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

TL;DR

This paper introduces a gauge-invariant, unified framework for computing electromagnetic responses across molecules, crystals, and heterogeneous media, enabling accurate, unit-consistent predictions from radio frequencies to ultraviolet.

Contribution

It develops a compact, gauge-invariant method that bridges length- and velocity-gauge formalisms, handling complex interfaces and finite temperatures with stable numerical evaluation.

Findings

Validated gauge invariance through numerical checks.

Achieved accurate predictions of dielectric and optical properties.

Enabled analysis of interfacial and thin film optics.

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 $(\alpha_{\rm fs})$. 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.