Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 37

Page 37 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 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.

Which torsions a molecule's own operations turn backwards. Every torsion orbit of the molecules here that have torsions: how many operations carry the torsion onto itself, how many of those are proper, and which operations reverse it. 7 of 9 orbits are reversed, and in every one the operation doing it is improper — the plane of a planar molecule, a mirror bisecting the torsion's bond, or a centre of inversion at that bond. Proper operations fix torsions too, benzene's twofold axes and hydrogen peroxide's among them, and never reverse one. Staggered ferrocene gives the same rows as eclipsed.

Five coordinates for six vibrations

A torsion is reversed by every improper operation that carries it onto itself and by no proper one, so adding torsions tests the orbit rule on a second kind of signed coordinate — and the rule holds on every molecule. It also moves hydrogen peroxide's count of totally symmetric vibrations from three to four, which a property of a molecule cannot do. Its five coordinates never spanned its six vibrations, and three of fifteen coordinate sets had been counting vibrations they did not describe.

7 figures Two levels of different symmetry, closest at 0.0399865. The lowest m = 0 level and the lowest |m| = 1 level of the zero-defect shell, in units of the s–p gap, against the field in the same units. Both fall. Their separation is smallest at 0.039986525, where it is 0.97648 of the zero-field gap, and they never meet. The field along z commutes with the angular momentum about z, the two levels belong to different values of it, and no element of the field connects them.

The crossing nothing couples

A tilted Stark shell's one avoided crossing settled at 0.04000 of the zero-field gap, and the natural guess was a ratio of angular integrals. With the quantum defect taken out the limit is 0.0399865, not four hundredths — and the two levels at that minimum belong to different symmetries about the field, which no element of the field connects. It was never an avoided crossing. The one minimum between levels that do interact sits at half the field, behind a level of the other kind.

6 figures One spectrum along three directions, and three different sets of estimates. The two-state estimate gap ÷ 2d for every pair of the shell's functions the field couples, with the field along z, along x and at the tilt the defect sweep used. Along z there are three distinct estimates, along x three different ones and at the tilt eight, none equal to any of the axial three. The exact spectrum is identical along all three directions, and its two minima are drawn as vertical lines: the one between coupled levels at 0.0196 and the tangency of uncoupled levels at 0.0400. An estimate is a property of the axes the functions were written along, and a feature is not.

Two levels cannot make a minimum

The one minimum between coupled levels in a Stark shell sits at 0.0196 of the s–p gap, and the nearest two-state estimate at 0.0192 — two per cent away, which reads as the estimates having been aimed at the right feature all along. They were not. Two coupled levels only ever separate, so no estimate can be where its own pair is closest. The minimum belongs to a third level, exists only while the d level sits within a quarter of the s–p gap, and meets the estimate by crossing it.

6 figures A fixed offset against a rising threshold. The offset at which a shell's coupled minimum disappears, against the principal quantum number, with the offset every shell's own d level actually has. The threshold is 2(n² − 4)/(5(n² − 1)) — zero at the second shell, a quarter at the third, and rising to two fifths. The shell's own offset is one fifth whatever the shell, because it comes from the reciprocal of l plus a half and carries no n at all. So the comparison is a constant against a curve, and it changes answer exactly once.

The quarter, generalised

A shell's one coupled minimum exists because its d level sits within a quarter of the s–p gap, and the quarter was found by bisecting a numerical search on one shell. It is exactly 2(n² − 4)/(5(n² − 1)) for every shell — zero at the second, a quarter at the third, two fifths in the limit — while the offset it is compared against is one fifth whatever the shell.

6 figures Water's two hydrogens are closer than platinum's chlorine. For each molecule, the longest pair that is a bond and the shortest pair that is not, on a logarithmic length axis. A cutoff on the length has to sit to the right of every filled mark and to the left of every open one, and it cannot: the longest bond in the collection is 2.3200 ångström and the shortest non-bond is 1.5144. The two populations overlap by a factor of 1.53, so the rule in use is not a rule with a badly chosen number in it — it is a rule with no number that works.

No length separates them

A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.

6 figures Only lithium's d level sits close enough to p. Each alkali's offset — how far its d level sits above p, in units of its s–p gap — at every shell from its valence shell to the thirtieth, computed from measured quantum defects, against the threshold below which a shell's s and p levels have a coupled minimum in a field. The screening model's offset of one fifth is drawn for comparison. Lithium sits below the threshold at every shell. Sodium, potassium, rubidium and caesium sit above it everywhere, by factors of 2.6 to 7.

Four alkalis the model cannot hold

A shell's s and p levels have a coupled minimum in an electric field when its d level sits within a threshold fraction of the s–p gap above p, and a screening model put every shell's d level at one fifth — inside the threshold from the third shell on, with more room the larger the defect. Computed from measured quantum defects, lithium keeps the minimum at every shell. Sodium, potassium, rubidium and caesium lose it at every shell, and the screening model has no member that resembles any of them.

6 figures The orbit identity holds on all nineteen, and the formula it replaced on eight. For each molecule in the census under the radius rule: its number of totally symmetric vibrations, the corrected count — symmetric orbits less symmetric redundancies — and the usual formula, orbits less redundancies. The corrected count lands on the molecule's own count every time. The usual formula is right for 8 of 19, and for ferrocene under the new bond list it predicts minus sixty-six.

The census a bond rule was hiding

Every internal coordinate, redundancy and totally symmetric count here is built on a bond list, and the bond list came from a length cutoff that gave five molecules no bonds. Rebuilt on the radius rule, the census reaches nineteen molecules instead of fifteen, the orbit identity holds on every newcomer, ferrocene's coordinates finally span all its vibrations — and a different gap appears: no bond rule can give a square-planar centre its two out-of-plane vibrations.

6 figures Two molecules whose coordinates reach seven of nine. What fraction of each molecule's vibrations its stretches and in-plane bends span, with bonds taken out to 2.5 ångström so that every molecule in the census raises coordinates at all. Most reach every one. Three are short because they need a torsion, which an earlier essay established. Two are short for a different reason and are the subject here: a square-planar centre has two out-of-plane vibrations that no angle in its plane changes to first order.

The two coordinates a square plane was missing

A square-planar centre has nine vibrations and its stretches and in-plane bends span seven of them: no angle in the plane changes to first order when the centre leaves it, or when the ligands pucker. The construction that closes the gap is two coordinates, one per pair of trans ligands, each the signed distance of the centre from its own pair's axis — and two coordinates raise the rank by exactly two. Why nothing measured this for three essays is the more useful half: both molecules with a square-planar centre have bonds longer than the census's default cutoff, so they were dropped before anything looked at them.

6 figures Half the distance from a coin to a decision. Every distortion whose symmetry species appears more than once, sorted by how concentrated the projection is across the occurrences of that species. The dashed line at a half is what a label that said nothing would give, since every one of these species appears exactly twice. The mean is 0.761 — so the label carries the answer about half the distance from a coin to a decision. Six of the twenty-four are above nine tenths and settled; ten are below six tenths and are not.

Half the distance from a coin

Every essay of this argument that priced a distortion by its symmetry species carried the same caveat: where a species appears more than once, the label cannot say which vibration the distortion goes into. With every coordinate set now complete or incomplete for a stated reason, the caveat can be measured against an actual projection on the five molecules that have force fields. The label carries the answer from 0.500 to 0.761 — about half the distance from a coin to a decision — settling six of twenty-four distortions and leaving ten essentially undecided, the worst of them ammonia's bond angles at 0.516.

6 figures One sentence, three displacements, three answers. For each symmetry-distinct internal coordinate of the five molecules, how concentrated the projection is across the occurrences of the species the label names — under the B row the published measurement used, under the displacement that changes the coordinate at the least kinetic cost, and under the one that changes this coordinate and no other. Ammonia's bond angle goes from 0.516, a coin, to 0.883 and 0.926. Boron trifluoride's angles are the worst case under one clean displacement and its bonds under the other, both at 0.573.

Three ways to stretch a bond

A symmetry label was measured at half the distance from a coin to a decision about which vibration a distortion goes into. The distortion was a B-matrix row, which moves a central atom as far as a ligand — and in the metric where the vibrations are orthogonal, three quarters of water's bond-angle distortion turns out to be the molecule moving rather than deforming. Two displacements that realise the same sentence and carry no rigid motion at all double what the label decides, from eleven coordinates of twenty-nine to twenty-two, and move the worst case from ammonia to boron trifluoride. Between them sits an exact identity that found a rounding error in a stored molecule.

6 figures The threshold belongs to the atom, not to the shell. For each alkali's valence shell (lithium's third, the first with a d level): its offset — where d sits above p, in units of the s–p gap — against the threshold below which the s–p separation has a minimum in a field, computed first with hydrogen's dipole elements and then with the atom's own, on a logarithmic axis. Lithium's own threshold is half as large again as hydrogen's. Sodium's is three times hydrogen's, and its offset of 0.705 now sits below it. Potassium's, rubidium's and caesium's fall, by factors from one and a half to thirty, so they stay above for a new reason.

The threshold belongs to the atom

Whether an alkali shell's s and p levels have a coupled minimum in a field was decided by comparing where its d level sits with a threshold built from hydrogen's dipole elements. Computed from each atom's own quantum defects instead — with integrals that reproduce three measured resonance lines to one per cent — the threshold is not a property of the shell at all. Lithium's rises by half, sodium's third shell regains the minimum the hydrogenic comparison denied it, and the heavier alkalis keep their verdict for a reason nobody had given: their p–d integral cancels, and in potassium it changes sign between two shells.

7 figures 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.

7 figures