Reducible Representations and Predicting Infrared and Raman Activity
Take the motions of a molecule as a basis, work out the characters of that basis under each symmetry operation, and reduce it into irreducible labels. The labels say directly which vibrations absorb infrared light and which scatter it.
What a learner can do afterwards
- Builds the reducible representation for a chosen basis by counting unshifted contributions
- Applies the reduction formula and subtracts translation and rotation to leave the vibrations
- Assigns each remaining label as infrared active, Raman active, both or neither
- Uses the predicted band count to choose between two candidate geometries
1 · Read
Shape comes first: electron domains push apart, two in a line, three in a plane, four toward tetrahedron corners. The point group is the full set of symmetry moves the resting shape obeys: rotation axes, mirror planes, and centers. Assign that group before anything else, because every later count hangs on it.
Next, build the reducible picture of the motions. The usual basis is three arrows, x, y and z, on every atom. Run through each class of symmetry move and count the atoms that stay put. Each unshifted atom adds 3 under the identity, -1 under a C2 rotation, 1 under a mirror plane and -3 under inversion. Those totals, class by class, are the reducible characters.
Feed the characters into the reduction formula. For each irreducible label, multiply the reducible character by the label's character for every operation, add the products, and divide by h, the number of operations in the group. That gives how many times the label appears. Subtract translation and rotation labels, and the survivors are the vibrations: 3N - 6 for a nonlinear molecule of N atoms, 3N - 5 for a linear one. Matching x, y or z means infrared active, matching a quadratic function means Raman active, and matching neither means silent.
Use the band count to judge shapes. Each candidate geometry predicts its own count, so compare the spectrum against each prediction and let it vote. Water shows the whole method. In C2v, with the molecule in the yz plane, the characters for E, C2, the xz mirror and the yz mirror are 9, -1, 1 and 3. They reduce to 3A1 + A2 + 2B1 + 3B2. Removing translations and rotations leaves 2A1 + B2: three vibrations, all infrared and Raman active.
Group first, count unshifted moves, reduce, strip translations and rotations, and read each surviving label as infrared, Raman, or silent.
Readings 'Water, worked through' and 'Carbon dioxide and mutual exclusion' are original to this dataset.
2 · Watch
En anglaisTake 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.