# Analysing the Normal Modes and Dynamics of a Cluster of Atoms

Mr_Allod
Homework Statement:
Analyse modes of motion of a cluster of 79-atoms after each atom in the cluster (which has Lennard-Jones interactions between the atoms) is displaced by a small amount and is allowed to vibrate about it's equilibrium.
a. Derive numerically the eigenvalues and eigenvectors of the dynamical matrix ##\tilde D## of the system.
b. Identify which eigenvalues and eigenvectors correspond to translational, rotational and vibrational motion.
c. Comment on nature of the vibrational modes (eg. degeneracy, type of motion etc.)
Relevant Equations:
##\omega^2 \vec x = \tilde D\vec x##
I am trying to analyse the dynamics of a cluster of 79 atoms.

The system can be described with:

##\omega^2 \vec x = \tilde D\vec x##

Where ##\omega^2## (the eigenvalues) are the squares of the vibration frequencies for each mode of motion, ##\tilde D## is the "dynamical matrix" which is a ##3N\times3N## (N = 79) square matrix, and ##\vec x## are the eigenvectors of the form:
$$\begin{bmatrix} x_{11} \\ x_{12} \\ x_{13} \\ x_{21} \\ x_{22} \\ x_{23} \\ \vdots \\ x_{237, 3} \end{bmatrix}$$
Where ##(x_{11}, x_{12}, x_{13})## correspond to the cartesian coordinates of the amplitudes of the first atom.

The eigenvalues and eigenvectors are all calculated numerically and the initial equilibrium positions of the atoms are known. Based on this I must identify which of the first 10 eigenvectors correspond to translation, rotational and vibrational modes, as well as the directions of translation, the axes of rotation and the nature of the vibrational modes.

Identifying which eigenvector is which is not difficult, the first 6 have eigenvalues ##\omega^2 = 0## which means they must be either translational or rotational. The translational can be identified by the repeating values in the eigenvector for each atom eg. ##(2, 1, 0, 2, 1, 0, 2, 1, 0, \dots etc.)##. The rotational modes have no effect on the position of the central atom therefore the first 3 values of the eigenvector are ##(0,0,0)##. The remaining modes then must be vibrational.

I also know how to derive the direction of translation however I am stuck on how to find the axis of rotation and what kinds of deductions I can make about the vibrational modes. If someone could give me some insight into this I'd appreciate it.