The quest for "chemical accuracy" is becoming more and more demanded in the field of structure and kinetics of molecules at solid surfaces. In this paper, as an example, we focus on the barrier for hydrogen diffusion on a α-Al2O3(0001) surface, aiming for a couple cluster singles, doubles, and perturbative triples [CCSD(T)]-level benchmark. We employ the density functional theory (DFT) optimized minimum and transition state structures reported by Heiden, Usvyat, and Saalfrank [J. Phys. Chem. C 123, 6675 (2019)]. The barrier is first evaluated at the periodic Hartree-Fock and local Møller-Plesset second-order perturbation (MP2) level of theory. The possible sources of errors are then analyzed, which includes basis set incompleteness error, frozen core, density fitting, local approximation errors, as well as the MP2 method error. Using periodic and embedded fragment models, corrections to these errors are evaluated. In particular, two corrections are found to be non-negligible (both from the chemical accuracy perspective and at the scale of the barrier value of 0.72 eV): the correction to the frozen core-approximation of 0.06 eV and the CCSD(T) correction of 0.07 eV. Our correlated wave function results are compared to barriers obtained from DFT. Among the tested DFT functionals, the best performing for this barrier is B3LYP-D3.