The enhancement of nuclear magnetic resonance (NMR) relaxation rates produced by paramagnetic solutes is physically rather different for electron spin S = 1/2 paramagnetic species than for S greater than or equal to 1 species due to the presence of zero-field splitting interactions in the electron spin Hamiltonians of the latter. When the zfs energy is larger than the electronic Zeeman energy, the electron spin precessional motion is spatially quantized with respect to the molecule-fixed principal axis system (PAS) of the zfs tensor rather than along the external laboratory magnetic field. An analytical theory of the orthorhombic zfs limit has been derived in which the motion of the electron spin variables is described in the zfs-PAS and that of the nuclear spin variables in the laboratory coordinate frame. The resulting theoretical expressions are simple in form and suggest a physically transparent interpretation of the experiment. The NMR relaxation enhancement R-1p results from additive contributions, R-1x, R-1y, and R-1z, arising from the molecular-frame Cartesian components of the time-dependent electron spin magnetic moment operator mu(r)(t). Each Cartesian component R-1r depends on the dipolar power density at the nuclear Larmor frequency that is produced by the corresponding Cartesian component of mu(r)(t). The theory displays the dependence of the relaxation enhancement on the variables of molecular structure in a very simple and physically transparent form: R-1r proportional to r(-6)[1 + P-2(cos theta(r))], where r is the interspin distance and cos theta(r) is the direction cosine of the interspin vector with the rth principal axis of the zfs tensor. New experimental data are presented for the model S = 1 complex [trans-Ni(II)(acac)(2)(H2O)(2)] (acac = acetylacetonato) in dioxane solvent. The magnetic held dependence of the proton T-1 of the axial water ligands has been measured over the range 0.15-1.5 T, the lower end of which corresponds to the zfs limit. The experimental data have been analyzed using the new analytical theory for the zfs-limit regime in conjunction with spin dynamics simulations in the intermediate regime. Dipolar density power plots are presented as graphical, devices which clearly exhibit the physical information In the experiment, and which permit a sapid differentiation of the sensitive and insensitive parameters of theory. The data analysis depends strongly on the zfs parameter \E\ and on the electron spin relaxation time tau(S,z) along the zfs-PAS z-axis, but only very weakly on the other parameters of theory. A fit of the data to theory provided the values \E\ = 1.8 +/- 0.1 cm(-1) and tau(S,z) = 8.0 +/- 0.3 ps. (C) 1997 American Institute of Physics.