Linear response calculation with nonlocal van der Waals density functionals

The linear-response approach with the nonlocal van der Waals density functionals is presented, considering three kinds of perturbations: Atomic displacements, uniform electric fields, and strain. The formulas to compute the response with respect to strain are derived for the van der Waals density functionals as well as for the generalized gradient approximation. The linear-response method is implemented within the ultrasoft pseudopotential scheme. The method is applied to weak-coupled layered materials, graphite and MoS2. The results support the validity of the derived formulas and demonstrate the utilities of the linear-response method for weak-coupled van der Waals systems.