$$<p>where $\phi_{A/B}(\vec{r}) =\left(\frac{1}{\pi a_0^3}\right)^{1/2} e^{-|\vec{r}-\vec{R}_{A/B}|/a_0}$ are electronic ground states for the hydrogen atom and $a_0 = \frac{ 4 \pi \epsilon_0 \hbar^2}{e^2 m}$ is the Bohr radius.</p>
<p>a) Derive explicitely (in terms of $R = |\vec{R}_A-\vec{R}_B|$, $a_0$ and the hydrogen atom groundstate energy $\epsilon_H$) the following expressions</p>
\begin{align}
S &= \int dr^3 \phi_A(\vec{r}) \phi_B(\vec{r}), \\
\epsilon_{AA} &= \int dr^3 \phi_A(\vec{r}) H \phi_A(\vec{r}) = \int dr^3 \phi_B(\vec{r}) H \phi_B(\vec{r}), \\
\epsilon_{AB} &= \int dr^3 \phi_A(\vec{r}) H \phi_B(\vec{r}),
\end{align}<p>which contribute to the final energy $\epsilon_\pm = \frac{1}{1 \pm S} (\epsilon_{AA}\pm \epsilon_{AB})$.</p>
<p>$\textit{Hint:}$ Make use of elliptical coordinates and define</p>
$$ \mu = \frac{1}{R} \left(|\vec{r}-\vec{R}_A| + | \vec{r} - \vec{R}_B| \right),$$$$ \nu = \frac{1}{R} \left(|\vec{r}-\vec{R}_A| - | \vec{r} - \vec{R}_B| \right),$$$$\int d^3r = \int_1^\infty d\mu \int_{-1}^1 d \nu \int_0^{2\pi} d \phi \frac{R^3}{8} ( \mu^2 - \nu^2).$$<p>b) Plot the final energies as a function of $R$. For the $\psi_+$ state numerically determine the interatomic distance $R_0$ and the corresponding energy $E_0$.</p>
<p>c) Is the Ansatz $\psi_+ = N_+ \left( \phi_A(\vec{r})+ \phi_B(\vec{r}) \right)$ reasonable for finding the groundstate energy and wavefunction in the $R \to \infty$ limit.</p>
<p>d) What about $R \to 0$?</p>
<p>$\textit{Hint:}$ When $R \to 0$ the electron problem (i.e. the Hamiltonian without the ion repulsion) is equivalent to the He$^+$ problem, which we know how to solve analytically (Serie 1, exercise 2). Compare the electron energy and wavefunction of the true groundstate to the one obtained with the Ansatz above (use the virial theorem from Serie 1, exercise 1 in your calculations).</p>
<p>$\newline$</p>
<p>Following the discussion in c) and d) we propose a generalized Ansatz redefining $\phi_{A/B}(\vec{r}) =\left(\frac{1}{\pi a^3}\right)^{1/2} e^{-|\vec{r}-\vec{R}_{A/B}|/a}$, where $a$ is now a variational parameter.</p>
<p>e) Recalculate the energy $\epsilon_{\pm}$ for the generalized Ansatz.</p>
<p>$\textit{Hint}:$ Use that the single hydrogen Hamiltonian acts as</p>
$$-\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} \right) - \frac{e^2}{4 \pi \epsilon_0 r}$$<p>on the radial part of $\phi_{A/B}(\vec{r})$.</p>
<p>f) Numerically determine $a$ that minimizes energy. What lower and upper bounds would you expect for $a$? Calculate the new interatomic distance $R_0$ and corresponding energy $E_0$.</p>
<p>$\textit{Remark:}$ The last exercise is an example of the $\textbf{variational method}$. We use the fact that any expectation value $\langle \phi(a) | H | \phi(a) \rangle$ is greater or equal than the groundstate energy, for any set of trial functions $|\phi(a) \rangle$ and variational paramter $a$. This means that one can minimize the expectation value by vaying $a$ in order to attain an energy estimate that is closer to the groundstate energy.</p>
$$\textit{In this exercise we will derive the Heisenberg model, commonly used to describe magnetism in systems with localized electrons.}$$<p>$$\textit{We will understand that the cause of magnetism is not actually magnetic but rather lies in electrostatic repulsion and the Pauli principle.}$$</p>
$$<p>a) Let $\epsilon_C(R)$, $\epsilon_{X}(R)$ be the Coloumb and exchange energies, while $S$ is the overlap integral, and $\epsilon_H$ is the groundstate energy of the hydrogen atom. Express the energies of the singlet and triplet state in the Heitler-London approximation.</p>
<p>$\textit{Hint}:$ The Heitler-London Ansatz is</p>
<p>where $\phi_{A/B}(\vec{r}) =\left(\frac{1}{\pi a_0^3}\right)^{1/2} e^{-|\vec{r}-\vec{R}_{A/B}|/a_0}$ are electronic ground states for the hydrogen atom and $a_0 = \frac{ 4 \pi \epsilon_0 \hbar^2}{e^2 m}$ is the Bohr radius. The states $\chi_s$ and $\chi_t$ (shorthand for all 3 triplet states) are the spin degrees of freedom.</p>
<p>Furthermore, the definition of the Coloumb/exchange energies and the overlap integral are:</p>
S &= \int dr^3 \phi_A(\vec{r}) \phi_B(\vec{r})
\end{align}<p>$\newline$</p>
<p>b) Write down the Hamiltonian $H$ in the basis of $\{|\psi_s \rangle, | \psi_t \rangle \}$.</p>
<p>c) Let us define the Heisenberg Hamiltonian</p>
$$H_{HM} = \epsilon_{HM} + J \vec{S}_i \cdot \vec{S}_j $$<p>where $\vec{S}_i$ is the usual spin operator acting on a spin$-1/2$ electron, $\epsilon_{HM}$ is a reference energy and $J$ is the so called coupling constant. Write the Hamiltonian in the basis $\{|\psi_s \rangle, |\psi_t \rangle \}$.</p>
<p>d) What are $\epsilon_{HM}$ and $J$ in terms of $\epsilon_C(R)$, $\epsilon_{exc}(R)$, $S$ and $\epsilon_H$ such that $H_{HM}$ and $H$ are exactly the same (on the subspace spanned by $\{|\psi_s \rangle, | \psi_t \rangle \}$).</p>
<p>e) Since we are dealing with electronic energies, $J$ is expected to be on the orders of eV (exact solutions give roughly 0.1 - 0.2 eV for $H_2$) . Compare this to $J_{magn} \vec{S}_i \cdot \vec{S}_j$, which expresses (part of) the magnetic energy when looking at the two electron spins as magnetic dipoles. Estimate the order of magnitude of $J_{magn}$ for the hydrogen atom.</p>
<p>$\textit{Hint:}$ The magnetic dipole-dipole interaction is usually modelled by the Hamiltonian</p>
$$ H = \frac{\mu_0}{4 \pi r^3} [ \vec{m}_1 \cdot \vec{m}_2 - 3(\vec{m}_1 \cdot \hat{r})(\vec{m}_2 \cdot \hat{r})] - \mu_0 \frac{2}{3} \vec{m}_1 \cdot \vec{m}_2 \delta(\vec{r})$$<p>where $\hat{r}$ is the normalized vector $\vec{r}$ connecting the two magnetic moments $\vec{m}_i$. For this exercise will ignore all $\hat{r}$ dependent terms and also the contact interaction given by the $\delta(\vec{r})$ contribution.</p>