The term a harmonic field cannot produce

A zero-point correction to a moment of inertia is usually computed one way: a sum over normal modes of the second derivative of a principal moment with respect to that mode’s coordinate, times the mode’s mean-square amplitude. That is what a harmonic force field can supply. A quadratic potential is symmetric about its minimum, so the mean displacement in any of its states is exactly zero, and the mean square is the only average there is.

A per-axis comparison leaves the anharmonic term out entirely, and a natural guess is that it works in one direction on every axis. A correction invented rather than computed is where the harmonic habit starts. The guess is wrong in an interesting way, and the reason is not the size of the anharmonicity. It is what a rotational constant averages.

What a constant averages #

A rotational constant is $B = h/8\pi^2 I$, and for a vibrating molecule the quantity a spectrum fits is not the constant at any one separation but its average over the vibrational state. The constant goes as the reciprocal of the moment, so what is averaged is $1/r^2$ and not $r^2$.

Write $x = \Delta r / r_e$ and expand:

$$\left\langle \frac{1}{r^2} \right\rangle = \frac{1}{r_e^2}\left(1 - 2\langle x\rangle + 3\langle x^2\rangle - \cdots\right)$$

so the moment the constant reports, $I_B = \mu/\langle 1/r^2\rangle$, sits above the equilibrium moment by $+2\langle x\rangle - 3\langle x^2\rangle$ to second order.

The first term is the mean displacement. In a symmetric well it is identically zero, which is why no harmonic calculation has ever contained it. The second is the harmonic term, and it is negative: a molecule that merely rattles symmetrically about its minimum reports a smaller moment than the rigid one, because the reciprocal is convex and the excursions to short separations count for more than the excursions to long ones.

So the two terms do not merely differ in size. They pull in opposite directions, and the one a harmonic calculation makes is the one that pulls the wrong way.

The numbers are not close. For hydrogen chloride the anharmonic term is +2.4008 per cent and the harmonic term −1.1335, so the exact answer of +1.3289 is a difference rather than a sum, and it has the sign of the missing term. Take the anharmonicity away and the correction changes sign.

How large the missing term is, everywhere #

Four molecules is a small sample and the pattern in it is uniform.

Carbon monoxide is the mildest case and still runs 0.6998 against −0.2717 per cent. Hydrogen fluoride is the largest, at 3.1920 against −1.6445. Deuterium chloride, which is the same potential with a heavier atom on it, is 1.7151 against −0.7977. In every case the term a harmonic field cannot produce is roughly twice the term it can.

The table’s last column is the honest one. Two terms are not the answer: the expansion has a third moment in it, and cutting the series after two leaves between 0.0053 and 0.1082 percentage points on the table — at worst a twentieth of the total. That is small enough that the reading above holds and large enough that quoting the two terms as the value would be wrong, which is why the exact column is computed as $1/(r_e^2\langle 1/r^2\rangle) - 1$ over the solved wavefunction with no expansion in it at all. The series is used to attribute the answer, never to produce it.

The ratio runs the wrong way #

The natural expectation is that a more anharmonic well is one in which the anharmonic term matters more. It is not.

Carbon monoxide has $\omega_e x_e/\omega_e = 0.61$ per cent, the mildest anharmonicity of the four, and the largest ratio at 2.575. Hydrogen fluoride has 2.17 per cent, the strongest, and the smallest ratio at 1.941. The order is monotone across all four.

Both terms grow with the anharmonicity, because both are properties of a wider excursion in a softer well. The mean square grows faster. For a Morse oscillator the mean displacement goes as the range parameter $a$ — which goes as $\sqrt{\omega_e x_e}$ — while the mean square carries the harmonic width, which is itself inflated by the anharmonicity at second order. The ratio therefore falls, slowly, and the fall is not large: 2.58 to 1.94 across a factor of three and a half in the anharmonicity constant.

The practical reading is more useful than the mechanism. The ratio is close to two for every well here, mild or otherwise, so a correction computed harmonically is wrong by roughly a factor of three and in the wrong direction, across a range of molecules wide enough that no molecule-specific excuse is available.

The same insensitivity is what makes the term worth carrying rather than estimating. A quantity that varied by an order of magnitude between molecules would be a nuisance parameter; one that sits near two everywhere is a systematic error with a known size, and a correction that could be applied.

What happens as the molecule gets hotter #

The ratio does move, and the direction is again the opposite of the obvious one.

At $v = 0$ the ratio is 2.118; at $v = 3$ it is 1.709. Climbing the vibrational levels makes a harmonic treatment less wrong rather than more, even though the state is exploring more of the anharmonicity and both terms have grown by a factor of seven.

The reason is the same as the reason the ratio falls across molecules, arriving in a different variable. The mean displacement grows nearly linearly with $v + \tfrac12$ for a Morse oscillator — 15.30, 46.9, 79.7, 113.9 milliångström for the four states — while the mean square grows as the width of a state that is both higher and, near dissociation, much broader. The square wins slowly.

A hot band therefore has a better-behaved decomposition than the ground state, which is an inversion of the usual advice that anharmonicity is a high-state problem. It is a high-state problem for the energies, where the anharmonicity is the whole of the departure from a series of equal steps. It is a ground-state problem for the reported moment, because that is where the term the model omits is the largest share of the answer.

Two moments from one wavefunction #

There is a second thing the reciprocal average does, and it is worth separating from the anharmonicity because it survives even in a perfectly harmonic well.

For hydrogen chloride the average separation implies a moment 2.415 per cent above the equilibrium one, and the constant reports 1.329 per cent. Those are the same wavefunction read two ways, and they differ by nearly a factor of two.

There are three bond lengths from one potential to compare — the minimum, the mean, and the one a rotational constant implies — and this is the same fact stated as a moment rather than as a length, where its size is easier to weigh against a computed correction. A structure quoted to a milliångström is quoted more precisely than the difference between two definitions of what it is.

Which of the two an experiment measures is decided by the experiment. Electron diffraction is near the first, being an average over a distribution of separations; microwave spectroscopy reports the second. Comparing a diffraction structure with a spectroscopic one, without saying which average each is, is comparing two numbers that differ here by 1.09 percentage points on a quantity whose whole vibrational correction is 2.4.

What deuteration does to the parts #

The substitution method’s per-axis cancellation runs to sixty-five per cent and worse. It is now possible to say what is cancelling.

Both terms shrink with the heavier atom, and — this is the part worth noticing — they shrink by almost the same fraction: 28.6 per cent for the anharmonic term and 29.6 for the harmonic one. The near-equality is not required by anything. It is what makes the substitution method work as well as it does, because two contributions of opposite sign that both fall by three tenths leave a total that also falls by three tenths, and the shape of the correction is preserved even though its size is not.

It also says exactly how the method fails. The cancellation is between the totals, and the totals are differences of larger quantities with opposite signs. Hydrogen chloride’s total of 1.3289 per cent is 2.4008 minus 1.1335, so a five per cent error in either part is a nine per cent error in the total. Anything that changes the balance between the two terms — a different reduced mass, a different well shape, a molecule where one bond is much softer than another — moves the total by more than it moves either part.

That is the general form of the substitution finding. The mismatch there was largest on water’s smallest moment, whose correction is smallest; here the amplification is visible in a system with one coordinate and no frame rotation to confound it. A quantity computed as a difference of two larger quantities does not inherit their stability, and both the per-axis mismatches and the sign changes are that arithmetic rather than anything about isotopes.

What was computed, and how #

Each molecule’s potential is a Morse curve built from its measured $\omega_e$, $\omega_e x_e$ and $r_e$ — those three numbers are quoted from the standard diatomic compilations and everything else is computed. The Morse relations give the well depth as $\omega_e^2/4\omega_e x_e$ and the range parameter as $\sqrt{2\mu\omega_e x_e}$, so no parameter is fitted here.

The states are found by diagonalising the Hamiltonian on a four-hundred-point grid from 0.5 to 8 bohr, and the computed levels are checked against the closed-form Morse spectrum to better than two per cent of the level spacing. The three expectation values — $\langle r\rangle$, $\langle r^2\rangle$ and $\langle 1/r^2\rangle$ — are sums over that same grid against the same normalised wavefunction, so the decomposition compares three averages of one object rather than three models of it.

The refusal is a well made nearly harmonic. Dividing the anharmonicity constant by four hundred divides the range parameter by twenty, which must divide the mean displacement by twenty and collapse the anharmonic term. It does: the ratio falls from 2.118 to 0.112, a factor of nineteen, so the term being called anharmonic is the anharmonicity and not an artefact of the grid or of the reciprocal average. A version of this that measured a grid error would not care what the well was shaped like.

Deuterium chloride appears twice in this collection’s tables and the two entries are not the same object. One uses its own measured constants; the other is hydrogen chloride’s potential with a heavier mass on it, which is what the Born–Oppenheimer separation says they share. The figures here use the measured constants, and the two differ in the third figure of the mean displacement — 10.930 against 10.962 milliångström — which is the size of the separation’s own error and is worth knowing before any conclusion is drawn from a difference smaller than that.

Where the model stops #

Everything here is one-dimensional, and a polyatomic molecule’s anharmonicity is not. Water has three coordinates and a cubic force field has ten independent constants in it, of which the diagonal stretching ones are the analogue of what is computed here; the cross terms have no counterpart in a diatomic and can carry either sign. So the factor of two above is a statement about a bond, not about a molecule, and applying it to a principal moment would require knowing how the modes project onto the bonds — which is what a normal coordinate refuses to be.

The Morse potential is a shape, not a measurement. It is the right shape near the minimum by construction, because its first two spectroscopic constants are the measured ones, and it is wrong further out; the third moment of the displacement is more sensitive to that than the first two are, which is one reason the residual column is quoted rather than the series being trusted.

And a rotational constant fitted to a real spectrum is not $\langle B\rangle$ over a vibrational state alone. It also carries centrifugal distortion and, in a polyatomic, Coriolis coupling, both of which are separate effects and neither of which is in the averages above. What is computed here is the vibrational average and nothing else, which is the term the harmonic treatment gets wrong, not the whole of what a fit reports. A bond length taken out of a spectrum inherits every one of them.

The generalisation #

The average of a function is not the function of the average, and which one an instrument reports is a property of the instrument. $\langle 1/r^2\rangle$ and $1/\langle r^2\rangle$ differ here by a percentage point on a quantity whose whole correction is two, and no amount of care about the wavefunction closes that gap, because the two are different questions. The same distinction applies to a rotational constant that reports a moment and not a shape and to four measures of an orbital’s size that span a factor of 2.66: a single object, several honest numbers, and a convention deciding which one gets quoted.

And a term that vanishes by symmetry in a model is not a small term. The mean displacement is exactly zero in a harmonic well, which reads as “the harmonic model gets it right to leading order” and means “the harmonic model cannot see it at all”. Zero by symmetry and small are opposite situations: a small term can be estimated from within the model and a symmetric zero cannot, because the model has no parameter that controls it. The same shape appears wherever a first-order term is forbidden — an exactly vanishing overlap is the benign version, where the zero is the answer, and this is the malignant one, where the zero is the model’s blindness to the largest term in the problem.

Who found it, and when #

That a rotational constant averages $1/r^2$ and therefore reports neither $r_e$ nor $r_0$ is standard, and the resulting $r_B$ has been distinguished from both since Herzberg. The expansion of $\langle 1/r^2\rangle$ in the displacement, and the observation that its leading term is the anharmonic one, is the elementary derivation behind the vibration–rotation interaction constant $\alpha_e$, whose dominant contribution in every textbook treatment is the cubic force constant.

So the physics is old. What is done here is to put the two terms side by side as numbers, computed from one wavefunction rather than from two expressions, and note the consequence: a harmonic zero-point correction to a moment is the smaller of the two terms and the one with the wrong sign. That is not a criticism of the harmonic force fields, which are doing what they can. It is a statement about which of the corrections computed here can be quoted and which cannot, and the answer is that the per-axis harmonic correction is a real quantity and is not the correction a spectroscopist needs.

Still open: water’s moments, and a measured test #

The obvious open question is the polyatomic version, which needs one cubic constant rather than a whole field. The diagonal cubic constant for a bond stretch can be estimated from that bond’s own Morse parameters, and water’s O–H stretches have measured anharmonicity constants — so a first-order anharmonic correction to water’s three principal moments is a sum over stretches of exactly the term computed here, projected onto each axis. It would say whether the anharmonic term also changes water’s a-axis correction from negative to positive, which is the one question the harmonic treatment leaves open.

The nearer question is $\alpha_e$ itself. The vibration–rotation constant is measured for all four of these molecules and is precisely the difference between $B_0$ and $B_e$ that the decomposition above computes. Comparing the computed difference with the published $\alpha_e$ is a test against a measurement rather than against another calculation, and it is stronger than any internal consistency test. If the computed and measured differences agree to a per cent, the two-term reading is established; if they do not, the residual is the third moment and is measurable.