The total energy against the distortion, at several gaps

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.

Two 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.

A distortion needs two states

The total energy against the distortion at five gaps, with the two-level problem solved exactly rather than to second order. The critical gap here is 1.00: the two curves below it have their minimum away from zero and the three above it do not. Nothing in any of these five cases is degenerate.

Every symmetry species of the octahedral group, and whether a T1u distortion can mix an excited state of that species into a closed A1g shell. One of the ten can. A molecule whose low-lying excited states are all of the other nine kinds cannot distort along that mode however small its gap is.

The critical gap at several couplings and stiffnesses. A weakly coupling mode on a stiff molecule needs an excited state almost on top of the ground state before it distorts; a strongly coupling mode on a soft one distorts with an excited state four times as far up.

Two distortions in one coordinate

The second-order picture: a closed-shell molecule’s energy along a coordinate, at five gaps. Above the critical gap the symmetric structure is the minimum and below it is not, and the boundary is where the closed form puts it. Everything in this essay happens above that boundary, where by itself nothing would happen at all.

The three curves. First order alone puts the minimum at 0.600; second order alone puts it at zero, because the gap is above critical; the two together put it at 1.075 and gain 0.18031, which is exactly twice what the first-order effect gains alone. The second-order term does nothing by itself and contributes half the answer.

How far the molecule distorts as the first-order coupling is turned up, with the second-order term on and off. The gap is held above critical throughout, so the second-order term never distorts anything by itself, and at every first-order coupling the two together go further. The gain grows with the first-order coupling, which is what “each makes the other larger” looks like as a curve.