A level no symmetry was protecting

Three orbitals have carried three arguments. Two of them, A and B, have no overlap and no coupling with each other and sit at the same energy, and both couple to a third, C. One of the trio’s levels sits at the free-atom energy exactly, at every energy of the third orbital, and the bond order between two orbitals that do not interact rests on it.

The reason given for that level was a symmetry. A and B are equivalent, so the combination $(A - B)/\sqrt{2}$ is alone in its representation, and a function alone in its representation has nothing to mix with. A symmetry holds or it does not tested it by raising A’s site energy, found the level leave at half the detuning with no protection at all, and drew the general moral: an exactness that comes from a symmetry is a discrete fact that is replaced rather than degraded when the symmetry goes. It ended by proposing a second way to make A and B inequivalent — change an overlap instead of an energy — and by naming the outcome that would be interesting: two first-order terms nearly cancelling, leaving the level looking protected without any symmetry to protect it.

The level is protected without any symmetry. It is not protected by a cancellation. And it was never protected by the symmetry in the first place.

One overlap changed #

The trio here is the one the earlier essays used: both A and B at −13.6 electronvolts, overlaps of 0.25 to C, the coupling taken from each overlap by the Wolfsberg–Helmholz rule with K = 1.75, and the overlap kept in the secular problem — the choice that makes an antibonding level rise further than the bonding one falls, and that ties every coupling to its overlap so that overlap and interaction cannot be chosen separately. Changing A’s overlap to C means moving A; it enters the overlap matrix and, through the rule, the coupling. B’s overlap to C stays at 0.25.

At a change of 0.2 the overlaps are 0.45 and 0.25, and nothing about the trio relates A to B any more. The lowest level has fallen from −16.26 to −17.07 electronvolts and the highest has risen from −8.02 to −2.78. The level that sat at the free-atom energy is at −13.60000 — and its largest departure from −13.6 across the whole sweep is 2.7 × 10⁻¹⁴ electronvolts, which is rounding.

That is not a feature of one choice of the third orbital. With C placed at seven energies from −24 to +2 electronvolts, the same change leaves the level at the free-atom energy at every one, to the same few parts in a hundred thousand billion.

What does move it, and what does not #

Three changes each break the equivalence of A and B, and they do not do the same thing.

Raising A’s site energy moves the level at half the change — the earlier result, first order and immediate. Giving A and B an overlap with each other moves it too, at 10.20 electronvolts per unit of overlap, which is exactly −(K − 1)α: the part of the new coupling that the new overlap does not cancel, weighted by the product of the level’s amplitudes on A and B. Both of those are first order, and both put a slope of one on the logarithmic plot.

Changing the A–C overlap puts nothing there. Its points lie at the arithmetic’s floor across two decades of change.

The earlier essay anticipated two first-order terms from such a change, one from the coupling and one from the overlap matrix, and both are indeed present: first-order theory in a non-orthogonal basis takes the change in H less the level’s energy times the change in S, and here that is (Kᾱ − α) times the change in overlap, where ᾱ is the mean of A’s and C’s energies. But that quantity sits only in the A–C element, and it multiplies the level’s amplitude on C. The level’s amplitude on C is zero. The first-order shift vanishes because the change acts where the level is not, and nothing needs to cancel.

That explains first order. It does not explain why the level is still at −13.6 at a change of a fifth, where every higher order would have had its chance.

A count, not a group #

The level sits at α whenever H − αS has a zero eigenvalue — whenever its determinant vanishes. Write out the matrix for this trio. A and B both have α on the diagonal, so their diagonal entries are zero; they have no coupling and no overlap with each other, so the A–B entries are zero; and each has one non-zero entry, in C’s column, equal to (Kᾱ − α) times its own overlap with C.

So the rows belonging to A and B are both zero except in one column. Two rows with a single non-zero entry in the same place are proportional, whatever those entries are, and a matrix with two proportional rows has a zero determinant. The level at α exists for any overlaps to C, any energy of C and any K.

The combination that sits there is the vector the two proportional rows annihilate: $S_{BC}A - S_{AC}B$, the one mixture of the pair for which C’s two couplings cancel. With the overlaps equal it is $(A - B)/\sqrt{2}$, which is why it looked like a symmetry’s function. With them unequal it is a different mixture, and still exact.

The solved level follows that formula at every step: half A and half B at the start, 23.6 per cent A and 76.4 per cent B at a change of 0.2, and exactly nothing on C throughout. As C sees more of A, the protected mixture leans towards B by just enough that C’s view of it stays empty.

This is a result chemistry already has, under another name. Allyl’s non-bonding orbital sits at α with zero amplitude on the central carbon and equal and opposite amplitudes on the ends, and Longuet-Higgins showed in 1950 that it does not need the two ends to be equivalent: in any alternant hydrocarbon the non-bonding orbitals are counted by how many more atoms one set has than the other, and their coefficients around each atom of the smaller set must sum to zero. The trio is allyl with its overlaps kept — two starred atoms, one unstarred — and the level is its non-bonding orbital, held by a count. Hückel theory draws that orbital for the symmetric molecule, where the count and the symmetry agree and nothing tells them apart. The same algebra holds a dark state in a three-level system in optics, where two states coupled to one excited state always leave a superposition that the light cannot reach, whatever the two couplings are.

Rank without symmetry, and symmetry without rank #

A count makes a sharper prediction than a symmetry does, and a fourth orbital tests it.

Add D at −10 electronvolts, coupled to both A and B and with a small overlap to C. Now the rows of A and B each have two non-zero entries, in C’s column and D’s. They are proportional — and the level stays exact — only if C and D see the pair in the same ratio.

With both orbitals’ couplings symmetric, the symmetry is intact and the level is exact. With C seeing A and B in the ratio 1.4 to one and D seeing them equally, the level moves by 0.0250 electronvolts. With C at 1.4 to one and D also at 1.4 to one — no symmetry anywhere, since neither orbital treats A and B alike — the level is exact again, at 7 × 10⁻¹⁵.

That separates the two explanations completely. The symmetric arrangement is exact under both. The second breaks the symmetry and the rank condition together, and the level goes. The third breaks the symmetry and keeps the rank condition, and the level stays. What protects the level is how many independent ways the pair is seen, not whether A and B are equivalent. The symmetry was one way of guaranteeing a single ratio, and never the thing itself.

The quantity that answers the other change #

A filled shell is not an empty statement found that the A–B bond order at six electrons, with every level full, is twice the off-diagonal entry of the inverse overlap matrix, one seventh for this trio, and independent of every energy in the problem. The earlier essay predicted from that a clean test: a site-energy change should move the filled-shell order by exactly nothing while it moves everything else first order.

It does. Raising A’s site energy moves the six-electron order at 1.1 × 10⁻¹² per electronvolt, which is nothing. Raising the A–C overlap moves it at 0.6531 per unit, the rate its closed form $2S_{AC}S_{BC}/(1 - S_{AC}^2 - S_{BC}^2)$ gives, from one seventh to 0.306 at a change of 0.2. The four-electron order, which depends on energies and overlaps both, moves under both changes: at −0.0347 per electronvolt and −0.1930 per unit of overlap.

Set side by side, the two quantities are mirror images. The level answers a site-energy change and a coupling between the pair, and ignores an overlap to C. The filled-shell order answers the overlap to C and ignores the site energy. Each is blind, to first order, to exactly the change the other responds to. That is the separation the earlier essays asked for between the part of a bond order that is physics and the part that is metric, and it turns out to have a counterpart in the spectrum: the exact level is a statement about energies and couplings within the pair, and the filled-shell order is a statement about how the functions overlap.

What the numbers were computed from #

Every level is an exact solution of the three-by-three generalised eigenvalue problem the trio essays use: site energies on the diagonal, couplings from the overlaps by Wolfsberg–Helmholz with K = 1.75, and the overlap matrix kept. The A–C overlap is changed by up to 0.2 in eight steps and every other number is held. The four-orbital problem is solved by symmetric orthogonalisation of its four-by-four overlap matrix and a diagonalisation of the result, with a check that the four functions are linearly independent.

First-order responses are central differences at a step of 10⁻⁴, and the bond orders use the definition the trio essays share, the one built from the eigenvector coefficients.

The checks are these. At seven energies of the third orbital, the level stays within 10⁻¹¹ eV of the free-atom energy for every overlap change up to a fifth, and its function matches $S_{BC}A - S_{AC}B$ to twelve figures. A coupling between A and B moves it first order at −(K − 1)α. With a fourth orbital, the level is exact when the arrangement is symmetric, moves by more than a millielectronvolt when the two outer orbitals see the pair in different ratios, and is exact when they see it in the same ratio with no symmetry. The filled-shell order does not move under a site-energy change and moves at its closed form’s rate under an overlap change. And the refusal: the same central differences must read a site-energy change as moving the level at one half, so that the zero they return for the overlap change is a measurement and not a blind spot.

Where the three orbitals stop #

The coupling rule decides the size of the other effects, not this one. Wolfsberg–Helmholz makes each coupling proportional to its overlap, and the rate at which an A–B overlap moves the level is (K − 1)α because of it. The exactness under an A–C change does not use the rule at all: any coupling that vanishes when the overlap does, and any value of C’s energy, leaves A’s and B’s rows proportional.

Equal site energies are required. The count holds the level at α only while A and B are both at α — the same kind of condition, an exactness from an absence, that a gap only a tetrahedron closes lost at the first degree of distortion. Change either energy and the level leaves at once, which is the earlier essay’s result, now read correctly: it measured the one change that removes the condition rather than the one that removes the symmetry.

One electron, and no repulsion. In a many-electron model the non-bonding orbital of an alternant system is where the spin density of a radical sits and where repulsion changes the ordering of states; nothing here addresses that. The statement is about the one-electron spectrum only.

And a molecule does not change one overlap while keeping another. Moving one ligand changes its distance to everything, including the other ligand, so a real distortion changes the A–B overlap too — and that is one of the changes that does move the level. The construction isolates a change a molecule would mix with others, which is what makes it a test of an explanation rather than a prediction about a geometry.

Test an explanation where it and its rival disagree #

The transferable point is about how an explanation that was never wrong can still be the wrong explanation.

The symmetry argument for the exact level is valid. A and B are equivalent in the undistorted trio, the antisymmetric combination is alone in its representation, and a function alone in its representation cannot mix. What it is not is necessary, and nothing tested on the symmetric trio, or on a site-energy change that destroys both the symmetry and the real condition together, could reveal that. The earlier essay’s general statement — an exactness that comes from a symmetry is a discrete fact that is replaced rather than degraded — is true, and it was attached to an exactness that does not come from one.

The move that exposes it is the one the fourth orbital makes explicit: find a change that removes the proposed cause and keeps the rival one, and another that keeps the proposed cause and removes the rival. When the result follows the rival in both, the rival is the explanation. Here the change of overlap was the first kind, proposed for a different reason, and it came back with a zero that the symmetry account could not produce. It is the same design a symmetry refused as the explanation for a set of cages needed, and the same lesson an exactly vanishing overlap teaches in reverse: a zero produced by counting states can look identical to one produced by a group.

There is a second point, about the particular shape of this protection. A level held by a count survives any change that leaves the count alone and falls to any change that alters it, and the changes that alter it are not the ones a symmetry would single out. So a molecule can keep a non-bonding level through a large, visibly unsymmetrical distortion and lose it to a small one that looks innocuous — a coupling between the two ends of an allyl fragment, say. Knowing which kind of protection a level has is knowing which distortions to worry about.

Who counted it first #

Longuet-Higgins’s rules for the non-bonding orbitals of alternant hydrocarbons date from 1950, and Coulson and Rushbrooke’s pairing theorem, which they rest on, from 1940. Wolfsberg and Helmholz’s rule for couplings is from 1952. The dark state of a three-level system was observed as coherent population trapping by Alzetta and colleagues in 1976. None of that is new.

What is new here is the application to a trio that had been explained by symmetry for three essays, the measurement that an overlap change leaves its level exact to rounding at every third-orbital energy, the four-orbital arrangement that separates rank from symmetry, and the mirror relation with the filled-shell bond order. The earlier essay deserves the credit for proposing the overlap change and for naming, in advance, the possibility that a level might look protected without a symmetry — which is exactly what happened, for a reason it had no way to see from the symmetric case.

Still open: what a coupling between the ends does, and the radical #

The obvious open question is the coupling between A and B, which is the one change here that both breaks the count and is unavoidable in a real molecule. It moves the level first order at a rate set by (K − 1)α, and whether a small A–B overlap of the size a bent allyl fragment would have shifts the non-bonding level by more or less than the site-energy differences a substituent brings is a comparison of two numbers the trio can supply directly.

The nearer question is the electron count the non-bonding orbital decides. At three electrons the trio is a radical with its unpaired electron in exactly this level, and its spin density sits on A and B in the proportions the protected mixture has — so the overlap change that leaves the level’s energy alone moves the spin from half and half to a quarter and three quarters. That is an observable the symmetry account would have said could not change without the energy changing too, and it is one calculation away.