In the seminal 1954 book “Dynamical Theory of Crystal Lattices” by Max Born and Kun Huang, much of the theory of lattice dynamics was established by considering the response of solids to long-wavelength atomic vibrations (i.e., acoustic phonons). In this talk, I will show that this classic technique, coupled with modern first-principles calculations based on density functional perturbation theory, can yield new insights. Specifically, I will consider a long-wavelength expansion of the electrical polarization response, order by order. At zeroth order in the wavevector, the polarization response gives the Born effective charges of the ions, which are crucial to understanding, e.g., ferroelectric polarization, phonon dispersions in ionic insulators, electron-phonon scattering, dielectric screening, electromechanical coupling, and optical spectra in the IR/THz regime. I will show that going beyond the adiabatic approximation usually used for lattice dynamics extends the definition of Born effective charges from insulators to conducting systems and relates them to a seemingly unrelated fundamental property of metals: the Drude weight . At first order in the wavevector, the polarization response an atomically-resolved piezoelectric response, or “dynamical quadrupoles.” I will show that these quantities can be crucial in calculations of phonon-limited electron mobility , and even the response of the lattice to magnetic fields. Finally, at second order in the wavevector, we obtain a description of the flexoelectric response of the materials, i.e., the polarization induced by a strain gradient . I will give examples of all of these effects in a variety of materials systems including perovskite oxides, semiconductors, and layered van der Waals materials.
 C. E. Dreyer, S. Coh, M. Stengel, arXiv:2103.044251
 V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, M. Bernardi, Phys. Rev. Lett. 125, 136602 (2020); J. Park, J.-J. Zhou, V. A. Jhalani, C. E. Dreyer, M. Bernardi, Phys. Rev. B 102, 125203 (2020).
 C. E. Dreyer, M. Stengel, D. Vanderbilt, Phys. Rev. B 98, 075153 (2018); A. Schiaffino, C. E. Dreyer, D.Vanderbilt, and M. Stengel, Phys. Rev. B 99, 085107 (2019).