Geometry index explained

In coordination chemistry and crystallography, the geometry index or structural parameter is a number ranging from 0 to 1 that indicates what the geometry of the coordination center is. The first such parameter for 5-coordinate compounds was developed in 1984. Later, parameters for 4-coordinate compounds were developed.

5-coordinate compounds

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

\tau _{5} = (\beta - \alpha)/(60 ^{\circ}) ≈ -0 . 01667 \alpha + 0 . 01667 \beta where: are the two greatest valence angles of the coordination center.

When is close to 0 the geometry is similar to square pyramidal, while if is close to 1 the geometry is similar to trigonal bipyramidal:

4-coordinate compounds

In 2007 Houser et al. developed the analogous parameter to distinguish whether the geometry of the coordination center is square planar or tetrahedral.^[2] The formula is:

\tau _{4} = (360 ^{\circ} - (\alpha + \beta))/(360 ^{\circ} -2 \theta) ≈ -0 . 00709 \alpha - 0 . 00709 \beta + 2 . 55 where: and are the two greatest valence angles of coordination center; is a tetrahedral angle.

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

\tau _{4'} = (\beta - \alpha)/(360 ^{\circ} - \theta) + (180 ^{\circ} - \beta)/(180 ^{\circ} - \theta) ≈ -0 . 00399 \alpha - 0 . 01019 \beta + 2 . 55 where: are the two greatest valence angles of coordination center; is a tetrahedral angle.

Extreme values of and denote exactly the same geometries, however is always less or equal to so the deviation from ideal tetrahedral geometry is more visible. If for tetrahedral complex the value of 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.^[4]

Read More

• A web application for determining molecular geometry indices on the basis of 3D structural files can be found here.

Notes and References

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. . 7 . 1349–1356 . 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, . Dalton Trans. . 9 . 955–64 . 10.1039/b617136b. 17308676 .
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 . 10.1016/j.poly.2015.01.035.
4. Rosiak . D. . Okuniewski . A. . Chojnacki . J. . 2018 . Novel complexes possessing Hg‒(Cl, Br, I)···O=C halogen bonding and unusual Hg₂S₂(Br/I)₄ kernel. The usefulness of structural parameter. . Polyhedron . 146 . 35–41 . 10.1016/j.poly.2018.02.016.