Balancing Overlap and Energy Mismatch

From WikidChem

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Slides 4-6: Bonding Shift with Mismatched Energies

This slide shows how we can geometrically determine the bonding shift when energy is mismatched. The trick is to orient the energy split caused by a certain amount of overlap with a perfect match (A-B, blue arrow) perpendicularly to the mis-match (A-C, black arrow). The diagonal of the resulting rectangle (orange arrow) is the splitting energy for the mismatch. The center must be shifted upwards a bit from the average of the A and C energies to account for the normalization that makes the change in energy from an anti-bond slightly greater than the change in energy from a bond.

A mathematical approach for this is-- Splitting = sqrt (AO energy difference^2 + AO overlap^2))

Note: The discussion above assumes that AC has the same overlap as AB. The trick with the diagonal of a rectangle provides a geometric way of visualizing the solution of a quadratic equation that arises in this problem. - JMM As expected, two new orbitals (one higher energy, one lower energy) arise from the overlap of A and C, just as it did with perfectly matching A and B. But here, the higher energy anti-bonding orbital is more like A (the higher energy original orbital) in shape and energy, and the lower energy bonding orbital is more like C (the lower energy original orbital).

You can also see what would influence the energy split for mismatched orbitals (orange arrow). The splitting is not very sensitive to the lesser contributor, either mismatch (black arrow) or overlap (blue arrow). If the overlap is very small, the resulting orbitals look a lot like the original orbitals, and the shift in energy between the original orbital and the new orbital would be very small. (The red arrows, up from A to the antibonding orbital and down from C to the bonding orbital, would be very short). This accounts for the fact that core electrons exist in orbitals that are effectively just pure atomic orbitals; the overlap is so small that there is no perceptible change in energy.

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Slides 8-9: Important Generalizations

• Mixing two orbitals gives one new orbital lower in energy than either parent and one higher in energy than either parent.

Mixing orbitals creates a lower energy bonding orbital and a higher energy antibonding orbital.

• The lower-energy combination looks mostly like the lower-energy parent, both in shape and in energy (and vice versa).

This should make sense. Please ask if you need clarification.

• For a given overlap, increasing energy mismatch decreases the amount of mixing and decreases the magnitude of energy shifts.

The energy shifts he's talking about are the red arrows from the original orbitals to the new orbitals. Visually, you can see that a longer black arrow would result in shorter red arrows. Furthermore, with increased energy mismatch, the bonding and antibonding orbitals would increasingly resemble the original HOMO & LUMO orbitals, so there will not be a significant shift in energy.

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Slide 23: Which bond is stronger?

Compared to what? The question "Which bond is stronger?" is important, but you've got to ask the more important question "Compared to what?" before you get a clear answer.

There are two scenarios here: the electrons in the bonding orbitals could come only from B or C (A contributes no electrons to the bond) or one electron could come from each (A contributes 1 electron and B or C contributes 1 electron).

(1) In the first case, when both electrons come from either B or C, the AB bond is stronger because it results in two electrons changing to a much lower energy state (change in their energy represented by the two blue arrows). The change in energy of the two electrons in C would be much smaller (shorter red arrows).

(2) But the situation changes when one electron comes from each atom. The change in energy of both electrons in the AB bond would stay the same (the blue arrows are the same length they were before). However, the electron contributed by A in the AC bond decreases in energy and exists in a much more stable state, even though the electron from C is only slightly more stable as a result of the bond. The total change in stability of these electrons in the AC bond (represented by the sum of long and short red arrows) is greater than the change in stability of the electrons in the AB bond, indicating that in this case, the AC bond is stronger.

Case (1) represents heterolysis or unsymmetrical cleavage. Case (2) represents homolysis or symmetrical cleavage.