1. Symmetry Properties of Molecules
Chemistry 330
Symmetry Properties of Molecules

2. Symmetry Elements
Some of the symmetry elements of a cube. The twofold, threefold, and fourfold axes are labelled with the conventional symbols.

3. Proper Rotations An NH3 molecule has a threefold (C3) axis
An H2O molecule has a twofold (C2) axis.

4. Mirror Planes
An H2O molecule has two mirror planes. They are both vertical (that is, contain the principal axis) and so are denoted v and ‘v.

5. Dihedral mirror planes
Dihedral mirror planes (d) bisect the C2 axes perpendicular to the principal axis.

6. The Inversion Centre
A regular octahedron has a centre of inversion (i).

7. Improper Rotation Axes
A CH4 molecule has a fourfold improper rotation axis (S4)
The staggered form of ethane has an S6 axis composed of a 60 rotation followed by a reflection.

8. The Inversion Centre
The presence of a twofold axis and a horizontal mirror plane jointly imply the presence of a centre of inversion in the molecule.

9. The Dn Point Group
A molecule with n twofold rotation axes perpendicular to an n-fold rotation axis belongs to the group Dn.

10. The Dnh Point Group
A molecule with a mirror plane perpendicular to a Cn axis, and with n twofold axes in the plane, belongs to the group Dnh.

11. Cubic Point Groups
Tetrahedral and octahedral molecules are drawn in a way that shows their relation to a cube: they belong to the cubic groups Td and Oh, respectively.

12. The Point Group Flow Diagram
A flow diagram for determining the point group of a molecule. Start at the top and answer the question posed in each diamond (Y = yes, N = no).

13. Molecules and Point Groups
A summary of the shapes corresponding to different point groups.

14. Equivalent Elements
Some symmetry elements are implied by the other symmetry elements in a group
Presence of an inversion means at least an S2 element
Hence, i and S2 are equivalent

15. Characters of Rotations
A rotation through 180 about the internuclear axis leaves the sign of a  orbital unchanged but the sign of a  orbital is changed.
The characters of the C2 rotation are +1 and -1 for the  and  orbitals, respectively.

16. Classes and Groups
Symmetry operations in the same class are related to one another by the symmetry operations of the group.

17. The px Orbitals and Symmetry
A px orbital on the central atom of a C2v molecule and the symmetry elements of the group.

18. Reflections
The two orbitals shown here have different properties under reflection through the mirror plane: one changes sign (character -1), the other does not (character +1).

19. SALC’s Two symmetry-adapted linear combinations
Each span a one-dimensional irreducible representation, and their symmetry species are different.

20. More SALC’s
Typical symmetry-adapted linear combinations of orbitals in a C3v molecule.

21. Symmetry Species and Orbital Overlap
Orbitals of the same symmetry species may have non-vanishing overlap
The a1 and the two components of the doubly degenerate e orbitals