Molecular Geometry & Bonding
what the shape decides
What is taught wrongly

Orbitals are not where the electron is

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.
11 min read 7 figures One electron onlyTaught confidently, and wrong

The hydrogen atom has exact orbitals. Helium does not, and neither does anything else.

That sentence is true, is not controversial among people who work on this, and is almost never said in a first course — where orbitals are introduced, filled with electrons and used for the rest of the subject as though they were places.

Orbitals at the 90 per cent contourSeveral orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density.one scale across the plate1s0 radial · 0 angular2s1 radial · 0 angular2pz0 radial · 1 angulareach level solved for separately90% · one electron
Fig. 1 Three hydrogenic orbitals at one enclosed fraction and one scale. These are exact solutions for one electron and one nucleus. For any atom with more than one electron they are not solutions of anything, and their role changes from description to basis.

What goes wrong with two electrons

The Schrödinger equation for helium contains a term for the repulsion between the two electrons, 1/r121/r_{12}, that depends on both positions at once.

That term is why the equation does not separate. With one electron the problem factors into radial and angular parts and is solved exactly. With two, the motion of each depends on where the other is, and there is no exact analytic solution — not because nobody has been clever enough, but in the same way that the three-body problem in gravitation has none.

So a two-electron wavefunction is a function of six coordinates that does not factor into two functions of three. Writing it as a product of orbitals is an approximation, and the name for it is the orbital approximation.

What the approximation assumes

It assumes each electron moves in the average field of the others rather than responding to their instantaneous positions.

That is the Hartree–Fock picture, and it is remarkably good — it recovers about 99 per cent of the total energy of a small atom. The remaining one per cent is called the correlation energy, and the name is exactly right: it is the part arising from electrons avoiding one another in ways an average field cannot capture.

One per cent sounds negligible and is not. Chemical bond energies are of that order, so the error the orbital approximation makes is comparable to the quantity chemistry cares about. Essentially all of modern computational chemistry is machinery for recovering it.

Why the picture works anyway

Given that, it is fair to ask why orbitals are so useful.

The shapes are robust. A carbon 2p orbital in a real calculation is contracted relative to hydrogen’s and has the same nodal structure, the same symmetry, and the same qualitative shape. Node counts are unchanged by any reasonable approximation.

Symmetry survives exactly. Whatever the radial details, the angular behaviour is fixed by the symmetry of the problem, and every symmetry-based conclusion on this site is therefore exact rather than approximate.

The ordering survives. Which orbitals lie below which, and therefore the structure of the periodic table, comes out of the approximation correctly.

So the picture gets the qualitative structure right and the quantitative details approximately, which is the ideal position for a teaching model and a dangerous one for a claim about reality.

What the picture does not license

Four conclusions people draw that the approximation does not support.

“This electron is in the 2p orbital.” Electrons are indistinguishable. A many-electron wavefunction is antisymmetric under exchanging any two of them, so no electron has an orbital of its own. The configuration 1s22s22p21s^2 2s^2 2p^2 describes a determinant built from those orbitals, not an assignment of electrons to boxes.

“The orbital energies add up to the total energy.” They do not, because summing them counts each electron–electron repulsion twice.

“An orbital energy is an ionisation energy.” Approximately, by Koopmans’ theorem, and it neglects the relaxation of the remaining electrons. Good enough for the methane argument and not exact.

“The orbital shapes are observable.” Various experiments have been reported as imaging orbitals, and what they measure is a density or a momentum distribution, both of which are observables that a chosen orbital contributes to. The orbital itself is basis-dependent and therefore not an observable.

The determinant, and why indistinguishability matters

Worth setting out, because it is the technical reason the box picture fails.

A many-electron wavefunction must change sign when any two electrons are exchanged — the Pauli principle in its general form. The simplest function with that property built from orbitals is a Slater determinant, a determinant whose rows are orbitals and whose columns are electrons.

Expanding it gives a sum over every way of assigning electrons to orbitals, with alternating signs. Every electron appears in every orbital. Asking which electron is in which is asking a question the mathematics has deliberately made meaningless.

That is a strong statement and it is not a subtlety. It is the reason the exclusion principle works, and it is incompatible with the picture of electrons occupying orbitals as places.

The one place it is visible

There is a case where the approximation’s failure shows up in a measurement that an introductory course does discuss, and it is worth having.

Helium’s first ionisation energy is 24.6 electron volts. Its second is 54.4. If the two electrons were independent, both in the same 1s orbital, the two energies would be equal — they are not, by more than a factor of two, and the reason is that removing the first electron changes the field the second one sees.

That is the orbital approximation failing in the simplest possible system, and failing by an amount nobody could call negligible. The two-electron problem cannot be reduced to two one-electron problems, and helium is where that becomes arithmetic rather than philosophy.

The 1s orbitalThe 1s 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.1sencloses 90% of the densitycontour at |ψ| = 3.94e-20 radial nodes0 angular nodescontour solved for by integrating the density1 shell · one electron
Fig. 2 The orbital both helium electrons are said to occupy. It is an exact solution for one electron and a hydrogen nucleus, and helium’s electrons are in neither of those situations — the nuclear charge is different and each electron is screened by the other.

The same point applies wherever an electron configuration is written down. The notation 1s22s22p61s^2 2s^2 2p^6 names a determinant built from chosen orbitals; it is a label for an approximate state and not a description of six electrons sitting in places.

What is real

Three things, against a long list of things that are not.

The total electron density is an observable. It is what X-ray diffraction measures, it is the same whichever basis is used, and it is a well-defined function of position.

The total energy is an observable.

The spectrum — the set of transition energies and intensities — is an observable.

Orbitals, orbital energies, hybridisation, bond order, atomic charges and resonance structures are none of them observables. They are constructs within a description, they are useful, and they are basis-dependent. The same point recurs throughout this subject in different clothes.

How to hold the picture

Not to abandon it. The suggestion is narrower.

Use orbitals for what they are good at: symmetry, nodal structure, qualitative ordering, and the enormous organising power they give to the periodic table and to bonding.

Be careful when a conclusion depends on orbitals being real rather than useful. That is where errors live, and it is a small list — mostly claims that a measurement will find something an orbital picture suggests.

And when an experiment appears to contradict an orbital picture, as methane’s spectrum does, the first question is which basis the experiment couples to. Usually the contradiction dissolves.

What this site does

The standing disclosure, stated here because this is the essay it belongs in.

Every wavefunction drawn on this site is hydrogenic: an exact one-electron solution. The contour levels, node counts, overlaps and shapes are all computed from those functions, and they are exactly right for hydrogen-like systems and qualitatively right elsewhere.

Where an essay discusses a real many-electron molecule, the geometry and symmetry are the site’s own computations and any electronic-structure numbers are quoted from the literature. No figure here is a many-electron calculation, and the caption line on every orbital figure says so.

Where it came from

Hartree proposed the self-consistent field method in 1927, the year after Schrödinger’s equation. Fock and Slater added antisymmetry in 1930, giving what is now called Hartree–Fock.

The correlation problem was recognised immediately and has occupied quantum chemistry ever since. Configuration interaction, coupled cluster, density functional theory and the rest are all ways of doing better than a single determinant, and the field’s Nobel Prize — Pople and Kohn in 1998 — was for making that practical.

So the limitation is not a recent discovery or a philosophical quibble. It is the central technical problem of the field, and it is odd that the introductory presentation of orbitals rarely mentions that the objects being introduced are exactly soluble for one atom.

What survives the approximation

Since the essay has been largely a list of what the orbital picture does not license, it is worth being explicit about the substantial part that it does.

Symmetry classifications are exact. Which orbitals may mix, which transitions are allowed, which molecules may be polar — all follow from the symmetry of the exact many-electron Hamiltonian, not from the orbital approximation. They are not approximate at all.

Node counts are robust. No reasonable approximation changes them, and they are fixed by the angular momentum and the number of radial functions.

The ordering, and therefore the periodic table. Which orbitals fill in which order comes out of the approximation correctly, and the structure of the table is its most successful prediction.

Orbitals at the 90 per cent contourSeveral orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density.one scale across the plate2s1 radial · 0 angular2pz0 radial · 1 angular3dz20 radial · 2 angulareach level solved for separately90% · one electron
Fig. 3 Three orbitals whose relative energies and shapes the approximation gets right, and whose radial details it gets approximately. Everything a first course does with these — filling order, bonding, spectra — rests on the robust part.
The radial function of 2sThe 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.node 2.00most probable radius 5.24 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function1 radial node · one electron
Fig. 4 The feature that explains the ordering: the inner peak of the 2s distribution, which penetrates the screening of the 1s shell. That is a many-electron argument made with one-electron functions, and it works.

So the picture is not to be abandoned. It is to be held as a very good approximation whose qualitative content is exact and whose quantitative content is not — which is a perfectly ordinary situation for a model, and unusual only in how rarely it is said out loud.

The functions themselves

Since the essay is about the status of these objects, it is worth looking at them once more with that status in mind.

The 2s orbitalThe 2s 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.2sencloses 90% of the densitycontour at |ψ| = 7.40e-31 radial node0 angular nodescontour solved for by integrating the density2 shells · one electron
Fig. 5 A 2s orbital at its ninety-per-cent contour: two nested shells, exactly soluble for hydrogen, and a basis function for everything else. The nodal structure survives every approximation; the radial scale does not.
The 3dz2 orbitalThe 3dz2 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.3dz2encloses 90% of the densitycontour at |ψ| = 3.60e-30 radial nodes2 angular nodescontour solved for by integrating the density1 shell · one electron
Fig. 6 A 3d orbital. The shape and the two angular nodes are fixed by symmetry and are exact in any atom; the size is hydrogenic and is not what a transition metal’s 3d orbital looks like.
Orbitals at the 90 per cent contourSeveral orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density.one scale across the plate1s0 radial · 0 angular2s1 radial · 0 angular2pz0 radial · 1 angular3dz20 radial · 2 angulareach level solved for separately90% · one electron
Fig. 7 Four of them at one scale. Everything qualitative in this plate — the relative sizes, the shapes, the node counts — carries over to real atoms. Everything quantitative is for hydrogen.

That division is the practical form of this essay’s argument, and it is worth carrying: the shapes and the counts are robust, and the numbers are not. A conclusion resting on the first kind is safe; one resting on the second needs the real functions.

Where the ladder goes next

The construction itself is what an orbital is.

The sharpest experimental illustration is hybridisation does not explain.

And the related confusion about position is where the electron is.

What the pictures here cannot show. Every orbital figure on this site is a one-electron function. A many-electron atom’s density is not any of them, and no figure here draws one — which is the point of this essay rather than a limitation of it.

Where this goes next

2pzencloses 90% of the densitycontour at |ψ| = 9.48e-30 radial nodes1 angular nodecontour solved for by integrating the density1 shell · one electron 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. HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms Hybridisation does not explain Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find. most probable radius 1.00 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function0 radial nodes · one electron 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.