Avoided crossing rule

The photochemical disrotatory closure of butadiene to cyclobutene has been described with a state-correlation diagram, like that shown in Figure 21.4. It is based on the familiar orbital-correlation diagram of Woodward and Hoffmann," from which the intended correlations indicated by the dashed lines can readily be deduced. The solid lines indicate that there is an avoided crossing, which is put in as a result of the quantum mechanical noncrossing rule. It says that two states of the same total symmetry cannot cross. Instead, as they approach each other in energy, they will mix and separate, as the solid lines indicate. 

A question of particular importance is whether the intersection of two configuration curves will result in a real crossing, as indicated in [6a], or an avoided crossing as indicated in [6b]. The general rule which governs the... 

To summarize, we note two key points. Firstly, that the favourable interaction between DA and D + A in allowed reactions does not preclude barrier formation. The CM model, by its very nature, emphasizes barrier formation through the avoided crossing of reactant and product configurations. Second, the CM model gives rise to the Woodward-Hoffmann rules through consideration of the symmetry properties of the DA and D+A-configurations. This is as it should be. As we have noted previously, a key test of the CM model is whether it blends in naturally with existing theories that focus on specific areas of reactivity. 4... 

Inspection of the orbital make-up of reactant and product states may imply intended correlations which would lead to state crossings. If the states involved may not cross by rule 3, an avoided crossing occurs. The intention to cross is often depicted by dashed lines. 

The Ay state arising from 3B1 and the Ax state arising from 1A1 may now mix, since the noncrossing rule applies.128 Thus in the presence of spin interactions there is an avoided crossing as indicated in Figure 5. This noncrossing implies that in the isolated CH2 molecule there is a finite transition probability between the nonstationary zero-order states. If we... 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

In this first approach we follow Kasha s rule, that is, we consider that photochemical reactivity starts from the minimum of the lowest excited-state surface and focus on how the products are formed from there. Later we will discuss how the excited-state minimum can be reached from the FC region. There the molecnle may go through a crossing between two excited states or a state switch along an avoided crossing. The concepts we introduce now will also be useful to discuss these issues. 

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