Rotational constant — where it appears
Named by 13 essays across one field — each of them below, with the objects they name alongside it.
The rotational spectrum is a moment of inertia
Every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. No bonding argument appears anywhere in it.
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 moment that is the sum of the other two
Three numbers computed from the coordinates and the masses decide whether a molecule is flat. For water, sulfur dioxide, formaldehyde, benzene and every other planar structure here the largest moment of inertia is the sum of the other two exactly; for ammonia it misses by 0.76 and for methane by 3.18.
The rotor that stretches
A rigid rotor's lines are evenly spaced, and a real molecule's are not — it pulls itself apart as it spins. How much is not a fitting parameter: it follows from the stretching frequency by one relation, and the prediction agrees with the measured constant to a few per cent across four molecules spanning three orders of magnitude.
The constant a spectrum cannot see
A symmetric top has two rotational constants and its microwave spectrum reports one of them. Not badly, not with difficulty: ammonia's A of 6.3406 wavenumbers appears in none of its lines at any J and any K, because the term it belongs to cancels exactly out of every transition. The molecule turns about that axis, the energy is real, and the measurement is blind to it.
The top that reports all three
A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.
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 correction that was invented
A standard error model puts the zero-point error in a rotational constant at a few tenths of a per cent, shared between the three moments by invented weights, with the parent and its deuterated form differing by five. Computed from a force field it is 1.88 per cent, one of the three shares is negative, and the mismatch is 35 — so the cancellation the substitution method rests on is worth a factor of 2.5 and not thirteen.
Two moments about two different lines
Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Moment of inertiaModel limitRigid-rotorConventionDegeneracyIsotope substitutionBond lengthHarmonic approximationZero-point energyApproximationLeast-squaresReduced mass