Lecture_8

# Lecture_8 - MIT OpenCourseWare http/ocw.mit.edu 5.04...

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Unformatted text preview: 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 . 5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 8: N-Dimensional Cyclic Systems This lecture will provide a derivation of the LCAO eigenfunctions and eigenvalues of N total number of orbitals in a cyclic arrangement. The problem is illustrated below: There are two derivations to this problem. Polynomial Derivation The Hückel determinant is given by, x 1 1 x 1 1 x O D N (x) = 1 O O O O O = 0 where β α E x − = O O O O O 1 O x 1 1 x From a Laplace expansion one finds, D N (x) = xD n–1 (x) – D N–2 (x) where D 1 (x) = x x 1 D 2 (x) = = x 2 − 1 1 x 5.04, Principles of Inorganic Chemistry II Lecture 8 Prof. Daniel G. Nocera Page 1 of 6 With these parameters defined, the polynomial form of D N (x) for any value of N can be obtained, D 3 (x) = xD 2 (x) – D 1 (x) = x(x 2 –1) – x = x(x 2 –2) D 4 (x) = xD 3 (x) – D 2 (x) = x 2 (x 2 –2) – (x 2 –1) and so on The expansion of D N (x) has as its solution, 2 π x = − 2cos j (j = 0,1, 2, 3...N − 1) N and substituting for x, 2 π E = α + 2 β cos j (j = 0,1, 2, 3...N − 1) N Standing Wave Derivation...
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Lecture_8 - MIT OpenCourseWare http/ocw.mit.edu 5.04...

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