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

TL;DR

The paper tackles the problem of unifying length-gauge molecular polarizabilities with velocity-gauge Kubo conductivities across molecules, crystals, and heterogeneous media in a gauge-invariant, unit-faithful way. It introduces a compact bridge that propagates the same microscopic inputs through a reduced dielectric matrix for the long-wavelength limit, enforces length–velocity equivalence by including the equal-time term, and treats finite temperature on the Matsubara axis with analytic continuation to real frequencies. The approach yields explicit SI prefactors via the fine-structure constant $\\alpha_{\\rm fs}$ and provides gauge checks (length–velocity overlay, $f$-sum) plus an interface-aware treatment using sheet boundary conditions, enabling predictions of observables from RF to UV such as penetration depth, dielectric logging, thin-film optics, and adsorption metrics. The work lays out a practical, transferable workflow from per-cell or molecular polarizabilities to macroscopic response functions, with immediate applicability to interfacial optics, RF heating, and digital-twin multiphysics, and outlines future directions for temperature-/salinity-aware kernels and operando diagnostics.

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 $(α_{\rm fs})$. 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 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.

Figures (4)

• Figure 1: Real (top) and imaginary (bottom) parts of $\varepsilon(\omega)$ for a single Lorentz oscillator, evaluated in two numerically independent ways. The solid curves use the length-gauge polarizability [Eq. \ref{['eq:eps_from_alpha']}], while the dashed curves use the velocity-gauge conductivity [Eq. \ref{['eq:eps_from_sigma']}] including both paramagnetic and equal-time (diamagnetic/contact) contributions. These two routes are numerically indistinguishable across the RF–UV window, demonstrating practical length–velocity equivalence in our implementation. For comparison, the dotted curves show the velocity result when the equal-time term is deliberately omitted (“no-dia”), leading to the expected low-frequency mismatch. All curves are generated from the test-oscillator parameters listed in Table \ref{['tab:allparams']}.
• Figure 2: Cumulative integral of the real part of the conductivity, $\mathcal{C}(\omega)=\int_0^\omega\mathrm{Re}\,\sigma(\Omega)\,d\Omega$, for the same single-oscillator model as in Fig. \ref{['fig:F1']}. The high-frequency plateau coincides with the longitudinal $f$-sum value $\pi n e^2/(2m)$ for this model, to within numerical accuracy. This provides a global check that our velocity-gauge implementation, including the equal-time term and SI prefactors, is internally consistent.
• Figure 3: Reflectance $R(\omega)$ and absorbance $A(\omega)$ for a conductive 2D sheet on a dielectric substrate, computed in two equivalent ways. Solid curves: sheet boundary conditions formulated directly in terms of the sheet conductivity $\sigma_{2\mathrm{D}}(\omega)$ in the $q\!\to\!0$ limit. Dashed/dotted curves: ultrathin-film description in which the same sheet conductance is represented by a uniform slab of thickness $d=1$ nm, with scalar $\varepsilon(\omega)$ obtained from a 3D conductivity $\sigma_{3\mathrm{D}}(\omega)=\sigma_{2\mathrm{D}}(\omega)/d$ and treated as a standard three-layer Fresnel problem. For $d\ll\lambda$ and this matching of conductances, the two descriptions are analytically equivalent to leading order, so the corresponding $R(\omega)$ and $A(\omega)$ curves lie almost on top of each other. Figure \ref{['fig:F3']} therefore serves as a numerical consistency test between the sheet and thin-film implementations, using the parameter set in Table \ref{['tab:allparams']}.
• Figure 4: Skin depth $\delta(\omega,T)$ as a function of frequency for two temperatures, computed from a phonon-limited Drude test model. At fixed temperature, $\delta(\omega,T)$ decreases approximately as $\omega^{-1/2}$; at fixed frequency, the sample with larger DC conductivity has a smaller skin depth. These trends are the expected behaviour of electromagnetic waves in good conductors and provide an RF–microwave sanity check for the conventions used in our $\sigma(\omega,T)$ and $\varepsilon(\omega)$ modules.