Molecular orbital and valence bond

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.

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.

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 a method 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.

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.

The two bases, and the density they share #

The claim that the two frameworks converge is usually made by argument: push each far enough and they meet. That is true and it is asymptotic, and there is a case where the meeting can be shown rather than reasoned about, at a place where both descriptions are exact.

Methane’s four bonding electron pairs can be written as four equivalent localised bonds — which is the valence bond picture, one bond to one hydrogen — or as one totally symmetric orbital plus three degenerate ones, which is the molecular orbital picture. The two sets are related by a four-by-four orthogonal matrix.

The density along a carbon–hydrogen line computed from the localised set and from the canonical set is one curve, with a largest disagreement of about 10⁻¹⁶ across every point sampled. That is the arithmetic rather than the physics, and it is the demonstration that the two descriptions are the same state.

That is the convergence, in the one case where it is exact rather than asymptotic. It is exact here because both descriptions span the same four-dimensional space, and a density built from an orthonormal set is $CC^{\mathsf T}$, which an orthogonal mixing leaves alone.

Where the two frameworks genuinely differ is in what they make easy, and the difference is visible in the same figures.

How many hydrogens each orbital effectively occupies differs by nearly a factor of four between the two sets, and what they produce does not differ at all. The descriptions disagree about where the electrons are and agree about everything measurable, which is the situation this essay is about.

A reader who wanted to know which framework is true has the answer in that pair of numbers: 1.07 and 4.000 describe the same eight electrons. The question was about the coordinate system.

Where the model stops #

Three limits.

Neither is a calculation. Both frameworks as described here are qualitative. Modern computational chemistry uses molecular orbital methods 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.

The convergence shown above is a statement about bases, not about methane. No Hamiltonian appears anywhere in it. The functions are hydrogen-like, carbon’s effective charge is Slater’s, and how much hydrogen enters each bond orbital is a free parameter that nothing here fixes. Every result is invariant to that parameter — it is checked at four separate values — and that invariance is precisely why the demonstration says nothing about the energy of methane. It says the two descriptions cannot be distinguished by any measurement of the density, which is what the essay claims and is not the same as calculating anything.

What it costs to choose between them #

Nothing, and that is the answer the question deserves.

The two frameworks demand the same information — a geometry and a set of atomic orbitals — and produce the same observables when each is carried to convergence. Choosing between them is choosing which expansion converges faster for the problem at hand, and the honest answer varies by problem rather than by framework.

Where a molecular orbital treatment converges quickly: anything delocalised, anything where the spectrum matters, anything where a computer is going to do the work. Its methods systematise, which is why essentially every production quantum chemistry program is built on it.

Where a valence bond treatment converges quickly: bond dissociation, where the molecular orbital wavefunction famously goes wrong at long separation and needs configurations added to repair it; and any argument where the chemistry is local, which is most of organic chemistry’s reasoning.

The cost that does exist is a cost in vocabulary. A reader fluent in one and not the other will read the two accounts of the same molecule as two claims about it, and will look for the experiment that decides. The figures above are what that experiment would return.

There is a second cost, and it is the one that does real damage in teaching. Each framework has a favourite failure to point at in the other, and both examples are true and both are usually presented without the repair. Molecular orbital theory dissociates hydrogen wrongly, predicting half the products as ions at infinite separation — and the repair, mixing in the doubly excited configuration, is standard and rarely mentioned in the same breath. Valence bond theory misses oxygen’s paramagnetism, and the repair is to include enough structures, which is equally standard and equally omitted. A student left with the two failures and neither repair concludes that both frameworks are broken, when what the pair of examples actually shows is that each converges slowly where the other converges fast.

What each says about a real molecule #

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

Water is the case a first course meets: valence bond theory gives two O–H σ bonds and two lone pairs; molecular orbital theory gives four occupied valence orbitals classified by symmetry, none of which is a bond and none of which is a lone pair. Both are correct and only one couples to a photoelectron spectrum.

The hybrid set the localised description uses is 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 — which is a fact about the experiment rather than about the molecule.

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.

The molecule where the two frameworks did disagree #

Both accounts of oxygen’s paramagnetism can be made correct is true and understates the case, because the two did not arrive at it on the same terms — and the difference is the clearest available illustration of what which expansion converges faster means in practice.

The measurement is unambiguous and old. Liquid oxygen is attracted into the field between the poles of a magnet strongly enough to hang there, and its magnetic moment corresponds to two unpaired electrons. Nothing about that is subtle or model-dependent.

The molecular orbital account produces it without adjustment. Fill the levels of a two-atom sixteen-electron system in order and the last two electrons arrive at a degenerate pair of antibonding π orbitals, one each by Hund’s rule. Two unpaired electrons, a bond order of two, and a triplet ground state — all of it from a filling diagram anybody can draw, with no knowledge that the answer was going to be paramagnetic.

The Lewis and valence bond account, as it is first written, gives the opposite. A double bond between two oxygens, with two lone pairs on each, uses every electron in a pair. It predicts a diamagnetic molecule, and it does so for the same reason it predicts a diamagnetic nitrogen and a diamagnetic ethene, which are both correct.

The framework can be repaired — a description built from two three-electron bonds rather than one double bond gives the right multiplicity — and the repair works. But it was constructed after the fact, for this molecule, and the ordinary version of the theory that a chemist draws does not produce it.

That is the honest form of the comparison and it is worth stating carefully. The two frameworks are equivalent when both are expanded fully, which is the essay’s argument. They are not equivalent at the level anybody actually uses them, which is one Lewis structure against one determinant — and which of the two is closer depends on the molecule.

Oxygen is the case where one determinant is nearly right and one Lewis structure is qualitatively wrong. A bond being pulled apart is the reverse: one Lewis structure remains sensible all the way to separated atoms, and one determinant becomes qualitatively wrong — the reverse failure, and the natural next case to study.

So the two frameworks fail in opposite places, and neither failure is a failure of the framework. It is a failure of the first term, and the two theories have different first terms.

Where to read on #

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.

A last practical note. A reader meeting the two frameworks in different courses, taught by people with different preferences, will encounter apparent contradictions that are none. Both accounts of benzene are correct, both accounts of oxygen’s paramagnetism can be made correct, and the disagreements in the literature are almost always about which expansion converges faster rather than about what the molecule is doing.

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.