A weight that depends on how it is weighed

A valence bond description of a two-electron bond is a sum of structures. One is covalent — each atom keeps an electron — and one is ionic, with both electrons on the same atom. The wavefunction is written as one plus a bit of the other, and the bit is quoted as a percentage: the bond is eighty per cent covalent and twenty per cent ionic.

The percentage is not a property of the wavefunction. It is the answer to a question about how to share out a quantity between two things that are not distinct, and there are three standard answers.

Why squaring the coefficients does not work #

The structures overlap each other. Built from atomic orbitals with an overlap S, the normalised covalent and ionic structures of a two-electron bond have

$$\langle \text{cov} | \text{ion} \rangle = \frac{2S}{1 + S^2}$$

and at the bond length of a hydrogen molecule, where S = 0.5865, that comes out at 0.873. The two structures are very nearly the same function.

So the coefficients in front of them are not shares of anything. Two vectors at an angle of thirty degrees can be combined with coefficients summing to more or less than the length of the result, and the squares of the coefficients do not add to one — they add to whatever the geometry says.

Every repair to that problem is a decision about where the overlap term belongs, and there are three in ordinary use.

Chirgwin–Coulson gives each structure its own squared coefficient plus half of every cross term it appears in. The weights sum to one exactly, and a weight can come out negative.

Löwdin orthogonalises the structures first — symmetrically, so that each new structure is as close as possible to the old one — and then squares. Every weight is positive by construction, and the structures being weighed are no longer the structures that were written down.

The inverse-overlap weight asks how much of the wavefunction only that structure can account for. It is positive, and it does not sum to one until it is renormalised.

What they give #

The second row is the sharper of the two. A wavefunction written with no ionic structure in it at all — the classical Heitler–London function, the covalent structure alone — is 25.59 per cent ionic under the Löwdin convention.

That is not a bug in the convention. Symmetric orthogonalisation replaces each structure by the nearest orthogonal one, and the nearest orthogonal covalent structure has some of the ionic structure subtracted from it — which means that the original covalent structure, expressed in the new basis, has some ionic character in it. The number is correct and it is a statement about the orthogonalisation.

It is also a number that would be quoted, in a table, as the ionic character of a bond.

Where they agree, and it is not nowhere #

The conventions are not arbitrary and the essay’s claim would be much weaker if they were.

At λ = 1 the wavefunction is the molecular orbital one — both electrons in the same delocalised orbital — and it contains the covalent and ionic structures in exactly equal measure. Symmetry then forces every convention to fifty per cent, and all three give it exactly. That is the fifty per cent ionic character every account of the molecular orbital picture complains about, and it is one number every convention agrees on.

At zero overlap the three are identical. That is the check the whole calculation rests on: the three routines are three ways of doing one thing, and the disagreement between them is created entirely by the structures overlapping.

And the disagreement grows monotonically with the overlap. So the size of the ambiguity is a computable property of the bond — small for a long weak bond, large for a short strong one, and largest exactly where the valence bond description is most often used.

The overlap that causes it, measured #

The whole difficulty is one number, and it can be computed from first principles rather than quoted.

So the ambiguity is not constant across chemistry. A long, weak bond has nearly orthogonal structures and the three conventions nearly agree; a short, strong one has structures that are almost the same function and the conventions disagree by factors.

The formula also says something about the limit. As the two atoms are pushed together S goes to one and the structure overlap goes to one as well — the covalent and ionic structures become the same function, and the question of how much of each a wavefunction contains stops having any content at all. Nothing goes wrong with the arithmetic on the way; the weights simply become a statement about an ever finer distinction.

That last observation is the one worth carrying. The overlap between two atomic orbitals is what makes a bond; the overlap between two structures built from them is what makes the bond’s description ambiguous. They are not merely related — the second is a function of the first, and it is a function that rises faster.

Where the mixing comes from when it is computed #

λ is a parameter here, and it is worth showing what it is when a model is small enough to give it exactly — because that model is the one where none of this difficulty arises.

Those numbers are what an ionic weight is when the structures do not overlap. They fall smoothly from a half — the molecular orbital value — toward nothing as the electrons are made to repel, which is exactly the behaviour the valence bond picture describes and the molecular orbital picture gets wrong at dissociation.

The λ of 0.25 used above corresponds to a Chirgwin–Coulson ionic weight of about nineteen per cent, which is in the range that model gives for a repulsion between 2t and 4t. So the case chosen is an ordinary one rather than an extreme.

What the orthogonal model cannot do is have overlapping structures, which is the whole subject here. Its sites are two orthogonal orbitals with a hopping between them, and that hopping is not an overlap — it is a matrix element between orthogonal functions. Real atomic orbitals on two atoms are not orthogonal, and no amount of care in solving the model repairs that, because the model was built without it.

The same problem in the other picture #

It would be comfortable to conclude that valence bond theory has a weighting problem and molecular orbital theory does not. It does not survive contact with a population analysis.

A Mulliken population divides the electron density between basis functions. Where two functions on different atoms overlap, the density in the overlap region belongs to neither, and Mulliken’s rule is to split it equally — which is Chirgwin–Coulson’s rule applied to a density instead of to a set of structures, halves of cross terms and all.

The consequences are the ones everybody knows. Mulliken charges depend strongly on the basis set, can exceed the number of electrons an atom brought, and go to nonsense in a large diffuse basis where a function centred on one atom describes density near another. Löwdin populations are the symmetric-orthogonalisation version and are better behaved and differently wrong.

So the two frameworks have the same difficulty at the same place, and it is not a difficulty about either of them. It is what happens when a quantity that lives in a whole wavefunction is shared out among non-orthogonal parts, and the wavefunction does not care how it is shared.

What this is and is not a criticism of #

It is not a criticism of valence bond theory. The wavefunction is a perfectly good wavefunction; its energy, density and every other observable are what they are, and none of them depends on how the structures are weighed. What is at stake is a description of the wavefunction, and descriptions are not observables.

It is not an argument for molecular orbital theory either. The corresponding numbers there — orbital populations, Mulliken charges — have exactly the same difficulty for exactly the same reason: they share out a density between basis functions that overlap. Mulliken’s population analysis is Chirgwin–Coulson’s rule applied to one-electron densities rather than to structures, halves-of-cross-terms and all, and it is famous for giving charges that depend strongly on the basis set.

It is a difficulty that keeps coming back. A resonance energy is measured from a reference somebody chose; a mode’s percentage composition depends on the coordinates; a structure’s weight depends on how the overlap is shared. In each case a construction produces a number, the construction is dropped, and the number is quoted as though it were measured.

What was computed, and what is a parameter #

The overlap S is computed: it is the 1s–1s overlap at the separation in question, from its closed form. At two bohr it is 0.5865 and the structure overlap that follows is 0.8728.

The mixing λ is a parameter, and nothing here computes it. It is the amount of ionic structure in the wavefunction, and the point of the essay is what the conventions do with it rather than what its value should be. The corresponding quantity is computed exactly in a model where the two structures are orthogonal by construction — which is the model where none of this arises.

The three weighting rules are quoted from the literature and implemented as stated. The symmetric square root needed for the Löwdin weights is the same operation used to orthogonalise a set of basis functions, which is why the two arrive together.

There is another quantity of exactly this kind in the shape field. The s character of a hybrid is a coefficient in a chosen basis rather than a measurement, its numbers are useful, and its value depends on the choice in the same way an ionic weight does. Both are worth quoting and neither is worth quoting without the convention attached.

What was checked #

At zero overlap all three conventions agree with the squared coefficient, to 10⁻¹², at four different mixings. That is the check that makes the rest meaningful: three routines that disagreed at zero overlap would be three different rules rather than three treatments of one problem.

Chirgwin–Coulson and Löwdin sum to one at every overlap tried. A weighting scheme whose weights did not sum to one would be doing something else.

At a real overlap the three disagree by more than five percentage points, and the disagreement grows with the overlap at every step of a sweep. Both halves matter: a spread that did not grow would be a numerical artefact rather than the consequence of non-orthogonality.

A purely covalent wavefunction has zero ionic weight under two conventions and more than a fifth under the third. That one is checked with a floor rather than a value, because it is the qualitative claim that matters.

The standing demonstration behind all of this is that the canonical and localised descriptions of methane give the same density to the last bit a double holds. Which basis a wavefunction is written in changes every coefficient in it and nothing that can be measured, and a weight is a coefficient.

Who proposed each, and when #

Chirgwin and Coulson’s rule is from 1950 and was proposed for exactly this problem, in exactly this system. It is the natural generalisation of squaring a coefficient — it reduces to that when the structures are orthogonal — and its one embarrassment, that a weight can be negative, was known when it was published.

Löwdin’s symmetric orthogonalisation is from 1950 as well and was not proposed as a weighting rule at all: it is a way of turning a non-orthogonal basis into an orthogonal one while moving each function as little as possible, and it is used throughout quantum chemistry for that. Weighting by squaring the coefficients after it is a later application, and it inherits the property that the functions being weighed have been changed.

The inverse-overlap weight is Norbeck and Gallup’s, from 1974, and is the most conservative of the three: it credits each structure only with what nothing else can account for. That is why it gives the smallest numbers, and why it does not sum to one until it is renormalised.

Three rules, three decades, and all three still in use — which is itself the argument. If one of them were right the others would have gone.

Three conventions, one piece of linear algebra #

The three rules look like three chemical opinions and they are three resolutions of one mathematical ambiguity, which is worth stating because it says exactly when they can be trusted and exactly how far apart they can get.

In an orthogonal basis, the component of a vector along a basis direction is unambiguous: project, square, done. Every convention agrees, because there is only one thing to compute.

In a non-orthogonal basis there is no such thing as the component along one direction, and the reason is that the answer depends on what the other directions are. Expanding a vector in a non-orthogonal set produces coefficients; asking how much of the vector lies along one basis element produces a different number; and the two are related through the overlap matrix rather than being equal. Linear algebra provides two natural pairings rather than one, and a weight is a choice of which to use.

The three rules are three such choices.

Chirgwin and Coulson’s pairs each coefficient with the overlap-weighted sum of the others — the coefficient against its dual — which is why it sums to one automatically and why it can go negative, since nothing constrains the cross terms to be positive.

Löwdin’s changes the basis first, into the nearest orthogonal one, and then squares. It cannot go negative and it is not weighing the structures that were asked about: it is weighing their orthogonalised replacements, which are different functions.

The inverse-overlap rule credits each structure only with the part of the vector that lies along the direction orthogonal to all the others, which is the most conservative choice available and is why it gives the smallest numbers and does not sum to one.

None is wrong. They answer three different well-posed questions, and no non-orthogonal basis makes one of the three canonical.

Two consequences follow, and both are quantitative.

The disagreement is controlled by the overlap. As the structures become orthogonal all three converge to the same answer, so the spread between them is a measure of how far the structures are from independent — the factor of six here is a statement about an overlap of 0.7071 rather than about the molecule.

And the ambiguity grows with the number of structures. With two structures there is one overlap; with five there are ten, and the room for three conventions to disagree grows with them. That is why a weight quoted for a benzene structure is a much weaker statement than a weight quoted for a two-structure bond, and it is the reason the many-structure case named below is an obstacle rather than a curiosity.

The framing also says what would remove the ambiguity, and it is not a better rule. Use structures that are orthogonal to begin with, and every convention returns the same number, because the question stops being ill-posed. What that costs is the thing valence bond structures are for: an orthogonal set of two-electron functions no longer corresponds to a bond here and a lone pair there, which is the entire reason a chemist wanted the decomposition. The interpretability and the uniqueness are in direct competition, and the three conventions are three ways of keeping the first while giving up different amounts of the second.

Still open: weights with five structures #

The natural open question is the many-structure case, where the difficulty stops being a curiosity and becomes an obstacle. Benzene’s valence bond description has two Kekulé structures and three Dewar ones, and their weights are quoted routinely — with five non-orthogonal structures, the overlap matrix has ten independent entries and the three conventions have far more room to disagree than they do here.

The nearer question is whether any convention-free statement about ionic character exists. There is one, and it is not a weight: the charge on an atom, computed from the density rather than from the structures. It has its own difficulty — the density has to be divided between atoms, and that division is another convention — but it is a statement about an observable rather than about a decomposition, and the two difficulties are not the same size. That is the subject of the essays on electronegativity.