Hwang-Freed model

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

Fig. 15. Calculated outer-sphere NMRD profiles for integer spins (S = 1,2,3) for a large static ZFS. The dashed line is obtained using the Hwang and Freed model for S = 2. From Kruk, D. Nilsson, T. Kowalewski, J. Phys. Chem. Chem. Phys. 2001, 5, 4907-4917. Reproduced by permission of the PCCP Owner Societies.

An important theoretical development for the outer-sphere relaxation was proposed in the 1970s by Hwang and Freed (138). The authors corrected some earlier mistakes in the treatment of the boundary conditions in the diffusion equation and allowed for the role of intermolecular forces, as reflected in the IS radial distribution function, g(r). Ayant et al. (139) proposed, independently, a very similar model incorporating the effects of molecular interactions. The same group has also dealt with the effects of spin eccentricity or translation-rotation coupling (140). 

Zager and Freed have also compared their experimental data against line shapes predicted by perturbational treatments of simple FT models [35]. To lowest order, such models predict (cf. also Hwang et al. [10]) that the original Lorentzian shape of a pure FRD for the isolated first body should be replaced by a modified function... 

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