## Molecular Structure: Vibration
and Rotation

#### Molecular lasers function differently
than atomic lasers because they have vibrational and rotational energies as well as
electronic energy. These molecular vibrations occur because the relative positions
and orientations of the atomic nuclei are not absolutely fixed within the molecule.
The molecular rotations occur because the individual molecules are free to spin and rotate
in space since they are in a gaseous state. The energies associated with molecular
vibrations and rotations are quantized just like the electronic energy. Transitions
between vibrational energy levels emit photons with wavelengths in the infrared region,
while transitions between rotational energy levels emit photons in the microwave
region.

#### The CO_{2} laser is based on the vibrational
and rotational transitions of the CO_{2} molecule. This molecule consists of
two oxygen atoms covalently bonded with a central carbon atom. Arranged in sp^{2}
hybrid orbitals, each oxygen atom forms a sigma and a pi bond with the carbon atom and
leaves two free electron pairs (Zumdahl, 639):

#### The dynamics of the carbon dioxide molecule are very
similar to the dynamics of diatomic molecules such as CO or HCl.

## Molecular Vibration:

#### In a diatomic molecule such as O_{2},
N_{2} or CO, the individual atoms are bound by a molecular binding force that
functions much like the spring constant k of a linear harmonic oscillator. When
excited, the two nuclei will vibrate much like two masses connected by a spring.
While real diatomic molecules are not perfect harmonic oscillators, their potential energy
functions approximate those of a harmonic oscillator for a certain value of inter-nuclear
separation. Although Carbon Dioxide is a triatomic molecule, it behaves much like a
simple diatomic molecule because its structure is linear. Such a linear triatomic
molecule has three normal modes of vibration, described as the asymmetric stretch mode,
the bending mode and the symmetric stretch mode.

#### Each one of these normal modes of
vibration for the CO_{2 }molecule is associated with a characteristic frequency of
vibration (w) as well as a ladder of allowed energy levels. Therefore we label the
three vibrational modes n_{1}, n_{2}, and n_{3}, with their
vibrational energy levels being quantized (n= 0, 1, 2, 3, . . .). To describe
the vibrational modes we use the notation ( n_{1}, n_{2}, n_{3}).
We can approximate the vibrational energy levels as the quantum mechanical simple harmonic
oscillator (h_{b}= h/(2pi)):

#### E_{v}=h_{b}w_{0}*(n+1/2), where
v_{0} is the classical vibration frequency

#### w_{0}= (k/u)^1/2_{ }

#### Notice that there will be a ground state vibrational level (corresponding
to n=0) with some positive energy E_{0}= h_{b}w/2

#### While in reality each normal mode is slightly
anharmonic, the specific equation for the vibrational energy of CO_{2} can still
be described by:

### E(n_{1}, n_{2}, n_{3}) = hcw_{1} (n_{1}+1/2)
+ hcw_{2} (n_{2}+1/2) + hcw_{3} (n_{3}+1/2)

#### The following diagram depicts the first few vibrational
energy levels (n1 n2 n3) of the CO_{2} molecule and also lists the normal modes'
characteristic frequencies in units of wave numbers (cm^{-1}).

#### Notice that the base frequency for the symmetric stretch mode is w_{1}=1288cm^{-1}
and for the bending mode w_{2}= 667cm^{-1}. For the asymmetric
stretch mode, the base frequency is much higher at w_{3}= 2349cm^{-1}.

## Molecular Rotation:

#### Again because CO_{2} is a linear triatomic
molecule and thus behaves very much like a diatomic molecule, it is considerably simple to
understand the dynamics of its rotation. To start, picture the molecule as a
classical dumbbell with masses on the ends connected by a rod. When the dumbbell then
rotates about its center of mass (carbon atom) it has a kinetic energy that is a function
of its moment of inertia and the length of the rod.

#### This model must be adjusted somewhat to describe a
diatomic molecule because the separation of the masses is not fixed. As the molecule
rotates the oxygen atoms are pulled farther apart by the centrifugal force, therefore
increasing the moment of inertia. It is even more complicated if the molecule is
experiencing a molecular vibration. Fortunately, for our purposes the rigid
dumbbell
approximation will suffice for describing the rotational energy of a diatomic molecule.
When the classical equation for rotational energy is put into a quantum mechanical
form, with the quantum number J describing the rotational states, we then have an
expression for the rotational energy of a diatomic molecule. Thanks to its linear
structure, the rotational energy spectrum of CO_{2} has the same character as that
for diatomic molecules and the rotational energy levels are thus approximated by:

### E_{J} = hcB_{e}J(J+1), J = 0, 1, 2, 3, ...

###

#### where the rotational constant Be for the CO_{2} molecule
is B_{e} = .39 cm^{-1}. The difference between energy levels is:

### Table of Contents:

- Introduction
- Molecular Spectra
- Data
- Apparatus