Molecular Geometry & Bonding
what the shape decides
Bonding models

Molecular orbital and valence bond

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.
10 min read 6 figures A basis is not a thingTaught confidently, and wrong

Two accounts of chemical bonding were developed at almost the same time, by people who disliked each other’s approach, and they are still taught as alternatives. They are better understood as two starting points for the same calculation.

Splitting goes with overlapFor each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.1s-1sσ overlapS = 0.39002pz-2pzσ overlapS = 0.45852px-2pxπ overlapS = 0.75291s-2pxsymmetry forbids itS = 0 exactlyno splittingoverlaps computed at 2.8 bohr, levels in proportionone electron
Fig. 1 The molecular orbital picture in its simplest form: atomic levels on the outside, combinations in the middle, split by an amount that goes with the computed overlap. Everything about the framework is in that diagram.

Molecular orbital theory

Start with the nuclei in place and ask what one-electron wavefunctions the whole framework supports. Those are molecular orbitals, they generally extend over the entire molecule, and electrons are fed into them from the bottom up.

Each molecular orbital is built as a linear combination of atomic orbitals, with the mixing decided by overlap and energy match. Two atomic orbitals give a bonding and an antibonding combination; many give many.

What it gets right immediately. The paramagnetism of O₂ — two unpaired electrons in degenerate antibonding orbitals — which valence bond theory got badly wrong for years. Ionisation energies, since the orbitals correspond to states of the ion. Electronic spectra. Bond orders in odd-electron species.

What it is awkward about. A localised bond. In this picture there is no orbital corresponding to “the C–H bond”: every occupied orbital has amplitude on many atoms, and the familiar sticks in a structural formula are nowhere to be found.

Valence bond theory

Start from atoms with electrons in atomic orbitals, and form a bond by pairing electrons in overlapping orbitals on adjacent atoms. Each bond is a local object involving two centres.

What it gets right immediately. The structural formula. Localised bonds, directionality, and the whole vocabulary of functional groups fall out naturally, which is why organic chemistry runs on it.

What it is awkward about. Anything delocalised. Benzene has to be described as a superposition of structures — resonance — which is a correct treatment presented in a way that misleads almost every student who meets it, because the structures are basis functions rather than things the molecule alternates between.

Two pictures of one bond

The clearest way to see the difference is to write both descriptions of the same simple case.

1s with 1s at 2.8 bohrThe two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero.1s · 1sS = 0.38997sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron
Fig. 2 Two hydrogen 1s orbitals at bonding distance. Molecular orbital theory forms their sum and puts both electrons in it; valence bond theory pairs one electron in each and lets the pair be shared. Both are describing this picture.
Splitting goes with overlapFor each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.1s-1sσ overlapS = 0.39002pz-2pzσ overlapS = 0.45852px-2pxπ overlapS = 0.75291s-2pxsymmetry forbids itS = 0 exactlyno splittingoverlaps computed at 2.8 bohr, levels in proportionone electron
Fig. 3 The molecular orbital bookkeeping: a bonding combination below the atomic level and an antibonding one above, split by an amount set by the overlap. Valence bond theory has no such diagram, because its objects are bonds rather than orbitals of the whole molecule.

The difference in what each makes easy is visible immediately. The level diagram answers “what happens if an electron is removed” without further work, because its objects are one-electron states of the molecule. It has nothing to say about “where is the bond”, because no orbital in it corresponds to one.

The valence bond description is the reverse. The bond is the object; the ionisation energy requires reconstructing something else.

That is the whole practical content of the choice, and it is why neither is more real than the other — they are two coordinate systems, each aligned with a different question.

Where they meet

Neither framework is complete as usually presented, and the corrections make them converge.

Molecular orbital theory in its simple form places both electrons of a bond in the same delocalised orbital, which lets both sit on the same atom too often. Correcting that requires configuration interaction — mixing in excited configurations — and the first correction has exactly the form of a valence bond term.

Valence bond theory with a single structure is too localised. Correcting it requires including ionic structures and additional resonance forms, and taken to completion it reproduces the molecular orbital answer.

Push both far enough and they arrive at the same wavefunction, because both are complete bases for the same space. They differ in where they start, and therefore in what they get right with the least work.

The clearest illustration

Hydrogen, the two-electron molecule, is the case where the whole argument can be seen at once.

Simple molecular orbital theory puts both electrons in the bonding combination. Expand that and it contains covalent terms — one electron on each atom — and ionic terms with both on the same atom, weighted equally. At long separation that is badly wrong: pulling the atoms apart should give two neutral hydrogen atoms, and this description says half the time it gives H⁺ and H⁻.

Simple valence bond theory pairs one electron on each atom, with no ionic terms at all. That is right at long separation and slightly wrong at bonding distance, where a little ionic character genuinely helps.

The correct answer has both, in a ratio that varies with separation. Each simple theory has one of the two limits right, and the full treatment interpolates.

Where each fails first

A fair comparison needs the failures as well as the strengths, and both frameworks have a characteristic one.

Molecular orbital theory fails at dissociation. Put both electrons of H₂ in the bonding orbital and pull the atoms apart. The description says half the time there are two neutral atoms and half the time an ion pair, which is badly wrong at long range — the energy comes out far too high. Fixing it needs configuration interaction, which is machinery a first course does not have.

Valence bond theory fails at delocalisation. Benzene requires a superposition of structures, and the resonance notation is misread by almost everybody who meets it. Extended systems need many structures, and the bookkeeping becomes unmanageable long before the description becomes wrong.

benzene — D6hThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHCCHCCHCCHHD6hprincipal axis C67 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates12 atoms
Fig. 4 The molecule that makes valence bond theory work hardest. Six equivalent bonds, D₆ₕ symmetry recovered from the coordinates, and a localised description that needs at least two structures to reproduce what one delocalised set of orbitals gives directly.

There is a pattern in the two failures. Each framework fails where the physical situation is furthest from its starting assumption — molecular orbital theory where the electrons should be apart, valence bond theory where they should be spread out. Neither failure is a defect of the physics; both are the cost of choosing a basis, and both are repaired by adding terms the other framework has from the start.

Which to use

The honest answer is: whichever makes the question easy, and being fluent in both.

Use molecular orbital thinking for spectra, ionisation, redox, magnetism, and anything involving the whole framework or an odd number of electrons.

Use valence bond thinking for structure, mechanism, sterics, and anything where locality is the point.

Chemists do this constantly and without ceremony, and the discomfort is confined to teaching, where the two are presented as competing accounts of what is true.

The historical quarrel

The dispute was real and was personal.

Heitler and London gave the valence bond treatment of hydrogen in 1927 — the first quantum-mechanical account of a chemical bond. Pauling developed it into a chemical framework through the 1930s, added hybridisation and resonance, and The Nature of the Chemical Bond made it the dominant picture.

Hund and Mulliken developed the molecular orbital approach over the same period. It was less popular among chemists for two decades, partly because it was less visual, and partly because Pauling’s advocacy was extremely effective.

The molecular orbital picture won the argument on evidence in the 1950s, largely through spectroscopy — and specifically through cases like O₂ where valence bond theory as then formulated gave the wrong answer about a measurable property.

There is a less edifying part of the history. Resonance theory was denounced in the Soviet Union in the early 1950s as idealist and incompatible with dialectical materialism, and Soviet chemists were pressured to abandon it. The episode is a reminder that “which description is real” can be made into a question with political consequences, when it is not a question about nature at all.

What both frameworks are for

Stepping back, the point that survives.

A molecule has a wavefunction. Both frameworks are ways of writing an approximation to it in a chosen basis, and a basis is not a thing. The delocalised canonical orbitals of a molecular orbital calculation and a set of localised bond orbitals are related by a transformation that changes no observable — so a molecule is not “really” delocalised or “really” made of local bonds.

What is real is the total density, the total energy, and the spectrum. Everything else is bookkeeping, and the useful question about a piece of bookkeeping is what it makes easy.

Where the model stops

Two limits.

Neither is a calculation. Both frameworks as described here are qualitative. Modern computational chemistry uses molecular orbital machinery because it is easier to systematise, and gets to accurate answers through methods that are recognisably neither of the pictures above.

Both are one-electron pictures at heart. Orbitals are approximations for more than one electron, and the language of putting electrons into orbitals is exactly that approximation.

What each says about a real molecule

Water is a convenient test, because both frameworks describe it and they describe it differently.

water — C2vThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHOC2vprincipal axis C22 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms
Fig. 5 Water, C₂ᵥ. Valence bond theory: two O–H sigma bonds and two lone pairs on oxygen. Molecular orbital theory: four occupied valence orbitals classified by symmetry, none of which is a bond and none of which is a lone pair.
sp3 hybridsThe directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates.sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron
Fig. 6 The hybrid set the localised description uses — four directions, two for bonds and two for lone pairs. The canonical orbitals are a rotation of the same space, and the spectrum couples to those rather than to these.

Both descriptions give the same total density, the same energy, the same geometry. They differ in what is easy: predicting that water is a hydrogen-bond donor and acceptor is easy in the localised picture; predicting its ionisation energies is easy in the canonical one.

A chemist uses whichever suits, usually without noticing the switch, which is the right practice and worth being conscious of.

Where the ladder goes next

The transformation that relates the two is set out in hybrids are a basis, and the experimental case that decides what a spectrum measures is hybridisation does not explain.

The case that valence bond theory finds hardest is delocalisation.

What the pictures here cannot show. An energy level diagram is a one-electron construction, and a molecule’s states are many-electron. The levels drawn are useful bookkeeping and are not a set of boxes containing electrons.

Where this goes next

sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron Hybrids are a basis An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers. 1s · 1sS = 0.38997sigma interactionseparation 2.8 bohrcontours at 50% of each densitythe signed product integrated over all spaceone electron Overlap decides Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it. HHCCHCCHCCHHD6hprincipal axis C67 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates12 atoms Delocalisation Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.