--- title: "Reducible Representations and Predicting Infrared and Raman Activity" description: "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 i" canonical: https://lightmysky.com/learn/science/reducible-representations-and-predicting-infrared-and-raman-activity-mt_DP4JfkS6c0 source: https://lightmysky.com/learn/science/reducible-representations-and-predicting-infrared-and-raman-activity-mt_DP4JfkS6c0.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Chemistry · Ages 22 to 23 Page: https://lightmysky.com/learn/science/reducible-representations-and-predicting-infrared-and-raman-activity-mt_DP4JfkS6c0 ## Ready when they can - 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 ## Lesson: From symmetry to band count 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. **Example.** 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. **Recap.** Group first, count unshifted moves, reduce, strip translations and rotations, and read each surviving label as infrared, Raman, or silent. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [NMR After Assignment: Relaxation, Distance and Exchange](https://lightmysky.com/learn/science/nmr-after-assignment-relaxation-distance-and-exchange-mt_dCV2EbNlF2) - [Character Tables and What Symmetry Predicts](https://lightmysky.com/learn/science/character-tables-and-what-symmetry-predicts-mt_L8cTQjeKwf) ## Optional connections - [Selection Rules and Why Some Transitions Never Appear](https://lightmysky.com/learn/science/selection-rules-and-why-some-transitions-never-appear-mt_oJDr81W4fW) ## Opens up - [Choosing Between Vibrational Methods: Infrared, Raman and Resonance Raman](https://lightmysky.com/learn/science/choosing-between-vibrational-methods-infrared-raman-and-resonance-raman-mt_YxLAvBIBAR)