Molecular Geometry & Bonding
what the shape decides
Concept

Polarisability — where it appears

How much an electron distribution is distorted by an electric field, as a dipole per unit field. For a hydrogenic one-electron atom it is exactly 9/2Z⁴, so it falls very steeply with nuclear charge and a compact ion is far stiffer than a diffuse one.

Named by 12 essays across 5 fields — each of them below, with the objects they name alongside it.

What holds matter together, per pair. Four kinds of interaction between two units of matter, each computed from the model named beside it, on a logarithmic energy scale. The range from top to bottom is a factor of several hundred, which is the number behind why a molecular solid melts hundreds of degrees below a covalent one.

What holds a solid together

Four kinds of interaction between two units of matter, each computed from a stated model and put on one logarithmic scale. The ordering is not the one usually taught — an ion pair at contact beats a shared pair, which is a fact about the comparison rather than about the numbers.

solids · Cohesion A control that ranked better than the mechanism. Rank correlations against the additivity shortfall, over eight ion pairs. The overlap of the two closed shells ranks at 0.8571 — but the cation's formal charge, which cannot be a mechanism, ranks at 0.9524, so the set is confounded: its eight pairs split four and four by charge and everything else rises with it. Held fixed within a charge group the overlap still ranks at 0.80 — and so does the softness, at -1.00. Four pairs cannot separate two candidates.

A control that outranked the mechanism

Ruling polarisation out left one candidate, and the closed-shell overlap ranks at 0.857 against the additivity shortfall — which looked like the answer until the control was read. The cation's formal charge, which cannot be a mechanism, ranks at 0.9524. Eight pairs split four and four by charge cannot separate anything, and within a charge group two candidates both rank perfectly.

orbitals · Contour 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 Two quantities, and the best straight line between the two kinds of pair. Every pair the four electronegativity tables all cover, by how much the two elements differ and by how much the tables disagree about them. Filled points are pairs whose sign the tables dispute. The line is the boundary that misclassifies fewest — 3 of 153, against 6 for the best rule using the difference alone. It slopes upward, which is the mechanism: more dispute buys a larger difference and still leaves the sign in doubt.

Two numbers caught what one could not

No single threshold on the electronegativity difference separates the bond pairs whose polarity the four tables dispute from those they agree on — the best misses six of a hundred and fifty-three. Adding how much the tables disagree halves that to three and catches every disputed pair. And three is what four coin-flips produce: the rule flags nineteen pairs, an eighth of which should look agreed for no reason at all, which is 2.375 against the three observed.

shape · Dipole 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 Every level decides what three levels could not. For each alkali's valence shell, the polarisability of the s level and of the p level with no angular momentum about the field, each summed over every s, p and d level. Where the p bar is longer the s–p separation falls at first in a field. It is longer in sodium, potassium, rubidium and caesium — by 365, 506, 779, 1484 atomic units — and shorter only in lithium, whose valence shell has no d level of its own. The three-level threshold called potassium's, rubidium's and caesium's valence shells shells without a minimum.

The shell was the wrong unit

Whether an alkali's s and p levels approach each other in a weak field was decided from three levels of one shell. The criterion that holds every level is a difference of two polarisabilities, and polarisabilities are measured. Summed over every s, p and d level with integrals that reproduce five measured ground states to two per cent, the difference keeps sodium's verdict at thirty times the margin and reverses potassium's, rubidium's and caesium's at every shell — because the d level beside p belongs to the shell below.

symmetry · Representation 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 With its measured Ritz series potassium's p level stops being the more polarisable at the thirty-fourth shell. Potassium's α(np₀)/α(ns), every s, p and d level summed, against the principal quantum number, with three readings of the defects. The four-decimal asymptotic defects put the turn below one at n = 36.93; the measured δ₀ alone at 36.99; the measured Ritz series, whose corrections at those shells are a ten-thousandth to a thousandth, at 33.72. The ratio falls by about 0.0074 a shell, which is why corrections that small move it three shells.

Potassium's turn was a fine-structure question

Summed over every level, potassium's p level stops being more polarisable than its s level somewhere past the thirtieth shell, and a thousandth in any quantum defect moved the turn by shells. With the Ritz series measured on ultracold ³⁹K in 2019 the inputs are fixed to a millionth, and the turn lands at the thirty-fourth shell, not the thirty-seventh. But the two fine-structure components of the p series differ by three thousandths: with either one in place of their average the turn is at the twenty-third or the forty-fifth. A sign decided by the spinless sum is decided by an interval a hundred and fiftieth of the s–p gap.

symmetry · Representation

Named alongside it

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

Model limitDepolarisation ratioDegeneracySymmetry breakingRaman spectroscopyIrreducible representationsNormal modeApproximationConventionMatrix elementPerturbation theoryQuantum defect

All concepts