How many coordinates, and how many motions

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

Eight 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.

More coordinates than motions

Six molecules, with one square per internal coordinate and a rule drawn at the number of vibrations. Three of them have more coordinates than motions and three do not, and which is which is decided by shape rather than by size: ammonia’s three angles are independent and boron trifluoride’s are not, and the only difference between them is that one is flat.

Methane’s redundancy, component by component. Every one of the four stretches carries exactly zero and every one of the six angles carries the same weight — one over the square root of six, 0.408248 — which is what “the six angles are not six independent things” looks like written as a vector. Nothing about tetrahedral geometry was put in; the vector is what the null space of B contains.

Formaldehyde’s redundancy, which is not equally weighted. The three angles at its planar carbon still sum to a full turn, so in radians the constraint treats them alike — but the force field works in a scaled coordinate where an angle is multiplied by the geometric mean of the two bond lengths it lies between, and the C=O bond is longer than the C–H bonds. So the constraint is 0.5696, 0.5696 and 0.5926 in the units the calculation uses.

Ten directions no frequency can see

The same count for both molecules, with what a random member of the flat space does. It moves the force constants by more than a whole unit and every frequency by a part in a hundred million.

Methane’s constants under each convention, with the one that cannot be formed marked. Every column that exists reproduces all nine frequencies exactly; the columns disagree by up to the whole of a constant.

The same four columns for boron trifluoride. Two of its constants change sign in magnitude ranking between conventions; none of its frequencies changes at all.

The forty-five that are fixed

Each of methane’s fifty-five constants, grouped by kind, with the fraction of it the spectrum fixes. The stretches are at one; the bends are at 0.5357.

Every block of methane’s constants, diagonalised. Ones are combinations fixed exactly, zeros are combinations no frequency can see, and the bending block has neither.

The same census on boron trifluoride: three stretches at one, three bends with nothing at either extreme, and three of nine stretch–bend combinations invisible.

A formula that predicts minus eleven vibrations

The formula against the answer. Where the answer is a small number and the redundancy count is large, the formula goes through zero and keeps going.

Boron trifluoride’s coordinates, coloured by orbit. Three bonds of one colour are one coordinate as far as the symmetry is concerned.

Benzene’s nine redundancies and their species. Two of the nine are totally symmetric; the other seven are combinations no totally symmetric vibration was ever going to be subtracted by.

Five coordinates for six vibrations

Every torsion orbit of the molecules here that have torsions: how many operations fix it, how many of those are proper, and what reverses it.

The operations of benzene’s group that carry one ring torsion onto itself, and what each does to its sign.

Hydrogen peroxide’s coordinates as built and with its torsion, what they span, and the vibrational representation from the Cartesian displacements.

The second molecule with a blind spot

Methane’s fifty-five independent force constants and boron trifluoride’s twenty-eight, split into what a spectrum determines and what it cannot touch.

The share of each force field a spectrum cannot reach.

Boron trifluoride’s constants by kind, with how many of each the spectrum determines.

Adding data made it worse

Methane’s force field refitted with the bend–bend constant restored, to the light molecule alone and to both isotopologues.

Methane’s nine frequencies before and after the force field is moved ninety times its own norm along a direction the spectrum cannot see.

The worst relative frequency change on each isotopologue when the field is moved along the flat direction.

The residual was a loop

Boron trifluoride’s six force constants sorted by symmetry block, against the distinct frequencies each block supplies.

Every force field that reproduces all six of boron trifluoride’s frequencies exactly, drawn in the stretch constant and the stretch–bend coupling.

The four constants the E′ block depends on, at every point around the family of exact fits.