--- title: "Symmetry-Adapted Orbitals for Polyatomic Molecules" description: "Symmetry-adapted combinations group ligand orbitals by symmetry. Matching labels identify permitted central-atom partners. Orbital overlap and energy differences affect the mixing. Water shows how to " canonical: https://lightmysky.com/learn/science/symmetry-adapted-orbitals-for-polyatomic-molecules-mt_llEYzYuutf source: https://lightmysky.com/learn/science/symmetry-adapted-orbitals-for-polyatomic-molecules-mt_llEYzYuutf.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`. # Symmetry-Adapted Orbitals for Polyatomic Molecules Symmetry-adapted combinations group ligand orbitals by symmetry. Matching labels identify permitted central-atom partners. Orbital overlap and energy differences affect the mixing. Water shows how to organise these blocks, fill a supplied energy order and interpret lone-pair orbitals. Subject: Science · Area: Chemistry · Ages 19 to 20 Page: https://lightmysky.com/learn/science/symmetry-adapted-orbitals-for-polyatomic-molecules-mt_llEYzYuutf ## Ready when they can - Forms water's hydrogen sum and difference combinations and assigns their symmetry labels in a stated axis convention - Matches ligand combinations with central-atom orbitals of the same symmetry and rejects mismatches - Organises water's minimal valence basis into symmetry blocks and fills a supplied MO energy order - Distinguishes water's strictly nonbonding MO from its other largely lone-pair MO in the stated minimal basis ## Lesson: Matching orbitals by symmetry You have used character tables to label motions. The same labels classify orbital functions. In the symmetry-adapted method, combine orbitals on outer atoms, called ligands, into symmetry-adapted linear combinations (SALCs), then match central-atom orbitals. Each SALC has one symmetry label. Different labels forbid mixing if symmetry is retained. Equal labels permit mixing; spatial overlap and orbital energy differences affect its size. For C2v water, put the molecule in yz with z bisecting H-O-H. Oxygen 2s and 2pz are a1, 2py is b2 and 2px is b1. The a1 hydrogen SALC joins both oxygen a1 orbitals in a three-function block, giving three MOs. The b2 SALC and oxygen 2py give two MOs. Oxygen 2px supplies one b1 MO. Thus four oxygen valence orbitals plus two hydrogen 1s orbitals give six valence MOs. Omit oxygen 1s. Let H1 and H2 name the hydrogen 1s functions. SALCs are proportional to H1 + H2 and H1 - H2; normalization factors are omitted. Signs mark wavefunction phase, not charge. C2 and σv(xz) exchange the hydrogens; σv(yz) leaves each fixed. In order E,C2,σv(xz),σv(yz), the sum has characters (1,1,1,1), so it is a1. The difference has (1,-1,-1,1), so it is b2. *(drawing: The difference changes sign under exchange. The sum remains H1 + H2. Neither expression alone is a finished molecular orbital.)* In the oxygen 2s/2p and hydrogen 1s basis, there is no hydrogen b1 SALC. Oxygen 2px forms the nonbonding 1b1 MO. Here nonbonding means no bonding or antibonding mixing with hydrogen in this basis, not an unchanged isolated-atom energy or a required occupation. The occupied 3a1 MO is also largely lone-pair-like, but has some O-H bonding character in a matched a1 block. The two Lewis lone pairs are not two symmetry-unmatched canonical MOs. For the schematic diagram, use low-to-high order 2a1, 1b2, 3a1, 1b1, 4a1, 2b2. Symmetry alone does not give this order. Water has eight valence electrons: 6 + 1 + 1. Put at most two opposite-spin electrons in each MO, starting at lowest energy. The first four each hold two; the antibonding 4a1 and 2b2 are empty. The omitted core MO is 1a1, which explains why the valence a1 numbering starts at 2. *(drawing: Read left to right from low to high energy. Each cell names one MO and its electron count. Cell spacing does not measure energy gaps.)* In tetrahedral methane, four hydrogen 1s orbitals give one a1 SALC and a t2 set containing three independent SALCs. Triply degenerate means the three share an energy; it does not reduce their number. Carbon 2s is a1 and its three 2p orbitals form t2. Four carbon orbitals plus four hydrogen orbitals give eight MOs: four bonding and four antibonding. Methane's eight valence electrons fill the four bonding MOs. **Recap.** Label ligand SALCs, match central orbitals by symmetry, and fill MOs using their energy order. In water's minimal basis, 1b1 is nonbonding; the largely lone-pair 3a1 MO has some bonding character. *Lesson, questions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Character Tables and What Symmetry Predicts](https://lightmysky.com/learn/science/character-tables-and-what-symmetry-predicts-mt_L8cTQjeKwf) ## Opens up - [Ligand field theory and pi bonding in complexes](https://lightmysky.com/learn/science/ligand-field-theory-and-pi-bonding-in-complexes-mt_Gq3JG7F7Nl)