1H NMRD Profile and ESR Lineshape Calculation for an Isotropic Electron Spin System withS= 7/2. A Generalized Modified Solomon–Bloembergen–Morgan Theory for Nonextreme-Narrowing Conditions

Abstract

A generalization of the modified Solomon–Bloembergen–Morgan (MSBM) theory is presented for electron-spin quantum numberS= 7/2 taking multiexponential electron-spin relaxation into account. The theoretical approach closely follows P.-O. Westlund [Mol. Phys.85, 1165–1178 (1995)]. In the nonextreme-narrowing regime for the electron spin system, three correction terms to the traditional MSBM equations arise. They are all derived in closed analytical form to give the generalized equations in a simple and convenient form: MSBM × (1 + correction). The ESR lineshape function forS= 7/2 is also given in closed analytical form. A number of protonT₁NMRD profiles representing different Gd(III) complexes are investigated, using a spectral density function of the transient ZFS interaction which allows for two relaxation times. For Gd–macromolecular complexes the largest deviation from the traditional MSBM theory may be about 15.2%, but for the parameter sets investigated the corrections were about 2–3%. A computer program is developed which calculates the NMRD profile and the ESR lineshape (X band or Q band) for the set of model parameters. These parameters describe the electron spin system in terms of Δ_t(the root-mean-square value of the transient ZFS interaction) and τ_v1, τ_v2, andS₀(correlation times and an order parameter of the transient ZFS correlation function). For the paramagnetically enhanced nuclear (I= $\text{1}\text{2}$) spin relaxation, the parameters τ_R(the reorientational correlation time),q(the number of fast exchanging water molecules in the first hydration shell), andr_IS(the point spin–dipole distance) are also used.