Simply false — page 1 of 5

“Solid-state physics and molecular bonding are separate subjects with separate methods.”

The band of a two-hundred-atom chain is produced here by exactly the calculation that gives ethene its bonding and antibonding pair: build the adjacency matrix, diagonalise it, fill the levels. Nothing is added at any size. The figures for n = 2 and n = 200 are the same calculation done twice.

Tested in A solid is a molecule that did not stop · the series on Bands in a solid

“A larger molecule has a wider spread of orbital energies.”

Every level of a chain of n sites lies in the interval from −2β to +2β for every n whatever. Measured across n = 2, 4, 8, 16 and 40, the band width stays at 4β to within a part in a thousand; what changes is the spacing between levels, which falls as 1/n.

Tested in A solid is a molecule that did not stop · the series on Bands in a solid

“Sulfur hexafluoride expands its octet by hybridising in two 3d orbitals.”

Sulfur's 3d lies about ten electron volts above its 3p and is far more diffuse, so the mixing that overlap over energy difference allows is negligible; calculated d participation is a few per cent. The three-centre four-electron account needs no d orbitals and gives the geometry.

Tested in Hypervalency without d orbitals · the series on hypervalency

“Each bond in a molecule has its own stretching vibration, and the spectrum shows one line per bond.”

Diagonalising water's mass-weighted Hessian gives two stretching modes, and the share of the motion in each O–H bond is 50% in both of them. Neither mode is a motion of one bond. The same computation gives methane four stretching modes, every one of them a quarter in each of the four C–H bonds.

Tested in Normal modes are not bond stretches · the series on normal mode

“A ligand field lowers the energy of the d orbitals, which is why complexes form.”

The five energies sum to five times the spherically averaged potential whatever the arrangement — Unsöld's theorem, reproduced to ten decimal places by the point-charge integral for octahedral, tetrahedral, cubic and square-planar fields. A field redistributes the levels about a centre of gravity it cannot move. Complexes form because of the metal–ligand bonding, which is a different calculation.

Tested in The splitting is a symmetry statement · the series on ligand field

“An orbital is the region of space in which the electron is found.”

A contour is a surface of constant |ψ| at a level somebody chose. The ninety per cent surface leaves a tenth of the density outside it, and the same function drawn at ninety-nine per cent is a different surface of the same orbital.

Tested in What an orbital is · the series on orbital

“Benzene resonates between two Kekulé structures.”

The Kekulé forms are basis functions in a description, not states the molecule visits. The computed π bond order is exactly 2/3 on all six bonds, which is one structure rather than an average of two.

Tested in Delocalisation · the series on delocalisation

“An overlap that symmetry forbids is very small.”

Computed, the 1s with 2px overlap at 2.8 bohr is −3.7e-17: arithmetic noise, not a small physical quantity. The same integrator reproduces the closed-form 1s–1s overlap to 6e-5, so the vanishing is not a defect of the method.

Tested in Exactly zero · the series on overlap

“The five positions of a trigonal bipyramid are equivalent.”

The angle spectrum of the minimised five-point arrangement has three distinct angles where four and six sites have one and two. That inequivalence is measured, not assumed, and it is what Berry pseudorotation exchanges.

Tested in Five sites are not alike · the series on VSEPR

“A force field has to be refitted for each isotopologue.”

Force constants are properties of the electronic energy surface, which the Born–Oppenheimer separation makes independent of nuclear mass. Water's field, fitted to H₂O and D₂O, gives HOD's three frequencies at 1445, 2824 and 3890 with nothing refitted — against quoted harmonic values of 1440, 2824 and 3890.

Tested in The isotope shift is arithmetic · the series on normal mode

“A band gets wider as the crystal gets larger.”

⟨x²⟩ is the mean coordination, which is bounded by how many neighbours an atom has and is unaffected by how many atoms there are. Measured on chains and rings of 40, 60 and 80 sites, the second moment agrees with the counted coordination to machine precision at every size.

Tested in The width of a band is a count of neighbours · the series on Bands in a solid

“A tetrahedral complex has a smaller splitting because it has fewer ligands — four instead of six, so two thirds as much.”

Two thirds is 0.667 and the computed ratio is 0.4444, in both models, to eight decimal places. The ligand count enters through the trace, which is exactly N times the σ parameter and is not the splitting; the splitting is a difference between eigenvalues and depends on the directions. A cube has eight ligands, more than an octahedron, and splits by eight ninths of one.

Tested in Two models, one ratio · the series on ligand field

“The density of states is what an absorption spectrum measures.”

An absorption spectrum measures transitions, which need an initial state, a final state and a non-zero moment between them. The density of states counts states at one energy and knows nothing about pairs. Two systems with identical densities of states and different symmetry give different spectra, and one of them can give no spectrum at all.

Tested in A density of states is not a spectrum · the series on Bands in a solid

“Selection rules stop mattering once a molecule is large enough to have bands.”

A vanishing transition moment vanishes for the same reason at any size — an integrand odd about a plane integrates to zero whether the molecule has four atoms or four thousand. What changes is that a forbidden transition in a solid removes a contribution from a continuum rather than removing a visible line, so the failure is a missing shoulder rather than a missing peak.

Tested in A density of states is not a spectrum · the series on Bands in a solid

“A 2s orbital is a single sphere, slightly larger than a 1s.”

n − l − 1 = 1 radial node, so the isosurface is a shell inside a shell. The shells are counted on the drawn surface and required to equal the radial node count plus one, and a 2s surface claimed as one shell is refused.

Tested in Nodes · the series on orbital

“Bands come from periodicity.”

Every band in this field is computed from a chain or ring with no periodicity assumed anywhere, and amorphous solids — which are periodic in no direction — have bands, gaps and densities of states much like their crystalline counterparts. What periodicity buys is a labelling of the states, not their existence.

Tested in A band with no structure in it · the series on Bands in a solid

“Eight-coordinate structures are cubic.”

Minimised, eight points on a sphere give the square antiprism — one face of the cube twisted by forty-five degrees — at a lower repulsion energy than the cube. Both arrangements are computed here and compared.

Tested in The shapes above six coordination · the series on VSEPR

“Electronegativity is a property of an atom, and the scales agree well enough.”

Four definitions in four sets of units. Measured across the table: twelve discordant pairs between Pauling and Mulliken out of a hundred and fifty-three, and six ordinary bonds — C–H among them — whose polarity points in opposite directions depending on the table consulted.

Tested in Electronegativity is not one quantity · the series on electronegativity

“A molecule with 3N−6 vibrations shows 3N−6 bands.”

Two separate reductions stand between the count and the spectrum. Degeneracy collapses benzene's thirty modes into twenty frequencies and methane's nine into four; selection rules then cut benzene's twenty to eleven observable in either experiment, with nine visible in neither. The count 3N−6 is an upper bound on a number of lines, and for a symmetric molecule it is a bad one.

Tested in How many frequencies, not how many modes · the series on spectrum

“Anything a molecule does vibrationally can be seen in the infrared or the Raman.”

A mode is infrared active when its species carries x, y or z and Raman active when it carries a quadratic function, and a species can carry neither. Nine of benzene's twenty frequencies belong to such species, as does one of sulfur hexafluoride's six and one of ethene's twelve. These are silent modes, forbidden in both experiments by the same vanishing-integral theorem that permits the others.

Tested in How many frequencies, not how many modes · the series on spectrum

“Molecular orbital theory and valence bond theory are rivals, and one of them is right.”

Each turns into the other when pushed: configuration interaction on one side, ionic structures on the other. They are two bases for one description, and the observable they are meant to disagree about is the same in both.

Tested in Molecular orbital and valence bond · the series on models

“A metal is a very large molecule, so its levels are very spread out.”

Every level of a ring of n atoms is 2cos(2πk/n), so all of them lie in [−2, 2] for every n. Adding atoms does not widen the range; it puts more levels inside a range that never changes. The width of a band is set by the neighbour interaction, not by the size of the crystal.

Tested in The band limit · the series on multicentre

“The density of states in a band is roughly uniform.”

Computed for a ring of 2000 and compared with the closed form 1/(π√(4−x²)), the density piles up at both band edges and diverges there. The histogram agrees with the closed form to better than 0.01 across the interior, which is the check the figure makes.

Tested in The band limit · the series on multicentre

“A molecular geometry has to be settled by diffraction; a spectrum can only confirm it.”

Linear XY₂ has four vibrations giving three frequencies, of which two are infrared active and one Raman active with no coincidence; bent XY₂ has three, all active in both, with three coincidences. Those predictions differ in the count alone. Carbon dioxide shows the first pattern and sulfur dioxide the second, and neither conclusion needs a frequency to be assigned or a force constant to be known.

Tested in Two structures, two spectra · the series on spectrum

“Both isomers would show the same number of bands, since both have the same number of atoms.”

3N−5 for a linear molecule against 3N−6 for a bent one is already a difference of one, and degeneracy and selection rules widen it: the linear case has one of its four modes doubly degenerate and its infrared and Raman spectra disjoint, so three frequencies give three observable bands split two and one across the two experiments, against the bent case's three and three.

Tested in Two structures, two spectra · the series on spectrum

“A double bond is two bonds, so it is twice as strong.”

Measured, C–C is 348 kJ/mol and C=C is 614 — a ratio of 1.76, not 2. Computed, the σ and π overlaps of two carbon 2p orbitals at the C=C bond length are 0.325 and 0.270: different integrals over different orbitals, with no reason to be equal and no reason for their contributions to add.

Tested in A double bond is not two single bonds · the series on overlap

“A strong ligand strengthens the whole complex, so a good donor makes all the metal–ligand bonds shorter.”

Two ligands trans to one another compete for one metal orbital. In a three-level model solved exactly, raising one ligand's interaction from 1.0 to 1.6 raises its own bond order from 0.6155 to 0.7807 and lowers the bond order to the ligand opposite from 0.6155 to 0.4880. The measured counterpart is a bond length: a Pt–Cl bond trans to a strong σ donor is longer than one trans to a chloride by up to a tenth of an ångström.

Tested in The trans influence is an overlap argument · the series on overlap

“A three-centre four-electron bond puts extra electrons on the central atom.”

Solved as a three-by-three matrix, the occupied non-bonding orbital has a coefficient of exactly zero on the central atom — required by the mirror symmetry of the arrangement, not by the parameters. The computed charges are 1.92 on each ligand and 0.17 on the centre: the centre has given charge away rather than taken it.

Tested in Three-centre bonding, computed · the series on hypervalency

“The band is missing, so that vibration is not there.”

Nine of benzene's twenty vibrational frequencies belong to species that carry neither a linear nor a quadratic function, so they cannot appear as fundamentals in an infrared or a Raman spectrum at all. The vibrations exist, at well-defined energies, and are reached by inelastic neutron scattering, by combination bands and by lowering the symmetry.

Tested in What an absence proves · the series on spectrum

“A symmetry-forbidden band is very weak.”

The integral that would give it vanishes identically: every contribution is paired with its own negative by an operation of the group, so the sum is zero rather than small. The same distinction is what makes an overlap forbidden by symmetry come out at arithmetic noise rather than at a small number — see the computation in `exactly zero`. What is weak, and what does appear, is the intensity a lower-symmetry environment or a vibronic mechanism restores.

Tested in What an absence proves · the series on spectrum

“A conjugated chain has equal bond lengths, like benzene.”

Benzene's equal bonds are protected by a closed π shell; a long chain's are not. Computed on a hundred-site chain, alternating the bonds by twelve per cent gains 0.0078β per site more than a spring constant of 1.6 costs, and the best alternation is at a non-zero value for every spring constant tested.

Tested in A chain cannot stay even · the series on Peierls distortion

“A small enough distortion always costs more than it gains, because the elastic term is quadratic and the electronic term is too.”

The electronic gain is not quadratic. Dividing the computed gain by δ² gives a ratio that rises as δ falls — 2.564, 3.194, 5.330 at δ = 0.08, 0.05 and 0.02 on a hundred-site chain — which is the logarithm showing itself. A quadratic gain would give the same number three times, and the same test applied to a genuinely quadratic sequence returns one.

Tested in A chain cannot stay even · the series on Peierls distortion

“A molecule with five equivalent-looking bonds gives one signal because the five atoms are equivalent.”

The five fluorines of phosphorus pentafluoride fall into two orbits under its D3h group — two axial and three equatorial — and orbit sizes 2 and 3 with stabilisers of order 6 and 4 both satisfy the orbit–stabiliser relation against a group of order 12. The atoms are not equivalent. One signal is observed because the molecule exchanges them faster than the measurement can resolve.

Tested in A spectrum counts environments, not atoms · the series on spectrum

“A copper(II) complex with six identical ligands is octahedral, because six identical ligands arranged by repulsion give an octahedron.”

The repulsion argument fixes the arrangement and the electrons overrule it. A d⁹ ion has three electrons in the doubly degenerate upper pair, and the energy gained by elongating along one axis is linear in the distortion while the elastic cost is quadratic — computed slope −7.32 against a cost that starts at zero curvature, so the best distortion is at a non-zero value for any stiffness. Copper(II) hexaaqua has four bonds near 1.97 Å and two near 2.30.

Tested in Copper is never quite octahedral · the series on Peierls distortion

“Any partly filled degenerate level makes a complex distort.”

Only an unequal occupation does. Computing the initial slope for six configurations gives −7.32 for d⁹ and high-spin d⁴, and 0, −4×10⁻¹⁴, 0 and −9×10⁻¹⁴ for d³, high-spin d⁵, low-spin d⁶ and d¹⁰ — arithmetic zeroes, because the level shifts cancel term by term over a symmetric occupation. A half-filled t2g set is degenerate and gains nothing.

Tested in Copper is never quite octahedral · the series on Peierls distortion

“The 4s orbital fills before the 3d because the 4s is lower in energy.”

Computed from Slater's screened charges, a scandium 3d electron feels 3.00 and a 4s electron feels 3.00 — but the hydrogenic energy goes as Z²/n², so the 3d comes out at −13.6 eV and the 4s at −7.7. By iron the gap is −59.1 against −12.0. Every one-electron estimate puts the 3d lower, at every element in the series.

Tested in The aufbau order is not a property of the atom · the series on approximation

“Once the 4s has filled, the 3d fills, so ionisation removes a 3d electron.”

Every transition metal ionises from the 4s. Fe²⁺ is [Ar]3d⁶ with no 4s electrons at all. A filling order that is reversed by removing one electron is not describing a fixed order of orbital energies.

Tested in The aufbau order is not a property of the atom · the series on approximation

“Seven-coordinate geometry follows from the same rule as six.”

Minimised at 1/r the seven-point arrangement is a pentagonal bipyramid with four distinct angles; at 1/r⁶ it is a different arrangement with six. There is no maximally symmetric arrangement of seven points on a sphere, so nothing pins the answer and the unstated force law decides it.

Tested in Which angles are symmetry and which are the model · the series on VSEPR

“A magnetic moment measures how many d electrons a metal has.”

It counts the unpaired ones. Iron(II) and iron(III) differ by one d electron and their high-spin moments are 5.4 and 5.9 Bohr magnetons, while iron(II) high-spin and low-spin have the same six d electrons and moments of 5.4 and zero. The formula √(n(n+2)) takes the unpaired count and nothing else, and reproduces five of nine measured first-row values to a hundredth.

Tested in A moment counts electrons, not orbitals · the series on magnetism

“A bigger orbital is a more loosely bound one.”

The 4s orbital of scandium is more than twice the size of its 3d by every radial measure and is filled first. What binds it is a peak at 0.73 bohr holding a few per cent of its density, which no measure of overall size records.

Tested in How big is an orbital · the series on contour

“Hydrogen sulfide and phosphine are sp³ centres with badly distorted angles.”

At 92.1° the relation gives an s fraction of 0.0353, which is sp²⁷: the S–H bonds are all but pure p. Phosphine at 93.3° gives 0.0544 per bond, so its three bonds spend 0.163 of the s function and its single lone pair holds 0.837 — eighty-four per cent s. Calling either an sp³ centre misdescribes the bonds by a factor of seven in s character and misplaces almost the whole s orbital.

Tested in The angle does not fix the hybridisation · the series on hybrids

“A carbon atom could hold four bonds at 120° if the geometry demanded it.”

Four bonds at 120° require an s fraction of 1/3 each, which is 4/3 of an s orbital, and there is one. The budget refuses the arrangement rather than costing it, and the refusal is checked, not assumed.

Tested in The angle does not fix the hybridisation · the series on hybrids

“More force constants give a better description of the molecule.”

Methane's six bond angles are not independent — five bending coordinates span every deformation there is — so a fit with both a bending and a bend–bend constant has a direction it can move along freely. Allowed to, it did: both came back at −97,596 mdyn Å⁻¹, equal to four figures and opposite in effect, reproducing all eighteen frequencies of CH₄ and CD₄ to half a per cent.

Tested in The force field is not in the spectrum · the series on normal mode

“A wide band means a small gap and a narrow band means a large one.”

Alternating a chain's bonds by fifteen per cent opens a gap of 0.60β while leaving the second moment — the exact measure of the band's scale — essentially unchanged at 2 + 2δ², a shift of four per cent in the wrong direction. Width and gap are set by different features of the same matrix and moved by different changes to it.

Tested in The gap is not the band width · the series on Peierls distortion

“Canonical molecular orbitals and localised bond orbitals describe different distributions of the electrons.”

The transformation is applied to four Löwdin-orthogonalised bond orbitals and the total density is unchanged to 1e-16, at four values of the free mixing parameter. A mixing that is not orthogonal changes it by a fifth, so the test can fail.

Tested in The localisation transformation, demonstrated · the series on hybrids

“A rotational spectrum tells you about the bonding in a molecule.”

Every line position follows from the masses and the coordinates alone: B = h/8π²cI with I a sum of mass times distance squared. Two molecules with identical mass distributions and completely different bonding would give identical spectra. What a rotational spectrum determines is a geometry, which is why it determines one so well.

Tested in The rotational spectrum is a moment of inertia · the series on rotation

“Every molecule has a rotational spectrum; it is just a matter of sensitivity.”

A pure rotational transition needs a permanent dipole, and whether a molecule has one follows from its point group with no reference to bonding at all. Carbon dioxide, benzene, methane and sulfur hexafluoride each have a perfectly well-defined set of rotational levels computed here, and none of them can show a line of it.

Tested in The rotational spectrum is a moment of inertia · the series on rotation

“A subgroup gives the same answers as the group, just with different labels.”

A subgroup has fewer operations, so it forbids less. Computed for D∞h in its working subgroup D2h: the species Δu is infrared forbidden in the full group and correlates onto Au ⊕ B₁u, of which B₁u carries z. The subgroup therefore calls infrared-active a mode the real group forbids, and the error is in the permissive direction.

Tested in An infinite group, worked in a finite one · the series on point group

“The octet rule and band filling are different ideas from different subjects.”

Both say a system is stable when no level is partly occupied. An octet fills the four valence orbitals of one centre; a filled band fills every level of an extended structure. The 8 − N rule connects them explicitly by predicting how many neighbours an element takes in order to reach the closed case, and it is right for groups four, five, six and seven.

Tested in Counting electrons in an extended structure · the series on metal

“A molecule with a rotation axis cannot be chiral.”

Chirality is forbidden by improper operations — mirrors, inversion, rotoreflections — and not by proper ones. Hydrogen peroxide belongs to C2 at every dihedral strictly between 0° and 180°, which is a genuine twofold rotation and no improper operation at all, and it is chiral throughout that range. The two angles at which it is achiral are the two at which a mirror plane or an inversion centre appears.

Tested in One coordinate, three point groups · the series on dipole

“A set of orbitals drawn at the same contour value is drawn on a common footing, so their sizes can be compared.”

At the level that encloses 90% of a 1s — |ψ| = 3.94×10⁻², computed by bisection on the integral — a 2s encloses 4.44%, a 3s 0.80%, a 4s 0.14% and a 3d nothing at all. A plate at that one level compares one orbital's ninety per cent with another's four, which is not a comparison of sizes but of how sharply each function peaks.

Tested in One level is not one comparison · the series on contour

“Drawing an orbital at a lower contour level shows more of its structure, so the shells and nodes that appear depend on how generously the picture is drawn.”

The number of closed shells the surface has is n − l at every fraction tested, from 30% to 99%: two for a 2s, three for a 3s, four for a 4s, one for every p and d here. A radial node is a sphere on which the wavefunction is exactly zero, and no contour at a positive level can cross it, so the topology of the picture is fixed and only its size moves.

Tested in One level is not one comparison · the series on contour

“In a two-orbital interaction, the bonding level goes down by as much as the antibonding level goes up.”

Solving the 2×2 secular determinant with the overlap kept at S = 0.25 gives a bonding level 0.800|β| below the mean and an antibonding level 1.333|β| above it. The equality is what happens when S is set to zero, and S is not zero — it is the reason the two orbitals interact at all.

Tested in The antibonding level goes up more · the series on overlap

“Two bonds of the same order between the same elements have the same stretching frequency.”

Carbon dioxide's two C=O bonds are related by symmetry, so they are the same bond by any definition, and they carry one force constant in the fitted field. The molecule has two stretching frequencies a thousand wavenumbers apart, because a frequency is a property of a mode and a mode involves both bonds and the atom between them.

Tested in The frequency is not the bond strength · the series on normal mode

“Low-spin tetrahedral complexes do not exist because a tetrahedral field offers no spin choice.”

The enumeration says otherwise. With two orbitals below three, four configurations — d³ to d⁶ — have distinct high-spin and low-spin fillings, being the octahedral four reflected by the particle–hole map. What makes them unreachable is a magnitude: a tetrahedral splitting is four ninths of an octahedral one, so even carbon monoxide's 34,000 cm⁻¹ becomes about 15,000, below every pairing energy in the table.

Tested in The pairing energy decides the moment · the series on magnetism

“A delocalised electron pair is spread thin, so it binds the molecule less than a localised pair would.”

One pair in a ring of n gives a total π bond order of exactly 2 and a π energy of exactly 4β, for every n from three to eight and beyond, because a ring's lowest level is 2β at any size. The per-bond order falls as 2/n and the total does not move. What is divided is the bonding among the bonds, not the amount of it.

Tested in What one pair can hold together · the series on multicentre

“A single impurity in a crystal of 10²³ atoms cannot change anything measurable.”

It changes the states, not just their number. One modified site in a hundred-and-sixty-site chain pulls a level out of the band entirely and binds a state to itself; that state is at an energy no state of the pure chain occupies, so it absorbs light the pure material does not. One part in 10⁷ of a dopant changes silicon's conductivity by orders of magnitude for exactly this reason.

Tested in A defect is a level in the gap · the series on defect

“A defect has to be strong enough to bind a state, like a potential well needing a minimum depth.”

In three dimensions, yes. In one dimension, no — measured by bisection on chains of 40, 80 and 160 sites, the apparent threshold for a mid-chain defect is 0.098, 0.049 and 0.025 β, halving with every doubling. It is the finite chain's own level spacing, and it goes to zero: an arbitrarily weak one-dimensional defect binds a state.

Tested in A defect is a level in the gap · the series on defect

“Sulfur's 3d orbitals are the wrong symmetry to help, so d participation is forbidden.”

Sulfur's 3d functions span eg ⊕ t₂g in Oh, and eg is exactly the species the ligand σ set has spare. Symmetry permits the interaction; what makes it negligible is size and energy, not symmetry. A free-atom 3d at an effective charge near one has a mean radius of 5.6 Å against an S–F distance of 1.56, and how much a molecular field contracts it is beyond a symmetry argument.

Tested in Six bonds and four orbitals · the series on hypervalency

“Nitrogen trifluoride has a larger dipole than ammonia, because N–F bonds are more polar than N–H bonds.”

The bond-vector sum is larger too — three N–F bonds at 102.2° sum to 1.316 of a bond unit against 1.080 for three N–H bonds at 107.8° — so a bond-dipole model predicts NF₃ to beat NH₃ by about a third. The measured moments are 0.24 D and 1.47 D, the wrong way round by a factor of eight, because the lone pair contributes a moment that adds in one molecule and subtracts in the other.

Tested in What a dipole cannot tell apart · the series on dipole

“Two molecules with the same dipole moment have similar charge distributions.”

A dipole is three numbers taken from a function of three variables, and the map is enormously many-to-one. Methane has no dipole and no quadrupole either — Td forbids both — while boron trifluoride has no dipole and a quadrupole of 0.437 in the same model. Zero dipole is compatible with distributions that differ at every higher moment, and the quadrupole is only the first place they part.

Tested in What a dipole cannot tell apart · the series on dipole

“Two magnetic ions interact because each one's magnetic field is felt by the other.”

The magnetic dipole–dipole energy of two moments of about two Bohr magnetons three ångströms apart is roughly 0.06 cm⁻¹. Measured exchange couplings in bridged copper and iron complexes run from tens to several hundred wavenumbers — three to four orders of magnitude larger. The interaction that produces them is electrostatic and kinetic: an exactly solved two-site model with no magnetic term in it at all gives a singlet–triplet splitting approaching −4t²/U.

Tested in What couples two spins · the series on magnetism

“The lower-energy orbital of a shell is the smaller one.”

At any shared effective charge the closed forms give ⟨r⟩ = 6/Z for a 2s and 5/Z for a 2p, so the 2s is larger by exactly a fifth at every element in the period — 2.308 against 1.923 bohr at boron, 1.026 against 0.855 at neon — and it is the 2s that lies lower. Size and energy are answers to different questions, and neither settles the other.

Tested in What the screening model cannot see · the series on orbital

“d–d transitions are weak because the d orbitals overlap poorly with the light.”

The intensity of an electric dipole transition is fixed by whether the product of the two states' representations with the dipole operator contains the totally symmetric representation. In Oh both d levels are even and the dipole is odd, so the product is odd and the integral is exactly zero — computed as a character product and a reduction, with nothing about overlap or size entering. The same d orbitals in a tetrahedral complex give an allowed transition and absorb about a hundred times more strongly.

Tested in Why a d–d band is weak · the series on representation

“The extra skeletal pair is a feature of the icosahedral geometry of B₁₂H₁₂²⁻.”

It is a property of any connected graph. The largest eigenvalue of a non-negative connected adjacency matrix is simple and its eigenvector has one sign throughout — Perron's theorem — so there is exactly one nodeless radial combination for every cage whatever its shape, and every other radial combination has a node. The octahedron, the pentagonal bipyramid and the icosahedron all give one, and so would a cage nobody has made.

Tested in A cage needs one pair more than it has corners · the series on multicentre

“A neon atom has eight electrons in dumbbell-shaped orbitals, so its electron density has lobes.”

The angular densities of the three 2p functions sum to 3/4π in every direction, and the departure from that constant over a grid of 768 directions is 1.1×10⁻¹⁶. Neon's density is exactly spherical. The lobes exist in one choice of basis for the shell and cancel completely when the shell is filled.

Tested in A filled shell has no shape · the series on contour

“The failure can be repaired by computing the band more accurately.”

The failure is not in the band. Every one-electron scheme, however carefully the average potential is constructed, assigns each electron a single orbital and lets it ignore where the others are. What is missing is a correlation between electrons on different sites, which no single-determinant description contains at all.

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

“Isotopic substitution shifts the bands of a spectrum without changing how many there are, since it does not change the bonding.”

Methane's nine vibrations give four distinct frequencies and CH₂D₂'s give nine, from the identical force field. The count is set by the degeneracies, the degeneracies are set by the point group, and the point group is set by which operations permute atoms of equal mass — 24 for CH₄, 6 for CH₃D, 4 for CH₂D₂. Substitution changes the group without moving a nucleus.

Tested in A spectrum that changes when only a mass does · the series on spectrum

“Lowering the symmetry of a molecule with a threefold axis can leave it with a fivefold one.”

Lagrange's theorem: the order of a subgroup divides the order of the group. Ammonia's C3v has order six, so its subgroups have orders 1, 2, 3 and 6 and nothing else, and the search finds exactly one of order three, three of order two, and the two trivial ones. A fivefold axis would need a subgroup of order five in a group of order six.

Tested in Every group a molecule can fall to · the series on point group

“A molecule that is not chiral cannot be made chiral by distorting it.”

Chirality is the absence of any improper operation, and it is a property of the subgroup the distortion leaves rather than of the parent. Methane's Td contains a subgroup of order twelve with no improper operation in it, and three more of order four, three of order three and three of order two. A distortion that lands in any of them gives a chiral structure from an achiral parent.

Tested in Every group a molecule can fall to · the series on point group

“Orthogonality of hybrids is a mathematical nicety with no physical content.”

Every quantity computed from a set of orbitals — a population, a bond order, an energy from a secular determinant — assumes either that the set is orthogonal or that the overlap has been kept. At an overlap of 0.625 the two assumptions differ by more than the quantities being computed. Löwdin orthogonalisation repairs the set and changes the directions of the orbitals while doing it, which is the price.

Tested in Hybrids that were never orthogonal · the series on hybrids

“A sixteen-electron complex is two electrons short and therefore unstable.”

It has the same number of filled orbitals its geometry offers, so it is closed by its own arithmetic. Wilkinson's catalyst, Vaska's compound, tetrachloridoplatinate and Zeise's salt are all sixteen-electron and all isolable. What is true is that the empty perpendicular orbital gives them a route to add a ligand that an eighteen-electron octahedron does not have, which is a statement about a rate rather than about stability.

Tested in Sixteen is also a count · the series on electron count

“A surface atom is the easiest place to trap a state, because it has a neighbour missing.”

Measured by bisection: a mid-chain site binds a state for any energy difference at all, while an end site needs its energy to differ by more than one β. The thresholds on chains of 40, 80 and 160 are 1.025, 1.013 and 1.006 at the end and 0.098, 0.049, 0.025 in the middle — one converging on one, the other on zero.

Tested in The end is the hardest place to bind · the series on defect

“A band gap of 1.1 eV means it takes 1.1 eV to break a bond in silicon.”

Silicon's Si–Si bond enthalpy is about 2.3 eV, twice the gap. The gap is the cost of moving one electron from a filled level to an empty one with the atoms staying where they are; the bond energy is the cost of removing a pair from a bonding level and letting two atoms separate to infinity. The two differ by the antibonding level's position, by the relaxation of the structure, and by every repulsion term the one-electron picture omits.

Tested in A band gap is not a bond energy · the series on Bands in a solid

“Carbon monoxide is high in the spectrochemical series because it is a strong sigma donor.”

Its sigma donor orbital is a weakly bonding lone pair on carbon and it is a mediocre donor by any measure. It is high in the series because it is a strong pi acceptor: its empty pi* orbitals lie close in energy to the metal t2g set, so the gap in the denominator is small and the interaction is large. This site orders the whole series by the pi parameter and by charge, and the pi parameter orders all of it.

Tested in Overlap is not interaction · the series on overlap

“Ring strain is a matter of bond angles.”

Cyclopentane's planar angle of 108° is within a degree and a half of tetrahedral, closer than cyclohexane's chair, and cyclopentane carries about 26 kJ/mol of strain against cyclohexane's nothing. The angles are not the problem there; the eclipsed hydrogens are. Two different quantities travel under one word, and the geometry computed here bounds only the first.

Tested in The angle a ring cannot have · the series on strain

“Contracting an orbital always lowers the energy, since the electron gets closer to the nuclei.”

Kinetic energy goes as the square of the exponent and nuclear attraction goes as the exponent, so contracting always costs more kinetic energy than it buys attraction at large enough exponents, and there is a minimum. Past about five bohr the balance tips the other way and the molecule's best exponent falls slightly below one, reaching 0.9951 at six bohr — an expansion, not a contraction.

Tested in The atom does not bring its own orbital · the series on orbital

“Electrons avoid each other because they repel, so a calculation without repulsion has them moving independently.”

At U = 0 the exact ground state of a two-site model has a neighbouring spin correlation of −0.125000000, not zero. The same number comes out of the one-electron density matrix as −½ρ², with no repulsion anywhere in the derivation. Two electrons of the same spin cannot occupy the same site whatever the interaction is, and that alone anticorrelates the spins.

Tested in The hole that is not repulsion · the series on correlation

“Dispersion forces are negligible.”

Computed for argon from the London expression at its nearest-neighbour separation, one pair is bound by 0.0099 eV — small against a covalent bond by a factor of 251, and large enough that argon is a solid below 84 K. Every molecular crystal, every protein fold and all of adhesion runs on the quantity being called negligible.

Tested in What holds a solid together · the series on cohesion

“A character table lists the representations somebody found useful; a group could have more of them.”

The number of inequivalent irreducible representations equals the number of conjugacy classes, exactly. C2v has four classes and four representations, and the sum of the squares of their dimensions is 1+1+1+1 = 4, which is the order of the group. A fifth representation of any dimension whatever would push that sum past four, and the sum is an identity rather than a bound.

Tested in Why a character table stops where it stops · the series on representation

“The characters in a table are measured or fitted quantities of some kind.”

They are traces of matrices, and the table's shape is settled before any of them is written down. Reducing the regular representation — character h at the identity and zero everywhere else, because multiplying by anything else moves every group element — returns each representation exactly as many times as its own dimension, in all nineteen tables here, with the multiplicities coming out as whole numbers or the reduction refusing.

Tested in Why a character table stops where it stops · the series on representation

“A photoelectron spectrum shows one band per bonding electron pair, so a molecule with four equivalent bonds shows four bands.”

Methane has four equivalent C–H bonds and two valence bands, at 12.7 and 23.0 eV. The four hydrogen 1s functions span a₁ ⊕ t₂ in Td, so the eight valence electrons occupy two symmetry species — one non-degenerate and one triply degenerate — and a spectrum counts species rather than pairs.

Tested in Fewer bands than electrons · the series on photoelectron

“Lowering a molecule's symmetry does not change how much information its spectrum carries, since the same electrons are there.”

Methane, ammonia and water all have eight valence electrons and give two, three and four bands. The valence basis spans A1 ⊕ T2 in Td, 2A1 ⊕ E in C3v and 3A1 ⊕ B1 ⊕ B2 in C2v — four occupied orbitals throughout, falling into two, three and four species. Every degeneracy the group permits is one band fewer.

Tested in Fewer bands than electrons · the series on photoelectron

“A lattice energy is a sum over neighbours, so it can be computed by adding up shells until the terms get small.”

Summing rock salt shell by shell gives partial sums of 6.000, −2.485, 2.134, −0.866, 9.867 and 0.069 for an answer of 1.7476 — oscillating with an amplitude larger than the answer and getting worse at the fifth shell. The terms do not shrink: a shell at distance r holds about r² ions each contributing 1/r, so every shell contributes about as much as the last. The series converges only conditionally, and a different order of summation gives a different sum.

Tested in The lattice sum that depends on the order of adding · the series on cohesion

“The tight-binding picture could be extended to ionic solids by including more neighbours.”

The tight-binding matrix works because the covalent interaction decays exponentially, so second neighbours contribute a few per cent and third essentially nothing. A Coulomb interaction decays as 1/r and never becomes negligible; there is no distance at which the list of neighbours can be truncated, and the whole structure of the method fails rather than becoming inaccurate.

Tested in The lattice sum that depends on the order of adding · the series on cohesion

“Drawing a set of orbitals at the same isovalue makes them comparable.”

It makes them incomparable in a particular way. At ψ = 0.02 atomic units, hydrogen's 1s encloses 96.2 per cent of its density, its 2s 51.8, its 3s 1.3 and its 4s 0.4. The picture of the 4s is a surface containing four electrons in a thousand and looks like an orbital; the picture of the 1s is a surface containing almost all of one and looks like an orbital too.

Tested in The isovalue nobody chose · the series on contour

“A ring with fixed bond lengths and fixed bond angles has a fixed shape.”

The rank of the closure constraints says otherwise. For six atoms the chair's constraint matrix has rank 9 out of 12, so its only zero modes are the three rotations and it is rigid; the twist-boat's has rank 8, leaving one direction that changes no bond and no angle. Following that direction traces thirty-six distinct conformers, every one with the same six bonds and the same six angles to a residual below 10⁻¹¹, and torsions spanning 120°.

Tested in The ring that cannot hold still · the series on strain

“Ligands split a d shell in proportion to how strongly they repel the d electrons, so more charge means a bigger splitting.”

Of the twenty pairs in the tabulated series that differ in charge, three are in the order a point-charge argument predicts and seventeen are not. The two ligands with the largest measured splittings, carbon monoxide at 34,000 cm⁻¹ and ammonia at 21,600, are electrically neutral, and iodide with a full negative charge is at the bottom of the list at 7,000.

Tested in The spectrochemical series is not electrostatics · the series on ligand field

“Loosening the tolerance can only move the answer up a chain of subgroups.”

A benzene stretched by 2 per cent along one axis is D2h up to a tolerance of 0.04 Å and C6h at 0.08. D2h has order 8 and C6h has order 12, and 8 does not divide 12 — so by Lagrange the first is not a subgroup of the second. What is FOUND does grow monotonically: the count of accepted operations rises from 7 to 15 to 19 across the same sweep. It is the classification of what was found that is not monotone.

Tested in The tolerance is a decision · the series on point group

“Molecular orbital theory and valence bond theory are two languages for the same answer, so the choice between them is a matter of taste.”

On an exactly solvable two-site, two-electron model they give different energies at every repulsion strength but one. At U = 0 the molecular orbital answer is exact and valence bond is 2t too high; at U = 32t valence bond is 0.12t too high and molecular orbital is 14.12t too high. They are equally wrong at exactly U = 4t, where both miss by 0.8284t, and the errors run in the same direction and at wildly different rates.

Tested in Where molecular orbital theory dissociates · the series on models

“A molecular orbital wavefunction describes a bond that weakens smoothly as the atoms separate.”

The doubly occupied bonding orbital gives a total double occupancy of exactly 0.5 at every value of the repulsion and therefore at every bond length — the two electrons are on the same atom half the time whatever the separation. The exact ground state gives 0.5 at U = 0, 0.1464 at U = 4 and 0.0039 at U = 32. The failure is not a matter of degree: the description contains a fixed fifty per cent ionic character it has no way to remove.

Tested in Where molecular orbital theory dissociates · the series on models

“Benzene's bonds are equal because delocalisation stabilises the π system.”

The π energy is LOWERED by alternating the bonds, at every ring size computed here. Fitting the gain against the distortion gives a coefficient of 5.96 for benzene, 13.19 for a ten-ring and 45.97 for cyclobutadiene, all positive — so a π system left to itself would alternate. What keeps the ring regular is the σ frame, which pays for every bond it stretches.

Tested in The hexagon is the frame's doing · the series on delocalisation

“A material with a half-filled band is a metal.”

The one-electron gap of a half-filled four-site ring is exactly zero at every value of the repulsion, because the repulsion is not in that matrix. The exact charge gap of the same ring — E(N+1) + E(N−1) − 2E(N), three separate diagonalisations — is zero at U = 0 and rises to 5.99 t at U = 8, with no sign of stopping. Nickel oxide has a half-filled d band and a gap of about four electronvolts.

Tested in The insulator band theory cannot see · the series on metal

“Adding a lone-pair moment repairs the additive bond-dipole model.”

For phosphorus trifluoride the bond sum comes out at −6.14 D against a measured total of 1.03, so the leftover would have to be at least 5.1 D — larger than the dipole of any small molecule, and far larger than any lone pair could supply. The additive model has failed before the lone pair is reached, and a correction refitted to every molecule it is applied to is not a correction.

Tested in The lone pair is not the missing term · the series on dipole

“The symmetry-adapted combinations of a degenerate representation are determined by the symmetry.”

The projector onto E1g applied to the p function on atom 1 gives (0.577, 0.289, −0.289, −0.577, −0.289, 0.289); applied to atom 2 it gives (0.289, 0.577, 0.289, −0.289, −0.577, −0.289). Both belong to E1g, both are correct, and they overlap by exactly 0.500 — they are two bases of one subspace, related by a rotation nobody chose. What is determined is the subspace, and the Hückel eigenvalue computed in either is 1.000000 to twelve decimal places.

Tested in The projector is unique, the basis is not · the series on representation

“Adding electrons to a conjugated system increases its π binding energy, since there is more bonding to go round.”

The cyclopropenyl cation's π energy is 4.0000β and the anion's, with two more electrons, is 2.0000β. The extra pair goes into a degenerate antibonding level at −1β apiece, so the total falls by 2β. Cycloheptatrienyl falls from 8.9879β at six electrons to 8.0978β at eight by the same mechanism.

Tested in The same ring, three charges · the series on aromaticity

“A ring with a delocalisation energy of zero has no π bonding in it.”

Square cyclobutadiene has a π energy of 4.0000β at two, four and six electrons alike, and a delocalisation energy of exactly zero at all three — because the reference state of isolated double bonds has the same energy, not because the π system is absent. The delocalisation energy is a difference against a stated reference and can be zero while both terms are large.

Tested in The same ring, three charges · the series on aromaticity

“Square-planar complexes are square planar because that arrangement minimises ligand repulsion.”

It maximises it among four-coordinate arrangements. A tetrahedron's angles are all 109.47° and a square plane has four at 90° and two at 180°, so the square plane is worse on every repulsion measure. The arrangement is adopted in spite of the repulsion, which is why the effect appears only where a ligand field term is large enough to pay for it.

Tested in VSEPR does not reach a transition metal · the series on VSEPR

“If a molecule's infrared and Raman spectra have no band in common, it has a centre of inversion.”

Eclipsed ferrocene is D5h, which contains no inversion. Its 57 vibrations span 4A1′ ⊕ A2′ ⊕ 6E1′ ⊕ 6E2′ ⊕ 2A1″ ⊕ 4A2″ ⊕ 5E1″ ⊕ 6E2″; the infrared-active species are E1′ and A2″, the Raman-active are A1′, E2′ and E1″, and the two lists share nothing. The rule runs one way only, and the counterexample needs a fivefold axis: the coordinates go to E1 and their products to E2, which cannot happen with a threefold or fourfold axis.

Tested in Mutual exclusion does not prove a centre · the series on spectrum

“Two identical impurities in a crystal give two identical levels at the same energy, since each sees the same environment.”

Two impurities of strength −2β in a chain of 61 give levels split by 0.535β at two sites apart, 0.0143 at six and 1.2×10⁻⁵ at fourteen. The splitting is never zero at any finite separation, because the two bound states overlap, and it falls geometrically with a ratio of 0.41421 per site — the same κ the isolated bound state decays with.

Tested in Two defects, and the level between them · the series on defect

“The correlation energy tells an attraction from a repulsion.”

At U and −U the correlation energy is identical to the last bit a double holds, and so are the natural occupations. What differs is the sign of the mixing coefficient: a repulsion of 4 gives an ionic weight of 0.146 and an attraction of 4 gives 0.854, and the two weights sum to exactly one. Every scalar measure of correlation is blind to a qualitative difference between two states.

Tested in Two kinds of correlation, and only one is small · the series on correlation

“The difference between them is a matter of interpretation.”

It is one number. The weight of the configurations with both electrons on the same atom is fixed at exactly 1/2 by the molecular-orbital function and at exactly 0 by the valence-bond function, whatever the repulsion. The exact value runs 0.500, 0.379, 0.276, 0.146, 0.053, 0.015 as U/t goes 0, 1, 2, 4, 8, 16 — never reaching either.

Tested in Two pictures, one plane · the series on models

“A calculation whose energy is accurate to a tenth of a per cent has a wavefunction accurate to about that much.”

The energy error is second order in the wavefunction error and an observable's is first, measured here at exponents of 2.00 and 1.01. A trial state overlapping the exact one by 0.9806 — a two per cent error in the wavefunction — has a double occupancy out by 53 per cent. Halving the wavefunction error divides the energy error by four and the observable's by two, so the ratio doubles at every halving without limit.

Tested in A better energy is not a better answer · the series on approximation

“A method that is accurate for one molecule is accurate for two of them.”

Truncating the expansion at one doubly occupied site is exact for one Hubbard dimer, because two electrons cannot doubly occupy two sites. For two independent dimers at U = 4t it misses 0.1928 of a hopping unit, and for three it misses 0.4853 — an error PER DIMER of 0, 0.0964 and 0.1618. The exact energies are additive to 10⁻⁹: −0.828427, −1.656854, −2.485281.

Tested in A method that is not additive · the series on correlation

“The error of a truncated method is a fixed percentage of the correlation energy.”

The fraction recovered falls as the system grows, and it falls faster where there is more to recover. With the repulsion switched off — the most correlated case in this model, since the exact state is then maximally ionic — the same truncation misses 1.1716 for two dimers and 2.5359 for three, against 0.1928 and 0.4853 at U = 4t. Both the error and its growth depend on the repulsion, so no single percentage describes the method.

Tested in A method that is not additive · the series on correlation

“A solid with more valence electrons is more strongly bound, since there is more bonding to go round.”

The occupied-level sum per site of a ring of 60 rises to 1.2721β at half filling and falls symmetrically to exactly 0 when the band is full. Beyond half filling every added electron enters an antibonding level and subtracts as much as an electron below it added. The measured cohesive energies of the transition series peak near the middle for the same reason.

Tested in Half filled is as bonded as it gets · the series on Bands in a solid

“A filled band still holds the solid together — the electrons are in bonding orbitals lower than the atomic level.”

The sum of every level of a symmetric band is exactly zero, because the trace of the hopping matrix is zero: each bonding level below the atomic energy is matched by an antibonding one exactly as far above. Computed here the full-band value is 0 to 10⁻¹⁵β per site, which is arithmetic noise rather than a small residue.

Tested in Half filled is as bonded as it gets · the series on Bands in a solid

“A character table determines the point group.”

Comparing character matrices and class sizes rather than names finds four sets of point groups sharing a table: {Cs, Ci, C2} at order 2, {C2v, C2h, D2} at order 4, {C4v, D4, D2d} at order 8, and {D5h, D5d} at order 20. Every entry agrees; only the class labels differ.

Tested in One table, three groups · the series on representation

“Two groups with the same character table make the same physical predictions.”

Of {C2v, C2h, D2}: one may be polar, one is centrosymmetric, one is chiral, and no two of them are the same. A z-polarised transition belongs to A1 in the first, Au in the second and B1 in the third. Every number in the three tables is identical.

Tested in One table, three groups · the series on representation

“A quenched orbital moment contributes nothing to a magnetic measurement.”

Spin-orbit coupling mixes the quenched ground state with excited d states, and the g-value shifts by −2λ times a sum of squared matrix elements over energy denominators. For copper(II) that is 8|λ|/Δ along the axis and 2|λ|/Δ across it, giving g∥ = 2.417 and g⊥ = 2.095 against measured 2.39 and 2.08.

Tested in The g-value is the orbital coming back · the series on magnetism

“A quadrature that agrees with itself at higher resolution has converged.”

A product rule over a box fourteen bohr across, checked by raising the number of points, is 9 per cent low at one bohr and 31 per cent high at six for two 2p orbitals in σ — and agrees with itself to five decimal places throughout, because the error is in the domain and the check varies the resolution.

Tested in Closer is not more overlap · the series on overlap

“A polyiodide is an iodide ion loosely holding some iodine molecules.”

That picture predicts short bonds where the I₂ units are and long ones between. The chain predicts strong bonds at the ENDS and weak in the middle — orders 0.789, 0.577, 0.577, 0.789 for five centres — and I₅⁻ is measured at 2.81, 3.17, 3.17, 2.81 Å. Short outside, long inside, which is the chain's order and the reverse of the other.

Tested in Hypervalency does not stop at three centres · the series on hypervalency

“A self-consistent field gives one answer for a molecule.”

It gives whichever answer the symmetry it was told to look for allows. Above a repulsion of about 2t the same iteration on the same four-site system finds a second solution with up electrons on alternate sites, and it is LOWER: at U = 16t the symmetric solution gives +11.53 and the polarised one −0.374, against an exact −0.583. The better mean field is the one describing a spin arrangement the exact ground state does not have.

Tested in Koopmans' theorem is exact for nothing · the series on photoelectron

“A structure with more neighbours per atom is more strongly bonded, in proportion to the number of neighbours.”

Binding per site goes 1.272, 1.611, 1.979 for coordinations of two, four and six — a rise of 56 per cent while the coordination triples. Binding per bond therefore falls from 0.636 to 0.330. More neighbours means more binding in total and less binding per bond, which is why a close-packed metal has bonds worth a fraction of a covalent one and a great many of them.

Tested in The bond that weakens as neighbours multiply · the series on cohesion

“A molecule with no dipole moment has no long-range electrostatic interaction.”

Benzene's quadrupole moment is −8.7 D Å. Two benzenes 5 Å apart interact by +8.8 kJ/mol stacked and −4.6 kJ/mol edge-to-face, from the quadrupole alone with no dispersion in the calculation at all. The sign changes between 50° and 60° of relative tilt, so the two arrangements are not merely different in strength.

Tested in Zero dipole is not no interaction · the series on dipole

“The momentum-space wavefunction contains different information from the position-space one.”

It is a unitary transform: nothing is lost and nothing is gained. Every moment here is computable from either. What changes is which facts are obvious — the kinetic energy is a curvature in position and a width in momentum, and ⟨p²⟩ computed by transforming agrees with 2⟨T⟩ from the virial theorem to three parts in a thousand for every orbital tested, which is the same number arrived at two ways.

Tested in The orbital in momentum space · the series on orbital

“A function that is nearly a duplicate is a harmless waste.”

A fifth Gaussian at 1.01 times an existing exponent lowers the energy by 3.4×10⁻⁹ hartree against a well-placed fifth function's 5.3×10⁻⁴ — a factor of 160,000 — while the smallest overlap eigenvalue falls to 4.6×10⁻⁶. At 1.00003 times, the set is refused: it is no longer a basis.

Tested in A function that is already there · the series on basis

“A surface costs one broken bond per surface atom, at the bulk value of a bond.”

A surface site keeps 91.24 per cent of the binding a bulk site has while keeping 83.33 per cent of its bonds. The one broken bond therefore costs 0.526 of what a bond is worth in the bulk — the estimate is nearly double the truth, at a coordination of six.

Tested in A surface is not a count of broken bonds · the series on cohesion

“The error is a small correction to a sound argument.”

The computed fraction is 91.24 per cent against √(5/6) = 91.29 per cent, agreeing to five hundredths of a point, and against 5/6 = 83.33 per cent, differing by eight points. The bond-counting estimate is not a first approximation to the answer; a different rule is, and it is not more complicated.

Tested in A surface is not a count of broken bonds · the series on cohesion

“A rotational spectrum determines a molecule's moments of inertia.”

For a symmetric top the transition energies are 2B(J+1) for every J and every K, to 10⁻¹², so the constant of the unique axis appears in no line. Ammonia's A is 6.3406 wavenumbers against B of 9.8601 — a difference of 36 per cent of the larger — and the spectrum reports only the second.

Tested in The constant a spectrum cannot see · the series on rotation

“An accidental degeneracy is an accident.”

It survives no departure from 1/r whatever. ⟨1/r²⟩ is 0.2500 for the 2s and 0.0833 for the 2p, in the ratio 3:1 exactly, so a correction going as 1/r² splits the shell at first order — and 5:1 across the n = 3 shell. Classically, the orbit closes for an inverse-square force to 0.003 degrees a turn and precesses by 17 to 20 degrees a turn for exponents of 1.9 and 2.1.

Tested in The degeneracy no group predicts · the series on representation

“The shape of a density of states is a fact about the chemistry.”

Every structure here is a graph with no chemistry in it at all: the matrix has ones where two sites are joined and zeroes elsewhere. The three shapes differ because the three dimensions differ, and nothing else varies between them.

Tested in Where the states pile up · the series on Bands in a solid

“The spread is a thermal effect and vanishes on cooling.”

Every number here is the ground state of the oscillator, at no temperature. The mean square displacement of a normal coordinate is ħ/2ω, which is what it is at absolute zero; warming the molecule only increases it. Hydrogen's own molecule is spread by 0.087 Å against a bond of 0.741 — 11.8 per cent — with no thermal contribution at all.

Tested in The atoms are not at the points · the series on approximation

“Repulsion is a correction to the two-level result.”

At a difference of 8 and no repulsion the transfer is 0.9701 of an electron; at the same difference and U = 16 it is 0.0247. A correction changes an answer by a fraction of itself. This changes it by a factor of thirty-nine, and the limit at large repulsion is zero transfer at every difference.

Tested in A difference does not make a transfer · the series on electronegativity

“A molecule needs a dipole moment to have an infrared spectrum.”

Carbon dioxide's dipole is exactly zero at every origin and three of its four modes have a non-vanishing dipole derivative — the bend at 673 cm⁻¹ twice over and the antisymmetric stretch at 2396. What a band needs is a dipole that changes as the molecule moves, which is a different quantity.

Tested in A dipole is not what an infrared spectrum sees · the series on dipole

“A more polar molecule has a stronger infrared spectrum.”

Of five molecules here, the three with no dipole at all — carbon dioxide, methane and boron trifluoride — account for fourteen of the nineteen active modes. Boron trifluoride's largest dipole derivative is 0.1676 against water's 0.1751, on molecules whose dipole moments are 0 and 0.1697.

Tested in A dipole is not what an infrared spectrum sees · the series on dipole

“Deciding which bands appear needs a calculation.”

It needs the point group. Every mode whose species carries none of the coordinate functions has a dipole derivative that vanishes — computed here from displacements and charges, with no group anywhere in the calculation, at 10⁻¹⁶ for exact coordinates and 10⁻¹⁰ for coordinates quoted to four decimal places. The two routes agree on all twenty-eight modes.

Tested in A dipole is not what an infrared spectrum sees · the series on dipole

“Cyclooctatetraene is aromatic if it is planar, since its delocalisation energy is 1.66β.”

1.6569β is what the isolated-bond reference gives, and it is 83 per cent of benzene's. Against the ring cut open the same planar structure gives 0.1393β, an eighth of benzene's 1.0121β and a fortieth of a beta per electron. The first number is measuring the four double bonds it was allowed to count, not the ring.

Tested in A stabilisation is measured from somewhere · the series on delocalisation

“A vibrational assignment's percentages are properties of the mode.”

Boron trifluoride's 1454 cm⁻¹ mode is 36.7 per cent B–F stretch by squared displacement, 49.9 per cent with the angles measured in radians instead of ångström, 97.9 per cent by the potential-energy distribution, and 94.6 per cent by its diagonal part. The mode is the same eigenvector in all four.

Tested in How much of a band is a bond stretch · the series on normal mode

“A potential-energy distribution is a distribution, so its entries are between nought and one.”

The entries include the off-diagonal terms of the force matrix, which belong to no single coordinate. Water's bend at 1649 cm⁻¹ comes out at 100.1 per cent of the energy with each of its two stretches at −0.03 per cent — small here, and negative, which no share of anything can be.

Tested in How much of a band is a bond stretch · the series on normal mode

“The conventions disagree, so the numbers mean nothing.”

Where a mode really is one coordinate every convention says so and says the same one: boron trifluoride's 888 cm⁻¹ band is 100.0 per cent B–F stretch under all four, water's 3833 is 100.0 per cent stretch under all four. The disagreement is confined to mixed modes, which is exactly where the percentage is being used to decide something.

Tested in How much of a band is a bond stretch · the series on normal mode

“A ligand's charge tells you how it behaves in this respect.”

Of the anionic ligands here, iodide, bromide, chloride and fluoride close at twelve and cyanide closes at eighteen. Of the neutral ones, water closes at twelve, carbon monoxide at eighteen, and ammonia at neither. The sign of eπ sorts all nine perfectly and the charge sorts none of them.

Tested in The count that is not always eighteen · the series on electron count

“The period of the distortion is a property of the lattice.”

The same 120-site ring, with the same hopping and the same elastic cost, chooses period two, three or four according to how many electrons are in it. Nothing about the structure changes between the three cases; only the electron count does.

Tested in The distortion the filling chooses · the series on Peierls distortion

“A basis set that gives the energy to five figures is accurate to five figures.”

The best six-Gaussian fit to hydrogen's 1s gives −0.499946 against −0.5, an error of 0.011 per cent. Its Compton profile at zero momentum is 0.84653 against 8/3π = 0.84883, an error of 0.27 per cent — twenty-five times larger, and larger still at other momenta.

Tested in The measurement a basis was not fitted to · the series on basis

“Improving a basis improves every property it computes at the same rate.”

Going from one Gaussian to six divides the energy error by 1,389 and the profile error by 43. The ratio between the two errors runs 0.8, 4.8, 24.9 across the three bases, so the two do not merely differ — they diverge.

Tested in The measurement a basis was not fitted to · the series on basis

“The error a fit makes is spread evenly over the property it computes.”

The relative error in the three-Gaussian profile crosses zero several times between q = 0 and q = 4 and reaches 16.7 per cent at its worst. A fit is a compromise, and a compromise buys agreement in one place by paying for it in another.

Tested in The measurement a basis was not fitted to · the series on basis

“Hydrogen's l-degeneracy is accidental.”

Six operators built from the shell's own dipole integrals close into an algebra — [Aᵢ, Aⱼ] = −εᵢⱼₖ Gₖ to 1.2 × 10⁻⁸, which is the integrator's noise — and their Casimir comes out at 3.0000000 against n² − 1 = 3. A degeneracy that follows from an algebra that closes is not an accident; it is the signature of a symmetry that is not a rotation of ordinary space.

Tested in The symmetry that is not a rotation · the series on representation

“A picture of a σ bond is two atomic orbitals overlapping.”

At two bohr the surface enclosing ninety per cent of the molecular orbital's density is a single closed surface with both nuclei inside, at |ψ| = 0.0359. The pair of atomic surfaces at the same stated fraction sits at 0.0394, has a waist where the molecular one does not, and encloses 91.70 per cent of the molecular orbital rather than ninety.

Tested in A bond is not two atoms overlapping · the series on contour

“Whether a bonding orbital looks like one object or two is a fact about the bond.”

The contour divides in two at 4.00 bohr if it encloses half the density, 6.76 bohr at ninety per cent, and 9.82 bohr at ninety-nine — a factor of 2.45, on the same orbital at the same separations. Nothing happens to the orbital at any of these; what changes is the level the caption asked for.

Tested in A bond is not two atoms overlapping · the series on contour

“A molecule with a non-degenerate ground state keeps its symmetric structure.”

The second-order lowering from mixing in an excited state goes as λ²Q²/Δ and the elastic cost as ½kQ². Both are quadratic, so the symmetric structure survives only while Δ exceeds 2λ²/k. At λ = 0.5 and k = 0.5 that is a gap of 1.00, and a molecule with a gap of 0.7 distorts to Q = 0.85 with nothing degenerate anywhere.

Tested in A distortion needs two states · the series on Peierls distortion

“Any low-lying excited state will do.”

The triple product of the two states and the distortion must contain the totally symmetric species. In an octahedron a T1u distortion acting on an A1g ground state can mix in exactly one of the ten species — T1u — so a molecule with no T1u excited state cannot distort along that mode at any gap whatever.

Tested in A distortion needs two states · the series on Peierls distortion

“Correlation energy is a small correction.”

For four electrons on four sites the mean field's error is 0.10 at U = t, 1.48 at U = 4t, 12.11 at U = 16t and 27.82 at U = 32t, while the exact energy over the same range shrinks from −3.58 to −0.29. At U = 32t the correction is ninety-four times the quantity it is correcting.

Tested in A mean field cannot get out of the way · the series on correlation

“How much correlation energy a system has is something only a calculation can say.”

The rate at which the error grows with the repulsion is n²/4N — 0.25, 1.00 and 1.50 for two electrons on four sites, four on four, and six on six. The measured slopes are 0.2477, 0.9820 and 1.4715. Nothing in the prediction is an energy: it is the number of times a uniform density puts two opposite spins on the same site.

Tested in A mean field cannot get out of the way · the series on correlation

“A valence bond wavefunction is so many per cent covalent and so many per cent ionic.”

The structures are not orthogonal — at hydrogen's bond length the covalent and ionic ones overlap by 0.873 — so squaring a coefficient is not a weight. The three standard repairs give 18.73, 34.74 and 5.88 per cent for one wavefunction, and each of them sums its weights to one.

Tested in A weight that depends on how it is weighed · the series on models

“The disagreement is a technicality about how the numbers are normalised.”

Take the wavefunction with no ionic structure in it at all. Chirgwin–Coulson and the inverse-overlap convention both report zero ionic weight; Löwdin reports 25.59 per cent, because symmetric orthogonalisation moves ionic character into the covalent structure before anything is squared.

Tested in A weight that depends on how it is weighed · the series on models

“Two structures that a point-group search calls by the same name are equally symmetric.”

Benzene with one atom moved 0.06 Å and benzene with alternating bonds both first come out as D6h at a tolerance of 0.08 Å. Measured against D6h itself, the alternating structure is 0.00962 and the one-atom structure 0.00558 — 72 per cent further, at the tolerance the search treats as identical.

Tested in How much symmetry is left · the series on point group

“Finding the nearest symmetric structure means searching over structures.”

It is an average. For a group of order h with the permutation each operation induces, the nearest structure with that symmetry is qᵢ = (1/h) Σ g⁻¹ r[g(i)], and applying any operation of the group to that returns it exactly — checked to 10⁻⁹ rather than argued for. No minimisation is involved.

Tested in How much symmetry is left · the series on point group

“A molecule has a bond length.”

Hydrogen chloride's potential has its minimum at 1.27455 Å. The average separation in its ground vibrational state is 1.28985, and the length implied by its rotational constant — which measures ⟨1/r²⟩ rather than ⟨r⟩ — is 1.28299. Three numbers from one potential, spanning fifteen milliångström.

Tested in The bond length that depends on the isotope · the series on approximation

“The zero-point motion smears a bond length out symmetrically.”

A harmonic well would. A real one is steeper inward than outward, so the average is displaced outward — 15.30 mÅ for HCl, 3.95 for carbon monoxide — and the displacement scales as the square root of the anharmonicity constant, which is checked here by changing that constant by a factor of four hundred and watching the shift change by twenty.

Tested in The bond length that depends on the isotope · the series on approximation

“Ring strain is angle strain.”

A flat cyclopentane has interior angles of 108.0°, which is 1.5° from tetrahedral, and an angle strain of 0.4 kJ mol⁻¹ by an ordinary bending constant. It is strained by 26 kJ mol⁻¹. The difference is torsional: a flat ring has every bond eclipsed, which costs one ethane barrier — 12.1 kJ mol⁻¹ — per bond, whatever the ring size.

Tested in The strain that is not in the angles · the series on strain

“The chair is the shape the bond angles pick out.”

Bond angles and ring closure leave a continuous family of six-rings — a result of counting constraints against freedoms — and every member has the same angle strain by construction. The best member a search over those constraints returns has a torsional energy of 23.9 kJ mol⁻¹ against the chair's 1.41. What picks the chair out is the torsional term and nothing else.

Tested in The strain that is not in the angles · the series on strain

“A distortion lowers a molecule to a subgroup that depends on the molecule.”

It depends on the species. Applying each of benzene's twelve species as a distortion on its own and recovering the point group from the coordinates gives D6h, C6h, D3d, C2h, C6v, D3h, Cs and C2 — one per species, and two species that describe no displacement of twelve atoms at all.

Tested in How far, and along which coordinate · the series on point group

“Counting levels below the band counts the defects.”

It does exactly, up to one impurity in ten, and then stops: at fifteen per cent a ring of a hundred and sixty carries 23.25 levels below the band for twenty-four impurities, and at twenty per cent 30.50 for thirty-two. Two impurities on neighbouring sites interact strongly enough to push one of their two levels back into the host band, and the shortfall is a measure of how often that happens.

Tested in One defect is a level, many are a band · the series on defect

“There is a concentration at which impurity levels start interacting.”

There is not. The width of the impurity band grows smoothly from zero — 0.014 at two and a half per cent, 0.29 at five, 1.45 at ten — so the levels were always interacting and the question is only when the interaction exceeds whatever resolution is being used to look.

Tested in One defect is a level, many are a band · the series on defect

“The lowest-energy conformer of a ring is the one with the best torsion angles.”

Of the twelve conformers here, the one with the lowest torsional energy — 44.5 kJ/mol, ten below its nearest rival — brings two carbons to 2.639 ångström of each other. The two-term account's favourite is not the roomiest structure, and on the quoted contact potential its cross-ring term alone is 122 kJ/mol.

Tested in The atoms that meet across a ring · the series on strain

“A magnetic moment is what a susceptibility measurement returns.”

A susceptibility measurement returns χ at each temperature. A moment is extracted by fitting a model, and for a coupled pair the model is wrong: fitting the same exact data over 80–150 K gives 3.590 Bohr magnetons and over 300–600 K gives 2.471, a spread of 45 per cent.

Tested in The moment a fit invents · the series on magnetism

“A good fit means the parameters are right.”

The 150–300 K fit reproduces its own data with r² = 0.99950 and returns a Weiss temperature of −78.1 K for a system whose coupling corresponds to −48.0 K in the mean-field conversion. The 300–600 K fit has r² = 0.99999 and returns −51.4. The quality of a fit measures agreement over the range fitted and says nothing about the extrapolation the parameters are.

Tested in The moment a fit invents · the series on magnetism

“The virial ratio checks that a calculation has converged.”

Scaling every exponent is a variation the optimiser is free to make, so at any stationary point the ratio is two identically. It is 2.0000000 for the six-function basis whose energy is right to a hundredth of a per cent and 2.0000001 for the one-function basis whose energy is 15 per cent wrong. A quantity that is true of a bad answer as well as a good one is not a diagnostic.

Tested in The property that gets worse · the series on basis

“Agreement with an exact property is evidence that a wavefunction is right.”

The optimal single Gaussian has ⟨r⟩ = 3/2Z exactly, which is hydrogen's exact value, because the stationarity condition in the exponent is that equation. The same function has ⟨1/r⟩ = 8Z/3π, which is 15.1 per cent below the exact Z, and a density at the nucleus 76 per cent too small.

Tested in The property that gets worse · the series on basis

“Electronegativity is the atom's pull on shared electrons, and that is all a bond needs.”

The pull is the first derivative of the atom's energy with respect to its electron count. Moving charge changes that pull, and the rate at which it changes is the second derivative — the hardness, which runs from 1.92 eV for potassium to 7.30 for nitrogen. The charge that flows is the ratio of the two, not the first alone.

Tested in The quantity no scale prints · the series on electronegativity

“The correlation energy is a property of a system.”

It is the difference between the exact energy and a mean-field one, and the second term is a choice. For four electrons on four sites at U = 32 the restricted reference gives −27.82 and the unrestricted one −0.107. The exact energy in both subtractions is the same number.

Tested in The reference decides the correlation · the series on correlation

“A material with states at the Fermi level is a metal.”

The states are there and each of them lives on ten sites out of two hundred. Growing the ring by a factor of four grows the clean chain's states by 3.8 and the disordered chain's by 1.27, so at that disorder a state has stopped knowing how large the system is — which is what makes it unable to carry anything from one end to the other.

Tested in The third way to be an insulator · the series on metal

“Disorder makes a gap.”

It does not. The band widens rather than splitting — from 4.00 to 6.80 in units of the hopping as the disorder goes from zero to four — and the density of states at the middle stays finite. Nothing is removed from the Fermi level; what changes is what the states there look like.

Tested in The third way to be an insulator · the series on metal

“A broad band means a poorly resolved one.”

The breadth is computed here from two measured potential curves and comes out as a distribution over the ion's vibrational states, with 0.263 of the intensity in the lowest and 0.318 in the next. The band is spread because the geometry changed, and its width in quanta is √S, the square root of the displacement parameter.

Tested in The width of a band is a bond length · the series on photoelectron

“Bonding and antibonding cannot be told apart in a spectrum.”

The sign of the bond-length change separates them and the vibrational spacing measures it. Removing nitrogen's 1πu electron lengthens the bond and lowers the vibrational frequency from 2359 to 1904 wavenumbers; removing the 2σu electron shortens it and raises the frequency to 2420. Both are read off the same spectrum.

Tested in The width of a band is a bond length · the series on photoelectron

“Hypervalency happens when an atom uses more than four orbitals.”

In every geometry here the number of ligand σ combinations that transform as one of the central atom's s and p orbitals is at most four — two for a linear ligand set, three for a trigonal planar one, four for tetrahedral, trigonal bipyramidal, square planar and octahedral. Six ligands and four matches leaves two combinations with no partner, and the electron count exceeds eight by exactly four.

Tested in Four is all that s and p can match · the series on hypervalency

“Hypervalent molecules are the ones with more than four ligands.”

Xenon difluoride has two ligands and is the archetype. The census makes hypervalency the condition n + L > 4, counting lone pairs as well as ligands, and the two mechanisms are different: sulfur hexafluoride's leftover pair has no central orbital of its symmetry, xenon difluoride's has one that is already holding a lone pair. Xenon tetrafluoride has one of each.

Tested in Four is all that s and p can match · the series on hypervalency

“The octet is broken in these molecules.”

The electron count around the centre exceeds eight and the electrons on the centre do not. In every row of the census the centre holds at most eight, in at most four orbitals, and the excess sits in ligand combinations with no central partner — which is why the ligands in a hypervalent molecule carry the negative charge.

Tested in Four is all that s and p can match · the series on hypervalency

“β is measured from a spectrum.”

A spectrum with two lines in it constrains two combinations of three parameters. Fitting benzene's two π ionisation energies exactly at neighbour overlaps from 0 to 0.30 gives resonance integrals from −3.05 to −7.66 electronvolts, a factor of 2.5, and every member reproduces both measurements to the last digit the solver carries.

Tested in One spectrum, a line of models · the series on models

“The bond orders are a structural prediction, so they do not depend on the parametrisation.”

The orbitals do not depend on it — the same six vectors come out at every overlap, pointing in the same directions to nine decimal places. The bond orders computed from them do: butadiene's terminal bond order is exactly twice its central one at zero overlap and 2.671 times it at 0.30, because each orbital is normalised against a different metric.

Tested in One spectrum, a line of models · the series on models

“A strong infrared band is a big motion.”

Across the active modes of five molecules the rank correlation between band strength and root-mean-square zero-point displacement is +1.00, +0.50, +0.20, −0.50 and −0.66. Two of the five run backwards, and boron trifluoride's strongest band has the third smallest amplitude of its five.

Tested in The mode that moves least radiates most · the series on dipole

“Heavier atoms moving means a stronger band.”

Every normal mode moves exactly the same weighted amount of mass — the sum of m|d|² is one for all of them, by the mass-weighted normalisation — so that quantity distinguishes nothing. What does vary is the reduced mass, from 1.03 to 20.3 across these molecules, and for boron trifluoride it ranks the intensities exactly backwards, at −1.00.

Tested in The mode that moves least radiates most · the series on dipole

“Infrared intensities are the part of a spectrum a simple model gets roughly right.”

They are the part it gets least right. Which bands exist is exact from the group alone and agrees with the computed derivatives on all twenty-eight modes; the positions come from a fitted force field and are good; the intensities need the electron density's response to a nuclear displacement, which a fixed-charge model does not contain at all.

Tested in The mode that moves least radiates most · the series on dipole

“The electron count depends on how the ligands are counted.”

It does not. Ten complexes counted both ways give the same total in every case — an anionic ligand brings one electron and leaves the metal neutral, or brings two and takes one from the metal, and the sum is unchanged by construction. The invariance is the useful part of the rule.

Tested in The same count, two oxidation states · the series on electron count

“The oxidation state is a property of the compound.”

Hexaamminecobalt(III) is cobalt(III) on the ionic convention and neutral cobalt on the other, and both counts give eighteen. For a nitrosyl the ionic convention itself does not settle it: counting NO as a cation, a neutral radical or an anion puts cobalt at −1, 0 or +1 in the same molecule, with a total of eighteen in all three.

Tested in The same count, two oxidation states · the series on electron count

“Since the total is the same, the choice of convention does not matter.”

Everything except the total differs, including the d count that a ligand-field diagram is labelled with. For three complexes with measured moments the ionic d count predicts 0.00, 4.90 and 2.83 Bohr magnetons against measurements of 0.00, 5.40 and 3.20; the group-number count predicts 1.73, 2.83 and 0.00, and is wrong by more than a Bohr magneton in each.

Tested in The same count, two oxidation states · the series on electron count

“If two atoms' orbitals do not overlap in the picture, they do not interact.”

Two ninety per cent contours of hydrogen 1s orbitals stop touching at 5.322 bohr. The overlap integral there is 0.0768 — larger than the overlap that produces a hydrogen bond — and it is still 0.0222 at seven bohr, where the two spheres are separated by a bohr and a half of apparently empty space.

Tested in The tenth that is not drawn · the series on contour

“The structure that fits all the measured constants is the right one.”

Fitting the deuterated molecule's B as well picks the member at a C=O length of 1.2495 ångström, a C–H of 1.0222 and an angle of 134.6 degrees, against an accepted structure of 1.2033, 1.1005 and 116.3. The measured constants are vibrational averages and no rigid structure reproduces them: the measured moments fail the planarity relation by 0.057 u Ų, which every structure in the family satisfies exactly.

Tested in Three numbers is not a structure · the series on rotation

“A Möbius ring is a different kind of system from a flat one.”

It is a flat ring at exactly half a flux quantum. The twisted eigenvalues 2cos((2k+1)π/n) and the flux expression 2cos(2π(k+½)/n) are the same list for every ring from three to ten, to the last digit the solver carries, because a twist and a half-quantum put the same half-integer into the label.

Tested in Two rules that share no arithmetic · the series on aromaticity

“The binding energy per bond is a transferable quantity.”

Divide the bindings above by four and the same spread remains. A bond in a structure with many short closed walks through it is worth less than the same bond elsewhere, because the binding is a sum over a whole distribution of levels and the distribution is not fixed by counting neighbours.

Tested in Two structures with the same neighbours · the series on cohesion

“Error cancellation is a property of the method.”

It is a property of the pair. The same mean field applied to the singlet and the triplet of one four-site chain makes errors of 12.11 and 0.17 at the same repulsion — a factor of seventy — because one of the two states is nearly a single determinant and the other is not.

Tested in Two wrong numbers and a right difference · the series on approximation

“A method that gets total energies badly wrong can still order two states correctly.”

From a repulsion of four times the hopping the mean field puts the triplet 0.53 below the singlet, where the exact answer has it 0.54 above. Not an inaccurate gap: the wrong ground state, and it gets worse as the repulsion rises.

Tested in Two wrong numbers and a right difference · the series on approximation

“More neighbours means a more Gaussian band.”

A chain reaching to its third neighbour has six neighbours like a cubic structure and a fourth moment of 3.389 against the cubic 2.500 — on the far side of a Gaussian's 3, where no hypercubic structure of any dimension can go. A hypercubic band approaches 3 from below and never reaches it.

Tested in A band becomes a bell curve · the series on Bands in a solid

“Two structures with the same coordination have the same band.”

A chain touching its second neighbours and a square net both have four neighbours and the same normalised fourth moment, 2.250 to nine figures. Their sixth moments are 6.719 and 6.250, and their binding per unit band width differs by 27 per cent.

Tested in A band becomes a bell curve · the series on Bands in a solid

“Which modes appear depends on how big the geometry change is.”

It depends on the group. The zeros above are zeros of a projection between a symmetric displacement and a non-symmetric mode, and they stay zero at any size of change. An unsymmetric change of the same size activates them: the same modes get factors up to 0.34.

Tested in A band is a filter on the modes · the series on photoelectron

“A molecule with several progressions in one band has a low-symmetry ion.”

It has more than one totally symmetric vibration. Sulfur dioxide is C2v and has two, and a symmetric change of geometry gives both of them intensity — 0.145 and 2.089 — while boron trifluoride, of much higher symmetry, has one and shows one.

Tested in A band is a filter on the modes · the series on photoelectron

“Every molecule's internal coordinates span its vibrations.”

Carbon dioxide's do not. Its bending coordinate has an identically zero row in the matrix that turns a displacement into a coordinate change, because an angle at exactly 180° is stationary, so three coordinates span two directions where the molecule has four vibrations. Both bending modes have no coordinate at all.

Tested in More coordinates than motions · the series on normal mode

“A force field fitted to every observed frequency of a molecule is determined by them.”

For a redundant coordinate set it is determined only up to a whole direction. Adding λ r rᵀ to the force constant matrix, for a redundancy r, changes the Cartesian Hessian by exactly zero — measured at 5.5 × 10⁻⁸ for methane at λ = 25, against 135 per cent for the same nudge along an ordinary coordinate.

Tested in More coordinates than motions · the series on normal mode

“A symmetric cage localises to symmetric bonds.”

The six-vertex cage's four skeletal pairs come out as two two-centre orbitals and two spread over five atoms, reaching a functional of exactly 11/6 from every start tried. The maximum breaks the octahedron's own symmetry, and it is the unique maximum.

Tested in One scale, from two centres to a cage · the series on multicentre

“A ninety per cent contour is a picture of how big an atom is.”

The surface two closed shells actually stop at encloses 99.73, 99.75 and 99.38 per cent of the outermost orbital's density for helium, neon and argon. A ninety per cent contour is far inside where the next atom is, and the two conventions differ by an amount that a picture cannot make small.

Tested in The radius that was tabulated · the series on contour

“Slater's rules and a Hartree–Fock fit are two ways of writing the same effective charge.”

They disagree by thirty-one per cent for neon's 2p — 5.85 against 4.4532 — and the disagreement is the difference between a contact distance 4.7 per cent too long and one 27.7 per cent too short. Helium, where the two agree to one part in a hundred, does not move.

Tested in The radius that was tabulated · the series on contour

“The secular determinant's overlap denominators move the maximum of the bonding away from the maximum of the overlap.”

They do not, and the calculation refused the expectation. At fixed atomic energies the stabilisation is a function of the overlap alone and a strictly increasing one, so it peaks where the overlap peaks: 4.16818 against 4.16817 bohr for a 1s with a 2s, and 7.58281 against 7.58281 for a 2s with a 2p.

Tested in The same overlap, a different bond · the series on overlap

“Doubling the overlap doubles the bonding.”

For two orbitals at the same energy the stabilisation is exactly 2(1−K)αS/(1+S), which saturates. Going from 0.05 to 0.1 buys 1.909 times the bonding; going from 0.4 to 0.8 buys 1.556.

Tested in The same overlap, a different bond · the series on overlap

“The spectrochemical series has no structural explanation.”

One computed overlap integral, squared, accounts for most of it. Across five chromium(III) complexes whose splittings vary by a factor of 1.96, the ratio of splitting to squared overlap varies by 1.31 — with the exponent imposed by the model rather than fitted, and with the bond lengths taken from structure determinations rather than adjusted.

Tested in The splitting against something structural · the series on ligand field

“A triple bond's bent components are a different kind of hybrid from a double bond's.”

They are the same hybrid. Both have exactly one sixth s character and both make an angle of 101.53696 degrees with one another. A double bond's carbon is sp², shares a third of an s orbital with the C–C region and divides it between two components; a triple bond's is sp, shares a half, and divides it between three. One third over two is one half over three.

Tested in Three bent bonds, and the same hybrid · the series on hybrids

“The angle between two equivalent hybrids depends on the molecule.”

It depends on their s character and on nothing else. Two hybrids √s|s⟩ + √(1−s)|p⟩ are orthogonal exactly when s + (1−s)cos θ = 0, so cos θ = −s/(1−s). At a quarter that is 109.47122 degrees, at a third 120, at a sixth 101.53696 — and the computed angles here reproduce all of them.

Tested in Three bent bonds, and the same hybrid · the series on hybrids

“The bent-bond and σ–π descriptions of a triple bond are rival accounts.”

They are two bases for one three-dimensional occupied space, related by an orthogonal mixing. Every mutual overlap between the three bent bonds is zero to 10⁻³, which is the quadrature's limit rather than the algebra's, and the charge each puts off the axis is the same 0.31519 ångström for all three.

Tested in Three bent bonds, and the same hybrid · the series on hybrids

“A dimensionless ratio between two molecules is a safe way to quote a comparison.”

It is immune to both exact symmetries of the model — moving the zero of energy and changing the unit leave it alone to 10⁻¹⁵ — and it is the single most reference-sensitive quantity in the table. Benzene's delocalisation energy over cyclobutadiene's is infinite against one reference, because cyclobutadiene's is exactly zero, and 2.14 against the other.

Tested in Which numbers carry a frame · the series on delocalisation

“An orbital energy is a quantity.”

It moves under both exact symmetries. Adding 0.37 to the zero of energy moves benzene's highest occupied level by 0.37 β and changing the unit by 1.83 moves it by 0.83 β, and neither change alters anything about the molecule. It is a coordinate, and only differences of coordinates are quantities.

Tested in Which numbers carry a frame · the series on delocalisation

“Every quantity a Hückel calculation prints depends on the parameters.”

Three of the nine tested do not depend on any of the four changes: the spread of bond orders, the ratio of two bond orders, and the largest charge on an atom. All three are properties of the eigenvectors, which are unchanged by any shift or scaling of the eigenvalues.

Tested in Which numbers carry a frame · the series on delocalisation