Molecular Geometry & Bonding
what the shape decides
Concept

Isotopologue — where it appears

A version of a molecule differing only in which isotopes it contains. It shares a potential energy surface exactly, so it has the same equilibrium structure and different vibrational frequencies, moments of inertia and average bond lengths.

Named by 15 essays across 3 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 ¹¹BF₃: what each mode is made of. ¹¹BF₃. Each row is one distinct frequency and each column one internal coordinate; the bar is the share of the motion in that coordinate. A mode whose largest share reaches nine tenths is a motion of one bond or one angle and is named for it. 1 of 4 here are.

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.

spectra · Normal mode Every force field the spectrum permits. 5 force fields for H₂O, each one refitted with rr:O:H,H held at the value on the axis, and every one of them reproducing all of the molecule's own frequencies. The heavy line is the error each makes on the isotopologue it was never fitted to. The spectrum cannot choose between these fields; the isotopologue can.

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.

wrong · 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 6 fitted force fields. Every valence force field fitted here, ordered by the size of its bond stretching constant, with the stretching frequencies of the molecule beside it. The two orders are not the same, which is the whole of what separates a force constant from a frequency. The last two columns say how many constants were fitted to how many observed frequencies, and a field with as many of the first as the molecule has distinct frequencies fits exactly and reports nothing.

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.

wrong · Normal mode methane, substituted 5 ways. CH₄, CH₃D, CH₂D₂, CHD₃, CD₄ — the same molecule with the same force constants, differing only in which nuclei are heavy. Every column is a consequence of which of the parent's operations survive the mass pattern: the surviving set is a subgroup, it is named from its own conjugacy classes, and the vibrations are reduced in it. The allowed and computed columns come from group theory and from the mass-weighted Hessian respectively, and they agree in every row.

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.

spectra · Spectrum 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 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 One per cent on the barrier is 3.6 per cent on the splitting. The ground inversion splitting of NH₃'s quartic well against the barrier height, both logarithmic, with the geometry and the reduced mass held at their measured values. The curve is visibly bent: its local slope is -3.56 at the published barrier and steepens either side, so a power law is a tangent to it rather than a description of it. The measured 0.7935 wavenumbers is reached at 2330, which is 15.3 per cent above the quoted 2020 — so a splitting wrong by a factor of 1.70 is a barrier wrong by a sixth. The same derivative read the other way is what makes a barrier quoted to ten per cent useless for predicting a splitting.

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.

shape · Inversion One isotopologue converges and two run away. Methane's force field refitted with the bend–bend constant restored, to the light molecule alone and to both isotopologues. The first converges to an ordinary field with a C–H constant of 5.42. The second walks along the direction no frequency can see until two constants reach minus ninety-seven thousand, equal to four figures and opposite in effect. Adding data made it worse.

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.

spectra · Normal mode Every exact fit lies on one closed curve. The one-parameter family of force fields that reproduce all six of boron trifluoride's frequencies exactly, drawn in the B–F stretch constant and the stretch–bend coupling. It is a closed curve: the stretch constant runs from 3.94 to 12.11 millidyne per ångström and the coupling from -0.36 to 6.71. The four fits from four starting points sit together in one small patch of it.

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.

spectra · Normal mode 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 term two of the four molecules have and two do not. The radial coefficient of the bond-conserving reduced mass for the four isotopologues: the sum of the ligand masses, and what is left after the asymmetric correction. Three ligands at a hundred and twenty degrees on a circle whose radius changes as the apex descends move their own horizontal centre of mass outward — unless the three masses are equal. A frame that does not translate has to subtract that motion, and the amount is half the sum of the squared mass differences over the total mass. It is zero at both ends of the series and the same number in the middle.

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.

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 massForce constantModel limitUnderdeterminationConventionLeast-squaresNormal modeBorn–Oppenheimer separationValence force fieldVibrational modesHarmonic approximationInversion splitting

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