d⁹: what a tetragonal distortion is worth

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.

Four 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.

Copper is never quite octahedral

The energy of a d⁹ complex against a tetragonal distortion: the electronic gain, the elastic cost, and their sum. The best distortion is at 0.141 of a bond length and is worth 0.537 in units of eσ. The flat line beneath is low-spin d⁶ in the same field, which gains nothing at all.

High-spin d⁴, which has one electron in the upper pair rather than three, against d³, which has none. The slopes are −7.32 and exactly zero: one electron in a degenerate pair and one hole in it are the same problem, and a half-filled lower set is not a problem at all.

A low-spin d⁴ ion, the last configuration with an unevenly filled degenerate pair. Its gain is the largest of the four drawn here, which is the ordering the labels alone would not have given — and what the distortion does to the labels is the ordinary consequence of dropping from Oh to D₄ₕ, where every degenerate pair must split because the smaller group has none.

The g-value is the orbital coming back

The Jahn-Teller distortion of a d⁹ ion, computed as an energy against the distortion coordinate. That distortion is what makes copper’s ground state a single non-degenerate orbital in the first place — and therefore what makes the perturbation treatment above legitimate, since a degenerate ground state would need the coupling handled exactly rather than as a correction.

A distortion needs two states

The first-order effect: the electronic energy of a d⁹ ion falls linearly with a tetragonal distortion and the elastic cost rises quadratically, so the minimum is off zero for any stiffness whatever. What makes it linear is a degeneracy — an unequal occupation of two orbitals that a distortion separates.

Two distortions in one coordinate

A first-order Jahn–Teller distortion: a degenerate level, split linearly by the coordinate, against the elastic cost. Everything to do with the first order in this essay is this picture; what is added is a second state, higher up, mixing into the lower branch.