# 10.4.1: Sigma Bonding in the Angular Overlap Model

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The pictures we considered previously explicitly addressed sigma overlap between the ligand donor orbital and the metal d acceptor orbital. The donor electrons drop in energy into the new sigma bonding combination and any electrons in the d orbital are raised into the new sigma antibonding combination. For example, when an axial ligand interacts with a metal dz2 orbital, the in-phase, bonding combination drops in energy by a quantity that we will call eσ. At the same time, the out-of-phase, antibonding combination is raised in energy by the same quantity, eσ. A real computational treatment would reveal that this is not exactly true; the amount of energy by which the bonding orbital is stabilized is slightly different from the amount by which the antibonding orbital is destabilized. However, the distinction is pretty minor for our purposes.

From the interactions we have already looked at, we know that this specific interaction is a strong one. Other metal-ligand interactions may involve less overlap than this one and will lead to smaller changes in energy. For this reason, other metal-ligand interactions are expressed as fractions of the quantity, eσ, found in the case of the dz2 orbital interacting with an axial ligand.

Our first task is to label the positions that will be occupied by ligands in a number of different geometries. To avoid a cluttered diagram of these positions, we can use three separate illustrations to illustrate the ligand positions in some common geometries. The first drawing below includes all the geometries in which ligands are found along Cartesian coordinates. The second drawing describes tetrahedral geometry, whereas the third one describes trigonal structures.

Given those ligand positions, we can assess the interactions that would occur if ligands were purely sigma donors. For reference, the dz2 orbital is assumed to lie along the axis between positions 1 and 6 and the magnitudes of the interactions are all scaled relative to the interaction of a ligand at position 1 or 6 with the dz2 orbital. We will not address the approach that was taken to arrive at these relative numbers; we will simply use the results tabulated here.

Sigma Interactions of Ligands with Metal d Orbitals (units of eσ)
Ligand Positions dz2 dx2-y2 dxy dxz dyz
1 1 0 0 0 0
2 1/4 3/4 0 0 0
3 1/4 3/4 0 0 0
4 1/4 3/4 0 0 0
5 1/4 3/4 0 0 0
6 1 0 0 0 0
7 0 0 1/3 1/3 1/3
8 0 0 1/3 1/3 1/3
9 0 0 1/3 1/3 1/3
10 0 0 1/3 1/3 1/3
11 1/4 3/16 9/16 0 0
12 1/4 3/16 9/16 0 0

In order to quantify the interactions between ligands and metal orbitals, we simply tally up the interactions for each orbital. For example, consider a complex like hexaamminecobalt(III) chloride, [Co(NH3)6]Cl3. It has six sigma donor ligands, forming an octahedral structure. The total interactions with each orbital in this geometry would be calculated by totaling the interactions at all the ligand positions for each orbital.

Looking under the dz2 column and totaling only the values for positions 1 through 6, we find:

dz2: (1 + 1/4 + 1/4 + 1/4 + 1/4 + 1) eσ = 3eσ

A similar approach using the dx2-y2 column leads to:

dx2-y2: (0 + 3/4 + 3/4 + 3/4 + 3/4 + 0) eσ = 3eσ

However, the remaining three d orbitals have no interactions with the ligands in these six positions:

dxy: 0; dxz: 0; dyz: 0

So, we find that the dz2 and dx2-y2 orbitals are both raised in energy by 3eσ. At the same time, we know that the ligand donor orbitals are stabilized by their interaction with the metal orbitals. To see how much, we can tally up the interaction for each ligand at its position. That is, for the ligand in position 1, we would add in its interaction with each of the five d orbitals to determine the amount by which it is stabilized by bonding. We simply add up the values across that row.

Ligand in position 1: - (1 + 0 + 0 + 0 + 0) eσ = - eσ

Ligand in position 2: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 3: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 4: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 5: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 6: - (1 + 0 + 0 + 0 + 0) eσ = - eσ

We have now generated a diagram that looks very much like what we saw from ligand field theory. We see the familiar d orbital splitting diagram for octahedral geometry as well as stabilization of the ligand donor orbitals in sigma bonds.

The results from the angular overlap model are somewhat simplified compared to the results from ligand field theory. For example, they completely neglect any interaction between ligand orbitals and metal s or p orbitals. However, this model allows a very straightforward calculation of the relative d orbital energy levels.

## Problems

1. Use the results of the calculation for the octahedral geometry to calculate the net stabilization due to bonding (in units of eσ) for the following complexes.

a) [Co(NH3)6]Cl3 (assume low spin) b) [Fe(NH3)6](NO3)3 (assume high spin)

c) [Ni(NH3)6]Cl2

2. Use the table of sigma interactions to calculate orbital energy stabilization or destabilization for the following geometries.

a) trigonal planar ML3 b) square planar ML4 c) trigonal bipyramidal ML5

Solutions

1.

2. a) Positions 2, 11, 12.

dz2: (1/4 + 1/4 + 1/4) eσ = 3/4 eσ

dx2-y2: (3/4 + 3/16 + 3/16) eσ = 18/16 = 9/8 eσ

dxy: (0 + 9/16 + 9/16) eσ = 18/16 = 9/8 eσ

dxz: 0

dyz: 0

Ligand in position 2: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 11: - (1/4 + 3/16 + 9/16 + 0 + 0) eσ = - eσ

Ligand in position 12: - (1/4 + 3/4 + 9/16 + 0 + 0) eσ = - eσ

b) Positions 2, 3, 4, 5.

dz2: (1/4 + 1/4 + 1/4 + 1/4) eσ = eσ

dx2-y2: (3/4 + 3/4 + 3/4 + 3/4) eσ = 3 eσ

dxy: 0

dxz: 0

dyz: 0

Ligand in position 2: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 3: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 4: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 5: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

c) Positions 1, 2, 6, 11, 12.

dz2: (1 + 1/4 + 1+ 1/4 + 1/4) eσ = 11/4 eσ

dx2-y2: (0 + 3/4 + 0 + 3/16 + 3/16) eσ = 18/16 = 9/8 eσ

dxy: (0 + 0 + 0 + 9/16 + 9/16) eσ = 18/16 = 9/8 eσ

dxz: 0

dyz: 0

Ligand in position 1: - (1 + 0 + 0 + 0 + 0) eσ = - eσ

Ligand in position 2: - (1/4 + 3/4 + 0 + 0 + 0) eσ = - eσ

Ligand in position 6: - (1 + 0 + 0 + 0 + 0) eσ = - eσ

Ligand in position 11: - (1/4 + 3/16 + 9/16 + 0 + 0) eσ = - eσ

Ligand in position 12: - (1/4 + 3/4 + 9/16 + 0 + 0) eσ = - eσ

This page titled 10.4.1: Sigma Bonding in the Angular Overlap Model is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Chris Schaller.