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4 s band tight binding lcao band theory tight lcao

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Unformatted text preview: is called a band . 4β “s” band Tight-binding (LCAO) Band Theory Tight LCAO Wavefunction Wavefunction Energy for LCAO Bands Energy Energy for LCAO Bands Energy Reduced Hamiltonian Matrix: Reduced Overlap Matrix: Reduced Overlap Matrix for 1-D Lattice Reduced Single orbital, single atom basis Reduced Hamiltonian Matrix for 1-D Lattice Reduced Single orbital, single atom basis Energy Band for 1-D Lattice Energy Single orbital, single atom basis LCAO Wavefunction for 1-D Lattice LCAO Single orbital, single atom basis LCAO Wavefunction for 1-D Lattice LCAO Single orbital, single atom basis k=0 k≠0 k=π/a k = 2πp /( Na ) LCAO Wavefunction for 1-D Lattice LCAO Single orbital, single atom basis lowest energy (fewest nodes) remember H2 ? H2 highest energy (most nodes) Bloch’s Theorem Bloch’s Translation of wavefunction by a lattice constant… …yields the original wavefunction multiplied by a phase factor Consistent that the probability density is equal at each lattice site Wavefunction Normalization Normalization Using periodic boundary conditione for a crystal with N lattice sites between boundaries… Counting Number of States in a Band Counting Combining periodic boundary condition… …with Bloch’s theorem… …yields a discrete set of k-vectors Within the 1st Brillouin Zone there are N states or 2N electrons Tight-binding and Lattice Wave Formalism Tight Electrons (LCAO) Lattice Waves...
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