Ab-initio stiffness values were computed in an automated way for two trigonal reference materials: dextro low quartz and calcite with respective space groups P3₂21 and R-3c. Both were computed in their IRE settings, namely with r faces indexed {10-11} for right-quartz and with obverse setting of the rhombohedral lattice in the hexagonal axes for calcite. The VASP total energy and stress engine was used for the calculations. Results in the order C₁₁, C₁₂, C₁₃, C₁₄, C₃₃, C₄₄ are 82, 2, 10, —20, 98, 63 GPa for quartz and 159, 70, 63, +20, 98, 39 GPa for calcite. Quartz results are in good agreement with printed literature values. Comparison with experimental values for calcite disclosed a reverse orientation of the rhombohedral lattice in the hexagonal axes for the measurements that had not been clearly reported but was still recoverable from one sufficiently detailed reference. In spite of its current limited accuracy on extracted elastic coefficients, ab-initio calculation of elastic coefficients may already be a reliable way to complement and assist the experiment in its most difficult aspects.