In materials science and engineering, scientists often assume that crystals respond locally to an externally applied perturbation such as a strain or an electromagnetic field. Microscopically, however, the effects of the perturbation always propagate over a neighborhood around the point of application. At the macroscopic level, this means that the material response depends on gradients of the applied field, which is known as spatial dispersion. While these effects are generally small, they have attracted increasing interest in the past few years. Notable examples are flexoelectricity, the electrical voltage generated by a flexural deformation, and natural optical activity, the rotation of transmitted light polarization by some crystals. Here, we establish a general and efficient quantum-mechanical formalism to address this broad class of problems.

Density-functional perturbation theory (DFPT) is nowadays the state-of-the-art method to accurately calculate from first principles how materials respond to external stimuli. Our new approach consists of incorporating the long-wave method, a mainstay of condensed-matter theory since the early days of quantum mechanics, into the modern tools of DFPT. This allows one to access a broad range of spatial-dispersion properties at a surprisingly small computational cost and with unprecedented accuracy. We demonstrate our method, which we have implemented in a publicly distributed package (abinit), by calculating the flexoelectric tensor and the “dynamical quadrupoles” (i.e., the quadrupolar moment of the charge-density response to an atomic displacement) of several materials. We obtain excellent agreement with earlier studies, whenever available.

Our “long-wave DFPT” significantly extends the scopes and capabilities of perturbative electronic-structure approaches and opens the door to the systematic exploration of a vast range of gradient-related physical properties.