Update docs/5_atoms_and_lcao.md

docs/5_atoms_and_lcao.md

@@ -359,11 +359,11 @@ It was previously mentioned that $V_0>0$.
 Using this, determine which energy corresponds to the bonding energy.
 
 
-### Exercise 3: Polarization of a hydrogen molecule
+### Exercise 3: Polarization of a hydrogen molecule in an electric field
+Applying an electric field can induce an electric dipole moment in a molecule, yielding a 'polarized' molecule. The reason is that the electric field redistributes the charge over the molecule by exerting a force on its charged constituents. Here, we analyze the electric-field-induced dipole moment of a molecule using the LCAO model.
 
-Consider a hydrogen molecule as a one-dimensional system with two identical nuclei at $x=-d/2$ and $x=+d/2$, so that the center of the molecule is at $x=0$.
-Each atom contains a single electron with charge $-e$.
-The LCAO Hamiltonian of the system is given by
+We consider a hydrogen molecule as a one-dimensional system with two identical nuclei at $x=-d/2$ and $x=+d/2$, so that the center of the molecule is at $x=0$.
+Each atom contributes a single electron with charge $-e$. The LCAO Hamiltonian of the system is given by
 
 $$
 H_{\textrm{eff}} = \begin{pmatrix}
@@ -372,29 +372,23 @@ H_{\textrm{eff}} = \begin{pmatrix}
 \end{pmatrix}.
 $$
 
-#### Question 1.
-Let us add an electric field $\mathcal{E} \hat{\bf{x}}$ to the system.
-Which term needs to be added to the Hamiltonian of each electron?
+1.Let us add an electric field $\mathcal{E} \hat{\bf{x}}$ to the system. Which term needs to be added to the Hamiltonian of each electron?
 
-??? hint "The electric potential is given by"
+    ??? hint "The electric potential is given by"
 
-    $$
-    V_{\mathcal{E}}=-\int \mathcal{E} d x
-    $$
+        $$
+        V_{\mathcal{E}}=- \int \mathcal{E} d x
+        $$
 
-#### Question 2.
-Compute the LCAO Hamiltonian of the system in presence of the electric field.
-What are the new onsite energies of the two orbitals?
-#### Question 3.
-Diagonalize the modified LCAO Hamiltonian. Find the ground state wavefunction $ψ$. Note: for simplicity assume that the electric field is small, such that $\gamma \ll t$, with $\gamma = e d \mathcal{E}/2$.
-#### Question 4.
-Find the polarization $P$ of the molecule in the ground state. Check if your answer makes sense when you set the electric field to zero.
+2. Compute the LCAO Hamiltonian of the system in presence of the electric field. What are the new onsite energies of the two orbitals?
+3. Diagonalize the modified LCAO Hamiltonian. Find the ground state wavefunction $ψ$. Note: for simplicity assume that the electric field is small, such that $\gamma \ll t$, with $\gamma = e d \mathcal{E}/2$.
+4. Find the induced dipole moment $p$ of the molecule in the ground state. Check if your answer makes sense when you set the electric field to zero.
 
-??? hint "Reminder: polarization"
-    The polarization $P$ of a molecule with $n\leq 2$ electrons at its ground state $|ψ⟩$ is:
+??? hint "Reminder: dipole moment"
+    The dipole moment $p$ of a molecule with $n\leq 2$ electrons in its orbital ground state $|ψ⟩$ is:
 
     $$
-    P = n e ⟨ψ|x|ψ⟩.
+    p = n e ⟨ψ|x|ψ⟩.
     $$
 
     Use that ground state you found in 3.2 is a linear superposition of two orthogonal orbitals centered at $-d/2$ and $+d/2$.