Two pictures, one plane

Molecular orbital and valence bond sets out two frameworks taught as rivals: one starts from orbitals spread over the whole molecule and localises afterwards, the other starts from electron pairs on bonds and delocalises afterwards. Its conclusion is that they are two coordinate systems rather than two theories.

Where molecular orbital theory dissociates then found the place the two genuinely part company: a molecular-orbital wavefunction for a two-electron bond puts both electrons on the same atom half the time, whatever the separation, so it dissociates to the wrong thing.

This essay puts numbers on the relation between them, because in a model small enough to solve completely there is nothing left to interpret.

The model, and why the space is only two-dimensional #

Two sites, one orbital each, two electrons, and a cost $U$ for putting both on the same site. That is the Hubbard dimer, and the smallest many-electron calculation establishes it as the smallest system in which repulsion does anything at all.

Its four configurations with one up and one down electron, which the solver keeps in full with no truncation anywhere, are: both on site 1, both on site 2, up on 1 and down on 2, and the reverse. So the space the exact ground state lives in has dimension four.

The ground state is a singlet and it is symmetric under exchanging the two sites, and those two facts cut the space to two. The antisymmetric combination of the two neutral configurations is the triplet; the antisymmetric combination of the two ionic ones has the wrong symmetry under the mirror that swaps the atoms. What is left is the symmetric neutral combination and the symmetric ionic combination — a plane.

Both textbook wavefunctions live in that plane:

The molecular-orbital function puts both electrons in the bonding combination $(\phi_1 + \phi_2)/\sqrt2$. Expanding, that is all four configurations with equal weight — half of it neutral and half of it ionic, whatever $U$ is.

The valence-bond function — Heitler and London’s, from 1927 — is one electron on each atom with their spins paired: the two neutral configurations and nothing else. No ionic weight at all, whatever $U$ is.

The overlap check that makes it a fact #

The claim “the exact answer is a mixture of the two pictures” is a familiar sentence and usually an assurance. Here it is a measurement: project the exact ground state, from the full four-dimensional diagonalisation, onto the plane the two functions span.

The result is $1$ to twelve decimal places at every repulsion tried, from $U = 0$ to $U = 32t$. Not approximately in the plane — in it.

That is a statement about this model and not about molecules, and the reason it is exact here is the symmetry argument above. In a real bond there are more configurations, the two functions still span a plane, and the exact state has a component outside it. The size of that component is what a correlated calculation is computing, and the hole that is not repulsion is where the part exchange already supplies is separated from the part it does not.

The one number they disagree about #

Both functions are normalised; both are singlets; both are symmetric. What separates them is a single quantity: the weight of the two configurations with both electrons on one atom.

The molecular-orbital function fixes it at $\tfrac12$. The valence-bond function fixes it at $0$. The exact answer computes it, and it is a function of the repulsion:

At zero repulsion the exact state is the molecular-orbital function: the ionic weight is exactly one half and the overlap is exactly one. That has to be so — with no repulsion the exact ground state is a single determinant, which is what the molecular-orbital function is.

At the other end the ionic weight falls towards zero and never arrives. At $U = 32t$ it is still $0.0039$, which is small and is not nothing: the valence-bond function is the exact answer only in a limit no molecule is in.

The triplet, which neither picture argues about #

There is a third state in the same model that both descriptions get exactly right, and it is worth a paragraph because it shows what the disagreement is not about.

Put the two electrons in with parallel spins. Now neither can hop, because the site it would land on already holds an electron of its own spin — the Pauli principle does what an infinite repulsion would do. The triplet’s energy is exactly zero for every $U$, in a model where the singlet’s runs from $-2t$ to nearly zero.

Both pictures give that state correctly and identically: the molecular-orbital description puts one electron in each of the bonding and antibonding orbitals, the valence-bond description puts one on each atom with spins parallel, and the two are the same determinant. There is nothing to disagree about, because the state has no ionic component to weight.

So the whole of the argument between the two frameworks lives in the singlet, and it lives specifically in how much of the singlet is ionic. That is worth knowing when reading a claim that one description “explains magnetism” or “explains the singlet–triplet gap”: the gap is a difference between a state the two agree about and a state they do not, so it is entirely a statement about the ionic weight.

Where they are equally wrong #

There is a crossing point, and it is worth a moment because it is where the argument about which picture is “better” is emptiest.

At $U = 4t$ the exact state’s overlap with the molecular-orbital function and with the valence-bond function are both $0.9239$, which is $\cos 22.5°$. The state sits exactly halfway between them in angle.

Real single bonds are in this region. The ratio $U/t$ for a $\sigma$ bond at its equilibrium length is a few, not zero and not infinite, which is why arguing that either picture is correct has always been possible and never been productive: both are about ninety-two per cent right, in different directions.

What the ionic weight is, in a real molecule’s terms #

The weight above has a name in each language and they are the same quantity.

In valence-bond language it is the coefficient of the ionic structures — H⁺H⁻ and H⁻H⁺ — that the Heitler–London function leaves out and that any serious valence-bond calculation puts back. In molecular-orbital language it is what configuration interaction adds: mixing the doubly excited configuration into the ground determinant reduces the ionic weight from one half towards its correct value.

Those are the same one-parameter improvement, approached from opposite sides, and in this model they meet exactly: mixing the ionic structure into the valence-bond function and mixing the double excitation into the molecular-orbital function both sweep out the same one-parameter family, which is the plane. So “valence bond with ionic structures” and “molecular orbital with configuration interaction” are not two roads to the same place — they are the same road, traversed from the two ends.

The overlap between the two, which is not zero #

One number in the picture above is easy to skip and is doing real work: the molecular-orbital and valence-bond functions overlap by $1/\sqrt2$, so they sit forty-five degrees apart rather than at right angles.

That is why the argument between their advocates could run for decades. Two descriptions at right angles would be visibly different — an experiment sensitive to one would be blind to the other. Two descriptions seventy-one per cent identical agree about most things, and about everything that depends only on the neutral configurations they share.

It also means the two coefficients in a mixture are not weights and do not add to one. Expanding the exact state as $c_{\text{MO}}$ times one plus $c_{\text{VB}}$ times the other, the two coefficients are large and of opposite sign at large $U$, because a non-orthogonal pair can represent a small vector as a difference of two big ones. Quoting “the wavefunction is 60 per cent valence bond” is therefore a statement whose meaning depends on a convention for splitting the overlap, and different conventions give different percentages for the same state.

The unambiguous quantity is the one this essay uses: the ionic weight, which is an expectation value of an operator and does not depend on how the state is written down. That is the same lesson hybrids are a basis draws about $s$ character — a coefficient in a non-orthogonal expansion is a bookkeeping choice, and an expectation value is not.

Which picture to use, and for what #

The two are not interchangeable in practice, and the model says why in a way that is not a matter of taste.

Near equilibrium, the molecular-orbital picture is easier and adequate. The ionic weight is genuinely large there, so a description that includes it from the start needs less correcting. And it extends: a single determinant of delocalised orbitals is what makes calculations on molecules of any size possible.

At dissociation it is qualitatively wrong, by the amount where molecular orbital theory dissociates computes — half the density on the wrong atom, an energy that is too high by a finite amount at infinite separation.

And the valence-bond picture is right at dissociation and awkward everywhere else — the same non-orthogonality that makes the localisation transformation need a Löwdin step before it can start — because its structures are not orthogonal and the number of them grows faster than the number of determinants.

Which is why the practical answer has been the same for fifty years and is not a compromise: use delocalised orbitals, and put the correlation back with configuration interaction. That is a molecular-orbital calculation whose first correction is the valence-bond insight.

What the argument looked like from inside it #

The dispute between the two frameworks ran for about fifty years, and the model above says something about why it was so hard to settle.

Both sides had the same data. The two functions overlap by $1/\sqrt2$, so any measurement mostly sensitive to what they share was consistent with either. The observables that separate them are the ones sensitive to the ionic weight — and there are few of those, because the ionic weight is a coefficient in a wavefunction rather than an expectation of anything obvious.

Both sides could correct their picture towards the other. Adding ionic structures to valence bond and configuration interaction to molecular orbital theory are the same one-parameter improvement, so each framework could absorb the other’s advantage without conceding anything.

And the practical question was decided by arithmetic rather than by physics. Molecular orbital theory won because its functions are orthogonal, which makes the integrals tractable, and because it scales. That is a fact about computation, not about which description is more nearly true — and the model here says neither is true, since the exact answer is a mixture at every finite repulsion.

The one thing the model does settle is that the question has an answer, and that the answer is a number rather than a preference. For a two-site system at $U = 4t$ the ionic weight is $0.146$: less than the molecular-orbital picture’s half and more than the valence-bond picture’s zero, and computable to as many decimal places as anybody wants.

Where the model stops #

One orbital per atom. A real hydrogen molecule has more, and their contribution is not small: even at equilibrium a minimal basis misses a few per cent of the binding, for reasons a Gaussian is the wrong shape is about.

On-site repulsion only. Two electrons on neighbouring sites do not repel in this model. Adding the neighbour term changes the numbers and not the structure of the argument, because it is diagonal in the same configurations.

No nuclear repulsion and no geometry. $U/t$ stands in for the separation — a longer bond means smaller $t$ and therefore larger $U/t$ — but nothing here computes a bond length, and the potential energy curves this argument is usually illustrated with cannot be drawn from it.

Two electrons. The plane is two-dimensional because of a symmetry argument that applies to this system. For four electrons in four orbitals the exact state has components no pair of textbook functions spans, and the whole subject of correlated methods is the size of those components.

One number in the table repays a second look. The ionic weight at $U = 2t$ is $0.276$, which is close to a quarter — and a quarter is what a state with equal amplitudes on the neutral and ionic pairs would have. That is a coincidence of this repulsion rather than a special point, and it is worth naming as one, because a number near a simple fraction invites an explanation it does not have.

The genuinely special points are two: $U = 0$, where the exact state is the molecular-orbital function exactly, and $U = 4t$, where the overlaps with the two pictures are equal. The first is a theorem and the second is where the state’s angle bisects theirs — a fact about this model’s parameters and not about any molecule.

The third function, which is a path rather than a point #

The plane contains the two textbook pictures and everything between them, and it is worth knowing that the family between them has a name and was written down for exactly this purpose.

Take the valence bond function and let each electron’s orbital leak a little onto the other atom: instead of a pure atomic function on each centre, use $\phi_A + \lambda\phi_B$ for one electron and $\phi_B + \lambda\phi_A$ for the other, with $\lambda$ a single number to be optimised.

At $\lambda = 0$ that is the valence bond function exactly. At $\lambda = 1$ the two orbitals become identical and it is the molecular orbital function exactly. In between it is neither, and it traces out the line between them in the plane drawn here.

So the third description is not a compromise between two rivals; it is one function with one parameter of which both rivals are limits, and the parameter is variational — it takes whatever value minimises the energy at each separation.

What it does is exactly what the model here measures. Near the equilibrium separation the optimum sits at or very near $\lambda = 1$: the molecular orbital function is the best available and there is nothing to gain by deforming it. Pull the atoms apart and the optimum leaves one, falling towards zero, so the orbitals localise onto their own atoms and the ionic contamination goes with them.

That converts the old argument between the two frameworks into a measurement. Asking which picture is right becomes asking what value does $\lambda$ take here, and the answer is a number between zero and one that depends on the separation and on nothing else. The two frameworks are the two ends of its range, and a molecule is somewhere inside.

What the numbers add #

The qualitative argument says the two frameworks are two coordinate systems; dissociation is the place they genuinely differ, and the error there can be measured.

This one writes both as vectors and measures the geometry: the plane they span contains the exact answer exactly, the angle between them is fixed by an overlap of $0.7071$, the exact state’s position in the plane is set by one number, and that number is the ionic weight — half for one picture, zero for the other, and something else for every molecule. The argument about which framework is right turns out to be an argument about the value of a single coefficient that neither framework computes.

The open question is what happens when the plane is not enough: four electrons, where the exact state leaves the span of every simple picture, and the leftover has to be given a name and a size.