Equations of motion with in the harmonic approximation

Mass-weighted Cartesian coordinates

The kinetic energy is

The potential energy is

The f_ij's are constants given by

and the set of coordinates q_i are mass-weighted Cartesian coordinates.

Newton's equations of motion can be written in the form

Substitution of the kinetic and potential energy yields

Solutions can have the form

By substituting these solutions into the linear differential equations a set of algebraic equations results

This set of equations forms a matrix known as the secular determinant.

Cartesian coordinates

The kinetic energy is

The potential energy is

The f_ij''s are constants given by

The secular equations in Cartesian coordinates are

An example of a three-body problem demonstrates the application of these equations.

Normal coordinates

The solutions represent oscillatory motion of each atom about its equilibrium position with amplitude A_ik, frequency l^k1/2/2p, and phase f. The amplitudes are different for each atom, but their frequencies and phases are identical. Thus, each atom reaches the maximum displacement and equilibrium position at the same time. Such a concerted motion is called a normal mode of vibration.

Normal coordinates are related the mass-weighted Cartesian coordinates by the linear equations

The coefficients are chosen so that the kinetic and potential energies have the form

The eigenvalues l_k are the same as in the mass-weighted Cartesian equations above. The coefficients l_ik have the property (l^-1)_ki = l_ik.

Internal coordinates

For non-linear molecules, six of the 3N coordinates can be removed by defining the problem in the center-of-mass coordinate system. There are three coordinates due to translation of the center of mass and three coordinates due to rotation.

Instead of using the three Cartesian coordinates to describe the displacement of each atom, it is convenient to introduce a vector r_a for each atom a whose components along the internuclear axis are the Cartesian displacements for each atom. This is known as a bond stretching internal coordinate. We can also define internal coordinates for valence angle bending, torsions, out-of-plane wags, and linear bends.

For small motions we can define 3N - 6 internal coordinates S_t in terms of Cartesian coordinates

The G matrix is the inverse mass matrix used to define the kinetic energy in internal coordinates. The G matrix is defined by the transformation

so that the kinetic energy is

The potential energy in internal coordinates is

The transformation from Cartesian to internal coordinates and the generation of the G matrix can be illustrated by the non-linear three body system.

As above for the other coordinate systems an assumed solution leads to a set of linear equations. In matrix form we can write this as

|F - G^-1l| = 0

or multiplying both sides by G we have

|GF - El| = 0.