  The role of internal motions and molecular geometry on $^1$H NMR relaxation
times $T_{1,2}$ in hydrocarbons is investigated using MD (molecular dynamics)
simulations of the autocorrelation functions for in{\it tra}molecular $G_R(t)$
and in{\it ter}molecular $G_T(t)$ $^1$H-$^1$H dipole-dipole interactions
arising from rotational ($R$) and translational ($T$) diffusion, respectively.
We show that molecules with increased molecular symmetry such as neopentane,
benzene, and isooctane show better agreement with traditional hard-sphere
models than their corresponding straight-chain $n$-alkane, and furthermore that
spherically-symmetric neopentane agrees well with the Stokes-Einstein theory.
The influence of internal motions on the dynamics and $T_{1,2}$ relaxation of
$n$-alkanes are investigated by simulating rigid $n$-alkanes and comparing with
flexible (i.e. non-rigid) $n$-alkanes. Internal motions cause the rotational
and translational correlation-times $\tau_{R,T}$ to get significantly shorter
and the relaxation times $T_{1,2}$ to get significantly longer, especially for
longer-chain $n$-alkanes. Site-by-site simulations of $^1$H's along the chains
indicate significant variations in $\tau_{R,T}$ and $T_{1,2}$ across the chain,
especially for longer-chain $n$-alkanes. The extent of the stretched (i.e.
multi-exponential) decay in the autocorrelation functions $G_{R,T}(t)$ are
quantified using inverse Laplace transforms, for both rigid and flexible
molecules, and on a site-by-site bases. Comparison of $T_{1,2}$ measurements
with the site-by-site simulations indicate that cross-relaxation (partially)
averages-out the variations in $\tau_{R,T}$ and $T_{1,2}$ across the chain of
long-chain $n$-alkanes. This work also has implications on the role of
nano-pore confinement on the NMR relaxation of fluids in the organic-matter
pores of kerogen and bitumen.
