boron trifluoride at 1454 cm⁻¹, shared out four ways

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

Two 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.

How much of a band is a bond stretch

One mode of boron trifluoride — the degenerate pair at 1454 cm⁻¹ — with each internal coordinate’s share of it computed four ways. Every bar in a group is the same eigenvector of the same matrix. The B–F stretch is 12.2, 16.6, 32.6 or 31.5 per cent of the mode per bond, which is 36.7, 49.9, 97.9 or 94.6 per cent in total.

Every mode of boron trifluoride shared out by what moves. Three of the six are unambiguous: the 719 cm⁻¹ mode is entirely the out-of-plane coordinate, the 888 is entirely the three stretches, and the 480 pair is entirely the three angles. The mixed one is the 1454 pair, and it is the only row any of this is about.

The same six modes shared out by the potential-energy distribution. Five rows are identical to the previous figure to within a tenth of a per cent. The 1454 row is not: what was 12.2 per cent per bond is now 32.6, and what was 21.1 per cent per angle is now 0.7.

More coordinates than motions

Boron trifluoride’s modes shared out among its internal coordinates. Every share here is a share of a set that carries a redundancy, so a combination of the columns can be added to any row without changing the mode it describes — which is the null space behind the percentage conventions, arriving as a property of the coordinate set rather than as a choice of convention.