Molecular Geometry & Bonding
what the shape decides
Concept

Moment of inertia — where it appears

The sum of each mass times the square of its distance from an axis. Three of them, about the principal axes, are all a rotational spectrum measures, and no bonding argument appears anywhere in that measurement.

Named by 15 essays across 2 fields — each of them below, with the objects they name alongside it.

carbonyl sulfide: a linear. The principal axes of carbonyl sulfide drawn at its centre of mass, with the moment of inertia about each. Which kind of top this makes it is decided by comparing three numbers, and the point group forces the same answer independently: an axis of order infinity means a linear molecule, with one moment at zero forbids an asymmetric top.

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.

spectra · Rotation 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 One dihedral, three point groups. Hydrogen peroxide built at 13 values of its dihedral angle, with the point group searched for from the coordinates at each. The group is C2v when the hydrogens eclipse, C2h when they are anti, and C2 at every angle strictly between. The rails below are what the group settles on its own: the molecule may be polar except at the anti arrangement, and it is chiral except at the two ends. No energy is computed anywhere, and the marked angle is the measured one rather than a minimum found here.

One coordinate, three point groups

Hydrogen peroxide has four atoms and one soft internal coordinate. Turning it from nought to a hundred and eighty degrees takes the molecule through C2v, C2 and C2h — so it is chiral at every angle but two, and forbidden a dipole at exactly one of them.

shape · Dipole 7 molecules, three moments each. The principal moments of inertia of water, sulfur dioxide, formaldehyde, boron trifluoride, ammonia, methane, hydrogen peroxide, in u Ų, with the inertial defect and the asymmetry parameter beside them. The defect vanishes exactly for a planar structure and does not for any other, so three numbers computed from the coordinates decide planarity with no model anywhere in the argument. Every classification is checked against the one the molecule's point group forces.

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.

spectra · Rotation H₂O: 3 distinct modes. The displacement of every atom in 3 normal modes of H₂O, drawn from the eigenvectors of the mass-weighted Hessian. Under each is the internal coordinate with the largest share of the motion and how large that share is; where no coordinate holds nine tenths, the mode is not a motion of one bond or one angle and is labelled so.

The six motions that are not modes

Three N minus six is quoted in every textbook and the six are almost never computed. Diagonalising a mass-weighted Hessian gives six eigenvalues at arithmetic noise, and projecting their vectors onto the three translations and three rotations written down from the geometry alone accounts for every one of them.

spectra · Normal mode A microwave constant predicted from an infrared one. For four diatomics: the rotational constant and the stretching frequency, both measured, and the centrifugal distortion constant predicted from them by 4B³/ω² — then the constant a microwave spectroscopist fits to the line positions. The prediction and the fit agree within a few per cent across three orders of magnitude in the quantity, and nothing connects the two measurements except the assumption that the bond stretching under rotation is the same bond that vibrates.

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.

spectra · Rotation ammonia's rotational levels, sorted by K. The rigid rotational levels of ammonia up to J = 4, each drawn at its computed energy and grouped by J. Within a group the levels are pushed apart by the second rotational constant, so it is plainly there in the level pattern.

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.

spectra · Rotation water: every level up to J = 4, from a matrix. The rotational levels of water at κ = -0.4322, each J diagonalised in the symmetric-top basis. A symmetric top would show one level per K with everything above K = 0 doubly degenerate; here every degeneracy is split, and the size of each splitting is what the third constant is measured from.

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.

spectra · Rotation Three parameters, two numbers, and a curve of answers. seven structures of formaldehyde, every one of which reproduces the measured rotational constants A and B exactly. The C=O length runs from 1 to 1.3 ångström, the C–H length from 1.56 down to 0.95, and the HCH angle from 74.58 to 164.47 degrees. The third constant is not a third number: for a planar molecule it is fixed by the other two, and it comes out at 1.14 for every member.

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.

spectra · Rotation Every sign, lost. Formaldehyde in its own principal axes. Open circles are the atoms where they are; filled ones are where Kraitchman's equations put them, from the change in the three moments when each atom in turn is made heavier. The two agree to 7.6e-8 ångström — the equations are an identity for a rigid structure — but they return the square of each coordinate, so the two hydrogens at b = ±0.9348 both come back at +0.9348 and land on the same point.

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.

spectra · Rotation The factor of thirteen was generous. An invented error model against the one computed from a force field. Its invented mismatch of five per cent made the correlation between the parent and the substituted species worth a factor of 12; the computed mismatch of 35 per cent makes it worth 2.5. The substitution structure went from being 2.5 times worse than a direct fit to 26, and at the computed size of the correction its worst coordinate is out by 10.7 per cent.

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.

spectra · Rotation One expression, four molecules, a factor of eleven. The computed zero-point correction to each molecule's moment of inertia against the expression A²·Σ(1/ν)/I, which has no fitted quantity in it. The dashed line is the mean dimensionless coefficient, 0.6343; the four points lie within 18 per cent of it, on corrections that span a factor of 11.4. The expression is evaluated from a moment of inertia and a list of wavenumbers, which is what a spectroscopist has before doing anything.

An expression for what was a warning

The zero-point correction to a moment of inertia comes out at 1.88 per cent where a few tenths had been assumed, and heavy molecules are safer. Written out, the correction is a mean curvature times the sum of reciprocal wavenumbers over the moment — one line, evaluated from things a spectroscopist has before starting. One coefficient serves four molecules whose corrections span a factor of eleven.

spectra · Rotation The coefficient each principal axis needs. The dimensionless coefficient the molecule-averaged expression requires, evaluated on each principal axis separately rather than on the mean of the three. Across the twelve axes the positive ones span a factor of 2.33, against the 1.36 the molecule-averaged version spans — and one of them is negative, which no positive constant can be.

The axis that goes the other way

One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.

spectra · Rotation The frame H₂O → HDO turns. H₂O → HDO: the parent's principal axes and the daughter's, drawn on the same nuclei. Two of the three turn by 21.12° and the third does not move at all, because it is the normal to a plane no substitution can tilt. A per-axis comparison between these two species is comparing moments about lines this far apart, which is a rotation of the frame rather than a correction to a number.

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.

spectra · Rotation 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

Named alongside it

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

Rotational constantModel limitHarmonic approximationConventionDegeneracyRigid-rotorApproximationIsotope substitutionZero-point energyBond lengthClosed formReduced mass

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