Which count closes a shell, ligand by ligand

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

Twelve essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The count that is not always eighteen

The level diagram for a carbonyl: six σ-bonding combinations at the bottom, the three t2g orbitals in the middle and the two eg* orbitals at the top. Filling from below closes at twelve electrons and again at eighteen. Which of the two is the deeper closure decides which count the chemistry is organised around.

The two gaps against the π parameter. They cross at exactly zero — not at a fitted value, not in a band, at the point where the π interaction vanishes. To the left of the crossing eighteen is the special number and to the right twelve is.

The level diagram for a ligand with no π interaction at all. The two gaps come out equal to the last bit a double holds, so neither count is special — and ammine complexes are duly found at every electron count from twelve to eighteen.

An integer nobody measured

Which count closes a shell depends on where the ligands put the antibonding pair. That is a statement about levels, and it survives everything in this essay for the same reason the count does.

The orbital a ligand cannot reach

The gap above each count against the π strength. One of the two curves is a straight line falling at four times the π strength; the other is flat, exactly, until a threshold.

The two gaps side by side, with the orbital that sets the square plane’s. The two columns of numbers agree from the threshold onward.

The four ligands and the orbital that cannot see them. d(z²) is symmetric about the plane and about the axis; no ligand π combination is.

The gap that only a tetrahedron closes

The gap above eight d electrons as four ligands fold out of a square plane towards a tetrahedron, at three π strengths. Both ends are π-independent, for two different reasons.

The six ligand–ligand angles at five points on the path, with the gap beside them. A square plane has four angles of ninety and two of a hundred and eighty; a tetrahedron has six of 109.47.

All five d levels along the path. The square plane’s three-low-then-two pattern becomes the tetrahedron’s two-below-three, and the level that changes places is d(z²).

The ligand the rule was waiting for

The gap above eight d electrons as one axial σ donor is brought in, and as two are, against their strength relative to the equatorial four.

Every d level along the approach of two axial donors. d(x²−y²) does not move at all; d(z²) rises to meet it.

Three points of the same approach, with all five levels at each and the gap beneath them.

The distortion that opens the gap

The gap above eight d electrons as two of the four ligands fold to the same side, against the fold angle.

Every d level along the fold. The top two approach, turn away from each other, and then separate.

The derivative of the gap with respect to the π strength along the fold.

The direction the gap cannot see

The gap along three directions out of the square plane: folding one pair, folding both equally, and folding one pair further while unfolding the other by as much.

The sixteen-electron gap with one pair of trans ligands folded by p and the other by q.

The gap’s two derivatives and the direction perpendicular to them, at each arrangement.

A blindness that is inherited

How far each quantity moves along the gap’s blind direction, at every distorted geometry in the family. The bars are the repulsion and the dots are the gap.

The two null directions in the plane of the two fold angles. They coincide exactly on the symmetric line and are unrelated off it.

Both quantities along the antisymmetric fold from ten degrees. The gap rises by three quarters of a per cent over five degrees; the repulsion falls by a tenth.

One was a symmetry and one was not

The two-fold family at four ratios of the two trans pairs’ bond lengths. Only the first has a symmetry that exchanges its pairs.

The gap’s slope along the antisymmetric fold, against the bond-length ratio. Zero at one by a symmetry, and non-zero at every other ratio tried.

The repulsion’s curvature along the fold, at six ratios. Negative at every one, shrinking by a factor of three rather than changing sign.

Four lifts and one that matters

At each fold where all four are lifted alike, the gap’s rate of change as each ligand alone is lifted further, as four bars.

The gap’s first-order change along eight distortions at a fold of ten degrees. Four of them do not change it.

The gap’s first-order change when one, two, three or all four ligands are lifted by a further degree, as a fraction of the change when all four are.

The blindness needed a single top level

The gap’s first-order change along three distortions whose lifts add to zero — the pair fold, three up with one down, and the tilt of a pair — against the common fold.

The change in the gap along the pair fold, from a fold of 15°, from exactly the crossing and from 20°.

The two highest filled levels and dz² at a fold of 20°, against the pair-fold amplitude.

The pair fold was the road to a tetrahedron

The flattening angle of the lowest-energy geometry against the weight given to the filled d levels’ energy relative to the repulsion, with the geometry sketched at four weights: a regular tetrahedron with no weight, a square beyond 0.0762.

The slopes of the filled levels’ energy and the repulsion along the common fold, per degree of fold.

The filled d levels’ energy and the repulsion along the road, from the square through the regular tetrahedron, each as its change from the square scaled to its own extreme.