Consider a triangular molecule with 3 carbon atoms at the corner of a regu- lar triangle of side 'a'. Assume 1 orbital per atom. The Hamiltonian matrix has Han= 0 (site energy), Hnm = -t if n and m are nearest neighbors, and zero otherwise. (a) What are the energy levels? Use Bloch's theorem since this is a 1D peri- odic solid. In other words, you have an analytical 1D E-k, and you have to impose periodic boundary conditions on the Bloch wavefunction exp (ikna) for site n, to get the allowed values of k. (b) What are the eigenfunctions? (c) Repeat for a square.

Consider a triangular molecule with 3 carbon atoms at the corner of a regu- lar triangle of side 'a'. Assume 1 orbital per atom. The Hamiltonian matrix has Han= 0 (site energy), Hnm = -t if n and m are nearest neighbors, and zero otherwise. (a) What are the energy levels? Use Bloch's theorem since this is a 1D peri- odic solid. In other words, you have an analytical 1D E-k, and you have to impose periodic boundary conditions on the Bloch wavefunction exp (ikna) for site n, to get the allowed values of k. (b) What are the eigenfunctions? (c) Repeat for a square.

College Physics

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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Question

Consider a triangular molecule with 3 carbon atoms at the corner of a regu-
lar triangle of side 'a'. Assume 1 orbital per atom. The Hamiltonian matrix
has Hnn = 0 (site energy), Hnm = -t if n and m are nearest neighbors, and
zero otherwise.
+
-t
+
(a) What are the energy levels? Use Bloch's theorem since this is a 1D peri-
odic solid. In other words, you have an analytical 1D E-k, and you have to
impose periodic boundary conditions on the Bloch wavefunction exp (ikna)
for site n, to get the allowed values of k.
(b) What are the eigenfunctions?
(c) Repeat for a square.
(d) Repeat for a square of dimers, i.e., opposite corner atoms same, neigh-
boring atoms different.

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