Molecular Symmetry and Group Theory
Lecture 5 — Chemical bonding

Lecture Summary
Applications of symmetry and group theory:
Chirality — a molecule may be chiral if it does not possess an improper rotation axis. As S_{1} ≡ σ and S_{2} ≡ i, a chiral molecule can possess no symmetry elements other than E and proper rotation axes C_{n}.
This excludes all molecules except those with point groups C_{1}, C_{n} or D_{n}.
Dipole moments — a molecule may not have a dipole moment:
 perpendicular to an axis of rotation;
 perpendicular to a mirror plane;
 or in any direction at all if the molecule possesses an inversion centre.
This excludes all molecules except those with point groups C_{1}, C_{n} or C_{nv}.
Orbital degeneracies — inspection of the appropriate character table can tell us which p or d orbitals will be degenerate in a given geometry e.g. p_{x} and p_{y} in BF_{3} (D_{3h})
Hybridization — Workshop 5
Molecular orbital theory — Workshop 5
Molecular vibrations — Workshop 6
The application of group theory to chemical problems can be summarized in the following three steps:
 use an appropriate basis to generate a reducible representation of the point group
 reduce this representation to its constituent irreducible representations
 interpret the results
Hybridization
We can use group theory to decide which atomic orbitals on a central atom can hybridize to give an appropriate set of orbitals for a given molecular geometry.
We know that the hybridization of the C 2s and 2p orbitals gives four tetrahedral sp^{3} hybrids, but how could we tell, for instance, that it is this set of orbitals that gives this particular geometry (Exercise 1)? Or which orbitals would have to combine to give a different geometry (Exercise 2)?
Molecular Orbital (MO) Theory
There are three major requirements for the formation of molecular orbitals from atomic orbitals:
 The atomic orbitals must have similar energy
 They must have significant overlap
 They must have appropriate symmetry
We can only judge whether the third requirement is met through the application of group theory (Exercises 3—6).