Every essay — page 5
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 the shape is for
Coordination compounds, where the shape decides a colour, a magnetic moment and a bond length — and where the one-electron picture used everywhere else in the collection stops being enough.
One was a symmetry and one was not
A square-planar complex has a direction the sixteen-electron gap does not move along, and the repulsion falls along it — so the distortion looked free in both senses at once. Both results came from four ligands at the same bond length, and the two behave completely differently when that is relaxed. Two per cent of a bond length gives the gap a first-order slope; the repulsion keeps falling out to a fifty per cent mismatch, and its curvature never changes sign. The blindness was a symmetry; the fall is a property of the arrangement.
Four lifts and one that matters
The direction a sixteen-electron gap cannot see was found in a plane of two folds, where it was the antisymmetric fold of the two trans pairs. That plane hid a choice: the ligands were moved in pairs and they were all alike, and nothing separated which of those the blindness needed. Let each of the four ligands lift by its own angle and the answer is plain. The gap's gradient has four equal parts, so every distortion whose lifts add to zero is blind — three directions, not one — and the pairing never mattered. What sorts the three is the second order, and there the two models agree: the pair fold is rewarded by both and the tilt of a pair is penalised by both.
The excitation that raises the spin
A susceptibility fit's upper separation — the coupling at which its monomer fraction and temperature-independent term decouple for the second time — was placed by a band that broke, and the lead it left was that at fixed ground spin the first excitation ordered it. Every connected cluster of four, five and six spin-½ centres answers that. Among twenty five-centre doublets the first excitation ranks the separations at 0.36 and fewer bonds does better; the first excitation that raises the total spin ranks them at 0.86, and its product with the separation stays within a factor of one and a half where the separation itself spans three and a half. The separation sits where the first higher-spin multiplet is about 355 cm⁻¹ above the ground.
The window sets the constant
A susceptibility fit's upper separation, multiplied by the gap to the first spin-raising multiplet, came to about 355 cm⁻¹ across forty-one clusters, and that was read as a position: the raising multiplet sitting 1.7 kT above the ground at the top of a window ending at 300 K. The reading predicted about 240 at a 200 K top and 475 at 400 K. Re-read under nine windows, the forty-one clusters give 243 and 463, and the raising gap collapses them equally well under every one. But the constant is not quite a multiple of kT at the top — it drifts from 1.81 to 1.64 as the top rises — and the drift is the window's width: windows that are scaled copies of one another give the same constant to a per cent, because the model has no energy scale of its own.
The spacing sets the constant
A susceptibility fit's upper separation, in units of kT at the window's top, was found to depend on the window's span — 1.81 at a span of 7.5, 1.64 at 25 — and every window read had forty temperatures evenly spaced, so a wider window was also a sparser one at its cold end. Holding the span and changing the design separates the two. The number of temperatures barely matters: twenty, forty and eighty converge on one value from above, and a design with the census's own step reproduces the forty-point constant at both ends of the range. How the temperatures are spaced matters a great deal. Spaced evenly in their logarithm, as many measurements are, the constant is twelve per cent lower and leans more, not less; spaced evenly in their inverse, lower again.
The blindness needed a single top level
Every distortion of a folded sixteen-electron complex whose four ligand lifts add to zero leaves its gap unchanged to first order — while the highest filled level is dz². Past a fold of 17.90° it is the degenerate dxz, dyz pair, and the pair fold, which splits that pair, closes the gap at 0.026 eσ a degree whichever way the pairs move; the tilts, which cannot split it, stay blind. Exactly at the crossing the tilts see too, seventeen times more faintly. And the energy of the filled levels sees none of it at first order: its own turn along the pair fold comes at 13.8°, before the crossing, where nothing in the gap changes.
The pair fold was the road to a tetrahedron
Past a fold of 13.8° both of the model's energies fall along the pair fold, and the question left was where a folded sixteen-electron complex goes. It goes nowhere from there, because a folded square is not at rest: both energies rise along the common fold. Applied to the flat square, the same pair fold is the road to a tetrahedron, and on it the minimum of any weighting of the two energies is a flattened tetrahedron that flattens continuously, as a square root, into the square at a weight of 0.0762. The d levels cannot tell the two roads apart.
Beyond the octet
Hypervalency without d orbitals, delocalisation, aromaticity, and the structures the first-year rules quietly cannot describe.
Hypervalency without d orbitals
Sulfur hexafluoride is not d²sp³ hybridised. The d orbitals are far too high in energy to contribute meaningfully, the bonding is three-centre four-electron, and the textbook account has been known to be wrong for fifty years.
Delocalisation
Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.
Conjugation, and its limits
Every extra double bond in a chain lowers the gap between the highest occupied and lowest empty orbital, which is why long conjugated molecules are coloured. The trend has a limit, and the limit is not where the arithmetic says it should be.
Where two-centre bonding stops
A bond between two atoms is a special case, not the general one. Rings, clusters and metals are held together by orbitals spread over many centres, and the arithmetic that describes them is the arithmetic already used for benzene.
The band limit
Every level of a ring of n atoms lies between −2β and +2β, however large n gets. The levels do not spread out as the molecule grows; they crowd into a fixed interval — and that crowding, computed, is a band with its density of states diverging at both edges.