Molecular Geometry & Bonding
what the shape decides
Concept

Enclosed probability — where it appears

The fraction of an orbital's density that lies inside a drawn surface, found by integrating. Stating it is what makes an orbital picture a claim rather than an illustration, and it is what almost no published picture states.

Named by 12 essays across one field — each of them below, with the objects they name alongside it.

The 2pz orbital. The 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

What an orbital is

Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.

orbitals · Orbital Choosing a contour for 3s. The fraction of the density enclosed by a contour, against the contour's level. Picking a level is picking a point on this curve, and the usual practice of picking one that looks right is picking a point without knowing which. Contours drawn: 3s at 50% of its density, |ψ| = 6.24e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3s at 99% of its density, |ψ| = 7.15e-4.

Say what it encloses

An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly different sizes with the same caption, and both be printed in good faith.

orbitals · Contour The radial function of 1s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

Where the electron is

The wavefunction is largest at the nucleus, the electron is most likely to be found a bohr out, and the ninety-per-cent contour is at 2.66. Three numbers, all correct, all answering different questions.

orbitals · Orbital Four measures of size for 6 orbitals. The most probable radius, the mean radius, the root-mean-square radius and the radius of the sphere holding ninety per cent of the density, for 1s, 2s, 2pz, 3s, 3dz2, 4s. All four are computed from the same radial function, all four are correct, and they are not the same number.

How big is an orbital

Four measures of size, all computed from the same radial function, all correct, and spanning a factor of two and a half for a 1s orbital. The one that governs chemistry is a fifth, and it is not a measure of size at all.

orbitals · Contour 1s, 2s, 2pz, 3s, 3dz2, 4s at one level and at one fraction. The orbitals 1s, 2s, 2pz, 3s, 3dz2, 4s, each with the contour level that encloses 90 per cent of its own density, and beside it what each one encloses when they are instead all drawn at 1s's level. The first column is a set of pictures that can be compared; the second is what a plate drawn at a single contour value actually shows. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

One level is not one comparison

A plate of orbitals drawn at a single contour value looks like a comparison and is not one. At the level that encloses ninety per cent of a 1s, a 2s encloses four per cent, a 3s under one, and a 3d has no surface at all — its wavefunction never reaches that value anywhere in space.

orbitals · Contour One isovalue, many fractions — and one fraction, many isovalues. What each of 4 conventional isovalues encloses, for 7 orbitals of hydrogen, and in the last column the level each one needs to enclose 90 per cent. At 0.02 atomic units the fractions run from 0.4 to 96.2 per cent, and the levels in the last column differ by a factor of 32. A plate of orbitals drawn at one value is not a comparison of sizes. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3pz at 90% of its density, |ψ| = 3.03e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

The isovalue nobody chose

Every program that draws an orbital asks for a number, and almost every user accepts the default. At 0.02 atomic units that default encloses 96 per cent of hydrogen's 1s, 52 per cent of its 2s and four tenths of one per cent of its 4s — three pictures drawn to one rule, meaning three completely different things.

orbitals · Contour The surface-honest curve and the page-honest curve. The 2pz orbital in its xz plane, drawn twice. The outer curve is the contour whose SURFACE encloses 90.0 per cent of the density in space — the claim every orbital figure here makes — and in this plane it encloses 96.7 per cent. The inner curve is the one that encloses 90 per cent in the plane, which is what a reader looking at a flat picture would take the caption to mean. The two levels differ by a factor of 1.87 and the curves differ visibly; the caption does not. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

A slice is not the surface

The page is flat, so every printed orbital is a slice through a contour rather than the contour itself — and a curve that leaves a tenth of the density outside it in space leaves about a thirtieth outside it on the page. Both claims are true of the same picture and only one of them is ever stated.

orbitals · Contour A σ contour at 90 per cent, and the two atomic ones. The section through both nuclei of the surface enclosing 90 per cent of the bonding orbital's density at 2 bohr, with circles marking where two atomic contours of the same stated fraction would be. The two pictures are different shapes and enclose different amounts, and the atomic pair encloses 91.70 per cent of the molecular orbital's density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2.

A bond is not two atoms overlapping

The surface enclosing ninety per cent of a σ orbital's density is one closed surface with both nuclei inside it, at a level of 0.0359. The two atomic surfaces usually drawn instead sit at 0.0394, are a different shape, and enclose 91.70 per cent of the same orbital — and whether the molecular one is one object or two is decided by the fraction the caption claims, anywhere between 2.1 and 5.2 ångström.

orbitals · Contour The part of the bond that is outside the picture. The share of the overlap integral between two 1s orbitals that lies outside both of their 90 per cent contours, against how far apart the atoms are. At a bond length it is 8.3 per cent; by 7 bohr it is 53.4. The two drawn surfaces stop touching at 5.32 bohr, where the overlap is still 0.08 — so the picture separates well before the interaction does.

The tenth that is not drawn

A ninety per cent contour of a hydrogen 1s orbital is a sphere of radius 2.661 bohr, and two of them stop touching at 5.322 bohr — where the overlap between the two orbitals is still 0.0768 and rising in importance. At a three-ångström contact, 36.9 per cent of the overlap integral lies outside both drawn surfaces, and holding nine tenths of it inside the picture would take a contour enclosing 97.28 per cent.

orbitals · Contour Where two He atoms stop, with no contact distance put in. The repulsion between two He atoms, computed from the overlap of their filled valence orbitals — four electrons in a bonding and an antibonding pair, of which the antibonding one rises further — against London's dispersion attraction from the measured polarisability and ionisation energy. The minimum is at 3.14 ångström where the tabulated van der Waals contact is 2.8, and nothing anywhere in the calculation is a length.

The radius that was tabulated

A van der Waals radius is a fitted number that every structural argument in chemistry uses. Computing it instead — a repulsion taken from computed overlap integrals, an attraction taken from two measured scalars, and no length anywhere — puts helium's contact at 3.140 ångström against a tabulated 2.80, neon's at 3.226 against 3.08 and argon's at 3.996 against 3.76. And the surface two atoms actually stop at encloses 99.4 per cent of the density, not ninety.

orbitals · Contour The fraction a table encloses is not one number. For each ion, the fraction of its own electron density that lies inside its tabulated radius. Along each isoelectronic series the answer runs over eight percentage points, where three neutral atoms at their contact distances spanned three tenths of one. And it peaks at the neutral rather than falling through it, because a van der Waals radius is fitted to the distance between two atoms that are not bonded and an ionic radius is one term of a sum fitted to the distance between two that are.

The surface a table draws

Three noble gases stop at a surface enclosing between 99.38 and 99.75 per cent of their density — a near-constant, and an argument that a contour is a real boundary. Charge the atoms and it collapses. Across ten electrons the tabulated radius encloses anything from 94.4 to 99.98 per cent, it peaks at the neutral rather than trending through it, and radii built at a fixed enclosure do not add up to a single measured separation.

orbitals · Contour A ninety per cent surface with a neighbour beside it. The contour enclosing 90 per cent of a one-electron ion's density at an effective charge of 1.6, drawn with no field as a circle and in a field of 0.05 atomic units as the closed curve. The surface moves out by 47.3 millibohr on the side the field pulls the density towards and in by the same amount on the far side — 2.84 per cent of its own radius. What the sphere encloses does not change to first order; only where the surface is does.

The surface a neighbour moves

An ion in a crystal sits in the field of the ion next to it, and that field moves its contour. The displacement has a closed form, it is checked against the polarisability it implies, and it turns out to be almost perfectly anti-correlated with the discrepancy it was proposed to explain — the pairs that need the most correction get the least.

orbitals · Contour

Named alongside it

The objects these essays reach for when they reach for this one.

Contour levelProbability densityIsosurfaceConventionRadial distributionModel limitOne-electron modelsWavefunctionEffective nuclear chargeIntermolecular forceNodeQuadrature

All concepts