The fourth constant chose a curve, not a depth

A Morse curve whose range bends linearly with the separation, held to a diatomic’s measured $\omega_e$ and $\omega_e x_e$, reached carbon monoxide’s measured $\alpha_e$ twice and the three hydrides’ not at all. The two carbon monoxide fits carry the same cubic and quartic coefficients and are the same curve wherever the lowest levels go, and they part completely beyond: one rises towards dissociation, the other turns over at 2.10 bond lengths and falls back to the bottom. They predicted different values of the next constant, $\omega_e y_e$ — 0.018 and 0.011 cm⁻¹ — and the question left was the sharpest one available. If a measured fourth constant chose the second, it would be choosing a potential that does not dissociate.

It does choose the second. That turns out to say more about the family than about the molecule, and the way to show it is to give the family a second parameter and ask the question again.

Read like with like #

The comparison has to be made on equal terms first. Every constant in this line of essays — from the stretch a spinning rotor shows onwards — is read from a potential the same way: solve for its lowest four levels, take the three spacings, and solve for the three constants $\omega_e$, $\omega_e x_e$ and $\omega_e y_e$ that reproduce them exactly. A published $\omega_e y_e$ is not that. It comes from a fit to many levels with more terms — carbon monoxide’s expansion runs to the sixth power of $(v + \tfrac12)$ over thirty-eight levels — and its $\omega_e y_e$ is the coefficient of one term in a longer series, with the higher terms carrying part of what four levels would assign to it.

So the measured value is read the same way the potentials are: the published expansion’s lowest four levels, three spacings, three constants. For carbon monoxide that gives 0.01101 cm⁻¹, against the tabulated 0.010511 — the higher terms fold about five per cent into it. For hydrogen chloride, whose expansion stops at the cubic term, the reading is the tabulated 0.22437 exactly; for deuterium chloride, with a quartic term, 0.05809 against 0.08649; for hydrogen fluoride, 0.7835 against 0.90. These four-level readings are the measured values everything below is compared with, and $\omega_e$ and $\omega_e x_e$ are held at the same four-level readings rather than at the tabulated ones, which differ from them by at most four tenths of a per cent.

Along the linear family, carbon monoxide’s $\omega_e y_e$ rises from zero at the Morse curve — a Morse curve’s level spacings are exactly linear, so its $\omega_e y_e$ vanishes — to a maximum near λ = −0.3 and falls again. The dissociating fit at λ = −0.2214 predicts 0.01817. The turning-over fit at λ = −0.4534 predicts 0.01100. The measurement is 0.01101: the second, to a tenth of a per cent, and the first is 65 per cent too high.

Taken at face value that is a strong statement. Four measured constants, every one of them reproduced, by a curve that has a barrier at 2.10 bond lengths and a second region of low energy beyond it — and the curve that behaves like a bond does not fit. If the linear family were the only way to bend a Morse range, the spectroscopy of carbon monoxide’s lowest four levels would be evidence that its potential does not dissociate.

A range that bends and stays bounded #

It is not the only way. The linear range $a(1 + \lambda\xi)$, with $\xi = (r - r_e)/r_e$, turns over because it keeps shrinking: with λ negative the exponent $a(1 + \lambda\xi)(r - r_e)$ reaches a maximum at $\xi = 1/2|\lambda|$ and comes back down. A range that bends near the minimum and then stops bending does not have to. The simplest one divides the bend by a second factor:

$$V(r) = D\left(1 - e^{-a\left(1 + \frac{\lambda\xi}{1 + \mu\xi}\right)(r - r_e)}\right)^2 .$$

At μ = 0 this is the linear family. With μ positive, the range starts out bending at the same rate λ and saturates at $a(1 + \lambda/\mu)$ far out. At μ = ½ it is, exactly, the one-term expanded Morse oscillator with the reduced coordinate $(r - r_e)/(r + r_e)$, a form spectroscopists already use for this purpose.

When does such a curve dissociate? The exponent must rise for ever, which means $\xi(1 + \lambda\xi/(1 + \mu\xi))$ must increase for every positive ξ. Its derivative is $1 + (\lambda/\mu)\bigl(1 - 1/(1 + \mu\xi)^2\bigr)$, and the bracket climbs from 0 to 1 as ξ grows, so the derivative stays positive everywhere exactly when $1 + \lambda/\mu > 0$. The curve dissociates if and only if μ > |λ|. The line μ = |λ| divides the plane into curves that are bonds and curves with a barrier, and it is drawn from the algebra rather than found by search. It is also checked, because a line like that is where a sign error would hide: a member at λ = −0.5 with μ = 0.45 must turn back down somewhere past the minimum on a grid out to fifty bond lengths, and one with μ = 0.55 must not — and they do and do not.

Every molecule reaches the third constant #

At every point of the plane D and a are solved so that $\omega_e$ and $\omega_e x_e$ are the measured ones, to a part in a hundred million, and the $\alpha_e$ and $\omega_e y_e$ that leaves are read from the same four levels. The measured $\alpha_e$ is then a line across the plane.

For hydrogen chloride the line exists, which is already a change: the linear family’s $\alpha_e$ topped out 4.3 per cent short of the measurement. With μ above about a quarter the measured value is reached, along two branches. The branch nearer the Morse curve — smaller |λ| — rises steeply into the dissociating half of the plane and stays there; the other runs off to larger |λ| beneath the diagonal, every member of it with a barrier. Deuterium chloride and hydrogen fluoride behave the same way, and carbon monoxide, which the linear family already reached, has the near branch all the way down to μ = 0 and the far one only at μ = 0 itself, where it is the turning-over fit.

So the hydrides’ ceilings, which the earlier essay could not assign, belong to the linear family’s way of reaching $\alpha_e$ and not to the requirement of reaching it. A range that bends and saturates carries the cubic coefficient the measured $\alpha_e$ needs without paying for it with a turnover.

The fourth constant picks a point #

Along the near branch, moving up in μ, each member has the measured $\omega_e$, $\omega_e x_e$ and $\alpha_e$, and its $\omega_e y_e$ falls steadily.

Each molecule’s falls through its measured value exactly once, and the crossing is a curve that dissociates. Carbon monoxide’s is at μ = 0.765, λ = −0.086: well above the diagonal, a bond in every sense, with all four of its measured constants reproduced — $\alpha_e$ to a part in ten million and $\omega_e y_e$ to a part in a million. Hydrogen chloride’s is at μ = 0.829, λ = −0.142; deuterium chloride’s at 1.119, −0.118; hydrogen fluoride’s at 0.757, −0.220.

That is the verdict on the question carbon monoxide’s two fits raised. Its fourth constant chose the turning-over fit because, in a family with one shape parameter, the only way to lower $\omega_e y_e$ while holding $\alpha_e$ was to move out to large |λ|, and large |λ| is where the linear range turns over. Give the family a second direction to move in and the same four constants are met by a bond. A measured constant can only choose among the curves on offer; what it chose was a member of a family, and the barrier came with the family.

For hydrogen chloride the far branch, with its barriers, also crosses the measured $\omega_e y_e$, at μ = 0.393 and λ = −0.657 — a curve that turns over at 2.47 bond lengths. So on that molecule the four constants are met twice, once by a bond and once by a curve that is not one, and nothing in the four distinguishes them.

What the four constants say about depth #

A curve that dissociates has a depth, and the bounded family’s depth is its parameter D: far out the exponent grows without limit and the curve rises to D. That gives the four-constant fits a prediction no earlier fit in this line could make, because the earlier fits either had no depth to speak of or were the Morse curve, whose depth $\omega_e^2/4\omega_e x_e$ is fixed by two constants and is not a fit at all. And depths are measured: the dissociation energy at zero kelvin plus the zero-point level from the same expansion gives 4.62 eV for hydrogen chloride (and deuterium chloride, on the same potential), 6.12 for hydrogen fluoride and 11.22 for carbon monoxide.

The fourth constant picks a depth, and it picks it too deep on every molecule. Carbon monoxide’s four-constant curve is 11.86 eV deep against 11.22, 5.7 per cent too deep; hydrogen chloride’s 6.04 against 4.62, 31 per cent; deuterium chloride’s 5.77, 25 per cent; hydrogen fluoride’s 7.70 against 6.12, 26 per cent.

The Morse curve, built from the first two constants and nothing else, does better on all four: 10.98 eV for carbon monoxide, 2.2 per cent shallow; 5.25 for the chlorides, 14 per cent deep; 5.91 for hydrogen fluoride, 3.5 per cent shallow. Two more measured constants, both reproduced exactly, carry the depth further from the measurement on every molecule — by a factor of two and a half for carbon monoxide, two for the chlorides and seven for hydrogen fluoride.

That is not a paradox once it is said what the constants are. They are the curvature of the well and its first few corrections at the bottom; the depth is the energy at the top. A family with two shape parameters has two dials to set those corrections with, and its depth is whatever those settings imply about a region no low level reaches. The Morse curve’s depth is better not because two constants know more than four but because the Morse shape happens to extrapolate these molecules’ wells more honestly than this family’s settings do, and a fit that matches more of the bottom with a less suitable shape can extrapolate worse.

Two isotopologues, one potential, two curves #

There is a direct test of whether a fitted curve is the molecule’s potential or only a summary of its constants, and this line of essays has used it before: an isotope changes the masses and not the electrons, so hydrogen chloride and deuterium chloride have one potential between them, and anything that is a property of the potential must come out the same for both. It is the same reasoning that lets a second isotopologue supply the second equation a bond length needs.

The two four-constant fits are different members of the family. Hydrogen chloride’s sits at μ = 0.829, deuterium chloride’s at 1.119; their depths are 6.04 and 5.77 eV. Near the bottom they are one curve — between 0.9 and 1.2 bond lengths, the stretch the lowest four levels of either molecule explore, they differ by at most 2.7 cm⁻¹ — and beyond it they part by more than a quarter of an electronvolt at the top. Each is a faithful summary of its own molecule’s four constants and neither is the potential the two molecules share; if either were, the other would have landed on it.

The deuterium chloride constants carry their own caveat. Its quartic level term was derived from hydrogen chloride’s by isotope relations rather than fitted independently, and it moves the four-level $\omega_e y_e$ by a third; a different quartic would move its fit. But that cuts the other way from rescue. A fitted curve whose depth shifts by tenths of an electronvolt when one higher term of one isotopologue’s expansion is revised is a curve whose depth the constants were never fixing.

How the plane was searched #

Every point is the same calculation as the linear family’s: a finite-difference Hamiltonian on four thousand points from 0.6 to 2.6 bond lengths, its lowest four levels and their rotational constants, damped Newton steps on D and a until the four-level $\omega_e$ and $\omega_e x_e$ match the targets to $10^{-8}$, and $\alpha_e$ and $\omega_e y_e$ read from what is left. The plane is first laid out on a grid of λ from 0 to −0.8 and μ from 0 to 2, keeping only members whose barrier, if they have one, lies past two bond lengths — the bound the linear scan to λ = −0.5 kept.

The measured-$\alpha_e$ line is traced at forty-one values of μ by walking λ down from zero in steps of 0.05 and refining every bracket by regula falsi, and the member with the measured $\omega_e y_e$ is found by regula falsi in μ along the near branch, each trial a fresh walk to the line. The depths are the fitted D in electronvolts; the measured ones are the zero-kelvin dissociation energies compiled by Huber and Herzberg — 4.4336, 5.869 and 11.09 eV — with the zero-point level from the same published expansions added.

The checks, made wherever these figures are drawn: every member of every plane holds $\omega_e$ and $\omega_e x_e$ to $10^{-8}$; carbon monoxide’s linear family has exactly two fits, the turning-over one within half a per cent of the measured $\omega_e y_e$ and the dissociating one more than half again above it; every molecule has a dissociating member with $\alpha_e$ to $10^{-7}$ and $\omega_e y_e$ to $10^{-6}$; every such member is too deep, and further from the measured depth than the Morse curve on at least three of the four; the two chlorides’ members differ by more than a tenth in μ and a tenth of an electronvolt in depth. The refusal is the dissociation boundary, which must hold numerically on both sides of the line the algebra draws.

What two parameters do not settle #

One more family. A range that saturates is one way to bend without turning over; a range with two saturating terms, or a long-range tail of the form a dispersion force has, would set the depth differently. Nothing here says that no family fitted to four constants gets the depth right — only that this one, the most direct repair of the one before it, does not, and that the constants gave it no reason to.

Four levels. Every constant here, measured or computed, is read from four levels, which is the honest comparison and also the limit of it. The published expansions were fitted to many more: carbon monoxide’s to thirty-eight levels, hydrogen fluoride’s to ten. A fit to those levels directly, rather than to constants read from four of them, would put the higher levels — which reach further out — to work on exactly the region where the depth is decided.

The masses as the only difference between isotopologues. The two chlorides share a potential to the accuracy of the Born–Oppenheimer separation, and the corrections to it shift levels by fractions of a wavenumber. They cannot account for a 0.27 eV disagreement in depth.

And the measured inputs. Hydrogen fluoride’s expansion has three terms past the quadratic and deuterium chloride’s quartic is borrowed from hydrogen chloride, and each moves the four-level reading. They move the fitted member with it; none of them makes it the potential.

Where the constants stop seeing #

Three constants fixed the bottom of a well and chose between two very different potentials not at all. A fourth fixes a little more of the bottom and chooses between members of whatever family is on offer — which, in a family that can only bend by turning over, looks like evidence about dissociation and is not. The chain of reasoning that runs from a Morse curve’s guessed cubic through two coefficients that are not a potential to here keeps arriving at the same place from new directions: spectroscopic constants are a description of a narrow skin around the minimum, as the rotational constant was a description of an average, and every property of the curve outside that skin — a barrier, a depth, the absence of either — is supplied by the shape the fit was given, not by the constants it was fitted to.

The surprising half is the depth. It is natural to expect a fit to improve as measured inputs are added, and every number here is reproduced exactly; but the fit improved where it was being measured and got worse where it was not, and the plain Morse curve, which knows only two constants, is the best of the depths computed. Three numbers are not a structure, and four are not a potential.

Who measured what #

The level expansions are those compiled by Huber and Herzberg, carbon monoxide’s from a fit to thirty-eight levels, hydrogen fluoride’s from Webb and Rao, and deuterium chloride’s quartic term from Rank, Eastman and co-workers by isotope relations; the dissociation energies are from the same compilation. The Morse curve is Morse’s of 1929, the relation between the constants and the potential’s coefficients Dunham’s of 1932, and the expanded Morse oscillator — of which one member of the family here is the simplest case — belongs to the direct-potential-fit methods developed by Le Roy and co-workers. The like-for-like reading of the measured constants, the bounded family’s dissociation boundary, the four-constant members, their depths and the isotope comparison are computed here.

Still open: the levels themselves, and a long-range tail #

The obvious open question is to stop reading constants at all. Every measured level of carbon monoxide up to v = 37 is implied by its published expansion, and the highest of them reach most of the way to dissociation; fitting the bounded family to the levels themselves, or to as many as it can reproduce, would put the depth under direct constraint for the first time in this line of essays. Whether the family can then reach 11.22 eV while holding the bottom, or whether its shape forbids it, is a question about the family that the four constants could not ask.

The nearer question is the shape far out. A real bond’s potential approaches its asymptote as an inverse power of the separation, not as an exponential, and that is where the depth is decided. A range that saturates gives the curve an exponential tail with a different decay length; one that bends the other way at large separation could mimic the inverse power over the region that matters. Whether a family with the right tail, held to the same four constants, lands nearer the measured depths would say whether the excess found here belongs to having too few constants or to having the wrong tail.