The inherent uncertainties in modern first-principles calculations are reviewed using geophysically relevant examples. The elastic constants of perovskite at lower-mantle temperatures and pressures are calculated using ab initio molecular dynamics. These are used in conjunction with seismic tomographic models to estimate that the lateral temperature contrasts in the Earth's lower mantle are 800 K at a depth of 1000 km, and 1500 K at a depth of 2000 km. The effect of Al(3+) on the compressibility of MgSiO(3) perovskite is calculated using three different pseudopotentials. The results confirm earlier work and show that the compressibility of perovskites with Al(3+) substituted for both Si(4+) and Mg(2+) is very similar to the compressibility of Al(3+)-free perovskite. Even when 100% of the Si(4+) and Mg(2+) ions are replaced with Al(3+), the bulk modulus is only 7% less than that for Al(3+)-free perovskite. In contrast, perovskites where Al(3+) substitutes for Si(4+) only and that are charge balanced by oxygen vacancies do show higher compressibilities. When corrected to similar concentrations of Al(3+), the calculated compressibilities of the oxygen-vacancy-rich perovskites are in agreement with experimental results.