  Atomistic molecular dynamics simulations are used to investigate $^1$H NMR
$T_1$ relaxation of water from paramagnetic Gd$^{3+}$ ions in solution at
25$^{\circ}$C. Simulations of the $T_1$ relaxivity dispersion function $r_1$
computed from the Gd$^{3+}$--$^1$H dipole--dipole autocorrelation function
agree within $\simeq 5$\% of measurements above $f_0 \gtrsim $ 5 MHz, without
any adjustable parameters in the interpretation of the simulations. The
agreement between simulated and measured $r_1$ above $f_0 \gtrsim $ 5 MHz (i.e.
$B_0 \gtrsim $ 0.1 T) shows potential for predicting $r_1$ in chelated
Gd$^{3+}$ contrast-agents used for clinical MRI, without any adjustable
parameters or models. Below $f_0 \lesssim $ 5 MHz the simulation overestimates
$r_1$ compared to measurements, which is used to estimate the zero-field
electron-spin relaxation time. The most strongly bound water molecule to the
Gd$^{3+}$ complex evinces an autocorrelation function consistent with the
mono-exponential decay used in the Solomon-Bloembergen-Morgan (SBM)
inner-sphere model.
