Water Molecular Orbitals by Symmetry

Introduction

Water's bent geometry makes its two O–H bonds equivalent but not collinear. A molecular-orbital description can be built systematically by combining the two hydrogen 1s functions and matching their symmetry to oxygen valence orbitals. This approach explains which interactions are permitted and why an oxygen-centred orbital can remain largely nonbonding in a sigma-only basis. The goal is an organised qualitative diagram, not an exact numerical energy spectrum.

Core explanation

Use the coordinate convention established earlier: water lies in the yz plane, z bisects the H–O–H angle, and x is perpendicular to the molecular plane. The C₂v operation order is E, C₂(z), σ(xz), σ(yz). The two hydrogen 1s orbitals form h₊ ∝ h₁+h₂ of A₁ symmetry and h₋ ∝ h₁−h₂ of B₂ symmetry. The normalisation constants depend on H–H orbital overlap, but the symmetry labels do not.

Oxygen 2s is unchanged under all operations, so it is A₁. Oxygen 2p z also has A₁ symmetry because z is unchanged by C₂ and both vertical mirrors. Oxygen 2p y is B₂: y reverses under C₂ and σ(xz) but remains under σ(yz). Oxygen 2p x is B₁: x reverses under C₂ and σ(yz) but remains under σ(xz). These transformation signs provide a direct check on any character-table lookup.

The six-function valence basis therefore splits into an A₁ block containing O 2s, O 2p z and h₊; a B₂ block containing O 2p y and h₋; and a B₁ block containing O 2p x alone. The full Hamiltonian matrix can be arranged into blocks of dimensions three, two and one. Matrix elements between different blocks vanish at ideal C₂v symmetry, although functions within the same block need not mix strongly. The oxygen 2p x function has no hydrogen 1s SALC of B₁ symmetry in this limited sigma basis, so it remains a largely oxygen-centred nonbonding function.

The A₁ block produces three molecular-orbital combinations after diagonalisation; the B₂ block produces two; B₁ produces one. Counting 3 + 2 + 1 = 6 ensures no basis function was lost or invented. The actual ordering depends on oxygen orbital energies, overlap and Hamiltonian couplings. Oxygen 2s is relatively low in energy and may mix less strongly with hydrogen functions than a simple all-equal sketch suggests. One A₁ occupied orbital has substantial oxygen lone-pair character in common qualitative descriptions, while the B₁ orbital is the clearer symmetry-unmatched lone-pair-like function.

Water has eight valence electrons: six from oxygen and one from each hydrogen. In a closed-shell ground-state picture, four spatial valence molecular orbitals are occupied by pairs of electrons. A complete MO diagram also contains antibonding unoccupied functions from the A₁ and B₂ blocks. Symmetry determines the blocks, not the exact electron density or the number of strictly localised lone-pair orbitals. A localised Lewis or hybrid-orbital description and a delocalised symmetry-labelled MO description are different representations of related electronic structure, not necessarily competing experimental facts.

The bent angle is important. If H–O–H became linear, the point group and orbital symmetry labels would change, and some interactions could be reclassified. Conversely, a solvent or external field can lower effective symmetry and induce small couplings forbidden in the isolated ideal model. The clean C₂v diagram remains a valuable starting point for understanding spectroscopy and bonding.

Step-by-step reasoning

Fix the yz-plane convention. Reduce the two-H basis to A₁ + B₂ and classify O 2s and three 2p orbitals from their coordinate signs. Group matching labels into blocks, count six total functions, and then assign eight valence electrons to the lowest four spatial orbitals only after considering a plausible energy order. Distinguish exact symmetry non-mixing from merely weak energetic mixing.

Visual explanation

Draw two H 1s circles with same-phase shading for h₊ and opposite-phase shading for h₋. Place O 2s and 2p z beside h₊ under A₁, O 2p y beside h₋ under B₂, and O 2p x alone under B₁. Draw horizontal dividing lines between blocks. An energy-level sketch can then show three A₁, two B₂ and one B₁ output lines.

Real-world analogy

Imagine instruments sorted into rehearsal rooms according to the musical part they can play. The A₁ and B₂ rooms contain both oxygen and hydrogen participants, so ensemble sound can form; the B₁ room has only an oxygen participant in this limited arrangement. Matching rooms permit interaction but do not determine how loud each instrument plays, just as symmetry matching does not fix orbital mixing strength.

Real-world example

The oxygen-centred occupied orbitals of water make it a Lewis base toward suitable acceptors and participate in hydrogen bonding. A symmetry-based MO description clarifies that these orbitals are not two perfectly identical, fixed sp³ lobes in the canonical delocalised basis. Localised lone-pair pictures can still be useful for chemical intuition, but their exact shapes depend on the chosen orbital representation.

Why?

Why does O 2p x lack a hydrogen 1s partner? Both hydrogen SALCs are unchanged by reflection in the molecular yz plane, because the H 1s functions lie in it. O 2p x changes sign under that reflection. Its interaction with either H SALC would therefore have an integrand that reverses sign and integrates to zero in the ideal symmetric model.

Common misconception

Calling an orbital nonbonding does not mean it has no energy or electrons; it means it makes little net bonding contribution in the specified interaction model. Also, an A₁ match does not guarantee oxygen 2s, oxygen 2p z and h₊ all mix equally. Different energies and overlaps shape the final molecular orbitals.

Worked example

Classify the six valence basis functions. The two H 1s functions reduce to A₁ + B₂. Oxygen adds 2s(A₁), 2p z(A₁), 2p y(B₂) and 2p x(B₁). Therefore the total representation is 3A₁ + 2B₂ + B₁ by dimension count. The corresponding block dimensions are 3, 2 and 1. Their sum is six, equal to the starting two H plus four O valence functions.

Quick check

1. Which oxygen 2p function is perpendicular to the molecular plane in this convention? Answer: Oxygen 2p x, which has B₁ symmetry and no H 1s partner in the sigma-only basis. 2. How many spatial valence orbitals are occupied by water's eight valence electrons in a closed-shell ground state? Answer: Four spatial orbitals are doubly occupied.

Exam focus

State the coordinate convention before naming B₁ or B₂. Show the H SALC reduction, classify O 2s and 2p functions, and count all six input and output orbitals. Explain why symmetry permits a block without asserting a numerical energy order from symmetry alone.

Advanced insight

Canonical molecular orbitals diagonalise a chosen one-electron operator and can be delocalised; unitary mixing among occupied orbitals can produce more localised bond and lone-pair pictures without changing the total occupied electronic subspace of a single-determinant model. This is why a Lewis-style picture and a symmetry-labelled canonical MO diagram can both be useful, provided their different purposes are clear.

Summary

In yz-plane C₂v water, H 1s functions form A₁ and B₂ SALCs. O 2s and 2p z join A₁, O 2p y joins B₂ and O 2p x forms a lone B₁ block in the sigma-only basis. Six input functions produce six MOs, four occupied by water's eight valence electrons. Symmetry fixes block structure, while energies and overlaps determine detailed bonding.

Practice questions

1. Why is an O 2p x–H 1s SALC interaction forbidden in ideal water under the stated coordinate convention? Answer: O 2p x is odd under reflection in the molecular plane, while both H 1s SALCs are even. Their Hamiltonian interaction integrand is not totally symmetric and cancels. 2. A student draws four A₁ molecular orbitals from the A₁ block of this six-function model. What is wrong? Answer: The A₁ block has only three independent basis functions: O 2s, O 2p z and h₊. It can produce only three independent A₁ molecular orbitals. 3. Does the A₁ label alone tell whether an orbital is bonding or antibonding? Answer: No. Several A₁ orbitals arise from the same block and differ in energy and phase pattern. The label states transformation symmetry, not bonding character or energy order.