Symmetric stress tensor in the local-density-functional framework using a separable nonlocal pseudopotential.

We present the explicit symmetric formulas for the nonlocal pseudopotential contribution to the stress tensor. The symmetric stress tensor can be derived by applying the scaling procedure of wave vector K, which satisfies the symmetry property of \ensuremath{\partial}${\mathit{K}}_{\ensuremath{\gamma}}$/\ensuremath{\partial}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\beta}}}$=\ensuremath{\partial}${\mathit{K}}_{\ensuremath{\gamma}}$/\ensuremath{\partial}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\beta}}\mathrm{\ensuremath{\alpha}}}$ for the symmetric strain ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\beta}}}$. We note that this scaling procedure gives rise to the same expression for the stress tensor as that derived using a semilocal or a separable nonlocal pseudopotential.