Problem 1
(a) What is the total Hamiltonian for a carbon monoxide CO molecule? (You don't have to write out all the electron-electron terms. Just make it clear how many terms there are and what form they have.)
(b) If we apply the Born-Oppenheimer approximation and then neglect the electron-electron interactions the Hamiltonian is simplified to a multielectron Hamiltonian that can be solved by the separation of variables (Trennung der Veränderlichen). After the separation of variables has been performed, the molecular orbital Hamiltonian for a single electron must be solved. What is the molecular orbital Hamiltonian for carbon monoxide?
(c) How could you determine the molecular orbitals of CO?
Problem 2
(a) SrTiO3 has a simple cubic Bravais lattice and a lattice constant of 3.905 Å. A x-ray diffraction experiment is performed using x-rays with a wavelength of 0.5 nm. How many diffraction peaks are observed?
(b) The electron density is a three dimensional periodic function. Express the electron density as a Fourier series in terms of the structure factors.
Problem 3
The stress tensor $\sigma$ of a crystal is related to the strain tensor $\epsilon$ by the compliance tensor,
Where $i,j,k,l$ sum over $x,y$ and $z$. For a cubic crystal, what components of the compliance tensor must be equivalent to $c_{xxyy}$? What is the interpretation of $c_{xxyy}$?
How could you calculate the compliance tensor from the microscopic states?
Problem 4
Germanium is an indirect semiconductor with an fcc Bravais lattice. The valence band maximum is at $k = 0$ but the conduction band minimum on the Brillouin zone boundary at L in the <111> direction. There is a minimum in the conduction band at Γ. The direct band gap at Γ is 0.8 eV. Draw the dispersion relation (energy vs. $k$) for energies near the top of the valence band and the bottom of the conduction band using the information given below. Energy should be measured in electron volts in the drawing. Include the chemical potential in the drawing.
Energy gap: $E_g=0.661$ eV
Effective longitudinal electron mass: $m_l= 1.6m_e$
Effective transverse electron mass: $m_t = 0.08m_e$
Effective heavy hole mass: $m_{hh} = 0.33m_e$
Effective light hole mass: $m_{lh} = 0.043m_e$
Split-off band energy: $E_{so} = 0.29$ eV
Effective hole mass in the split-off band $m_{so} = 0.084m_e$
Here $m_e$ is the mass of an electron.