One metal orbital, two ligands competing for it

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.

One essay draws 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 trans influence is an overlap argument

Metal–ligand bond orders in a three-orbital model as one ligand’s interaction is turned up. Its own bond order rises from 0.6155 at equal strengths to 0.8422; the bond order to the ligand opposite falls from 0.6155 to 0.4211. Both curves come from the same filled orbital of the same exactly solved problem.

The control, drawn: the same scan, over the same range, with the right-hand interaction set to zero. The left-hand bond order rises to 0.93 — higher than before, because it is no longer sharing — and the line for the bond trans to it is flat at zero from one end to the other. Flatness is the claim, not smallness: a channel other than the shared metal orbital would show here as a slope, and there is nothing in the calculation that could produce one.

The same competition with the ligand level at −1.2, close to the metal’s zero. Across a scan from 0.8 to 1.6 the trans bond order falls from 0.65 at equal strengths to 0.51 — a drop of 0.14, and the closer in energy two orbitals are the more thoroughly they mix, which is what overlap decides says about a two-body interaction seen here through a third body.