An expression for what was a warning

The zero-point correction to four molecules’ moments of inertia is larger than the error model it replaced: 1.88 per cent for water against a few tenths assumed, with one of the three moments shrinking rather than growing. It comes with an observation and a request:

Boron trifluoride’s correction is a sixth of water’s, methane’s is comparable with water’s, and the difference is the amplitude — which goes as the reciprocal square root of the mass and of the frequency. Writing the correction as a function of those two would turn heavy molecules are safer into an expression a spectroscopist could evaluate.

The expression is one line and there is almost nothing in it.

The line #

A mode’s mean-square amplitude in mass-weighted coordinates is $A^2/\nu$ with $A$ the amplitude constant used throughout, and the second derivative of a moment of inertia with respect to a mass-weighted coordinate is dimensionless and of order one. So the correction is a mean of those curvatures times a sum of amplitudes over the moment:

$$\frac{\Delta I}{I} \;\approx\; c \cdot A^2 \cdot \frac{\sum_k 1/\nu_k}{I}$$

Nothing in it is fitted except $c$, and $c$ is one number for all four molecules. The amplitude constant $A$ is the one used wherever a zero-point motion appears, unchanged. Everything else — the moment, the list of wavenumbers — is what a spectroscopist has before doing anything at all.

Evaluated #

The coefficient is 0.6343, and the four molecules require 0.518, 0.647, 0.667 and 0.705 — a spread of 1.36 against a spread of 11.4 in the thing being predicted.

There is a way to read those four numbers that makes the pattern in them visible. They rise with the number of modes — water three at 0.518, ammonia six at 0.647, boron trifluoride six at 0.705, methane nine at 0.667 — which is roughly what a mean over more samples should do if the true distribution of per-mode curvatures is the same everywhere and water is simply an unlucky draw of three. Four points cannot establish that, and it is the reading to test next rather than one to rely on.

The claim is the ratio of spreads and not the coefficient. A number fitted to four points is not a discovery; a number that is the same for four molecules whose corrections differ elevenfold is a statement that the two factors in the expression are the two that matter.

It is worth saying what kind of expression this is, because it is not a fit and it is not a derivation.

A fit would take the four corrections and adjust as many constants as it needed. A derivation would produce the coefficient from the geometry. This is neither: the form comes from an argument — an amplitude is $A^2/\nu$, a curvature is dimensionless — and the one number left over is measured once and reused. That is the shape of every useful estimate in spectroscopy, and it is worth distinguishing from the two things it resembles, because its failure mode is its own: the form can be right and the coefficient can vary, which is exactly what is being tested.

The forty-five combinations a spectrum fixes is the same distinction seen from the other end — there, what is determined is a subspace and what is quoted is a set of numbers; here, what is determined is a form and what is quoted is one number.

Which factor is doing the work #

The expression has a vibrational factor and a structural one, and across these four they pull in opposite directions.

Boron trifluoride is the softest molecule of the four. Its sum of reciprocal wavenumbers is 8.06 × 10⁻³ against water’s 1.12 × 10⁻³, a factor of 7.2 — its modes are slower, its amplitudes are larger, and every part of the usual intuition says it should have the larger correction.

It has the smallest, by a factor of eleven, because its moment of inertia is 65.2 amu Ų against water’s 1.18 — a factor of 55.

The comparison is worth doing in both directions. Boron trifluoride’s softness is 7.2 times water’s, which alone would make its correction 7.2 times larger; its moment is 55 times water’s, which alone would make it 55 times smaller. The product is a factor of 7.7 smaller, and the computed ratio is 5.6 — the difference being the coefficient, which is 36 per cent larger for boron trifluoride than for water. Two effects of a factor of fifty and a factor of seven, pulling against each other, with the answer decided by the larger one.

So heavy molecules are safer is true, and not for the reason it is usually given. The safety is the moment of inertia and the frequencies work against it: a heavy molecule is slow, and slow means large amplitudes. What saves it is that its moment is large enough that a large amplitude is a small fraction of it.

Why the curvature is of order one #

The dimensionless coefficient is worth a paragraph, because of order one is the part of the argument that could have failed.

A moment of inertia is $\sum_i m_i r_i^2$, and a mass-weighted normal coordinate is $\sqrt{m}\,x$. So $\partial^2 I/\partial Q^2$ is a sum of terms like $m (\partial r/\partial x)^2 / m$, which is a pure number — a geometric factor of how much a mode’s displacement pattern moves mass away from the axis, between zero for a mode that does not and about one for a mode that moves every atom radially.

The measured values, 0.518 to 0.705, are exactly that: between a half and three quarters of the atoms’ displacement is radial, on average over the modes. A molecule whose modes were all tangential would have a coefficient near zero and a correction near nothing, and no molecule here is close to that. That the four values cluster so tightly is the same kind of statement a depolarisation ratio of exactly three quarters makes and a weaker one: there, a geometric average is forced to a value by symmetry; here, it is merely constrained to a range by the fact that molecules are made of stretches and bends.

That is why one coefficient serves. It is not an empirical constant with a value to be looked up; it is a geometric average that cannot be far from a half for any molecule whose modes are a mixture of stretches and bends.

Where the expression is worst #

Water, at 22 per cent, and the reason points at the expression’s limit.

Water has three modes. A mean curvature over three numbers is not much of a mean, and water’s three are unusually unequal — the two stretches move hydrogens almost radially and the bend moves them almost tangentially, so the average is dragged low. Methane has nine modes and ammonia six, and both sit within five per cent.

There is a second reason water is the hard case, and it is the one found in a different currency: water is the most asymmetric of the four, its three moments are 0.62, 1.16 and 1.77 amu Ų, and dividing a single correction by their mean is a coarser thing to do than for a molecule whose three moments are equal. Methane’s are all 3.18 and boron trifluoride’s are two equal and one double; only water has three genuinely different numbers to average.

So the expression’s accuracy is a statement about how many modes there are to average over, which is a useful thing for a user to know: it is good for a molecule of five atoms and rough for one of three. That is the opposite of the usual pattern, where a rule of thumb is trusted most on the smallest system.

What this replaces #

The finding was that the received error model for a substitution structure is wrong in both of its numbers — a zero-point inflation of 1.88 per cent where a few tenths were assumed, and a mismatch between isotopologues of 35 per cent where five were. That is a warning and a spectroscopist can do nothing with it except be careful.

It is worth being fair to the received wisdom here. Heavy is safer is not merely a slogan that happens to be right: a heavy molecule has a large moment more or less by definition, since a moment is a mass times a squared distance and the mass is the heavy part. What the expression adds is that the frequencies — which the slogan is usually justified by — push the other way, so the rule survives only because the factor it does not name is much larger than the factor it does.

This is a number they can compute. Before attempting a substitution structure, take the moment, take the wavenumbers, evaluate one expression, and know to within a fifth how large the correction being neglected is. If it is a fifth of a per cent the structure is worth attempting; if it is two per cent, the bond lengths that come out are not the bond lengths.

What a spectroscopist would do with it #

The practical shape of the thing is worth setting out, since making it usable was the request.

Before attempting a substitution structure, the inputs are a trial geometry — which gives the moment — and a vibrational spectrum, which gives the wavenumbers. Both exist before any isotopologue is measured. Evaluating the expression takes a sum of reciprocals and a division.

What comes out is a fraction, and the fraction is compared against what the structure needs. A bond length determined from a moment to a thousandth of an ångström needs the moment to about a tenth of a per cent, so a correction of 1.6 per cent is sixteen times too large to neglect and a correction of 0.15 is not quite.

And what the expression cannot do is correct anything. It says how big the neglected term is, not what it is — which is the right division of labour, because the correction itself needs a force field and the estimate does not. The available force fields cover six molecules; the estimate works for anything with a spectrum.

That last point is why the expression is worth more than the four computed corrections it was built from. The corrections are exact and available for four molecules. The expression is approximate and available for all of them.

What is quoted, and what is computed #

The observed frequencies are quoted, for each molecule and one isotopologue, and they are what the force fields were fitted to. Nothing else is.

The corrections themselves are computed directly: displace along each normal mode by a stated step, recompute the principal moments, take the second difference, and weight by the mode’s mean-square amplitude. The moments are exact for the structure supplied.

The expression’s two ingredients are read off the same objects — the moment before displacement and the fitted frequencies — and the coefficient is the mean of the four ratios. There is no second fit: the coefficient is not adjusted to minimise anything, it is an average, and the errors quoted are what that average leaves.

What would test the coefficient properly #

One coefficient serving four molecules whose corrections span a factor of eleven is a strong result, and the way to strengthen it further is cheap and worth stating, because a coefficient fitted to four things is exposed to whatever those four have in common.

The four are small molecules of light atoms. So a fifth chosen to be unlike them is worth more than a fifth chosen at random: a molecule with a heavy atom, or with a very low-frequency mode, or with a moment of inertia an order of magnitude larger.

The expression predicts what should happen in each case, which is what makes them tests rather than additions. A larger moment divides the same sum of reciprocal wavenumbers by more, so the correction should fall; a molecule with an unusually soft mode contributes a large reciprocal wavenumber and the correction should rise. Both are directions the expression commits to before the molecule is measured.

A coefficient that holds across a case chosen to break it is a coefficient worth quoting. One that holds across four molecules of the same kind is a coefficient that may be reporting what those four have in common — which is the same caution the collection applies to any fit over a small and homogeneous set.

What this cannot say #

The four are the four with fitted fields. Everything here needs a normal-mode analysis, which needs a force field, and this collection fits six of which four have the isotopologue data the corrections were computed against.

Four molecules is four points. They are the four with force fields here, they are all small and all built around one central atom, and the coefficient’s range across them is a fact about those four. A molecule with a heavy atom off the symmetry axis would have a different geometry of displacement and could easily sit outside 0.5 to 0.7.

The correction is harmonic. Everything here is second order in the amplitude, and a real vibration–rotation constant has a cubic term of comparable size with the opposite sign — so the true correction is smaller than the one this expression estimates, by an amount not computed here. What the expression gives is the harmonic part, which is the part that can be had without a cubic field.

The amplitude constant is a convention as much as a constant. $A$ converts a wavenumber into a root-mean-square normal coordinate, and it carries the choice of mass-weighting and of units that the whole of this file makes. Any other convention would change $A$ and change the coefficient by the same factor, leaving every predicted correction where it is — so the value 0.6343 is not comparable with a number from anywhere else, and only the spread is.

And the mean over the three moments hides a sign. Water’s three corrections are −0.28, +1.88 and +1.07 per cent of their moments: one of them shrinks. The expression predicts a single number for the molecule and cannot say which moment does what, which is exactly the information a substitution structure needs. It is an estimate of the size of the problem rather than a correction to be applied.

What was checked #

The four corrections span more than a factor of eight, which is the range the expression is being asked to cover and would make the agreement trivial if it were small.

One coefficient serves all four to within a factor of 1.6, and the expression is within a quarter of every computed correction — two claims rather than one, because the first can hold while the worst case is bad.

The molecule with the softest modes has the smallest correction. That is the finding about which factor decides, checked as a conjunction of two inequalities so that it fails if either half stops being true.

And the refusal is the frequencies. Doubling every wavenumber must halve the predictor exactly, since the expression is linear in the sum of their reciprocals — a test that costs one evaluation and that a routine computing the predictor from anything else would fail.

Still open: whether the harmonic estimate is a bound #

The obvious open question is the cubic term the expression leaves out. Its sign is known — opposite to the harmonic one for a stretching coordinate — so the harmonic estimate is an upper bound, and a bound is more useful than an estimate for deciding whether to attempt a structure. Establishing that it really is a bound, rather than usually one, needs a cubic force field, and there is a cheaper route: the difference between the harmonic estimate and the measured vibration–rotation constants, which are tabulated for water and methane and would give the ratio directly.

The nearer question is the per-axis version. The expression predicts one number for a molecule and the three axes behave differently — one of water’s shrinks. Splitting the sum by axis costs nothing, because the second differences are already computed per axis, and it would say whether the same coefficient serves all three or whether the axis that shrinks needs its own. That is the version a substitution structure actually needs, since a structure is fitted to individual moments rather than to their mean.