Structural Chirality and Natural Optical Activity across the $α$-to-$β$ Phase Transition in SiO$_2$ and AlPO$_4$ from first-principles

Natural optical activity (NOA), the ability of a material to rotate the plane of polarized light, has traditionally been associated with structural chirality. However, this relationship has often been oversimplified, leading to conceptual misunderstandings, particularly when attempts are made to directly correlate structural handedness with optical rotatory power. In reality, the relationship between chirality and NOA is more nuanced: optical activity can arise in both chiral and achiral crystal structures, and the sign of the rotation cannot necessarily be inferred from the handedness of the space group. % In this work, we conduct a first-principles investigation of natural optical activity in SiO$_2$ and AlPO$_4$ crystals, focusing on their enantiomorphic structural phase transition from high-symmetry hexagonal ($P6_422$ or $P6_222$) to low-symmetry trigonal ($P3_121$ or $P3_221$) space groups. This transition, driven by the condensation of a zone-center $Γ_3$ phonon mode, reverses the screw axis type given by the space group symbol while leaving the sign of the optical activity unchanged. By following the evolution of the structure and the optical response along the transition pathway, we clarify the microscopic origin of this behavior. We demonstrate that the sense of optical rotation is determined not by the nominal helicity of the screw axis given in the space group symbol, but by the atomic-scale helicity of the most polarizable atoms of the structure.

💡 Research Summary

This paper presents a comprehensive first‑principles investigation of natural optical activity (NOA) in two prototypical chiral oxides—silica (SiO₂, quartz) and aluminum phosphate (AlPO₄, berlinite)—focusing on the α‑to‑β phase transition that reverses the screw‑axis handedness of the crystal’s space group. Using density‑functional theory (DFT) with the PBEsol functional and density‑functional perturbation theory (DFPT) as implemented in ABINIT, the authors first relax the high‑symmetry β‑phase structures (P6₄2₂ for SiO₂ and P6₄2₂ for AlPO₄) and then identify an unstable Γ₃ zone‑center phonon mode in each material. The imaginary frequencies (≈ 61 cm⁻¹ for SiO₂ and ≈ 63 cm⁻¹ for AlPO₄) indicate a displacive instability that drives the transition to the low‑symmetry α‑phase (P3₁2₁ for SiO₂, P3₁2₁ for AlPO₄). Condensation of the Γ₃ eigenvector lowers the total energy by ~‑60 meV per 9‑atom SiO₂ cell and ~‑155 meV per 18‑atom AlPO₄ cell, and it transforms the 6₄ (or 6₂) screw axis into a 3₁ (or 3₂) screw axis while preserving the overall chiral network of corner‑sharing tetrahedra.

To connect structural changes with optical activity, the authors compute the zero‑frequency limit of the rotatory power tensor η_xyz using a recent linear‑response DFPT approach (Zabalo & Stengel). The quantity ρ̄(0) = η_xyz/(ℏc)², which is proportional to the optical rotation per unit length, is evaluated for light propagating along the crystallographic c‑axis. The calculations reveal that both the β‑ and α‑phases of SiO₂ possess a positive η_xyz, meaning that light traveling along +z rotates clockwise (positive rotatory power). In contrast, both phases of AlPO₄ exhibit a negative η_xyz, indicating counter‑clockwise rotation despite sharing the same enantiomorphic space‑group symbols. These results are consistent with experimental data for quartz and with earlier theoretical work on berlinite.

A key insight emerges from a series of “atom‑swap” computational experiments: when Si atoms are placed on the Al/P sites of the AlPO₄ structure (and vice‑versa), the sign of η_xyz reverses. This demonstrates that the sign of NOA is governed not by the nominal handedness of the screw axis encoded in the space‑group symbol, but by the helicity of the most polarizable sublattice—essentially the arrangement of the atoms that contribute most strongly to the electronic polarizability (Si in quartz, Al/P in berlinite). Consequently, the reversal of the screw‑axis handedness across the α‑to‑β transition does not alter the sign of the optical rotation; the underlying atomic‑scale helicity remains unchanged throughout the distortion path.

The evolution of ρ̄ as a function of the normalized Γ₃ distortion amplitude ξ is plotted for both materials. As ξ varies from 0 (pure β‑phase) to 1 (pure α‑phase), ρ̄ changes modestly (≈ 10 % variation) but retains its sign, confirming that the transition is essentially a continuous rotation of rigid SiO₄ (or AlO₄/PO₄) tetrahedra without flipping the microscopic helicity that determines NOA.

Overall, the study delivers several important conclusions: (i) structural chirality is a sufficient but not necessary condition for NOA; achiral crystals can also exhibit optical activity. (ii) The handedness of a space‑group screw axis cannot be used as a reliable predictor of the sign of optical rotation. (iii) First‑principles DFPT‑based linear‑response methods are capable of accurately capturing NOA in complex inorganic solids. By elucidating the microscopic origin of optical activity in SiO₂ and AlPO₄, the work clarifies a long‑standing conceptual confusion and provides a robust framework for interpreting and predicting NOA in other chiral and even achiral materials.