Orbitals are not where the electron is

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.

What goes wrong with two electrons #

The Schrödinger equation for helium contains a term for the repulsion between the two electrons, $1/r_{12}$, 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 a set of methods 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 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 $1s^2 2s^2 2p^2$ 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.

What survives when no electron has an orbital #

If a determinant makes the question which electron is in which orbital meaningless, it is fair to ask what does remain well defined. The answer is sharp, it is not the orbitals, and it is the cleanest available statement of what the approximation is approximating.

Integrate the many-electron wavefunction against itself over every coordinate but one, keeping phase rather than only modulus, and what is left is the one-particle density matrix. That object is basis-independent in the way the total density is: it does not depend on which orbitals were used to build the state it came from. Diagonalise it. Its eigenvectors are the natural orbitals and its eigenvalues are their occupation numbers.

For a single determinant those occupation numbers are exactly two and zero. That is what a determinant is, restated in a language that never mentions a basis, and it is why the label $1s^2 2s^2$ is a faithful summary of a Hartree–Fock state and a misleading summary of anything else.

For an exact wavefunction they are neither. Helium’s ground state has a leading natural occupation of about $1.98$ rather than $2$, with the missing two-hundredth spread across an infinite tail of further natural orbitals, none of which any first course names. Nothing was assumed to obtain that number; it is a property of the exact two-electron function, read off an operator that exists whether or not anybody has chosen a basis.

So the departure of the occupation numbers from integers is the correlation described above, measured in a form that survives every change of basis. It also puts the one-per-cent figure on a footing that does not depend on a total energy: helium is about ninety-nine per cent one configuration, which is why the picture works so well, and about one per cent not, which is why what it leaves out is the size of a bond.

The number is not a constant of nature, and watching it move is the useful part. Hydrogen molecule occupations near the equilibrium bond length are roughly $1.96$ and $0.04$; pull the atoms apart and they go to $1$ and $1$ — two half-filled natural orbitals and no dominant configuration at all. That is where molecular orbital theory dissociates written as a pair of eigenvalues, and it shows the orbital approximation degrading continuously along a coordinate rather than holding until it snaps.

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 same point applies wherever an electron configuration is written down. The notation $1s^2 2s^2 2p^6$ 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 the orbital pictures here are #

It is worth being exact about what an orbital picture here is, because this is the essay where that matters.

Every wavefunction drawn here 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 a real many-electron molecule is discussed, the geometry and symmetry are computed directly 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.

The one thing that survives being a bad approximation #

There is a class of statement in this subject that does not depend on the orbital approximation at all, and separating it out is the most useful thing this essay can do — because it is the difference between an argument that a better calculation could overturn and one that nothing could.

Symmetry conclusions follow from the exact Hamiltonian. A molecule’s point group is a symmetry of the full many-electron problem, not of a one-electron caricature of it. So when the group forbids a dipole moment, the moment is zero for the real molecule with all its electrons and all their correlation — not zero within an approximation. When an overlap vanishes by symmetry, the corresponding matrix element of the true Hamiltonian vanishes too.

That is why symmetry arguments can be stated without hedging while everything else carries a caution. They are not better approximations. They are not approximations.

The distinction has a practical edge. A calculation that predicts a small dipole moment for benzene has a bug, and no amount of improving the basis set will fix it, because the correct answer is zero for a reason that has nothing to do with basis sets. A calculation that gets a bond length wrong by two per cent may simply need a better method. Knowing which kind of quantity is which tells a reader whether a disagreement is interesting.

Two more classes belong in the same box. Degeneracies required by symmetry are exact: methane’s three t₂ orbitals are exactly degenerate, and a calculation that splits them has broken the symmetry somewhere. And counts — how many vibrational modes a molecule has, how many distinct orbital energies its ligand orbitals can produce, how many bands a spectrum can show — are integers fixed by the group, and an integer is not something a better method improves.

What remains approximate is everything with a magnitude in it: energies, lengths, angles, intensities, charges. That is a large category and it is the one the orbital picture serves, and the caution at the foot of every orbital figure here is aimed squarely at it.

What it costs to keep saying so #

The disclosure above is repeated on every orbital figure here — a short line in the caption strip reading one electron — and it is worth explaining why a caption is spent on it rather than a single note somewhere central.

A figure is seen alone. It is scrolled past, screenshotted, linked to and quoted out of the page it was drawn for, and a caveat that lives three sections up travels with none of those. The cost of carrying it is a dozen characters in a strip that already exists; the cost of not carrying it is a picture that implies a many-electron calculation was performed.

That principle has a second half which is less obvious and matters more. A single picture can only vouch for what it shows. A surface drawn at one contour level cannot support a claim that compares two levels; one frame of a sequence cannot support a claim about the whole range, because a frame is seen alone. Claims of that shape need a comparison that sees every frame at once, and they belong in the argument rather than under any one figure.

So the division of labour is: a caption states what its own picture shows, and the argument states what the pictures show together. Neither can do the other’s work, and a caption is the only part of a figure that survives being separated from everything around it.

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.

The radial picture is the one that shows why, because the argument is about where the density is rather than about what shape it has.

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 d functions make the same division sharper still, because there the shape is fixed by symmetry and the size is fixed by nothing the atom in question supplies.

Put one of each on a single scale and the whole compromise is on one page: four shapes that are exact statements about angular momentum, drawn at four sizes that are statements about hydrogen and about nothing else.

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 to read on #

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.

The position this leaves a reader in is more comfortable than it sounds. Orbitals are an excellent approximation with a known error, they get the qualitative structure of the subject exactly right, and their limits are stated rather than lurking. What is not available is the thing the pictures invite — treating them as places electrons occupy — and giving that up costs nothing that was ever earned.

What the pictures here cannot show. Every orbital figure here 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.