Concept

Anharmonicity — where it appears

The departure of a bond's potential from a parabola, which crowds its vibrational levels together as they rise. It makes a bond's average length grow with vibrational excitation, and in a diatomic it is measured through the vibrational and vibration–rotation constants — terms a harmonic force field cannot produce.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Also named here as morse potential — the same set of essays touches all of them, so they are one junction rather than several.

The two terms in a vibrationally averaged rotational constant. A rotational constant averages 1/r², not r², so the moment it reports carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. The two have opposite signs in every molecule here, and the anharmonic one — which is exactly zero in any symmetric well and therefore absent from every harmonic force field — is larger by a factor of 1.94 to 2.58.

The term a harmonic field cannot produce

The usual zero-point correction to a moment of inertia comes from a harmonic force field, which contains the mean square displacement and nothing else. A rotational constant does not average that. It averages one over r squared, whose leading correction is the mean displacement — zero in any symmetric well — and which enters with the opposite sign and about twice the size.

spectra · Rotation $A Morse curve's αₑ falls 4 to 15 per cent short of the measurement. The vibration–rotation constant αₑ of each diatomic, averaged over the states of a Morse curve built from its measured ωₑ, ωₑxₑ and rₑ, as a fraction of the tabulated value. H³⁵Cl: 0.27747 against 0.3072 cm⁻¹, ×0.903; D³⁵Cl: 0.10240 against 0.1133 cm⁻¹, ×0.904; ¹²C¹⁶O: 0.01674 against 0.0175 cm⁻¹, ×0.957; H¹⁹F: 0.68164 against 0.798 cm⁻¹, ×0.854. Every one is short, and HCl and DCl — one potential with two masses on it — are short by the same fraction.$

The cubic a Morse curve guesses

The two terms in a vibrationally averaged rotational constant were computed on Morse curves built from measured constants, and the vibration–rotation constant αₑ is the measurement that tests them. In all four molecules the Morse curve's αₑ is short, by four to fifteen per cent. The averaging is not the error — it matches the closed form to four parts in ten thousand. The curve's cubic is, and the measurement asks for more of exactly the term a harmonic field cannot produce.

spectra · Rotation $The measured cubic and quartic overshoot what Morse fell short of. Each molecule's αₑ as a fraction of the measured value, from its Morse curve and from a quartic potential built with the cubic and quartic coefficients the measured αₑ and ωₑxₑ imply. Every Morse curve is short, by 4 to 15 per cent. Every quartic is over, by 1 to 5 per cent: the repair moves αₑ past the measurement rather than onto it.$

Two coefficients are not a potential

A Morse curve built from measured constants gets the vibration–rotation constant αₑ short by four to fifteen per cent, and the measured αₑ and anharmonicity imply the cubic and quartic a real potential should have. Built with exactly those two coefficients and solved, the potential overshoots instead. The reason is that the Morse curve's own series, cut after its quartic, moves αₑ by a sixth to a half of the error being repaired — so a quartic cannot tell whether the shortfall was the cubic.

spectra · Rotation $Three molecules the family cannot reach, and one it reaches twice. Along a family of potentials that all hold the measured ωₑ and ωₑxₑ, the vibration–rotation constant αₑ as a fraction of the measured one, against the parameter λ that bends the Morse curve's range; λ = 0 is the Morse curve. Hydrogen chloride and deuterium chloride rise to 0.957 and 0.957 and fall again; hydrogen fluoride to 0.909. None reaches one. Carbon monoxide crosses one twice.$

Three constants, two potentials

A Morse curve falls short of the measured vibration–rotation constant, and a quartic built from the two measured coefficients overshoots it. A Morse curve whose range bends with the separation has one more shape parameter, and fitted to ωₑ, ωₑxₑ and αₑ together it should say whether 'the shortfall is the cubic' was ever a well-posed statement. For carbon monoxide it is: two members of the family fit all three constants, and both carry the leading-order cubic to five figures. For hydrogen chloride, deuterium chloride and hydrogen fluoride no member fits at all — and every member that gets close has stopped being a curve that dissociates.

spectra · Rotation

Carbon monoxide's fourth constant picks the fit that turns over. Along the Morse curves with a linearly bent range that hold carbon monoxide's measured ωₑ and ωₑxₑ, the ωₑyₑ read from the lowest four levels, against the bend λ. The two members that also have the measured αₑ are marked: the one at λ = −0.221 dissociates and predicts 0.0182 cm⁻¹; the one at λ = −0.453 turns over at 2.10 bond lengths and predicts 0.0110. The measured value, read from the published level expansion the same way, is 0.0110 — the second, to 0.0 per cent.

The fourth constant chose a curve, not a depth

Read from its lowest four levels the way every potential here is read, carbon monoxide's measured ωₑyₑ is 0.01101 cm⁻¹ — and of the two bent-range Morse curves that fit its first three constants, the one that predicts 0.01100 is the one that never dissociates. Let the range bend and stay bounded and every molecule, the hydrides included, is fitted to all four constants by a curve that does dissociate. The depths those curves carry are 6 to 31 per cent too deep, further from the measurement than the plain Morse curve's on all four, and hydrogen chloride and deuterium chloride — one potential — are fitted by two different ones.

spectra · Rotation

Named alongside it

The objects these essays reach for when they reach for this one.

Model limit Morse potential Rotational constant Expectation value Isotope substitution Perturbation theory Underdetermination Zero-point energy Approximation Bond length Born–Oppenheimer separation Convention

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