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

• Mr_Allod
In summary, the conversation discusses the analysis of a cluster of 79 atoms by using a dynamical matrix and the corresponding eigenvalues and eigenvectors. The first 10 eigenvectors are identified as either translational, rotational, or vibrational modes based on their eigenvalues. The direction of translation can be derived, but the axis of rotation and deductions about the vibrational modes are still being worked on. The conversation also discusses the components of the eigenvectors for atoms lying on the axis of rotation and in the plane orthogonal to it, as well as the expected components for a normal mode corresponding to the C=O stretch in acetone.

#### 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.

Mr_Allod said:
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?
Mr_Allod said:
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?

Twigg

## 1. What is the purpose of analysing the normal modes and dynamics of a cluster of atoms?

The purpose of this analysis is to understand the behavior and movement of atoms in a cluster, which can provide valuable insights into the physical and chemical properties of the cluster. It can also help in predicting the stability and reactivity of the cluster.

## 2. How is the normal mode analysis of a cluster of atoms performed?

The normal mode analysis involves solving the equations of motion for each atom in the cluster, taking into account their interactions with each other. This can be done using various computational methods, such as molecular dynamics simulations or quantum mechanical calculations.

## 3. What information can be obtained from the normal mode analysis?

The normal mode analysis can provide information about the vibrational frequencies and amplitudes of the atoms in the cluster, as well as their collective motion. It can also reveal any potential energy barriers or stable configurations of the cluster.

## 4. How does the dynamics of a cluster of atoms affect its properties?

The dynamics of a cluster of atoms, which include their movements and interactions, can significantly influence the properties of the cluster. For example, it can affect the melting point, surface tension, and reactivity of the cluster.

## 5. What are the applications of analysing the normal modes and dynamics of a cluster of atoms?

This analysis has various applications in materials science, chemistry, and nanotechnology. It can help in designing and optimizing new materials, understanding chemical reactions and catalysis, and developing nanoscale devices and sensors.