Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 32

Page 32 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

What a spectrum settles

A spectrum is a list of positions, and how many there can be is decided by the shape before any of them is measured. Counting them, and reading a structure back out.

The reading is a function of direction, with a sixty-degree period. The departure of a depolarised band from three quarters, for distortions of equal magnitude pointing all the way round the plane of stretches that sum to zero. It is not one number: it runs from 8.177e-6 to 1.617e-5, a factor of 1.978. The minima are at one bond against another and the maxima at two bonds against one, and the pattern repeats every sixty degrees.

One number was a direction too

The depolarisation probe is exactly blind to the totally symmetric direction, which suggests it reads the distortion's component outside that species. What is left is a plane, and over a circle in it the reading varies by a factor of 1.98 — smallest for one bond against another, largest for two bonds against one, repeating every sixty degrees.

6 figures The sum is flat and the maximum is not. Both readings round the circle, each divided by its own average so the two can be drawn together. The maximum varies by 98 per cent; the sum over all four depolarised bands varies by 0.038. A sum cannot be changed by reordering, so if each band's departure is an isotropic quadratic form the sum must be constant — and to four parts in ten thousand it is.

The sum was flat all along

The depolarisation reading varies by a factor of two round a circle in the non-symmetric plane, and the anisotropy may belong to taking a maximum over four bands rather than to the physics. Summing the four instead gives a reading constant to four parts in ten thousand, against a maximum that varies by ninety-eight per cent.

6 figures Reversing the distortion changes the answer, which settles the order. How much the sum changes when the distortion is reversed — the same direction taken backwards — at each angle in the plane. Along a basis direction it changes by two parts in ten million, which is nothing. Halfway between them it changes by 7.36e-4, which is the same size as the isotropy residual itself at 5.24e-4. A response with only even powers of the amplitude cannot do that, so the leading anisotropic term is the cubic one and not the quartic.

The suspect that did not fit

The sum over depolarised bands is flat to four parts in ten thousand rather than exactly, and the quartic term is the obvious suspect. Two tests say otherwise. The residual scales as amplitude to the 1.248, which is neither candidate — and reversing a distortion changes the sum by as much as the residual is, which only an odd power can do. The leading term is the cubic, and the suspect was wrong by one order.

7 figures A Morse curve's αₑ falls 4 to 15 per cent short of the measurement. The vibration–rotation constant αₑ of each diatomic, averaged over the states of a Morse curve built from its measured ωₑ, ωₑxₑ and rₑ, as a fraction of the tabulated value. H³⁵Cl: 0.27747 against 0.3072 cm⁻¹, ×0.903; D³⁵Cl: 0.10240 against 0.1133 cm⁻¹, ×0.904; ¹²C¹⁶O: 0.01674 against 0.0175 cm⁻¹, ×0.957; H¹⁹F: 0.68164 against 0.798 cm⁻¹, ×0.854. Every one is short, and HCl and DCl — one potential with two masses on it — are short by the same fraction.

The cubic a Morse curve guesses

The two terms in a vibrationally averaged rotational constant were computed on Morse curves built from measured constants, and the vibration–rotation constant αₑ is the measurement that tests them. In all four molecules the Morse curve's αₑ is short, by four to fifteen per cent. The averaging is not the error — it matches the closed form to four parts in ten thousand. The curve's cubic is, and the measurement asks for more of exactly the term a harmonic field cannot produce.

6 figures The two molecules with a direction no spectrum can see. Methane's fifty-five independent force constants and boron trifluoride's twenty-eight, split into the combinations a spectrum determines and the ones it cannot touch. Both molecules have a redundant coordinate set — methane's six angles at a tetrahedral centre are five coordinates' worth, and boron trifluoride's three angles at a planar centre are two — so both have a flat space, and the smaller molecule's is the larger share of its field.

The second molecule with a blind spot

Methane's fifty-five force constants have ten directions no spectrum can touch, and the forty-five that are fixed were worth a table. Boron trifluoride is the only other molecule here with a redundant coordinate set, and it is not the same case: eight kinds of constant rather than five, a quarter of its field invisible rather than a fifth, and an out-of-plane coordinate the redundancy cannot reach.

5 figures One isotopologue converges and two run away. Methane's force field refitted with the bend–bend constant restored, to the light molecule alone and to both isotopologues. The first converges to an ordinary field with a C–H constant of 5.42. The second walks along the direction no frequency can see until two constants reach minus ninety-seven thousand, equal to four figures and opposite in effect. Adding data made it worse.

Adding data made it worse

A projection leaves a residual spread across four starting points, and an earlier essay attributed it to the eigenvalue problem rather than to the coordinate set — an attribution rather than a measurement. Refitting with two isotopologues was the proposed test. Run, it produces a runaway to minus ninety-seven thousand that one isotopologue alone does not, and the reason is that the direction it walks along is flat at every mass.

5 figures Every exact fit lies on one closed curve. The one-parameter family of force fields that reproduce all six of boron trifluoride's frequencies exactly, drawn in the B–F stretch constant and the stretch–bend coupling. It is a closed curve: the stretch constant runs from 3.94 to 12.11 millidyne per ångström and the coupling from -0.36 to 6.71. The four fits from four starting points sit together in one small patch of it.

The residual was a loop

Projecting away boron trifluoride's flat direction left four fits disagreeing by a tenth in the B–F stretch constant, and the disagreement was put down to the search or to the arithmetic. It is neither. Two frequencies cannot fix three constants in one symmetry block, so the exact fits form a closed curve along which the stretch constant runs from 3.9 to 12.1, and the four fits are four points on a short arc of it.

6 figures The measured cubic and quartic overshoot what Morse fell short of. Each molecule's αₑ as a fraction of the measured value, from its Morse curve and from a quartic potential built with the cubic and quartic coefficients the measured αₑ and ωₑxₑ imply. Every Morse curve is short, by 4 to 15 per cent. Every quartic is over, by 1 to 5 per cent: the repair moves αₑ past the measurement rather than onto it.

Two coefficients are not a potential

A Morse curve built from measured constants gets the vibration–rotation constant αₑ short by four to fifteen per cent, and the measured αₑ and anharmonicity imply the cubic and quartic a real potential should have. Built with exactly those two coefficients and solved, the potential overshoots instead. The reason is that the Morse curve's own series, cut after its quartic, moves αₑ by a sixth to a half of the error being repaired — so a quartic cannot tell whether the shortfall was the cubic.

6 figures Each half has an integer exponent and the mixture does not. On logarithmic axes against the amplitude: the combined residual, the even half's anisotropy and the odd half relative to the mean. The even half is a straight line of slope 2.005. The odd half has slope 0.981 below 0.04 and then bends over. The combined residual, fitted as one power, gives 1.248 — an average of two integers weighted by where the sweep happens to sit, with the halves crossing at 0.084.

Two integers made one exponent

A sum over boron trifluoride's depolarised bands is nearly isotropic, and its residual scaled as the amplitude to the 1.248 — no integer, and a two-term fit left a pattern it could not remove. Averaging each distortion with its reverse splits the residual exactly into an even half that scales as the square, to 2.005, and an odd half that scales as the first power until a fifth-order term turns it over. And the sum was a stand-in: what an unresolved pair of bands would actually show is twenty times less flat.

6 figures The same three stretches, the same axis, one mirror plane apart. Boron trifluoride and ammonia put through the identical probe: a circle of distortions in the plane of stretch combinations that sum to zero, with the largest departure from three quarters taken over the depolarised bands at each direction. The two molecules share a three-fold axis and differ by a horizontal mirror. The first four rows are the same for both and the last three are not.

The shape was the group and the size was not

A depolarisation probe swept round a circle of distortions gives a reading with a sixty-degree period and a largest-over-smallest of about two, and the residual left in the sum over bands scaled as the amplitude to the 1.248. Run the identical probe on ammonia — the same three stretches, the same three-fold axis, the mirror plane gone — and the period and the ratio come back within three per cent, while the departure is a hundred and sixty-nine times larger, the residual twenty-three times, and the exponent is 0.816. The angular shape is the group's. Nothing else is, and the pyramid's residual is not even a power law.

6 figures No substitution that keeps the symmetry can tell the two fields apart. Ten isotopologues of boron trifluoride, real and hypothetical, each computed under the fitted force field and under the second field a boron-10 substitution cannot exclude, with the largest difference between their frequencies on a logarithmic axis. The six that keep all three fluorines alike give the same spectrum under both fields to a millionth of a wavenumber, the precision of the calculation. The four that replace one or two fluorines differ by four to seven tenths of a wavenumber.

The third mass that cannot help

Boron trifluoride's E′ block has three force constants and two frequencies, so its exact fits form a closed curve, and a boron-10 substitution cuts that curve to two fields. The obvious remedy is a third isotopologue, and it was named as the first data that could pick one of the two. It is not, if the third substitution keeps the molecule's symmetry: every such isotopologue gives the same spectrum under both fields to a millionth of a wavenumber, because its sum rule is one plane written in two reciprocal masses and its product rule is Teller and Redlich's. Only a substitution that breaks the symmetry adds an equation — and it separates the two fields by half a wavenumber.

6 figures The direction survives until the lines are as wide as the bending pair is split. How much the probe's reading varies round the circle of distortions — largest over smallest — against the line width of the spectrum it is read from, on a logarithmic axis. Resolved, it varies by 1.96. It holds that until the width passes about half a wavenumber, falls as the width approaches the bending pair's 2.8 cm⁻¹ splitting, and by a width of ten wavenumbers varies by one per cent. The stretching pair, split by half a wavenumber, merges first and costs almost nothing, because the largest departure was never its.

What a linewidth leaves of the probe

The depolarisation probe had two limits. Read on each band it varies by a factor of two round a circle of distortions, and read on each pair blended it varies by two thirds of a per cent. A spectrum sits between them, and where it sits can now be computed. For boron trifluoride distorted by two hundredths of an ångström the direction survives until the line width approaches the bending pair's 2.8 cm⁻¹ splitting, not the stretching pair's 0.53. A pair drawn as one peak is still not read as its blend: that takes widths several times the splitting. And the resolution was always the easy half, because the splitting is first order in the distortion and the signal second.

6 figures