Four centres, and the pair that will not localise

The occupied orbitals of a molecule are fixed as a space, not as a set. Any unitary mixing of them leaves the density, the energy and every observable exactly as they were, and this collection demonstrates that on methane — four equivalent bond orbitals and one $a_1$ with three $t_2$ give densities agreeing to $10^{-16}$.

The freedom is usually spent on making the orbitals as local as possible, because local orbitals look like bonds and bonds are how chemists think. The interesting question is what happens when the freedom is spent and the answer is still not a bond.

The molecule #

Methyllithium is not a monomer. In the solid and in hydrocarbon solution it is a tetramer: four lithiums at the corners of a tetrahedron, with four methyl groups capping the four faces, each carbon sitting above the centre of a triangle of metals.

Count the electrons available to hold that framework together. Each lithium brings one valence electron; each methyl brings one, through the carbon orbital pointing at the face. Eight electrons, four pairs.

Count the contacts. Six lithium–lithium edges and twelve carbon–lithium contacts: eighteen.

Four pairs for eighteen contacts. That is what electron deficient means as a count, and the arithmetic is the same arithmetic the boranes run on.

The question, and how to answer it #

A count says there are not enough pairs for the contacts. It does not say what the pairs are doing. The stronger version of the claim is: no unitary mixing of the four occupied orbitals produces two-centre bonds, and that is a statement about the orbital space which can be tested by searching it.

The functional maximised is Pipek and Mezey’s — the sum over orbitals of the sum over atoms of the squared atomic population. It is largest when each orbital sits on as few atoms as possible, and it is the same kind of unitary freedom hybrids exploit — a change of basis inside a fixed space. In an orthonormal one-orbital-per-atom basis the population is the squared coefficient, so the whole thing is a sequence of two-by-two rotations with a closed-form optimum, which is the same Jacobi sweep this collection’s eigensolver uses, on a different objective.

The sweep must move the orbitals and it does: the functional rises from its canonical value, so the delocalised set is not already the most local one and there is something to find.

The control, which is the point of the exercise #

A routine that always returned four centres would prove nothing, and there is no way to tell from the cluster alone whether it has. So the same sweep is run on a system whose answer is known.

Two ethenes side by side have two-centre bonds, and the sweep finds them: each localised orbital has amplitude on exactly two atoms, and the participation number — one over the sum of the squared populations — comes out at 2.000.

That is the check the whole result rests on. The routine can find a two-centre bond when there is one to find.

What it finds instead #

Run on the tetramer, every one of the four localised orbitals has real amplitude on four atoms: one carbon, and the three lithiums of the face that carbon caps.

Nothing put the faces in. The sweep sees an adjacency matrix and a set of coefficients; it does not know which atoms are metals, which are carbons, or that the framework has faces at all. It maximises a sum of fourth powers and comes back with the tetrahedron’s own geometry.

What the delocalised description looks like #

The canonical orbitals — the ones that come straight out of the diagonalisation — are worth looking at before the mixing, because they are what a calculation actually produces and they look nothing like bonds.

The framework’s levels come out at 4.854, three at 0.618, three at −1.618 and one at −1.854 in units of the resonance integral, and the eight electrons fill the four lowest. So the occupied set is one strongly bonding orbital, well separated, and a triply degenerate set above it — which is $a_1 \oplus t_2$, the representation the tetrahedron’s own symmetry demands.

The lowest of those spreads over all eight atoms with the same sign everywhere. It is a bonding orbital for the whole cluster and it is not a bond in any sense a chemist would use: it has no direction, it belongs to no pair of atoms, and drawing it would give a picture of a cage rather than of a framework.

Both descriptions are equally correct. What the localisation buys is not truth but recognisability, and the finding is that even after buying as much as the space allows, what comes out is still not a two-centre bond.

That is the point at which electron deficient stops being a count and becomes a description. A molecule is electron deficient when the most local honest description of it is not a set of two-centre bonds — and the only way to establish that is to look for one and fail.

The shares are not equal #

The number that says how many centres an orbital effectively spans is the participation number, and it comes out at 2.54 rather than at four.

That is not a contradiction and it is the most useful thing in the calculation. The four shares are one large and three small: fifty-nine per cent of each pair on its carbon, about twelve per cent on each of the three lithiums, and three per cent on the fourth. An orbital with amplitude on four atoms but most of its weight on one has a participation number near two and a half, and that is what a lopsided distribution over four centres looks like.

A bond over four centres is not a bond divided into four. The pair belongs mostly to the carbon and is lent to the metals, which is exactly what one would expect from the electronegativity difference — and it is a description arrived at without any electronegativity being supplied.

The comparison against the control is what makes 2.54 meaningful. The ethenes give 2.000 and the tetramer gives 2.54, so the tetramer’s orbitals are measurably less local than a two-centre bond even by a measure that weights the largest share most heavily.

The density does not move #

Every localisation of this kind has to be checked in one place, and it is the place where an error would be invisible. A unitary mixing preserves the density; a mixing that is not unitary does not; and a coding error that produces a non-unitary transformation gives orbitals that look plausible and describe a different molecule.

The charge on every atom is computed before and after, and they agree to $10^{-9}$: 0.776 on each lithium and 1.224 on each carbon, in both descriptions.

That check found a real error. The first version of the sweep built the second row of each rotated pair from the wrong source row — the sine term taken from the row being replaced rather than from its partner — which is not a rotation at all. It produced orbitals that looked entirely reasonable and a charge density that had moved by three parts in a hundred. Nothing about the orbitals themselves would have shown it.

The theorem the sweep relies on is demonstrated elsewhere on a molecule where both descriptions are familiar: methane’s four equivalent bond orbitals and its one totally symmetric orbital give densities that agree to the last bit a double holds. Nothing about the tetramer is special in that respect, which is exactly why the sweep is allowed to look for a localised description at all — if the rotation moved the density, choosing a set of orbitals would be choosing a state.

Why four and not three or six #

The result has a symmetry reading, and it is worth having because it says why the answer is stable.

The framework has tetrahedral symmetry. Its four occupied orbitals span a representation, and that representation is $a_1 \oplus t_2$ — one totally symmetric orbital and a set of three. Any localised description is a mixing of those four, and a mixing of $a_1$ with $t_2$ produces four equivalent orbitals related to each other by the operations of the group.

Four equivalent orbitals in a tetrahedral cluster with four faces are, by symmetry, one per face. There is nothing else for them to be: three would not be a closed set under the group, and six would need more orbitals than there are.

So the number four is fixed by the symmetry, and what the calculation supplies is the distribution — the 0.59 on the carbon, the 0.12 on each lithium, the 0.03 leaking across. The symmetry argument gives the shape of the answer and the arithmetic gives its contents, which is the ordinary division of labour when a group is available.

The same statement, from the count #

It is worth putting the two arguments side by side, because they are independent and they agree.

From the count: four pairs, eighteen contacts. If every pair were a two-centre bond there would be four bonds, and fourteen of the contacts would be holding nothing. A tetrahedral cluster with four of its eighteen contacts bonded and fourteen not is not a structure anybody would propose, and the symmetry forbids it outright — the four bonds would have to pick out four contacts from a set the group treats as equivalent.

From the orbitals: localise as hard as the space permits, and the best available orbitals span four atoms each.

The two are different in kind. The first is arithmetic plus a symmetry argument and needs no calculation; the second is a search over a continuum of unitary transformations. Getting the same answer from both is what makes the conclusion structural rather than an artefact of a functional.

This is also why the count is worth teaching first. Where two-centre bonding stops is a threshold that can be located by counting, and the orbital analysis is the confirmation rather than the discovery.

What this cannot say #

A graph, not a molecule. The calculation is a Hückel treatment of the framework: one orbital per atom, connections where there are contacts, no distinction between a lithium and a carbon in the site energies. The real cluster’s carbon is much more electronegative than its lithiums, which is why the localised orbitals sit mostly on carbon in the real molecule — and here that lopsidedness comes out of the connectivity rather than out of an electronegativity, since each carbon has three neighbours and each lithium has six.

One orbital per methyl. The methyl group is a single site here, standing for the carbon orbital pointing at the face. Its three C–H bonds and its own hybridisation are outside the model.

A localisation criterion is a choice. Pipek–Mezey maximises atomic populations; Boys minimises orbital spreads; Edmiston–Ruedenberg maximises self-repulsion. They agree on most molecules and disagree on some — famously about whether a double bond is two bent bonds or a σ and a π. The result above should be read as this criterion, applied to this framework, and its robustness is the fact that four is fixed by symmetry rather than by the criterion.

What a localised picture is worth #

A last word on why any of this is done at all, because the whole procedure invites the objection that it is decoration.

Canonical orbitals are what a calculation produces and they are what spectroscopy sees: a photoelectron spectrum counts canonical orbitals, not localised ones, and the number of bands in it is a property of the delocalised set. So the localised description is not the one an experiment reports.

What it is good for is transferability. A localised orbital on one C–H bond of one molecule looks very like a localised orbital on a C–H bond of another, which is why bond energies are approximately additive and why a chemist can reason about a fragment. That is a property of the localised description and not of the canonical one, where changing anything about a molecule changes every orbital.

So the question how local can these orbitals be made is the question how transferable is any fragment of this molecule, and the answer here is: not very. There is no piece of the tetramer that can be lifted out and reused, because the smallest self-contained unit is a carbon with three metals, and that unit shares each of its metals with two other units.

An electron-deficient cluster is therefore not merely unusual in its bonding — it is unusual in resisting the decomposition that makes most of chemistry portable.

The comparison worth making is with a cage. The radial orbitals of a four-vertex deltahedron have the same shape of spectrum — one strongly bonding combination well below the rest — and that pattern is what makes a cage a cage rather than a set of edges. The tetramer is the molecular version of it, and its resistance to localisation is the same fact restated in the other basis.

The lithiums, and what holds them to each other #

One loose end. Six of the eighteen contacts are lithium–lithium edges, and nothing above has said whether those are bonds.

The localised orbitals answer it. Each has twelve per cent of a pair on each of three metals, so a lithium–lithium edge carries amplitude from the two orbitals whose faces share it — and the amount is small. There is no localised orbital sitting on a Li–Li edge, and no pair to assign to one.

That is the right answer chemically. The metal–metal distances in alkyllithium tetramers are close to those in lithium metal, and their bonding is best described as whatever is left over from the framework orbitals rather than as six two-centre bonds. Calling them bonds would need six more pairs, and there are none.

It is also a case where the count and the picture agree in a useful way: the count says there are not enough pairs for the metal edges, and the localisation says no orbital is on one. The framework is held together by the carbon-capped faces, and the metals are near each other as a consequence rather than as a cause.

The structure the four-centre answer requires #

A localisation returning four centres is a statement about a computed wavefunction, and it makes a structural prediction that a diffraction measurement can refuse.

If each pair holds one carbon and the three lithiums of a face together, then the carbon must sit over the face rather than at one vertex or along one edge, and its three distances to those lithiums must be equal — not approximately, but by the same symmetry that makes the four pairs equivalent.

That is what the tetramer’s structure is. Four lithiums at the corners of a tetrahedron, four methyl groups capping the four faces, each carbon equidistant from the three lithiums beneath it at about 2.3 ångström, and the whole arrangement of the highest symmetry the composition allows.

The alternative a two-centre description would require is quite different: four Li–C bonds, one carbon per lithium, with each methyl attached to one metal and much further from the others. Nothing about the measured structure resembles it.

So the localisation’s answer and the crystallography agree about a number — three lithiums per carbon — arrived at from a computed wavefunction on one side and from a set of distances on the other, with nothing passing between them.

That agreement is worth more than it looks, because a localised description is otherwise unfalsifiable: a basis for the occupied space can be chosen many ways and no measurement singles one out. Here the number of centres the best localisation reaches is matched by the number of atoms the geometry places within bonding distance, and a molecule whose carbons sat on edges rather than faces would have refused it.

Still open: the hexamer, and one scale for multi-centre bonds #

The obvious open question is the other alkyllithiums. The tetramer is the aggregate for methyllithium and for many others; tert-butyllithium is also a tetramer, and n-butyllithium is a hexamer in hydrocarbons, with an octahedral metal framework and six faces to cap — which by the same symmetry argument would need six localised orbitals over four centres each, and there is no reason the participation numbers should be the same.

The other open question is one that several arguments circle from different sides. The three-centre two-electron bond of diborane, the four-centre bond here, and the delocalised cage orbitals of a closo borane are three points on one scale, and what changes along it is how many centres one pair is asked to hold. A single treatment that produced all three, with the number of centres as an output rather than as a label, would be a real unification — and the localisation above is the method that could do it, since it measures the number of centres rather than assuming one.