Cubic groupsIcosahedral groupLinear groupsImmediate Consequences of SymmetryDefinitionsInterpretation of the Character TableGroup Theory and Molecular SymmetryMolecular SymmetrySymmetry Elements and OperationsIdentity element – E- Apply E to object and nothing happens. Object is unmoved.Rotation axis – Cn- Rotation of object through 360/n yields an indistinguishable object.C a x i s2C a x i s3C a x i s4- 2 2C C E , 3 3 3C C C E  , etc …- The Cn axis with the highest n is an object’s principal axis.Reflection plane - - vertical - v- plane is parallel to principal axis- 1,1,1 trichloroethane (eclipsed or staggered) is molecule with only vCl HCl HCl HClHClHClH- horizontal - h- plane is perpendicular to principal axis- ethane (eclipsed) is molecule with h- dihedral - d- plane is parallel to principal axis and bisects angle between two C2 axes.- staggered ethane is molecule with d- Examples of composite operations- E , d 2C E  , 4 V 2C C Center of inversion – i- turns molecule “inside out”- make following replacements x x , y y  and z z - staggered ethane has i, eclipsed ethane does not.Improper rotation axes – Sn1- reflection followed by rotation through 360/n- object may have Sn without having h or Cn separately.- Allene is an example.C C CHHHHSymmetry Groups- all symmetry operations on a molecule form a closed group called a symmetry group.Example: C2vC2vE C2vv'E E C2vv'C2C2E v' vvvv' E C2v' v' vC2EOHHvv’C2Special GroupsC1 – has no symmetryN CCH ClHHHOOCi – has only inversion symmetryBrClBrClCs – has only mirror planeClBrBrClCn GroupsCn – has E and (n-1) Cn symmetry elements2CCHOHHOOHHH C3Cnv – has E, (n-1) Cn and n v symmetry elementsNH HH C3vCnh – has E, (n-1) Cn, (n-1) Sn, h, et. al. symmetry elementsHClClH C2hDn Groups – similar to Cn, but has n C2 axes perpendicular to Cn axisDn – has E and (n-1) Cn, n C2 axes  principal axis, et. al. symmetry elementsHHHHHH D3Dnh – has E, (n-1) Cn, n v, h et. al. symmetry elementsRu D5hDnd – has E, (n-1) Cn, v, d, et. al. symmetry elementsHHHHHH D3dSn groupsSn – has E, (n-1) Sn symmetry elements3neither staggered nor eclipsedCubic groupsT, Td, Th – tetrahedral groupsSiFF FFTdO, Oh – octahedral groupsCoClCl ClClClClOhIcosahedral groupIh – icosahedral groupBuckminsterfullereneLinear groupsCv- Heteronuclear diatomic molecules are most important example.Dh- Homonuclear diatomic molecules are most important example.Immediate Consequences of Symmetry1. Polarity- Molecule cannot have dipole moment perpendicular to Cn axis.- Molecule may have dipole moment parallel to principal axis if no C2 axes are perpendicular to principal axis- Molecule with Sn axis may not have dipole moment.- Thus only C1, Cs, Cn and Cnv molecules can have dipole moment.2. Chirality- Molecule without Sn axis may be chiral- A chiral molecule cannot have center of inversion.Representations of Groups4Recall that the symmetry elements for a C2v molecule form a group that, by definition, is closed under multiplication.C2vE C2vv'E E C2vv'C2C2E v' vvvv' E C2v' v' vC2EOHHvv’C2Rather than relying on our ability to visualize the spatial changes that occur during a pair of symmetry operator, we would like to use numerical representations for the group.Consider the following vector to represent the nine coordinates that describe the position of a water molecule.111222333HxHyHzOxv OyOzHxHyHz� �� �� �� �� �� �� �=� �� �� �� �� �� �� �If we wanted to represent the identity element with a matrix, the following matrix E would be a sensible choice. Note the effect of multiplying our coordinate vector v by the matrix E. Indeed, E is an identity element.1 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 0E0 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1� �� �� �� �� �� �� �=� �� �� �� �� �� �� � 1 11 11 12 22 22 23 33 33 3Hx Hx1 0 0 0 0 0 0 0 0Hy Hy0 1 0 0 0 0 0 0 0Hz Hz0 0 1 0 0 0 0 0 0Ox Ox0 0 0 1 0 0 0 0 0E v Oy Oy0 0 0 0 1 0 0 0 0Oz Oz0 0 0 0 0 1 0 0 0Hx Hx0 0 0 0 0 0 1 0 0Hy Hy0 0 0 0 0 0 0 1 0Hz Hz0 0 0 0 0 0 0 0 1� � �� �� � �� �� �� �� �� �� �� �� �� �� �� ��= � =� �� �� �� �� �� �� �� �� �� �� �� �� �� � ���� �� �� �� �� �� �� �� �� �� �� ��5With the C2 rotation, H1 and H3 switch places and the x and y directions change sign. Thefollowing matrix represents the C2 rotation.20 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 10 0 0 1 0 0 0 0 0C0 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0-� �� �-� �� �� �-� �� �=-� �� �� �-� �-� �� �� �131313222 2222333HxHx0 0 0 0 0 0 1 0 0HyHy0 0 0 0 0 0 0 1 0HzHz0 0 0 0 0 0 0 0 1OxOx0 0 0 1 0 0 0 0 0C v OyOy0 0 0 0 1 0 0 0 0OzOz0 0 0 0 0 1 0 0 0HxH1 0 0 0 0 0 0 0 0Hy0 1 0 0 0 0 0 0 0Hz0 0 1 0 0 0 0 0 0--� �� �� �� �--� �� �� �� �� �� �--� �� �� �� ��= =--� �� �� �� �� �� �--� �� �-� �� �� �� �� �� �111xHyHz� �� �� �� �� �� �� �� �� �� �� �-� �� �� �Thus vs and v�s also have 9  9 matrix representations.v0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 10 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0� �� �-� �� �� �� �� �s =-� �� �� �� �-� �� �� � 13131322v 22223133HxHx0 0 0 0 0 0 1 0 0HyHy0 0 0 0 0 0 0 1 0HzHz0 0 0 0 0 0 0 0 1OxOx0 0 0 1 0 0 0 0 0v OyOy0 0 0 0 1 0 0 0 0OzOz0 0 0 0 0 1 0 0 0HxHx1 0 0 0 0 0 0 0 0HyHy0 1 0 0 0 0 0 0 0Hz0 0 1 …