Analysing the Normal Modes and Dynamics of a Cluster of Atoms

  • #1
Mr_Allod
42
16
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.
 

Answers and Replies

  • #2
TeethWhitener
Science Advisor
Gold Member
2,434
1,981
I am stuck on how to find the axis of rotation
What can you say about the eigenvector components of an atom that lies on the axis of rotation? Slightly off of it? In the plane orthogonal to it?
what kinds of deductions I can make about the vibrational modes
It’s not 100% clear what exactly this is asking, but for instance, if you were looking at the normal mode corresponding to the C=O stretch in acetone, what would you expect the hydrogen components of the eigenvector to look like?
 

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