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.
What a learner can do afterwards
- 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
1 · Read
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.
This is orbital bookkeeping by symmetry block, not a drawing of orbital shapes or energy levels.
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.
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.
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.
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.
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What a symmetry block tells you
Choose the atomic orbital functions before constructing molecular orbitals. Replacing those functions with independent linear combinations changes the basis, not its size. Six independent input functions still produce six MOs within that model.
Symmetry separates the basis into blocks. Functions in different irreducible symmetry types cannot mix when the molecular symmetry is retained. Functions with matching labels are permitted to interact, but their spatial overlap and energy separation affect the interaction. Equal labels do not guarantee strong mixing.
For water in yz with z bisecting H-O-H, one hydrogen a1 SALC joins oxygen 2s and 2pz. Those three functions produce three a1 MOs. The hydrogen b2 SALC and oxygen 2py produce two b2 MOs. Oxygen 2px is the remaining b1 function and produces one MO in this minimal basis.
This accounts for all six valence MOs. It does not calculate their energies or electron counts. Those are separate parts of building and filling the diagram.
Two choices of orbital basis in methane
A carbon hybrid and a hydrogen SALC combine different starting functions. Carbon sp3 hybrids combine carbon's own 2s and three 2p functions. Hydrogen SALCs combine functions centred on different hydrogen atoms. These are mathematical descriptions, not separate physical events that atoms must undergo before bonding.
Take methane's tetrahedral geometry as given. Its four hydrogen 1s functions form one a1 SALC and a t2 set with three components. Carbon 2s has a1 symmetry. The three carbon 2p functions together have t2 symmetry. The matching blocks contain two a1 functions and six t2 functions.
The resulting valence diagram contains eight MOs: one bonding a1 orbital, three bonding t2 orbitals and their four antibonding partners. Methane has eight valence electrons, so its ground state fills the four bonding orbitals with two electrons each.
The three members of a t2 level are degenerate: they share an energy. They remain three independent orbitals. Also, classifying functions at a tetrahedral geometry does not by itself prove that this geometry has the lowest energy. That comparison needs energy information.
Following water's eight valence electrons
Place water in yz and let z bisect the H-O-H angle. Oxygen 2s and 2pz are a1, 2py is b2 and 2px is b1. The hydrogen functions supply a1 and b2 SALCs but no b1 SALC.
Within this minimal valence basis, oxygen 2px therefore gives the nonbonding 1b1 MO. Its lack of a hydrogen partner does not require its energy to equal the isolated oxygen orbital's energy. The molecular environment still matters.
Use the schematic increasing-energy order 2a1, 1b2, 3a1, 1b1, 4a1, 2b2. Oxygen contributes six valence electrons and the two hydrogens contribute one each. Eight electrons fill the first four MOs with two opposite-spin electrons per orbital. The last two are empty. Oxygen's core 1a1 MO is omitted from this valence diagram, but retained in the numbering.
The occupied 1b1 and 3a1 MOs are associated with the lone-pair description. Only 1b1 is strictly nonbonding in this basis. The 3a1 MO belongs to a matched a1 block and has some O-H bonding character. Two local lone pairs in a Lewis structure need not correspond to two pure, unmatched canonical MOs.
A difference combination is not automatically antibonding
Write the hydrogen 1s functions as H1 and H2. Their symmetry-adapted combinations are proportional to H1 + H2 and H1 - H2. Normalization factors are omitted here. Plus and minus describe relative wavefunction phase, not electric charge.
For water in yz with z bisecting H-O-H, exchanging the hydrogens leaves the sum unchanged. The difference becomes H2 - H1, which is the negative of the original expression. Its characters in order E, C2, σv(xz), σv(yz) are (1,-1,-1,1), giving b2 symmetry.
Oxygen 2py also has b2 symmetry. Mixing it with the hydrogen b2 SALC produces a bonding and an antibonding b2 MO. The minus sign in the hydrogen difference alone does not identify either finished MO. The relative phases between the oxygen and hydrogen contributions matter too.
Nor does the b2 label establish a pi bond. Point-group labels describe transformation under molecular symmetry operations. Sigma and pi classifications concern behaviour relative to a bond axis. Water's hydrogen difference participates in its O-H bonding framework; it is not evidence for a separate pi bond.
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