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 8 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

Named alongside it

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

Force constantMode compositionNormal modeVibrational modesHarmonic approximationValence force fieldIsotopologueReduced massDegeneracyModel limitBond lengthLeast-squares

All concepts