2. ( 40 points) In class we talked about a linear chain of equidistant H atoms, with lattice constant "a" as a simplified model to understand band structures. However, most chemists would tell you that the H atoms would not be equidistant, but instead would be more stable as a chain of dimerized H2 molecules. Let's derive a new band structure for this linear chain of H2 molecules. (Physicists would call this o "Peirels distortion", chemists might call this a "Jahn-Teller" distortion). a) Draw what you would expect the repeating unit cell to be for this new chain structure. b) The new 10 lattice constant for the H2 unit cell we will call "and". Do you expect a and42 to be approximately double, half, or the same as the lattice constant " a " for the equidistant chain of atoms? c) Based on your answer in b, do you expect the horizontal length of the 1tt Brillouin zone edge for the H2 chain (k=/a42) to be double, half or the same as that for the linear H chain (k=/a)?(Hint,k-space is referred to as "reciprocal" space) d) How many 1 s orbital bands should appear in the H2 band diagram from k=0 to k=/ara2 ? In a single unit cell, draw the different molecular orbitals (linear combinations of atomic orbitals), you would expect to see. e) Draw what you would expect the phases of the crystal orbital wavefunction(s) would look like for each band at the Gamma point (k=0) by shading or unshading them in appropriately. (Considering your answer in d, it is possible that not every diagram below is needed). Assume that the size of each orbital is the same. f) Draw what you would expect the phase of the crystal orbital wavefunction(s) would look like for each band at the Brillouin zone edge (ker//ana) by shading or unshading them in appropriately. (Considering your answer in d, it is possible that not every diagram below is needed). Assume that the size of each orbital is the same. g) Draw the band diagram for the H2 that you would expect based on the bonding and antibonding interactions of the 1s orbitals. State (or draw again) which crystal orbital wavefunctions that you drew in (e) and (f) correspond to the intersection of a band with the k=0 or k=/an. h) To the right of the band diagram above, draw the density of states plot. In both diagrams draw a horizontal line corresponding to the Fermi level