Title: ETH_Sem6 / ACIV / Section5

Q: Indicate the lattice and the basis in the 2D structure
A: Lattice:
Basis:

Q: Indicate basis, lattice vectors, crystal system and stoichiometry
A: Perovskite \((\ce{ABO3}) \, \ce{BaTiO3}\)
Crystal system: cubic
Centering: Primitive
Cell Basis: Ba at origin, Ti at (1/2, 1/2, 1/2), 3O at (1/2, 1/2, 0), (1/2, 0, 1/2), (0, 1/2, 1/2)

Q: Basis, lattice vectors, crystal system and stoichiometry
A: Calcium Fluoride \(\ce{CaF2}\)
FCC lattice of Ca cations with F anions in all \(T_d\) voids
Basis: Ca at origin, 2 F at (1/4, 1/4, 1/4), (1/4, 3/4, 1/4)

Miller indicies

Q: How many bands do you expect in the band diagram of graphene? Consider the unit cell
A: Valence atomic orbitals are the \(2s\) and the three \(2p\) orbitals. As there are two (1 + 4 x 1/4 = 2) atoms per unit cell, a total of \(8\) (2 atoms per unit cell x 4 orbitals per atom) bands is expected.

Q: Band diagram for an infinite chain of \(\ce{C=C}\) double bonds, spaced by distance \(d\)
A: One carbon in the unit cell gives four bands (-> four orbitals): two degenerate \(\pi\) bands derived from the \(p_x\) and \(p_y\) orbitals and two \(\sigma\) bands derived from the \(p_z\) and \(s\) orbitals. The \(\sigma\) bands repel each other and form \(\sigma, \sigma^*\). The \(\pi\) bands do not interact with each other and are energetically ordered as shown below due to symmetry considering \(e^{i\vec k \cdot \vec x}\).

Q: MO diagram for an unfilled-unfilled, a filled-unfilled, and a filled-filled interaction in the MO picture, categorize
A: Unfilled-unfilled: no change in energy
Filled-unfilled: stabilizing
Filled-filled: destabilizing

Q: MO diagram for an unfilled-unfilled and a filled-filled interaction in the band picture
A: Filled orbitals that are above the Fermi level donate into empty orbitals that are below the Fermi level, this can cause formally unfilled-unfilled and filled-filled interactions to be bonding in character.