6 fitted force fields

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

Four 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.

The isotope shift is arithmetic

The fitted fields, with the three the product rule is tested on at the top. The last column is the count that matters: four constants against eighteen frequencies for methane, five against twelve for ammonia, four against six for water — and three against three for sulfur dioxide and carbon dioxide, which is no test at all. The identity is satisfied to a part in a billion by every one of them, and would be satisfied just as exactly by five quite different fields.

Group frequencies, and where they stop

The six fitted fields, ordered by bond force constant. Nothing in this ordering predicts which molecules have localised modes: methane’s constant is the smallest here and it has no localised mode, boron trifluoride’s is in the middle and it has one, carbon dioxide’s is the largest and it has one. Localisation is a symmetry property, not a strength property.

The force field is not in the spectrum

Five force fields for water. Each one has its stretch–stretch interaction constant held at the value on the axis and the other three refitted, and each reproduces every one of water’s three frequencies to better than a hundred-thousandth. The thin lines are the other constants, which move a great deal. The heavy line is the error each field makes on deuterium oxide, which none of them was fitted to — and there the family disagrees with itself by more than a per cent.

Three members of the family at closer spacing, so the constants can be read off. Every one of them reproduces all three of water’s frequencies exactly — the interaction constant runs over a range and the spectrum does not move at all, which is the whole of what “not in the spectrum” means.

The same family sampled along a different direction: the stretch–bend interaction held at each value, the other three constants refitted. Every field again reproduces H₂O exactly, and the deuterium oxide error again has a minimum — this time a shallower one, at about 0.2, because this constant matters less to the substituted molecule than the stretch–stretch coupling does.

The frequency is not the bond strength

Six fitted valence force fields, ordered by the size of the bond stretching constant, with the molecule’s own stretching frequencies beside it. The two orders are not the same. Sulfur dioxide sits above water in force constant and far below it in frequency; boron trifluoride sits below water in force constant and below it in frequency for a quite different reason.

The carbon–oxygen bonds alone, across every molecule here that has one. Carbon dioxide’s two C=O bonds are related by an operation of the molecule, so they are the same bond by any definition available and the fitted field gives them one constant — and that constant sits far from where the frequency ordering would put it.

The O–H bonds, which are the highest-frequency stretches on the site and not the strongest. Their frequency is high because hydrogen is light; their force constant is middling. Every entry in this column makes the essay’s point on its own, and the column makes it without any comparison across bond types.