Molecular Geometry & Bonding
what the shape decides
Concept

Depolarisation ratio — where it appears

The ratio of scattered light polarised across the incident polarisation to that polarised along it, for one Raman band. It is fixed at three quarters for any mode that is not totally symmetric, which makes it one of the few intensities symmetry decides outright.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

How fast the three quarters goes, along each coordinate. The coefficient of the square, for every non-symmetric mode of three molecules, against the mode's own frequency. A single measurement takes one of these — along a bond stretch — and reports the exponent rather than the size. The sizes span a factor of thirty within methane alone, and the coordinates a molecule is softest along are not systematically the most sensitive: ammonia's stiff pair is five times more sensitive than its soft one.

One number was one direction

The departure of a depolarisation ratio from three quarters goes as the square of a distortion, and the exponent was first measured along one bond stretch. Computed along every non-symmetric coordinate the exponent is always two and the coefficient is not: it spans a factor of thirty inside methane. And the ratio is not preferentially sensitive to the coordinates a molecule is soft along — in two molecules of three the stiffest coordinate is the most sensitive.

spectra · Spectrum 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.

spectra · Spectrum 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.

spectra · Spectrum 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.

spectra · Spectrum 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.

spectra · Spectrum 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.

spectra · Spectrum 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.

spectra · Spectrum 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.

spectra · Spectrum Ammonia's direction goes at a narrower width than boron trifluoride's. How much the probe's reading varies round the circle of distortions — largest over smallest — against the spectrum's line width, for the pyramid and the plane, both distorted by 0.02 Å. Both start near two. The plane's falls to half its excess at 3.48 cm⁻¹, set by its bending pair's 2.82 cm⁻¹ splitting; the pyramid's at 2.56 cm⁻¹, set by its stretching pair's 2.12 — although both the pyramid's pairs are split more than either of the plane's.

The pyramid trades its pairs

Ammonia's depolarisation probe was expected to survive a wider line than boron trifluoride's, because both of its degenerate pairs are split more by the same distortion. It survives a narrower one. On the plane the bending pair carries the reading; on the pyramid the stretching pair does, thirty times over, and its splitting is the smaller of ammonia's two — so the direction halves at 2.6 cm⁻¹ instead of 3.5. What the pyramid keeps is the size. The contrast a measurement has to resolve is 178 times the plane's, and at a ratio precision of one per cent ammonia's direction becomes readable from 0.046 Å of distortion, where boron trifluoride's is unreadable at 0.16.

spectra · Spectrum

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitPolarisabilitySymmetry breakingDegeneracyNormal modeRaman spectroscopyIrreducible representationsRaman activitySelection rulesApproximationConventionHarmonic approximation

All concepts