Solutions to lecture 5: LCAO model

Solutions to the LCAO model

src/5_atoms_and_lcao_solutions.md

+# Solutions for LCAO model exercises
+
+### Question 1
+
+1. See lecture notes.
+2. The atomic number of Tungsten is 74:
+  $$
+  1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^66s^24f^{14}5d^4
+  $$
+3. $$
+  \begin{align}
+  \textrm{Cu} &= [\textrm{Ar}]4s^23d^9\\
+  \textrm{Pd} &= [\textrm{Kr}]5s^24d^8\\
+  \textrm{Ag} &= [\textrm{Kr}]5s^24d^9\\
+  \textrm{Au} &= [\textrm{Xe}]6s^24f^145d^9
+  \end{align}
+  $$
+
+### Question 2
+1. $$
+    \psi(x) =
+    \begin{cases}
+        &\sqrt{κ}e^{κ(x-x_1)}, x<x_1\\
+        &\sqrt{κ}e^{-κ(x-x_1)}, x>x_1
+    \end{cases}
+$$
+
+  Where $κ = \sqrt{\frac{-2mE}{ħ^2}} = \frac{mV_0}{ħ^2}$.
+
+  The energy is given by $ϵ_1 = ϵ_2 = -\frac{mV_0}{ħ^2}$
+
+  The wave function of a single delta peak is given by
+
+  $$
+      \psi_1(x) = \frac{\sqrt{mV_0}}{ħ}e^{-\frac{mV_0}{ħ^2}|x-x_1|}
+  $$
+
+  $\psi_2(x)$ can be found by replacing $x_1$ by $x_2$
+
+2. $$
+    H = -\frac{mV_0^2}{ħ^2}\begin{pmatrix}
+        1/2+\exp(-\frac{mV_0}{ħ^2}(x_2-x_1)) &
+        \exp(\frac{mV_0}{ħ^2}(x_2-x_1))\\
+        \exp(-\frac{mV_0}{ħ^2}(x_2-x_1)) &
+        1/2+\exp(+\frac{mV_0}{ħ^2}(x_2-x_1))
+    \end{pmatrix}
+  $$
+
+3. $$
+    ϵ_{\pm} = \beta(3/2+\cosh{2α}+2\cosh{α}\pm \cosh{α})
+   $$
+
+   Where $\beta = -\frac{mV_0^2}{ħ^2}$ and $α = \frac{mV_0}{ħ^2}(x_2-x_1)$
+
+### Question 3
+
+1.
+
+$$
+    H_{\mathcal{E}} = eR\mathcal{E},
+$$
+where R is the distance between the negatively charged electrons and the positive charged nuclei.
+
+2.
+$$
+    H_{eff} = \begin{pmatrix}
+    E_0 - \gamma & -t\\
+    -t & E_0 + \gamma
+    \end{pmatrix}
+$$
+Where $\gamma = e d \mathcal{E}/2$ and where we have used that $$⟨1|H_{eff}|1⟩ = -e d \mathcal{E}/2⟨1|1⟩ = e d \mathcal{E}/2$$
+
+3.
+
+The eigenstates of the Hamiltonian are given by:
+$$
+    E_{\pm} = E_0\pm\sqrt{t^2+\gamma^2}
+$$
+The ground state wave function is:
+$$
+    \begin{split}
+        |\psi⟩ &= \frac{t}{\sqrt{(\gamma+\sqrt{\gamma^2+t^2})^2+t^2}}\begin{pmatrix}
+        \frac{\gamma+\sqrt{t^2+\gamma^2}}{t}\\
+        1
+        \end{pmatrix}\\
+        |\psi⟩ &= \frac{\gamma+\sqrt{t^2+\gamma^2}}{\sqrt{(\gamma+\sqrt{\gamma^2+t^2})^2+t^2}}|1⟩+\frac{t}{\sqrt{(\gamma+\sqrt{\gamma^2+t^2})^2+t^2}}|2⟩
+    \end{split}
+$$
+
+4.
+$$
+    P = -\frac{2\gamma^2}{\mathcal{E}}(\frac{1}{\sqrt{\gamma^2+t^2}})
+$$