2.Symmetry and group theory (Notes 2).pdf - Notes on...

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Notes on Symmetry and Group Theory Let's go back and review the characteristics of character tables, using the C 2 v point group as an example: C 2 v E C 2 σ v (xz) σ v (yz) A 1 1 1 1 1 z x 2 , y 2 , z 2 A 2 1 1 –1 –1 R z xy B 1 1 –1 1 –1 x, R y xz B 2 1 –1 –1 1 y, R x yz II I III IV As we briefly mentioned on Friday, these tables compile data for all of the irreducible representations of a given point group ( C 2 v in this case). There are 4 main areas of the table: I: Characters of irreducible representations (remember these indicate the "trace" or "character" of the matrices describing each operation for each irreducible representation. The value under E is always the "dimensionality" or "degeneracy" of the irreducible representation, since E will always leave every element unchanged. II: Names of irreducible representations (Mulliken symbols). I've included info about how these are assigned, but the most important thing is that A/B are for singly degenerate irreducible representations (meaning that each constituent matrix is 1×1), E (not the same as the symmetry operation) is for doubly degenerate irreducible reps (come from 2×2 matrices), and T is for triply degenerate irreducible reps (3×3 matrices). III: Symbols representing x, y, and z coordinates/axes and rotations about each axis * These can be grouped together for higher-dimensional irreducible reps if symmetry operations mix them * R x , R y , and R z can be visualized as curved arrows around x, y, and z axes IV: Quadratics squares and binary products of x, y, and z can also form bases for irreducible representations * We'll get to the utility of these (mostly because they relate to d orbitals) when we discuss MO theory
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