Where molecular orbital theory dissociates

Two descriptions of a two-electron bond have been in use since the late 1920s and they disagree about something concrete.

The molecular orbital description builds a bonding combination from the two atomic functions and puts both electrons in it. Multiplying that product out gives four terms of equal weight: one electron on each atom, twice, and both electrons on the same atom, twice. So the description carries fifty per cent ionic character, built in.

The valence bond description writes the two electrons on different atoms and antisymmetrises. It carries no ionic terms at all.

Neither is right, and the interesting question is not which is closer but where each fails and by how much. That question has an exact answer on a model small enough to solve completely.

The model, and why it is enough #

Two sites, one orbital each, two electrons. Electrons hop between the sites with amplitude $t$, and two electrons on the same site cost $U$. That is the Hubbard dimer, and it is the smallest system in which the competition between delocalisation and repulsion exists at all.

There are four states — both up-spin arrangements aside, the singlet sector holds three and the triplet one — and the Hamiltonian in that basis is a small matrix that can be diagonalised exactly. The ground-state energy has a closed form:

$$E_0 = \frac{U - \sqrt{U^2 + 16t^2}}{2}$$

and it is checked against the diagonalisation at every point drawn here.

The correspondence with a real bond is the ratio $U/t$. A short bond has large hopping and the ratio is small; stretching the bond shrinks $t$ while leaving $U$ alone, so increasing $U/t$ is dissociating the molecule. The limit $U/t \to \infty$ is two separated atoms.

What the model leaves out is everything else: no nuclear repulsion, no overlap, no long-range attraction, one orbital per atom. It is not a calculation of a dissociation curve. It is a calculation of the one feature the two descriptions disagree about, with everything they agree about removed.

What each description gives #

Both descriptions can be evaluated on this model exactly, which is the point of using it.

Molecular orbital. Put both electrons in the symmetric combination. The energy is $-2t + U/2$: the hopping term is exactly right and the repulsion term is $U$ times the double occupancy, which is $1/2$.

Valence bond. Take the covalent singlet with one electron on each site. The energy is $0$: it never pays the repulsion, and it gets no benefit from hopping either, because hopping takes it to an ionic arrangement that the state does not contain.

Exact. The closed form above.

At no repulsion the molecular orbital answer is exact and valence bond is wrong by $2t$, which is the whole of the bonding. At large repulsion valence bond approaches the exact answer while the molecular orbital energy runs off to $+\infty$, describing a state whose energy rises without limit as a molecule is pulled apart.

The two are equally wrong at exactly $U = 4t$, where both miss by $0.8284t$. That crossing is the honest form of the statement that neither description is generally better.

The crossing point is worth one more sentence, because it is not an artefact of the numbers chosen. Setting the molecular orbital error equal to the valence bond error means $-2t + U/2 = -E_0$ with $E_0$ negative, and the closed form makes that condition $U = 4t$ exactly, independently of the units. On one side of it the delocalised description wins and on the other the localised one does, and the boundary is a single ratio with no fitted quantity in it.

The half that never goes away #

The energies are one way of seeing the failure and the double occupancy is a sharper one, because it is a property of the wavefunction rather than of a number derived from it.

Double occupancy is the probability that both electrons are on the same site. Computed from the exact ground state it runs $0.5000$, $0.4380$, $0.3787$, $0.2764$, $0.1464$, $0.0528$, $0.0149$, $0.0039$ across the repulsions in the figure. At large $U$ it heads to zero, which is what two separated atoms must give: an electron each.

The molecular orbital value is $0.5000$ at every one of those points. It is not approximately a half; it is a half by construction, because the doubly occupied bonding orbital is a product state and the product does not know about $U$.

So the molecular orbital description of a dissociating bond says the two electrons are on the same atom half the time no matter how far apart the atoms are — in the case of hydrogen, that the products are fifty per cent H⁺ and H⁻. That is a large, specific, physically wrong claim, and it is the reason molecular orbital and valence bond treats the two as descriptions with different failure modes rather than as rivals.

What suppresses the ionic terms is a cost, and the cost is computable: the insulator band theory cannot see plots the energy of moving one electron from one site to another against the repulsion — nothing at U = 0, and 5.99t by U = 8. That is the quantity the molecular orbital description has no representation of anywhere, which is why its ionic fraction cannot respond to it.

What is missing has a name #

The gap between the molecular orbital energy and the exact one is called correlation energy, and this model shows exactly what it is doing.

A single product of orbitals says the two electrons are placed independently. They are not: when one is on the left site, the other is more likely to be on the right, because being on the same site costs $U$. The exact wavefunction correlates their positions and the product cannot.

The correction is not small where it matters. At $U = 4t$ it is $0.83t$ against a total binding of $2t$ at zero repulsion — forty per cent of the bond. At $U = 8t$ it exceeds the entire bonding energy of the uncorrelated case.

The standard repair is to add the doubly excited configuration, in which both electrons occupy the antibonding orbital, with a coefficient chosen to minimise the energy. In this two-site model that repair is not an approximation: the singlet sector is two-dimensional, so the bonding-squared and antibonding-squared configurations span it completely, and mixing them is the exact solution. The smallest configuration-interaction calculation there is recovers the whole of the correlation energy here, and the reason is that there is nothing else for it to miss.

The one-electron half of the story is settled elsewhere and is worth keeping separate: with the overlap kept, the antibonding level rises more than the bonding one falls — overlap decides computes the asymmetry. That is not correlation and it survives at every repulsion, because it comes from the overlap rather than from the interaction.

Where each description is the right one to use #

The comparison above is not an argument for either. It is an argument for reading the ratio.

At short bond lengths, near equilibrium, molecular orbital theory is close to right and is enormously more convenient. The hopping dominates, the exact ground state is genuinely delocalised, and its double occupancy is not far from a half. Every band-structure argument about solids is built on that regime, and a solid is a molecule that did not stop carries the same one-electron treatment to two thousand atoms without apology.

At long bond lengths, and in any system where repulsion beats hopping, it is not. The insulator band theory cannot see is the extended version of this failure: a half-filled band that band theory calls a metal and that is an insulator because the electrons cannot afford to be on the same site.

And the crossover is at $U \approx 4t$, which is not a large number. It is met in transition-metal oxides, in stretched bonds, and in every bond-breaking process — which is why the description that works for a molecule at rest is the wrong one for the molecule coming apart.

In the large-repulsion limit the two-site problem stops being an electronic problem and becomes a spin one, with an antiferromagnetic coupling that goes as −4t²/U — the smallest many-electron calculation plots U times the splitting settling on −4t² exactly. That limit is the one place the two descriptions agree about everything, because there is nothing left for either to be wrong about.

The overlap that both descriptions share #

One quantity is common to both and is worth putting beside the model, because it is the thing $t$ stands for.

The hopping integral scales with the overlap and both fall exponentially with separation, while $U$ — an integral over one centre — barely changes. So $U/t$ grows exponentially as a bond is stretched, and a molecule passes from the left of the table to the right of it over a range of a few tenths of an ångström.

The overlap both descriptions share falls off exponentially with separation — closer is not more overlap computes the curve for two hydrogen 1s functions and checks it against the closed form — and at 4.5 bohr there is very little density left between them. That vanishing region is what makes the hopping small, and a small hopping against a fixed repulsion is the whole of the ratio this essay is about.

The same failure at four sites and at six #

Two sites is the smallest case and it is fair to ask whether the fifty-per-cent figure is an artefact of it. It is not, and the generalisation is a quarter rather than a half.

An independent-electron description of $n$ sites at half filling puts each electron on each site with probability $1/n$ and treats the two spins as independent, so the chance of finding both an up-spin and a down-spin electron on one particular site is $1/4$ — per site, at every $n$, at every interaction strength.

The exact ground states say otherwise as soon as the repulsion is switched on:

Every row starts at about a quarter, which is the check that the exact solver reproduces the independent-electron answer where the two must agree, and every row collapses. By $U = 16t$ the double occupancy is a twenty-fifth of its uncorrelated value in each of the three.

The ring of four is the interesting row, because it starts above a quarter at $0.2562$ — the degenerate half-filled shell that makes cyclobutadiene the awkward case in aromaticity as a computed shell closure also makes the uncorrelated ground state ambiguous, and the exact solution resolves it in a way the independent-electron count does not anticipate.

The solver refuses more than six sites, which is the honest boundary: exactness is the claim being made, and the dimension of the many-electron space grows fast enough that keeping the claim means keeping the system small.

What this model is not #

Three limits, since the model is small enough that its limits can be stated exactly.

It is not a dissociation curve. There is no nuclear repulsion, no kinetic energy of the nuclei, and no long-range attraction, so nothing here has a minimum at a bond length. The axis is $U/t$ and reading it as a distance is a translation, not a calculation.

It has one orbital per atom. A real dissociating hydrogen molecule polarises, its orbitals contract toward the atomic ones, and none of that is in the model.

It is exact where it applies, and that is the point. The smallest many-electron calculation makes the case for using a model small enough to solve completely rather than a large one solved approximately, and this essay is that argument applied to the oldest disagreement in the subject. The claims made here can be checked by hand.

One consequence for the rest of the site is worth stating plainly. Every Hückel calculation here is a molecular orbital calculation, so every one of them carries the fifty-per-cent ionic term computed above — which is why their energies are quoted in units of β rather than in kilojoules, and why a stabilisation is measured from somewhere insists on naming the reference.

What the failure costs in a number a chemist quotes #

The half-ionic wavefunction at infinite separation is a statement about a description, and it has a price in the one quantity anybody actually wants from a bond calculation.

Take hydrogen. Its dissociation energy is measured at 4.75 electronvolts. A Hartree–Fock calculation at the basis-set limit — one determinant, both electrons in one spatial orbital, everything else done as well as it can be done — returns 3.64.

The missing 1.11 electronvolts is, by definition, the correlation energy of the hydrogen molecule. And essentially all of it is the defect computed here: at the equilibrium separation the determinant is a decent description and the error is modest, while at large separation the same determinant insists on half-ionic character that the molecule does not have, so the calculated energy at dissociation is far too high and the difference between the two ends is far too small.

That is a twenty-three per cent error in the single most quoted property of the simplest bond in chemistry, from a method that is otherwise excellent — and it is an error whose whole origin is visible in a two-site model with one parameter.

The size also explains why the failure is more damaging than the equilibrium numbers suggest. A method’s error at one geometry can be absorbed into a comparison; an error that grows along a coordinate cannot, because it changes the shape of the curve. Everything read off a potential surface — a dissociation energy, a barrier height, a reaction energy where a bond is broken — inherits it.

Two repairs are standard and both have a price worth naming.

Let the two spins have different spatial orbitals. Beyond a certain separation a solution appears in which one electron sits mostly on one atom and the other mostly on the other, and it dissociates correctly. What it gives up is spin: the resulting function is not an eigenfunction of the total spin at all, it is a mixture of a singlet and a triplet, and the energy curve acquires a kink at the separation where the new solution appears. The molecule has no kink there, so a feature has been introduced that is a property of the method.

Or use more than one determinant. Adding the doubly excited configuration to the ground one recovers exactly what is missing — the two configurations mix, the ionic contamination cancels at large separation, and the curve is right at both ends with no spin contamination and no kink. That is the correct repair and it is the expensive one: the number of configurations needed grows with the number of bonds being broken, so a molecule dissociating one bond needs two and a molecule with several needs many.

Which is the practical shape of the whole subject. The cheap method is qualitatively wrong exactly where bonds break, and the method that is right there costs more the more bonds there are. The two-site model does not solve that; what it does is show that the difficulty is not a deficiency of any particular program or basis, but a property of describing two electrons with one orbital, present in the smallest system where two electrons and two atoms exist.

It is worth noticing which of the two repairs the model itself performs. Diagonalising the two-site problem exactly is the second one — the exact ground state is a mixture of the two configurations, with a weight that varies smoothly with the repulsion — and the smoothness is why nothing in these curves has a kink in it. A model small enough to solve completely never needs the cheap repair, and a molecule large enough to be interesting can never afford the expensive one, which is a reasonable statement of why the subject has the shape it does.

Who found it, and when #

The molecular orbital description is Friedrich Hund’s and Robert Mulliken’s, from 1927 onward. The valence bond description is Walter Heitler and Fritz London’s, from 1927 exactly — their paper on H₂ is usually taken as the first quantum-mechanical calculation of a chemical bond.

The disagreement was noticed almost immediately, and the standard resolution is older than it looks: adding ionic terms to valence bond and adding excited configurations to molecular orbital theory converge on the same wavefunction, a result usually credited to Slater in the early 1930s. What kept the two schools apart for decades was not the mathematics but which correction was cheap enough to do.

The exactly-solvable model came from elsewhere entirely. John Hubbard wrote his down in 1963 for electrons in narrow bands in solids, with no molecular application in mind, and the two-site case has been rediscovered as a teaching example many times since. That a model built for transition-metal oxides settles a 1927 argument about hydrogen is the kind of thing that keeps happening in physics, and the reason is that both problems are the same problem: what happens when hopping and repulsion are comparable.

The same competition elsewhere #

Put side by side, the two descriptions describe one thing. This essay shows the one thing they describe differently, and measures the difference on a system where the answer is known.

The thread continues wherever repulsion and delocalisation compete. In what couples two spins the same dimer supplies the exchange coupling; in a half-filled band is not always a metal the same competition decides a material’s conductivity; and in delocalisation is not always stabilising it decides whether a ring gains anything by conjugating at all. Each is the same axis with a different quantity plotted against it.