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W2013CHM2311 Part 3b Notes(1)

W2013CHM2311 Part 3b Notes(1) - CHM2311 Introduc0on to...

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CHM2311’ Introduc0on’to’Structure’&’Bonding’ Part’3’ Symmetry’and’Group’Theory’
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Recall’that’this’ set of characters ’gives’a’shorthand’version’of’the’transforma0on’ matrix’representa0ons’(symmetry’opera0ons).’ The’complete’set’of’irreducible’representa0ons’for’a’given’point’group’is’called’the’ character’table for’that’point’group.’Each’point’group’has’its’own’unique’character’ table.’ Character’Tables’ C 2v E C 2 σ v (xz) v (yz) coordinate 1 -1 1 -1 x 1 -1 -1 1 y 1 1 1 1 z Γ 3 -1 1 1
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What’Do’Character’Tables’Tell’Us?’ C 2v E C 2 σ v (xz) v (yz) coordinate 1 -1 1 -1 x 1 -1 -1 1 y 1 1 1 1 z Γ 3 -1 1 1 The’character’for’a’given’coordinate’symbolizes’the’result’of’a’par0cular’opera0on’on’ that’coordinate.’ e.g.’For’the’x’coordinate’in’a’C 2v ’point’group:’ ’’ E’has’no’e±ect’on’x:’character’is’ 1’ C 2 ’causes’a’sign’change’in’x:’character’is’ -1 σ v (xz)’has’no’e±ect’on’x:’character’is’ 1 σ v (yz)’causes’a’sign’change’in’x:’character’is’ -1’ RECALL that the symmetry of the atomic orbitals is related to the coordinates!
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Symmetric’ vs ’ An0symmetric’ Note: ’ An0-symmetric’ is’ NOT’ the’ same’ as’ non-symmetric’ or’ asymmetric.’ An0- symmetric’just’means’that’the’ sign of’the’coordinate’changes.’ A’character’of’–1’indicates’that’the’given’coordinate’is’ an±-symmetric with’ respect’ to’ the’ opera0on.’ The’ opera0on’ results’ in’ a’ sign’ change’ in’ that’ coordinate.’For’example,’the’x’coordinate’is’an0-symmetric’with’respect’to’C 2 rota0on. A’character’of’1’indicates’that’the’given’coordinate’is’ symmetric with’respect’ to’ the’ opera0on.’ The’ opera0on’ does’ not’ result’ in’ a’ sign’ change’ in’ that’ coordinate.’For’example,’the’x’coordinate’is’symmetric’with’respect’to’reflec0on’ through’the’ xz’plane.’
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Character’Tables’and’Orbital’Symmetry’ C 2v E C 2 σ v (xz) v ´ (yz) Sample functions A 1 1 1 1 1 z A 2 1 1 1 1 R z B 1 1 1 1 1 x, R y B 2 1 1 1 1 Y, R z Symmetry opera0ons Characters of irreducible representa0on for each symmetry opera0on. This is the trace (sum of diagonal terms) of a transforma0on matrix. Function is symmetric wrt n-fold rotation about main axis. Function is anti-symmetric wrt n-fold rotation about main axis. Function is symmetric (or anti) wrt C 2 perpendicular to main axis, or (if missing) a vertical mirror.
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W2013CHM2311 Part 3b Notes(1) - CHM2311 Introduc0on to...

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