Normal Modes of Vibration

Polyatomic molecules can be considered as a set of coupled harmonic oscillators. Although this is a classical model we shall see that it can used to interpret spectra using the quantum-mechanical harmonic oscillator wave functions. The collective motions of the atoms in a molecule are decomposed into normal modes of vibration within the harmonic approximation. The normal modes are mutually orthogonal. That is they represent linearly independent motions of the nuclei about the center-of-mass of the molecule.

Our starting point is to consider the potential and kinetic energy in Cartesian coordinates:

We will assume that the potential energy is expanded about the equilibrium position and thus the first derivative terms are equal to zero. The term V0 is an arbitrary energy offset and it will also be set equal to zero. These expressions can be greatly simplified using mass-weighted coordinates:

where

The equations of motion for the collection of atoms in the molecule are:

The trial solutions have the form:

When substituted into the equations of motion we have:

for i = 1 to 3N. This equation is a matrix equation

I is the identity matrix. The general form of the matrix equations is:

There is a trivial solution in which all of the terms in the h0 column vector are zero. The interesting solution, however, is the solution for which the determinant of the matrix |A - lI| is equal to zero.

Transformation to normal coordinates

The potential energy is written in the general form:

or in matrix form

where hT is the transpose of h.

A is a symmetric matrix, but it is not diagonal. In fact, the procedure of finding det |A - lI| is a matrix diagonalization of A. To perform this diagonalization we transform to normal coordinates Qi where:

for I = 1 … 3N.

In matrix form Q = LTh.

where LT is the transpose of L. L is a unitary matrix; its inverse is equal to its transpose L-1 = LT. The matrix L will diagonalize A.

Because L is unitary Q = LTh = L-1h and h = LQ. Thus,

The eigenvalues are

The kinetic and potential energies are given by

The uncoupled equations of motion are now represented by

Each normal mode oscillates independently about the center-of-mass of the molecule.

Internal Coordinates

Cartesian coordinates are less convenient than a coordinate system defined in terms of the bonds, angles, etc. of the molecule. Such a coordinate system is called an internal coordinate system. Only motions relative to the center-of-mass are included and thus there are 3N - 6 internal coordinates for a non-linear polyatomic with N atoms. For linear polyatomic molecules there are 3N - 5 internal coordinates. The internal coordinates are

  1. Stretch Dr
  2. Bend Dq
  3. Torsion Dt
  4. Wag Dw

The advantages of this coordinate system are:

  1. Translation and rotation are eliminated.
  2. Force constant are defined in terms of bond stretches, valence angle bends, torsions, and wags. These quantities can be related to bond strengths and barriers for internal rotation.

For example, for CO2 we have the following internal coordinates.

The internal coordinates are obtained from the Cartesian coordinates by R = Bx. The B matrix is not square and has dimensions 3N - 5 by 3N. For non-linear polyatomic molecules the dimensions of the B matrix are 3N - 6 by 3N.

The potential energy is

This can be written as:

2V = hTAh = RTFR

where F is the force constant matrix in internal coordinates.

The kinetic energy is:

G is the geometry matrix. The coupled linear equations can be written

The matrix diagonalization proceeds in a fashion analogous to the procedure described above for mass-weighted Cartesian coordinates.