Molecular vibrations can be expressed in two very different ways. The
simpler way to describe a vibration is as a simple harmonic motion of the atoms
in a molecule. For example, in the case of the diatomic nitrogen molecule,
N_{2}, the normal mode of vibration consists of the two nitrogen atoms
moving together and apart. This is the classical way of describing a
vibration. The other way is to describe molecular vibrations in terms of
quantum-theoretical wavefunctions. For the N_{2} normal mode, the
wavefunction would have a positive value in the region of space around one of
the nitrogen atoms, and a negative value in the region of space around the other
nitrogen atom, that is, there would be a phase-change in going from one atom to
the other. To understand the significance of this, be aware that quantum
theory predicts that the probability of finding an atom at any given point in
space can be obtained from the square of the vibration wavefunction at that
point.

The quantum theoretical description is definitely more complicated than the
classical picture, and involves some phenomena that have no equivalent in the
real world, the world we live in. A simple example of such a
phenomenon can be provided by comparing the classical and quantum descriptions
of the vibration of an N_{2} molecule. In the classical
description of the vibration, each nitrogen atom moves away from its equilibrium
position, slows down, stops, then reverses direction. So each atom spends
most of its time at the extremes of the vibration and a smaller amount of time
at in the region of the equilibrium position. This is characteristic of
simple harmonic motion. The quantum description is that the two nitrogen
atoms spend most of their time in the region of the extremes of the classical vibration. This,
too, is similar to the classical description. But the quantum and
classical descriptions differ when the probability of finding the two atoms in
the region of the equilibrium position is calculated. In the quantum
picture, the probability of finding the atoms near to the equilibrium position
drops to exactly zero. This is a direct consequence of the phase-change in
the wavefunction in v=1.

Within MOPAC, the vibrational analysis uses a mixture of quantum and classical descriptions. Quantum theory is used in calculating the vibrational frequencies and their wavefunctions, and classical theory is used in mapping out the trajectories of the atoms in each vibration. Concepts such as atom position, velocity, acceleration, simple harmonic motion, etc., are classical. Eigenstates (vibrational frequencies), eigenfunctions (normal modes), orthonormality of the modes, probability distributions, etc., are quantum theoretical.

In the following derivations the first part uses mainly quantum theory, and the second part uses mainly classical theory. A common drawback in many descriptions of vibrational phenomena is that relating algebraic expressions to real quantities is often difficult. To reduce the effort required in relating expressions and numerical values, these relationships will be given when appropriate. A good starting point for all numerical work is a list of the Fundamental and Secondary constants used in this analysis.

Vibrational frequencies are calculated using a Hessian matrix;
this is a 3N x 3N matrix of second derivatives of the energy with respect to
Cartesian coordinates. In its original form, the Hessian matrix is in kcal
mol^{-1} Å^{-2} , but the usual form for representing the
force matrix is in (millidynes Å^{-1}), so this conversion is made
before the matrix is used (see Hessian Matrix;).
The Hessian matrix is then mass-weighted. Diagonalization of this matrix
produces normal modes and eigenvalues that can be converted into vibrational
frequencies and other quantities.

Steps in modeling the vibration of a nitrogen molecule are described. Nitrogen was chosen as the example system because it's a stable, diatomic, closed-shell molecule. Quantities calculated include:

- Force constants
- Reduced masses
- Vibrational energies as cm
^{-1}(frequencies or wavenumbers), kcal mol^{-1}, Joules, and ergs - Initial velocity in centimeters per second
- Period of oscillation in femtoseconds
- Angular frequency in reciprocal seconds
- Zero point energy
- Travel in Ångstroms.
- Vibrational transition dipole (system used is HCl)

Equations for each step are given, and then applied to the N_{2}
or HCl
molecule.

Nitrogen does not have a vibrational transition dipole, so for that calculation a molecule of hydrogen chloride was used.

A discussion of the relationship of quantum and classical vibrational quantities is presented. The main differences are that the energies of vibrations predicted using quantum theory are quantized, and the ground-state energy predicted by quantum theory is non-zero.