Theory of chemically induced dynamic spin polarization. 5. Orientation-dependent effects

Abstract

The Pedersen-Freed theory for chemically induced dynamic spin polarization CIDN(E)P is generalized to include the effects of anisotropic reactivities and anisotropic exchange interactions on the radical-pair mechanism. Detailed results are given for the simple case in which only one radical exhibits anisotropy that is approximated by a cosine distribution, and the rotational and translational motions are described by Brownian diffusion models. The primary effect upon CIDNP is the reduction in A, the reaction probability for the full collision. This effect can be rather accurately approximated by the use of an “effective” spherically symmetric specific rate constant, which depends, to some extent, on the rotational diffusion coefficient due to the effect of rotational relaxation. Thus CIDNP effects are rather well approximated by a spherically symmetric theory with a renormalized A. In the absence of reactivity, CIDEP effects for our model are reasonably well approximated by a spherically symmetric theory with a renormalized exchange interaction, especially for the asymptotic polarizations for large exchange interactions. When, however, there are orientation-dependent reactivities present with substantially greater orientation dependence than the exchange interaction, then significant deviations from a spherically symmetric theory are predicted even for asymptotic polarizations. The relationship to recent experiments is briefly discussed in this light. Anisotropic initial conditions are typically found to be relaxed by the effective rotational diffusion before they can substantially affect the CIDN(E)P observables. I. Introduction The existing theories of chemically induced dynamic spin polarization were all developed for spherically symmetric interactions between the radical pairs.’?’ Nevertheless, most interacting radicals are expected to display anisotropic features both in their ability to react and in their exchange features. In particular, the rate of reaction between nonspherical or bulky radicals is expected to be reduced by steric hindrance and other structural characteristics, including those of the solvent. Such effects have been confirmed experimentally in a recent CIDNP st~dy,~ and are otherwise well-known in many kinetic studies. The matter of orientation-dependent effects upon radical (or molecule) reactivities has lead to several theoretical studies. The general problem was first discussed by Solc and Stockmayer4 in terms of a combined translational and orientation-dependent diffusion equation with reactive boundary conditions. They showed how the problem could be solved numerically in terms of Bessel function expansions (in radial space) and spherical Harmonic expansions (in orientation space). Freed and PedersenPa suggested an approach closely related to that of Solc and Stockmayer but differing mainly in that finite differences are used in radial space to more easily deal with the short-range interactions that are so important in the CIDN(E)P phenomenon. Some later models were developed by other^^,^ *to treat simplified cages, but the more realistic the model the greater is the reliance on numerical treatment^^^,^ or else considerable simplification of the model is req~ired~”~ (e.g., simple two-state kinetic models for the orientationdependent reactivity) and/or appeal to reencounter probabilistic arguments6 rather than direct solution of the diffusion equation. In this work, following the suggestions of Freed and Pedersen, we include the spin-dependent nature of the interacting radicals, i.e., their spin-dependent reactivity and exchange interactions, together with the diffusive aspects through the stochastic-Liouville formalism (SLE) in a combined approach by utilizing finite-differences (in radial space) and eigenfunction expansions (in orientation space). Thus, we are able to extend the theory of Pedersen