A property of the model, mistaken for one of the molecule

“Each electron in an atom occupies one of the atom's orbitals.”

A many-electron atom has no exact orbitals. Every orbital drawn here is a hydrogen-like one-electron solution, and the difference between that picture and the truth is the correlation energy — a quantity with a name because it is what the orbital picture leaves out.

Tested in Orbitals are not where the electron is · the series on approximation

“The gap keeps falling as a conjugated chain lengthens, so a long enough polyene conducts.”

Hückel's gap does fall towards zero as the chain grows, and the observed gap does not. Bond alternation — the Peierls distortion — opens a gap that a matrix of ones and zeroes cannot see, because the matrix has no geometry in it.

Tested in Conjugation, and its limits · the series on delocalisation

“Whether the electrons are really in hybrid orbitals is a question about the molecule.”

The hybrid matrix is orthogonal, so the change is unitary and no observable moves. The question is which coordinate system to write the same state in, and nature does not prefer one.

Tested in Hybrids are a basis · the series on hybrids

“The energy of the first d–d absorption is Δ, the ligand field splitting.”

That identification is exact only for a d¹ ion and its d⁹ hole counterpart, where one electron moves between two one-electron levels. For every other filling the initial and final states differ in how the remaining electrons repel each other, and the band energy contains interelectronic repulsion terms that a one-electron splitting does not include. A one-electron model computes the splitting and cannot compute those terms.

Tested in Where a d–d band falls · the series on ligand field

“Aromatic rings are the ones that obey Hückel's 4n+2 rule.”

Every ring from three to ten is filled with its own electrons and asked whether the highest occupied shell came out full. The rings that close are exactly the 4n+2 counts, so the rule is an output of the filling and the calculation that produces it does not know the phrase.

Tested in Aromaticity as a computed shell closure · the series on aromaticity

“px, py and pz are the p orbitals.”

They are real combinations of the complex eigenfunctions, chosen because they point along axes. The complex set is what an applied magnetic field distinguishes, and both sets span the same subspace.

Tested in Complex harmonics against real ones · the series on orbital

“Effective nuclear charge is a property of an atom.”

It is a property of an atom AND a shell AND a set of rules. Iron's 3d electron feels 6.25 and its 4s electron feels 3.75 at the same instant in the same atom, and both numbers come from a fit published in 1930 rather than from a measurement.

Tested in What an electron actually feels · the series on orbital

“The structure obtained is the molecule's structure.”

What comes out is the r₀ structure, the one reproducing ground-state moments, and a ground-state moment is an average over zero-point motion. Averaging 1/r² is not the same operation as taking 1/⟨r⟩², so the recovered length is systematically not the equilibrium one — the inversion reproduces a structure it is given to twelve figures, and disagrees with the equilibrium values by thousandths of an ångström when given real measurements.

Tested in A bond length out of a spectrum · the series on rotation

“Orbital energies are physical quantities, since they can be measured.”

An orbital energy belongs to a chosen set of orbitals, and the set is not unique: the canonical and localised descriptions of methane give identical electron densities to the last bit a double can hold, and completely different orbital energies. No measurement can distinguish descriptions that agree about the density, so no measurement can be a measurement of an orbital energy.

Tested in What a photoelectron spectrum measures · the series on photoelectron

“A material with a partly filled band is a metal.”

It is a metal in a one-electron model, and that model omits the cost of putting two electrons on the same atom. Where that cost exceeds the band width, the electrons localise one per site and the material is an insulator with an unchanged half-filled band. Nickel oxide, cobalt oxide and the undoped copper-oxide superconductors are all in that class, and band theory predicts every one of them to be a metal.

Tested in A half-filled band is not always a metal · the series on metal

“Square cyclobutadiene has two electrons in a degenerate pair, so by Hund's rule it is a triplet.”

Hund's rule is a statement about electron repulsion and Hückel theory contains none, which is visible in the exact calculation: at U = 0 the singlet and the triplet of a four-site ring are exactly degenerate, so the one-electron model has no preference to express. Switch the repulsion on and the singlet drops below the triplet for every U above zero — 0.0037 t at U = 0.25, rising to 0.33 at U = 8 — which is Lieb's theorem for a half-filled bipartite lattice.

Tested in The smallest many-electron calculation · the series on correlation

“The eighteen-electron rule tells you how many metal–metal bonds a cluster has.”

For thirteen carbonyls the leftover count (18n − total)/2 matches the crystallographic bond count exactly for every cluster of one to five metals — 1 for Mn₂(CO)₁₀, 3 for Fe₃(CO)₁₂, 6 for Co₄(CO)₁₂, 9 for Os₅(CO)₁₆. Both six-metal clusters tested give 11 where the octahedron has 12.

Tested in The bonds are what is left over · the series on electron count

“A molecule has a point group.”

The point group is the set of operations available without crossing a barrier. Ethane's is D3d, of order 12; the group of operations actually available to it at room temperature has 36 elements, because a single methyl group may turn independently of the other and no rigid motion does that. Ammonia's doubles from 6 to 12 when the inversion barrier is crossed, which it is 24 billion times a second.

Tested in The group of a molecule that will not hold still · the series on point group

“A bulkier substituent goes equatorial because there is more room there.”

Given a longer bond and nothing else, the repulsion model puts the odd substituent axial at every length tested, by a margin growing from 0.003 at a ratio of 1.1 to 0.016 at 1.6. The reason is arithmetic: a site pushed further out interacts with everything less, so what is left to decide is the arrangement of the four sites behind it, and three equatorial with one axial is the lower-energy set of four — 3.853 against 3.906 in the same units.

Tested in The sites are not the same size · the series on VSEPR

“The trigonal bipyramid has bond angles of exactly 90 and 120 degrees.”

With one equatorial bond 1.4 times the others, the axial angle relaxes to 168.4° and the axial-to-equatorial angles to 87.0°. Any inequality of length bends them: a bond 0.8 times the others gives 176.8°. Exactly 180° requires exact equality, and phosphorus pentafluoride — where the axial bonds are 1.577 Å and the equatorial 1.534 — does not have it.

Tested in The sites are not the same size · the series on VSEPR

“The difference is a detail of how the picture was made.”

A volume integral weights the far tail by r² and an area integral weights it by r, so the shell of low-amplitude density that a three-dimensional surface leaves outside it is largely absent from any plane through the nucleus. The gap is 5.4 to 6.9 points for every orbital tested — nearly the same for a compact 1s and a diffuse 4s — which is what a result that comes from the measure rather than from the shape looks like.

Tested in A slice is not the surface · the series on contour

“A double bond is either a σ and a π or two bent bonds, and one of the two pictures is the correct one.”

The two are related by an orthogonal mixing of the occupied space, so the density matrix is unchanged at every mixing angle — one on the diagonal and zero off it, to 10⁻¹⁵, at 0°, 30°, 45° and 71.3°. A mixing that is not orthogonal changes it, which is what makes the invariance a statement rather than an identity.

Tested in Two bent bonds, or a σ and a π · the series on hybrids

“Mulliken and Löwdin populations are two approximations to the metal's charge.”

They are two points on a continuum with no distinguished member. Splitting each shared pair in the ratio λ to the ligand gives −3.040, −2.524, −2.009, −1.494 and −0.979 at λ = 0, ¼, ½, ¾ and 1; Mulliken is the middle one by construction and Löwdin is −2.369, which is not on the line at all.

Tested in An integer nobody measured · the series on electron count

“The cage has ten localised descriptions.”

It has at least fourteen. Ten is what a search with a hundred and eighty starts and a three-batch stopping rule found; four hundred and eighty starts find four more, three of which are reached by fewer than nine starts each. The count is a property of how long the search ran.

Tested in The answer a search is most likely to give · the series on multicentre

“A half-filled ring of 4m carries one pair of carriers at every temperature, however cold.”

Only if it is held undistorted. Allowed to relax against its own free energy at an elastic constant of 1.6, a ring of forty alternates by 0.12317, opens a gap of 0.49267, and carries 2 × 10⁻⁸ carriers where the held-flat ring carries 1.000000.

Tested in The carriers a distortion was hiding · the series on metal

“The orphan count is constant along the Bailar twist, because the twist keeps D3 throughout.”

The premise holds and the conclusion has no truth value here. At every angle strictly between the ends the finder assembles exactly six operations — E, 2C₃, 3C₂ — with a worst residual of 4.0 × 10⁻¹⁶ Å, which is D3 and nothing else. The nineteen character tables the calculation held did not include D3, so the ligand representation could not be reduced and no count was produced at any of those angles. The prediction is not wrong; it is unanswerable with those tables, and writing the missing one makes it answerable.

Tested in The group nobody wrote a table for · the series on hypervalency

“Dinitrogen is a weak π acceptor, so a model that gets cyanide and carbon monoxide right should place it just below zero.”

The model cannot place it at all. Dinitrogen is homonuclear, so its π and π* have equal coefficients by symmetry and its two overlaps differ by 1.21 rather than by the 3.7 and 4.4 of the polarised ligands. Its net parameter crosses zero at a metal level of −8.25 eV, inside the admissible range, so whether it is a donor or an acceptor is decided entirely by the one quantity nothing here measures.

Tested in The channel that points at the metal · the series on ligand field

“Every shell above the second has a coupled minimum, because its d level sits at one fifth of the s–p gap above p.”

One fifth is the screening model's small-defect limit. With measured defects the offset at the valence shell is 0.103 for lithium, 0.705 for sodium, 1.071 for potassium, 0.997 for rubidium and 0.953 for caesium, against thresholds of 0.25 to 0.37. Only lithium is below its threshold, and it is below it at every shell to the thirtieth; the other four are above theirs at every shell by factors of 2.6 to 7.

Tested in Four alkalis the model cannot hold · the series on representation

“A singlet ground state has no separation coupling in the low-coupling regime.”

True at two per cent for all nine singlet clusters. The octahedron and the fan of six acquire two at six per cent, six more singlets at eight per cent, and the ring of four at sixteen. The ground-spin account of where separations lie is an account of reasonably pure samples.

Tested in Purity renames the poles · the series on magnetism

“A one-dimensional model with the right mass should need one barrier for both isotopologues.”

No construction, well or ordering manages it. Fitted separately, ND₃ needs a barrier 4.70 per cent lower under the usual mass, 4.58 with the bonds held fixed, 3.42 and 4.25 in a well shaped to both lines, and between 4.58 and 4.64 under five orderings. A potential that absorbs the zero-point energy of the other vibrations is not mass-independent, so the gap is a statement about the reduction.

Tested in Deuterium cannot tell the masses apart · the series on inversion

“A spin density on overlapping atoms depends on how the overlap population is divided.”

In general it does. Here it does not: Mulliken and Löwdin partitions agree to every digit, because the singly occupied orbital has no amplitude on C and A and B do not overlap, which makes it an eigenvector of the overlap matrix with eigenvalue one. Every partition built from the overlap matrix returns the same split.

Tested in The spin the count does not hold · the series on overlap

“Placing a substituent on the site it does not prefer gives a higher-energy isomer.”

At five, yes: every placement at every length is a minimum. At seven a bond of 1.05 or longer set on the axis falls when nudged, by 1.4 × 10⁻⁵ at 1.05 and 1.7 × 10⁻² at 1.2, sliding into the equator; a bond of 0.7 to 0.9 set in the equator falls too. The disfavoured seven-coordinate placement is not an isomer.

Tested in The long bond goes to the crowded site · the series on VSEPR