  An angular time-dependent probability density function describing Brownian or
anomalous rotational dynamics of fixed-length atom-to-atom vectors is
presented. The probability density function, which fully incorporates angular
boundary conditions, is applied to aqueous ion complexes. The rotational
dynamics of ion-$^1$H vectors are shown by molecular dynamics (MD) simulation
to be Brownian. A Brownian shell model is presented which yields a closed form
expression for the frequency-dependent nuclear-magnetic-resonance spin-lattice
relaxation rate $T_1^{-1}(\omega)$ based on a distance parameter and time
constant. Appropriate combinations of shell and/or continuum models are shown
to provide excellent fully-quantitative fits to experimental $T_1^{-1}(\omega)$
dispersion curves from aqueous manganese(II), iron(III) and copper(II) chloride
solutions. The distance parameters and time constants obtained from the fits
are in good agreement with independent experimental and MD data in the
literature. The Brownian shell model is a significant enhancement to existing
particle-particle models that describe the rotational correlation function as a
single exponential and are unable to provide the correct distance dependence
for a shell of $^1$H spin density preventing a match to experiment without an
arbitrary scaling factor.
