Optimal Control of Bimolecular Scattering

We can readily extend bimolecular control to superpositions composed of more than two states. Indeed, we can introduce a straightforward method to optimize the reactive cross section as a function of am for any number of states [252], Doing so is an example of optimal control theory, a general approach to altering control parameters to optimize the probability of achieving a desired goal, introduced in Chapter 4. 

Consider scattering from incident state E, q, i 0) to final state , q, f 0). The. label / includes the angles 9, f into which the products scatter. Hence summations . below over / imply integrations over these angles. In accord with Eq. (7.2), the tprobability P(f, q i, q) of producing product in final state i , q, f 0) having. .started in the initial state E, q, i 0) is 

To simplify the notation we introduce the matrix it = with elements 

Here t denotes the Hermitian conjugate, and the q subscript on the S indicates that we are dealing with the submatrix of the S matrix associated with scattering into the product manifold defined by the q quantum number. 

One can optimize scattering into arrangement channel q, with the normalization constraint Kl2 = 1, by requiring 

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