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.

15 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.

The two coordinates a square plane was missing

What fraction of each molecule’s vibrations its stretches and in-plane bends span. Two are short for a reason no torsion fixes.

The two out-of-plane vibrations, drawn side on. In the first the centre leaves the plane; in the second the ligands alternate and the centre does not move.

The construction. Each coordinate is the signed distance of the centre from one trans pair’s axis, along the normal the two axes define.

Half the distance from a coin

Every ambiguous distortion, sorted. The dashed line is what a label that said nothing would give; the mean is 0.761.

The six least concentrated, with the named species’ share split between its two occurrences. The worst are ammonia’s bond angles.

Each molecule’s mean, with the chance level marked. The two three-atom molecules are the informative ones.

The third mass that cannot help

Ten isotopologues, each computed under both fields, with the largest difference between the two predicted spectra on a logarithmic axis.

The sum of squared E′ frequencies fitted as a linear function of the two reciprocal masses on three isotopologues, and its error on two more.

The fitted field and its partner, constant by constant.

Three ways to stretch a bond

How much of each coordinate’s B-row displacement is the whole molecule translating and rotating, in the metric where the vibrations are orthogonal.

For each distinct coordinate, the share on the favoured occurrence of the species the label names, under the three displacements. The dashed line is a coin.

Every one of the twenty-nine internal coordinates, by what its label decides under each displacement.

The worst case was the loosest constant

How much worse each fit gets as the stretch–bend constant is held away from its fitted value, for water and for ammonia. The shaded band is where the fit cannot tell the fields apart.

The five directions of ammonia’s fit, loosest first, with how fast the misfit grows along each; the last row is the direction the walk actually moves.

Ammonia’s E-species concentrations for every field in the walk: the angle and the bond under the least-cost displacement, and both under the B row.

Two Coriolis constants are one number

$\zeta_3$ (the E stretch), $\zeta_4$ (the E bend) and their sum on every member of ammonia’s family of fields, against the held stretch–bend constant. The shaded band is where the fit cannot tell the fields apart.

How far each observable moves between the two edges of the band, relative to its own size, on a logarithmic scale: the four distinct frequencies of each isotopologue, and NH₃’s two Coriolis constants.

Ammonia seen down its axis, with the path each atom traces when an E vibration’s two partners run a quarter-period apart, at the two edges of the band. The field changes the hydrogens’ ellipses in shape, not in size.

The Coriolis constant was an isotope shift

$\zeta_3$ of ¹¹BF₃ around the loop of exactly fitting fields, against the B–F stretch constant. It runs from −1 to +1; the fitted field’s value meets the loop at exactly two points.

The axial Coriolis constants of boron trifluoride’s E′ motions: the translation, the stretch and the bend, for ¹¹BF₃ and ¹⁰BF₃. The two vibrations’ constants are equal and opposite.

On every field of the loop, $\zeta_3$ against the boron’s share of the stretch’s motion in the plane. Every point lies on one straight line.