# Geometry index

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{{short description|Parameter used to characterize molecular geometry}}
In [coordination chemistry](/source/coordination_chemistry) and [crystallography](/source/crystallography), the '''geometry index''' or '''structural parameter''' ({{mvar|τ}}) is a number ranging from 0 to 1 that indicates what the [geometry](/source/Molecular_geometry) of the coordination center is. The first such parameter for 5-coordinate compounds was developed in 1984.<ref name="Addison"/> Later, parameters for 4-coordinate compounds were developed.<ref name="Houser"/>

==5-coordinate compounds==

<div class="thumb floatright">
File:Tau5 parameter for selected geometries.gif
</div>

To distinguish whether the geometry of the coordination center is trigonal bipyramidal or square pyramidal, the {{math|''τ''<sub>5</sub>}} (originally just {{math|''τ''}}) parameter was proposed by Addison ''et al.'':<ref name="Addison">{{cite journal |last1 = Addison |first1 = A. W. |last2 = Rao |first2 = N. T. |last3 = Reedijk |first3 = J. |last4 = van Rijn |first4 = J. |last5 = Verschoor |first5 = G. C. |year = 1984 |title = Synthesis, structure, and spectroscopic properties of copper(II) compounds containing nitrogen–sulphur donor ligands; the crystal and molecular structure of aqua[1,7-bis(''N''-methylbenzimidazol-2′-yl)-2,6-dithiaheptane]copper(II) perchlorate |journal = J. Chem. Soc., Dalton Trans. |issue = 7 |pages = 1349–1356 |doi = 10.1039/dt9840001349}}</ref>
:<math>\tau_5 = \frac{\beta-\alpha}{60^\circ} \approx -0.01667\alpha + 0.01667\beta </math>
where: {{math|''β'' > ''α''}} are the two greatest valence angles of the coordination center.

When {{math|''τ''<sub>5</sub>}} is close to 0 the geometry is similar to square pyramidal, while if {{math|''τ''<sub>5</sub>}} is close to 1 the geometry is similar to trigonal bipyramidal:

{{Gallery
|title=Extreme values of {{math|''τ''<sub>5</sub>}}
|align=center
|File:Square-pyramidal-3D-balls.png
 |alt1=Square pyramidal geometry
 |Square pyramidal geometry <br> ({{math|''β'' {{=}} ''α'' {{=}} 180°}}) <br> {{math|''τ''<sub>5</sub> {{=}} 0}}
|File:Trigonal-bipyramidal-3D-balls.png
 |alt2=Trigonal bipyramidal geometry
 |Trigonal bipyramidal geometry <br> ({{math|''β'' {{=}} 180°}}, {{math|''α'' {{=}} 120°}}) <br> {{math|''τ''<sub>5</sub> {{=}} 1}}
}}

==4-coordinate compounds==

<div class="thumb floatright">
File:Tau4 and tau4prime parameters for selected geometries.gif
</div>

In 2007 Houser ''et al.'' developed the analogous {{math|''τ''<sub>4</sub>}} parameter to distinguish whether the geometry of the coordination center is square planar or tetrahedral.<ref name="Houser">{{cite journal |last1 = Yang |first1 = L. |last2 = Powell |first2 = D. R. |last3 = Houser |first3 = R. P. |year = 2007 |title = Structural variation in copper(I) complexes with pyridylmethylamide ligands: structural analysis with a new four-coordinate geometry index, {{math|''τ''<sub>4</sub>}} |journal = Dalton Trans. |issue = 9 |pages = 955–64 |doi = 10.1039/b617136b|pmid = 17308676 }}</ref> The formula is:
:<math>\tau_4 = \frac{360^\circ - (\alpha + \beta)}{360^\circ - 2\theta} \approx -0.00709\alpha - 0.00709\beta + 2.55</math>
where: {{math|''α''}} and {{math|''β''}} are the two greatest valence angles of coordination center; {{math|''&theta;'' {{=}} cos<sup>−1</sup>(− {{frac|1|3}}) ≈ 109.5°}} is a tetrahedral angle.

When {{math|''τ''<sub>4</sub>}} is close to 0 the geometry is similar to square planar, while if {{math|''τ''<sub>4</sub>}} is close to 1 then the geometry is similar to tetrahedral. However, in contrast to the {{math|''τ''<sub>5</sub>}} parameter, this does not distinguish {{math|''α''}} and {{math|''β''}} angles, so structures of significantly different geometries can have similar {{math|''τ''<sub>4</sub>}} values. To overcome this issue, in 2015 Okuniewski ''et al.'' developed parameter {{math|''τ''<sub>4</sub>′}} that adopts values similar to {{math|''τ''<sub>4</sub>}} but better differentiates the examined structures:<ref>{{cite journal |last1 = Okuniewski |first1 = A. |last2 = Rosiak |first2 = D. |last3 = Chojnacki |first3 = J. |last4 = Becker |first4 = B. |year = 2015 |title = Coordination polymers and molecular structures among complexes of mercury(II) halides with selected 1-benzoylthioureas |journal = Polyhedron |volume = 90 |pages = 47–57 |doi = 10.1016/j.poly.2015.01.035}}</ref>
:<math>\tau_4' = \frac{\beta - \alpha}{360^\circ - \theta} + \frac{180^\circ - \beta}{180^\circ - \theta} \approx -0.00399\alpha - 0.01019\beta + 2.55</math>
where: {{math|''β'' > ''α''}} are the two greatest valence angles of coordination center; {{math|''&theta;'' {{=}} cos<sup>−1</sup>(− {{frac|1|3}}) ≈ 109.5°}} is a tetrahedral angle.

Extreme values of {{math|''τ''<sub>4</sub>}} and {{math|''τ''<sub>4</sub>′}} denote exactly the same geometries, however {{math|''τ''<sub>4</sub>′}} is always less or equal to {{math|''τ''<sub>4</sub>}} so the deviation from ideal tetrahedral geometry is more visible. If for tetrahedral complex the value of {{math|''τ''<sub>4</sub>′}} parameter is low, then one should check if there are some additional interactions within coordination sphere. For example, in complexes of mercury(II), the Hg···''π'' interactions were found this way.<ref>{{cite journal |last1 = Rosiak |first1 = D. |last2 = Okuniewski |first2 = A. |last3 = Chojnacki |first3 = J. |year = 2018 |title = Novel complexes possessing Hg‒(Cl, Br, I)···O=C halogen bonding and unusual Hg<sub>2</sub>S<sub>2</sub>(Br/I)<sub>4</sub> kernel. The usefulness of {{math|''τ''<sub>4</sub>′}} structural parameter. |journal = Polyhedron |volume = 146 |pages = 35–41 |doi = 10.1016/j.poly.2018.02.016}}</ref>

{{Gallery
|title=Selected geometries and corresponding values of {{math|''τ''<sub>4</sub>}} and  {{math|''τ''<sub>4</sub>′}}
|align=center
|File:Square-planar-3D-balls.png
 |alt1=Square planar geometry
 |Square planar geometry <br> ({{math|''β'' {{=}} ''α'' {{=}} 180°}}) <br> {{math|''τ''<sub>4</sub> {{=}} ''τ''<sub>4</sub>′ {{=}} 0}}
|File:Seesaw-3D-balls.png
 |alt2=Seesaw geometry
 |Seesaw geometry <br> ({{math|''β'' {{=}} 180°}}, {{math|''α'' {{=}} 120°}}) <br> {{math|''τ''<sub>4</sub> ≈ 0.43}}, {{math|''τ''<sub>4</sub>′ ≈ 0.24}}
|File:AX4E0-3D-balls.png
 |alt3=Tetrahedral geometry
 |Tetrahedral geometry <br> ({{math|''β'' {{=}} ''α'' {{=}} ''θ'' ≈ 109.5°}}) <br> {{math|''τ''<sub>4</sub> {{=}} ''τ''<sub>4</sub>′ {{=}} 1}}
}}

==References==
{{Reflist}}

==External links==
{{Commons category|Geometry index}}

Category:Crystallography
Category:Coordination chemistry
Category:Chemical structures

==Read more==
*A web application for determining molecular geometry indices on the basis of 3D structural files can be found [http://kchn.pg.gda.pl/geom/ here].

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Adapted from the Wikipedia article [Geometry index](https://en.wikipedia.org/wiki/Geometry_index) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Geometry_index?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.