Concept

Born–Oppenheimer separation — where it appears

The separation of nuclear from electronic motion, on the grounds that the nuclei are thousands of times heavier. It is what makes a potential energy surface exist at all, and it is why two isotopes share one surface.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

H₂O and its D isotopologue. Every frequency of H₂O joined to the frequency the same force field gives when every H is replaced by D, with the ratio on each join. The force constants were not refitted and could not be: they do not depend on mass. The product of all the ratios is fixed by the masses and the moments of inertia alone, and is checked against that identity while this figure is drawn.

The isotope shift is arithmetic

Replace hydrogen with deuterium and every frequency drops. The usual rule says by a factor of the square root of two — and of water's three modes, not one of them does that. What is exact is a different identity, and the force constants cancel out of it.

spectra · Normal mode

A structure out of a spectrum. Two rotational constants and two bond lengths, three times over, from three pairs of carbonyl sulfide isotopologues. Above them, the same inversion run on moments computed from a known structure, which returns it to twelve figures. The measured pairs disagree with one another by 7.4 milliangstrom, which is the difference between a ground-state average and an equilibrium geometry.

A bond length out of a spectrum

A linear triatomic has two bond lengths and one moment of inertia, so one measurement cannot determine it. Substituting an isotope gives a second measurement on the same structure, and two equations in two unknowns have a solution — which comes out differently depending on which isotope is used.

spectra · Rotation $water, drawn with its nuclei the size they are. The molecule with a disc round each nucleus whose radius is the computed root-mean-square displacement of that nucleus in the vibrational ground state. The hydrogens' discs are a substantial fraction of the bond length, and this is at no temperature at all.$

The atoms are not at the points

Every structure in this collection is a set of points, and every bond angle it argues about is a property of that set. Computed from the fitted force fields it already has, water's hydrogens are 0.094 Å from where they are drawn and its bond angle has a spread of 8.9 degrees — larger than the difference between 104.5 and the tetrahedral value that half the essays here are about. This is at no temperature at all.

shape · Approximation

H³⁵Cl: the well, its states and their averages. The Morse potential built from H³⁵Cl's measured vibrational constants, with the lowest four states drawn at their computed energies and the average separation of each marked. Every average lies to the right of the minimum, because the well is not symmetric — and they move outward as the state rises.

The bond length that depends on the isotope

Hydrogen chloride and deuterium chloride have the same potential energy curve, and their measured equilibrium lengths agree to three hundredths of a milliångström. Their average bond lengths differ by 4.34 mÅ — a hundred times more — because a lighter atom explores more of a well that is not symmetric.

wrong · Approximation

A correction computed at 3 bohr and used everywhere. Three binding curves for H₂⁺ in 2 Gaussians a centre: uncorrected, properly counterpoise corrected at every separation, and corrected once at 3 bohr with that value subtracted throughout. The frozen curve is the uncorrected one shifted down by a constant, so its minimum sits at 2.2270 bohr — exactly where the uncorrected minimum is, and 4.2 millibohr from where the full correction puts it. The depth moves and the structure does not.

A correction computed at one length

The counterpoise correction is expensive, so it is evaluated once at a reference geometry and subtracted across a whole potential surface. A constant does not move a minimum — so a frozen correction returns the uncorrected bond length exactly, at every reference geometry and in every basis, and everything the correction does to a structure is the part that has just been thrown away.

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

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

Every prediction of the isotope ratio overshoots, and the mass decides nothing. The ratio of NH₃'s ground inversion splitting to ND₃'s, predicted six ways, against the measured 14.94. At the published barrier the usual mass gives 15.77 and the bond-conserving mass 17.79. With a quartic fitted to NH₃'s splitting they give 19.26 and 19.00; with a well whose shape is fitted to both of NH₃'s lines, 17.62 and 18.48. The bond-conserving mass is nearer the measurement in one of the three pairs and further in two, and the difference within any pair is smaller than the distance of either from the measurement.

Deuterium cannot tell the masses apart

A reduced mass built by holding ammonia's bonds rigid predicts a deuterated molecule differently from any constant mass, and ND₃'s splitting is measured. Run as a test, it cannot choose. Every well and every mass needs a barrier for ND₃ several per cent lower than for NH₃, every prediction of the isotope ratio from a well fitted to NH₃ overshoots by eighteen to twenty-nine per cent, and the two masses differ by less than either misses — in opposite directions in the two wells.

shape · Inversion

The covariant operator is BenDaniel–Duke plus this. The difference between the Laplace–Beltrami operator — the one a one-dimensional manifold with metric μ(x) distinguishes, carried across to the flat measure by the unitary map ψ ↦ μ^(¼)ψ — and the BenDaniel–Duke ordering, divided by the function it was applied to, at forty-one positions inside the molecule's own range. The curve drawn through the marks is the two-function fit every ordering is a combination of, and it passes through them to a part in ten million.

The ordering a manifold picks

A position-dependent mass leaves the kinetic energy with no unique quantum form, and an earlier sweep of the five orderings in use found half a per cent between them. A one-dimensional reduction is a one-dimensional manifold, a manifold has a distinguished Laplacian, and carrying it to the flat measure lands on exactly one of those five — not the one with no extra potential, and not the one anybody reaches for.

shape · Inversion

Named alongside it

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

Reduced mass Isotopologue Model limit Zero-point energy Bond length Harmonic approximation Closed form Expectation value Convention Inversion splitting Least-squares Normal mode

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