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.