Problem 1
A linear combination of atomic orbitals used to find the molecular orbitals of a He2 molecule contains four atomic orbitals,
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 $\overline{\Gamma X}= \frac{2\pi}{a}$.
(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 $\pi_i$,
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?