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10.4: Angular Overlap

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    The angular overlap model is an approach to quantifying the interaction between metal and ligand orbitals in different geometries, with a focus on the metal d orbitals. Although developed in the 1970's, this approach is still used as a starting point for theoretical calculations using advanced computational chemistry methods available today.1

    The core concept of the angular overlap model is that different ligand orbitals will overlap with metal d orbitals to different extents because of the variety of angles at which these orbitals would approach each other. Stronger overlap leads to greater interaction. That means both greater stabilization of the ligand-centered bonding orbital and greater destabilization of the metal-centered antibonding orbital.


    Let's look at some examples. Consider the overlap of the \(d_{z^2}\) orbital with an axial ligand in octahedral geometry, with the assumption that the \(d_{z^2}\) orbital lies in the axial direction. There should be considerable overlap between the metal and ligand orbitals, leading to both significant stabilization of the ligand donor electrons and similar destabilization of an electrons in the metal \(d_{z^2}\) orbital. By comparison, an equatorial ligand would have significantly less overlap with the \(d_{z^2}\) orbital. Instead of overlapping with the substantial lobe along the z axis, the ligand would be interacting with the minimal \(d_{z^2}\) toroid in the xy plane. The bonding orbital would be stabilized to a significantly lesser extent compared to the axial ligand. The antibonding orbital would be destabilized by a correspondingly smaller amount.

    In the case of a tetrahedral ligand, there is essentially no overlap with the \(d_{z^2}\) orbital because its on-axis lobe is too far away from the cubic corner positions occupied by ligands in a tetrahedral array. There is really no bonding or antibonding in this case. On the other hand, the dxz orbital reaches a little closer to that corner position, allowing for some overlap with the ligand orbital. Consequently, there is some stabilization of the bonding electrons and destabilization of the antibonding d orbital. At first glance, the amount of overlap in this case, and the amount of stabilization or destabilization, appears much more similar to the case of the equatorial ligand with the \(d_{z^2}\) orbital than the axial ligand with the \(d_{z^2}\) orbital.

    We will not go into the mathematics that explore the exact extent of overlap expected in each case. Instead, we will go straight to a summary of the results in the next section.

    Exercise \(\PageIndex{1}\)

    Estimate the degree of interaction between the ligand and metal orbital: large, medium, or none.




    a) none b) medium c) medium (although it may appear greater than in (b), and it is) d) large


    1. For example: Chilkuri, V. G.; DeBeer, S.; Neese, F. Ligand Field Theory and Angular Overlap Model Based Analysis of the Electronic Structure of Homovalent Iron–Sulfur Dimers. Inorg. Chem. 2020, 59(2), 984-995.

    This page titled 10.4: Angular Overlap is shared under a CC BY-NC 3.0 license and was authored, remixed, and/or curated by Chris Schaller.

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