The atoms are not at the points

This field is called where the atoms go, and every essay in it computes an arrangement of points: four points at 109.47°, five points in two inequivalent sets, six points at right angles. Every one of those numbers is right about the thing it computes, and the thing it computes is the minimum of a potential.

A molecule does not sit at the minimum of its potential. It cannot, and the amount by which it cannot is computable from a harmonic force field.

The gap between those two sentences is the whole of this essay, and it is not a philosophical one. It is a number, it is computable from a fitted force field, and it is larger than most of the differences this field argues about.

The number, for the simplest case #

Hydrogen’s molecule has one vibration, at 4,401 wavenumbers, and a bond of 0.741 Å. Its reduced mass is 0.504 atomic mass units.

The mean square displacement of a harmonic oscillator in its ground state is ħ/2ω. In the units this collection works in — displacements in ångström, masses in atomic mass units, frequencies in wavenumbers — that is

$$\sqrt{\langle Q^2\rangle} = \frac{4.1058}{\sqrt{\tilde\nu}}$$

for the normal coordinate, and for a diatomic the normal coordinate is the change in bond length times the square root of the reduced mass. So the bond length of H₂ has a root-mean-square spread of 0.0872 Å, which is 11.8 per cent of the bond itself.

Not 11.8 per cent at room temperature. At absolute zero, in the lowest state the molecule has. There is nothing to cool away.

It is worth pausing on how little that calculation needed. A frequency, a mass and a bond length — all three quoted from measurement, none of them computed here — and one line of arithmetic. The reason the number is not better known is not that it is hard to get.

The same calculation for a polyatomic #

For a molecule with several modes the arithmetic is the same done several times. Each normal coordinate has its own spread, set by its own frequency; the modes are independent in the harmonic approximation, so their contributions to any one nucleus’s displacement add in quadrature.

The force fields this needs are already here. The force field is not in the spectrum is the essay about how they were obtained and about what they do and do not determine, and every one of them was fitted to measured frequencies of two isotopologues.

The pattern is the one the formula predicts: a spread goes as the inverse square root of the frequency and of the mass, so light atoms in soft modes are furthest from their points. Hydrogen is light and everything else in these molecules is not.

There is one more feature of the formula worth drawing out, because it explains something about the table that would otherwise look like an accident. Carbon dioxide’s carbon is spread by 0.059 Å — more than twice its oxygens’ 0.030 — even though it is the heavier atom by a quarter. The reason is the bending modes: they sit at 673 wavenumbers, the lowest frequency in any of these molecules, and in a bend of a linear triatomic it is the central atom that moves most. A low frequency and a large share of the motion beat a mass advantage easily.

The coordinates are what matter, not the atoms #

A nucleus’s displacement is partly a motion of the whole molecule about its centre of mass, which changes no shape at all, and partly a change of shape. Only the second is an error in the structure, so the useful quantity is the spread of each internal coordinate rather than of each atom.

Seven per cent of a bond length is 0.068 Å for water’s O–H and 0.077 Å for methane’s C–H. Those are not small on the scale this collection works at: the double bond a ring cannot hold turns on a difference of 3.3 degrees between a computed limit and a measured structure, and the trans influence is a bond length that grows by a tenth of an ångström.

The angle is the number this field argues about #

The row that should stop a reader is the angle. Water’s H–O–H coordinate has a root-mean-square spread of 8.9 degrees.

Water’s bond angle is 104.5°. The tetrahedral angle is 109.47°. The difference between them — 4.97 degrees — is the subject of why water is bent, and of a great deal of argument elsewhere about lone pairs and about how much more they repel than bonding pairs. What a lone pair is worth puts a number on that repulsion and fits it to exactly this angle.

The molecule’s angle wanders by nearly twice that difference, in its ground state, before anything is warmed.

That does not make the argument wrong, and it is important to say why not. The angle being fitted is the mean — or more precisely the minimum of the potential, which is close to the mean for a nearly harmonic well — and a mean can be well defined to a hundredth of a degree while the distribution about it is many degrees wide. Averaging is exactly what makes the number meaningful.

It helps to be exact about why. The ground state has a definite expectation value for the angle, as every state has for every coordinate, and that value does not become uncertain because the distribution is wide: the width and the mean are separate properties of one state. A measurement that averages over many molecules, each bending tens of trillions of times a second, recovers the mean and is blind to the width unless it is designed to see it.

What it does make wrong is a picture: the picture of a molecule as a rigid object with a definite shape that a measurement reveals. The mean is definite. The shape is not.

Deuterium halves nothing, and shrinks this by a third #

The spread goes as the inverse square root of the mass as well as of the frequency, and both of those change on substituting deuterium — which makes the isotope effect on the spread a good test of whether the arithmetic is being done rather than assumed.

A naive expectation would be that doubling the mass shrinks the amplitude by √2. It does not, because the frequency changes too: the frequency itself falls by roughly √2, and the two effects work against each other in the amplitude, which goes as (mν̃)^(−1/2). Doubling the mass and dividing the frequency by √2 gives a net factor of 2^(−1/4) ≈ 0.84.

So D₂O’s hydrogens are spread by about sixteen per cent less than H₂O’s rather than by thirty per cent less, and the same cancellation is behind the isotope shift is arithmetic, where the naive √2 rule for frequencies fails for water’s three modes and an exact identity holds in its place.

What a measurement of a structure measures #

The distinction has consequences for what published structures are, and they are not academic.

A rotational spectrum gives a moment of inertia, and a moment of inertia is an average of mass times distance squared over the molecule’s motion. So a bond length out of a spectrum is an average of r² rather than a value of r, and the two differ by exactly the spread computed here — at second order, which for a seven per cent spread is a few tenths of a per cent in the length.

That is why the literature has several different bond lengths for the same molecule with different subscripts on them: one from the minimum of the potential, one from the ground-state average, one from an average at some temperature. They differ by amounts that are small but larger than the precision each is quoted to, and the difference is the subject of this essay rather than an inconsistency in the measurements. The same care is behind the two moments in the moment that is the sum of the other two: the planar sum rule holds exactly for a set of points and holds approximately for a real molecule, and the size of the discrepancy is the out-of-plane amplitude.

The same drawing for a molecule with four identical light atoms shows the mass dependence with the mode dependence taken out.

And for one where three hydrogens and a heavier centre are arranged so that the lowest mode moves all four of them, which puts the nitrogen’s spread up rather than down.

What survives the molecule not holding still #

It would be easy to read all of this as an argument that the structures in this collection are not worth computing, and that reading would be wrong in an instructive way. Some statements survive intact and some do not, and the division between them is sharp.

Symmetry statements survive completely. Water’s C₂ axis is an exact symmetry of the potential, so it is an exact symmetry of the vibrational wavefunction: the distribution of the two O–H lengths is identical, however wide each of them is. Every statement in symmetry forbids a dipole and its neighbours is about the group of the potential and is untouched by the amplitudes.

Statements about a mean survive, with care. A bond angle of 104.5° is the minimum of the potential and is well defined however wide the distribution about it is.

Statements about a single geometry do not survive. Anything of the form “at this geometry the overlap is such-and-such, therefore the molecule does such-and-such” is a statement about one point in a distribution that is several degrees wide, and its status depends on how strongly the quantity varies across that width.

The last is the one to watch for, and it is the one made most often.

Why the model that produced this cannot fully judge itself #

The calculation above is harmonic — every potential is a parabola, every mode independent, every amplitude the ground state of an oscillator. That is an approximation being used to measure the error of a different approximation, and its own error runs in a known direction.

A real bond stretching potential is not a parabola: it is steeper on the compression side and flatter on the extension side, because a bond can be broken and cannot be crushed. So the ground state of a real bond is displaced outwards from the minimum, and its distribution is skewed rather than symmetric. The harmonic number computed here is the width of a symmetric distribution standing in for an asymmetric one, and the true mean bond length is slightly longer than the potential minimum.

The direction of that correction is the same as the direction of the effect being computed, so nothing here is overstated by using the harmonic model. What is understated is the asymmetry, which the harmonic model has none of.

Where this sits among the approximations #

The subject rests on layers of approximation, and this is a different layer from the ones usually discussed.

Orbitals are not where the electron is is about the electronic wavefunction: a many-electron atom has no exact orbitals, and the orbital picture is a basis for an approximation. A better energy is not a better answer is about what a variational calculation optimises and what it therefore gets wrong.

Both of those hold the nuclei fixed while they argue about the electrons, and holding the nuclei fixed is itself the approximation. It has a name — the electrons are assumed to follow the nuclei instantly, because they are lighter by three orders of magnitude — and the size of its error is set by the fourth root of the mass ratio, which for hydrogen is about 0.15. That is the same order as the eleven per cent measured above, arriving from a completely different direction, and it is not a coincidence.

So the layers are: the nuclei have positions (this essay); the electrons around them have orbitals (the first essay named above); those orbitals are approximations to an energy (the second). The first is the one nobody mentions, and it is the one every drawn structure depends on most directly.

What a drawn structure should be understood to show #

It is worth saying plainly what a drawn structure is, now that the number is in hand.

A structure drawn here is the arrangement of nuclei that minimises the potential energy surface — the geometry every calculation converges to, the one every published structure is compared against, and the one that makes bond angles comparable between molecules. It is a well-defined object and it is what the word structure means in this subject.

It is not a photograph of a molecule at an instant, and the difference is a tenth of a bond length for anything with a hydrogen in it. A reader who takes the picture literally will over-trust small differences between geometries and under-trust the idea that a molecule is a floppy object held near a shape.

The honest reading is the one the figures in this essay give: a set of centres, each with a cloud round it whose size is computable, and a set of internal coordinates each of which has a mean and a width. Most structure figures draw the centres and omit the clouds, which is the right choice for what those figures are about and is a choice.

How much of that width reaches a computed number #

A spread of nine degrees sounds ruinous for a collection whose arguments turn on differences of five. Whether it is ruinous depends on the quantity, and one line of calculus separates the quantities it wrecks from the ones it barely touches.

Any quantity $f$ computed from a coordinate has an average over the distribution rather than a value at its mean, and expanding about the mean gives

$$\langle f \rangle \approx f(\theta_0) + \tfrac{1}{2} f''(\theta_0)\,\sigma^2.$$

The first-order term vanishes because the distribution is symmetric about $\theta_0$. So the error is second order in the width, and the coefficient is the curvature of whatever is being computed — not of the potential.

Take water’s dipole moment, which is the clearest case because it has a closed form: two bond moments at the bond angle give $\mu = 2\mu_{\text{OH}}\cos(\theta/2)$. Then $f'' = -\tfrac{1}{4}\cos(\theta/2)$, and the correction is a clean fraction,

$$\frac{\langle \mu \rangle}{\mu(\theta_0)} \approx 1 - \frac{\sigma^2}{8}.$$

At $\sigma = 8.9° = 0.155$ radian that is a reduction of 0.30 per cent — about 0.006 D on water’s 1.85. A nine-degree wander moves the dipole by less than the last digit most tables quote it to.

The same smallness holds for the bond-length spread. A quantity falling off exponentially with a decay length $d$ averages to $\exp(\sigma^2/2d^2)$ times its value at the mean, and for water’s O–H spread of 0.068 Å against an overlap decay length near 0.53 Å that is an enhancement of 0.8 per cent.

So the reassuring half of the answer is that most computed structural numbers are robust, and robust for a stated reason: they are smooth functions being averaged over a symmetric distribution, and smooth beats wide.

The other half is sharper, and it is the case where the width supplies not a correction but the whole value. A quantity that symmetry forces to be exactly zero at the mean geometry has no leading term for the width to correct. Its average is $\tfrac{1}{2}f''\sigma^2$ and nothing else, so the entire quantity is manufactured by the molecule not holding still. Every vibrationally allowed but electronically forbidden transition is of that kind, which is why forbidden bands appear at all, and it is why the distinction drawn above between symmetry statements and magnitude statements has a third case sitting between them: a magnitude that symmetry sets to zero is the one place a nine-degree spread matters infinitely more than a one-degree one.

What is left #

The section above takes the effect of the spread on a computed quantity only as far as one worked case and one general expansion. What it does not do is apply that expansion across every computed quantity: every number here that is a function of the geometry has a distribution rather than a value, and the curvature that decides the size of the correction is different for a Hückel energy, an overlap integral and a moment of inertia.

That sweep is computable from what is here — the amplitudes are known, the functions are known, and the average is one more integral — and it would say which computed numbers are robust to the molecule not holding still and which are not. It is the natural next question and it is not attempted here.

The other absence is temperature. Everything above is the ground state, which is the right place to start because it cannot be argued away. At three hundred kelvin the bending modes of these molecules are partly excited — water’s bend is at 1,649 wavenumbers, which is about eight times kT, so barely; ammonia’s umbrella at 1,020 is about five times — and the spreads grow accordingly. The zero-point figure is a floor rather than an estimate.