First-principles calculations of dynamical Born effective charges, quadrupoles, and higher order terms from the charge response in large semiconducting and metallic systems

Within the context of first-principles techniques, we present a theoretical and computational framework to quickly determine, at finite momentum, the static self-consistent (longitudinal) charge response to an external perturbation that enters the determination of the scattering cross section of inelastic scattering processes such as EELS. We also determine the (transverse) charge response computed in short-circuit conditions. The all-order quasimomentum expansion of the transverse charge response to an atomic displacement are identified with dynamical Born effective charges, quadrupoles, octupoles, etc. Theoretically, we demonstrate that the transverse charge response can be related to the longitudinal one via a well-defined static long-range dielectric function, going beyond the random phase approximation. Our theoretical advancements allow for an efficient use of perturbation theory in the computational implementation. Due to its more favorable scaling, our method provides an interesting alternative to the use of the $2n+1$ theorem, especially for the study of semiconductors and metals with large unit cells. For semiconductors, we apply our developments to the computation of the piezoelectric properties of a large cell solid solution of semiconducting hafnium oxide containing 96 atoms. We here show that the clamped ion piezoelectric response, which is determined solely by dynamical quadrupoles, can be decomposed into real-space localized contributions that mostly depend on the chemical environment, paving the way for the use of machine-learning techniques in the material search for optimized piezoelectrics. We further apply our methodology to determine the density response of metals. We here find that the leading terms of the charge expansion are related to the Fermi energy shift of the potential if admitted by symmetry, and by Born effective charges which do not sum to zero over the atoms. These terms are then linked to the leading order expansion of the macroscopic electron-phonon coupling in metals. We apply our developments to the TEM-EELS spectroscopy of lithium intercalated graphites, where we find that approximating the density response via the use of the atomic form-factor in the long-wavelength limit does not take into account the anisotropy of the atomic chemical bonding in the crystal.