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The translational-diffusion coefficient $D_T$ and the spin-rotationcontribution to the $^1$H NMR relaxation time $T_{1J}$ for methane (CH$_4$) areinvestigated using MD (molecular dynamics) simulations, over a wide range ofdensities $\rho$ and temperatures $T$, spanning the liquid, supercritical, andgas phases. The simulated $D_T$ agree well with measurements, without anyadjustable parameters in the interpretation of the simulations. A minimizationtechnique is developed to compute the angular-velocity for non-rigid sphericalmolecules, which is used to simulate the autocorrelation function $G_{\!J}(t)$for spin-rotation interactions. With increasing $D_T$ (i.e. decreasing $\rho$),$G_{\!J}(t)$ shows increasing deviations from the single-exponential decaypredicted by the Langevin theory for hard spheres, and the deviations arequantified using inverse Laplace transforms of $G_{\!J}(t)$. $T_{1J}$ isderived from $G_{\!J}(t)$ using the kinetic model "km" for gases($T_{1J}^{km}$), and the diffusion model "dm" for liquids ($T_{1J}^{dm}$).$T_{1J}^{km}$ shows better agreement with $T_1$ measurements at higher $D_T$,while $T_{1J}^{dm}$ shows better agreement with $T_1$ measurements at lower$D_T$. $T_{1J}^{km}$ is shown to dominate over the MD simulated $^1$H-$^1$Hdipole-dipole relaxation $T_{1RT}$ at high $D_T$, while the opposite is foundat low $D_T$. At high $D_T$, the simulated spin-rotation correlation-time$\tau_J$ agrees with the kinetic collision time $\tau_K$ for gases, from whicha new relation $1/T_{1J}^{km} \propto D_T$ is inferred, without any adjustableparameters.

NMR Spin-Rotation Relaxation and Diffusion of Methane