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Is it possible to have tetragonal distortion in tetrahedral complexes? Because distortions in octahedral complexes take place along the C4 axis to reduce the symmetry to D4h, but there is no C4 operation in the Td point group, I was inclined to think there cannot be z-in/z-out as such?

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Since $D_{h4}$ is not a subgroup of $T_d$, the z-in/z-out distortion you talk about does not give rise to a tetragonal distortion for a tetrahedral structure. It is possible to have tetragonal distortions in tetrahedral complexes, and has been observed in some cases (Bates and Chandler 1975) and (Sharnoff 1965). There is an illustration in Figure 1 of (Virot, Hayn, and Boukortt) which shows what turns out to be the easiest and most common tetragonal distortion for a tetrahedron, the one to the point group $D_{2d}$. So while z-in/z-out is associated with octohedral groups (and is usually the easiest way for them to have a tetragonal distortion), there are other distortions which occur in tetrahedrons.

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  • $\begingroup$ So, wouldn't it be kind of wrong to call it a tetragonal distortion? Shouldn't it simply be a J-T distortion which of course would arise for any non-linear electronically degenerate molecule? I'll look up the first two references as soon as I can. Thank you so much! $\endgroup$
    – Sagnik
    Feb 27, 2018 at 6:46
  • $\begingroup$ I think it depends on what you mean by "tetragonal distortion." I guess the way I am thinking about it that a Jahn Teller distortion that lowers the point group symmetry to a tetragonal point group is a "tetragonal distortion", even if it is not z-in/z-out and I think that is how some of the references talk about it. However, the vast majority of the time, "z-in/z-out" and "tetragonal distortion" are synonymous. At least this is my interpretation . . . $\endgroup$
    – C_F
    Feb 27, 2018 at 23:24
  • $\begingroup$ I see. What would be the defining criterion for a tetragonal point group? A fourfold rotational axis? $\endgroup$
    – Sagnik
    Feb 28, 2018 at 13:42
  • $\begingroup$ Ashcroft and Mermin has a good explanation of this in Chapter 7. But, the short answer is, you take your seven Bravais lattice crystal systems and reduce the symmetry to derive your 32 crystallographic point groups. Then, a crystallographic point group is in the category (of the Bravais lattice point group) it was originally derived from until all its symmetry elements can be found in a Bravais lattice crystal system group of lower symmetry. A fourfold axis means a point group is at least tetragonal, though it could be cubic. Since $D_{2d}$ has S4, it is not orthorhombic so it is tetragonal. $\endgroup$
    – C_F
    Feb 28, 2018 at 15:55

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