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A quasi-analytical theoretical method is devised, and implemented in the  Crystal  program, for calculation of the direct “proper” piezoelectric tensor of periodic systems including both the clamped-nuclei electronic and nuclear relaxation contributions. It is based on using the analytical Coupled-Perturbed-Hartree–Fock/Kohn–Sham (CPHF/KS) procedure to obtain dipole derivatives with respect to lattice deformations as well as internal coordinates. The sole numerical step required involves building the Hessian matrix through differentiation of analytical energy gradients. Two prototypical piezoelectric, non-ferroelectric, crystals, namely ZnO and  α -quartz, are used to demonstrate the accuracy and computational efficiency of our new scheme, which significantly improves upon the commonly used numerical Berry phase approach.

Direct Piezoelectric Tensor of 3D Periodic Systems through a Coupled Perturbed Hartree–Fock/Kohn–Sham Method