Gauge-Invariant Long-Wavelength TDDFT Without Empty States: From Polarizability to Kubo Conductivity Across Heterogeneous Materials

Abstract

Electromagnetic response is commonly computed in two languages: length-gauge molecular polarizabilities and velocity-gauge (Kubo) conductivities for periodic solids. We introduce a compact, gauge-invariant bridge that carries the same microscopic inputs-transition dipoles and interaction kernels-from molecules to crystals and heterogeneous media, with explicit SI prefactors and fine-structure scaling via $({\alpha }_{\mathrm{fs}})$. The long-wavelength limit is handled through a reduced dielectric matrix that retains local-field mixing, interfaces and 2D layers are treated with sheet boundary conditions (rather than naïve ultrathin films), and length-velocity equivalence is enforced in practice by including the equal-time (diamagnetic/contact) term alongside the paramagnetic current. Finite temperature is addressed on the Matsubara axis with numerically stable real-axis evaluation (complex polarization propagator), preserving unit consistency end-to-end. The framework enables predictive, unit-faithful observables from radio frequency to ultraviolet-RF/microwave heating and penetration depth, dielectric-logging contrast, interfacial optics of thin films and 2D sheets, and adsorption metrics via imaginary-axis polarizabilities. Numerical checks (gauge overlay and optical $(f)$-sum saturation) validate the implementation. Immediate priorities include compact, temperature- and salinity-aware kernels with quantified uncertainties and \emph{operando} interfacial diagnostics for integration into multiphysics digital twins.

References in corpus (9)

• Phonons and related properties of extended systems from density-functional perturbation theory
• Universal Dynamic Conductivity and Quantized Visible Opacity of Suspended Graphene
• A First-Principles Approach to Insulators in Finite Electric Fields
• A new and efficient approach to time-dependent density-functional perturbation theory for optical spectroscopy
• Gauge covariances and nonlinear optical responses
• Linear and nonlinear optical response of crystals using length and velocity gauges: Effect of basis truncation
• Gauge invariance of light-matter interactions in first-principle tight-binding models
• Gauge invariance of excitonic linear and nonlinear optical response
• Nonlinear optical response of doped mono- and bilayer graphene: length gauge tight-binding model