Structural parameter

In structural chemistry and crystallography geometry index or structural parameter ( $τ$ ) is the number in a range 0...1 that indicates what geometry of the coordination center is. First such parameter for 5-coordinate compounds was developed in 1984. Later parameters for 4-coordinate compounds were developed.

5-coordinate compounds

To distinguish wether the geometry of coordination center is rather trigonal bipyramidal or square pyramidal the $τ 5$ (originally just $τ$ ) parameter was proposed by Addison et al.:^[1]

\tau _{5}={\frac {\beta -\alpha }{60^{\circ }}}\approx -0.01667\alpha +0.01667\beta

where: $β > α$ are the two greatest valence angles of coordination center.

When $τ 5$ is close to 0 then geometry is similar to square pyramidal, while if $τ 5$ is close to 1 then geometry is similar to trigonal bipyramidal:

Extreme values of

τ 5

Square pyramidal geometry
$τ 5 = 0$
Trigonal bipyramidal geometry
$τ 5 = 1$

4-coordinate compounds

In 2007 Yang et al. have developed analogic $τ 4$ parameter to distinguish wether the geometry of coordination center is rather square planar or tetrahedral.^[2] The formula is:

\tau _{4}={\frac {360^{\circ }-(\alpha +\beta )}{360^{\circ }-2\theta }}\approx -0.00709\alpha -0.00709\beta +2.55

where: $β > α$ are the two greatest valence angles of coordination center; $θ = cos−1(− .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄3) ≈ 109.5°$ is a tetrahedral angle.

When $τ 4$ is close to 0 then geometry is similar to square planar, while if $τ 4$ is close to 1 then geometry is similar to tetrahedral. However, in contrast to $τ 5$ parameter, this one does not distinguish $α$ and $β$ angles, so structures of significantly different geometries can have similar $τ 4$ value. To overcome this issue in 2015 Okuniewski et al. have developed parameter $τ 4'$ that adopts values similar to $τ 4$ , however better differentiates the examined structures:^[3]

\tau _{4}'={\frac {\beta -\alpha }{360^{\circ }-\theta }}+{\frac {180^{\circ }-\beta }{180^{\circ }-\theta }}\approx -0.00399\alpha -0.01019\beta +2.55

where: $β > α$ are the two greatest valence angles of coordination center; $θ = cos -1 (- 1 ⁄ 3) \approx 109.5°$ is a tetrahedral angle.

Extreme values of $τ 4$ and $τ 4'$ denote exactly the same geometries, however $τ 4'$ is always less or equal to $τ 4$ , so the deviation from ideal tetrahedral geometry is more visible.

Extreme values of

τ 4

and

τ 4'

Square planar geometry
$τ 4 = τ 4' = 0$
Tetrahedral geometry
$τ 4 = τ 4' = 1$

References

^ Addison, A. W.; Rao, N. T.; Reedijk, J.; van Rijn, J.; Verschoor, G. C. (1984). "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". J. Chem. Soc. Dalton Trans.: 1349. doi:10.1039/dt9840001349.
^ Yang, L.; Powell, D. R.; Houser, R. P. (2007). "Structural variation in copper(I) complexes with pyridylmethylamide ligands: structural analysis with a new four-coordinate geometry index, τ₄". Dalton Trans.: 955. doi:10.1039/b617136b.
^ Okuniewski, A.; Rosiak, D.; Chojnacki, J.; Becker, B. (2015). "Coordination polymers and molecular structures among complexes of mercury(II) halides with selected 1-benzoylthioureas". Polyhedron. 90: 47–57. doi:10.1016/j.poly.2015.01.035.

[1] Addison, A. W.; Rao, N. T.; Reedijk, J.; van Rijn, J.; Verschoor, G. C. (1984). "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". J. Chem. Soc. Dalton Trans.: 1349. doi:10.1039/dt9840001349.

[2] Yang, L.; Powell, D. R.; Houser, R. P. (2007). "Structural variation in copper(I) complexes with pyridylmethylamide ligands: structural analysis with a new four-coordinate geometry index, τ₄". Dalton Trans.: 955. doi:10.1039/b617136b.

[3] Okuniewski, A.; Rosiak, D.; Chojnacki, J.; Becker, B. (2015). "Coordination polymers and molecular structures among complexes of mercury(II) halides with selected 1-benzoylthioureas". Polyhedron. 90: 47–57. doi:10.1016/j.poly.2015.01.035.

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