Overlap is not interaction

Two orbitals brought together produce a bonding combination below both of them and an antibonding combination above. How far the bonding one falls is the quantity a chemist reasons with, and the rule of thumb attached to it is that more overlap gives more of it.

That rule is half of the answer, and the missing half is often the larger one.

Solve the two-level problem. With diagonal energies $\alpha_A$ and $\alpha_B$ and an off-diagonal coupling $\beta$, the levels are

$$E_\pm = \frac{\alpha_A + \alpha_B}{2} \pm \sqrt{\left(\frac{\alpha_A - \alpha_B}{2}\right)^2 + \beta^2}$$

and when the gap $\Delta = |\alpha_A - \alpha_B|$ is large compared with $\beta$ the lower level falls below the lower of the two originals by approximately

$$\frac{\beta^2}{\Delta}.$$

Two factors, and only one of them is the overlap. The coupling $\beta$ is roughly proportional to the overlap; the gap is a property of the two atoms and has nothing to do with how well their orbitals fit together in space.

The numbers, and how badly the rule of thumb can fail #

Take the exact expression and vary the gap with the coupling held at one. Measuring the stabilisation of the lower combination relative to the lower of the two originals:

The perturbation estimate is good from a gap of about two upwards, and the exact answer falls by a factor of eight while the gap grows by eight.

Now compare two pairs of orbitals with different couplings and different gaps. A coupling of $1.5$ across a gap of $8$ gives $0.272$. A coupling of $1.0$ across a gap of $2$ gives $0.414$.

The pair with the larger coupling is stabilised a third less. Increasing the overlap by half was not enough to overcome a gap four times worse, and it never is — the coupling enters squared and the gap enters linearly, so a factor of four in the gap needs a factor of two in the coupling merely to break even.

What “energy match” means physically #

The gap is not an arbitrary parameter. It is the difference between what two atoms want to do with an electron, and it is set by their ionisation energies.

An orbital’s diagonal energy is, to a good approximation, minus the ionisation energy of the electron in it. Two atoms with similar ionisation energies have well-matched orbitals; an electropositive atom and an electronegative one do not.

That is where electronegativity re-enters, and it re-enters in a role different from the one it usually plays. In the bond-polarity story an electronegativity difference says which end of a bond is negative. Here it says something about the strength of the interaction as well: a large difference means a large gap, which means a small interaction between the two orbitals however well they overlap.

The consequence is a genuine tension. A large electronegativity difference makes a bond ionic, which is stabilising in its own way — but it also makes the covalent interaction weaker, because the orbitals no longer talk to each other efficiently. Electronegativity is not one quantity sets out how badly defined the scale itself is; this is a separate point about what a difference on it does.

Where the electron pair ends up #

The stabilisation is not the only thing the gap decides. It also decides where the electrons are, and the two are connected.

The bonding combination of two orbitals with diagonal energies $\alpha_A$ and $\alpha_B$ is not an equal mixture unless the two are equal. The coefficient ratio comes out of the same secular problem, and for a large gap the lower combination is almost entirely the lower orbital with a small admixture of the upper one — of order $\beta/\Delta$.

So a badly matched pair produces a bond in which the pair is mostly on one atom, which is what “polar” means expressed in coefficients rather than in charges. And the two statements are the same statement: the interaction is weak because the mixing is small, and the bond is polar because the mixing is small. Polarity and weakness of the covalent interaction have one cause.

That is worth holding against the usual account, in which a polar bond is described as strong because of the electrostatic attraction between the partial charges. Both are true, and they are pulling in opposite directions on the covalent term. Which wins is a question about magnitudes that neither this site nor a qualitative argument can settle, and the honest version is that a polar bond has a small covalent stabilisation and an electrostatic one that a covalent picture does not contain.

The dipole is not a sum of bonds and what a dipole cannot tell apart treat the observable end of the same phenomenon. This is the orbital end of it.

Two cases where the rule of thumb gets it backwards #

Fluorine as a pi donor. A fluorine attached to carbon has filled 2p orbitals perpendicular to the bond, and they can in principle donate into the carbon’s empty pi system. The overlap is fine: the two are in the same shell, at a short bond distance, and this site’s integrator gives a 2p–2p pi overlap comparable with what two carbons manage.

What is not fine is the match. Fluorine’s 2p level sits several electronvolts below carbon’s — that is what its ionisation energy of 17.4 eV against carbon’s 11.3 eV means — so the gap is large and the interaction is small.

The observable consequence is that fluorine’s pi donation into an aromatic ring is real but weak, far weaker than the overlap alone would suggest, and much weaker than nitrogen’s or oxygen’s from the same shell. A ranking by overlap would put all three together; a ranking by $\beta^2/\Delta$ puts them in the order the chemistry does.

Carbon monoxide in the spectrochemical series. Carbon monoxide sits at the top of the series, producing the largest ligand-field splittings of any common ligand, and the reason usually given first is that it is a strong donor. It is not: its sigma donor orbital is a weakly bonding lone pair on carbon, and by any measure of donor strength it is unremarkable.

What it is, is a superb acceptor, and for the reason this essay is about. Its empty pi* orbitals lie close in energy to a metal’s filled $\mathrm{t}_{2g}$ set — a small gap — so the interaction between them is large even though the overlap is nothing special. The spectrochemical series is not electrostatics shows that ordering the whole series by charge gets three of twenty pairs right and ordering it by the pi parameter gets all thirty-five, and the pi parameter is precisely the small-gap interaction.

What the overlap being kept does to the arithmetic #

Everything above sets the overlap integral $S$ to zero in the secular determinant while letting the coupling $\beta$ stand for it, which is the usual approximation and is worth separating from the argument.

The antibonding level goes up more works the case with $S$ kept, and the result is that the two levels move asymmetrically: the antibonding one rises by more than the bonding one falls, by an amount proportional to $S$. Keeping the overlap therefore changes the sizes of both shifts and does not change the structure of the argument here, since the gap enters the denominator either way.

What it does change is one conclusion at the margin. With $S$ kept, filling both levels is net destabilising — which is why helium does not form a diatomic molecule, and which the $S = 0$ treatment cannot show at all. So an interaction between two filled orbitals is repulsive rather than merely useless, and the $\beta^2/\Delta$ estimate has the wrong sign for it.

That matters for one of the cases above. A pi-donor ligand’s filled orbitals meeting a metal’s filled $\mathrm{t}_{2g}$ set is a filled-filled interaction, and its effect on the splitting comes from the four-electron repulsion rather than from a stabilisation. The sign works out the same way — the metal orbital goes up — but the mechanism is the one that needs $S$, and describing it as a small stabilisation would be describing it wrongly.

Where the overlap does dominate #

It would be a poor essay that only argued one way, and there is a large class of cases where the gap is roughly constant and the overlap decides everything.

Down a group. Comparing a C–C bond with an Si–Si bond, both partners move together, so the gap between the two partners stays near zero in both cases and the interaction is governed by the coupling alone. The coupling falls because the orbitals are larger and more diffuse, and the bond is weaker — exactly as the rule of thumb says.

Across a bond length. Overlap decides computes the overlap as a function of separation for several pairs, and pulling two identical atoms apart leaves the gap at zero and reduces the coupling monotonically. Again the rule of thumb is right, because there is only one factor in play.

Between symmetry-equivalent partners. Any two orbitals related by a symmetry operation have exactly the same diagonal energy, so the gap is exactly zero and the stabilisation is exactly $|\beta|$ — the case at the left-hand end of the table above. Every homonuclear diatomic and every symmetric combination in a symmetric molecule is in this class.

So the rule of thumb is not wrong; it is a special case, and it is the special case that most introductory examples happen to be in.

Why the square, and where the estimate fails #

The square in the numerator is worth one paragraph because it is the part that makes the trade-off steep.

Second-order perturbation theory gives the shift of a level as a sum over the states it mixes with, each contributing $|\langle a|\hat{H}|b\rangle|^2 / (E_a - E_b)$. The matrix element appears squared because the interaction has to act twice — once to mix $b$ into $a$, and once to bring it back — and the energy denominator appears because the amount of mixing is itself inversely proportional to the gap.

That is also where the estimate fails. When the gap approaches the coupling, the mixing is not small and the perturbation series is not converging; the exact expression above is then the only thing to use, and it saturates at $|\beta|$ rather than diverging. The table shows the crossover: the estimate is 20 per cent high at a gap of two and 1.6 per cent high at a gap of eight.

For chemistry that matters because the two regimes correspond to two familiar pictures. Small gap: a covalent bond, with the electron pair shared roughly equally, and the stabilisation of order $\beta$. Large gap: a donor–acceptor interaction, with the pair still mostly on the donor and the stabilisation of order $\beta^2/\Delta$. Those are different kinds of bond and the difference is the ratio of two numbers rather than a difference of kind.

Two well-matched pairs are the case where the whole stabilisation is the coupling, because the gap is zero by symmetry and there is nothing else for it to depend on. That is the case the rule of thumb was formed on, and everything in this essay is about what happens when the two atoms are not the same.

The same arithmetic inside a ligand field #

The whole of ligand-field theory is this expression applied five times, and putting it that way makes the two models this site runs look less different than they do.

An angular-overlap treatment writes each metal d orbital’s shift as a sum of terms $e_\sigma$ and $e_\pi$ weighted by geometric factors, and the parameters are quoted rather than computed. What they are is exactly $\beta^2/\Delta$ for the relevant metal–ligand orbital pair: a coupling squared over an energy denominator, with the coupling carrying the overlap and the denominator carrying the match.

That is why $e_\pi$ can be negative. A pi-donor ligand has filled pi orbitals below the metal’s, so the interaction pushes the metal orbital up; a pi-acceptor has empty ones above, and pushes it down. Same expression, opposite sign of the denominator, opposite effect on the splitting — and the ordering of the spectrochemical series from pi-donors at the bottom through pure sigma donors to pi-acceptors at the top is a list sorted by that sign and magnitude.

The point-charge model, which this site runs alongside and which agrees with the angular-overlap one on every ratio, contains none of this. It has no orbitals, no couplings and no gaps; it integrates an electrostatic potential over an angular density. That the two agree on the ratios between geometries — four ninths for a tetrahedron, eight ninths for a cube — and disagree entirely about what sets the magnitude is the cleanest statement of what each model is for.

Separating the two factors experimentally #

The formula has two ingredients and the comparisons above vary both at once, which is what makes the result surprising. Chemistry supplies a family in which one of them is held fixed, and the resulting series is one of the most-used tables in the subject.

The family is the tertiary phosphines. Every one of them binds a metal through a phosphorus lone pair, so the donor orbital is the same kind of function on the same element at very nearly the same distance — the overlap is essentially constant across the series. What changes is the substituents on the phosphorus, and what they change is the energy of that lone pair: electron-withdrawing groups pull it down, electron-releasing ones push it up.

So a comparison across the series varies the energy denominator alone, and any variation in how strongly a phosphine binds is a measurement of the energy match with no overlap term in it.

That variation is large and it is tabulated. Substituting the phosphorus with fluorines rather than alkyl groups changes the donor strength by an amount comparable to the whole range of ordinary ligands, and the ordering is measured — through the carbonyl stretching frequency of a standard complex, which reports how much density the metal has left to give away once the phosphine has finished donating.

Two things follow.

The series is a pure energy-match series, and its success is evidence that the denominator matters as much as the formula says. A model with only overlap in it would predict that all these phosphines behave alike, since their overlaps are alike.

And it is why phosphines are the ligand of choice for tuning a catalyst. A chemist wanting to change how strongly a ligand binds, without changing the geometry it enforces or the distance it sits at, changes the substituents — which moves one factor of the formula and holds the other.

That is the experimentalist’s version of the same point. The two ingredients are separable, one of them is adjustable by ordinary synthesis, and an entire field of catalyst design is built on varying the one that a maximum-overlap argument would say is irrelevant.

What the energy gap adds to overlap #

The overlap argument computes the integral, shows which integrals vanish exactly by symmetry, shows that the antibonding level rises more than the bonding one falls, and applies the whole thing to a trans influence. All of that treats the overlap as the quantity that decides.

This essay says it is one of two, and that the other one is frequently larger. The consequence is a habit rather than a formula: before comparing two interactions by their overlaps, check whether their energy gaps are the same. Where they are — same shell, same element, symmetry-equivalent partners — the comparison is sound. Where they are not, the overlap ranking and the interaction ranking can differ and often do.

The next case is three or more levels, where the sum over partners matters and a single orbital can be stabilised by several at once. That is what an angular-overlap treatment of a coordination complex already does, and two models, one ratio runs it — but it runs it with the gaps folded into a parameter rather than made explicit. Pulling them back out would turn the ligand-field parameters into products of an overlap and an energy denominator, which is what they have always been.