Problem 1

(a) The molecular orbitals are found by solving the molecular orbital Hamiltonian. What is the molecular orbital Hamiltonian for CH₄?

(b) The molecular orbital Hamiltonian can be solved using the method of Linear Combination of Atomic Orbitals. Suppose we assume that we assume that the molecular orbitals for CH₄ can be written as a linear combination of the carbon 1s, 2s, and 2p and the hydrogen 1s atomic orbitals. How many molecular orbitals would be calculated?

(c) Write down the ground state many-electron wavefunction for CH₄.

Problem 2
Given that you know the mass of a gold atom, how could you calculate the mass density of gold from the phonon density of states?

Problem 3
The electron density of states of a free electron gas in two dimensions is $D(E)=\frac{m}{\pi \hbar^2}\, \left[\frac{1}{\text{J}\,\text{m}^2}\right]$ for $E > 0$ and $D(E)=0$ for $E < 0$. Calculate the electronic contribution to the specific heat. How does the electronic contribution of the specific heat compare to the phonon contribution at low temperature?

Problem 4
Sketch the electronic band structure $E(\vec{k})$ of an insulator. Include the chemical potential in the diagram. How could you calcuate the chemical potential from the electron density of states? How could you measure the band gap of an insulator experimentally?