This document discusses diatomic molecules and their behavior as simple anharmonic oscillators. It begins by defining diatomic molecules as molecules composed of two atoms that can be the same or different elements. It then describes molecular vibration, noting that diatomic molecules have one normal mode of vibration involving periodic motion of the atoms. The motion is approximated as simple harmonic motion. However, real molecules are anharmonic oscillators as the restoring force is not perfectly proportional to displacement from equilibrium. The document explores vibrational energy levels and selection rules for infrared spectroscopy. It concludes by relating the potential energy curve of an oscillator to the period of oscillations at different energies.

1. DIATOMIC MOLECULES AS A
SIMPLE ANHARMONIC
OSCILLATOR

2. Made By:
Anita Malviya

3. Diatomic Molecules
 Diatomic molecules are molecules composed of only
two atoms, of either the same or different chemical
elements. The prefix di- is of Greek origin, meaning
"two".
 Ifa diatomic molecule consists of two atoms of the
same element, such as hydrogen (H2) or oxygen (O2),
then it is said to be homonuclear.
 If a diatomic molecule consists of two different atoms,
such as carbon monoxide (CO) or nitric oxide (NO),
the molecule is said to be heteronuclear.

4. Molecular vibration
 A molecular vibration occurs when atoms in a molecule are
in periodic motion while the molecule as a whole has constant
translational and rotational motion. The frequency of the periodic
motion is known as a vibration frequency, and the typical
frequencies of molecular vibrations range from less than 1012 to
approximately 1014 Hz.
 A diatomic molecule has one normal mode of vibration. The
normal modes of vibration of polyatomic molecules are
independent of each other but each normal mode will involve
simultaneous vibrations of different parts of the molecule such as
different chemical bonds.
 the motion in a normal vibration can be described as a kind
of simple harmonic motion. In this approximation, the vibrational
energy is a quadratic function (parabola) with respect to the atomic
displacements and the first overtone has twice the frequency of the
fundamental.

5. Anharmonic Oscillator
 A harmonic oscillator obeys Hooke's Law and is an
idealized expression that assumes that a system
displaced from equilibrium responds with a restoring
force whose magnitude is proportional to the
displacement. In nature, idealized situations break
down and fails to describe linear equations of motion.
Anharmonic oscillation is described as the restoring
force is no longer proportional to the displacement.
 Anharmonic oscillators can be approximated to a
harmonic oscillator and the anharmonicity can be
calculated using perturbation theory.

6. Vibrational Energy Levels
J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.
Selection Rules:
1) Must have a change in dipole moment (for IR).
2) Dv = 1

8. Potential energy from period of
oscillations
 Let us consider a potential well . Assuming that the
curve is symmetric about the -axis, the shape of the
curve can be implicitly determined from the period of
the oscillations of particles with energy according to
the formula:
 

U
EU
dEET
m 0
)(
22
1
x(U)