Lecture_2_GroupTheory

# Lecture_2_GroupTheor - Chem 310 Lecture Module 2 Symmetry of Complexes and Character Tables A REVIEW Symmetry and Group Theory Review of Chem 212

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Chem 310 Lecture Module 2 Symmetry of Complexes and Character Tables: A REVIEW

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Symmetry and Group Theory: Review of Chem 212 ymmetry: certain geometrical transformations leave an object indistinguishable Symmetry: certain geometrical transformations leave an object indistinguishable. The geometric objects (axes, planes, centres of inversion) are called symmetry elements The symmetry operations are the transformations themselves. The set of all symmetry elements associated with a molecule obeys the properties associated with a mathematical group so the mathematics of group theory can be applied to molecules. This math allows us to characterize and label some properties of molecules such as molecular orbitals or spectroscopic transitions. 3 classes of common symmetry operations: 1. inversions 2. rotations 1 + 2 lead to Point Groups translations +2+3leadto pace Groups 3. translations 1 + 2 + 3 lead to Space Groups
Symmetry Operations and the Elements that Define Them . Inversion 1. Inversion i The signs of the Cartesian coordinates are made negative leaving the object invariant in appearance (absolutely identical). (x, y, z) Î (-x, -y, -z) 2. Rotations C n An angular motion of 360º/n clockwise (by convention) about an axis (the axis may be defined in the symmetry element) special cases: 0 o rotation is given the symbol E and acts as the identity element in group theory C is an infinitely small rotation - appropriate for linear molecules For any given C n axis, there are n -1 different rotations or operations: C n , C n 2 , C n 3 , . .., C n n-1 3. mirrors or planes of symmetry eflection through a plane that leaves an object invariant Reflection through a plane that leaves an object invariant All mirrors are composite symmetry elements - combinations of i and C 2 σ xy = i × C 2z i tii b d( l l) ib t b d (l l ) σ v = mirror containing bonds (usually); σ d = mirror between bonds (usually) σ h = mirror perpendicular to highest rotation axis Improper rotations 4. Improper rotations S n A rotation followed by a mirror perpendicular to the rotation: S n = C n × σ h Note that S 2 = i and S 1 = σ

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Proper Rotations C 3 1 C 3 2 E C 3 3 = E Note that all objects can be operated upon by the identity operator (E), as it leaves the molecule unchanged (identical)
Reflections If a reflection of all parts of a molecule (or lattice) through a lane produces an indistinguishable configuration, the operation plane produces an indistinguishable configuration, the operation is one of reflection and the symmetry element is the mirror plane. here is one reflection operation for each reflection symmetry There is one reflection operation for each reflection symmetry element.

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Improper Rotations An improper rotation (rotator-reflection), S n , involves rotation about 360 ° /n, followed by reflection through a plane that the perpendicular to the rotation axis. In the diagram pp g above, there are a possibility of three S 4 operations: S C S S E) S 4 1 ; S 4 2 (= C 2 ); S 4 3 ; S 4 4 (=E)
Point Groups The set of all symmetry elements associated with a molecule is called the point group.

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## This note was uploaded on 02/28/2011 for the course CHEM 310 taught by Professor Nazar during the Fall '09 term at Waterloo.

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Lecture_2_GroupTheor - Chem 310 Lecture Module 2 Symmetry of Complexes and Character Tables A REVIEW Symmetry and Group Theory Review of Chem 212

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