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# NormalModesText - Classical Normal Mode Analysis Harmonic...

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Classical Normal Mode Analysis: Harmonic Approximation The vibrations of a molecule are given by its normal modes. Each absorption in a vibrational spectrum corresponds to a normal mode. The four normal modes of carbon dioxide, Figure 1, are the symmetric stretch, the asymmetric stretch and two bending modes. The two bending modes have the same energy and differ only in the direction of the bending motion. Modes that have the same energy are called degenerate. In the classical treatment of molecular vibrations, each normal mode is treated as a simple harmonic oscillator. Symmetric stretch Asymmetric stretch Bend Bend Figure 1. Normal Modes for a linear triatomic molecule. In the last bending vibration the motion of the atoms is in-and-out of the plane of the paper. In general linear molecules have 3N-5 normal modes, where N is the number of atoms. The five remaining degrees of freedom for a linear molecule are three coordinates for the motion of the center of mass (x, y, z) and two rotational angles. Non-linear molecules have three rotational angles, hence 3N-6 normal modes. The characteristics of normal modes are summarized below. Characteristics of Normal Modes 1. Each normal mode acts like a simple harmonic oscillator. 2. A normal mode is a concerted motion of many atoms. 3. The center of mass doesn’t move. 4. All atoms pass through their equilibrium positions at the same time. 5. Normal modes are independent; they don’t interact. In the asymmetric stretch and the two bending vibrations for CO 2 , all the atoms move. The concerted motion of many of the atoms is a common characteristic of normal modes. However, in the symmetric stretch, to keep the center of mass constant, the center atom is stationary. In small molecules all or most all of the atoms move in a given normal mode; however, symmetry may require that a few atoms remain stationary for some normal modes. The last characteristic, that normal modes are independent, means that normal modes don’t exchange energy. For example, if the symmetric stretch is excited, the energy stays in the symmetric stretch. The background spectrum of air, Figure 2, shows the asymmetric and symmetric stretches and the bending vibration for water, and the asymmetric stretch and bending vibrations for CO 2 . The symmetric stretch for CO 2 doesn’t appear in the Infrared; a Raman spectrum is needed to measure the frequency of the symmetric stretch. These absorptions are responsible for the vast majority of the greenhouse effect. We will also use CO 2 as an example, below. The normal modes are calculated using Newton’s equations of motion. 1-4 Molecular mechanics and molecular orbital programs use the same methods. Normal mode calculations are available on-line. 5 +

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2 Colby College 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Single Beam 500 1000 1500 2000 2500 3000 3500 4000 Wavenumbers (cm-1) Figure 2. The Infrared spectrum of air. This spectrum is the background scan from an FT-IR spectrometer.
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