Two coefficients are not a potential

The vibration–rotation constant $\alpha_e$ is how fast a diatomic’s rotational constant falls as the molecule is vibrationally excited, and a Morse curve built from measured constants gets it short in every molecule tried: by 9.7 per cent for HCl, 9.6 for DCl, 14.6 for HF and 4.3 for CO. Dunham’s expansion of the vibration–rotation energy turns $\alpha_e$ into a measurement of the potential’s cubic coefficient, and once the cubic is known the anharmonicity $\omega_e x_e$ is a measurement of the quartic. The Morse curve’s cubic is too shallow, and its quartic — tied to the cubic by the curve’s single shape parameter — is too small as well.

So there is an obvious repair. Build a potential with the measured cubic and the measured quartic, solve it on the same grid, and see whether the shortfall goes. If it does, the whole of the Morse curve’s error was its shape near the minimum; if part remains, something higher matters. That is the test the calculation of the cubic ended by proposing, and it is a clean one.

The repair goes past the measurement #

The quartic is $V = a_0\xi^2(1 + a_1\xi + a_2\xi^2)$ in $\xi = (r - r_e)/r_e$, with $a_0 = \omega_e^2/4B_e$ fixing the harmonic frequency, $a_1$ from the measured $\alpha_e$ and $a_2$ from the measured $\omega_e x_e$. For HCl the coefficients are −2.3646 and 3.6652, against the Morse curve’s −2.2330 and 2.9086.

Solved, the quartic overshoots in every molecule. HCl’s $\alpha_e$ comes out at 0.3215 against a measured 0.3072 — 4.7 per cent over, where the Morse curve was 9.7 per cent under. DCl is 3.3 per cent over, HF 3.4 and CO 0.8. The repair has moved $\alpha_e$ across the measurement by a third to a half of the distance it started from, which is not what repairing the only error would do.

The anharmonicity does not return either. HCl’s quartic gives $\omega_e x_e$ = 50.55 against the 52.82 its quartic coefficient was chosen to reproduce, and HF’s gives 69.28 against 89.88. Only carbon monoxide’s comes back within two per cent.

So a potential with two measured coefficients does not reproduce the two measurements it was built from. Either the solver is wrong, or the relations used to choose the coefficients are not what they appear to be.

The Morse curve’s own series says which #

The solver is the easier suspect to clear. On the complete Morse curve it reproduces the closed-form levels to better than half a wavenumber, returns $\omega_e x_e$ to a part in ten thousand, and gives the same short $\alpha_e$ as a dense diagonalisation and as Pekeris’s formula. A harmonic well gives an anharmonicity of 0.01 wavenumbers, which is the grid’s resolution.

That leaves the relations, and the cleanest test of them uses no measurement at all. A Morse curve has its own Dunham coefficients, and it satisfies both relations exactly: Pekeris’s $\alpha_e$ and the Morse anharmonicity are what the leading-order formulas give for its cubic and quartic. So take the Morse curve, expand it in powers of the displacement — $x^2(1 - x + \tfrac{7}{12}x^2 - \tfrac14 x^3 + \tfrac{31}{360}x^4 - \dots)$ with $x = \beta(r - r_e)$ — and cut the series after the quartic. The cubic and quartic are untouched; only the higher terms are gone.

Cut after the quartic, the Morse curve’s own $\alpha_e$ moves by several per cent: up by 0.7 per cent for CO, 3.1 for DCl, 5.4 for HCl and 7.4 for HF, with its cubic and quartic exactly what they were. Kept to the sextic the errors fall to at most 1.8 per cent, and kept to the octic every molecule is within 0.1. The series converges, and it converges slowly for exactly the molecules whose shortfall was largest.

That is the whole explanation of the overshoot, and it means the relations were never a statement about the cubic and quartic alone. They are the leading terms of an expansion whose next terms involve the coefficients past the quartic, and a potential that has those coefficients — a Morse curve, or a real molecule — satisfies the leading terms because the higher ones are present and arranged to fit. Remove them and the leading relations stop holding by an amount the higher coefficients set.

The size of the effect against the size of the question #

The test the quartic was supposed to perform was whether the Morse shortfall is all cubic. A quartic can perform it only if cutting the series changes $\alpha_e$ by much less than the shortfall. It does not.

Measured against the observed $\alpha_e$, the truncation alone moves the answer by 0.7 per cent for CO, 2.8 for DCl, 4.9 for HCl and 6.3 for HF. The shortfalls being repaired are 4.3, 9.6, 9.7 and 14.6. The truncation is between a sixth and a half of the thing it was meant to measure. A test whose own systematic error is half the effect cannot say whether the effect is all cubic.

The order across the four molecules is exact and it has a reason. The terms past the leading order scale with $(\beta r_e)^2 B_e/\omega_e$: the Morse shape parameter squared, times the ratio of the rotational constant to the vibrational one. That combination is 0.0061 for CO, 0.0127 for DCl, 0.0177 for HCl and 0.0217 for HF, and the truncation’s effect on $\alpha_e$ rises in precisely that order. Carbon monoxide is stiff and heavy, so its zero-point motion samples very little of the potential beyond the quartic; hydrogen fluoride is light and anharmonic, and its motion samples a great deal. The same ordering ran through the calculation of the cubic, where HF missed most and CO least, and it is the same physics: how far from the minimum a molecule’s ground state reaches.

Where the overshoot actually comes from #

$\alpha_e$ here is read from the three lowest rotational constants as $B_0 - B_1 + 2\gamma_e$, with $\gamma_e = (B_0 - 2B_1 + B_2)/2$ taking the quadratic dependence on the vibrational quantum number out — the same definition the Morse comparison used, so that the numbers are comparable. The definition hides something that turns out to be the point.

On HCl’s complete Morse curve, $B_0 - B_1$ is 0.2802 and $2\gamma_e$ is −0.0027. On the quartic with measured coefficients, $B_0 - B_1$ is 0.2506 — smaller than the Morse curve’s, not larger — and $2\gamma_e$ is +0.0709, twenty-six times the Morse value and of the opposite sign. The quartic’s $\alpha_e$ is above the measurement entirely because of $\gamma_e$. The mean displacement, which is what the cubic coefficient was supposed to correct, is only one per cent larger than the Morse curve’s: $\langle\xi\rangle$ = 0.01214 against 0.01202.

The vibrational levels tell the same story in their own third constant. The complete Morse curve’s $\omega_e y_e$ is zero to the grid’s resolution, a few ten-thousandths of a wavenumber, because a Morse curve’s levels are exactly quadratic in $v + \tfrac12$. Every quartic has an $\omega_e y_e$ between 0.4 and 14 wavenumbers — the Morse series cut after $x^4$ and the quartic with measured coefficients alike — and cutting after the sextic brings it down by an order of magnitude, after the octic by another.

So the truncation does not sit quietly in the leading constants. It puts a curvature into the level spacings and into the rotational constants that the complete curve does not have, and the leading constants, read from a handful of levels, inherit it. The constants that are usually treated as small corrections — $\omega_e y_e$ and $\gamma_e$ — are where a missing piece of the potential shows first.

What a picture of the three potentials shows #

Drawn over the region the lowest levels occupy, the three potentials are one curve up to about five thousand wavenumbers, which is where the ground state and most of the first excited state live. They part only on the outer wall, above that: the complete curve rises most steeply, the quartic with measured coefficients least. A difference that shifts $\alpha_e$ by several per cent is nearly invisible where the molecule spends its time.

That is not a defect of the drawing; it is what these constants are sensitive to. A rotational constant is an average of $1/r^2$ over a wavefunction, and its change with vibrational excitation is a small difference between two such averages. The same shape of problem made the mean-displacement term dominate the harmonic one: quantities that are differences of nearly equal averages respond to parts of the potential that the averages themselves barely weight.

The row with the measured cubic and the Morse curve’s quartic is the one that isolates the cubic, and it is the most instructive failure of the six. Its $\alpha_e$ overshoots by 6.7 per cent, more than the quartic with both coefficients, and its anharmonicity is 71.8 against a measured 52.8. A Morse curve ties its cubic and quartic together through one shape parameter, so moving the cubic alone breaks the one constant the curve had right; the measured quartic is what restores it, and it restores it only to leading order. For hydrogen fluoride the same row gives an anharmonicity of 139.6 against 89.9. The cubic cannot be corrected on its own, which is the practical form of the finding that two numbers that are each wrong can make a right difference: the Morse curve’s cubic and quartic are both wrong, and wrong in the proportion that keeps $\omega_e x_e$ exact.

The table puts every number on one page. The complete curve and its octic cut agree on every constant to a tenth of a per cent or better. The sextic cut already disagrees on $\gamma_e$’s sign. The two quartics — one with the measured cubic and the Morse quartic, one with both measured — overshoot $\alpha_e$ by 7 and 5 per cent and disagree with each other about $\omega_e x_e$ by twenty wavenumbers, because the quartic coefficient that was chosen to fix $\omega_e x_e$ at leading order is fighting a cubic that the leading order does not fully describe.

How it was solved #

Each potential is put on a uniform grid of four thousand points from 0.6 to 1.9 times the equilibrium length — 2.6 times for the complete Morse curve, whose wall is softer — and the finite-difference Hamiltonian, which is tridiagonal, is diagonalised by Sturm bisection for the four lowest levels with inverse iteration for their wavefunctions. The rotational constant of each level is $\langle 1/r^2\rangle/2\mu$.

The vibrational constants come from the three spacings between the four levels, solved for $\omega_e$, $\omega_e x_e$ and $\omega_e y_e$ together. That detail matters more than it looks. An earlier reading of the same quartics took $\omega_e x_e$ from two spacings with $\omega_e y_e$ assumed zero, and because a quartic’s $\omega_e y_e$ is several wavenumbers, it reported HCl’s quartic anharmonicity as 19.5 instead of 50.6. A number read through a model of the reading inherits that model’s assumptions, and here the assumption was exactly the thing under test.

The measured constants are the ones the Morse comparison used: Huber and Herzberg’s $\omega_e$, $\omega_e x_e$, $r_e$, $B_e$ and $\alpha_e$ for the four molecules, with atomic masses for the reduced mass.

The checks, run wherever these figures are drawn: the solver reproduces each Morse curve’s closed-form levels and anharmonicity; each Morse curve’s $\alpha_e$ is short of the measurement; each quartic with measured coefficients overshoots it; each Morse series cut after the quartic moves $\alpha_e$ by more than a sixth of the shortfall; every quartic’s $\omega_e y_e$ is over a thousand times the complete curve’s; kept to the octic every series returns the complete curve’s $\omega_e x_e$ to a per cent and its $\alpha_e$ to half of one; and the truncation’s effect is ordered exactly as $(\beta r_e)^2B_e/\omega_e$. The refusal is a harmonic well, which must report no anharmonicity to the grid’s resolution — and does.

What the four molecules leave open #

The potential beyond the quartic is not measured by these constants. $\alpha_e$ and $\omega_e x_e$ fix a cubic and a quartic at leading order; the fifth- and sixth-order coefficients that decide how well the leading order holds appear first in $\gamma_e$, $\omega_e y_e$ and the next Dunham terms, which are tabulated for these molecules and not used here.

The truncation error is not a property of the potential alone. HCl and DCl share one Born–Oppenheimer curve, and their measured constants agree about its cubic to a part in ten thousand. Their truncation errors do not agree: 4.9 per cent of the measured $\alpha_e$ for HCl and 2.8 for DCl. The controlling combination $(\beta r_e)^2B_e/\omega_e$ has the same shape parameter for both and a ratio $B_e/\omega_e$ that scales as the inverse square root of the reduced mass, so it is 1.39 times larger for the lighter molecule — which is $\sqrt{1.904/0.980}$ to three figures. A quartic fitted to one isotopologue and used for the other would carry a different error for each, on a potential they share.

A quartic is the worst possible extension. It is the only polynomial with the two measured coefficients and nothing else, which is why it was the test proposed, and it is not the only way to carry them. A potential with a shape — a Morse curve whose range varies with the separation, or a Rydberg–Klein–Rees curve built from level energies directly — would carry higher coefficients chosen by the data rather than set to zero.

And the grid and the levels are finite. Four levels give three spacings, so $\omega_e y_e$ is determined and nothing higher is; the constants read from them are the constants of those four levels, which is what a spectroscopist fitting low-lying bands would also get.

A leading order is a claim about the rest #

The general lesson is about leading-order relations. They are usually stated as though they held between the quantities they name — this measured constant, that coefficient — and they hold only in the presence of everything they do not name. A Morse curve satisfies Dunham’s relations to the digit and its own quartic does not, and nothing in the relations says so. Inverting a leading-order relation to get a coefficient is safe; building a model that contains only the coefficients it gives is not, because the relation was never about a model that small.

The same caution applies to the harmonic term a spectrum is so often reduced to, which worked for the centrifugal distortion constant because Kratzer’s relation is exact for any potential’s harmonic part. The difference is whether a relation is exact or leading-order, and it is worth finding out before building on it.

Still open: a shaped potential, and water #

A potential that carries the higher terms by construction is the way past this. A Morse curve whose range parameter varies linearly with the separation has one more shape parameter, and fitted to three measured constants it would have a cubic and quartic chosen by the data and higher terms that are not zero.

The obvious open question has a known first step and an unknown second one: fit the family to $\omega_e$, $\omega_e x_e$ and $\alpha_e$ together, then check whether the cubic it lands on is the leading-order inversion or something else. The answer decides whether “the shortfall is the cubic” was ever a well-posed statement, or only a statement about a truncation.

The nearer question is water, which was the reason for building a potential from measured constants at all. Its O–H stretches have measured anharmonicities and its rotational constants are measured on every axis, and its bond coordinates already carry a zero-point correction whose size depends on exactly this mean displacement. The result above says what not to do with them: a quartic in each bond coordinate will move the mean-displacement term by an amount comparable to the correction being sought. Whether a bond coordinate’s $(\beta r_e)^2B/\omega$ is closer to carbon monoxide’s or to hydrogen fluoride’s decides how much that matters, and it can be computed before anything is solved.