The Tolman Cone Angle

It had long been recognised that by changing the substituents on a phosphorus ligand, the behaviour of that ligand and the organometallic compounds associated with it were also changed. However, it wasn’t until the 1970’s that steric effects were shown to be at least as important as the electronic effects. It should be noted that the steric and electronic effects are closely related and thus difficult to separate completely. Generally electronic effects are discussed using the parameter n (based on CO stretching frequencies) and steric effects are discussed using the Tolman cone angle (θ). This research work was only concerned with the determination of cone angles.

The Tolman cone angle was introduced upon the discovery that phosphorus ligands competed for coordination at a Ni(0) centre and that the decreased binding ability of certain ligands could not be fully explained in terms of their electronic character and was in fact related to increased congestion [of the substituents] around the phosphorus atom.

Tolman originally calculated the cone angle using direct physical measurement from space-filling CPK models. The cone angle is defined as the apex angle of a cylindrical cone centred 2.28 Ǻ from the centre of the P atom and just touching the van der Waals radii of the outermost atoms of the molecule (see Figure 1, below).

Definition of Tolman Cone Angle

Figure 1 : Diagrammatic representation of the Tolman Cone Angle.1


However, this method requires the phosphine to be symmetrical which is not always the case, therefore, for measuring unsymmetrical phosphines half angles are utilised (see Figure 2, below) and the full cone angle is calculated using Equation 1 (below).


Definition of Tolman Cone Angle using half angles

Figure 2 : Diagrammatic representation of half cone angles.1



Mathematical Definition of Cone Angle

Equation 1: Mathematical description of cone angle using half angles.1


Tolman discovered that for ligands with few internal degrees of freedom θ could be measured to within ± 2 º. With more complex ligands it was much harder to determine when the cone was at a minimum therefore making accuracy in measurements much poorer1, despite this the Tolman cone angle is elegant in its simplicity and remains the standard parameter for discussing ligand size.

The importance of steric effects has led to many alternative methods to the original proposal by Tolman in order to quantify them. Methods used to derive ligand steric parameters have included molecular mechanics models based on the calculation of intermolecular van der Waals energies in model compounds, the calculation of molecular volumes and the application of relative rates of reactions.

Müller and Mingos surveyed the cone angles of various phosphines in transition metal complexes on the Cambridge Crystallographic Database using a method that was geometrically equivalent to Tolman’s. Each phosphine examined had between 100 and 2000 different structures on the database, and although a wide range of cone angles was found, the average angle calculated was seen to be very close to the Tolman value2. This can be used to show that the Tolman model is remarkably accurate.

Other groups have used various quantum mechanics calculations, such as AM13 and molecular mechanics MM24 methods to calculate cone angles, all of which are time consuming and relatively complicated. With the introduction of computer and profile analysis these calculations have become more available but it was of interest to find out if a simpler method was available and accurate. To this end the project aimed to determine whether it was possible to use the molecular mechanics minimisation function of the computer program PC Spartan Pro to determine the cone angle and test how accurate the answer was in relation to crystallographic values and the published Tolman cone angles.

The cone angle has important uses in predicting how bulky phosphine ligands will be and whether transition metal complexes formed using these ligands will adopt the cis- or trans- configuration in a square planar complex. The cone angle may also be of use in determining whether a phosphine could be useful in promoting reductive elimination, a reaction that is favoured by bulky phosophines.

It was found that although PC Spartan Pro gave cone angles that were significantly different from those published by Tolman, the trend followed by the Spartan values was remarkably similar to that followed by the Tolman values (see Figure 3, below). When the calculated cone angles were compared with those determined from crystallographic data, the values were similar and, again, followed the same trend as the Tolman angles.

radial graph of results

Figure 3 : Radial Graph of Results showing the trends found.


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  1. C. A. Tolman, Chem. Rev., 1977, 77, 313.
  2. T. E. Müller, D. M. P. Mingos, Transition Met. Chem., 1995, 20, 533.
  3. M. Chin, G. L. Durst, S. R. Head, P. L. Bock, J. A. Mosbo, J. Organomet. Chem., 1994, 470, 73.
  4. A. I. Polosukhin, A. Y. Kovalevskii, K. N. Gavilov, Russian Journal of Coordination Chemistry, 1999, 25, 758.