Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 36

Page 36 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.

6 rings fused two ways. Two catacondensed chains of 6 hexagons: the linear one, where every fusion continues the line, and the angular one, where the fusions alternate. They have the same formula and the same number of bonds and they are not the same graph. Ring centres are numbered; in the linear molecule two rings k steps apart have centres 1.7321k units apart and in the angular one they do not, which is the whole reason this pair can be asked the question.

Neither of the two separations

A linear acene cannot pose the question, because the number of fusions between two rings and the distance between their centres are the same variable there. Bending the molecule pulls them apart — and the response follows neither. Two pairs of rings the same distance apart differ by two thirds, and the larger one is at the greater distance.

7 figures Every crossover a shell has. The field at which each coupled pair's linear behaviour returns — the gap between the two levels divided by twice the dipole joining them — on a logarithmic axis. The n = 3 shell has 3 distinct fields rather than four, because its m = ±1 half is a two-level ladder with one coupled pair. All three are below the n = 2 shell's single one.

Three events, and a ratio of two dipoles

The m = 0 half of a shell has two crossovers because it has three levels and two coupled pairs. The other half has two levels and one, so the whole shell has three distinct fields rather than four — and two of the three share a gap exactly, which makes the ratio between them a ratio of two dipoles, 2/√3.

6 figures The end pair against the deepest pair, at each separation. For each separation, the response of the pair that touches an end divided by the response of the pair at the same separation sitting deepest in the molecule. A straight chain is below one at every separation and a zigzag is above one at every separation, so the end effect has opposite signs on the two shapes. The third chain — two straight arms meeting at one angular ring — is above two at three separations and below one at the fourth, which is a third behaviour and not an intermediate one.

An end effect with two signs

Neither of two separations accounts for the scatter in a fused ring system's response, and the natural guess is the end: pairs with more molecule outboard should behave differently from pairs at an edge. They do. In a straight chain an end pair responds a third less than an interior one, in a zigzag a quarter more, and in a chain of two straight arms meeting at one angular ring the anomaly is in the middle.

6 figures One turned fusion, moved along the chain. Chains of 9 rings differing in one integer: which fusion's direction is turned. Turned at the first or last fusion, that leaves one angular ring beside an end; anywhere between, it leaves two adjacent angular rings whose turns cancel. The widest ratio between two pairs at the same separation, against which fusion is turned. A straight chain gives 1.520 and every bent one gives more — from 3.052 to 4.600. The two ends of the curve are the one-ring members; every interior point is a two-ring step.

One integer, and everything it changes

Eight molecules with the same rings, the same carbons and the same graph distance between every pair, differing in which fusion's direction is turned — one angular ring when the turned fusion is at an end, two adjacent ones anywhere else. The scatter within a separation class runs from three to four and a half times, against a straight chain's one and a half — and the two ends of one molecule disagree by up to a factor of four.

6 figures Tilting the field raises the count, and then lowers it again. How many distinct fields the shell has an event at, against the angle between the field and the z axis. Along either axis there are 3; at a general tilt every one of the 6 coupled pairs has its own field and there are 6. In between the count comes back down at four angles where two events coincide, and at forty-five degrees two separate coincidences happen at once.

Four angles the shell chooses

A field along one axis gives a shell of nine functions three fields with an event, and tilting the field should separate the coincident ones and raise the count towards the number of coupled pairs. It does — from three to six. But not monotonically: at four angles two events collide again, and every one of those angles is the arctangent of a ratio of the shell's own angular integrals.

6 figures One crossing, and the six fields that are not one. Open marks: every two-state crossover field, at each tilt. Filled line: the field at which the exact spectrum's one avoided crossing actually sits. The estimates scatter over a factor of four to twenty; the real crossing moves by a factor of 1.34 across the whole ninety degrees, and passes the four coincidence angles — the dashed verticals — without any feature at all.

None of the six was a crossing

Counting events in a tilted field finds the count rising from three to six, dropping again at four angles that are exact arctangents of the shell's own integrals. Every one of those statements is true of the two-state estimates. Diagonalising the five-level problem exactly finds one avoided crossing at every tilt, in a field that moves by a third across ninety degrees, and no feature whatever at any of the four angles.

6 figures A thirty-four per cent variation that is entirely the truncation. The field at which the one avoided crossing sits, against the tilt, computed in the five functions a field in the xz plane couples and in the whole nine-function shell. The truncated answer runs from 1.6435e-5 to 2.2014e-5 — a factor of 1.34. The whole shell's is 2.0876e-5 at every direction, and equals the truncated answer at zero tilt, where the truncation is exact because the field is along z and the excluded functions genuinely do not couple.

The variation was the basis

Solved in the five functions a field in one plane couples, the tilted Stark problem has one avoided crossing at every tilt and a crossing field that moves by a third across ninety degrees. Solved in the whole nine-function shell the field does not move at all — the same number at every direction, to eleven decimal places — and the thirty-four per cent was the truncation.

5 figures One and eight, over two decades of defect. The number of avoided crossings the whole shell has, against the number of two-state crossover fields its coupled pairs supply, as the quantum defect is swept towards zero. The question is whether the estimated count falls to meet the exact one as the l degeneracy closes. It does not move: one against eight at every defect tried, from 0.02 down to 0.0002, with the estimates spanning a factor of 7.08 throughout.

Consistently wrong is not a limit

Does the two-state picture of a tilted Stark shell become right as the quantum defect closes the l degeneracy? Swept over two decades it does not move: one avoided crossing against eight estimates at every defect. The reason is that every dimensionless quantity settles — the crossing sits at 0.04000 of the zero-field gap and the nearest estimate at 0.9067 of the crossing, and neither is heading anywhere.

6 figures 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