Chapter 1 Symmetry

Chapter 1 Symmetry

1. Line of symmetry

   • a line on which a figure can be folded so both sides can match

2. Types of symmetry in a plane

   • Rotation
     • rotate a figure around a centre & must hv angle
   • Translation
     • sliding a figure to a direction & hv a distance
   • Reflection
     • flip the figure over a mirror line & produce a mirror image
   • Glide reflection
     • reflection then translation along the direction of mirror line

3. Symmetry operation

   • action done on an object & action leave the object same as b4 although some
   • Identity, E
     • Do nothing / rotate 360
   • n-fold rotation, Cn
     • rotation 360°/n
       • C2: rotate 180 C3: rotate 120 C4: rotate 90 C5: rotate 72 C6: rotate 60
     • principal axis(axis with highest value of n)
     • C2 axes perpendicular to principal Cn is called C2'
   • a plane symmetry, σ
     • reflection at a mirror plane
       • vertical plane,σ v
         • σ // & containing principal axis
       • horizontal plane, σ h
         • σ perpendicular to principal axis
       • dihedral plane, σd
         • σ // to principal axis &bisecting 2 C2' axes
   • Inversion centre, i
     • (x,y,z) -> (-x,-y,-z)
   • improper rotation, Sn
     • rotation by 360/n then reflection at mirror plane vertical to axis of rotation

4. Symmetry elements

   • geometrical entity with respect to 1/> symmetry operation are carried out
     • point
     • line
     • plane
     • line & plane

5. Group

   • a set containing h operation, q1, q2, q3,... which obey the properties of a group
     • condition: 1) contain identity, E with Eq1=q1E =q1
     • 2) contain inversion, with q1q'=q'q1=E
     • 3）q1(q2q3) = (q1q2)q3
   • Point group
     • at least a point in molecule X move during symmetry operation
     • the axis with highest order is the z axis & is vertical
     • if molecule is planar & z axis is in the plane, x is vertical to plane
     • If molecule is planar & z axis is vertical to plane, x axis is in the plane
     • y axis follow right hand rule
   • Steps in symmetry method:
     • 1) identify the correct structure
     • 2) identify all symmetry elements
     • 3) determine x,y & z axis
     • 4) determine the group
     • 5) use character table to solve problem
   • C1, Ci, Cs group
     • C1 : only E
     • Ci : E & i
     • Cs : E & σ
   • Cn, Cnv, Cnh group
     • Cn : E & Cn
     • Cnv : σv & Cn
     • Cnh : σh & Cn
   • Dn, Dnh, Dnd
     • Dn : E, Cn & C2 perpendicular to Cn
     • Dnh : E, Cn, C2 perpendicular to Cn & σh
     • Dnd : E, Cn, C2 perpendicular to Cn & σd
   • Sn group
   • Cubic group
     • Tetrahedron group, Td
     • Octahedron group, Oh

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