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Closed-form expressions for level-averaged electron spin relaxation times outside the Zeeman limit: application to paramagnetic NMR relaxation.

Paramagnetic enhancement of NMR relaxation (NMR-PRE) depends on thermal relaxation of the electron spin system. Most previous analyses of experimental NMR-PRE data have relied on Bloembergen--Morgan (B--M) theory to describe the magnetic field dependence of electron spin relaxation in liquid samples. However, B--M theory assumes a Zeeman-limit situation and is not physically appropriate to the common case of S > or = 1 transition metal ions which possess a permanent zero-field splitting (zfs) that is comparable to or larger than the Zeeman splitting. Theory has been needed which (1) includes the effects of the zfs interaction, thus providing a realistic description of the magnetic field dependence of the NMR-PRE outside the Zeeman limit, and (2) describes electron spin relaxation phenomena at a comparable level of complexity to that of B--M theory, i.e., with two magnetic field-dependent electron spin relaxation times, tau(S1) and tau(S2), defined in the laboratory coordinate frame. Theory of this kind is developed. Expressions derived in a previous study (R. R. Sharp and L. L. Lohr, J. Chem. Phys. 115, 5005 (2001).) for level-specific relaxation rates have been averaged over spin eigenstates to give level-averaged quantities, tau(S1,2). This kind of averaging leads to a great simplification in the mathematical form of the results. Simple zfs-limit molecular-frame and laboratory-frame expressions are given for electron spin S=1, 3/2, 2, and 5/2. General expressions, valid for S > or = 1 and for arbitrary magnitudes of the Zeeman and zfs energies, are derived for level-averaged electron spin relaxation times defined in both the laboratory- and the molecule-fixed coordinate frames. The new theory coincides with B--M theory in the Zeeman limit.