A bond is not two atoms overlapping

Every contour so far has been drawn round one nucleus. The method that draws them is a root-find along each ray outward: the density falls monotonically along any direction from the centre, so there is exactly one radius at which it crosses the level, and the surface is that radius as a function of direction.

A bonding orbital is not shaped like that. Along a line from one nucleus towards the other the density falls, reaches a minimum somewhere between, and rises again — so a ray can cross the level twice, or not at all, and the one-centre root-find does not apply.

That is a small technical fact with a large consequence, because the picture drawn instead of the real one is drawn everywhere.

What is usually drawn #

Open any account of a σ bond and the picture is two atomic orbitals, each a sphere or a lobe, overlapping in the middle. It is drawn that way because it is how the orbital was constructed — a linear combination of two atomic functions — and the construction is faithfully reproduced in the drawing.

The orbital is not two atomic orbitals. It is one function, and the surface enclosing a stated fraction of its density is one surface.

The two pictures differ in three ways, and each is a number.

The level is different. The molecular contour sits at $|\psi| = 0.0359$ and the atomic one at $0.0394$. Nothing forces those to agree — the two functions are normalised over different regions and have different shapes — and a plate drawing both at one level is a comparison that is not one.

The shape is different. The molecular surface is convex along the axis between the nuclei; the union of two spheres has a re-entrant waist where they intersect. That waist is where the bonding density is.

The amount enclosed is different. The two atomic spheres, drawn at the level that encloses ninety per cent of an atomic density, together enclose 91.70 per cent of the molecular orbital’s density. Not ninety, which is what the caption on such a figure invariably says.

That last number is the sharpest of the three, because it is the one a reader would use. A picture captioned the ninety per cent contour which encloses 91.70 is wrong in the way this whole site was built to complain about — and the error is not large, which is what makes it worth measuring rather than assuming.

How the contour is solved for #

The method has to change because the geometry has changed, and what replaces it is simpler rather than harder.

A σ orbital is axially symmetric, so its density is a function of two coordinates: the distance ρ from the axis and the position z along it. At any fixed z the density falls monotonically outward in ρ, so the region inside the contour is a disc, and its radius is one bisection. The enclosed fraction is then

$$F(v) = 2\pi \int \mathrm{d}z \int_0^{\rho^*(z)} |\psi|^2\, \rho\, \mathrm{d}\rho$$

with the inner integral over a finite interval and the outer one over the whole line by the mapped rule this file uses everywhere — no box, which is the defect that was found and repaired in the overlap integrator and is not about to be reintroduced.

Solving for the level is then a bisection on $F$, and the level that comes out is verified by recomputing the enclosed fraction at a different resolution — half again as many points in each direction — which is the check a bisection cannot make on itself.

The two pictures, measured against each other #

The differences listed above can be given numbers, and the numbers are more interesting than the list.

At a nucleus the molecular orbital’s amplitude is 0.3596 against the atomic function’s 0.5642. The bonding combination is one function spread over two centres, so its peak is lower by very nearly the factor $1/\sqrt{2+2S}$ that normalises it — and the level enclosing ninety per cent of it is correspondingly lower, 0.0359 against 0.0394, a ratio of 0.912.

At the midpoint the molecular surface is 2.690 bohr wide — measured as the distance from the axis at which the density falls to the level — while the pair of atomic spheres has a waist of 2.468 there. The molecular contour is fatter across the bond by nine per cent, which is the bonding density showing up as a shape.

Along the axis the molecular surface reaches 3.304 bohr from the centre and the atomic pair reaches 3.663. So the pair of spheres is longer and narrower and the molecular surface is shorter and fatter, at equal claimed fractions.

That pattern is worth stating as a sentence, because it is the shape of every bonding orbital and not a quirk of this one: the molecular contour takes material from the outer ends and puts it in the middle. Which is what a bond does, and is precisely the difference the two-sphere picture erases — the two-sphere picture puts the material back at the ends and digs a waist where the bonding is.

Why the levels cannot be the same #

There is a temptation, when two contours are being compared, to draw them at the same level so that the comparison is fair. It is the wrong instinct here and the reason is worth separating from the arithmetic.

A contour level is a value of the wavefunction, and a wavefunction’s values depend on how it is normalised — which is to say, on how much space it is spread over. Two functions with the same shape and different extents have different values everywhere, and the level enclosing a given fraction of each is different. That is why this collection states the fraction rather than the level in every caption: the fraction is comparable and the level is not.

For the case here the difference is only nine per cent, which sounds small and is not: it is the difference between a picture whose caption is true and one whose caption is out by nearly two percentage points in the quantity it names. And in the other direction — drawing both at one level — the discrepancy is what the plate at a single contour value already measured, and it runs to a factor of twenty across the orbitals of one atom.

Where the picture stops being one object #

The interesting behaviour appears when the two nuclei are pulled apart.

On the axis between the two nuclei the density has a minimum at the midpoint. A contour drawn at a level above that minimum is two separate surfaces; one drawn below it is a single surface with both nuclei inside. So there is a separation at which the picture divides, and the separation is where the midpoint density crosses the level.

The level is set by the fraction. So the separation is set by the fraction.

Half the density divides at 4.00 bohr, ninety per cent at 6.76 and ninety-nine per cent at 9.82 — a factor of 2.45 between the extremes, on one orbital, decided entirely by a number the illustrator chose. In ångström those are 2.12, 3.58 and 5.19. A reader shown a picture of a bonding orbital as one object at 3 Å and as two objects at 4 Å has been shown a fact about the contour level and no fact whatever about the molecule.

The honest statement of what the picture divides at is therefore: the contour enclosing f of the density divides at the separation where the midpoint density equals its level, which is a sentence about a drawing. Nothing in the orbital, its energy, its overlap or its bond order does anything at that separation.

What is between the nuclei #

The other quantity a picture of a bond is supposed to show is the density piled up between the nuclei, and it can be measured directly rather than inferred from a shape.

At two bohr the bonding combination holds 0.5284 of its density between the nuclei and the antibonding one 0.2580 — a factor of 2.05, which is the accumulation every account of bonding describes. The antibonding orbital has a nodal plane exactly at the midpoint, so it is not merely holding less there, it is holding exactly none at the plane.

The convergence at large separation is worth reading carefully, because it is the opposite of what the picture suggests. At eight bohr the two combinations hold 0.5043 and 0.4956 — practically the same. That is not the bonding disappearing; it is the region between the nuclei becoming so empty that the difference between having a node in it and not having one stops mattering. The overlap at that separation is 0.0102, and overlap is what decides.

Why the wrong picture is drawn, and when it is right #

The two-sphere picture is not carelessness. It has a job, and it does it well.

Its job is to show how the orbital was built: two atomic functions, in phase, on two centres. That is a statement about the construction, and the construction is the whole content of the LCAO method — which combination of which functions, with what signs. A drawing that shows two spheres in phase communicates that immediately, and a drawing of the true contour communicates none of it, because the true contour has no seam in it to show where one atom ended and the other began.

So the two pictures answer different questions, and the trouble is only that they are drawn identically and captioned identically. A figure showing two overlapping spheres labelled the ninety per cent contour of the σ orbital is making a claim about a surface, and the surface it draws is not that one.

There is also a case where the two pictures very nearly agree, and it is the case the picture is usually first met in: a long bond. At six bohr the true contour is still a single surface but its waist is deep, and by eight bohr it has divided into two pieces that are nearly the atomic pair. So the two-sphere picture becomes accurate exactly where the bonding has become negligible — which is a poor recommendation for it as a picture of bonding.

The one level in this whole anchor that nobody chose #

Every contour question so far has been about a level somebody picked and failed to state. The two-centre case supplies the exception, and it is worth separating out, because it is the only contour here whose value is decided by the function rather than by a caption.

Along the axis between the nuclei the density falls, reaches a minimum and rises again. In the full three-dimensional picture that minimum is not a minimum: it is a saddle, lowest along the axis and highest across it, and its existence is what makes the topology question askable at all. Call the density there $\rho_b$.

Every contour drawn above $\rho_b$ is two separate surfaces, one round each nucleus. Every contour drawn below it is a single surface with both nuclei inside. The transition is not gradual and it is not a matter of taste: at exactly $\rho_b$ the two pieces touch at one point, and one side of that value gives a bond drawn as two atoms and the other gives it drawn as one object.

So the essay’s finding — that whether the ninety-per-cent surface is one object or two depends on the separation, changing over somewhere between 2.1 and 5.2 ångström — is the statement that the chosen level crosses $\rho_b$ somewhere in that range. The arbitrary number is the ninety; the thing it is being compared against is not arbitrary at all.

That quantity has a life outside this collection. The saddle between two bonded nuclei is the bond critical point of the topological analysis of molecular charge densities, its density is used as a measure of bond strength, and it is obtainable from an experimental charge density as well as a computed one — which makes it one of the very few numbers in this whole subject that is both a property of the density alone and a thing an X-ray measurement returns. Its values are informative: a few tenths of an electron per cubic bohr for an ordinary covalent bond, and a few hundredths for a hydrogen bond, which is a factor of ten separating two things usually described with the same word.

Which gives contour levels a rule they have not had. Every other level is a convention and has to be stated. This one is a measurement, and a picture drawn at it is drawn at the boundary between the two topologies rather than at some fraction of the way through the density.

It is not a replacement for the enclosed-fraction convention, because it exists only where there is a saddle — a single atom has none, and neither does a non-bonded pair. What it is, is the one place in this collection where the question what level should the contour be at has an answer that does not begin with somebody’s preference.

What this model is, and is not #

Three things have to be said plainly, because the calculation above is deliberately the simplest one that makes the point.

This is the LCAO orbital, not the exact one. The one-electron diatomic has an exact solution — it separates in confocal elliptic coordinates — and the combination used here is the two-term approximation to it. The exact orbital is more contracted along the axis, so its contour is slightly different and every conclusion above is the same. The approximation is stated rather than hidden, and the reason for using it is that it is the function the picture being criticised is a picture of.

There is one electron. Every statement here is about a one-electron orbital, and a real bond has two electrons in it that repel. The smallest calculation with repulsion in it uses a different model entirely; nothing here has repulsion.

The drawings are sections. What is drawn is the intersection of the surface with a plane through both nuclei, which is not the surface — a curve enclosing a tenth of the density on the page is a surface enclosing rather less in space. The enclosed fractions quoted are always the three-dimensional ones, computed for the surface and not for the curve.

What that difference amounts to has been measured: a section is a fair picture of a surface’s shape and a misleading picture of its content. Every fraction quoted in this essay is a property of the surface rather than of any plane through it.

The energies that go with these shapes belong to another essay and are worth naming rather than redrawing: two atomic levels, the two combinations they form, and a splitting proportional to the computed overlap. Every contour in this essay is a picture of the lower of those two combinations, at one separation and one stated fraction.

What was checked #

The orbital is normalised, integrated over the whole of space by the same rule the enclosure uses, at a level of zero. That is the check that the closed-form overlap in the denominator is the right one — a wrong normalisation would leave every enclosed fraction wrong by the same factor and nothing would look odd.

The solved contour encloses what it claims, to five parts in a thousand, and encloses it again under a rule with more points in both directions, which is the resolution check that a bisection against a single rule cannot supply.

A contour three times too high encloses visibly less, which is the requirement that the test can fail. Without it, an enclosure routine that returned the target fraction whatever it was handed would pass every check above.

The bonding combination holds more density between the nuclei than the antibonding one, checked as an inequality rather than as a number, because the number depends on the separation and the inequality does not.

Still open: the π bond #

Two directions, and the first is close.

A π bond is not axially symmetric, so its contour is a genuinely three-dimensional problem: the density depends on all three coordinates and the region inside a level is not a disc at any fixed z. That needs a different method — a marching scheme, or a root-find in a rotated frame — and it would be the first contour here that needs a grid.

The second is the same question asked of a real molecule rather than of a model. Every contour here is drawn from an analytic function. A contour drawn from a calculation is drawn from values on a grid, and the enclosed fraction of a grid-interpolated function is not the enclosed fraction of the function — which is a source of disagreement between programs that all claim the same percentage, and is exactly the kind of thing the case for stating a contour’s level was written to complain about.