\textit{Ab initio} \textit{GW}-BSE theory of optical activity in $α$-quartz

Abstract: We present an ab initio many-body theory of optical activity in solids within the GW-BSE framework. Dielectric spatial dispersion is formulated in two complementary forms: exciton envelope modulation and sum-over-exciton-states expansion. Our application to $α$-quartz reveals that the envelope-modulated formulation captures the low-frequency region, whereas the sum-over-exciton-states formulation is essential to reproduce the correct full frequency dependence. Comparisons with the independent-particle approximation and simple local-field corrections further highlight the decisive role of excitonic many-body effects in shaping the spectral dispersion of optical activity in solids.

• The paper introduces an ab initio GW-BSE-SOXS framework that overcomes limitations of IPA and empirical methods in predicting α-quartz optical activity.
• Utilizing high-precision DFT, GW, and BSE calculations, the study achieves quantitative agreement with experiments, including a static rotation of 5.1 deg/[mm (eV)²].
• The work emphasizes that excitonic many-body effects are crucial for accurately modeling the frequency-dependent optical rotatory dispersion in chiral materials.

Ab Initio GW-BSE Theory of Optical Activity in $\alpha$-Quartz

Introduction and Context

The paper presents a comprehensive ab initio many-body theoretical formulation for optical activity in crystalline solids, specifically targeting $\alpha$-quartz—a benchmark chiral material. Historically, despite the long-standing experimental interest and diagnostic applications of optical activity in $\alpha$-quartz, rigorous first-principles predictions of its full frequency-dependent optical activity have been lacking. Previous theoretical approaches based on independent-particle approximations (IPA), ad hoc scissors corrections, and empirical local-field corrections (LFCs) have proved qualitatively and quantitatively insufficient, particularly in predicting optical rotatory dispersion across the frequency spectrum. Existing ab initio methods for molecules cannot be trivially extended to solids due to the subtleties associated with the position operator in periodic boundary conditions.

Theoretical Framework

Optical activity in solids is fundamentally described via the spatial dispersion of the dielectric response tensor $\epsilon_{ij}(\omega, \mathbf{q})$, whose first-order expansion in wavevector encodes the optical rotation and circular dichroism. The work extends previous electronic-structure-based approaches by formulating the dielectric spatial dispersion within the Bethe–Salpeter equation (BSE) framework, built atop many-body $GW$ quasiparticle corrections.

Two complementary approaches for incorporating excitonic effects into the theory of optical activity are introduced:

1. Exciton envelope modulation, wherein the $\mathbf{q}$-dependence of the exciton envelope is considered, paralleling the molecular multipole expansion.
2. Sum-over-exciton-states (SOXS) expansion, which inserts a complete basis of excitonic states, yielding a closed-form optical activity tensor fully self-consistent with the underlying $GW$-BSE Hamiltonian.

The formalism clarifies that, for 3D bulk crystals, the group velocity contribution $$\partial_\mathbf{q} \omega_\lambda(\mathbf{q})|_{\mathbf{q}=0}$$ vanishes near the zone center, focusing exclusive attention on the $\mathbf{q}$-dependence of the oscillator strength. Transition moments and multipolar contributions (magnetic dipole, electric quadrupole) are systematically constructed from the antisymmetrization and symmetrization of corresponding tensors.

The theory navigates technical challenges associated with the velocity operator’s dependence on the underlying Hamiltonian. Both DFT-based and $GW$-corrected velocity matrix elements are computed, with an additional “optimal” scissors-shifted scheme to match excitonic gaps.

Numerical Results

The theoretical framework is implemented for $\alpha$0-quartz, employing high-precision DFT, $\alpha$1, and BSE calculations. The DFT and $\alpha$2 corrections yield a direct band gap at $\alpha$3 of 6.3 eV and 10.0 eV, respectively, while the BSE predicts an optical gap of 8.9 eV—fully consistent with established benchmarks.

The calculated static limit optical rotation demonstrates several critical observations:

• The IPA yields a sign inconsistent with experiment and severely underestimates the magnitude of optical rotation.
• Inclusion of LFCs brings the computed value into rough agreement with experiment, but the dependence on the chosen electronic band gap and crystal structure is pronounced.
• Within the $\alpha$4-BSE formalism, the SOXS formulation achieves quantitative agreement with experiment both in static and frequency-dependent regimes.
• The optimal scissors-shifted method achieves the best correspondence with observed values, notably yielding a static optical rotation of 5.1 deg/[mm (eV)$\alpha$5] vs. the experimental 4.6±0.1 deg/[mm (eV)$\alpha$6].

The frequency-resolved optical rotatory dispersion (ORD) calculated via the SOXS method accurately reproduces the experimentally measured spectral lineshape over a broad energy range. The envelope modulation approach, even with optimal velocity matrix elements, fails to capture the correct frequency dependence, underlining the necessity of the SOXS expansion for a consistent excitonic description.

The significance of these results lies in their demonstration that excitonic many-body effects, fully treated in the GW-BSE-SOXS scheme, are decisive not only for the magnitude but also for the frequency dispersion of optical activity in wide-gap chiral insulators.

Implications and Future Directions

The presented ab initio GW-BSE theory rectifies prior deficiencies of effective single-particle and semi-empirical treatments, offering a predictive route for optical activity in chiral crystals. This establishes the essential role of consistent many-body excitonic effects in chiroptical phenomena beyond the static limit, suggesting immediate applicability for rational chiroptoelectronic material design.

The formalism is rigorously general, accommodating further extensions to complex band topologies and nontrivial geometrical electronic structure, such as systems characterized by large Berry curvature, quantum metric, or topological phases, where intraband geometry-dependent corrections may become significant. Future work might incorporate explicit geometric-phase and band-structure singularity effects and extend the approach to low-dimensional or topologically nontrivial materials.

On a practical level, the theory can guide the ab initio identification, screening, and engineering of functional chiral materials for photonic, optoelectronic, and quantum applications, where control of circular dichroism, optical rotation, and other nonreciprocal optical responses is crucial.

Conclusion

This work establishes a predictive and internally consistent ab initio many-body framework for the optical activity of solids within the GW–BSE formalism. By addressing key theoretical and numerical issues—in particular, the formulation of the exciton oscillator strength and frequency-dependent ORD—the study resolves the persistent gap between ab initio calculation and experiment in $\alpha$7-quartz. The approach is generalizable, positioning it as a foundation for future research in chiral light–matter interactions in solids and the computational design of optically active materials.