By Jasek FH. - Own work, [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/3.0 "Creative Commons Attribution-Share Alike 3.0"), [Link](https://commons.wikimedia.org/w/index.php?curid=2875238)
1. How many normal modes does this molecule have assuming the motion in 1D? How many normal modes does it have if the atoms may move in all three dimensions?
2. For simplicity, we only consider 1D motion of atoms. Write down the Newton's equations of motion of the atoms.
3. Consider a *symmetric* mode when the displacements of oxygen atoms are equal in magnitude and have an opposite direction. Find the eigenfrequency of this mode.
4. Now consider the antisymmetric mode when the oxygen atoms move in phase and have the same displacement. Find the ratio between the displacements of carbon and oxygen atoms that keeps the center of mass at rest.
5. Compute the eigenfrequency of antisymmetric mode.
??? hint
Compare your answers with [Wikipedia](https://en.wikipedia.org/wiki/Triatomic_molecule).
## Exercise 2: Lennard-Jones potential
A simple model approximating the interaction between a pair of noble gas atoms such as Argon is the [Lennard-Jones potential](https://en.wikipedia.org/wiki/Lennard-Jones_potential) in which the potential energy as a function of interatomic distance is
where r is the distance between two atoms, $\epsilon$ is the depth of the potential well, and $\sigma$ is the distance at which the inter-particle potential is zero.
1. Sketch U(r) as a function of interatomic distance and mark the regions of repulsive and attractive forces acting between atoms.
2. Find the distance, $r_0$ (bond length) at which the potential energy is minimal and find the value of the potential energy at this distance (binding energy of the molecule).
3. Expand U(r) in a Taylor's series around $r_0$ up to second order. By considering harmonic potential around the minimum ($r_0$), find an expression for the spring constant, $K$, in terms of $\epsilon$ and $\sigma$.
4. In the harmonic potential approximation of Lennard-Jones potential of argon, find the ground state energy of the molecule. What is the energy required to break the molecule apart?
5. What is the approximate number of phonons can occupy this mode before the potential becomes anharmonic?
## Exercise 3: Numerical simulation
For this exercise, use Matlab (or any other programming environment you are comfortable with).
1. Define a matrix that relates forces to displacements in a linear 1D chain containing $N=5$ atoms. Repeat for $N=200$.
??? hint
In Matlab use the function [`diag`](https://nl.mathworks.com/help/matlab/ref/diag.html).
2. Using numerical diagonalization ([`eig`](https://nl.mathworks.com/help/matlab/ref/eig.html)), compute the eigenfrequencies of this atomic chain. Plot a histogram of these eigenfrequencies.
3. Make the masses of every even atom different from the masses of every odd atom. Compute the eigenfrequencies of this atomic chain and plot a histogram.
