Ionisation energy — where it appears
Named by 28 essays across 4 fields — each of them below, with the objects they name alongside it.
Hybridisation does not explain
Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.
The aufbau order is not a property of the atom
Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.
What a photoelectron spectrum measures
The bands of a photoelectron spectrum are routinely read off as orbital energies. They are ionisation energies, which is a different quantity — and the identification rests on two errors of about an electronvolt each that happen to have opposite signs.
Water's lone pairs are not a pair
Every course draws two equivalent lone pairs on water, pointing away from the hydrogens like a pair of ears. Its photoelectron spectrum shows the two bands they would produce at 12.6 and 14.7 electronvolts, two point one apart, in different symmetry species.
A band gap is not a bond energy
Silicon's gap is 1.1 electronvolts and its Si–Si bond is 2.3. Both are quoted in the same units, both describe the same material, and neither is convertible into the other — one is the cost of promoting an electron and the other is the cost of taking two atoms apart.
Fewer bands than electrons
Methane has eight valence electrons and two photoelectron bands. Ammonia has eight and three; water has eight and four. The count is not of electrons, not of bonds and not of orbitals — it is of the symmetry species the occupied orbitals fall into, and it falls as the symmetry rises.
Koopmans' theorem is exact for nothing
Reading an ionisation energy off an orbital energy neglects two things that pull in opposite directions, and the cancellation between them is quoted as the reason it works. Compute all three energies in a model where the exact answer is available and the cancellation is real, partial, and gone by the time the repulsion is twice the hopping.
The quantity no scale prints
Every electronegativity table is half of a calculation. The other half is the hardness — half the difference of the same two measurements the Mulliken scale is half the sum of — and it varies by a factor of 3.8 across eighteen elements. Put both halves in and B–F, with an electronegativity difference of 6.12 electronvolts, moves less charge than lithium iodide, whose difference is 3.75.
The width of a band is a bond length
Nitrogen's three photoelectron bands are one sharp line, a progression of five, and a line with a shoulder. Computed from the measured bond lengths of the three states of the ion, the intensities come out at 0.917, 0.263 and 0.880 in the first line of each — because removing a weakly bonding electron lengthens the bond by 18.7 thousandths of an ångström, a strongly bonding one by 77.2, and an antibonding one shortens it by 23.7.
More bands than there are orbitals
A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.
The value that only exists in the bond
A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.
A hundred lines and no way to sort them
A spectrum with a hundred lines has six fundamentals in it somewhere. Sorting by height works until the tallest satellite is as tall as the shortest band, and sorting by position works until satellites start arriving between the bands — and on a ring of six both stop working at the same repulsion.
A mean that is low rather than right
Sanderson's rule takes the geometric mean of the atoms' electronegativities, and it works better than the plain average. Computed against the exact equalised answer for twelve molecules it is better by erring downwards — and two of the twelve have exact answers above the plain average, where no geometric mean can go at any parameter.
A parameter that never finds a value
Show a two-parameter model six measurements instead of two and it stops being underdetermined and starts being wrong. Adding the third parameter improves the fit by one part in eighty, moves the resonance integral by a factor of four, and never finds a best value at all — because nine tenths of the error is a term the model does not have.
The boundary belongs to the gap
A satellite stops being tellable from a fundamental somewhere, and it can be located on one ring at one filling. Four systems put it at repulsions of 2, 4 and 8 — ordering exactly with each one's own one-electron gap and not with its band width — and the fourth, whose gap is zero, has no boundary at all: its satellites are indistinguishable at every repulsion including none.
Two systems a model cannot tell apart
Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.
Where a closed form stops being one
What happens when the quadratic energy is not enough? A cubic makes the equalisation condition a quadratic with two roots, and something has to choose between them. The choice is easy and the finding is somewhere else — a cubic fitted through the dication gives lithium an electronegativity of −7.907 eV, a capacity of 0.073 of an electron, and a molecule of two alkali metals no solution at all.
The pair that is not a tie
An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.
A capacity that is largest where there is none
A cubic through four charge states implies a largest amount of electron an atom can accept, and it is natural to ask whether the idea survives its own model. It does not. The three atoms whose anions are not bound — beryllium, magnesium and nitrogen — have the largest capacities in the set, two of them infinite; the three smallest capacities all belong to atoms whose anions are bound. And among the fourteen where the test cannot bite, the ordering is sensible.
The gap that would have to be smaller
An angular overlap parameter is an overlap squared over an energy denominator, and the usual fit folds the denominator away. Put the measured ionisation energies back in and the denominator alone over-predicts the trend down the halide group at every metal level a donor permits — the smallest it can give is 1.67 against a fitted 1.43. So the overlap has to shrink down the group, which is the opposite of the usual expectation.
A satellite that never loses its place
The repulsion at which a satellite stops being tellable from a fundamental orders exactly with the one-electron gap across four systems. Changing the gap by the filling instead is the sharper test, and the ordering does not survive it: a six-site chain has a larger gap at half filling and a smaller boundary. Below half filling there is no boundary at all, at any repulsion up to sixty-four times the hopping.
A denominator that fails both ways
The energy gap an e_π folds away over-predicts the halide trend at every metal level a donor allows, and the acceptors look out of reach because a π* is not an atomic level. It is measurable — a slow electron is captured by it — and on that side the same denominator under-predicts. No metal level fixes either, and the two want it moved in opposite directions.
A ratio of exactly one is a tie
Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.
A correlation is not an account
A quantity called the chemical capacity is ordered against the thing it was supposed to predict and correlates instead with the second ionisation energy, which invites asking how much of it that accounts for — on the reasoning that a quantity which is ninety per cent of one input has a simpler name than the one it carries. It is three per cent of it. Nitrogen and potassium sit two electronvolts apart in the second ionisation energy and differ two hundredfold in capacity.
The worst of the six was the one we asked about
The capacity's correlation with the second ionisation energy explains 2.9 per cent, and the natural next step is the same pair test against every other candidate input. Every input fails it — but the second ionisation energy fails it by a factor of twenty-four more than the best, and the test itself had to be repaired first, because the version the question implied reports the capacity failing to be a function of itself.
Six of fifteen change verdict
The chemical capacity is finite or infinite according to the sign of a coefficient that is one sixth of a second difference of three measurements, and the pair test asked how much of its reported ordering is the quantity and how much is the fit's resolution. Adding a fifth point changes the verdict on six of fifteen atoms — and the five it unbounds are the five it had largest.
The channel that points at the metal
Two ligands in the spectrochemical series carry a fitted π parameter no quotient of squared overlaps can produce, because it is negative. Giving the derivation the second interaction it lacks makes both of them negative at every metal level — and the reason is not the energy denominators, which favour the donor channel in all three cases. It is where each orbital keeps its amplitude.
Four quantities go and one question stays
The exact theory says an atom's energy against electron count is straight segments between integers, and that model gets the alkali metals right for free — which is exactly where the fitted curves give lithium a negative electronegativity. What it costs is the hardness, the capacity, the electronegativity and equalisation, and what it leaves is a question whose answer is an integer.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitConventionElectron affinityElectronegativityApproximationPhotoelectron spectroscopyPhotoelectron spectrumClosed formExact diagonalisationHubbard modelKoopmans theoremLeast-squares