--- title: "Molecular Symmetry: Elements, Operations and Point Groups" description: "Symmetry operations return a fixed molecular geometry to an indistinguishable arrangement. Their full set defines its point group. This group constrains orbital mixing, chirality and the directions in" canonical: https://lightmysky.com/learn/science/molecular-symmetry-elements-operations-and-point-groups-mt_KcofGO11tm source: https://lightmysky.com/learn/science/molecular-symmetry-elements-operations-and-point-groups-mt_KcofGO11tm.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`. # Molecular Symmetry: Elements, Operations and Point Groups Symmetry operations return a fixed molecular geometry to an indistinguishable arrangement. Their full set defines its point group. This group constrains orbital mixing, chirality and the directions in which a permanent dipole is allowed. Subject: Science · Area: Chemistry · Ages 18 to 19 Page: https://lightmysky.com/learn/science/molecular-symmetry-elements-operations-and-point-groups-mt_KcofGO11tm ## Ready when they can - Tests rotations, reflections, inversion and improper rotations by comparing atom positions - Uses a stated decision branch to distinguish the C2v, C3v and D3h molecular examples - Applies the absence of all improper rotations as the chirality test for a fixed geometry - Distinguishes a symmetry-forbidden permanent dipole from a symmetry-allowed dipole ## Lesson: Moves that leave molecules unchanged You have used symmetry to test orbital mixing. A symmetry operation returns a fixed geometry to an indistinguishable arrangement: each atom reaches a position occupied by the same kind of atom. Identical atoms may exchange positions. E is the identity operation, which leaves every position unchanged. An element is an operation's axis, plane or point. A Cn rotation turns through 360/n degrees about an axis. Reflection, σ, reverses the coordinate perpendicular to a mirror plane. Inversion, i, sends (x,y,z) to (-x,-y,-z) through a centre. An improper rotation, Sn, combines a 360/n-degree rotation with reflection through the plane perpendicular to its axis. Test the combined result against the original atom positions. S1 is reflection; S2 is inversion. *(drawing: An Sn test follows this order. The cells show the procedure, not molecular geometry.)* For the three examples here, find the highest-order rotation axis. Look for C2 axes perpendicular to it. If present, a horizontal mirror plane leads to Dnh. If absent, vertical mirror planes lead to Cnv. Horizontal means perpendicular to the main axis; vertical means containing it. Other point groups need other branches. **Example.** Water is bent. Its C2 axis passes through oxygen along the H-O-H angle bisector. A 180-degree turn exchanges the hydrogens. One mirror plane contains the molecule; another contains C2 and is perpendicular to the molecular plane. There are no perpendicular C2 axes. The operations are E, C2 and two vertical reflections: C2v. Water has no C3 axis. **Example.** Ammonia is trigonal pyramidal. Its C3 axis passes through nitrogen and the centre of the hydrogen triangle. Turns of 120 and 240 degrees cycle the hydrogens. Three mirror planes contain this axis, each containing one N-H bond. No perpendicular C2 axes or horizontal mirror plane exist. Its operations are E, two C3 turns and three reflections: C3v. *(drawing: This decision path applies to trigonal planar boron trifluoride.)* **Example.** BF3 has three equal B-F bonds at 120 degrees in one plane. C3 is perpendicular to that plane; three C2 axes lie along the bonds. The molecular plane is σh. Its operations are E, two C3 turns, three C2 turns, σh, two S3 operations and three vertical reflections. This is D3h. The three equal bond dipoles cancel, making BF3 nonpolar. A fixed geometry is chiral when its mirror image cannot be superimposed by rotations and translations. This happens exactly when no Sn operation of any order exists. Mirror planes, S1, and inversion centres, S2, therefore forbid chirality. A geometry with only proper rotations besides E is chiral. Flexible molecules may interconvert between mirror-related geometries; this test describes one geometry. A permanent dipole must survive every symmetry operation. Ammonia has C3 and vertical mirrors but no perpendicular C2 axes: the Cnv branch with n=3 gives C3v. C2v and C3v allow a dipole along the main axis. Water's and ammonia's bond dipoles do not cancel, so both are polar. In D3h, C3 forbids an in-plane component and σh forbids a perpendicular component. Permission alone does not prove a nonzero dipole. **Recap.** Compare atom positions after each operation. Use the point group to test chirality and find which dipole directions symmetry allows. *Lesson, questions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Heteronuclear Diatomics and Polarity in Orbital Terms](https://lightmysky.com/learn/science/heteronuclear-diatomics-and-polarity-in-orbital-terms-mt_thdxFqyHJV) ## Optional connections - [Hybridisation, conjugation and delocalisation in organic molecules](https://lightmysky.com/learn/science/hybridisation-conjugation-and-delocalisation-in-organic-molecules-mt_Anv0MSQ4hq) ## Opens up - [Character Tables and What Symmetry Predicts](https://lightmysky.com/learn/science/character-tables-and-what-symmetry-predicts-mt_L8cTQjeKwf)