  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.
