Gauge-Invariant Long-Wavelength TDDFT Without Empty States: From Polarizability to Kubo Conductivity Across Heterogeneous Materials
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 $(\alpha_{\rm fs})$. 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 na\"ive 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.