Molecular Geometry & Bonding
what the shape decides
Concept

Wavenumber — where it appears

The reciprocal of a wavelength, in which vibrational and electronic energies are usually quoted. It is proportional to energy, so a difference in wavenumbers is a difference in energy without any conversion.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

Normal modes are not bond stretches

Water has two stretching frequencies and two O–H bonds, and it is almost irresistible to pair them off. Computed, each mode is exactly half in one bond and half in the other, and neither frequency belongs to a bond at all.

spectra · Normal mode 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 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 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 boron trifluoride at 1454 cm⁻¹, shared out four ways. One mode of boron trifluoride, with each internal coordinate's share of it computed by four conventions. Each convention is a bar in every group; the groups are the coordinates. A number quoted for this band without saying which bar it is has not said much.

How much of a band is a bond stretch

Boron trifluoride's 1454 cm⁻¹ band is 36.7 per cent B–F stretch, or 49.9, or 97.9, depending on which of four standard ways of sharing a mode out among internal coordinates is used. All four are defensible, all four sum to one, and the spread between them is sixty-one percentage points on one band of one molecule.

spectra · Normal mode NH₃: four states in a well the molecule does not sit at the bottom of. The umbrella coordinate of NH₃ — the signed distance of the N atom from the plane of its three H atoms — with the quartic well that has its minima at the measured 0.3816 ångström and its barrier at the quoted 2020 wavenumbers. The lowest 4 states are drawn at their computed energies. The lowest sits 587 wavenumbers above the bottom, which is 29.1 per cent of the way up the barrier, so the state is far from the harmonic bottom that a drawing of a pyramid implies.

A barrier is not what a splitting measures

Ammonia's inversion barrier is quoted everywhere as 2020 wavenumbers. Put that number into the simplest double well its own measured geometry allows and the ground-state splitting comes out at 1.3508 against a measured 0.7935, and the excited one at 68.37 against 35.81. Both are too large because a splitting is an area under a barrier and a height is only one of its two dimensions.

shape · Inversion Two pyramids, and only one of them has ever shown a line. What ammonia and phosphine each bring to the same one-dimensional model. The barrier is six times larger and the apex sits twice as far out, so the action under the barrier is seven times larger and the splitting is twenty-four orders of magnitude smaller. What is not six times anything is the transverse zero-point difference between a molecule and its deuterated twin: the modes that carry it are the ligands' own, the ligands are the same three hydrogens, and the two numbers are within thirty per cent of each other.

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.

shape · Inversion Three bands, and the isotope difference is not shared as they are. Ammonia's three transverse bands, each drawn twice: the zero-point energy it carries in NH₃ and the one it carries in ND₃, both from measured fundamentals. The pale bar is the light molecule and the solid one the heavy; the gap between them is that band's share of the isotope difference the correction has to come out of. The degenerate stretch carries half of it because it is doubly degenerate and high; the bend carries a quarter.

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.

shape · Inversion Two patterns, one shape, and a slope between them. How far the logarithm of each splitting falls as one hydrogen at a time is replaced by deuterium, for the ground doublet and for the first excited one. Neither falls in equal steps — the first step is half again the last — and the two patterns are the same shape: the excited step is a fixed fraction of the ground step at every stage, within a few per cent. The well is fitted to both of NH₃'s measured lines and nothing is fitted to any deuterated molecule.

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.

shape · Inversion

Named alongside it

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

Force constantIsotopologueReduced massMode compositionModel limitNormal modeVibrational modesHarmonic approximationValence force fieldInversion splittingZero-point energyDegeneracy

All concepts