Molecular Geometry & Bonding
what the shape decides
Concept

Electron affinity — where it appears

The energy released when an atom binds an extra electron, measured by photodetachment where the anion is stable. Half its sum with the ionisation energy is the Mulliken electronegativity and half their difference is the hardness, so an element whose anion is unbound has neither quantity measured.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

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.

wrong · Bands in a solid Two derivatives of the same energy, and only one is tabulated. The 18 elements of the electronegativity tables, placed by their chemical potential — half the sum of the ionisation energy and the electron affinity, which is the Mulliken electronegativity — against their hardness, half the difference of the same two numbers. Hardness runs from 1.92 to 7.3 electronvolts, a factor of 3.8, and does not follow the horizontal axis. Ringed points are the three elements whose anion is not bound, so whose affinity is not a measurement.

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.

bonding · Electronegativity seven sets of parameters, one spectrum. Benzene's π levels from seven sets of the three Hückel parameters, each fitted to reproduce the two measured ionisation energies exactly. The two occupied levels sit at the same energy in every column, because that is what was fitted. The empty level moves from -3.15 to 4.69 electronvolts across the family, and the resonance integral from -3.05 to -7.66.

One spectrum, a line of models

Hückel theory has three parameters and benzene's photoelectron spectrum supplies two numbers, so the fit has a curve of solutions rather than a point. Along it the resonance integral runs from −3.05 to −7.66 electronvolts, the empty level moves by eight, the terminal-to-central bond order ratio in butadiene goes from 2.000 to 2.671 — and the delocalisation energy is 6.100 electronvolts in every member.

bonding · Models Carbon is not one number. The charge equalisation puts on carbon in the four fluoromethanes, and on hydrogen in the three that have one. Carbon's runs from 0.07 to 0.31, and hydrogen's changes sign between methane and fluoromethane, so the same C–H bond is polarised one way in one molecule and the other way in the next. A table with one number for carbon is printing the value an iteration starts from.

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.

bonding · Electronegativity Four unweighted rules against the answer the hardnesses give. Twelve molecules, each drawn with the exact equalised electronegativity as a filled mark and the four unweighted means of the free atoms' values as open ones. The geometric mean — Sanderson's rule — is the closest on average, at 0.0706 eV against the arithmetic mean's 0.1666, and it is the closest on only 4 of the twelve. Every rule below the arithmetic mean is constrained to sit below it, and 2 of the exact answers do not.

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.

wrong · Electronegativity A fourth data point, and 3 negative electronegativities. Each element's electronegativity from three points on its energy curve, the cubic coefficient a fourth point adds, and what the fourth point leaves. The cubic coefficient is one sixth of a second difference of the ionisation series, so it is largest where that series has a kink — and the alkali metals, whose second electron comes out of a closed shell, are pushed to Li -7.91, Na -3.42, K -1.49 eV. The last column is where the two roots of the fixed-point equation collide, beyond which the atom has no solution at all.

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.

wrong · Electronegativity The capacity, against whether the anion exists. Each atom at its electron affinity and the capacity the cubic model gives it — the largest amount of electron the model says it can accept. A negative affinity is an anion that is not bound, which is the statement that the true capacity at the integer is zero. The three atoms with negative affinities are beryllium, magnesium and nitrogen, and they have the largest capacities in the set: two of them infinite and the third 31.5. The three smallest capacities all belong to atoms whose anions are bound.

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.

wrong · Electronegativity The capacity against the quantity it was supposed to be. The chemical capacity of 14 atoms against their second ionisation energy, with the capacity on a logarithmic axis because it spans nearly three orders of magnitude. The suspected relation is not there: N and K are 2.02 electronvolts apart in the second ionisation energy and differ by a factor of 193 in capacity, which no function of one variable can produce.

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.

wrong · Electronegativity The input the question named is the worst of the six. For each candidate input, the largest capacity ratio between two atoms that are neighbours in that input. A quantity the capacity were a function of would have a small bar. The dashed line is the floor — neighbours in the capacity itself still differ by ×5.0, because fourteen atoms spread over a factor of four hundred cannot do better. The second ionisation energy is ×193, which is 39 times that floor.

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.

wrong · Electronegativity Adding one measurement, and six of fifteen change verdict. Each atom's chemical capacity from a cubic fitted through four electron counts and from a quartic fitted through five, on a logarithmic axis, with an unbounded capacity drawn at the right-hand margin. Five atoms go from a finite capacity to an unbounded one and one goes the other way — and the five are exactly the five largest the cubic reported. The seven that stay finite keep their order and change their values.

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.

wrong · Electronegativity Straight segments, and the curve fitted through their ends. Cl's energy against the charge it carries. The exact theory says the energy is straight between integers, with a kink at each one: the slope below the neutral atom is the electron affinity and the slope above it is the first ionisation energy, and the two are different numbers. Every quantity in this argument comes from the smooth curve fitted through those points instead — and the curve's second derivative, which is the hardness, is a property the segments do not have at all.

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.

wrong · Electronegativity

Named alongside it

The objects these essays reach for when they reach for this one.

Ionisation energyElectronegativityModel limitConventionClosed formPartial chargeHardnessCharge transferChemical hardnessApproximationLeast-squaresMulliken scale

All concepts