Molecular Geometry & Bonding
what the shape decides
Figure

H₂O: 3 modes, computed

A stick at every computed vibrational frequency of H₂O, as tall as the mode is degenerate. A solid stick is infrared active, an outline is Raman active only, and a dotted stub is neither. The force field is fitted; the mode shapes and species are not. The observed frequencies are marked beneath, the worst disagreement 0.01%.
H₂O: 3 modes, computed. A stick at every computed vibrational frequency of H₂O, as tall as the mode is degenerate. A solid stick is infrared active, an outline is Raman active only, and a dotted stub is neither. The force field is fitted; the mode shapes and species are not. The observed frequencies are marked beneath, the worst disagreement 0.01%.

One of the figures on spectra: How many bands there can be, where they sit, and what an absent one proves.

Five essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

How many frequencies, not how many modes

The three counts on one figure. Boron trifluoride has six vibrations; the sticks show four distinct frequencies, with the degeneracy of each marked at its foot; and the drawing distinguishes the three that are infrared active from the one that is Raman only. Six, four, three — and every one of those numbers is the answer to a different question.

Two structures, two spectra

Carbon dioxide, computed. Three frequencies from four modes, the bend doubly degenerate; solid sticks are infrared active and the outline is Raman active only. The symmetric stretch at 1,354 is invisible in the infrared, the other two are invisible in the Raman, and no frequency appears in both.

Sulfur dioxide, computed on the same axis. Three frequencies, none degenerate, and every one of them active in both experiments. Superimposing the infrared and Raman spectra of this molecule gives three coincidences; doing it for carbon dioxide gives none.

Planar XY₃, computed: boron trifluoride in D3h, six modes at four distinct frequencies, with a solid stick for an infrared-active mode and an outline for one that is Raman active only. The crucial line is the a₁′ symmetric stretch, which carries a quadratic function and no linear one — so it is an outline and not a solid stick, Raman active and infrared forbidden. That single absence is the test, and it is visible here as a gap in one of the two spectra.

A spectrum that changes when only a mass does

CH₂D₂ from the same constants. The symmetry has dropped again — to C₂ᵥ, which has no degenerate species at all — so every one of the nine modes has its own frequency and the spectrum has nine lines where methane has four. Not one of them was fitted to anything.

Methane’s own spectrum: nine vibrations at four frequencies, with the measured positions beside the computed ones. The force field was fitted to these and to CD₄’s, so their agreement is a fit rather than a prediction — which is why everything else in this essay is about the isotopologues that were not fitted.

CH₃D from the same constants. Six lines where methane has four, with the two lowest at 1204 cm⁻¹ arising from a t₂ mode that has split. No frequency here was fitted to anything.

The one intensity symmetry does fix

Sulfur dioxide’s three bands, whose measured frequencies the force field reproduces. Two of them are totally symmetric and one is not, so a polarisation measurement separates them without any assignment being argued for — which is a stronger footing than the percentages an assignment table prints stand on — which is the practical value of the whole argument and does not depend on the intensities being right.

The table that could not have mattered

What the intensities look like as a spectrum. Every height in it is a magnitude, and every height in it moves with the table; the order they come in does not, and the order is what the argument was about.

Every figure · Every orbital, by what it encloses · All essays