AB_INITIO ANALYTICAL CONSTRUCTION OF PERMITTIVITY

This work presents a systematic, first-principles theoretical framework that explicitly bridges microscopic polarization in atoms with the macroscopic permittivity of dielectric materials. Motivated by the need for conceptual transparency beyond the Clausius–Mossotti approximation, the study develops a fully analytical approach based on a one-dimensional atomic crystal subjected to an external electric field. Each atom is modeled through δ-function and δ-gradient potentials that represent atomic binding and local field interactions respectively, and electrons are treated quantum mechanically, with thermodynamic effects incorporated via Fermi-level population. The model yields exact, non-perturbative expressions for both polarization and field-dependent permittivity, remaining constant up to 10¹⁰ V/m and valid up to 10¹¹ V/m where nonlinearities arise. This framework unites mathematical rigor and physical clarity, providing a transparent and quantitative foundation for understanding dielectric response from first principles.