  We study the effects of atomic vibrations on the solid-state chemical
shielding tensor using first principles density functional theory calculations.
At the harmonic level, we use a Monte Carlo method and a perturbative
expansion. The Monte Carlo method is accurate but computationally expensive,
while the perturbative method is computationally more efficient, but
approximate. We find excellent agreement between the two methods for both the
isotropic shift and the shielding anisotropy. The effects of zero-point quantum
mechanical nuclear motion are important up to relatively high temperatures: at
500 K they still represent about half of the overall vibrational contribution.
We also investigate the effects of anharmonic vibrations, finding that their
contribution to the zero-point correction to the chemical shielding tensor is
small. We exemplify these ideas using magnesium oxide and the molecular
crystals L-alanine and beta-aspartyl-L-alanine. We therefore propose as the
method of choice to incorporate the effects of temperature in solid state
chemical shielding tensor calculations the perturbative expansion within the
harmonic approximation. This approach is accurate and requires a computational
effort that is about an order of magnitude smaller than that of dynamical or
Monte Carlo approaches, so these effects might be routinely accounted for.
