  It is the purpose of this paper to present a computational framework for
reliably determining the frequency-dependent intermolecular and intramolecular
NMR dipole-dipole relaxation rate of spin $1/2$ nuclei from MD simulations. The
approach avoids alterations caused by well-known finite-size effects of the
translational diffusion. Moreover, a procedure is derived to control and
correct for effects caused by fixed distance-sampling cutoffs and periodic
boundary conditions. By construction, this approach is capable of accurately
predicting the correct low-frequency scaling behavior of the intermolecular NMR
dipole-dipole relaxation rate and thus allows the reliable calculation of the
frequency-dependent relaxation rate over many orders of magnitude. Our approach
is based on the utilisation of the theory of Hwang and Freed for the
intermolecular dipole-dipole correlation function and its corresponding
spectral density [J. Chem. Phys. 63, 4017 (1975)] and its combination with data
from molecular dynamics (MD) simulations. The deviations from the Hwang and
Freed theory caused by periodic boundary conditions and sampling distance
cutoffs are quantified by means of random walker Monte Carlo simulations. An
expression based on the Hwang and Freed theoryis also suggested for correcting
those effects. As a proof of principle, our approach is demonstrated by
computing the frequency-dependent inter- and intramolecular dipolar NMR
relaxation rate of the $^1$H nuclei in liquid water at $273\,\mbox{K}$ and
$298\,\mbox{K}$ based on simulations of the TIP4P/2005 model. Our calculations
are suggesting that the intermolecular contribution to the $^1$H NMR relaxation
rate of the TIP4P/2005 model in the extreme narrowing limit has previously been
substantially underestimated.
