Let us consider the 1s orbitals on three different atoms. The atoms can be arranged in...

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Let us consider the 1s orbitals on three different atoms. The atoms can be arranged in a linear chain or in an equilateral triangle. The interatomic distance is a. The nearest-neighbor transfer matrix element is −t < 0. Take the on-site binding energy equal to zero. (a) Give the Hamiltonian in matrix form for the two different configurations of atoms. (b) Find the eigenenergies for the two different systems and compare the total energy assuming that each atom contributes one electron. Include spin in the considerations (feel free to use programs such as wolfram alpha). (c) What are the wavefunctions for the triangular system. (d) The equilateral triangle is a periodic system and the eigenenergies can be obtained in momentum space. Recalculate the eigenenergies and wave- functions from the exact expressions. Let us consider the 1s orbitals on three different atoms. The atoms can be arranged in a linear chain or in an equilateral triangle. The interatomic distance is a. The nearest-neighbor transfer matrix element is −t < 0. Take the on-site binding energy equal to zero. (a) Give the Hamiltonian in matrix form for the two different configurations of atoms. (b) Find the eigenenergies for the two different systems and compare the total energy assuming that each atom contributes one electron. Include spin in the considerations (feel free to use programs such as wolfram alpha). (c) What are the wavefunctions for the triangular system. (d) The equilateral triangle is a periodic system and the eigenenergies can be obtained in momentum space. Recalculate the eigenenergies and wave- functions from the exact expressions.

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aTo construct the Hamiltonian for the system described we can consider the tightbinding model for a onedimensional linear chain and a twodimensional equilateral triangle In the tightbinding model we a... View the full answer