The antibonding level goes up more

Two atomic orbitals interact and produce two molecular orbitals, one lower and one higher. Every introduction draws that picture, and almost every one draws it symmetrically: the bonding level drops by some amount and the antibonding level rises by the same amount.

The symmetry is false, and it is false in a way that matters. It is what remains after the overlap integral has been set to zero for convenience — the same overlap integral that is the reason the two orbitals interact at all.

The determinant, written out #

Two orbitals $\phi_A$ and $\phi_B$, with $H_{AA} = H_{BB} = \alpha$, $H_{AB} = \beta$, and — this is the term at issue — an overlap $S_{AB} = \int \phi_A\phi_B\,\mathrm{d}\tau = S$.

Because the basis is not orthogonal, the eigenvalue problem is the generalised one, and the secular determinant is

$$\begin{vmatrix} \alpha - E & \beta - ES \\ \beta - ES & \alpha - E \end{vmatrix} = 0.$$

The $ES$ terms are the whole story. Expanding gives $(\alpha - E)^2 = (\beta - ES)^2$, and the two roots are

$$E_+ = \frac{\alpha + \beta}{1 + S}, \qquad E_- = \frac{\alpha - \beta}{1 - S}.$$

Set $S = 0$ and they collapse to $\alpha \pm \beta$: two shifts of equal size, in opposite directions, which is the symmetric picture. Keep $S$ and the denominators differ. With $\alpha = 0$ and $\beta$ negative, the bonding level lies $|\beta|/(1+S)$ below and the antibonding level $|\beta|/(1-S)$ above.

At $S = 0.25$ that is 0.800 against 1.333. The upper level moves two thirds further than the lower one.

Where the asymmetry comes from #

The denominators are not an algebraic accident. They are the normalisation.

The bonding combination $\phi_A + \phi_B$ has a squared norm of $2 + 2S$, because the cross term is the overlap. The antibonding combination $\phi_A - \phi_B$ has a squared norm of $2 - 2S$. Normalising divides each by its own norm, so the bonding orbital is spread over a larger effective volume than a bare sum would be, and the antibonding orbital over a smaller one.

The physical statement underneath is that the antibonding orbital has a node between the nuclei, so its density is squeezed out of the internuclear region and piled onto the far sides. Squeezing costs, and the more the two orbitals overlap the more there is to squeeze. That is why the asymmetry grows with $S$: at $S = 0.1$ the shifts are 0.909 and 1.111, at $S = 0.5$ they are 0.667 and 2.000.

Two electrons and four #

With two electrons the pair goes into the bonding level and the molecule is bound by $2 \times 0.800 = 1.600$ in units of $|\beta|$. The overlap has made the bond slightly less strong than the symmetric picture would suggest — $2|\beta|$ becomes $1.6|\beta|$ — but the sign is unchanged and the conclusion is the same.

With four electrons the story changes completely. Both levels are filled, and the total is

$$2 \times (-0.800) + 2 \times (+1.333) = +1.067,$$

which is positive. Four electrons in two overlapping orbitals are higher in energy than four electrons in two separate orbitals. He₂ does not fail to bond; it is pushed apart.

That distinction is the one the symmetric diagram cannot make. With equal shifts, four electrons come out exactly level with where they began, and the honest reading of that picture is that helium is indifferent to helium — which is wrong, and wrong in a way that leaves closed-shell repulsion unexplained.

The rule that follows, and how far it goes #

Once the asymmetry is admitted, a short list of consequences follows that between them cover a surprising amount of chemistry.

Two closed shells repel. Every pair of filled orbitals on two approaching molecules is a four-electron interaction, and every one of them is destabilising. The repulsion grows as the overlap does, which means roughly exponentially with decreasing distance, which is why the steep wall in every intermolecular potential is where it is. Nothing electrostatic is needed to produce it.

A filled orbital and an empty one attract. Two electrons in the lower level and none in the upper is the two-electron case above: net stabilising, and stabilising by more the larger the overlap. That is the frontier-orbital picture of a Lewis acid meeting a Lewis base, and of very nearly every reaction step drawn with a curly arrow.

Bond orders count the difference, not the sum. The conventional bond order — half the number of bonding electrons minus antibonding — is the symmetric picture’s arithmetic, and it gives He₂ a bond order of zero. The asymmetric arithmetic gives it a positive energy, which is a stronger and more useful statement: a bond order of zero predicts indifference and the molecule is repelled.

And the effect is largest exactly where the diagram is used most. At the overlaps typical of a real bond, 0.4 to 0.8, the asymmetry is not a correction. At $S = 0.75$ the upper level rises four times as far as the lower falls.

Where S actually comes from #

$S$ is not a parameter in the sense $\beta$ is. It is an integral over two known functions at a known separation, and here it is computed rather than quoted.

For two hydrogen 1s orbitals it even has a closed form,

$$S = e^{-R}\left(1 + R + \tfrac{R^2}{3}\right),$$

which is what the numerical integrator here is checked against — the two agree to six parts in a hundred thousand across every separation drawn. At H₂’s bond length of 1.4 bohr the overlap is 0.753, which is very far from zero, and at 2.8 bohr it is 0.390.

So the usual justification for dropping $S$ — that it is small — is not available for the case the diagram is usually drawn about. It is dropped because it makes the algebra one line instead of three.

What happens as the atoms separate #

The overlap is a function of distance, so the whole diagram is, and watching it as two atoms are pulled apart makes the argument concrete.

At 1.4 bohr — hydrogen’s bond length — the 1s–1s overlap is 0.753, and the two levels are split enormously, with the antibonding one far above. At 2.8 bohr the overlap has fallen to 0.390, and at 5.6 bohr to 0.063. By then the asymmetry has all but vanished: the shifts are 0.941 and 1.067, within thirteen per cent of each other, and the symmetric picture is a good approximation.

So the diagram usually drawn is the correct one for two atoms that are far apart, which is the situation in which nothing interesting is happening. It becomes wrong exactly as the interaction becomes strong enough to be worth drawing.

What was computed, and how #

Three separate computations meet in this essay, and only one of them contains a fitted number.

The secular determinant is solved in closed form for the 2×2 case, with $\alpha$, $\beta$ and $S$ as free inputs. Its output is checked against a claim that can fail: with the overlap kept, the upper shift must exceed the lower, and with the overlap dropped the two must be equal to within the last bit of a double. A version of the arithmetic that had lost the $ES$ terms would pass the second test and fail the first.

The overlap integrals are Gauss product quadratures over a 14-bohr box at ninetieth order, cached between builds and re-verified at fortieth order on every restore. They are checked against the closed form above at five separations, and separately against the requirement that a symmetry-forbidden overlap come out at arithmetic noise rather than merely small — which is the subject of exactly zero. Every contour drawn beside them states the fraction of the density it encloses, for the reason say what it encloses gives: two pictures at different levels are not a comparison.

$\beta$ is not computed and has no number. Every energy in this essay is in units of $|\beta|$, and $\beta$ is a parameter whose value depends on what it was fitted to. That is why the figures print ratios and differences rather than kilojoules: the asymmetry 1.333 against 0.800 is a property of the arithmetic, and any conversion of it into an energy would be a calibration with a hidden choice in it.

Where the model stops #

This is a one-electron argument used to discuss four electrons, which is a real gap rather than a formality. The levels are one-electron energies, and adding up the energies of four electrons in them ignores the repulsion between those electrons entirely. The sum is not the total energy of He₂ and should not be read as one.

The real closed-shell repulsion is exchange, not electrostatics. The account here — the antibonding orbital is destabilised more than the bonding one is stabilised, so filling both costs — reproduces the right sign and the right dependence on overlap, and it is a molecular-orbital restatement of what is more fundamentally the Pauli principle. The two are not in conflict; the first is the second expressed in a basis.

$\alpha_A = \alpha_B$ is assumed here, which is the homonuclear case. The unequal case is the more common one and its behaviour is different in an instructive way: as the two levels separate, the interaction weakens as $\beta^2/\Delta\alpha$, which is the origin of the rule that orbitals interact strongly only when they are close in energy — the rule hypervalency without d orbitals uses to dispose of sulfur’s 3d orbitals.

The two orbitals are assumed to be the whole basis. A real diatomic has more orbitals than two, and the ones left out mix in — which is what molecular orbital and valence bond is about when it describes the two frameworks converging.

Nothing here says where the equilibrium bond length is. The levels fall as the atoms approach, without limit, and what stops them is nuclear repulsion, which is not in this model at all.

The generalisation #

The habit this leaves is a specific suspicion: when a diagram is symmetric, ask what was set to zero to make it so.

Symmetry in a picture is usually either a consequence of a symmetry in the situation — in which case it is exact and can be stated without hedging — or a consequence of a term having been dropped. The two look identical on a whiteboard and they behave completely differently when the model is pushed.

Here the symmetry was the second kind, and the term dropped was the one carrying the entire explanation of why matter has volume. The same question is worth asking of the symmetric level patterns in Hückel theory, where the pairing of levels about $\alpha$ is the first kind — a consequence of the graph being alternant, exact, and provable — but where the neglect of overlap is the second kind and is doing quiet work of its own.

Distinguishing them is not a matter of taste. In the Hückel case the pairing survives any change to $\beta$; in this case the equality of the shifts survives nothing at all.

A note on orthogonalised bases #

There is a move that makes the asymmetry disappear again, and it is worth naming so that it is not mistaken for a refutation.

Löwdin orthogonalisation replaces the two overlapping atomic orbitals with two orthogonal combinations that resemble them as closely as possible. In that basis $S$ is zero by construction, the secular determinant is the ordinary one, and the two shifts come out equal.

Nothing physical has changed. The orthogonalised orbitals are not the atomic orbitals: they carry small negative tails on the neighbouring centre, and the price of orthogonality is paid inside them rather than in the determinant. The total energy, the density and every observable are the same either way — which is exactly the lesson of the localisation transformation, where a unitary change of basis leaves the density identical to the last bit of a double.

So “the shifts are equal in an orthogonal basis” is true and is not an argument that helium bonds. The four-electron destabilisation reappears in the orthogonalised picture as the cost of the orthogonalisation itself. Two descriptions, one answer, and the arithmetic is easier to follow in the one where the term is visible.

The molecule that exists because the antibonding level costs more #

Helium having no molecule is the standard consequence, and it is a negative one — an absence, which is weak evidence. There is a positive test available on the same two atoms, and it is decisive.

Take a helium dimer and remove one electron. Two electrons remain: both bonding levels filled, one antibonding electron gone. If the two levels were symmetric, as the textbook diagram draws them, the resulting ion would be bound by half of nothing much — a bond order of a half, a weak association, easily broken.

The diatomic helium cation is bound by about 2.4 electronvolts, with a bond length of 1.08 ångström. That is a real bond, comparable in strength to many ordinary single bonds, in a molecule made of two atoms that will not bond at all when the extra electron is present.

The asymmetry is the whole of the difference. Two electrons in the bonding level gain something; the third, in the antibonding level, loses more than one of them gained; and the fourth loses more again, which takes the neutral dimer past zero. Remove the fourth and the balance is positive by a wide margin — 2.4 electronvolts of it.

So the sequence across the four electrons is not two steps down and two steps up. It is down, down, up-by-more, up-by-more, and the molecule survives at three electrons and dies at four.

The same arithmetic accounts for the other well-known thing helium does. Excited helium dimers exist, are bound, and are the light source in helium lasers — because promoting one electron out of the antibonding level and into a higher one has the same effect as removing it. The bonding pair is left unopposed, and a pair of atoms that repel in the ground state attract in the excited one.

Two positive results, on the one pair of atoms whose failure to bond is the standard illustration, both of them following from a level diagram being asymmetric rather than symmetric.

The general form is worth extracting, because it applies wherever two closed shells meet. A filled level and a filled antibonding level do not cancel; they leave a net repulsion, and the size of the repulsion grows with the overlap. That is why two closed-shell atoms or molecules push each other apart at short range at all — the repulsion between two argon atoms, or between two non-bonded parts of one molecule, is this asymmetry and not an electrostatic effect.

It is also why the repulsion falls off so steeply. The asymmetry is second order in the overlap where the bonding is first order, so a repulsion between filled shells dies away faster than a bond does — which is the shape of every short-range repulsive term in every force field, arrived at from a level diagram rather than fitted to a curve.

Who found it, and when #

The generalised eigenvalue problem with overlap is in the earliest molecular-orbital work — Lennard-Jones’s 1929 paper sets it out — and the asymmetry is explicit in Mulliken’s writing through the 1930s.

The connection to closed-shell repulsion, and the phrase “four-electron destabilising interaction” that goes with it, belongs to the frontier-orbital tradition of the 1960s and 1970s, particularly Fukui’s and Hoffmann’s. Their formulation is the one that made the asymmetry a working tool rather than an algebraic remark: a filled orbital meeting a filled orbital is repulsive, a filled orbital meeting an empty one is attractive, and the size of both depends on overlap over energy gap.

The habit of dropping $S$ traces to Hückel’s 1931 papers, where it was one of several drastic simplifications made deliberately and stated as such. What has happened since is that the simplification has outlived the statement of it, and the symmetric diagram is now drawn in places where the reason for the symmetry is not mentioned at all.

Still open: two different kinds of orbital #

This essay takes the two-orbital problem apart with nothing dropped, and finds that keeping one integral changes the conclusion for closed shells completely. The obvious next question is what happens when the two orbitals being combined are not the same kind of orbital — where the σ and π overlaps of the same pair of atoms give bonds of very different character.