Four quantities go and one question stays

Every quantity in this family is a derivative of a curve. The electronegativity is the slope of an atom’s energy against its electron count, the hardness is its curvature, the capacity comes from the next derivative up, and equalisation sets the first of them equal across a molecule. The curve is fitted through the energies at integer electron counts — the anion, the neutral atom, the cation, and in the later essays the dication and the trication.

The exact theory says there is no curve. An atom’s energy against electron count is straight segments between integers, with a derivative discontinuity at each one.

What the segments are, and why they are right #

The slope of the energy just below the neutral atom is the electron affinity and the slope just above it is the first ionisation energy, and those are different numbers — 3.61 and 12.97 electronvolts for chlorine. A smooth curve through the integers has one slope at the neutral atom, somewhere between them, and the quadratic model’s choice is their mean.

The piecewise shape is not a refinement. It is what the exact theory gives for a system at zero temperature with a fractional average electron count, which is a statistical mixture of integer states rather than a state with a fractional number of electrons in it — so the energy is the straight interpolation between the two nearest integers, and the kink at each integer is the ionisation energy meeting the affinity.

And it fixes for free the thing the fitted models get worst. The cubic gives lithium an electronegativity of −7.907 electronvolts and a capacity of 0.073 of an electron, and a molecule of two alkali metals no solution at all. On the segments there is nothing to give: lithium’s slope below zero charge is 0.618 and above it is 5.392, both positive, both measured, no fit anywhere, no negative anything.

So the obvious question is what electronegativity theory looks like written this way, and it is the one route named earlier and never taken.

What it costs #

The hardness is a second derivative. A function made of straight segments has a second derivative of zero inside every segment and an undefined one at every integer. There is no hardness — not a small one, not a different one.

The capacity is built from a third derivative and goes the same way, twice over: it needs a curvature to divide and a cubic coefficient to divide by, and neither exists.

The electronegativity becomes a pair. The chemical potential is a one-sided slope, and there are two of them at every integer. The quadratic model’s electronegativity is their mean, and the mean is a choice rather than a quantity — which is the same complaint the first of these essays made about the four scales, arriving from a different direction.

And equalisation has nothing to set equal. Its whole content is that one number is the same across a molecule; with a pair at every integer and a flat slope in between, the condition is either trivially satisfied or unstatable.

So four quantities go. What stays is a question with an integer answer: does moving one whole electron from one atom to another lower the energy?

On isolated atoms the answer is no, everywhere #

Moving an electron from A to B costs A’s first ionisation energy and returns B’s electron affinity, so the cost is their difference. Every bond in the set has a positive cost in both directions.

The cheapest bond this collection prices is potassium–bromine at 0.977 electronvolts. The cheapest pair available anywhere among these seventeen atoms is potassium to chlorine at 0.729 — the least tightly bound electron in the set going to the most avid acceptor in it, which is as favourable as the periodic table’s own extremes allow.

Positive. So on isolated atoms the exact model predicts every bond non-polar, and it predicts it without a parameter to blame.

The separation at which even the cheapest becomes favourable is 19.75 ångström, which is where the Coulomb attraction of the resulting ion pair first pays for the transfer. No bond is anywhere near that, which is the honest statement of how badly isolated atoms mislead here.

Where the threshold sits, and what it is made of #

The nineteen ångström is worth unpacking, because it is the whole distance between the two verdicts.

An ion pair at separation R attracts with 14.40 electronvolt-ångströms divided by R, so the transfer becomes favourable when the cost falls below that — at R equal to 14.40 over the cost. For the cheapest pair in the set that is 19.75 ångström; for the cheapest bonded pair, potassium and bromine at 0.977, it is 14.74.

Those are enormous distances for a chemical question. At an ordinary bond length of two ångström the attraction is 7.2 electronvolts, which pays for a transfer costing anything up to that — and eight of the seventeen atoms have first ionisation energies above it, so the threshold is not a formality that every bond clears.

What makes the isolated answer so different from the bonded one is therefore a factor of ten in a length, and it is worth saying which way the surprise runs. The surprise is not that isolated atoms do not transfer; nothing does at nineteen ångström. It is that the model’s whole content is a comparison of two numbers, one of which is eight electronvolts larger than the other at the distances chemistry cares about — so the verdict is decided by the geometry far more than by the periodic table, and two atoms with a large transfer cost and a short bond come out the same as two with a small cost and a long one.

With the attraction in, the answer is a whole electron #

An atom in a molecule is not isolated, and the piece the isolated calculation omits is the largest one: two ions at a bond length attract each other. At the measured separations that attraction runs from 5.10 electronvolts in potassium bromide to 15.71 in hydrogen fluoride.

Against transfer costs of 0.977 to 10.23, the attraction wins for five of the eight bonds. Hydrogen fluoride, hydrogen chloride, lithium fluoride, sodium chloride and potassium bromide come out with a whole electron moved; hydrogen bromide, hydrogen iodide and chlorine fluoride come out with none.

And that is the whole of what the model can say. There is no intermediate. A fractional transfer is not a state the piecewise energy has — that was the point of the straight segments — so the prediction is one electron or zero, and the cases sit on either side of a threshold that is a bond length and an energy difference.

And every measurement is a fraction #

A dipole moment divided by a bond length is a charge, in units of the electron, and every one of the eight is a fraction: 0.415 for hydrogen fluoride, 0.181 for hydrogen chloride, 0.122 for hydrogen bromide, 0.058 for hydrogen iodide, and 0.842, 0.794 and 0.784 for the three alkali halides.

Not one of them is near zero and not one is near one. The alkali halides come closest to a whole electron at four fifths, and four fifths is not one — the same twenty per cent of an electron the equalisation models have been arguing about for six essays, and a difference does not make a transfer was the second essay’s way of saying it.

So the model is wrong about every number. It has no parameter to adjust and no adjustment would help, because the disagreement is not in a magnitude but in a kind: the prediction is an integer and the measurement is not.

It classifies better than it has any right to #

The fair comparison is kinder, and it is worth making because the unfair one is available and misleading.

Round each measured charge to the nearest integer. The four hydrogen halides and chlorine fluoride round to zero; the three alkali halides round to one. Compare against the prediction: six of the eight agree.

The two that do not are hydrogen fluoride, predicted a whole electron and measuring 0.415, and hydrogen chloride, predicted a whole electron and measuring 0.181. Both are cases where the attraction wins by a modest margin — 5.5 and 1.3 electronvolts — so both are near the threshold, which is the right place for a two-valued model to be wrong.

Six of eight, from a model with no fitted parameter of any kind, using only measured ionisation energies, affinities and bond lengths. That is better classification than the quadratic model’s fractional answers give, since a fraction has to be compared against a measurement to be right or wrong and an integer is right or wrong on its own.

So the verdict has two halves and neither is the obvious one. The exact energy shape sorts bonds into ionic and covalent with no parameters and gets six of eight; and it cannot say how ionic any of them is, at all, ever. Replacing a model that gives underdetermined fractions with one that gives determined integers is not an improvement and is not a regression. It is a different instrument.

What was computed, and how #

Every energy is quoted: seventeen first ionisation energies, seventeen electron affinities, eight bond lengths and eight dipole moments. The transfer cost is a subtraction, the attraction is the Coulomb energy of two unit charges at the measured separation, and the charge implied by a dipole is the dipole divided by the length in the units the measurement is quoted in. There is no fit and there is nothing to converge.

Fourteen things are checked. That no bond in the set has a favourable integer transfer on isolated atoms. That the cheapest pair available anywhere among the seventeen is positive, and is about three quarters of an electronvolt. That the separation at which even the cheapest bonded pair becomes favourable is longer than any bond. That with the attraction in, most of the bonds with measured dipoles do come out favourable. That the attraction at each measured length is between five and sixteen electronvolts, so none of them is an extrapolation. That every measured dipole implies a fraction between a twentieth and 0.95. That the fractions are spread across the range, so no single integer is nearly right. And that rounded to the nearest integer the prediction agrees with most but not all, with the disagreements named.

The two-part claim about the rounding is the one that keeps the essay honest. Claiming only that the values are wrong would make the model sound worse than it is; claiming only that six of eight classify correctly would make it sound better. Both are checked, and the second with an upper bound as well as a lower one — most and not all — so a version of the model that agreed with everything would fail the check rather than pass it.

Where this stops #

The Coulomb term is a caricature and it is the whole of the molecular physics here. Two point charges at the bond length is the crudest possible account of what holds an ion pair together; a real calculation has overlap, polarisation and the repulsion of the closed shells in it. The threshold computed here is therefore soft, and the two bonds that classify wrongly are the two nearest it.

The piecewise energy is exact for the total energy of a system at zero temperature with a fractional average electron count, and it is not a licence to read every consequence off it. In particular the segments are a property of the atom in isolation, and an atom in a molecule is coupled to another — which is where the fractional charges actually come from. A charge of 0.415 is not an atom holding a fractional electron; it is an integrated density in a region whose boundary is a convention, and that convention is worth a factor of several.

So the comparison is between two things that are not quite the same quantity. The model predicts how many electrons transfer; the dipole measures a charge distribution’s first moment. Those agree exactly only for two point charges, which is the caricature above. The disagreement between an integer and 0.415 survives that reservation because it is a disagreement in kind; a disagreement of twenty per cent would not.

The generalisation #

The habit is to ask what a model loses when it is made more nearly exact, and to count the losses rather than assume there are none.

A more faithful model is usually taken to be strictly better, and the reasoning is that whatever the cruder one got right it still gets right. That reasoning fails whenever the crude model’s quantities are derivatives of something the faithful one represents differently, because a derivative is not a property that survives a change of functional form. Here four quantities are derivatives of a curve and the faithful model has no curve, so four quantities have no counterparts — and each of them was doing work.

The corollary is about what to do when the faithful model answers a different question. The piecewise energy does not give a worse answer to how much charge moves; it declines the question and answers does charge move instead. Those are both useful and neither substitutes for the other, and the mistake available here is to grade the second as a poor attempt at the first. A model that changes the question has to be graded on the question it changed to, and on that question it gets six of eight with no parameters.

Who found it, and when #

The piecewise-linear energy is Perdew, Parr, Levy and Balduz, 1982, and it is the reason a fractional electron count is a mixture rather than a state. The quadratic model it displaces is Iczkowski and Margrave, 1961. The ionisation energies, affinities, bond lengths and dipole moments are all quoted. What is computed here is the transfer cost for every bond this collection prices, the attraction at each measured length, the integer prediction and its comparison with the charge each dipole implies.

The number worth carrying is not six of eight. It is 0.729 electronvolts: the cost of the most favourable electron transfer the periodic table’s own extremes allow, in a model whose whole prediction is whether that number is positive.

Still open: the two atoms, coupled #

The obvious open question is the coupling the segments do not have. The piecewise energy is exact for one system with a fractional average electron count, and a bond is two systems sharing electrons — so the right object is not two atomic energy curves but a single curve for the pair, whose own segments are in the total count and whose fractional charges come from the pair’s ground state being a superposition rather than a mixture. That is a different calculation and the means for it are already here on a two-site model: a Hubbard dimer’s ground state has a fractional double occupancy at every repulsion, and its charge on each site is fractional for exactly the reason the piecewise argument cannot see. Running the transfer question on that model rather than on two isolated atoms is the version that could produce a fraction.

The nearer question is the threshold’s softness. Two bonds classify wrongly and both are within a few electronvolts of the point where the attraction stops paying, which invites asking how far the threshold would have to move to fix them and whether the corrections the point-charge picture omits are that large. Closed-shell repulsion at a bond length is computable here and is of the right order; adding it raises the transfer cost and would push hydrogen fluoride and hydrogen chloride towards no transfer, which is the direction the measurements want. Whether it pushes them far enough without pushing an alkali halide across too is one number per bond and it decides whether six of eight is the model’s score or an accident of which term was left out.