A Appendix Monte Carlo sampling

After we discuss transition state theory we will be in a better position to handle this issue. Pun intended. 

The Monte Carlo sampling of initial conditions in the method of classical trajectories is illustrated here for the particular problem of computing the reaction cross-section ctr given in terms of the opacity function P b) by, Eq. (3.14), 

Now P(b) is the probabihty of reaction at impact parameter b. Let us therefore run N(b) classical trajectories, all of which have the same initial parameter b. Not all of these trajectories will necessarily lead to reaction. The reason is that they will differ in some other initial conditions (e.g., the phase of vibration or rotation of the diatomic reagent). Let Nf,(b), Nf,(b) N b), be the number of trajectories that do exit in the products valley. Then if N(b) is large enough, P(b) = N i,(b)/N b), and so we need to evaluate the integral 

Let us sample initial b values with some attention to tiie physics. In otiier words, like the public opinion pollsters let us prepare a representative sample. Higher b values are more heavily weighted in the cross-section, via the annulus Inb 6b. Therefore, if we plan to run a grand total of N trajectories, let us allot the number A( )) A ), where 

Integrating Eq. (A.5.3), over b from 0 to B, we verify that Nis the total number of trajectories 

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