Reaction hypercube

Reaction hypercube A generalization of the Albery-More O Ferrall-Jencks diagram to n dimensions. The two-dimensional case is a square, the three-dimensional case is a cube, and so on. For any reaction h3fpercube energy could be included as an additional dimension orthogonal to the others but this is difficult to draw except in the two-reaction-dimension case... 

One-thing-at-a-time process A process corresponding to progress along one of the edges of a reaction hypercube. Comer species have progressed by 0 or 1 along each of the reaction dimensions... 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

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