Isotopologue — where it appears
Named by 19 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
Group frequencies, and where they stop
A carbonyl band sits near 1,700 wavenumbers in every ketone anybody looks at, and that regularity is real. Computed for a set of small molecules, four of twenty-one distinct frequencies belong to a single internal coordinate — and the four are exactly the coordinates symmetry leaves alone.
The force field is not in the spectrum
Water has three vibrational frequencies and its force field has four constants, so the spectrum cannot determine the field. A whole family of quite different fields reproduces all three frequencies exactly, and only a second isotopologue can tell them apart.
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.
The frequency is not the bond strength
Sulfur dioxide's S–O force constant is larger than water's O–H constant, and its stretching bands sit at a third of the frequency. A vibrational frequency carries a mass as well as a force, and the two cannot be separated by looking at a spectrum.
A spectrum that changes when only a mass does
Methane has four distinct vibrational frequencies and two infrared bands. Replace two of its hydrogens with deuterium and it has nine and eight — with every force constant identical, every nucleus where it was, and the potential energy surface unchanged.
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.
Three numbers is not a structure
Formaldehyde's rotational spectrum gives three constants, of which a planar molecule's are only two independent numbers, and its structure has three parameters. Seven structures are computed here that reproduce A and B to the last digit the solver carries: the C=O length runs from 1.000 to 1.300 ångström, the C–H length from 1.557 down to 0.952, and the HCH angle from 74.6 degrees to 164.5.
The coordinate an isotope reports
Kraitchman's equations return an atom's position from the change in the moments when that atom alone is made heavier, and for a rigid structure they are an identity — formaldehyde's four atoms come back to a part in ten million. What they return is the square of each coordinate, so both hydrogens at b = ±0.9348 come back at +0.9348; every out-of-plane coordinate comes back imaginary at a moment error of one part in a hundred thousand; and the famous error cancellation, measured at a factor of thirteen, still leaves the answer two and a half times worse than a direct fit.
The exponent that runs both ways
How hard does a splitting depend on a barrier? Locally, as the power −3.5628 — and the local slope runs from −2.53 to −6.15 across the same sweep, so there is no power law. What is exact is stranger: rescaling the equation forces the mass exponent to be one below the barrier's and the geometry exponent to be twice the mass's, so the model's three sensitivities are one number and the arithmetic reproduces both identities to six decimals.
Adding data made it worse
A projection leaves a residual spread across four starting points, and an earlier essay attributed it to the eigenvalue problem rather than to the coordinate set — an attribution rather than a measurement. Refitting with two isotopologues was the proposed test. Run, it produces a runaway to minus ninety-seven thousand that one isotopologue alone does not, and the reason is that the direction it walks along is flat at every mass.
The residual was a loop
Projecting away boron trifluoride's flat direction left four fits disagreeing by a tenth in the B–F stretch constant, and the disagreement was put down to the search or to the arithmetic. It is neither. Two frequencies cannot fix three constants in one symmetry block, so the exact fits form a closed curve along which the stretch constant runs from 3.9 to 12.1, and the four fits are four points on a short arc of it.
The third mass that cannot help
Boron trifluoride's E′ block has three force constants and two frequencies, so its exact fits form a closed curve, and a boron-10 substitution cuts that curve to two fields. The obvious remedy is a third isotopologue, and it was named as the first data that could pick one of the two. It is not, if the third substitution keeps the molecule's symmetry: every such isotopologue gives the same spectrum under both fields to a millionth of a wavenumber, because its sum rule is one plane written in two reciprocal masses and its product rule is Teller and Redlich's. Only a substitution that breaks the symmetry adds an equation — and it separates the two fields by half a wavenumber.
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.
The two that are not on the line
Ammonia and its fully deuterated twin are two points, and two points cannot show a curve. Putting the partly deuterated molecules between them needs a term neither symmetric one has — three ligands of unequal mass move their own centre of mass sideways as the apex descends — and it moves the prediction by half a per cent, which is what a whole change of mass construction was worth.
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.
The same fraction at six times the barrier
Ammonia's two fitted barriers leave a gap that an isotope-dependent term would have to fill, and the term was sized at five per cent of the transverse zero-point difference. Phosphine can be asked the same question and cannot answer it the same way: its splitting is twenty-four orders of magnitude below anything the solver can represent, and the solver does not fail cleanly — it reports values that rise as the barrier rises. Calibrated where a splitting is still visible, the action carries the question across, and the same fraction moves phosphine's isotope ratio by sixteen per cent against ammonia's twenty-one.
The band that cannot supply what it changes
A missing transverse zero-point term was sized at five per cent of the difference between two molecules' frozen frequencies, as though the five frozen modes were one thing. They are three bands carrying 26, 50 and 24 per cent of that difference, so the requirement is not one fraction but three, spanning a factor of two — and the band a flattening pyramid changes most is the one that would have to move furthest. What survives the choice is the barrier: every version implies the same bare electronic value to one and a third per cent, because every band of a pyramidal hydride has the same isotope frequency ratio.
The steps were a square root of a mass
Replacing ammonia's hydrogens one at a time does not lower its splitting in equal steps, and the obvious readings are the potential's shape or the tunnelling arithmetic. Both lines — the ground doublet and the first excited one, which nothing in this argument has ever been fitted to — come out straight in the square root of the reduced mass and bent against the count, by a factor of eight, so the unequal steps are the shape of a square root drawn against the wrong variable. And the excited line is the one test the missing correction cannot pass by construction: it improves it threefold and over-shoots it.
Named alongside it
The objects these essays reach for when they reach for this one.
Reduced massModel limitUnderdeterminationForce constantConventionInversion splittingNormal modeWavenumberZero-point energyLeast-squaresValence force fieldBorn–Oppenheimer separation