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SYMMETRY - Molecular symmetry, group theory, and chemical
bonding
Dr. Claire Vallance
Michaelmas Term - Second Year 8 lectures
The first part of the course is an introduction to the systematic
classification of molecular symmetry, which will allow you
to recognise the symmetry elements (rotation axes, planes
of reflection etc.) possessed by a molecule. This will be
followed by a fairly comprehensive introduction to the mathematical
treatment of molecular symmetry known as group theory. There
are many applications of group theory in chemistry, and we
will consider several of examples. These will include symmetry
aspects of bonding in diatomic and polyatomic molecules; the
investigation of molecular motions, and in particular molecular
vibrations; and the way in which symmetry considerations govern
the interaction of light with molecules to induce spectroscopic
transitions.
A detailed breakdown of the course is as follows:
1. Introduction
2. Symmetry operations and symmetry elements
3. Symmetry classification of molecules – point groups
4. Symmetry and physical properties
4.1. Polarity
4.2. Chirality
5. Combining symmetry operations: ‘group multiplication’
6. Constructing higher groups from simpler groups
7. Mathematical definition of a group
8. Review of Matrices
8.1. Definitions
8.2. Matrix algebra
8.3 Direct products
8.4. Inverse matrices and determinants
9. Transformation matrices
10. Matrix representations of groups
10.1. Example: a matrix representation
of the C3V point group
(the ammonia molecule)
10.2. Example: a matrix representation
of theC2V point group
(the allyl radical)
11. Properties of matrix representations
11.1. Similarity transforms
11.2. Characters of representations
12. Reduction of representations I
13. Irreducible representations and symmetry species
14. Character tables
15. Reduction of representations II
15.1 General concepts of orthogonality
15.2 Orthogonality relationships in
group theory
15.3 Using the LOT to determine the
irreps spanned by a basis
16. Symmetry adapted linear combinations
17. Determining whether an integral can be non-zero
18. Bonding in diatomics
19. Bonding in polyatomics - constructing molecular orbitals
from SALCs
20. Calculating the orbital energies and expansion coefficients
21. Solving the secular equations
21.1 Matrix formulation of a set of
linear equations
21.2 Solving for the orbital energies
and expansion coefficients
22. Summary of the steps involved in constructing molecular
orbitals
23. A more complicated bonding example – the molecular
orbitals of H2O
23.1 Matrix representation, characters
and SALCs
24. Molecular vibrations
24.1 Molecular degrees of freedom
– determining the number of normal vibrational modes
24.2 Determining the symmetries of
molecular motions
24.3 Atomic displacements using the
3N Cartesian basis
24.4 Molecular vibrations using internal
coordinates
25. Summary of applying group theory to molecular motions
26. Spectroscopy – interaction of atoms and molecules
with light
26.1 Electronic transitions in molecules
26.2 Vibrational transitions in molecules
26.3 Raman scattering
27. Summary
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