On the Cauchy–Born approximation at finite temperature

• We derive a closed form expression for the first Piola–Kirchhoff stress using asymptotic expansion.
• A derivation is presented for systems under periodic boundary conditions.
• The issue of potential smoothness is addressed.
• We clarifies the removal of the zero phonon modes.

We address several issues regarding the derivation and implementation of the Cauchy–Born approximation of the stress at finite temperature. In particular, an asymptotic expansion is employed to derive a closed form expression for the first Piola–Kirchhoff stress. For systems under periodic boundary conditions, a derivation is presented, which takes into account the translational invariance and clarifies the removal of the zero phonon modes. Also revealed by the asymptotic approach is the role of the smoothness of the interatomic potential. Several numerical examples are provided to validate this approach.