Geometry index

In coordination chemistry and crystallography, the geometry index or structural parameter ($τ$) 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 $τ_{5}$ (originally just $τ$) parameter was proposed by Addison et al.:
 * $$\tau_5 = \frac{\beta-\alpha}{60^\circ} \approx -0.01667\alpha + 0.01667\beta $$

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

When $τ_{5}$ is close to 0 the geometry is similar to square pyramidal, while if $τ_{5}$ is close to 1 the geometry is similar to trigonal bipyramidal:

4-coordinate compounds


In 2007 Houser et al. developed the analogous $τ_{5}$ parameter to distinguish whether the geometry of the coordination center is square planar or tetrahedral. The formula is:
 * $$\tau_4 = \frac{360^\circ - (\alpha + \beta)}{360^\circ - 2\theta} \approx -0.00709\alpha - 0.00709\beta + 2.55$$

where: $β = α = 180°$ and $τ_{5} = 0$ are the two greatest valence angles of coordination center; $β = 180°$ is a tetrahedral angle.

When $α = 120°$ is close to 0 the geometry is similar to square planar, while if $τ_{5} = 1$ is close to 1 then the geometry is similar to tetrahedral. However, in contrast to the $τ_{4}$ parameter, this does not distinguish $τ_{4}$ and $α$ angles, so structures of significantly different geometries can have similar $β$ values. To overcome this issue, in 2015 Okuniewski et al. developed parameter $&theta; = cos^{−1}(− 1/3) ≈ 109.5°$ that adopts values similar to $τ_{4}$ but better differentiates the examined structures:
 * $$\tau_4' = \frac{\beta - \alpha}{360^\circ - \theta} + \frac{180^\circ - \beta}{180^\circ - \theta} \approx -0.00399\alpha - 0.01019\beta + 2.55$$

where: $τ_{4}$ are the two greatest valence angles of coordination center; $τ_{5}$ is a tetrahedral angle.

Extreme values of $α$ and $β$ 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. If for tetrahedral complex the value of $τ_{4}$ 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.

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