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src/6_bonds_and_spectra.md

@@ -211,7 +211,7 @@ DOS_finite_electron_chain(300)
 
 ## Exercise 1: Linear triatomic molecule
 
-Consider carbon dioxide (C0$_2$) which is a linear triatomic molecule shown below 
+Consider carbon dioxide (C0$_2$) which is a linear triatomic molecule shown below
 
 <img src="https://upload.wikimedia.org/wikipedia/commons/a/af/Carbon-dioxide-3D-vdW.svg" width="50%" alt="carbon dioxide"></img>
 
@@ -236,9 +236,9 @@ A simple model approximating the interaction between a pair of noble gas atoms s
 
 $$ U(r) = 4\epsilon\big[\big(\frac{\sigma}{r}\big)^{12}-\big(\frac{\sigma}{r}\big)^6\big] $$
 
-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. 
+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. 
+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?
@@ -252,7 +252,7 @@ For this exercise, use Matlab (or any other programming environment you are comf
 
     ??? hint
 
-        In Matlab use the function [`diag`](https://nl.mathworks.com/help/matlab/ref/diag.html). 
+        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.