Chapter 1

1. Chapter 1 Point Group Symmetry

2. Symmetry Elements E: Identity operation Cn: Proper rotation

3. Symmetry Elements i: Inversion sh: Horizontal Mirror Plane sv: Vertical Mirror Plane

4. Symmetry Elements Sn: Improper rotation: combination Cn and sh S2 is equivalent to inversion (i)

5. Symmetry Groups • Groups with no proper rotation axis • C1: Only E (i.e. no symmetry elements) • Cs: E and s • Ci: E and i • Sn: E, Sn (S1 = Cs; S2 = Ci) • Groups with one proper rotation axis • Cn: E, Cn only • Cnv: E, Cn, and n sv (linear unsymmetrical molecules are C∞v) • Cnh: E, Cn, and sh • Dihedral Groups: Groups with n C2 axes  to Cn • Dn: E, Cn, and n C2 axes  to Cn • Dnh: E, Cn, n C2 axes, and sh(linear symmetrical molecules are D∞h) • Dnd: E, Cn, n C2 axes, and n sv • Cubic Groups: Groups with more than one Cn (n ≥ 3) • Td: symmetry of a regular tetrahedron: 4 C3 • Oh: symmetry of a regular octagon: 6 C4 • Ih: symmetry of a regular icosahedron: 12 C5

6. Yes No No Cn Cnh sh? sv? Yes No Yes More than one Cn(n ≥ 3) Cubic T, O, I S2n colinear w/ Cn? C∞v or D ∞h Cnv Yes No Find principal axes Cn is the principal axis? n vertical mirror planes No No nC2 to Cn? S2n Linear? Yes Yes None No Yes Dnd Cs, Cior C1 sh? sv? Yes No Dnh Dn Symmetry Decision Tree Physical Chemistry, Joseph H. Noggle, 2nd ed., Scott Foresman & Co, Glenview, IL, 1996, pg 840.