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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.
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Figure 2. The Infrared spectrum of air. This spectrum is the background scan from an FT-IR
spectrometer.

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