Character Tables and What Symmetry Predicts
A character table is a compact record of how things behave under a molecule's symmetry operations. Reading one tells you how many vibrations a molecule has, which are infrared active and which are Raman active, without solving anything.
What a learner can do afterwards
- Reads a character table and states what each row and each column records
- Reduces a reducible representation to its irreducible parts
- Counts the vibrational modes of a small molecule and assigns each a symmetry label
- Uses the table to predict infrared and Raman activity and applies the rule of mutual exclusion
1 · Read
A character table fits a molecule's whole symmetry story on one page. Each row stands for one irreducible symmetry type, and each column stands for one class of symmetry operations. The entries record how each type responds to each operation, so the table tells you at a glance how any motion or orbital is allowed to behave.
To count vibrations, you reduce the full molecular motion into these labelled row parts and keep the vibrational ones. A shortcut gives the total first: a nonlinear molecule has 3N minus 6 vibrations and a linear one has 3N minus 5, where N counts the atoms. Carbon dioxide is linear with 3 atoms, so it has 4 modes; sulfur dioxide is bent with 3 atoms, so it has 3.
Each label then predicts the spectrum. A vibration that changes the dipole moment shows up in the infrared, and one that changes the polarizability shows up in Raman scattering. Carbon dioxide's symmetric stretch keeps the dipole at zero, so it is Raman only, while its antisymmetric stretch and its bends change the dipole and appear in the infrared.
In a molecule with a centre of symmetry, no vibration can be both infrared and Raman active: this is the rule of mutual exclusion. Read the table row by row, check the infrared and Raman columns for each label, and treat a missing peak as information rather than failure.
Rows name the symmetry types, columns name the operations, and the labels tell you how many vibrations exist and where each one appears.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.