updates error of solutions last exercise

src/5_atoms_and_lcao_solutions.md

@@ -56,12 +56,18 @@ $$
 
 2.
 $$
-    H_{eff} = \begin{pmatrix}
+    \hat{H} = \begin{pmatrix}
+    E_0  & -t\\
+    -t & E_0 
+    \end{pmatrix} +\begin{pmatrix}
+    ⟨1|ex\mathcal{E}|1⟩ & ⟨2|ex\mathcal{E}|1⟩\\
+    ⟨1|ex\mathcal{E}|2⟩ & ⟨2|ex\mathcal{E}|2⟩
+    \end{pmatrix} = \begin{pmatrix}
     E_0 - \gamma & -t\\
     -t & E_0 + \gamma
-    \end{pmatrix}
+    \end{pmatrix},
 $$
-Where $\gamma = e d \mathcal{E}/2$ and where we have used that $$⟨1|H_{eff}|1⟩ = -e d \mathcal{E}/2⟨1|1⟩ = e d \mathcal{E}/2$$
+where $\gamma = e d \mathcal{E}/2$ and have used $$⟨1|ex\mathcal{E}|1⟩ = -e d \mathcal{E}/2⟨1|1⟩ = -e d \mathcal{E}/2$$
 
 3.