Closer is not more overlap

Overlap decides is the first essay on overlap and it says what it says: two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which combination is lower. Everything after it is a refinement of that sentence.

Every curve drawn there falls monotonically from contact outwards, and so does every curve in every textbook, because the pairs everyone draws are pairs of nodeless functions. This essay is about what happens when one of the functions has a node in it, which is the ordinary case for everything past the first row.

The distinction being drawn is not a technicality about integrals. Which combination of two orbitals lies lower, how strongly two fragments couple, whether pushing them together helps — all of it is read off a single number, and the rule for how that number behaves has been generalised from the one pair everybody draws.

What a node does to an integral #

An overlap integral is the volume integral of a product of two functions. If both functions keep one sign everywhere, the product does too and the integral is a sum of contributions that all push the same way: bring the centres closer, the product gets larger everywhere at once, the integral grows.

A function with a radial node has a positive region and a negative one. Its product with something else therefore has regions of both signs, and the integral is a difference between them. Whether that difference grows or shrinks as the centres approach is not settled by anything general, because both parts are growing.

The consequence of that node is easiest to see as two separations that ought to coincide and do not.

The 1s with the 2s: smallest at contact #

And here is the integral itself for the pair that shows it most plainly, computed at thirty-seven separations rather than argued about.

At half a bohr the overlap is −0.0158. At 4.17 bohr it is −0.2820. That is a factor of eighteen, and it runs the wrong way round.

The reason is the node, and it can be read off the radial functions. At contact the 1s sits under both lobes of the 2s at once and the two contributions nearly cancel — the near-cancellation is why the number is so small, not because the functions are far apart. Pulling the centres apart moves the 1s out of the 2s’s inner lobe and into its outer one, so the cancellation stops and the integral grows. Only when the separation exceeds the reach of the functions does it start to fall.

The near-cancellation at contact has a name in a different context: it is the orthogonality of a 1s and a 2s on the same centre, which is exact. At zero separation the integral is exactly zero by that orthogonality, and everything at small separation is a perturbation of that zero.

Two p orbitals head-on: a change of sign #

This is the more startling of the two, because it is not a change in size but a change in kind. Below 5.06 bohr the combination that is lower in energy is one; above it, the other. The bonding and antibonding combinations swap.

The mechanism is the same one. Each 2p has a nodal plane through its nucleus, so each function is positive on one side and negative on the other. At short separation the two centres’ inner lobes dominate the product and the integral takes one sign; at long separation what remains of the product is the tails, and the tails’ relative arrangement is the other way round. Nodes is the essay about counting them; this is what one of them does once there are two centres in the problem.

A closed form is available for this pair and is what the numbers above are checked against. For two Slater 2p functions with equal exponents ζ, the σ overlap is

$$-e^{-p}\left(1 + p + \tfrac{1}{5}p^{2} - \tfrac{2}{15}p^{3} - \tfrac{1}{15}p^{4}\right), \qquad p = \zeta R,$$

and the polynomial in the bracket changes sign at p = 2.53, which for a hydrogenic 2p with ζ = ½ is R = 5.06 bohr. The computed root and the closed form’s agree to the precision the bisection was run to.

Two more nodal pairs, and one that behaves #

The 2s–2s pair is the useful control. Both functions have nodes, so the argument above says the integral is a difference of two contributions; but the two functions are identical, so their positive and negative regions coincide and the product is positive nearly everywhere. The result is monotonic after all — and remarkably flat, falling by only half over eight bohr, because these are diffuse functions.

So a node does not guarantee a non-monotonic overlap. What it does is remove the guarantee that the overlap is monotonic, and whether it is depends on how the nodes of the two functions are arranged relative to one another.

The rule this repairs #

The statement in overlap decides is not wrong; it is a statement about the value of the overlap and not about how the value varies with distance. What is wrong is a corollary almost everyone draws from it: that pushing atoms together strengthens their interaction.

For nodeless functions that corollary is true and the pictures show it. For anything with a node it holds only within a range, and the range has to be stated. The best coupling between two 2p orbitals in σ is at contact for magnitude and is not at contact for sign; the best coupling between a 1s and a 2s is at four bohr and a sixth, which is far outside any bond length.

The practical version of the caution is: the sign of an overlap integral between orbitals of different principal quantum number is not something to be read off a picture of two lobes. It has to be computed, and it can flip.

What this does to a two-level diagram #

The consequence for the picture everyone draws is worth working through, because a sign change in an off-diagonal element is not a small perturbation of a level diagram.

Two orbitals with a coupling β produce a pair of levels split by 2|β|, one down and one up, and which combination is which is decided by the sign. Where the overlap changes sign, the in-phase combination stops being the lower one and the out-of-phase combination takes its place. The splitting passes through zero at the crossing and reopens on the other side with the labels exchanged.

Nothing about the magnitudes is unusual — the splitting shrinks and grows the way it does at every separation — and everything about the identification changes. A reader who has learnt that the in-phase combination is the bonding one has learnt a fact about nodeless functions.

A quadrature that agrees with itself can still be wrong #

The curves above need an integrator that does not truncate space, and the reason is worth space in its own right because the failure is the kind every numerical result is exposed to.

The obvious rule integrates over a box: a Gauss–Legendre product rule over a cube fourteen bohr on a side, with the number of points as the accuracy knob, checked by recomputing at a different number of points and requiring agreement.

A hydrogenic 2p at unit nuclear charge has a mean radius of five bohr. A fourteen-bohr box — seven bohr each way — throws away a real part of it. So the computed overlap is wrong: for two 2p orbitals in σ, 9 per cent low at one bohr and 31 per cent high at six, measured against the closed form above.

And the check cannot see it, because the error was in the domain and the check varied the resolution. Forty points over a fourteen-bohr box and ninety points over a fourteen-bohr box agree to five decimal places. They agree about the wrong number.

The repair is to have no domain to get wrong. Each axis is mapped onto the whole real line — the substitution u = Lt/(1 − t²) takes the Gauss–Legendre interval to all of it — so nothing is truncated, and the map’s scale L becomes a parameter the answer barely depends on: anything between two and eight gives the same number to six decimals. A check then varies both the point count and the map, and in the direction that makes it the better rule of the two. Against the two closed forms available — 1s–1s and 2p–2p — the new rule agrees to about a part in a hundred thousand.

Past the root, at seven bohr, the sign has changed and the two 2p orbitals are anti-bonding in the σ arrangement that binds them at two. Nothing about either function has altered; what has altered is which regions of the product dominate, and the outer lobes now win.

What this says about self-checking #

The general lesson is worth extracting because it is not about quadrature.

A check that varies a parameter can only see errors that depend on that parameter. The old check varied the thing that was right and held fixed the thing that was wrong, and it did so for a reason that felt like rigour: the number of points is the obvious accuracy knob for a quadrature, and the box was chosen once, at the beginning, for the 1s orbitals that were the only pairs then drawn.

That is the shape the site’s other found defects have had. The tolerance is a decision is about a threshold chosen once and then treated as a fact; a shared function returning an empty array for a reversed axis was invisible because every check asked whether a label fitted rather than whether it existed. In each case something was fixed at the outset for a case that later stopped being the only one.

The practical form: a self-check should vary the parameter that was chosen for the first use case, not the parameter that looks like the accuracy knob.

It is worth noting what did work. The site’s rule that a cached value must be re-verified rather than trusted did its job perfectly — every stored number was recomputed and agreed. The verification was faithful and the thing it verified was the wrong claim, which is a failure of the check’s design rather than of the discipline around it.

What changed in the collection, and what did not #

The correction moves numbers that appear in figures on several essays, and it is worth saying which.

Every 2p overlap drawn at unit nuclear charge changes: the σ pair at 2.8 bohr goes from 0.459 to 0.535, the π pair at the same separation from 0.753 to 0.830. Nothing in any essay’s prose quoted those numbers, so no argument moves — but the curves in overlap is not interaction and the antibonding level goes up more are drawn differently now than they were.

Nothing at carbon’s screened charge changes at all, which is the useful diagnostic. At Z = 3.25 a 2p orbital is a bond-length-sized object and fits comfortably inside the old box: a double bond is not two single bonds computes its σ and π overlaps at that charge and its numbers move in the fifth decimal place. The error was exactly where the physical reasoning says it should have been — in the diffuse functions — which is some evidence that the account of it above is right.

And every overlap that symmetry forbids still vanishes: 1s with 2p perpendicular, 2s with 2p perpendicular, 2p with 2p perpendicular, all at 10⁻¹⁶ or below under the new rule as under the old. Exactly zero is untouched, because a cancellation that is exact by symmetry is exact under any rule that respects the symmetry, whatever its domain.

Where a chemist meets this #

Two places, and neither of them is a bond length.

The first is the interaction between orbitals of different shells, which is what a core orbital’s contribution to bonding is. A carbon 1s and a neighbouring carbon 2s are a pair with a node between them, and their overlap at a bond length is on the rising part of the curve rather than the falling part: pushing the atoms together does not increase it in the way the picture suggests.

The second is the long tail, and it is the one that decides how weakly interacting fragments behave. Two closed-shell molecules at a van der Waals separation are several ångström apart — well past the roots computed above, on the atomic scale that a diffuse orbital sets — and the sign of every σ overlap between their p functions is the opposite of what the short-range picture gives. What holds a solid together puts four kinds of interaction on one scale and the weakest of them lives entirely in that region.

Where the turnover comes from, and how to predict it #

The maximum in a 1s–2s overlap has a mechanism that is worth stating, because it says where the maximum will be for any such pair without computing the integral.

A 2s function has a radial node: it is positive close to the nucleus, crosses zero, and is negative beyond. So an overlap with it is a difference between two contributions — the other function’s amplitude in the inner, positive region, minus its amplitude in the outer, negative one.

Bring the two centres very close and the second function sits mostly in the inner region: the positive contribution dominates and the overlap is small only because the inner lobe is small. Pull them apart and the second function moves into the outer, negative region, and the negative contribution grows. Somewhere between, the two are in the ratio that makes the difference largest.

That locates the maximum. It sits at a separation of the order of the node’s position, which for a hydrogenic 2s at effective charge $Z$ is $2a_0/Z$ — a computable length, requiring no integral.

The prediction is checkable in the direction that matters, which is how the maximum moves. A more highly charged centre has its node closer in, so its overlap should turn over at a shorter separation; a diffuse function on a lightly screened atom should turn over further out. Both are statements about a computed length rather than about the shape of an integral.

It also explains the head-on 2p case in the same terms. Two 2p functions pointed at each other meet lobe to lobe, and the sign of the product changes when the separation is large enough that the near lobe of one has passed the far lobe of the other — again a length set by where the amplitude sits rather than by anything about the integral’s shape.

So the rule closer is more overlap is exactly the rule for functions with no nodes, and for everything else the correct rule is the overlap is largest when the second function sits over the first function’s largest lobe. For a nodeless function that is at contact; for a noded one it is somewhere out.

What is left #

The pairs here are hydrogenic, which is what makes the sign change so far out. A real 2p on a real atom is contracted by the nuclear charge it sees, and the sign change moves in proportionally: at carbon’s effective charge of 3.25 the σ overlap changes sign at about 1.6 bohr, which is well inside any carbon–carbon bond. What an electron actually feels is the essay about that contraction, and the correction it applies to these curves is a rescaling of the horizontal axis rather than a change to their shape. So the phenomenon is real and its location is not — a molecule never sits where the flip happens for the atoms it is made of.

Where it does matter is the long-range tail, which is what decides how two closed shells push each other apart and how a weak interaction between distant fragments behaves. There the sign is the wrong way round from what a picture of two lobes suggests, and nothing in the picture warns of it.

A ninety per cent contour of a diffuse orbital extends several bohr, so two atoms at a bond length overlap through the inner parts of their functions rather than through the tails everybody draws. That is why the sign of the product is decided by the radial node rather than by how close the pictures look.

The other thing left undone is the corresponding statement for the interaction rather than the overlap. Overlap is not interaction already separates the two — a coupling half again as large with a gap four times worse buys a third less stabilisation — and the non-monotonic overlaps here would feed through that argument to produce non-monotonic stabilisations, which is a curve nobody has drawn.