Solutions for LCAO model exercises¶

Question 1¶

See lecture notes.
The atomic number of Tungsten is 74: $$ 1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^66s^24f^{14}5d^4 $$
\[ \begin{align} \textrm{Cu} &= [\textrm{Ar}]4s^23d^9\\ \textrm{Pd} &= [\textrm{Kr}]5s^24d^8\\ \textrm{Ag} &= [\textrm{Kr}]5s^24d^9\\ \textrm{Au} &= [\textrm{Xe}]6s^24f^{14}5d^9 \end{align} \]

Question 2¶

$$ \psi(x) = \begin{cases} &\sqrt{κ}e^{κ(x-x_1)}, xx_1 \end{cases} $$ Where $κ = \sqrt{\frac{-2mE}{\hbar^2}} = \frac{mV_0}{\hbar^2}$. The energy is given by $ϵ_1 = ϵ_2 = -\frac{mV_0^2}{2\hbar^2}$ The wave function of a single delta peak is given by $$ \psi_1(x) = \frac{\sqrt{mV_0}}{\hbar}e^{-\frac{mV_0}{\hbar^2}|x-x_1|} $$ $\psi_2(x)$ can be found by replacing $x_1$ by $x_2$
\[ H = -\frac{mV_0^2}{\hbar^2}\begin{pmatrix} 1/2+\exp(-\frac{2mV_0}{\hbar^2}|x_2-x_1|) & \exp(-\frac{mV_0}{\hbar^2}|x_2-x_1|)\\ \exp(-\frac{mV_0}{\hbar^2}|x_2-x_1|) & 1/2+\exp(-\frac{2mV_0}{\hbar^2}|x_2-x_1|) \end{pmatrix} \]
$$ ϵ_{\pm} = \beta(1/2+\exp(-2\alpha) \pm \exp(-\alpha)) $$ Where $\beta = -\frac{mV_0^2}{\hbar^2}$ and $α = \frac{mV_0}{\hbar^2}|x_2-x_1|$

Question 3¶

1.

\[ H_{\mathcal{E}} = ex\mathcal{E}, \]

2. $$ \hat{H} = \begin{pmatrix} E_0 & -t\ -t & E_0 \end{pmatrix} +\begin{pmatrix} ⟨1|ex\mathcal{E}|1⟩ & ⟨1|ex\mathcal{E}|2⟩\ ⟨2|ex\mathcal{E}|1⟩ & ⟨2|ex\mathcal{E}|2⟩ \end{pmatrix} = \begin{pmatrix} E_0 - \gamma & -t\ -t & E_0 + \gamma \end{pmatrix}, $$ 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.

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. $$ P = -\frac{2\gamma^2}{\mathcal{E}}(\frac{1}{\sqrt{\gamma^2+t^2}}) $$