Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 11

Page 11 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

Bonding models

Valence bond, molecular orbital, and hybrids — three descriptions of one thing, related by transformations that change no observable.

Carbon is not one number. The charge equalisation puts on carbon in the four fluoromethanes, and on hydrogen in the three that have one. Carbon's runs from 0.07 to 0.31, and hydrogen's changes sign between methane and fluoromethane, so the same C–H bond is polarised one way in one molecule and the other way in the next. A table with one number for carbon is printing the value an iteration starts from.

The value that only exists in the bond

A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.

7 figures Two molecules, one eigenvalue, and 1.27 eV between them. Six alternant hydrocarbons placed by the Hückel eigenvalue of their highest occupied level — computed by diagonalising each molecule's own adjacency matrix — against the measured first π ionisation energy. Ethene and benzene share an eigenvalue of exactly 1 and their measurements differ by 1.27 eV; butadiene and naphthalene share 0.618 and differ by 0.94. A model that reads only the eigenvalue is a function of it, so it must give each pair one answer, and the two vertical pairs are the whole of its error.

A parameter that never finds a value

Show a two-parameter model six measurements instead of two and it stops being underdetermined and starts being wrong. Adding the third parameter improves the fit by one part in eighty, moves the resonance integral by a factor of four, and never finds a best value at all — because nine tenths of the error is a term the model does not have.

5 figures The hybrids do not point at the atoms. For each cycloalkane, the angle between its two ring hybrids — fixed by orthogonality once the measured H–C–H angle has said how much s character the hydrogens take — against the angle between its carbons. Cyclopropane's differ by 45.5°, so each hybrid points 22.75° outside the bond it makes; cyclohexane's agree to 0.035°, which is the control.

The hybrids that point outside the bonds

Coulson's relation says two equivalent hybrids sharing an s orbital are orthogonal only between ninety and a hundred and eighty degrees. Cyclopropane's carbons make sixty, so its ring hybrids cannot point at the atoms they bond to — and the same relation says by how much they miss: 22.75 degrees each, falling to 0.03 in cyclohexane.

7 figures The shared bond's response falls and reverses, at every coupling. How far the relaxation moves the most interior cross bond of an acene, against the number of rings, at three couplings. Every series falls monotonically and every one changes sign — at λ = 0.2 between 3 and 4 rings, at λ = 0.4 between 3 and 4 rings, at λ = 0.6 between 4 and 5 rings. Where it crosses is a property of the coupling; that it crosses is not. The natural reading — that the effect grows with the number of rings feeding a bond — is refused by every one of these curves.

An anomaly that is not the first of a series

Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.

6 figures What each of these predictors cannot see. Four predictors built elsewhere in this collection, each audited by its ties: the share of the variation in what was measured that the predictor demonstrably cannot account for, because two systems it assigns the same number to were measured to differ by that much. the spin-only moment leaves 23 per cent; the VSEPR angle leaves 92 per cent; the highest occupied Hückel eigenvalue leaves 41 per cent. The control has no ties at all and the instrument returns nothing for it, which is what it must do. No fitting anywhere: a tie is a claim a model of that form cannot escape.

Two systems a model cannot tell apart

Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.

7 figures A functional with no interior maximum. The Boys functional against the mixing angle for a carbonyl, evaluated directly at a hundred and eighty-one angles. It is a quadratic in cos²θ − ½ with no linear term, so it is symmetric about forty-five degrees and its maximum is at the middle or at the ends and nowhere else. Here the coefficient is positive, so the best is at 0° — the canonical σ and π. There is no angle to search for, however unsymmetrical the molecule is.

The angle that does not have to be searched for

A carbonyl's two bent components have no symmetry making them equivalent, so the mixing that best localises them looks like something to search for and their s characters look like two different numbers. Neither happens. The localisation functional is a quadratic with no linear term, so its maximum is at forty-five degrees or at the ends — bent bonds or canonical ones, decided by one inequality, with nothing in between.

6 figures Two instruments, and they do not agree. Each of this collection's four predictors under both tests. The exact-tie instrument reports the share of the variation a predictor demonstrably cannot account for, and it ranks VSEPR worst and has nothing at all to say about the ring-strain control. The near-tie instrument reports how much steeper any model must be somewhere than the set is overall, and it ranks the control worst and cannot speak about VSEPR, whose predictor is a label rather than a number. Neither instrument is the general one, and a predictor that passes one has not been audited.

The pair that is not a tie

An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.

6 figures The filled orbital at 3 bohr, with a node in the middle of the bond. The combination the two-level model puts lower at 3 bohr, drawn as contours of the wavefunction with the two signs in the two colours. The overlap here is 0.4825, so the interaction is of the sign that fills the plus combination. There is a nodal plane through the midpoint: the pair the model calls bonded has no density at all between its nuclei.

A bond with nothing in the middle

Two head-on 2p functions have an overlap that changes sign at 5.03 bohr, and the picture of what that means is worth drawing. Below that separation the combination the model fills has a nodal plane through the midpoint of the bond, so two electrons in it put exactly nothing between the nuclei; above it the same model fills the other one. Which picture a bonding orbital has is decided by a separation.

6 figures The inequality, decided twice. The quantity the localisation criterion compares — the centroid separation over twice the off-axis dipole — computed with hydrogenic radial functions and with Slater ones at the same effective charges. Below one the localised description is a pair of bent components; above it, σ and π. The hydrogenic answer is 1.8090 and the Slater answer is 0.3937, and they are on opposite sides.

The node that decided a picture

A hydrogenic calculation cannot say whether formaldehyde's localised description is two bent components or σ and π, because the integral that decides it came out at 0.00718 where it should be the largest in the molecule. Replacing the hydrogenic radial functions with nodeless Slater ones moves that overlap by a factor of ninety-seven, leaves the π overlap identical to the last bit, and flips the answer: the description is bent.

6 figures One heteroatom, and how far the relaxation carries it. The change every ring's bonds undergo when one carbon of the end ring is given a site energy, on a chain of 12 fused hexagons with its gap held open. The upper curve measures each molecule's couplings from its own mean bond order, which is what this collection's relaxation has always done; it stops falling at 4.48e-5 and stays there. The lower curve measures both from the same mean and keeps falling to 7.89e-9. Nothing about the molecule differs between them.

The floor was in the bookkeeping

A heteroatom in one ring of a fused chain is a perturbation with a place, and the relaxation carries it along the molecule. Measured directly, the response stops falling after five rings and sits at four hundredths of a millionth for ever. That floor is not the molecule. It is the relaxation measuring each system's couplings from its own mean, and removing it recovers nine orders of magnitude.

6 figures What a rotation group buys over a single angle. The Boys functional at the canonical orbitals, at the best mixing of the two bonds with the lone pair left alone, and at the maximum over the whole three-dimensional rotation group. The first step is 11.70 per cent and the second, which no two-orbital treatment can reach, is a further 2.50.

An interior maximum a third orbital allows

Two orbitals related by a mirror plane give a Boys functional with no linear term, so it has no interior maximum and no angle is ever searched for. Add the oxygen lone pair and the search over a three-dimensional rotation group finds one — eight and a half degrees of lone pair mixed into two bent bonds, worth a further two and a half per cent that no pair of orbitals could reach.

6 figures A heteroatom on ring 5, and the profile in both directions. The response of each ring to a heteroatom placed on ring 5 of 12, on a logarithmic scale. Two rings share the peak, because an interior ring's outermost carbon belongs to two rings at once, and the profile falls away at the same rate on both sides — which is the answer wanted: the reach is the molecule's and not the end's. Every point is a mean over the six bonds of its ring.

The reach is the molecule's

Every measurement of the relaxation's reach so far puts the heteroatom on the end ring, because that gives the longest run to measure a decay over — and the response of a molecule to a perturbation at its end is not the response to one in its middle. Moved inward, the decay is the same in both directions and the same as the end's. The amplitude is not: it halves, and splits across two rings.

6 figures