# Use of geometry index for the determination of coordination environment

Geometry index $\tau$ is supposed to resolve proper geometry for coordination numbers (C.N.) 4 and 5 based on its extreme values ($0$ or $1$). There is also a web app Geom which handles both cases for a structure in XYZ format.

I'd like to summarize the questionable topics regarding proper and efficient usage of this method:

Q1. I'm not sure how to address the intermediate values. Say, for $\tau_5 = 0.33$: is it a square pyramidal geometry with a character of trigonal bipyramid? Or one can just name this coordination environment square pyramid and call it a day?

Q2. Are there similar algorithms developed for the higher C.N.:

• capped trigonal prism vs pentagonal bipyramid (C.N. 7);
• cube vs square antiprism (C.N. 8)?
• Based on the abstract of the original paper that defined the geometry index (pubs.rsc.org/en/Content/ArticleLanding/1984/DT/DT9840001349) your first description in question 1 seems to be more accurate. It seems to suggest thinking of the parameter as measuring the percentage of trigonal distortion. Jul 30 '17 at 19:32

For your first question, the original paper (J. Chem. Soc., Dalton Trans. 1984, 1349–1356) that described the geometry index $$\tau_5$$ defined it as an "index of trigonality". For example, they write for a compound with $$\tau_5=0.48$$

By this criterion, the irregular co-ordination geometry of $$\ce{[Cu(bmdhp)(OH2)]2+}$$ in the solid state is described as being $$48\%$$ along the pathway of distortion from square pyramidal toward trigonal bipyramidal.

For your second question, while I haven't been able to find a $$\tau_7$$ or $$\tau_8$$ used in the literature, it seems possible to define such parameters under the right conditions. To devise a $$\tau_8$$, we can see that for a regular cube $$\ce{MX_8}$$, there can only be bond angles of $$70.5^\circ$$ (between adjacent $$\ce{X}$$ in the same square) and $$109.5^\circ$$ (between opposite corner $$\ce{X}$$ of the same square or between corner $$\ce{X}$$ of different squares). However, an antiprism instead has an angle of $$99.6^\circ$$ separating the $$\ce{X}$$ of different squares. (Image obtained from Inorganic Chemistry by Miessler and Tarr)

This suggests using a formula reminiscent of $$\tau_5$$ to define $$\tau_8$$ as the antiprismatic distortion index. One possibility is $$\tau_8=\frac{\beta-\alpha}{9.9^\circ}$$ where $$\beta > \alpha$$ are the two largest valence angles and $$9.9^\circ$$ is a normalization factor to make it between $$0$$ and $$1$$. So when $$\alpha=\beta=109.5^\circ \to \tau_8=0 \to$$ cubic geometry and $$\alpha=109.5^\circ$$ $$\beta=99.6^\circ \to \tau_8=1 \to$$ antiprismatic geometry.

This will only work if the structure is a regular antiprism (i.e an anticube). The same is true for defining $$\tau_7$$ between a pentagonal bipyramid and a monocapped trigonal prismatic. This is because the angles for these will vary if all the attached groups are not the same and so a consistent scheme based on the angles would not suffice.

I also imagine that $$\tau_7$$ would be harder to define in this way because I don't think there is pair of angles that on its own could describe the distortion between the two geometries.

• Yep, these "capped" geometries ($\tau_7$) are tricky, thankfully one can relatively easy spot one by cross-sectioning the crystal space with Voronoi-Dirichlet polyhedra. Thank you for the great answer! Aug 11 '17 at 19:58