First-principles calculation of third-order elastic constants via numerical differentiation of the second Piola-Kirchhoff stress tensor

Third-order elastic constants (TOECs) of materials are difficult to measure experimentally and
produce large errors. Computational methods are needed for overcoming these difficulties. Previous
methods to calculate TOECs are based on fitting energy-strain and/or stress-strain curves
calculated from density functional theory (DFT). These methods rely on symmetry relationships, and
for this reason, so far they have been applied mainly to cubic and hexagonal crystals. In this
paper, we present a novel method to calculate TOECs that is applicable to any system, regardless
of its symmetry and dimensionality. This method relies on second-order numerical differentiation
of the second Piola-Kirchhoff stress tensor. In this work, we combine this method to a plane-wave
DFT approach to calculate the TOECs of aluminum, diamond, silicon, magnesium, graphene, and
graphane. A comparison to experimental results shows that our new method is valid and accurate.