  A first-principles method is presented to calculate elastic constants up to
the fourth order of crystals with the cubic and hexagonal symmetries. The
method relies on the numerical differentiation of the second Piola-Kirchhoff
stress tensor and a density functional theory approach to compute the Cauchy
stress tensors for a minimal list of strained configurations of a reference
state. The number of strained configurations required to calculate the
independent elastic constants of the second, third, and fourth order is 24 and
37 for crystals with the cubic and hexagonal symmetries, respectively. Here,
this method is applied to five crystalline materials with the cubic symmetry
(diamond, silicon, aluminum, silver, and gold) and two metals with the
hexagonal close packing structure (beryllium and magnesium). Our results are
compared to available experimental data and previous computational studies.
Calculated linear and nonlinear elastic constants are also used, within a
nonlinear elasticity treatment of a material, to predict values of volume and
bulk modulus at zero temperature over an interval of pressures. To further
validate our method, these predictions are compared to results obtained from
explicit density functional theory calculations.
