecule in space and thus its rotational motion. This leaves 37V - 6 vibrational modes. For a linear molecule, there are just two rotational degrees of freedom, which leaves 3N -5 vibrational modes. For example, the nonlinear molecule H2O has three vibrational modes, while the linear molecule CO2 has four vibrational modes. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and nonrotating. These are called the normal modes. Each normal mode has a characteristic resonance frequency Vj (expressed in cm ), which is usually determined experimentally. To a reasonable approximation, each normal mode behaves as an independent harmonic oscillator of frequency u . The normal modes of H2O and CO2 are shown in Figs. 14.2 and 14.3. A normal mode will be infrared active only if it involves an oscillation of the dipole moment. All three modes of H2O are... 

The generalization of a force constant, k, and reduced mass, fi, from a one dimensional harmonic oscillator to the normal mode oscillators of a polyatomic molecule is accomplished by the F, G matrix methods of Wilson, et al., (1955). For the present discussion it is sufficient to know that a force constant and a reduced mass may be uniquely defined for each of 3N — 6 linearly independent sets of internal coordinate displacements in a polyatomic molecule (see also Section 9.4.12).] The harmonic oscillator Hamiltonian... 

The decomposition of coupled harmonic oscillators into a collection of independent oscillators is known as a normal mode expansion and the independent oscillators are called normal modes. Normal modes are defined as modes of vibration where the respective atomic motions of the atoms are in harmony , i.e., they all reach their maximum and minimum displacements at the same time. These normal modes can be expressed in terms of bond stretches and angle deformation (termed internal coordinates) and can be calculated by using a procedure called normal coordinate analysis. 

It is widely accepted that vibrational dynamics of atoms and molecules are reasonably well represented by harmonic force fields. The resolution of the secular equation transforms a set of (say N) coupled oscillators into (N) independent oscillators along orthogonal (normal) coordinates. Eigenvalues of the dynamical matrix are normal frequencies and eigenvectors give atomic displacements for each normal mode [3,4,13]. If band intensities cannot be frilly exploited, as it is normally the case for infrared and Raman spectra, these vectors are unknown and force fields refined with respect to observed frequencies only are largely underdetermined. For complex systems, symmetry consideration or/and isotopic substitutions may remove only partially this under determination. 

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). 

The number of degrees of freedom equals the number of normal modes of vibration. The normal modes, also called fundamental modes, are a set of harmonic motions, each independent of the others and each having a distinct frequency. It is possible for two or more of the frequencies to be identical, and the corresponding modes are said to be degenerate. However, the total number of modes in the individual degenerate states are counted separately and still total 3A — 6 for nonlinear and 3 A — 5 for linear molecules. A set of coordinates can be defined, each of which gives the displacement in one of the normal modes of vibration. The normal coordinates can be expressed as combinations of the x-, y-, and z-coordinates of the individual nuclei. 

---