Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 28

Page 28 of 36, 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.

Six distortions, and the one the ratio cannot see at all. How far a depolarised band of boron trifluoride comes off three quarters when the bonds are stretched in six different combinations, all at the same size. Five move it. The one that stretches every bond by the same amount moves it by nothing whatever — not a small amount, exactly none — because that distortion keeps every symmetry operation the molecule had, and the band sits at three quarters by its species.

The distortion the ratio cannot see

A depolarisation ratio can be made into a structural probe: break a molecule's symmetry and a band fixed at three quarters comes off it by an amount that depends on the distortion. The question is whether two distortions the ratio can see separately might cancel into one it cannot. They do — and the cancellation is exact, at any size, because the blind direction is the one that keeps the symmetry.

6 figures The same three fillings, on a ring and on a chain. The intensity contrast against the on-site repulsion at two, four and six electrons, for both geometries. Two of the ring's three curves flatten onto exactly one and stay there — the hollow marks, where the rank cut falls between two degenerate lines. The chain's corresponding curve approaches the same value from above without reaching it, because a chain of six has no exactly degenerate removal lines at any repulsion at all.

The number the tie got right

On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.

6 figures 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

What symmetry decides

A molecule's point group follows from its coordinates, and it settles whether the molecule can be polar or chiral with no reference to bonding.

ammonia — C3v. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Point groups from coordinates

A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.

8 figures