H₂O: what each mode is made of

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

Six 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.

Normal modes are not bond stretches

Water’s three modes against its three internal coordinates. The bend is entirely a bend — 100% in the H–O–H angle — and neither stretching mode is more than half in either bond. There is no way to read a bond off either frequency, because there is no bond in either frequency.

HOD’s compositions, against the same three coordinates. All three modes are now localised — the bend in the angle, one stretch in each bond — where H₂O had one localised mode and two that could not be. Substituting one atom of a symmetric pair is the only kind of isotopic substitution that can do this.

The isotope shift is arithmetic

Water’s three frequencies joined to deuterium oxide’s, with the ratio on each join. The force constants are identical on both sides and could not be otherwise. At the foot is the product of all three ratios against the value an exact identity requires, which the arithmetic agrees with to two parts in ten billion.

Methane and its fully deuterated form. Nine modes, four distinct frequencies, and four ratios: 1.3086 for the t₂ bend, 1.4137 for the e bend, 1.3654 for the a₁ stretch and 1.3654 for the t₂ stretch. Only one of the four reaches √2, and the reason it does is visible in the mode rather than in the bond.

A substitution on the heavy atom of a linear molecule rather than on a hydrogen. The shifts are much smaller — a part in fifty rather than a factor of the square root of two — and the product rule holds to the same precision, because the rule is about the determinant of a mass matrix and does not care how large the change in it is.

Group frequencies, and where they stop

Boron trifluoride’s four distinct frequencies against its seven internal coordinates. Exactly one of them is a single coordinate: the mode at 719 wavenumbers is 100 per cent the boron leaving the plane of its fluorines. The other three are spread over three bonds or three angles at a time, because those three bonds and those three angles are interchanged by the molecule’s own symmetry operations.

Water, for contrast. One localised mode and two that cannot be, and the two that cannot be are the two whose coordinates are exchanged by the molecule’s own twofold axis. Nothing about the O–H force constant enters this conclusion; it holds for any force field whatever, including a wrong one.

Ammonia, where neither condition holds for anything. All three bonds sit in one orbit and so do all three angles, so no mode is more than a third in any coordinate; and the stretches carry 7 per cent of bending character each because the a₁ stretch and the a₁ umbrella are of the same species. A correlation table has nothing to say about this spectrum that is true of it.

When a mode becomes a bond stretch

Water’s three modes with their compositions. The bend is 99.8 per cent in the angle. The two stretches are exactly half and half in the two bond coordinates, and no fitting could change that.

Heavy water. Every frequency has moved — the bend from 1,649 to 1,206 wavenumbers and the stretches from 3,833 and 3,943 to 2,764 and 2,889 — and every composition is where it was. The two stretches are still half and half.

HOD: one hydrogen replaced, the other left. The bend is still the bend. The two stretches are no longer shared at all — one is 99.5 per cent in the O–H coordinate and the other 99.7 per cent in the O–D, which is as localised as a normal mode of a real molecule gets.

How much of a band is a bond stretch

Ammonia under the most common convention — squared displacements with Wilson’s angle scaling. The umbrella mode is 9 per cent stretch here and 42 per cent by the diagonal energy, and both numbers are in the literature for the same band.

The coordinate an isotope reports