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Lecture_9 - MIT OpenCourseWare http/ocw.mit.edu 5.04...

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MIT OpenCourseWare http://ocw.mit.edu 5.04 Principles of Inorganic Chemistry II �� Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 9: Band Theory in Solids The LCAO method for cyclic systems provides a convenient starting point for the development of the electronic structure of solids. At very large N, as the circumference of the circle approaches , the cyclic problem converges to a linear one, ∞± Qualitatively, from a MO energy level perspective, ∞± 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 1 of 10
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More quantitatively, in moving from cyclic to linear systems, instead of describing orbital (atom) positions angularly, the position of an atom is described by m a , where m is the number of the atom in the array and a is the distance between atoms. Thus, the θ of the N-cyclic derivation becomes m a , ψ k = e ik(m a ) φ m ψ j = e i j θ φ m m m E j = α + 2 β cos 2 π j E k = α + 2 β cos 2 π j a multiplied by a / a N N a = α + 2 β cosk a 2 π j where k= N a A few words about k. It is: a measure of the number of nodes an index of wavefunction and accordingly symmetry of wavefunction a “quantum number” for a given ψ k a measure of length, related to wavelength λ –1 from DeBroglie’s relation, λ = h , therefore k is also a wave vector that measures momentum p Returning to the foregoing discussion, note that k parametrically depends on a . Since a is a lattice parameter of the unit cell, there are as many k’s as there are unit cells in the crystal. In the linear case, the unit cell is the distance between adjacent atoms: there are n atoms n unit cells or in other terms – there are as many k’s as atoms in the 1-D chain. π Let’s determine the energy values of limits, k = 0 and k = : a at k = 0: ψ 0 = e i0(m a ) φ m = φ m m m π i π a (m a ) at k = : ψ π = e φ m = e im π φ m = ( 1) m φ m a n m m a 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 2 of 10
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