Hausdorf dimension

However, there is another typ>e of confinement that can be imposed on a reactive system, namely, by a reduction in the effective dimensionality. The simplest examples are those in which the motions of the reactive species are confined to a flat surface or a one-dimensional chain. However, in many systems the connectivity of the configuration space is such that it has effectively a fractal dimension d. The Hausdorf dimension is defined from the behavior of the pair distribution function at sufficiently large R, which varies as that is, the probability of finding the pair with a separation between R and R + dR is proportional to dR. The reduction of the encounter problem from d dimensions to the one dimension R is studied in Section VII A. The important case of reactions on surfaces is considered separately in Section VIIB. 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

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