513.001 Molecular and Solid State Physics
29.01.2016

Problem 1
A linear combination of atomic orbitals used to find the molecular orbitals of a He2 molecule contains four atomic orbitals,

\[ \begin{equation} \psi(\vec{r})= c_1\phi^{Z=2}_{\text{1s}}\left(\vec{r}-\vec{r}_A\right)+c_2\phi^{Z=2}_{\text{1s}}\left(\vec{r}-\vec{r}_B\right)+c_3\phi^{Z=2}_{\text{2s}}\left(\vec{r}-\vec{r}_A\right)+c_4\phi^{Z=2}_{\text{2s}}\left(\vec{r}-\vec{r}_B\right). \end{equation} \]

What is the integral that needs to be evaluated to determine the matrix element $H_{12}$ in the Hückel model for this molecule? What is the integral that needs to be evaluated to determine the matrix element of the overlap matrix $S_{13}$ in the Roothaan equations? This integral is easy to evaluate. What is $S_{13}$?

Problem 2
A simple approximation for the electron density of an atom is the atomic number times a delta function $Z\delta (\vec{r})$. The atomic number $Z$ is the number of electrons that an atom has. In this approximation, the electron density of a crystal is,

$n(\vec{r}) = \sum \limits_{i,l,m,n}Z_i\delta(\vec{r}_i+l\vec{a}_1+m\vec{a}_2+n\vec{a}_3)$,

where $i$ sums over the atoms in the unit cell and the translation vector $\vec{T}_{lmn}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ repeats the unit cell everywhere in the crystal.

(a) Write down the general expression for a 3-D periodic function in terms of a Fourier series.

(b) CsCl has a simple cubic Bravais lattice. $Z_{Cs} = 55$, $Z_{Cl} = 17$. What are the structure factors $G_{000}$ and $G_{100}$?

Problem 3
(a) Draw the electron dispersion relation along L-Γ-X for aluminum (an fcc metal). Start with the empty lattice approximation and explain how this should be modified to include the periodicity of the lattice. For fcc, $\overline{\Gamma L}= \frac{\sqrt{3}\pi}{a}$ and $\overline{\Gamma X}= \frac{2\pi}{a}$.

(b) Draw the phonon dispersion relation along L-Γ-X for aluminum (an fcc metal). Start with the empty lattice approximation and explain how this should be modified to include the periodicity of the lattice. Indicate the Debye frequency in your drawing.

(c) If the electronic band structure of aluminum is known, how would you calculate the chemical potential?

Problem 4
Pyroelectricity describes how the electric polarization changes as the temperature changes. The pyroelectric coefficients form a rank 1 tensor $\pi_i$,

\[ \begin{equation} \pi_i=\frac{\partial P_i}{\partial T}. \end{equation} \]

The generating matrix of the point group m is,

\begin{equation} \sigma_h=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix}\right]. \end{equation}

(a) Give the independent tensor elements for this point group.

(b) How could the pyroelectric coefficient be calculated from the microscopic quantum states of the crystal?