The cubic a Morse curve guesses

The term a harmonic field cannot produce took apart the moment of inertia a rotational constant reports. A constant averages 1/r², and to second order the moment it implies sits above the equilibrium moment by twice the mean displacement less three times the mean square displacement, both in units of the equilibrium length. The first term is zero in any symmetric well, so no harmonic force field contains it; computed on Morse curves built from each molecule’s measured $\omega_e$ and $\omega_e x_e$, it came out about twice the size of the second and of the opposite sign.

That essay closed by naming the test. Every one of those numbers was computed on a potential nobody had measured, and there is a measured quantity that is precisely the change it describes:

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.

The test is run here. The Morse curves fail it, in all four molecules, in the same direction — and the way they fail says the decomposition was right about which term dominates and wrong about how much.

What the vibration–rotation constant measures #

A spectrum fitted state by state gives a rotational constant for each vibrational level, and they fall off as $B_v = B_e - \alpha_e(v + \tfrac{1}{2}) + \gamma_e(v + \tfrac{1}{2})^2$. So $\alpha_e$ is how fast the reported constant falls as the molecule is excited, and half of it is how far the ground state’s constant sits below the equilibrium one.

Dunham’s expansion of the potential about its minimum, in powers of $\xi = (r - r_e)/r_e$, gives that constant in closed form:

$$\alpha_e = -\frac{6B_e^2}{\omega_e}\,(1 + a_1),$$

where $a_1$ is the cubic coefficient of the potential in $\xi$. The two pieces are the two terms of the moment decomposition. The 1 is the harmonic spread, which makes a molecule’s average of $1/r^2$ larger than $1/r_e^2$ and so raises B. The $a_1$ is the mean displacement: a negative cubic tilts the well outwards, the average separation grows, and B falls. Because the spread’s contribution is always positive and the displacement’s negative, $\alpha_e$ is a measurement of $a_1$, offset by one.

A Morse curve has an $a_1$ too, fixed by the same two numbers that fix everything else about it. So the comparison is between the cubic a measurement implies and the cubic a Morse curve guesses.

The contrast with the other rotational constant a vibrational spectrum predicts is worth having in view. The rotor that stretches found the centrifugal distortion constant predicted from the stretching frequency to within a few per cent in four molecules, and the reason is that Kratzer’s relation, $D_e = 4B_e^3/\omega_e^2$, contains no anharmonicity at all: stretching under rotation is a harmonic response. $\alpha_e$ is the first rotational constant that cannot be had from a harmonic frequency, which is exactly why it is the one that can test a shape.

Short in all four #

Averaged over the first three solved states of each Morse curve, with $\gamma_e$ taken out so that the number compared is $\alpha_e$ itself, the vibration–rotation constants are 0.27747 cm⁻¹ for HCl, 0.10240 for DCl, 0.01674 for CO and 0.68164 for HF. Huber and Herzberg’s tabulated values are 0.3072, 0.1133, 0.0175 and 0.798.

The ratios are 0.903, 0.904, 0.957 and 0.854. Every Morse curve’s $\alpha_e$ is too small, by between four and fifteen per cent, and the order is suggestive before anything else is done: the hydrides miss by most, the lightest and most anharmonic of them by the most, and carbon monoxide, the stiffest and most nearly harmonic, by the least.

Each curve does reproduce its molecule’s measured $B_e$, to a part in ten thousand or better — 10.59345 against 10.5934 for HCl — so the comparison is not contaminated by a wrong equilibrium length. The rotational constant at the bottom of the well is right, and what is wrong is how it changes on the way up.

Not the averaging #

The first suspect is the arithmetic, since the averages are sums over a four-hundred-point grid of wavefunctions that are only as good as the grid.

Pekeris solved the rotating Morse oscillator in 1934 and gave its vibration–rotation constant as $6\sqrt{\omega_e x_e B_e^3}/\omega_e - 6B_e^2/\omega_e$, with nothing in it but the curve’s own parameters. The averages over the solved states agree with that formula to 0.04 per cent for HCl and HF, 0.03 for DCl, and to four figures for CO. Their distance from the measurement is a hundred to four hundred times larger. Doubling the resolution does not close it either: with six hundred points HCl’s average moves by under half a per cent.

So the solver and the formula describe the same curve, to a precision far beyond the effect. What misses the measurement is the curve.

The cubic the measurement asks for #

Inverting Dunham’s relation turns each $\alpha_e$ into a cubic coefficient. The Morse curves imply $a_1$ = −2.2330 for HCl, −2.6232 for CO and −2.0710 for HF. The measurements imply −2.3646, −2.6967 and −2.2533. In every molecule the real potential is more asymmetric about its minimum than the Morse curve that shares its $\omega_e$ and $\omega_e x_e$.

The reason a Morse curve gets this wrong is structural rather than accidental. A real potential’s shape near its minimum has independent cubic and quartic coefficients, and the anharmonicity constant $\omega_e x_e$ depends on both — on the square of the cubic and on the quartic together. A Morse curve has one shape parameter, its range, so matching $\omega_e x_e$ fixes its cubic and quartic in a single combination, and there is no reason for the cubic that falls out to be the molecule’s. $\alpha_e$ depends on the cubic alone, which is why it is the measurement that exposes the compromise.

One potential, two masses #

A shortfall in four molecules could still be a shortfall in four sets of tabulated constants. There is a way to tell that needs no further data.

HCl and DCl share one Born–Oppenheimer potential, so they share one cubic coefficient, while their $\omega_e$, $\omega_e x_e$, $B_e$ and $\alpha_e$ are all different because the mass is. From the measured constants the cubic comes out at −2.3646 for HCl and −2.3644 for DCl, the same to one part in ten thousand, from inputs that differ by factors of 1.4 to 2.7. That is a clean check on the tabulated data and on the Dunham relation at once — the same kind of check one potential with two bond lengths on it supplied for the average separation, and the one a model of ammonia and its deuterated form could not pass with its fitted barrier.

The Morse curves agree with each other as well, −2.2330 and −2.2334, because a Morse range parameter built from $\omega_e x_e$ and the reduced mass does not depend on the mass. And they miss by the same fraction, 0.903 and 0.904. So the shortfall moves with the potential and not with the molecule: it belongs to the Morse shape, identically for both masses on it.

Which molecules miss most, and why it is not the mass #

The four shortfalls order themselves — hydrogen fluoride by 14.6 per cent, the two hydrogen chlorides by 9.7, carbon monoxide by 4.4 — and the natural reading is that lighter molecules explore more of their wells and so expose more of a wrong shape. The isotopologue pair rules that reading out: DCl has twice HCl’s reduced-mass-weighted spread relative to its range, a harmonic share of 0.76 per cent of $B_e$ against HCl’s 1.06, and misses by the same fraction to the fourth figure.

What orders the four is how far each real potential sits from its Morse curve, which is a property of the potential and not of what is vibrating on it. The measured cubic is steeper than the Morse one by 0.18 for hydrogen fluoride, 0.13 for both hydrogen chlorides and 0.07 for carbon monoxide, and that is the order of the shortfalls. A bond to hydrogen is not more badly served by a Morse curve because hydrogen is light; hydrogen fluoride’s and hydrogen chloride’s potentials are simply less Morse-like near their minima than carbon monoxide’s, in a way the average bond length of an isotopologue pair cannot see and $\alpha_e$ can.

The same measurement also fixes what a structure from a rotational constant is exposed to. A bond length taken out of a spectrum from $B_0$ rather than $B_e$ carries the ground state’s shift, and that shift is half of $\alpha_e$ — so a Morse estimate of the correction to a hydride’s $r_0$ is short by the same ten per cent, in the same direction, before any zero-point motion of the other atoms is considered.

Why six per cent becomes ten #

The measured cubic is only 5.9 per cent steeper than the Morse one for HCl, and yet its $\alpha_e$ is 10.7 per cent larger. That gap is not a mistake; it is the offset in Dunham’s relation doing what a difference always does.

$\alpha_e$ is proportional to $-(1 + a_1)$, so for HCl it goes as 2.3646 − 1 against 2.2330 − 1: 1.3646 against 1.2330, a ratio of 1.107. The harmonic spread contributes the 1 and cancels part of the mean displacement’s contribution, and a change in the larger term shows up magnified in what is left. The same arithmetic runs through carbon monoxide, where a 2.8 per cent steeper cubic becomes a 4.5 per cent larger $\alpha_e$, and hydrogen fluoride, where 8.8 per cent becomes 17.

It is the pattern the moment decomposition itself warned about — a quantity computed as a difference of two larger quantities does not inherit their stability — seen here from the side of the measurement. A constant that is a small difference is a sensitive probe of the larger term in it, and $\alpha_e$ probes the cubic with a gain of nearly two in the hydrides.

What it does to the term a harmonic field misses #

The decomposition’s conclusion was that the mean-displacement term — the one a harmonic force field cannot produce — is about twice the harmonic term and of the opposite sign. The measurement does not weaken that. It strengthens it.

In Dunham’s form the harmonic spread lowers the reported moment by an amount set by $B_e/\omega_e$ alone, 1.063 per cent of $B_e$ for HCl, and that part a Morse curve gets exactly right because it gets the harmonic frequency right. The mean displacement’s contribution is 2.373 per cent on the Morse curve and 2.513 per cent with the measured cubic, a factor of 1.059 larger; for DCl the factor is 1.059, for carbon monoxide 1.028, and for hydrogen fluoride 1.088. The ratio of the two terms, which the decomposition put at a little over two, is 2.36 for HCl once the cubic is the one $\alpha_e$ asks for.

So every harmonic zero-point correction to a rotational constant is off by more than the Morse calculation said: it omits a term that is larger than the one it includes, and larger again than a Morse curve’s estimate of it. The per-axis corrections two moments about two different lines compared and the error model the substitution method inherited are both exposed to that term, and neither contains it.

How the constants were computed #

Each molecule’s Morse curve takes $\omega_e$, $\omega_e x_e$ and $r_e$ from Huber and Herzberg’s compilation and nothing else; the well depth is $\omega_e^2/4\omega_e x_e$ and the range follows from $\omega_e x_e$ and the reduced mass. The first three vibrational states are found by diagonalising the Hamiltonian on four hundred points between 0.5 and 8 bohr, and each state’s rotational constant is ⟨1/r²⟩ over its wavefunction divided by twice the reduced mass. $\alpha_e$ is $B_0 - B_1 + 2\gamma_e$ with $\gamma_e = (B_0 - 2B_1 + B_2)/2$, so the quadratic term in the vibrational dependence does not leak into the number compared.

The checks are these. Each curve reproduces its molecule’s measured $B_e$ to a part in a thousand. The averaged $\alpha_e$ agrees with Pekeris’s closed form to a per cent. Each falls short of the measured $\alpha_e$ by more than four per cent. The measured constants of HCl and DCl imply the same cubic to a part in a thousand, and their Morse curves miss by the same fraction to a per cent. Six hundred grid points move HCl’s $\alpha_e$ by under half a per cent. And the refusal: the same calculation on a nearly harmonic well — HCl’s curve with its anharmonicity divided by four hundred — must give a negative $\alpha_e$, since with no mean displacement the spread alone makes the constant rise with excitation. It gives −0.206 cm⁻¹. A calculation whose sign was built in could not.

What four diatomics cannot settle #

Dunham’s relation is a leading-order one. It connects $\alpha_e$ to the cubic coefficient with corrections of relative order $B_e/\omega_e$, which is a third of a per cent for HCl and smaller for the others — far below the shortfall, and the reason the cubic coefficients above are quoted to four figures and not five.

The tabulated constants are fits. $\alpha_e$ and $\omega_e x_e$ come from fitting level energies and line positions to power series, and the series are truncated where the data stop. The agreement between HCl and DCl is the evidence that those fits are consistent with one potential; it is not evidence about their absolute accuracy.

A diatomic is one coordinate. The polyatomic question the decomposition raised — whether a cubic term changes the sign of water’s smallest moment’s correction — needs three coordinates and cross terms no diatomic has. What transfers is the direction of the error: a potential fitted only to its frequencies and anharmonicities underestimates the mean displacement’s share, and the share is already the larger one.

And Born–Oppenheimer is assumed. The two isotopologues’ agreement to a part in ten thousand bounds how much a mass-dependent correction to the potential could change the cubic coefficient here; it does not remove the assumption.

A test locates what it tests #

The transferable point is about what a measurement can say about a model with several parts.

A comparison of a computed $\alpha_e$ with a measured one tests everything that went into the computation: the potential’s shape, the numerical average, the relation between the average and the constant, and the tabulated data. When the comparison fails, the useful work is to test each part separately against something that shares only that part. Here the closed form tested the averaging and cleared it; the isotopologue pair tested the data and the potential’s mass-independence and cleared them; what was left was the Morse shape, and the Dunham relation named the one parameter of it that $\alpha_e$ depends on.

The second half is about which direction the answer moved. A test against measurement is often expected to shrink a surprising result back towards convention. This one enlarged it. The decomposition’s claim that the omitted term dominates was a claim about Morse curves; the measurement says real potentials are more asymmetric than Morse curves, so the omitted term dominates more. An expression for what was a warning turned a zero-point correction into something a spectroscopist could evaluate; the evaluation now needs $\alpha_e$ as well as the harmonic frequencies, and a spectroscopist has that too.

Who measured it #

Morse proposed his potential in 1929 and Dunham gave the expansion of vibration–rotation energies in powers of the displacement in 1932; Pekeris solved the rotating Morse oscillator in 1934. The constants are Huber and Herzberg’s 1979 compilation, and the four molecules have been measured many times since with consistent results at the precision used here.

The comparison of exact Morse averages with measured $\alpha_e$, the isotopologue check on the cubic coefficient, and the consequence for the decomposition of a reported moment are computed here. The moment decomposition deserves the credit for naming the test and saying in advance what a mismatch would mean — though the mismatch turned out to live in the potential rather than in a higher moment, which is a more useful place for it to be.

Still open: the potential with its measured cubic, and water #

The obvious open question is the potential itself. Dunham’s relations give the cubic from $\alpha_e$ and the quartic from $\omega_e x_e$ once the cubic is known, so a curve with both measured coefficients can be built and solved on the same grid. It would say whether the whole of the Morse shortfall is the cubic or whether the quartic the Morse curve assumes also needs correcting, and it would give the moment decomposition its terms on a potential with no guessed shape parameter in it.

The nearer question is the one the decomposition left about water. Its O–H stretches have measured anharmonicity constants and its vibration–rotation constants are measured for every axis, so the same inversion — measured constants to cubic coefficients to the mean-displacement term — can be run bond by bond. Whether the measured cubic changes the sign of the a-axis correction that the harmonic treatment made negative is now a question with its inputs already tabulated.