Problem 1

In the ground state of a hydrogen molecule H₂, both electrons are in the lowest molecular orbital and have opposite spin.

(a) How do the two electrons occupy the molecular orbitals in the first excited state of H₂ and what are the possibilities for the spins?

(b) The first excited state splits in a spin singlet and a spin triplet. Explain what the singlet and triplet states are.

1. (c) What is the energy difference between the singlet and triplet states and how is this energy calculated?

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Problem 2
The conventional unit cell and the lattice parameters of the superconductor YBa₂Cu₃O₇ is shown below.

(a) What is the Bravais lattice of YBa₂Cu₃O₇?

(b) What are the primitive lattice vectors in reciprocal space?

$\vec{b}_1=$

$\vec{b}_2=$

$\vec{b}_3=$

(c) Could YBa₂Cu₃O₇ be piezoelectric? Why or why not?

(d) Sketch the phonon dispersion relation of YBa₂Cu₃O₇ along $X - \Gamma - R$. The distances to the Brillouin zone boundaries are,

(e) Above the superconducting transistion temperture YBa₂Cu₃O₇ is a low electron density metal. How does the specific heat of YBa₂Cu₃O₇ compare to a good metal like silver?

Problem 3
(a) Draw approximately the electron dispersion relation and the density of states for the one-dimensional potential, $V(x) = \cos\left(\frac{2\pi x}{a}\right)$ eV.

(b) There are 3 electrons per primitive unit cell, draw the position of the chemical potential in the plots of the dispersion relation and the density of states.

(c) Is this material a metal, a semiconductor, or an insulator? Why?

Problem 4

Tight binding is a method to calculate the electronic band structure of a crystal. The tight-binding wavefunction is,

Here $N$ is the number of unit cells in the crystal; $h$, $j$, and $l$ are integers that are used to label all the unit cells in the crystal; $\vec{k}$ is a wave vector; $\vec{a}_1$, $\vec{a}_2$, and $\vec{a}_3$ are primitive lattice vectors in real space, and $\psi_{\text{unit cell}}(\vec{r})$ is a trial wavefunction that is constructed from the valence orbitals $\phi_{i}$ of all of the atoms in a primitive unit cell,

Sodium has a bcc Bravais lattice and the valence orbitals are the 3s atomic orbitals.

(a) What is $\psi_{\text{unit cell}}$ for sodium? How many coefficients $c_i$ are there?

(b) How do you determine the coefficients $c_i$ for sodium?

(c) If the coefficients $c_i$ are known, how can you construct the dispersion relation for sodium?

(d) If the dispersion relation for sodium is known, how can you construct the density of states? Sketch approximately the electron density of states for sodium.

(e) If the density of states for sodium is known, how can you calculate the chemical potential? Describe how this calculation depends on the number of valence orbitals that are included in the tightbinding calculation.