Overlap — the series · Molecular Geometry & Bonding

1s with 1s at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 1s at 50% of its density, |ψ| = 1.48e-1.

Overlap decides

Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in energy. It is one integral, and almost everything about bonding follows from it.

part 1 · bonding

1s with 2px at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 2px at 50% of its density, |ψ| = 3.16e-2.

Exactly zero

Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

part 2 · bonding

Two orbitals, 4 electrons, S = 0 and S = 0.25. Two interacting orbitals with 4 electrons in them, drawn twice: once with the overlap set to zero and once with it kept at 0.25. Dropping the overlap makes the two shifts equal, which is the picture usually taught; keeping it makes the upper level rise by more than the lower falls, which is why four electrons in two orbitals is a repulsion.

The antibonding level goes up more

The two-level diagram every course draws is symmetric, and the symmetry is an artefact of setting the overlap to zero. Keep it, and the upper level rises further than the lower one falls — which is why helium has no molecule and why closed shells push each other apart.

part 3 · bonding

Overlap against separation, Z = 3.25. How the overlap integral falls as two atoms are pulled apart, for several pairs of orbitals. Where a closed form exists it is drawn over the computed curve, so the integrator is checked rather than trusted.

A double bond is not two single bonds

Carbon's single bond is 348 kilojoules a mole and its double is 614, which is not twice anything. The two halves are different integrals over different orbitals with different distance dependence, and computing them shows why no arithmetic could have made them add.

part 4 · wrong

One metal orbital, two ligands competing for it. Metal–ligand bond orders in a three-orbital model as the left-hand ligand's interaction is turned up. Its own bond order rises and the bond order to the ligand opposite falls, from 0.62 at equal strengths to 0.42 at the strongest. Nothing else in the model can carry the effect: switch the second bond off and it vanishes exactly.

The trans influence is an overlap argument

Two ligands on opposite sides of a metal both bond through the same metal orbital, and there is only one of it. Strengthen one and the bond order to the other falls — computed exactly on three levels, and measured as a bond length that grows by a tenth of an ångström.

part 5 · applied

Two orbitals, 2 electrons, S = 0 and S = 0.2. Two interacting orbitals with 2 electrons in them, drawn twice: once with the overlap set to zero and once with it kept at 0.2. Dropping the overlap makes the two shifts equal, which is the picture usually taught; keeping it makes the upper level rise by more than the lower falls, which is why four electrons in two orbitals is a repulsion.

Overlap is not interaction

Two orbitals interact by the square of their coupling divided by the distance between them in energy. A coupling half again as large, with a gap four times worse, buys a third less stabilisation — so the pair that overlaps best is often not the pair that bonds best.

part 6 · applied

Overlap does not always fall as the atoms are pulled apart. The overlap integral of two pairs of orbitals against the separation between their centres, with the turning point and the sign change of each found by bisection on the integral itself. A pair with a radial node in it does not fall monotonically and does not keep one sign.

Closer is not more overlap

Two 1s orbitals overlap more the closer they are, and every curve drawn from that pair says the same thing. Put a radial node into one of them and the rule fails: a 1s with a 2s overlaps by 0.016 at contact, by 0.282 near four bohr, and less again beyond — and two 2p orbitals head-on change sign at 5.06 bohr and are more strongly coupled at eight bohr than at four.

part 7 · bonding

The bonding follows the overlap over, and turns where it turns. The stabilisation of three pairs of orbitals against how far apart they are, each scaled to its own largest, with the coupling taken from the computed overlap. Two orbitals with no radial node are stabilised less at every separation further out; the two with one turn over — and the maximum of the bonding is at exactly the separation of maximum overlap, to five decimal places, because a coupling proportional to the overlap makes the stabilisation a strictly increasing function of it.

The same overlap, a different bond

A 1s and a 2s change their overlap by two thirds between two and seven bohr, and change what they are bonded by by 0.154 per cent. The overlap turns over and the bonding turns over with it, at exactly the same separation to five decimal places — the arithmetic refused the expectation that the secular denominators would move it — and what separates one pair from another is not where the maximum is but how little of it there is.

part 8 · wrong

The pair's regime is not a property of the pair. How much the pair's bonding responds to its own overlap — the ratio of what it is bonded by at an overlap of 0.4 to what it is bonded by at 0.1 — with and without a third orbital coupled to both. Alone it is 11.83, which is the regime in which bonding tracks overlap. With a third orbital present it falls to 1.46, 0.92, 0.74 — and two of those are below one, meaning a fourfold increase in the overlap between the two atoms buys them less bonding rather than more. The coupling comes from the overlap by the Wolfsberg–Helmholz rule with K = 1.75, which is fitted rather than derived. Every stabilisation here inherits that; the shape of the curve against separation does not, because K is a constant.

A regime that belongs to the neighbours

Two orbitals at the same energy are bonded in proportion to their overlap and two far apart are barely bonded at all — two regimes, and the natural question is whether the regime is a property of the pair. It is not. Put a third orbital beside them and the pair's response to its own overlap falls from twelvefold to less than one: more overlap buys less bonding.

part 9 · wrong

The filled orbital at 3 bohr, with a node in the middle of the bond. The combination the two-level model puts lower at 3 bohr, drawn as contours of the wavefunction with the two signs in the two colours. The overlap here is 0.4825, so the interaction is of the sign that fills the plus combination. There is a nodal plane through the midpoint: the pair the model calls bonded has no density at all between its nuclei.

A bond with nothing in the middle

Two head-on 2p functions have an overlap that changes sign at 5.03 bohr, and the picture of what that means is worth drawing. Below that separation the combination the model fills has a nodal plane through the midpoint of the bond, so two electrons in it put exactly nothing between the nuclei; above it the same model fills the other one. Which picture a bonding orbital has is decided by a separation.

part 10 · bonding

A pair with no overlap, and a third orbital swept past it. The three levels of a trio in which the two outer orbitals have exactly no overlap with each other, as the third orbital's energy is swept. The middle line is at -13.6 at every point — the antisymmetric combination of the two, which has no partner of its own symmetry and cannot mix with anything. The other two move, so the pair is split by an orbital it has no direct contact through.

A bond order between atoms that do not interact

A diatomic held where its overlap changes sign has no interaction between its two orbitals at all — which is what the sign change of its overlap means. Put a third orbital beside it and the pair is still split, one line sits exactly at the free-atom energy at every third-orbital energy, and the bond order between the two runs to −0.9999. Three measures of the same bond disagree completely.

part 11 · wrong

One geometry, four electron counts, four answers. The bond order between two orbitals with exactly no overlap and no resonance integral, as the third orbital's energy is swept, at every count the trio can hold. With none it is identically zero. With two it is positive and rises past one. With four it is negative and reaches -0.954. With six it is a horizontal line — the third orbital's energy stops mattering entirely.

A filled shell is not an empty statement

A bond order of −0.954 between two orbitals with no overlap and no resonance integral invites the prediction that at six electrons — every level occupied, the sum over a complete set — it would be exactly zero. It is exactly one seventh, and the reason is that a complete set in a non-orthogonal basis sums to the inverse of the overlap matrix, which has entries where the overlap has none.

part 12 · wrong

What happens to the level that was exact. The three levels of the trio as one of the two outer orbitals is raised. At zero detuning the middle one sits at -13.6 exactly — it is the antisymmetric combination, and nothing of its symmetry exists for it to mix with. The moment the two are made inequivalent that statement is gone: the level leaves linearly, and the other two barely move by comparison.

A symmetry holds or it does not

One level of a three-orbital trio sits at the free-atom energy exactly, at every third-orbital energy, because the antisymmetric combination of the pair has nothing of its own symmetry to mix with. Detuning one of the two by a twentieth of an electron volt moves it by half of that — first order, immediately, with no protected regime at all.

part 13 · wrong

The A–C overlap changes by a fifth and the exact level does not move. The three levels of the trio as the overlap between A and C is raised from 0.25 by up to 0.2, with B's overlap to C held and both site energies at -13.6 eV. The two outer orbitals stop being equivalent at the first step. The lowest level falls by 0.80 eV and the highest rises by 5.24, and the middle one stays at -13.6 eV — its largest departure over the whole sweep is 2.7×10⁻¹⁴ eV, which is rounding.

A level no symmetry was protecting

A three-orbital trio keeps one level at the free-atom energy exactly, and the reason given was that its two outer orbitals are equivalent. Make them inequivalent by changing one overlap rather than one energy and the level does not move at all — not to first order, not to any order, at any energy of the third orbital. It was never the symmetry. It is allyl's non-bonding orbital, held by a count.

part 14 · bonding

The unpaired electron's level stays put while its spin moves to the far end. Above, the trio's three levels as the overlap of A with C is raised from 0.25 to 0.45 while B's stays at 0.25. The lowest falls from -16.26 to -17.07 eV and the highest rises from -8.02 to -2.78 eV; the middle one, which holds the radical's unpaired electron, stays at −13.6 eV. Below, the spin on A and on B over the same change: from a half each to 0.236 on A and 0.764 on B, with the ratio of squared overlaps drawn as open circles on top.

The spin the count does not hold

Three orbitals in a row keep one level at the free-atom energy however the overlap of one end is changed, and at three electrons that level holds the radical's unpaired electron. Its energy does not move. Its spin does: from half on each end to 0.236 and 0.764 as one overlap goes from 0.25 to 0.45, exactly the squared ratio of the two overlaps, at every energy of the middle orbital. A coupling between the ends moves the level and cannot move the spin. The energy and the spin are answering to different things.

part 15 · bonding