[Hg_3Se_2]^2- cluster drives giant optical anisotropy and broad infrared transparency

Domain: physics
Published: 2026-01-06
Editorial outcome: major_revision

Abstract
Optical anisotropy, a fundamental physical property for polarization control, has long presented a critical consideration in the development of optical materials, particularly in terms of its modulation mechanisms and performance optimization. In the mid-to far-infrared region, simultaneously achieving large birefringence and broad transparency within a single material remains a major challenge. Herein, we report the synthesis of Hg18Ga8Se8Cl32 (HGSC), a crystalline material featuring linear [Hg3Se2] structural units. HGSC demonstrates a large birefringence of 0.871 at 546 nm, accompanied by the broadest transparency window among Hg-based chalcogenide single crystals (0.4 to 25 µm). Theoretical calculations reveal that the significant birefringence of HGSC originates from the well-aligned linear [Hg3Se2]2- clusters, which exhibit the highest polarizability anisotropy (δ = 430) among all reported birefringence-active functional units. The demonstration of [Hg3Se2]2- clusters as an effective bifunctional unit offers new opportunities for designing infrared photonic materials.

Introduction
Birefringent materials achieve polarization separation by exploiting the angular difference in total internal reflection between ordinary (o) and extraordinary (e) light, have been widely used as polarization modulators in optical imaging, remote sensing and laser communication systems1–5. Over the past few decades, a variety of birefringent materials have been developed, including commercially available crystals such as YVO4, TiO2, CaCO3, MgF2 and α-BaB2O4 6–10. However, the intrinsic absorption beyond 5 μm limits their applications in the mid- to far-infrared (IR) regions. In addition, polarization devices requiring high extinction ratios, excellent stability and compact integration demand materials with pronounced optical anisotropy11–14.

Despite these advances, achieving both broad IR transparency and large birefringence within a single material remains challenging, as these properties originate from fundamentally different physical mechanisms15. The IR optical transparency of a material is largely determined by its lattice vibrational characteristics, particularly phonon modes in the mid- to far-IR regions, whereas birefringence originates from anisotropy in its electronic structure and polarizability response16–19. Consequently, designing structural units that simultaneously suppress phonon-induced absorption and amplify electronic anisotropy represents a promising strategy for high-performance birefringent materials.

To achieve large birefringence, various strategies have been developed, such as introducing planar π-conjugated units20,21, linear units22–28 or elements containing stereo-electronic lone pairs (As3+, Sb3+, Sn2+)29,30. Mercury (Hg) is a distinctive element whose pronounced relativistic effects lead to significant contraction of the 6s orbital, thereby facilitating electronic rearrangements and enabling the formation of linear multinuclear metal clusters such as Hg22+ and Hg32+. Among these, the linear Hg22+ cluster has been experimentally demonstrated to serve as an effective structural unit capable of significantly enhancing optical anisotropy31,32. Our group revealed that the polarization anisotropy of linear units is markedly superior to that of conventional functional building blocks, with [Hg2I2] exhibiting the highest value among known birefringence-active clusters33. Kong et al. reported linear [Hg2O2] as a birefringence-active structural unit, successfully constructing a layered structure with significant optical anisotropy. This strategy effectively mitigated the mutual cancellation of anisotropic effects, resulting in the compounds Hg4(Te2O5)(SO4) and HgI2HgII(Te2O5)(HPO4) exhibiting notable birefringence of 0.542 and 0.444 at the wavelength of 546 nm, respectively34,35. Our group has revealed that the incorporation of well-aligned linear [Hg2S2]2- clusters significantly enhances the optical anisotropy of the Hg4InS2Cl5 compound, achieving the largest experimental birefringence of 0.35 for Hg-based thiohalides at a wavelength of 546 nm36.

In addition to their ability to form highly anisotropic linear units, Hg-based multinuclear clusters also provide a promising strategy for enhancing IR transparency. As a heavy metal element, Hg forms relatively soft metal-ligand bonds with lower stiffness compared to typical transition metals, resulting in reduced optical phonon energies and consequently suppressing high-frequency IR absorption. To circumvent the near-IR absorption typically associated with oxygenated acid salts, chalcogenides were selected in this study as the preferred anionic framework for structural design. A survey of the ICSD [Version 5.4.0 (build 20250403-0947)] database reveals that, to the best of our knowledge, there are no existing reports on the synthesis or properties of the [Hg3Se2] unit (Q = S or Se). The rational design and synthesis of multinuclear clusters with Hg32+ represents a promising direction for further exploration.

In this paper, a Hg-based thiohalide compound, Hg18Ga8Se8Cl32 (HGSC), was successfully synthesized, which contains linear [Hg3Se2] units. Millimeter-sized single crystals were grown using the chemical vapor transport method (CVT). As a structural unit with linear geometry, the [Hg3Se2]2- anionic multinuclear cluster exhibits the highest polarization anisotropy (δ = 430) to date, thereby producing significant birefringence gain. However, birefringence measurements in the IR region typically rely on large, optically homogeneous crystals, which are difficult to obtain at the early stage of material exploration. To address this, we proposed and validated an alternative approach that enables reliable evaluation of birefringence in small-sized crystals. The HGSC crystal demonstrates large birefringence of 0.871 at 546 nm and approximately 0.4 in the mid-IR region. Theoretical calculations indicate that the pronounced optical anisotropy in HGSC originates from the well-aligned linear [Hg3Se2] units. Furthermore, HGSC also demonstrates a broad IR transparency range from 0.4 to 25 µm, which is the widest reported among Hg-based chalcogenide single crystals. Benefiting from its linear coordination and multinuclear cluster configuration, the [Hg3Se2] unit also leads to additional physical behaviors such as reversible thermochromism. This work reveals the structure-property relationship of the linear [Hg3Se2] unit while addressing the intrinsic challenges of birefringence characterization in millimeter-sized crystals in the mid- to far-IR region.

Results
The crystal structure of HGSC was analyzed and refined using single-crystal X-ray diffraction (SXRD) data. HGSC crystallizes in the monoclinic space group C2/c (No. 15). In the asymmetric unit, there are five independent crystallographic Hg atoms, two Ga atoms, two Se atoms, and eight Cl atoms (Supplementary Tables 1–5), and the bond valence method was employed to determine the oxidation states of the various metal elements (Supplementary Table 4). The Hg atoms exhibit three distinct oxidation states: Hg0, Hg+, and Hg2+, with a total composition corresponding to Hg0 HgI2 HgII16 Se8 (GaCl4)8. The crystal structure of HGSC consists of three distinct linear units that are interconnected through shared Se atoms, forming a pseudo-tridymite-like topology. This topology preserves the open-framework architecture characteristic of tridymite-like structures, while exhibiting more diverse channel morphologies and a framework constructed from linear chain units rather than conventional tetrahedral building blocks (Supplementary Fig. 1). The Hg-Se bond length in [Hg3Se2] is 2.610 Å, which is longer than the average values in [Hg2Se2] (2.558 Å) and [HgSe2] (2.448 Å). This increase in bond length can be attributed to the larger induced dipole moments of the multinuclear clusters [Hg3Se2]2- and [Hg2Se2]2- compared to the [HgSe2] unit. The Hg-Hg-Hg bond angle is 174.57°, consistent with the known bond angle of Hg32+ cluster. Both the [Hg3Se2] and [HgSe2] units adopt geometries that closely resemble those found in previously reported Hg-based thiohalides (Fig. 1a). Through bridging Se atoms, [Hg3Se2] and [HgSe2] assemble into one-dimensional helical channels extending along the b-axis (Fig. 1b). Meanwhile, vertex-sharing between [Hg3Se2] and [HgSe2] units gives rise to chair-like Hg8Se6 conformations (Fig. 1c). These helical chains and chair-like units alternate in an antiparallel stacking sequence along the c-axis, generating two distinct channels of different sizes. The larger channels are defined by [Hg3Se2], while the smaller ones are enclosed by [Hg2Se2]2- clusters (Fig. 1d). Relatively weak electrostatic interactions within the multinuclear cluster structure stabilize [GaCl4] within the cavities of the framework, and the entire three-dimensional framework is formed through axial rotational symmetry around the central Hg atoms (Fig. 1e). The coordination geometry around Ga atoms is nearly ideal tetrahedral, as indicated by Cl-Ga-Cl angles of 104.7° to 114.2° and Ga-Cl bond lengths consistent with those previously reported. Furthermore, owing to its complex structure, the information complexity parameters, including information content (IG), total information content (IG, total), maximum structural information content (IG, max), and information density (ρinf.), are 5.569 bits/atom, 6014.520 bits/unit-cell, 7.044 bits/atom, and 1.678 bits/Å3, respectively37. The information complexity of HGSC is greater than that of most inorganic chalcogenides and may give rise to interesting optical, electrical, and thermal properties.

High-purity HGSC polycrystalline powder samples were synthesized via a stoichiometric solid-state reaction method. The powder X-ray diffraction (PXRD) pattern of the sample matches well with the theoretical one (Supplementary Fig. 2). Single crystals of HGSC were grown using the chemical vapor transport (CVT) method in sealed quartz tubes at 950 °C. Figure 2a presents a schematic representation of crystal growth via CVT. During the growth process, Hg2Cl2 served dual functions as both the transport agent and reagent. Under high-temperature conditions, Hg(I) species in Hg2Cl2 readily undergo disproportionation into Hg(II) and Hg(0). Meanwhile, the excess Hg source maintains a Hg-rich environment, thereby stabilizing the system and preventing decomposition of the desired phase. Millimeter-sized single crystals were obtained, including the (001) and (010) planes (Supplementary Fig. 3). The chemical composition of the obtained single crystals was verified by energy-dispersive X-ray spectroscopy (EDS), which confirms the presence of Hg, Ga, Se, and Cl (Supplementary Fig. 4). Inductively coupled plasma (ICP) quantitative analysis confirms the molar ratio of Hg: Ga: Se as 18.19: 7.82: 8.27 (Supplementary Table 6), consistent with the results from single-crystal refinement. To further investigate the effect of this chemical composition on the stability of HGSC crystals, experimental studies were conducted. Under atmospheric conditions (temperature: 25 ± 2 °C, relative humidity: 20 ± 5%), HGSC crystals were able to maintain a stable state for three days, after which significant hygroscopic degradation occurred.

The optical band gap of HGSC was determined to be 2.559 eV using UV-Vis-NIR diffuse reflectance spectroscopy of polycrystalline powder (Fig. 2b). A steep absorption edge at 423 nm further indicates efficient light absorption in the UV region. The photovoltaic conversion ability of HGSC crystals was evaluated. Two copper wires (diameter: 50 μm) were symmetrically fixed on the surface of the HGSC single crystal using highly conductive silver paste to construct a dual-electrode measurement configuration. The sample was illuminated by a monochromatic light source of 350 nm (incident power density: 7 mW/cm2), and the photocurrent-time (I-t) response was recorded using a high-precision digital source meter (model: Keithley 2450, Supplementary Fig. 5). This result suggests that HGSC crystals can effectively generate, separate, and transport photogenerated carriers under UV irradiation, thus providing experimental evidence for their potential application in UV photodetection.

The transmission spectrum of (001)-plane HGSC single crystals reveals an IR transparency extending up to 25 μm (Fig. 2c), covering a broad spectral range that includes the critical atmospheric windows at 3–5 μm and 8–14 μm. This wide IR transparency originates from the electronic configuration and bonding characteristics of the material. First, the fully filled 5d orbitals of Hg effectively prevent d-d electronic transitions, thereby avoiding absorption in the visible regions38. Second, the valence states of Hg in the [Hg3Se2] and [Hg2Se2] units, typically assigned as 0 or +1, introduce a high proportion of heavy atoms into the lattice, which effectively suppresses multiphonon absorption in the near-IR region. X-ray photoelectron spectroscopy (XPS) was employed to verify the presence of multiple valence states of Hg in HGSC. The high-resolution spectra revealed six different components, located at 105.3, 104.8, and 103.8 eV for the Hg 4f 5/2 levels, and at 101.2, 100.7, and 99.7 eV for the Hg 4f 7/2 levels. These peaks correspond to the oxidation states of Hg0, Hg+, and Hg2+ (Fig. 2d)39–41. Furthermore, the wide-range elemental composition of the sample was determined through XPS survey scans (Supplementary Fig. 6). The photoluminescent (PL) properties of HGSC was further investigated under excitation wavelengths ranging from 300 to 485 nm. The PL spectra recorded at room temperature (298 K) exhibited a prominent emission peak at ~332 nm and a broad green emission band centered at 501 nm (Supplementary Fig. 7). This band spans nearly the entire visible spectrum, highlighting its significant potential for applications in spectral lighting and advanced optoelectronic devices42.

In addition to its unusual valence states, the HGSC compound exhibits reversible thermochromic behavior (Fig. 2e). Upon cooling from 300 K to 77 K, the crystal undergoes a reversible color change from pale-yellow to colorless, which remanied stable over more than 100 thermal cycles (Supplementary Fig. 8), demonstrating the robustness and repeatability of the thermochromic behavior. To explore the origin of this behavior, temperature-dependent single-crystal XRD measurements were conducted from 80 K to 300 K in 15 K increments (Fig. 2f). The results confirm that HGSC retains its crystallographic symmetry throughout this temperature range without undergoing any structural phase transition (Supplementary Tables 7–11). Furthermore, the lattice constants display continuous and anisotropic thermal expansion (αa = 3.177 × 10−5, αb = 2.383 × 10−5, αc = 1.955 × 10−5), suggesting enhanced atomic vibrational amplitudes along specific crystallographic directions due to the ordered alignment of linear building units within the crystal lattice. Among the various mechanisms potentially contributing to thermochromic behavior, phonon dynamics, particularly phonon anharmonicity, can induce band structure renormalization and thereby modulating the material’s optical properties. This occurs as thermally activated lattice vibrations perturb the electronic states, especially near the conduction and valence band edges, resulting in band edge shifts and spectral broadening43. This mechanism is especially relevant in compounds containing heavy metal elements, where the intricate phonon spectrum and strong mass contrast often give rise to enhanced lattice anharmonicity. In electron-phonon coupling, only vibrational modes that induce changes in transition polarizability contribute significantly to the optical response, and these modes can be identified via Raman spectroscopy44. Since HGSC crystals belong to the C2h point group symmetry, the irreducible representations of their phonon modes are given in Eq. (1).

Γ = 49Ag + 48Au + 50Bg + 48Bu (1)

Among these, the Ag and Bg vibrational modes are Raman-active. In HGSC crystals, nine distinct Raman peaks were observed (Fig. 2g). Based on the simulated vibrational eigenmodes, the Ag peaks at 137.10 cm−1 and 215.89 cm−1 correspond to lattice motions primarily influenced by the [Hg2Se2] unit, whereas the Ag mode at 110.70 cm−1 involves substantial contributions from the [Hg3Se2] units (Supplementary Fig. 9a). It is important to emphasize that these assignments do not imply localized molecular-like vibrational modes, but instead reflect collective lattice dynamics predominantly governed by the bonding environments within the Hg-based structural units. In situ variable-temperature Raman spectroscopy was performed under 633 nm laser excitation (Fig. 2h) revealing several phonon modes (ωA, ωB, ωC) with pronounced temperature-sensitive frequencies and intensities, in contrast to temperature-invariant modes (ωD, ωE). To isolate the intrinsic anharmonic contributions to phonon frequency shifts, phonon mode calculations at 143 K were conducted for HGSC, thereby eliminating the confounding effects of thermal expansion (Supplementary Fig. 9b). Moreover, phonon mode calculations indicate that the temperature-sensitive modes originate from the cooperative lattice vibrational modes dominated by the Hg32+ and Hg22+ clusters (Supplementary Fig. 10). This directly links the thermochromic effect to dynamic distortions within the Hg multinuclear clusters, suggesting that the reversible thermochromic behavior observed in HGSC arises from an electron-phonon coupling-dominated mechanism, in accordance with previous reports on InSeI43.

To investigate the optical anisotropy of HGSC crystal, detailed observations were conducted using a ZEISS AxioScope 5 polarizing microscope. Figure 3a presents a schematic diagram of the polarizing microscope setup. When linearly polarized light is incident on the HGSC crystal, its polarization state becomes elliptical polarization due to its birefringence. This birefringence manifests as periodic variations in brightness under cross-polarized conditions as the sample is rotated (Fig. 3b). At a polarization angle of 0°, the crystal reaches an extinction state. As the angle increases, the transmitted light intensity rises, reaching a maximum at 45°; continued rotation of the crystal produces periodic changes in interference colors with a period of 45°. The pale-yellow color of HGSC at room temperature hinders accurate identification of interference fringes. Fortunately, HGSC exhibits pronounced thermochromic properties: as the temperature drops below the critical point (140 K), the crystals become colorless and transparent. This phenomenon effectively eliminates chromatic interference and improves the resolution of interference color fringes. The birefringence of HGSC crystal sample (thickness: 14.030 μm) was characterized at 140 K (Fig. 3c). Using a compensator with both forward and reverse rotations, the optical path difference was measured to be 11756.14 nm at the characteristic wavelength of 546.1 nm (Fig. 3d, e). According to the formula R = Δn × d, the birefringence at 546.1 nm was calculated to be 0.837. The measurement was repeated at 90 K, yielding an increased optical path difference of 12224.52 nm, corresponding to a birefringence of 0.871 (Supplementary Fig. 11). These results indicate that the birefringence of HGSC crystals increases with decreasing temperature.

As an IR birefringent material, further characterization of the long-wavelength properties of HGSC is necessary. Nevertheless, such characterization is challenging because newly synthesized crystals are usually limited to millimeter-scale dimensions, which restricts the use of conventional mid-IR birefringence measurements that typically require large and optically homogeneous samples. To address this limitation, we experimentally determined the birefringence of HGSC in the IR region through analysis of the polarization modulation induced by the phase delay between two orthogonal polarization components. In our experiment, b(x)-plane crystals were employed to measure nz - ny (Supplementary Fig. 12). Given that ny and nx are very close (difference of <0.05), the measured nz - ny can be reasonably taken as an approximation of the birefringence nz - nx of the crystal (i.e., Δn = nz - nx ≈ nz - ny). The schematic orientation of the HGSC crystal with respect to the laboratory coordinates is shown in Supplementary Fig. 13. Linearly polarized lasers with three typical near-IR wavelengths (1064 nm, 1540 nm, and 3500 nm) were used for the birefringence measurement, enabling a systematic investigation of the polarization evolution during propagation through the as-grown HGSC samples (Fig. 4a). Two HGSC samples with thicknesses of 12.9 μm and 14.0 μm were used for the measurements. The input beam with horizontal polarization was focused and normally incident into the samples. Upon transmission through the crystal, the polarization state of the laser was modulated (Fig. 4b–d), the crystal was functionally equivalent to a retarder. The experimental results reveal that HGSC exhibits large birefringence of 0.517, 0.468, and 0.453 at wavelengths of 1064 nm, 1540 nm, and 3500 nm, respectively (Supplementary Figs. 14, 15 and Supplementary Tables 12, 13). A comprehensive survey of reported birefringence values for Hg-based thiohalides and chalcogenides indicates that HGSC ranks among the highest in terms of experimentally measured birefringence (Supplementary Tables 14, 15). These findings verify that HGSC crystal exhibits large and spectrally stable birefringence over a broad IR range, thereby providing critical optical parameters for the design and engineering of polarization-dependent photonic devices. The laser damage behavior of HGSC single crystals was investigated using a 1064 nm picosecond-pulse laser. As shown in Supplementary Fig. 16, no detectable damage was observed in the HGSC crystal when the laser output energy ranged from 0.4 to 2.4 mJ. However, when the energy is increased to 2.8 mJ, microcracks begin to appear at the edges of the crystal. Further increasing the energy to 4.0 mJ results in noticeable cracks within the crystal. Experimental results based on picosecond laser irradiation indicate that HGSC crystals exhibit excellent resistance to laser damage, exceeding that of AgGaS2 by a factor of 3.5.

To elucidate the structural-property relationship of HGSC, the electronic structure was investigated through first-principles calculations. As shown in Fig. 5a, the valence band maximum (VBM) and conduction band minimum (CBM) of HGSC are located at different high-symmetry points, indicating that HGSC is an indirect band gap semiconductor. Owing to the well-known underestimation of bandgaps in Density Functional Theory (DFT) calculations caused by exchange-correlation energy discontinuities, the calculated bandgap is 2.110 eV, which is lower than the experimental value of 2.559 eV. Figure 5b illustrates the total density of states (TDOS) and partial density of states (PDOS) of HGSC. The Hg 5d orbitals are primarily localized in the energy range of −8 and −6 eV, whereas the valence band region near the Fermi level (−5 to 0 eV) is predominantly occupied by Hg 6s/6p, Ga 4p, Se 4p, and Cl 3p orbitals. The conduction band region, spanning from the Fermi energy level to 4.0 eV, is mainly occupied by Hg 6s/6p and Se 4s/4p orbitals, with a minor contribution from Cl 3s orbitals. Therefore, the optical properties of HGSC are primarily determined by the Hg 6s/6p, Se 4p, and Cl 3p orbitals. Based on the formula ε(ω) = ε1(ω) + iε2(ω) and n2(ω) = ε(ω), the calculated birefringence is 0.808 at 546 nm and 0.443 at 1064 nm (Fig. 5c). Given that birefringence depends on both the intrinsic properties and spatial arrangement of functional units45, the vertically aligned linear clusters in HGSC effectively reduce the cancellation of anisotropic polarizability, resulting in enhanced birefringence. To better elucidate the origin of birefringence, the real-space atom-cutting method was employed to quantitatively assess the contributions of individual linear building units to the optical anisotropy. As depicted in Fig. 5d, the contribution of [Hg2Se2] unit to birefringence is approximately 23.8%, while the contribution of the [Hg3Se2] unit is about 65.9%. As shown in Supplementary Fig. 17, the direction of the largest refractive index in the HGSC crystal aligns with the orientation of the linear [Hg3Se2] units, confirming that the significant optical anisotropy arises from their highly oriented arrangement. At the microscopic level, the polarization anisotropy of all linear units in the Hg-based chalcogenides was calculated using Gaussian0946. To make a more comprehensive comparison, several representative linear birefringence-active units were also evaluated (Fig. 5e). Among them, the [Hg3Se2] unit exhibit the highest polarization anisotropy (δ = 430) to date. Moreover, the Electronic Localization Function (ELF) of HGSC was visualized through first-principles calculations (Supplementary Fig. 18). In linearly coordinated units, the charge distribution around atoms is highly asymmetric, and the distortion of the electron clouds enhances electron localization, thereby amplifying electron displacement polarization. As a result, the [Hg3Se2]2- cluster emerges as one of the most promising functional units for achieving large birefringence. To fully elucidate the potential of HGSC as a birefringent material in the IR region, we compared the transparency ranges of previously reported Hg-based chalcogenide single crystals (Fig. 6a, Supplementary Table 16). HGSC exhibits a broad transparency range covering the atmospheric windows in the mid- to far-IR, representing the widest range reported among all Hg-based chalcogenide single crystals to date. In addition, Fig. 6b compares the birefringence of HGSC with that of representative commercial materials and state-of-the-art birefringent crystals reported in the literature. HGSC exhibits large birefringence (0.453 @ 3.5 μm), nearly twice that of TiO2 (0.236 @ 3.5 μm), positioning it at the forefront of IR birefringent materials. These findings highlight the potential of HGSC for advanced optical applications in the IR region.

Methods
Synthesis and crystal growth
The reagents including Hg2Cl2 (Aldrich, 99.9%), HgCl2 (Aexchem, 99.99%), HgSe (Aldrich, 99.99%), Ga2Se3 (Aldrich, 99.999%) are analytical grade and obtained from commercial sources without further purification. (Notice: Due to the toxicity of Hg2Cl2, HgCl2, and HgSe, all handling processes should be conducted in a glove box to ensure operational safety.) The millimeter-sized HGSC single crystals were grown using the chemical vapor transport method (CVT). A mixture of HgCl2 (0.1936 g, 3 mmol), Hg2Cl2 (0.5610 g, 5 mmol), HgSe (0.0664 g, 1 mmol), and Ga2Se3 (0.1789 g, 2 mmol) was prepared. A Hg-rich environment was created within a sealed tube by employing an excess of Hg2Cl2. The raw material mixture was placed into a quartz tube, which was sealed under high vacuum (~10−3 Pa) using an oxygen-methane torch. The sealed tube was then transferred to a computer-controlled furnace, and the temperature distribution was as follows: The furnace temperature was gradually increased to 400 °C within 15 h. It was then held at 400 °C for 10 h, followed by an increase to 950 °C within 48 h, and kept at this temperature for 10 days. Subsequently, the temperature was reduced to 300 °C at a rate of 1 °C/h, and finally gradually cooled down to room temperature within 24 h. Through the above process, a pale yellow pseudo-tridymite-like topological structure can be obtained. Hg in Hg2Cl2 is in the +1 oxidation state and readily undergoes disproportionation, rendering Hg2Cl2 a potential reductant and a reservoir of Hg species under CVT conditions. By contrast, Hg in HgCl2 is in the +2 oxidation state, where Hg(II) acts as a soft Lewis acid that preferentially coordinates with selenide, thereby facilitating Hg-Se bond formation. Under high-temperature conditions, Hg(I) undergoes a disproportionation reaction, converting into Hg(II) and Hg(0).

Characterization
High-optical quality crystal (HGSC, 0.091 × 0.068 × 0.092 mm3) was selected under a polarizing microscope and used for single-crystal X-ray data collection. Single-crystal XRD data were collected using a Bruker D8 VENTURE single-crystal X-ray diffractometer with Mo Kα radiation (λ = 0.71073 Å) at room temperature (298 K) and low temperature (143 K). Data integration, cell refinement, and absorption corrections were performed using the program SADABS47. The structure was solved by direct methods and refined on F2 by full-matrix least-squares techniques using the program suite SHELXTL48. Solutions were checked for missed symmetry using PLATON49. The crystal data yielded Rint = 0.0605, and the structure solution parameters are R1 = 0.0462, wR2 = 0.1114, GOF = 1.095. Powder XRD data were collected with a Bruker D2 PHASER diffractometer (Cu Kα radiation with λ = 1.5418 Å, 2θ = 5 to 70°, scan step width = 0.02°, and counting time = 1 s/step). IR transmittance spectrum was recorded on a Shimadzu IR Affinity-1 Fourier transform IR spectrometer in the range 400–4000 cm−1. X-ray photoelectron spectroscopy of HGSC powder samples was collected on a Thermo Scientific ESCALAB 250Xi spectrometer equipped with a monochromatic Al Kα X-ray source. Samples were analyzed under high vacuum (pressure < 10−9 mbar). Survey scans and high-resolution scans were collected, with binding energies of all scans were corrected according to the C1s binding energy at 284.8 eV. The optical diffuse-reflectance spectra of HGSC were recorded at room temperature on a Shimadzu SolidSpec-3700DUV spectrophotometer. The measured wavelength range was 200–2600 nm. To figure out the experimental band gap, the diffuse reflection data were converted into absorption data by the Kubelka-Munk function F(R) = K/S = (1−R)2/2R, where R represents the reflection coefficient, K is the absorption value, and S is the scattering coefficient. To verify the presence of Hg32+ and Hg22+, selected HGSC crystals were placed on a glass slide and irradiated with a 633 nm laser. Raman spectra were recorded in the range of 40–1000 cm−1 using an iHR550 Raman spectrometer equipped with a 633 nm He-Ne laser (5 mW), and the temperature during measurements was controlled using a Linkam THMS600 hot-cold stage system. To determine the chemical compositions, the EDS spectra and mappings of the compounds were tested on a field emission scanning electron microscope (FE-SEM, JEOL JSM-7610F Plus, Japan) equipped with an energy-dispersive X-ray spectrometer (Oxford, X-Max 50). Under standard testing conditions, the photovoltaic conversion ability of HGSC crystals was evaluated. Two copper wires (diameter: 50 μm) were symmetrically fixed on the surface of the HGSC single crystal using highly conductive silver paste to construct a dual-electrode measurement configuration. The sample was illuminated by a monochromatic light source of 350 nm (incident power density: 7 mW/cm2), and the photocurrent-time (I-t) response was recorded using a high-precision digital source meter (Keithley 2450). The birefringence of HGSC was measured using a polarizing microscope (ZEISS AxioScope.5 pol) equipped with a Berek compensator under a light source with an average wavelength of 546.1 nm. The interference color during the measurement was classified as first-, second-, or higher-order to ensure measurement accuracy, as their boundary lines were relatively clear compared to subsequent interference colors. Crystals with high optical quality were selected for scanning under the polarizing microscope. The LIDT of HGSC single crystals was measured using a picosecond-pulsed laser (1064 nm, 29 ps). The input laser intensities were raised from 0.4 mJ until damage occurred, and the laser energy was recorded.

Computational methods
The electronic structures, density of states, and birefringence of the title compounds were evaluated using the plane-wave pseudopotential method implemented in the CASTEP software based on DFT50. HGSC was optimized by the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional of generalized gradient approximation (GGA)51, while the following orbital electrons were treated as valence electrons: Hg 6s2 5d10, Ga 4s2 4p1, Se 4s2 4p4, and Cl 3s2 3p5, respectively. The plane-wave cutoff energy for HGSC was set to 800 eV during the calculation, and Monkhorst-Pack k-point meshes (2 × 3 × 1) with a density of less than 0.04 Å−1 in the Brillouin zone were adopted. The default values of the CASTEP code were used for the other calculation parameters and convergence criteria52. A scissor operator correction of 0.449 eV was applied in the Raman spectra calculations; the plane-wave cutoff energy was set to 800 eV; and a (2 × 2 × 1) Monkhorst-Pack k-point mesh was employed to ensure adequate convergence of the phonon frequencies and electronic structure. To calculate the birefringence, the complex dielectric function, ε(ω) = ε1(ω) + iε2(ω) was computed from the PBE wave functions. The imaginary component of the dielectric function ε2 can be estimated from the electronic structures, and its real part is determined using the Kramers-Kronig transform, which is employed to derive refractive indices and ultimately obtain the birefringence Δn; the optical properties were scissor-corrected. The anisotropy of the polarizability (Δα) was calculated using the Gaussian 09 revision D.01 program. The GaussView 5 software was used to visualize the output file53.

Δα = [1/2 ((αxx − αyy)2 + (αxx − αzz)2 + (αyy − αzz)2 + 6(αxy2 + αxz2 + αyz2))]1/2

Where αij (i, j = x, y, z) represents an independent element of the polarization tensor.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, the authors report a birefringent material with a novel Hg2Se3 cluster, which could be published in Nature Communications. However, several important issues must be addressed before being considered for acceptance：\n(1) The authors claim that this represents the first reported compound featuring Hg3Se2 cluster within a selenide framework. It is recommended that the authors further discuss the potential unique physical properties arising from the incorporation of Hg32+ into the selenide to enhance understanding of this material's characteristics. Additionally, a more comprehensive discussion on the unique features of this system, especially the influence of weak interactions on its structure and potential strategies for designing new frameworks, would greatly enhance the scientific impact of this work.\n(2) I note that the similar Hg polynuclear cluster compounds reported in References 43 and 44 exhibit poor stability under ambient conditions. Given the potential application of the reported material in mid-infrared lasers, it is important for the authors to experimentally verify whether HGSC suffers from similar stability issues.\n(3) Figure 5b presents a favorable comparison between HGSC and commercial birefringent materials. However, the comparison includes typical infrared nonlinear optical crystals such as AgGaS2, which are generally not employed in polarization modulation applications. To more convincingly highlight the advantages of HGSC in terms of birefringence, the authors are encouraged to include comparisons with recently reported advanced birefringent materials in the mid- to far-infrared region.\n(4) As a relatively rare structural motif, the synthetic chemistry underlying the formation of polynuclear Hg3 clusters warrants a more systematic discussion. A deeper exploration of this topic would enhance the completeness of the study and strengthen its relevance to the broader field of materials chemistry.\n(5) The authors employed Jones matrix analysis to verify the birefringence of the HGSC compound at 1064 nm. While χ and ψ are identified as key parameters characterizing the polarization state, the specific numerical values are not provided. It would be valuable to include these quantitative data as supplementary information. In addition, the authors suggest that HGSC holds potential as a birefringent material for mid- to far-infrared applications. To substantiate this claim, it is recommended that the authors provide birefringence data at longer wavelengths beyond the currently reported 1064 nm.", "reviewer_2": "Designing optical materials with large birefringence has long been a challenging pursuit for material scientists. In this manuscript, Prof. Pan and colleagues have developed an intriguing optical material, Hg9Ga4Se4Cl16, which exhibits both large birefringence and a broad transparency range. This compound demonstrates a birefringence of 0.808 at 546 nm and a wide transmission window spanning from 0.4 to 25 µm. Its substantial optical anisotropy has been experimentally verified through polarized light microscopy, with detailed theoretical calculations further corroborating the experimental findings. From a structural chemistry standpoint, the authors have developed a novel linear Hg3Se2 birefringent functional unit, which represents a significant advancement in both birefringent materials and structural chemistry. Given its high scientific merit and potential to engage a broad readership of Nature Communications, I think this manuscript is worthy of acceptance.\nI only have a few minor suggestions for the authors to consider:\n1. In the structural description section, the authors claim that Hg9Ga4Se4Cl16 crystallizes in the P21/c space group, while the information provided in Supporting Information Table 1 indicates a C2/c space group. The authors need to confirm the correct space group for this compound.\n2. In Fig. 3, the images and text descriptions do not correspond. Moreover, it would be better to provide the calculated values for the infrared band in the birefringence diagram.\n3. Given that the Hg3Se2 unit exhibits the highest polarization anisotropy among all known birefringence-active units to date, where does the birefringence of Hg9Ga4Se4Cl16 stand among Hg-based compounds? Is it the highest? The authors should consider including a comparative table about the birefringence of Hg-based compounds in the manuscript.\n4. In the manuscript, the authors mention that “the birefringence properties were calculated, resulting in refractive indices n(010) > n(100) > n(001)”, while in Fig. 3e, the notations used are nx, ny, nz.\n5. In the Supporting Information Table 4, the authors are advised to provide the oxidation states of the metals through bond valence calculations.\n6. Given that Hg-containing compounds like Hg2Cl2 and HgCl2 are highly toxic, the authors should consider including some hazard warnings in the manuscript.\n7. Hg9Ga4Se4Cl16 is a compound containing mixed-valence mercury. It is recommended that the authors distinguish the oxidation states of mercury and redefine the molecular formula, similar to the nomenclature of the mixed-valence mercury compound HgI2HgII(Te2O4)2(HPO4)2 mentioned in the introduction.\n8. In the check cif file, the calculated molecular formula does not match the reported one. The authors are advised to revise the Z value. Additionally, it is recommended to name the compound based on its molecular formula instead of using the designation “30r5-006”.", "reviewer_3": "This manuscript reports combined experimental and theoretical investigations of the IR birefringent semiconductor Hg9Ga4Se4Cl16 (HGSC). Birefringence represents a crucial optical property for numerous applications, particularly in optical imaging systems, remote sensing technologies, and laser communications - making this study potentially relevant to both fundamental researchers and applied engineers. While the research topic holds significant merit, the current manuscript appears incomplete, with several unresolved issues that substantially weaken its scientific rigor and overall persuasiveness.\n1. The most critical issue in this paper lies in the fact that the birefringence value of HGSC reported by the authors (Δn = 0.808 at 546 nm) is derived from theoretical calculations, whereas the actual experimental measurement yields a significantly smaller value of approximately 0.05. Moreover, the referenced data for the comparative sample MgF2 are experimental values, rendering such a comparison scientifically unsound and fundamentally unfair. This problematic practice recurs throughout the manuscript, as exemplified in Figure 5b. The authors must: (i) provide a rigorous justification for the substantial discrepancy between theoretical and experimental results; (ii) conduct cross-validation using multiple independent measurement methods to verify the experimental accuracy; (iii) fully characterize the uncertainty, including temperature dependence (dn/dT) and wavelength dispersion properties.\n2. The tridymite-like topology is typically defined by interconnected tetrahedral units forming regular channels with hexagonal symmetry. However, the structure of HGSC in this manuscript adopts a monoclinic framework constructed from three distinct linear units, lacking characteristic tridymite-like features. To aid reader understanding, please add a schematic diagram comparing the structures of HGSC and the classical tridymite-like topology, with a detailed explanation of their similarities and differences.\n3. In HGSC, there are multiple Hg with different coordination environments, and XPS verifies that there are three different valence states of Hg. Please assign the valence states of Hg in these different coordination environments and it can also be determined with the assistance of bond valence sum (BVS) calculations.\n4. The text mentions \"This makes the HGSC with retained linear mercury atom clusters promising candidates for more stable low-temperature superconducting materials.\" Then, is it possible to measure its low-temperature superconducting properties?\n5. Correct the inconsistency between the figure captions (Fig. 3a–3j) and the presented content (Figs. 3a–3g). Verify all figure labels and ensure numbering alignment.\n6. The theoretical birefringence (Δn = Δnmax = nx – nz = 0.443 at 1064 nm) differs significantly from the experimentally measured Δn = ny – nx = 0.052. Justify why ny – nx was measured instead of Δnmax and clarify how the measured value aligns with the theoretical prediction. Why was this choice made? Please explain the reason and supplement in this manuscript.\n7. Lines 276–278 claim agreement between experimental (ny – nx = 0.052) and theoretical (ny – nx = 0.046) values for the (001) plane. However, based on the refractive index calculations in Figure 3e, ny – nx < 0. Please explain the reason and supplement.\n7. Replace \"φ\" with \"Δφ\" for phase retardation to adhere to standard notation (φ = phase, Δφ = phase retardation).\n8. In the manuscript, the measured average value of the phase retardation φ = 2.79 + m·2π = 2π·(ny – nx)·z/λ (where m is an integer). Then, how to determine the value of m? Please add relevant explanations. Please carefully check and confirm that the experimental value of ny – nx = 0.052.\n9. What wavelengths correspond to the birefringence values of all compounds in Figure 5a? Are these experimental or theoretical values? It is suggested to add a table to summarize these values and label this information for reader convenience. For Figure 5b, explicitly label whether values are experimental or theoretical.\n10. The authors emphasize in the article that the birefringence contribution primarily originates from Hg₃Se₂. However, the structure contains three types of linear groups, and it is essential to calculate their respective contributions to accurately determine the magnitude of each group's effect.", "reviewer_4": "The manuscript “Unprecedented [Hg3Se2]2- cluster drives giant optical anisotropy and broad infrared transparency”, authored by Ren et al., reports the design and synthesis of Hg9Ga4Se4Cl16 (HGSC) single crystals. These crystals are composed of linearly arranged [Hg3Se2] clusters bridged by [Hg2Se2] and [HgSe2] units to form a tridymite-like structure, with tetrahedral GaCl4 filling the interstitials. The authors verify the existence of [Hg3]2+ and the multi-valency of Hg with a series of electron and phonon structure analysis, and resolve the transmittance window between the electron excitation and the phonon excitation with optical reflectance and transmittance measurements. To understand the optical anisotropy of HGSC, the authors take orthogonally polarized images of the HGSC crystal and analyze the birefringence with a standard polarization analyzer. The authors also carried out theoretical calculations on the electron structure of HGSC to support their claim of large birefringence and large polarization anisotropy of the [Hg3Se2] clusters, and compared the theoretical optical anisotropy with other commercially available birefringent crystals to highlight the large optical anisotropy of HGSC. Although there is novelty in synthesizing new [Hg3Se2] clusters with specific linear configurations, which cannot be found in the open source databases, the experiments proving the authors’ claims are poorly analyzed and sometimes inadequate to reach the conclusions the authors came to. I recommend that the authors complete the analysis of these experiments and add additional experiments when appropriate, or only draw logical conclusions in the discussion and leave objective anticipations to the perspectives. I regret that I do not recommend this manuscript for publication in Nature Communications at the current stage. The following is a list of concerns about the validity of the manuscript, sequenced by priority:\n1. The author claimed the “giant optical anisotropy” in the title, but the experimental evidence on the optical anisotropy is inadequate. The only quantitative analysis on the birefringence is through the detection of the incoming and transmitted polarization states at 1064 nm in Fig. 5, while most mentions of the HGSC birefringence are based on theoretical calculations. The authors should consider measuring the refractive index dispersion (as a function of wavelengths), as most works in Fig. 5b did, to make direct comparisons with other birefringent crystals.\n2. Even at 1064 nm, the authors omitted or disguised how they get to m=3 in the measured retardance “φ=2.79 + m·2π”, and directly assumed that a large birefringence, but not too large, best match with their theoretical calculations is the only correct solution. The authors should not only clarify their misleading deduction, but also consider carrying out in-depth polarimetry, resolving all nx, ny, and nz at 1064 nm. Examples could be: a. fit more polarization angles to get more accurate retardance “φ=φ0 + m·2π”; b. measuring multiple crystal tilting angles to approximate the thickness-dependency, thus eliminating m in “φ=φ0 + m·2π”; c. carry out ellipsometry in the reflectance mode.\n3. The manuscript claimed a space group of P21/c while the CIF file shows C2/c. Besides that, clarification is needed when the refined structure has relatively large refinement residuals and atomic displacement parameters. Without a direct representation of the diffraction pattern, such as precession maps along hk0, h0l, 0kl reciprocal planes, one cannot discriminate crystal disorder from the diffraction data quality. Especially, the large atomic displacement parameters of Hg and Cl atoms near z=0.75 can be a hint of symmetry lowering or disorders while forming domains.\n4. The 0 to 30 to 60 to 90 degrees angle dependence of transmitted images under orthogonally polarized light in Fig.4 is a basic property of any birefringent crystal, regardless of how large the birefringence is. At 0 and 90 degrees, polarization is parallel to the fast or slow axis of the crystal, thus maintaining the incoming polarization. Between 0 and 90 degrees, incoming light is decomposed into the fast and slow axes of the crystal and creates an elliptically polarized light which, in part, passes through the orthogonal polarizer. Such works for any birefringent crystal, but of a different amount for different retardances. Therefore, Fig. 4 is a qualitative argument that HGSC is optically anisotropic, and demands quantitative modeling to discriminate whether the optical anisotropy is large or small.\n5. The assignment of Raman peaks lacks clarification. The references cited also indicate large uncertainty of which Raman shift [Hg3]2+ and [Hg2]2+ has, dependent on the exact compound and the physical form. The Ga-Cl and Hg-Se also need clarification, especially Hg-Se, which can hardly be treated as a functional unit with a characteristic vibrational mode.\n6. Fig. 3 misses a few figures according to their caption.\n7. The authors made unsubstantiated speculations on perspective properties in a decisive tone multiple times, such as on the superconductive, thermoelectric, and optoelectronic properties of HGSC. The authors should try rationalizing such claims if they want to stick to ambitious speculations.\n8. The fitting of the XPS doesn’t match well with the experimental data at the shoulders of the peak. Might explain that.\n9. Quantitative analysis is commonly needed in EDS.\nAside from the above, the authors might benefit from professional assistance in enhancing the clarity of the language."}, "author_response": "Aug 1th, 2025\nPoint-by-Point Responses to the Reviewers’ Comments\nDear reviewers：\nWe sincerely thank the reviewers for the insightful comments concerning our\nmanuscript. These comments are very valuable for further improving the quality of this\nmanuscript. In the revised manuscript, we have tried our best to address all the concerns\nraised by the reviewers, and the point-by-point responses are listed below. The changes\nto the manuscript are indicated in RED in the marked version of the revised manuscript.\nWe hope that the quality of our manuscript has met the publication standard after this\nrevision.\nReviewer # 1 (Remarks to the Author):\nIn this manuscript, the authors report a birefringent material with a novel Hg Se\n3 2\ncluster, which could be published in Nature Communications. However, several\nimportant issues must be addressed before being considered for acceptance：\nQuestion 1: The authors claim that this represents the first reported compound\nfeaturing Hg Se cluster within a selenide framework. It is recommended that the\n3 2\nauthors further discuss the potential unique physical properties arising from the\nincorporation of Hg 2+ into the selenide to enhance understanding of this material's\n3\ncharacteristics. Additionally, a more comprehensive discussion on the unique features\nof this system, especially the influence of weak interactions on its structure and\npotential strategies for designing new frameworks, would greatly enhance the scientific\nimpact of this work.\nResponse:\nThank you for your insightful comments and constructive suggestions. We agree\nthat a more in-depth discussion on the unique physical properties associated with the\nincorporation of the Hg 2+ cluster into the selenide framework would greatly enhance\n3\nthe understanding and impact of our work. After a detailed investigation of the physical\nproperties of HGSC, we found that the compound exhibits a reversible thermochromic\nbehavior (Fig. R1a). Upon cooling from 300 K to 77 K, the crystal undergoes a\nreversible color change from pale-yellow to colorless, which remains stable over more\nthan 100 thermal cycles (Fig. R1b), demonstrating the robustness and repeatability of\nthe thermochromic behavior.\nFig. R1 a Thermochromic behavior observed upon cooling from 300 K to 77 K. b The\nphenomenon of thermal cycling of HGSC crystals for 100 cycles.\nFig. R2 Temperature variation of the lattice parameters (a, b, and c axis).\nTo explore the origin of this behavior, temperature-dependent single-crystal X-ray\ndiffraction measurements were conducted from 80 K to 300 K in 15 K increments (Fig.\nR2). The results confirm that HGSC retains its crystallographic symmetry throughout\nthis temperature range without undergoing any structural phase transition. Furthermore,\nthe lattice constants display continuous and anisotropic thermal expansion (α =\na\n3.177×10-5, α = 2.383×10-5, α = 1.955×10-5), suggesting enhanced atomic vibrational\nb c\namplitudes along specific crystallographic directions due to the ordered alignment of\nlinear building units within the crystal lattice.\nAmong the various mechanisms potentially contributing to thermochromic\nbehavior, phonon dynamics, particularly phonon anharmonicity, can induce band\nstructure renormalization and thereby modulate the material’s optical properties. This\noccurs as thermally activated lattice vibrations perturb the electronic states, especially\nnear the conduction and valence band edges, resulting in band edge shifts and spectral\nbroadening [Ref. 1 Adv. Mater. 36, 2312597 (2024)]. This mechanism is especially\nrelevant in compounds containing heavy metal elements, where the intricate phonon\nspectrum and strong mass contrast often give rise to enhanced lattice anharmonicity. In\nelectron-phonon coupling, only vibrational modes that induce changes in transition\npolarizability contribute significantly to the optical response, and these modes can be\nidentified through Raman spectroscopy [Ref. 2 Adv. Photonics Res. 6, 2400149 (2025)].\nSince HGSC crystals belong to the C point group symmetry, The irreducible\n2h\nrepresentations of their phonon modes are given in Equation R1.\nAmong these, the A𝛤𝛤g =an4d9 B𝐴𝐴g g v+ib r4a8ti𝐴𝐴ouna +l m 5o0d𝐵𝐵egs a+re 4R8a𝐵𝐵muan-active. In the H(G𝑅𝑅S1C)\ncrystals, nine distinct Raman peaks were observed (Fig. R3a), Based on the simulated\nvibrational eigenmodes, the A peaks at 137.10 cm-1 and 215.89 cm-1 correspond to\ng\nlattice motions primarily influenced by the [Hg Se ] unit, whereas the A mode at\n2 2 g\n110.70 cm-1 involves substantial contributions from the [Hg Se ] units (Fig. R3b). It is\n3 2\nimportant to emphasize that these assignments do not imply localized molecular-like\nvibrational modes, but instead reflect collective lattice dynamics predominantly\ngoverned by the bonding environments within the Hg-based structural units. In situ\nvariable-temperature Raman spectroscopy was performed under 633 nm laser\nexcitation (Fig. R3c) revealing several phonon modes (ꞷ , ꞷ , ꞷ ) with pronounced\nA B C\ntemperature-sensitive intensities, in contrast to temperature-invariant modes (ꞷ\nD\n, ꞷ\nE\n).\nNotably, to isolate the intrinsic anharmonic contributions to phonon frequency shifts,\nphonon mode calculations at 143 K were conducted for HGSC, thereby eliminating the\nconfounding effects of thermal expansion. Moreover, phonon mode calculations\nindicate that the temperature-sensitive modes originate from the cooperative lattice\nvibrational modes dominated by the Hg 2+ and Hg 2+ clusters. This directly links the\n3 2\nthermochromic effect to dynamic distortions within the Hg multinuclear clusters,\nsuggesting that the reversible thermochromic behavior observed in HGSC arises from\nan electron-phonon coupling-dominated mechanism, in accordance with previous\nreports on InSeI [Ref. 1 Adv. Mater. 36, 2312597 (2024)].\nFig. R3 a Comparison between experimental and calculated Raman spectrum (λ =\n622 nm). b The lattice vibrational A modes occur at 215.89, 137.10, and 110.70 cm-1,\ng\nThe blue, and yellow balls represent Hg and Se atoms, respectively. The red arrows\nrepresent the direction of atomic displacement. c Temperature-dependent Raman\nspectra measured from 83 K to 423 K.\nThese new insights underscore the potential of incorporating heavy-metal\nmultinuclear clusters such as Hg 2+ as active thermochromic centers in selenide\n3\nframeworks, opening new pathways for designing responsive optoelectronic materials.\nTo address the concerns regarding the physical properties mentioned in the comments,\nall discussions related to this have been included in the “Results and Discussion”\nsection of the revised manuscript, with marked revisions indicated on page 7-8.\nFurthermore, Figures R1-3 have been added to the revised manuscript.\nQuestion 2: I note that the similar Hg polynuclear cluster compounds reported in\nReferences 43 and 44 exhibit poor stability under ambient conditions. Given the\npotential application of the reported material in mid-infrared lasers, it is important for\nthe authors to experimentally verify whether HGSC suffers from similar stability issues.\nResponse: We sincerely appreciate the constructive comments provided by the\nreviewers. The chemical stability of HGSC was evaluated. Under atmospheric\nconditions (temperature: 25 ± 2 °C, relative humidity: 20 ± 5 %), HGSC crystals were\nable to maintain a stable state for three days, after which significant hygroscopic\ndegradation occurred. Compared to the Hg AsF crystals mentioned in the literature\n3-δ 6\n[Ref. 3 Phys. Rev. 45, 4836 (1992)], which can only exist within a glovebox, HGSC\ncrystals demonstrate a marked improvement in stability under environmental\nconditions. The findings further indicate that, HGSC crystals need to be stored in dry\nconditions to ensure long-term stability of the material.\nTo address the concern regarding material stability in this comment, all the above\ndiscussions have been added to the section “Results and Discussion” in the revised\nmanuscript with revision traces, see Page 5.\nQuestion 3: Figure 5b presents a favorable comparison between HGSC and\ncommercial birefringent materials. However, the comparison includes typical infrared\nnonlinear optical crystals such as AgGaS , which are generally not employed in\n2\npolarization modulation applications. To more convincingly highlight the advantages\nof HGSC in terms of birefringence, the authors are encouraged to include comparisons\nwith recently reported advanced birefringent materials in the mid- to far-infrared\nregion.\nResponse: We appreciate the reviewers’ comments. In response to this suggestion, we\nhave expanded the scope of the comparative analysis in the revised manuscript to\ninclude several recently reported advanced birefringent materials specifically designed\nfor the mid-IR region. These materials include CsICl , BaSbBS , Hg (Te O )(SO ),\n2 4 4 2 5 4\nCs S , and BaTeSeS . [Ref. 4 J. Am. Chem. Soc. 147, 26133-26138 (2025), Ref. 5 Chem.\n2 6 2\nSci. 16, 12577-12586 (2025), Ref. 6 J. Am. Chem. Soc. 146, 7868-7874 (2024), Ref. 7\nAngew. Chem. Int. Ed. 62, e202303711 (2023), Ref. 8 Angew. Chem. Int. Ed.\ne202508555] Additionally, Figure R4 has been incorporated into Figure 6b of the\nrevised manuscript.\nFig. R4 Comparison of the birefringence of HGSC with those of representative\ncommercial birefringent crystals and recently reported state-of-the-art materials.\nQuestion 4: As a relatively rare structural motif, the synthetic chemistry underlying the\nformation of polynuclear Hg clusters warrants a more systematic discussion. A deeper\n3\nexploration of this topic would enhance the completeness of the study and strengthen\nits relevance to the broader field of materials chemistry.\nResponse: We sincerely appreciate the reviewer’s valuable comments. In our\nstudy, the HGSC compounds were successfully synthesized using chemical vapor\ntransport (CVT). A Hg-rich environment was created within a sealed tube by employing\nan excess of Hg Cl . The reaction involved several precursor materials, including HgCl ,\n2 2 2\nHg Cl , HgSe, and Ga Se , which played a critical role in the formation of the Hg 2+\n2 2 2 3 3\ncluster. In this system, Hg in Hg Cl exists in the +1-oxidation state, which allows it to\n2 2\ndonate electrons during the reaction, while Hg in HgCl exists in the +2-oxidation state,\n2\nexhibiting strong oxidizing properties that promote its coordination with selenium.\nUnder high-temperature conditions, Hg(I) undergoes a disproportionation reaction,\nconverting into Hg(II) and Hg(0). This not only allows different oxidation states of Hg\nto coexist but also provides the necessary atomic and electronic sources for the\nformation of the [Hg Se ] unit.\n3 2\nMoreover, when we substituted the selenium-based precursor with sulfur-based\nreagents, no compounds containing the [Hg S ] unit were obtained. According to the\n3 2\nstudy by Melnick et al. [Ref. 9 J. Am. Chem. Soc. 132, 647-655 (2010)], the\ncoordination of Hg with selenium is more stable than with sulfur. Therefore, the\nselenophilicity of Hg played a significant role in the synthesis.\nWe have incorporated all the above discussions into the “Synthesis and crystal\ngrowth” section of the revised manuscript, along with revision traces. Specifically,\nplease refer on Page 15. We hope that this more in-depth analysis of the synthesis\nmechanism will further enhance the completeness of the study and its relevance to the\nbroader field of materials chemistry.\nQuestion 5: The authors employed Jones matrix analysis to verify the birefringence of\nthe HGSC compound at 1064 nm. While χ and ψ are identified as key parameters\ncharacterizing the polarization state, the specific numerical values are not provided. It\nwould be valuable to include these quantitative data as supplementary information. In\naddition, the authors suggest that HGSC holds potential as a birefringent material for\nmid- to far-infrared applications. To substantiate this claim, it is recommended that the\nauthors provide birefringence data at longer wavelengths beyond the currently reported\n1064 nm.\nResponse: We sincerely thank the reviewer for this insightful comment. Notably, the\nrevised manuscript adopts an improved analytical approach that eliminates the need for\nthe unknown variable θ, significantly simplifying the overall calculation. Based on an\noptimized measurement approach, have added the detailed polarization ellipse\nparameters (including ψ, χ and ellipticity tanχ) for all measured wavelengths (1064 nm,\n1540 nm, and 3500 nm) and both crystal thicknesses. These results have been included\nin the Supplementary Information (Supplementary Table 15), providing a more accurate\nand comprehensive basis for the Jones matrix analysis. This addition addresses the\nreviewer’s request for explicit numerical data to support our Jones matrix analysis and\nenhances the transparency of the birefringence evaluation.\nFurthermore, to support the claim that HGSC holds promise as a birefringent\nmaterial for mid-to far-IR applications, we extended our measurements beyond\n1064 nm. Specifically, we performed polarization-resolved transmission experiments at\n1540 nm and 3500 nm, as now reflected in both Figure R6 and Tables R1. These results\ndemonstrate that HGSC crystal exhibit large and spectrally stable birefringence over a\nbroad IR range, thereby providing critical optical parameters for the design and\nengineering of polarization-dependent photonic devices. The detailed process for\nmeasuring the IR birefringence of HGSC is provided below:\nThe birefringence of HGSC in IR region was experimentally determined by\nanalyzing the polarization modulation induced by the phase delay between two\northogonal polarization components. According to theoretical analysis, the relationship\nbetween the crystallographic axes and the principal dielectric axes of this crystal is a ≈\ny, b = x, and c ≈ z. In our experiment, b(x)-plane crystals were employed to measure n -\nz\nn . Given that n and n are very close (difference of <0.05), the measured n -n can be\ny y x z y\nreasonably taken as an approximation of the birefringence n -n of the crystal (i.e., Δn\nz x\n= n -n ≈ n -n ). The schematic orientation of the HGSC crystal with respect to the\nz x z y\nlaboratory coordinates is shown in Figure R5. Linearly polarized lasers with three\ntypical near-IR wavelengths (1064 nm, 1540 nm, and 3500 nm) were used for the\nbirefringence measurement, enabling a systematic investigation of the evolution of\npolarization as it propagated through the as-grown HGSC samples (Fig. R6a). Two\nHGSC samples with thicknesses of 12.9 μm and 14.0 μm were used for the\nmeasurements. The input beam with horizontal polarization was focused and normally\nincident into the samples. Upon transmission through the crystal, the polarization state\nof the laser was modulated (Figs. R6b-d), the crystal was functionally equivalent to a\nretarder. The experimental results reveal that HGSC exhibit large birefringence values\nof 0.517, 0.468, and 0.453 at wavelengths of 1064 nm, 1540 nm, and 3500 nm,\nrespectively (Table R1). These findings verify that HGSC crystal exhibit large and\nspectrally stable birefringence over a broad IR range, thereby providing critical optical\nparameters for the design and engineering of polarization-dependent photonic devices.\nFig. R5. Schematic illustration of the crystal orientation. The observed plane\ncorresponds to the b(x) surface of the HGSC samples, with the laser beam normally\nincident along the b(x)-axis. Axes H and V denote the horizontal and vertical directions\nin the laboratory frame, respectively. The crystal’s y- and z-axes lie in the H–V plane,\noriented at ±45 ° with respect to the horizontal direction. The orientation of the\ncrystallographic axes (a, b, c) was determined by a single-crystal X-ray diffractometer,\nand the approximate angles between the optical principal axes (x, y, z) and the\ncrystallographic axes were established based on theoretical calculations. The precise\ndirections of the y- and z-axes were further verified by the linearly polarized laser. This\nconfiguration enables equal projections of the refractive indices n and n along the\ny z\npolarization axis, facilitating birefringence-induced phase retardation, and ensures that\nthe major and minor axes of the resulting polarization ellipse are always aligned along\n±45 ° directions. The handedness of the elliptical polarization was determined using\nquarter-wave plates. It should be noted that ellipticity is defined as the ratio of the\nelectric field amplitude along the minor axis to that along the major axis. The sign of\nthe ellipticity indicates the handedness of the polarization.\nFig. R6. a Schematic illustration showing the polarization state change of the linearly\npolarized light propagating through the HGSC crystal. b, c, d Polarization-dependent\nintensity of the input light (blue curve) and output light for HGSC crystals with different\nthicknesses: red for the 12.9 μm sample and yellow for the 14.0 μm sample. Dots\nrepresent experimental data, and solid lines denote fitting curves.\nTable R1. Birefringence of HGSC crystals under different wavelengths.\nWavelength Thickness Birefringence\n12.9 μm\n1064 nm 0.517\n14.0 μm\n12.9 μm\n1540 nm 0.468\n14.9 μm\n12.9 μm\n3500 nm 0.453\n14.9 μm\nWe have incorporated all the above discussions into the “Results and Discussion”\nsection of the revised manuscript, along with revision traces. Specifically, please refer\nto Page 10.\nReviewer #2 (Remarks to the Author):\nDesigning optical materials with large birefringence has long been a challenging\npursuit for material scientists. In this manuscript, Prof. Pan and colleagues have\ndeveloped an intriguing optical material, Hg Ga Se Cl , which exhibits both large\n9 4 4 16\nbirefringence and a broad transparency range. This compound demonstrates a\nbirefringence of 0.808 at 546 nm and a wide transmission window spanning from 0.4\nto 25 µm. Its substantial optical anisotropy has been experimentally verified through\npolarized light microscopy, with detailed theoretical calculations further corroborating\nthe experimental findings. From a structural chemistry standpoint, the authors have\ndeveloped a novel linear Hg Se birefringent functional unit, which represents a\n3 2\nsignificant advancement in both birefringent materials and structural chemistry. Given\nits high scientific merit and potential to engage a broad readership of Nature\nCommunications, I think this manuscript is worthy of acceptance.\nI only have a few minor suggestions for the authors to consider:\nQuestion 1: In the structural description section, the authors claim that Hg Ga Se Cl\n9 4 4 16\ncrystallizes in the P2 /c space group, while the information provided in Supporting\ncorrect space group for this compound.\nResponse: We appreciate the reviewer for pointing out this important issue. Upon\nverification, we confirm that due to an oversight during our proofreading process, the\nspace group of the compound HGSC was incorrectly labeled as P2 /c in the original\n1\nmanuscript. The correct space group for this compound should be C2/c. We have made\nthis correction in the revised manuscript and have accordingly updated all affected\nstructural descriptions and tables.\nQuestion 2: In Fig. 3, the images and text descriptions do not correspond. Moreover, it\nwould be better to provide the calculated values for the infrared band in the\nbirefringence diagram.\nResponse: We sincerely appreciate the valuable feedback provided by the reviewer.\nRegarding the inconsistency between the images and the textual descriptions in Figure\n3, we have carefully revised both the figure and its corresponding text to ensure\ncomplete alignment. Additionally, we have included the calculated values for the IR\nregion in the birefringence analysis section (Fig. R7), and all relevant revisions have\nbeen incorporated into the revised manuscript.\nFig. R7 Calculated birefringence of HGSC.\nQuestion 3: Given that the Hg Se unit exhibits the highest polarization anisotropy\n3 2\namong all known birefringence-active units to date, where does the birefringence of\nHg Ga Se Cl stand among Hg-based compounds? Is it the highest? The authors\n9 4 4 16\nshould consider including a comparative table about the birefringence of Hg-based\ncompounds in the manuscript.\nResponse: We thank the reviewers for their comments. A comprehensive survey of\nreported birefringence values for Hg-based thiohalides and chalcogenides indicates that\nHGSC ranks among the highest in terms of experimentally measured birefringence.\n(page 11). Correspondingly, the revised manuscript includes Tables R2-R3\n(Supplementary Tables 12-13).\nTable R2. The structural features and birefringence values for Hg-based thiohalides.\nCrystal Space\nCompound Birefringence Basic Building Units\nSystem Group\na0.871@546 nm [Hg Se ] [Hg Se ]\n3 2 2 2\n18 8 8 32\na0.453@3500 nm [HgSe 2 ]\n3 4 3 3\n3 4 3 3\n3 4 3 3\n3 4 3 3\n4 2 5 2 2 2\n1 2\n1 1 1 2\n4 2 5 1 2 2 2\n2 2 1 2\n1 2\n3 2 4 3 2\n3 2 8 1 2 3\n3 2 8 1 2 3\n3 2 8 1 2 3\n3 2 2 1 2\n3 2 2 1 2\na Experimental value, b Calculated value.\nTable R3. The structural features and birefringence values for Hg-based chalcogenides.\nCrystal Space\nCompound Birefringence Basic Building Units\nSystem Group\na0.871@546 nm [Hg Se ] [Hg Se ]\n3 2 2 2\na0.453@3500 nm [HgSe 2 ]\n2 4 1 2\n2 2 6 4\n4 1 4\n4 4\n4 4\n2 2 2\n4 4\n6 6 1 3 2\n10 β-BaHgSnS Orthorhombic Ama2 b0.22@1064 nm [HgS ]\n4 4\n4 1 3 4\n4 4\n6 6 1 3 2\n4 3 4\n3 4\n2 1 3 2\n4 4\n4 4\n4 4\n2 4 4\n4\n4 1 4\n4 4\n[HgSe ] [HgSe ]\n2 3\n7 2 12\n[HgSe ]\n4\n4 4\n4 4\n2 2 8 4\n3 2 7 1 2\n4 3 4\n4 1 1 1 3 4\n2 4 4\n2 4 1 4 2\n4 2 7 4\n4 5 12 4\n3 2 8 4\n4 2 7 4\n4 3 9 1 4\n4 5 12 4\n4 5 12 4\n2 4 4\n4 5 12 4 4\na Experimental value, b Calculated value.\nQuestion 4: In the manuscript, the authors mention that “the birefringence properties\nwere calculated, resulting in refractive indices n > n > n ”, while in Fig. 3e,\n(010) (100) (001)\nthe notations used are n , n , n .\nx y z\nResponse: We thank the reviewer for pointing out this important inconsistency. In the\noriginal manuscript, we mistakenly president n > n > n , which implies a\n(010) (100) (001)\ndirect correspondence between the crystallographic axes and the optical principal axes.\nHowever, in monoclinic systems, the crystallographic axes (a, b, c) and the dielectric\nprincipal axes (x, y, z) are not necessarily aligned.\nUpon careful re-examination, the correct order of the principal refractive indices\nwas determined to be n > n > n , which is consistent with the notation used in\nz y x\nFigure R8. In the HGSC crystal, the crystallographic and optical axes exhibit the\nfollowing approximate relationships: b = x, a ≈ y and c ≈ z, with angular deviations of\n~2° for a/y and 5° for c/z.\nThe corresponding description has been corrected in the revised manuscript (page\n13).\nFig. R8 Diagrammatic sketch of functional modules in an optical indicatrix for HGSC\n(N > N > N ).\nz y x\nQuestion 5: In the Supporting Information Table 4, the authors are advised to provide\nthe oxidation states of the metals through bond valence calculations.\nResponse: We appreciate the valuable comments provided by the reviewers. In\nresponse to the reviewer’s suggestions, the bond valence method was employed to\ndetermine the oxidation states of the various metal elements (Table. R4). (page 3).\nAdditionally, Table R4 has been added to the revised Supplementary Information as\nSupplementary Table 4.\nTable R4. Bond lengths for Hg Ga Se Cl (298 K).\n18 8 8 32\nAtom Atom Length/Å Bond-valence BVS\nHg1 Se11 2.4718(15) 1.0555\n2.0528\nHg1 Se2 2.4498(15) 0.9973\nHg2 Se1 2.4716(15) 1.0583\n2.0530\nHg2 Se2 2.4493(15) 0.9946\nHg3 Hg32 2.5595(11) /\n0.7819\nHg3 Se1 2.5600(15) 0.7819\nHg4 Hg5 2.5755(6) /\n0.6868\nHg4 Se2 2.6087(15) 0.6868\nGa1 Cl1 2.180(4) 0.8640\nGa1 Cl2 2.124(5) 0.7094\n3.0694\nGa1 Cl3 2.198(4) 0.7549\nGa1 Cl4 2.173(4) 0.7408\nGa2 Cl5 2.167(4) 0.7428\nGa2 Cl6 2.181(4) 0.7735\n3.0614\nGa2 Cl7 2.165(4) 0.7756\nGa2 Cl8 2.163(4) 0.7693\n11/2+X,-1/2+Y,+Z; 21/2-X,3/2-Y,1-Z\nQuestion 6: Given that Hg-containing compounds like Hg Cl and HgCl are highly\n2 2 2\ntoxic, the authors should consider including some hazard warnings in the manuscript.\nResponse: We appreciate the valuable suggestions provided by the reviewer. In\naccordance with the reviewer's recommendation, we have included a safety statement\nin the revised manuscript (Materials part) to emphasize the potential hazards associated\nwith Hg Cl , HgCl , and HgSe. The statement is as follows:\n2 2 2\nNotice: Due to the toxicity of Hg Cl , HgCl , and HgSe, all handling processes should\n2 2 2\nbe conducted in a glove box to ensure operational safety. (page 15).\nQuestion 7: Hg Ga Se Cl is a compound containing mixed-valence mercury. It is\n9 4 4 16\nrecommended that the authors distinguish the oxidation states of mercury and redefine\nthe molecular formula, similar to the nomenclature of the mixed-valence mercury\ncompound HgI HgII(Te O ) (HPO ) mentioned in the introduction.\n2 2 4 2 4 2\nResponse: We sincerely appreciate the valuable suggestions provided by the reviewers.\nThe following statement has been added to the “Results and Discussion” section to\nclarify the valence state distribution of Hg in the compound: “The Hg atoms exhibit\nthree distinct oxidation states: Hg0, Hg+, and Hg2+, with a total composition\ncorresponding to Hg0 HgⅠ HgⅡ Se (GaCl ) .” (page 3).\n2 8 8 8 4 8\nQuestion 8: In the check cif file, the calculated molecular formula does not match the\nreported one. The authors are advised to revise the Z value. Additionally, it is\nrecommended to name the compound based on its molecular formula instead of using\nthe designation “30r5-006”.\nResponse: We sincerely appreciate the reviewers’ valuable comments. We have\nadjusted the Z-value in Olex2 to ensure consistency between the molecular formula\ncalculated in the check CIF file and the reported formula. Furthermore, we have\ncorrected the compound names in the check CIF file to align with the nomenclature\nused in our study.\nReviewer #3 (Remarks to the Author):\nThis manuscript reports combined experimental and theoretical investigations of the IR\nbirefringent semiconductor Hg Ga Se Cl (HGSC). Birefringence represents a crucial\n9 4 4 16\noptical property for numerous applications, particularly in optical imaging systems,\nremote sensing technologies, and laser communications - making this study potentially\nrelevant to both fundamental researchers and applied engineers. While the research\ntopic holds significant merit, the current manuscript appears incomplete, with several\nunresolved issues that substantially weaken its scientific rigor and overall\npersuasiveness.\nQuestion 1: The most critical issue in this paper lies in the fact that the birefringence\nvalue of HGSC reported by the authors (Δn = 0.808 at 546 nm) is derived from\ntheoretical calculations, whereas the actual experimental measurement yields a\nsignificantly smaller value of approximately 0.05. Moreover, the referenced data for the\ncomparative sample MgF are experimental values, rendering such a comparison\n2\nscientifically unsound and fundamentally unfair. This problematic practice recurs\nthroughout the manuscript, as exemplified in Figure 5b. The authors must: (i) provide\na rigorous justification for the substantial discrepancy between theoretical and\nexperimental results; (ii) conduct cross-validation using multiple independent\nmeasurement methods to verify the experimental accuracy; (iii) fully characterize the\nuncertainty, including temperature dependence (dn/dT) and wavelength dispersion\nproperties.\nResponse: We sincerely thank the reviewer for this critical and constructive comment.\nIn the initially submitted manuscript, only HGSC crystals with the (001) plane\nwere available. The previous measurement reflected the refractive index difference\nof n -n rather than the theoretically predicted birefringence Δn = n -n , resulting in a\ny x z x\nsignificantly lower experimental value. In the revised manuscript, we successfully\nobtained HGSC single crystals with an exposed (010) plane and remeasured the\nbirefringence. The updated results exhibit a strong consistency with the theoretical\ntrend. Moreover, we have removed the inappropriate comparison presented in the\noriginal manuscript and instead performed a more reasonable comparison between the\nnewly measured experimental birefringence of HGSC and those of other representative\ncompounds. Accordingly, Figure 5b has been replaced to reflect these revised data.\nBelow, we address each of the reviewer’s concerns in detail:\nResponse (i): In the original version of the manuscript, only (001) plane crystals were\navailable for measurement. Since the [001] direction is nearly aligned with the dielectric\nprincipal axis z, the measured birefringence represented an approximation of n -n\ny x\nrather than the birefringence Δn = n -n . As a result, the experimentally obtained values\nz x\nappeared significantly smaller than the theoretical predictions. In the revised\nmanuscript, (010) plane crystals were successfully prepared. As the [010] direction\ncoincides with the x-axis of the dielectric principal system, the birefringence measured\nalong this orientation corresponds to n -n . Given that n and n are very close (with a\nz y x y\ndifference of ＜0.05), it is reasonable to approximate the birefringence as Δn ≈ n -n .\nz y\nThe birefringence was experimentally determined to be 0.517 at 1064 nm, 0.468 at\n1540 nm, and 0.453 at 3500 nm, confirming the presence of large birefringence in the\ncrystal. The following describes the method for measuring birefringence of HGSC：\nThe birefringence of HGSC in IR region was experimentally determined by\nanalyzing the polarization modulation induced by the phase delay between two\northogonal polarization components. According to theoretical analysis, the relationship\nbetween the crystallographic axes and the principal dielectric axes of this crystal is a ≈\ny, b = x, and c ≈ z. In our experiment, b(x)-plane crystals were employed to measure n -\nz\nn . Given that n and n are very close (difference of <0.05), the measured n -n can be\ny y x z y\nreasonably taken as an approximation of the birefringence n -n of the crystal (i.e., Δn\nz x\n= n -n ≈ n -n ). The schematic orientation of the HGSC crystal with respect to the\nz x z y\nlaboratory coordinates is shown in Figure R9. Linearly polarized lasers with three\ntypical near-IR wavelengths (1064 nm, 1540 nm, and 3500 nm) were used for the\nbirefringence measurement, enabling a systematic investigation of the evolution of\npolarization as it propagated through the as-grown HGSC samples (Fig. R10a). Two\nHGSC samples with thicknesses of 12.9 μm and 14.0 μm were used for the\nmeasurements. The input beam with horizontal polarization was focused and normally\nincident into the samples. Upon transmission through the crystal, the polarization state\nof the laser was modulated (Figs. R10 b-d), the crystal was functionally equivalent to a\nretarder. The experimental results reveal that HGSC exhibit large birefringence of 0.517,\n0.468, and 0.453 at wavelengths of 1064 nm, 1540 nm, and 3500 nm, respectively (see\nthe “Birefringence Measurement of HGSC” section in the Supplementary Information).\nThese findings verify that HGSC crystal exhibit large and spectrally stable\nbirefringence over a broad IR range, thereby providing critical optical parameters for\nthe design and engineering of polarization-dependent photonic devices.\nFig. R9 Schematic illustration of the crystal orientation. The observed plane\ncorresponds to the b(x) surface of the HGSC samples, with the laser beam normally\nincident along the b(x)-axis. Axes H and V denote the horizontal and vertical directions\nin the laboratory frame, respectively. The crystal’s y- and z-axes lie in the H–V plane,\noriented at ±45 ° with respect to the horizontal direction.\nThe orientation of the crystallographic axes (a, b, c) was determined by a single-\ncrystal X-ray diffractometer, and the approximate angles between the optical principal\naxes (x, y, z) and the crystallographic axes were established based on theoretical\ncalculations. The precise directions of the y- and z-axes were further verified by the\nlinearly polarized laser. This configuration enables equal projections of the refractive\nindices n and n along the polarization axis, facilitating birefringence-induced phase\ny z\nretardation, and ensures that the major and minor axes of the resulting polarization\nellipse are always aligned along ±45 ° directions. The handedness of the elliptical\npolarization was determined using quarter-wave plates. It should be noted that\nellipticity is defined as the ratio of the electric field amplitude along the minor axis to\nthat along the major axis. The sign of the ellipticity indicates the handedness of the\npolarization.\nFig. R10 a Schematic illustration showing the polarization state change of the linear\npolarized light propagating through the HGSC crystal. b, c, d Polarization-dependent\nintensity of the input light (blue curve) and output light for HGSC crystals with different\nthicknesses: red for the 12.9 μm sample and yellow for the 14.0 μm sample. Dots\nrepresent experimental data, and solid lines denote fitting curves.\nThe experimental results reveal that HGSC exhibit significant birefringence values\nof 0.517, 0.468, and 0.453 at wavelengths of 1064 nm, 1540 nm, and 3500 nm,\nrespectively. The results are summarized in Table R5, where we also include the\ncorresponding theoretical birefringence values for direct comparison. The experimental\nvalues show good consistency with the theoretical predictions across all tested\nwavelengths, thereby resolving the discrepancy mentioned in the initial submission.\nTable R5. Experimental and calculated birefringence of HGSC crystals at different\nwavelengths.\nWavelength Experimental value Calculated value\n1064 nm 0.517 0.443\n1540 nm 0.468 0.403\n3500 nm 0.453 0.379\nResponse (ii): We sincerely appreciate the reviewers' expert suggestions regarding our\nresearch. In this revision, we synthesized the HGSC crystal with the (010) plane and\nfurther introduced the compensator method based on polarized light microscopy as a\nsecond independent measurement technique, conducting a quantitative assessment of\nits birefringence at the characteristic wavelength of 546.1 nm. The pale-yellow color of\nHGSC at room temperature hinders accurate identification of interference fringes.\nFortunately, HGSC exhibits pronounced thermochromic properties: as the temperature\ndrops below the critical point (140 K), the crystals become colorless and transparent\n(Fig. R11).\nFig. R11 Thermochromic process of HGSC from 300 K to 77 K.\nWe first performed single-crystal structural analysis of the HGSC crystal at low\ntemperature (130 K) to eliminate potential effects of structural phase transitions. The\nresults of the low-temperature X-ray single-crystal diffraction tests are presented in\nTable R6, confirming that the HGSC crystal retains the C2/c space group at 130 K, with\nno observed structural phase transition. This provides a reliable structural foundation\nfor subsequent studies on the temperature dependence of its optical properties. The\nbirefringence of an HGSC crystal sample (thickness: 14.030 μm) was characterized at\nthe optical path difference was measured to be 11756.14 nm at the characteristic\nwavelength of 546.1 nm (Figs. R12c, d). According to the formula R = Δn × d, the\nbirefringence at 546.1 nm was calculated to be 0.837.\nFig. R12 a Photograph of HGSC crystal for the measurement of birefringence (130K).\nb The thickness of HGSC for the birefringence measurement c, d The positive and\nnegative rotation of compensatory (130K).\nTable R6. Crystal data and structure refinement for Hg Ga Se Cl (130 K).\n18 8 8 32\nEmpirical formula Hg Ga Se Cl\n18 8 8 32\nFormula weight (Da) 5934.46\nTemperature (K) 143.00\nCrystal system Monoclinic\nSpace group C2/c\na (Å) 12.3166(9)\nb (Å) 7.0848(5)\nc (Å) 42.018(3)\nα (°) 90\nβ (°) 93.714(2)\nγ (°) 90\nVolume (Å3) 3658.8(4)\nZ 2\nDensity (calc.) (g/cm3) 5.387\nAbsorption coefficient (mm-1) 45.642\nF (000) 5008.0\nCompleteness to θ (%) 99.6\nRadiation MoKα (λ = 0.71073)\nTheta range for data collection 2.914 to 50.696\nIndex ranges -14 ≤ h ≤ 14, -8 ≤ k ≤ 7, -48 ≤ l ≤ 50\nReflections collected 20622\nIndependent reflections 3619 [R = 0.0434, R = 0.0312]\nint sigma\nData / restraints / parameters 3619 / 0 / 151\nGoodness-of-fit on F2 1.095\nFinal R indexes [I ≥ 2σ (I)] a R = 0.0494, wR = 0.1328\n1 2\nFinal R indexes [all data] a R = 0.0535, wR = 0.1361\n1 2\nLargest diff. peak / hole (e/Å-3) 2.27 / -2.13\na R = Σ||F | - |F ||/Σ|F | and wR = [Σw (F 2 – F 2)2 / Σw F 4]1/2 for F 2 > 2σ( F 2)\n1 o c o 2 o c o o o\nMeanwhile, to further expand the experimental data across different wavelength\nbands, we supplemented our study with birefringence measurement based on the laser\nphase delay effect, covering the wavelength ranges of 1064 nm, 1540nm, and 3500 nm.\nThe specific results are presented in Table R7. The birefringence measured for the\nHGSC crystal at the aforementioned wavelengths are 0.517, 0.468, and 0.453,\nrespectively.\nTable R7. Birefringence parameters of HGSC crystals under different wavelengths.\nWavelength Thickness Birefringence\n1064 nm 12.9 μm 0.517\n14.0 μm\n12.9 μm\n1540 nm 0.468\n14.9 μm\n12.9 μm\n3500 nm 0.453\n14.9 μm\nAlthough wavelength and sample thickness differed between the two methods, the\nresults aligned well with theoretical values, confirming the strong optical anisotropy of\nHGSC crystals in the vis-IR range. We believe that this multi-principle and wide-band\ncross-validation strategy significantly enhances the scientific credibility of the\nexperimental data. Future investigations will incorporate advanced characterization\ntechniques, including ellipsometry and conical light methods, to enable systematic\nstudies of the optical tensor, dispersion behavior, and thermo-optic properties of the\nHGSC crystal. We would like to express our sincere gratitude to the reviewers for their\nconstructive comments and guidance.\nResponse (iii): We sincerely thank the reviewer for the valuable suggestion regarding\nthe need to fully characterize the uncertainties, including the temperature dependence\n(dn/dT) and wavelength dispersion properties.\nAs noted, the HGSC crystals used in this study were grown via the chemical vapor\ntransport (CVT) method. Such crystals typically exhibit a plate-like morphology with\na limited thickness (generally less than 1 mm), which poses significant challenges in\npreparing sufficiently thick, prism-shaped samples required for accurate refractive\nindex measurements across a broad temperature and spectral range. Unfortunately,\novercoming the limitations in large-size crystal growth remains technically challenging\nin the short term.\nNevertheless, we fully agree with the reviewer that evaluating temperature-\ndependent optical properties is crucial. In this work, we focused on the pronounced\noptical anisotropy of the HGSC crystal. Importantly, the temperature-dependent\nbirefringence can still be effectively assessed despite the sample thickness constraints.\nCombined with the thermochromic behavior of HGSC, the variation in birefringence\nwith temperature becomes even more intriguing.\nThe procedure for measuring the birefringence of (001)-plane HGSC crystals at\ndifferent temperatures is outlined as follows:\nThe birefringence of an HGSC crystal sample (thickness: 14.030 μm) was\ncharacterized at 130 K (Figs. R13a, b). Using a compensator with both forward and\nreverse rotations, the optical path difference was measured to be 11756.14 nm at the\ncharacteristic wavelength of 546.1 nm (Figs. R13c, d). According to the formula R =\nΔn × d, the birefringence at 546.1 nm was calculated to be 0.837. The measurement\nwas repeated at 90 K, yielding an increased optical path difference of 12224.52 nm,\ncorresponding to a birefringence of 0.871 (Figs. R13e-g). These results indicate that the\nbirefringence of HGSC crystals increases with decreasing temperature. This behavior\nis similar to that reported in literature for typical chalcogenide compounds such as\nBaGa\n7\nand AgGaSe\n2\n. [Ref. 10 Inorg. Chem. 63, 10042-10049 (2024) and Ref. 11\nAppl. Opt. 44, 2673-2677 (2005)], further validating the temperature-driven tunable\noptical anisotropy characteristics of HGSC materials.\nFig. R13 a Photograph of HGSC crystal for the measurement of birefringence (130K).\nb The thickness of HGSC for the birefringence measurement c, d The positive and\nnegative rotation of compensatory (130K). e Photograph of HGSC crystal for the\nmeasurement of birefringence (90K). f, g The positive and negative rotation of\ncompensatory (90K).\nAlthough it remains technically challenging to overcome the crystal growth\nlimitations of HGSC and obtain precise refractive index dispersion curves in the short\nterm, we believe this does not compromise the core discussion of the intrinsic\nbirefringence properties presented in this study. Experimental Δn have been obtained\nthrough various methods at fixed wavelengths and have revealed the structure-\nperformance relationship through theoretical simulations. All discussions related to this\nhave been included in the “Results and Discussion” section of the revised manuscript,\nwith marked revisions indicated on pages 7-9. Furthermore, Figures R9-13 and Tables\nR5-7 have been added to the revised manuscript.\nWe sincerely appreciate the valuable suggestions provided by the reviewer. Future\nefforts will focus on optimizing the compositional ratio, adjusting the growth\ntemperature range, or employing alternative crystal growth strategies to obtain thicker\nsamples suitable for precise dispersion measurements. This will further enrich the\noptical properties database of HGSC materials.\nQuestion 2: The tridymite-like topology is typically defined by interconnected\ntetrahedral units forming regular channels with hexagonal symmetry. However, the\nstructure of HGSC in this manuscript adopts a monoclinic framework constructed from\nthree distinct linear units, lacking characteristic tridymite-like features. To aid reader\nunderstanding, please add a schematic diagram comparing the structures of HGSC and\nthe classical tridymite-like topology, with a detailed explanation of their similarities\nand differences.\nResponse: We sincerely appreciate the reviewers' comments. The reviewer is\nabsolutely correct that classical tridymite-like topology refers to three-dimensionally\ninterconnected tetrahedral networks forming regular channels with hexagonal\nsymmetry. In contrast, the HGSC structure adopts a monoclinic framework constructed\nfrom three distinct and nearly parallel linear building units ([HgSe ], [Hg Se ] and\n2 2 2\n[Hg Se ]), and it does not possess the canonical open-channel architecture\n3 2\ncharacteristic of traditional tridymite-like networks.\nIn the original manuscript, the term “tridymite-like” was used descriptively to\nreflect the superficial resemblance in structural alignment and layered motif, rather than\nimplying a strict topological equivalence. To avoid confusion and ensure structural\naccuracy, we have revised the terminology throughout the manuscript to refer to the\nstructure as “pseudo-tridymite-like”， which more appropriately captures the geometric\nsimilarity without suggesting true topological identity.\nTo further enhance reader understanding, we have added a schematic comparison\n(Fig. R14) between the HGSC structure and a representative tridymite-type framework.\nThe classification is divided into two distinct categories based on coordination\nenvironments and structural units:\nTridymite-like Topological Structures：\n(i) K Ba Li Ga S : This compound exhibits a three-dimensional (3D) open\n2 3 6 28 48\nframework constructed by [GaS ]and [LiS ] tetrahedral units. Along the c-axis, two\n4 4\ntypes of hexagonal channels can be observed, with K+/Ba2+ cations embedded in the\nchannels at varying ratios. These features collectively form a hexagonal tridymite-like\ntopology (Fig. R14b) [Ref. 12 Mater. Horiz. 8, 3394-3398 (2021)].\n(ii) Hg AsSe I: In this structure, planar triangular [HgSe ] and [AsSe ] units\n3 4 3 3\ninterconnect through shared Se atoms, forming hexagonal tridymite-like layers. I-\nanions are located in the interlayer spaces (Fig. R14c) [Ref. 13 Mater. Horiz. 10, 2876-\n2882 (2023)].\nPseudo-tridymite-like Topological Structures：\n(i) [Hg As] (CdI ): The crystal structure consists of linear [HgAs ] units assembled into\n2 2 4 2\na 3D cationic framework, forming hexagonal channels along the b-axis. [CdI ]\n4\ntetrahedra serve as anionic guests occupying these channels (Fig. R14f) [Ref. 14 J. Solid\nState Chem. 180, 805-811 (2007)].\n(ii) HGSC: Linear [HgSe ], [Hg Se ], and [Hg Se ] units construct two differently sized\n2 2 2 3 2\nchannels. Weak electrostatic interactions in the polynuclear cluster structure stabilize\n[GaCl ]- anions within the framework cavities, thereby completing the 3D architecture\n4\n(Fig. R14e).\nBased on the diverse topological structures formed by different coordination units,\nwe conducted an in-depth investigation of the similarities and differences between\ntridymite-like and pseudo-tridymite-like structures. In terms of similarities, both\nstructures exhibit complex multidimensional topological features, forming open 3D\nframeworks through the interconnection of functional units. This framework not only\npossesses high spatial filling efficiency but also effectively accommodates ions or\nmolecules, thereby providing a solid foundation for the various functional properties of\nthe materials across multiple domains, including optics, electronics, and\nthermodynamics.\nRegarding their differences, several key aspects can be highlighted: (i) Channel\nmorphology: tridymite-like frameworks form uniformly shaped hexagonal channels,\nwhile pseudo-tridymite-like frameworks present more diverse channel geometries. (ii)\nBuilding units: tridymite-like structures are dominated by tetrahedral or planar\ntriangular unit; tridymite-like structures (including pseudo-types) rely primarily on\nlinear chains. (iii) Interaction mechanism: Stability in tridymite-like networks hinges\non cation incorporation, whereas pseudo-tridymite-like structures rely more on weak\nhost–guest electrostatic interactions for structural stabilization.\nThrough the above supplementary discussion and schematic analysis, we hope to\npresent a clearer depiction of the structural uniqueness of HGSC and its rational\nclassification within the framework of topological structures. We once again sincerely\nthank the reviewer for their valuable comments. All discussions related to this have\nbeen included in the “Supplementary Information” section of the revised manuscript.\nFig. R14 a Schematic diagram of the tridymite-like topological structure. b Crystal\nstructure of K Ba Li Ga S . c Crystal structure of Hg AsSe I. d Schematic diagram\n2 3 6 28 48 3 4\nof the pseudo-tridymite-like topological structures. e Crystal structure of HGSC. f\nCrystal structure of [Hg As] (CdI ).\n2 2 4\nQuestion 3: In HGSC, there are multiple Hg with different coordination environments,\nand XPS verifies that there are three different valence states of Hg. Please assign the\nvalence states of Hg in these different coordination environments and it can also be\ndetermined with the assistance of bond valence sum (BVS) calculations.\nResponse: We thank the reviewer for their comments. In response to the reviewer’s\nsuggestions, we performed bond valence calculations to analyze the oxidation states of\nHg (Table R8). Additionally, Table R8 has been added to the revised Supplementary\nInformation as Supplementary Table 4.\nTable R8. Bond lengths and BVS for Hg\n32\n(298 K).\nAtom Atom Length/Å Bond-valence BVS\nHg1 Se11 2.4718(15) 1.0555\n2.0528\nHg1 Se2 2.4498(15) 0.9973\nHg2 Se1 2.4716(15) 1.0583\n2.0530\nHg2 Se2 2.4493(15) 0.9946\nHg3 Hg32 2.5595(11) /\n0.7819\nHg3 Se1 2.5600(15) 0.7819\nHg4 Hg5 2.5755(6) /\n0.6868\nHg4 Se2 2.6087(15) 0.6868\nGa1 Cl1 2.180(4) 0.8640\nGa1 Cl2 2.124(5) 0.7094\n3.0694\nGa1 Cl3 2.198(4) 0.7549\nGa1 Cl4 2.173(4) 0.7408\nGa2 Cl5 2.167(4) 0.7428\nGa2 Cl6 2.181(4) 0.7735\n3.0614\nGa2 Cl7 2.165(4) 0.7756\nGa2 Cl8 2.163(4) 0.7693\n11/2+X,-1/2+Y,+Z; 21/2-X,3/2-Y,1-Z\nQuestion 4: The text mentions \"This makes the HGSC with retained linear mercury\natom clusters promising candidates for more stable low-temperature superconducting\nmaterials.\" Then, is it possible to measure its low-temperature superconducting\nproperties?\nResponse: We sincerely thank the reviewer for raising this important point. After the\ninitial synthesis of the HGSC compound, we observed a weak repulsion between the\ncrystal and a magnet when the sample was immersed in liquid nitrogen (as shown in\nthe newly provided Video 1). This phenomenon, combined with previous reports of\nlow-temperature superconductivity in linear Hg 2+ cluster-based compounds such as\n3-δ 6\nmight also exhibit superconducting behavior at cryogenic temperatures.\nHowever, during the manuscript revision process, we conducted resistivity\nmeasurements of HGSC over a temperature range of 0-300 K. The results revealed that\nthe electrical conductivity remained in the megaohm (MΩ) range across the entire\ntemperature span (Fig. R15), with no indication of a superconducting transition. These\nfindings confirm that HGSC does not exhibit superconductivity under the investigated\nconditions.\nWe truly appreciate the reviewer’s insightful suggestion, which prompted us to\nperform this additional experimental verification. Accordingly, we have removed the\nrelated speculative statements from the revised manuscript to avoid any potential\nmisinterpretation.\nFig. R15 Temperature-dependent resistivity of the HGSC crystal measured in the 0-\nQuestion 5: Correct the inconsistency between the figure captions (Fig. 3a–3j) and the\npresented content (Figs. 3a–3g). Verify all figure labels and ensure numbering\nalignment.\nResponse: We sincerely appreciate the valuable feedback provided by the reviewers.\nRegarding the inconsistencies between the image and the textual description in Figure\n3, we have carefully revised both the figure and the corresponding text to ensure\ncomplete alignment. The updated version has been incorporated into the revised\nmanuscript.\nQuestion 6: The theoretical birefringence (Δn = Δn = n – n = 0.443 at 1064 nm)\nmax x z\ndiffers significantly from the experimentally measured Δn = n – n = 0.052. Justify why\ny x\nn – n was measured instead of Δn and clarify how the measured value aligns with\ny x max\nthe theoretical prediction. Why was this choice made? Please explain the reason and\nsupplement in this manuscript.\nResponse: We sincerely thank the reviewer for this insightful comment. In the original\nversion of the manuscript, only HGSC single crystals with the (001) plane were\navailable, which limited our ability to measure the birefringence (Δn = n -n ).\nz x\nDuring the revision process, we successfully obtained a small number of HGSC\ncrystals with an exposed (010) plane, which enabled us to directly measure the\nbirefringence Δn. The newly obtained experimental values are presented in Table R9 of\nthe revised manuscript. These results clearly demonstrate the strong optical anisotropy\nof the HGSC crystal and are in good agreement with the theoretical predictions. We\nappreciate the reviewer’s valuable suggestion, which significantly improved the\ncompleteness of our experimental analysis.\nTable R9. Experimental and calculated birefringence of HGSC crystals at different\nwavelengths.\nWavelength Experimental value Calculated value\n546 nm 0.837(130 K) 0.808\n1064 nm 0.517 0.443\n1540 nm 0.468 0.403\n3500 nm 0.453 0.379\nQuestion 7: Lines 276–278 claim agreement between experimental (n -n = 0.052) and\ny x\ntheoretical (n -n = 0.046) values for the (001) plane. However, based on the refractive\ny x\nindex calculations in Figure 3e, n -n < 0. Please explain the reason and supplement.\ny x\nResponse: We sincerely appreciate the valuable comments provided by the reviewers.\nIn response to the issues highlighted in Figure 3e, we have conducted a thorough review\nof the original data and confirmed that the abnormal result n -n < 0 for the refractive\ny x\nindex difference arose from a labeling error. We deeply apologize for this oversight and\nhave corrected the figure in the revised manuscript to ensure the consistency and\naccuracy of the presented data (Fig. R16).\nFig. R16 Calculated refractive indices and birefringence of HGSC.\nQuestion 8: Replace \"φ\" with \"Δφ\" for phase retardation to adhere to standard notation\n(φ = phase, Δφ = phase retardation).\nResponse: We sincerely appreciate the reviewer’s meticulous attention to the\nconsistency of notation. In accordance with the suggestions, we have replaced all\ninstances of “φ” referring to phase delay in the manuscript with “Δφ” This modification\nbetter adheres to standard notation conventions. The changes have been implemented\nin the text, figures, and equations (now updated in revised Supplementary Equations 2-\n4).\nQuestion 9: In the manuscript, the measured average value of the phase retardation φ\n= 2.79 + m·2π = 2π·(n – n )·z/λ (where m is an integer). Then, how to determine the\ny x\nvalue of m? Please add relevant explanations. Please carefully check and confirm that\nthe experimental value of n – n = 0.052.\ny x\nResponse: We sincerely appreciate the constructive comments provided by the\nreviewers. In our original work, the birefringence calculation was significantly\ncomplicated due to the uncertainty in the orientation of principal dielectric axes in the\nmonoclinic system. At that time, we could only confirm that the incident plane\ncorresponded to the (001) plane, while the directions of the two remaining principal\ndielectric axes perpendicular to the propagation direction were not well defined. As a\nresult, in addition to the phase retardation Δφ, the Jones vector analysis required the\nintroduction of two additional variables, the crystal orientation angle θ and the angle\nbetween the major axis of the ellipse and the V-axis ψ, which greatly increased the\ncomplexity of the calculation. Unfortunately, this complexity also led to inadvertent\nerrors in our previous computation, rendering the previous results (φ = 2.79 + m·2π)\nunreliable. Therefore, an improved measurement method was adopted in this study,\nreplacing that used in the original manuscript. The unknown variables of the crystal\nrotation angle θ and the ellipse orientation angle ψ can be eliminated (ψ is fixed at\n±45 °), allowing for a simplified and more robust calculation.\nUnder this improved framework, we remeasured the refractive index difference of\na (001) plane to determine the value of (n −n ). Although the (100) and (001) planes\ny x\nare not aligned with the principal dielectric axes, the angular deviation is small (~2 °\nand 5 °, respectively), allowing the crystal to be reasonably approximated as\northorhombic for the purpose of analysis. The measurement results are as shown in\nFigure R17. The measured parameters ψ, χ and tanχ, are summarized in Table R10.\nFig. R17 Polarization-dependent intensity of the input light (blue curve) and output\nlight for HGSC crystals with different thicknesses: red for the 44.5 μm sample and\nyellow for the 62.7 μm sample. Dots represent experimental data, and solid lines denote\nfitting curves.\nTable R10. Measured parameters ψ, χ, tanχ φ\nWavelength Thickness ψ χ φ\n0\n(tanχ)\n44.5μm 0.0058 -0.5309 -0.5871 4.2034\n1064nm\n62.7μm 0.0104 0.2911 0.2996 2.5593\nIn these measurements, the experimentally obtained phase retardation takes the\nform Δφ = φ + m 2π. Due to the inherent periodicity of phase, only the principal value\n0\nof φ\n0\nwithin a sing⋅le 2π cycle can be directly measured, defined within the range 0 ≤ φ\n0\n< 2π. To determine the birefringence, we calculated a series of candidate (n −n ) values\ny x\nbased on the relation\n𝜑𝜑0 +𝑚𝑚⋅2π\nWhere (n −n ) denotes the (c𝑛𝑛a𝑦𝑦nd−id𝑛𝑛a𝑥𝑥te) 𝑚𝑚(n=−n ) values co𝜆𝜆rresponding to each assu(m𝑅𝑅2ed)\ny x m y x\n2π𝐿𝐿\ninteger m and L is the crystal thickness. Owing to the periodicity of phase retardation,\nthe Equation R2 yields multiple Δn solutions for each measurement. Because the\nthickness appears in the denominator, the same 2π phase increment results in different\n(n −n ) intervals for samples of different thicknesses.\ny x m\nBy comparing two such sets of (n −n ) values for three obtained from HGSC\ny x m\ncrystals with thicknesses of 44.5 μm and 62.7 μm, we identified a pair of values that\nclosely agree, thereby uniquely determining the correct integer m and the birefringence\n(Table R11). As a result, the birefringence (n −n ) was determined to be 0.040.\ny x\nTable R11. Comparison of (n −n ) for derived from two HGSC samples with different\ny x m\nthicknesses to determine the integer m.\n1064 nm\nThickness 44.5 μm: (n −n ) = 0.0160 + m×0.0239\ny x m\nThickness 62.7 μm: (n −n ) = 0.0069 + m×0.0170\ny x m\n(n −n ) (44.5 μm) with m (n −n ) (62.7 μm) with m Difference\ny x m y x m\n0.0159 0 0.0238 1 -0.0079\n0.0398 1 0.0408 2 -0.0010 √\n0.0637 2 0.0578 3 0.0059\n0.0876 3 0.0747 4 0.0129\n0.1115 4 0.0917 5 0.0198\n0.1354 5 0.1086 6 0.0268\nQuestion 10: What wavelengths correspond to the birefringence values of all\ncompounds in Figure 5a? Are these experimental or theoretical values? It is suggested\nto add a table to summarize these values and label this information for reader\nconvenience. For Figure 5b, explicitly label whether values are experimental or\ntheoretical.\nResponse: We sincerely appreciate the valuable comments provided by the reviewers.\nIn accordance with their suggestions, we have systematically compiled the\nbirefringence, single-crystal transmission range, and bandgap parameters of the\ncompounds listed in Figure 6a into Table R12. Experimental values and theoretical\ncalculations have also been designated with the superscripts “a” and “b”, respectively.\nAdditionally, based on the collective feedback from all reviewers, we have revised the\ncomparative format of Figure 6b (Fig. R18). Similarly, to ensure clarity, we have\nmarked experimental values and theoretical values using “Exp.” and “Cal.”,\nrespectively. These revisions have been incorporated into the updated manuscript to\nenhance data transparency and reader comprehension (page 14).\nTable R12. The comparison of birefringence and IR transmission ranges of Hg-based\nchalcogenide single crystals.\nCrystal Space Transmission Birefringence Band\nCompound\nSystem Group ranges (IR) gap (eV)\n16 8 8 32\n3 4\n3 4\n7 2 12\n3 2 8\n3 2 8 1\n3 2 8 1\n3 2 8 1\n4\n4 1\n2 2 8\n2 4\na Experimental value, b Calculated value.\ncommercial birefringent crystals and recently reported state-of-the-art materials.\nQuestion 11: The authors emphasize in the article that the birefringence contribution\nprimarily originates from Hg Se . However, the structure contains three types of linear\n3 2\ngroups, and it is essential to calculate their respective contributions to accurately\ndetermine the magnitude of each group's effect.\nResponse: We thank the reviewer for this insightful comment. A quantitative analysis\nwas carried out using the real-space atom-cutting method to evaluate the respective\ncontributions of the linear [Hg Se ], [Hg Se ] and [HgSe ] units to the optical\n3 2 2 2 2\nanisotropy. To better elucidate the origin of birefringence, the real-space atom-cutting\nmethod was employed to quantitatively assess the contributions of individual linear\nbuilding units to the optical anisotropy. As depicted in Figure R19, the contribution of\nthe [Hg Se ] unit to birefringence is approximately 23.8 %, while the contribution of\n2 2\nthe [Hg\n2\n] unit is about 65.9 %. Moreover, cutting the [HgSe\n2\n] unit results in an\nincrease in birefringence by 12.4 %, primarily due to a further decrease in the principal\nrefractive index n . This reduction amplifies the disparity among the principal refractive\nx\nindices, thereby enhancing the birefringence. (Page 13). Correspondingly, the revised\nmanuscript includes Figure 5d (Fig. R19).\nFig. R19. The real-space atomic cutting method is used to calculate the specific\ncontributions of [Hg Se ], [Hg Se ], and [HgSe ] units.\n3 2 2 2 2\nReviewer #4 (Remarks to the Author):\nThe manuscript “Unprecedented [Hg Se ]2- cluster drives giant optical anisotropy and\n3 2\nbroad infrared transparency”, authored by Ren et al., reports the design and synthesis\nof Hg Ga Se Cl (HGSC) single crystals. These crystals are composed of linearly\n9 4 4 16\narranged [Hg Se ] clusters bridged by [Hg Se ] and [HgSe ] units to form a tridymite-\n3 2 2 2 2\nlike structure, with tetrahedral GaCl filling the interstitials. The authors verify the\n4\nexistence of [Hg ]2+ and the multi-valency of Hg with a series of electron and phonon\n3\nstructure analysis, and resolve the transmittance window between the electron\nexcitation and the phonon excitation with optical reflectance and transmittance\nmeasurements. To understand the optical anisotropy of HGSC, the authors take\northogonally polarized images of the HGSC crystal and analyze the birefringence with\na standard polarization analyzer. The authors also carried out theoretical calculations\non the electron structure of HGSC to support their claim of large birefringence and\nlarge polarization anisotropy of the [Hg Se ] clusters, and compared the theoretical\n3 2\noptical anisotropy with other commercially available birefringent crystals to highlight\nthe large optical anisotropy of HGSC.\nAlthough there is novelty in synthesizing new [Hg Se ] clusters with specific linear\n3 2\nconfigurations, which cannot be found in the open source databases, the experiments\nproving the authors’ claims are poorly analyzed and sometimes inadequate to reach the\nconclusions the authors came to. I recommend that the authors complete the analysis\nof these experiments and add additional experiments when appropriate, or only draw\nlogical conclusions in the discussion and leave objective anticipations to the\nperspectives. I regret that I do not recommend this manuscript for publication in Nature\nCommunications at the current stage. The following is a list of concerns about the\nvalidity of the manuscript, sequenced by priority:\nQuestion 1: The author claimed the “giant optical anisotropy” in the title, but the\nexperimental evidence on the optical anisotropy is inadequate. The only quantitative\nanalysis on the birefringence is through the detection of the incoming and transmitted\npolarization states at 1064 nm in Fig. 5, while most mentions of the HGSC birefringence\nare based on theoretical calculations. The authors should consider measuring the\nrefractive index dispersion (as a function of wavelengths), as most works in Fig. 5b did,\nto make direct comparisons with other birefringent crystals.\nResponse: We sincerely thank the reviewer for highlighting the importance of\nexperimental validation of optical anisotropy. As correctly pointed out, the “giant\noptical anisotropy” emphasized in the title refers primarily to the large birefringence\n(Δn) of the HGSC crystal. In the revised manuscript, we have provided direct\nexperimental evidence of the birefringence through polarization-resolved transmission\nmeasurements at three IR wavelengths—1064 nm, 1540 nm, and 3500 nm. These\nresults are presented in Figure 4 of the revised manuscript, with detailed analysis\nprovided in the “Birefringence Measurement of HGSC” section of the Supplementary\nInformation. The experimentally obtained birefringence values are as large as\n0.517@1064 nm, 0.468@1540 nm, and 0.453@3500 nm. Moreover, leveraging the\nunique thermochromic behavior of the HGSC crystal, its birefringence at 546 nm was\nevaluated using a polarizing microscope, yielding a value of 0.837 at 130 K.\nWe fully agree with the reviewer that wavelength-dependent refractive index\ndispersion measurements would allow a more comprehensive comparison with other\nbirefringent materials. At the current stage of crystal development, the available HGSC\nsamples are very small in size and not yet comparable to fully grown optical crystals.\nTherefore, traditional refractometric methods such as the minimum-deviation technique\nor prism-based measurements cannot be applied. To overcome this limitation, we\nemployed a polarization modulation method based on phase retardation analysis, which\nallowed us to determine birefringence across multiple wavelengths even in such small-\nsize samples. While the precision of this method may be lower than that of traditional\ntechniques applied to large crystals, the results remain scientifically informative and\noffer useful reference data. We have clarified this point in the revised manuscript and\nSupplementary Information, and we hope the reviewer will find the added explanation\nand data sufficient to support the claim of “giant optical anisotropy.”\nAlthough direct measurement of the refractive index dispersion was not feasible,\nwe constructed a birefringence dispersion curve with the experimentally measured Δn\nvalues [Ref. 15 SPIE. 13050, 117-125 (2024)], which has been added into Figure R20\nto facilitate comparison with other known birefringent materials.\nFig. R20 Comparison of the birefringence of HGSC with those of representative\ncommercial birefringent crystals and recently reported state-of-the-art materials.\nQuestion 2: Even at 1064 nm, the authors omitted or disguised how they get to m=3 in\nthe measured retardance “φ=2.79 + m·2π”, and directly assumed that a large\nbirefringence, but not too large, best match with their theoretical calculations is the\nonly correct solution. The authors should not only clarify their misleading deduction,\nbut also consider carrying out in-depth polarimetry, resolving all nx, ny, and nz at 1064\nnm. Examples could be: a. fit more polarization angles to get more accurate retardance\n“φ=φ0 + m·2π”; b. measuring multiple crystal tilting angles to approximate the\nthickness-dependency, thus eliminating m in “φ=φ0 + m·2π”; c. carry out ellipsometry\nin the reflectance mode.\nResponse: We sincerely thank the reviewer for the valuable comment and suggestion.\nIn our original manuscript, the birefringence of the crystal was measured using only a\nsingle sample. As the phase retardation is inherently periodic, the resulting Δn values\nwere ambiguous due to the m·2π uncertainty, hence we acknowledge that this limitation\nwas not clearly addressed in the original version.\nIn the revised manuscript, inspired by the reviewer’s suggestions, we have adopted\nan improved approach to resolve this ambiguity. Specifically, we performed\nmeasurements using two HGSC crystals with different thicknesses. This dual-thickness\nor multi- thickness method can effectively eliminate the m·2π phase ambiguity then\nenables us to accurately determine the integer m, thereby yielding a unique and reliable\nvalue for the birefringence. In the improved measurements, the measured Δφ = φ +\n0\nm 2π is defined within the range 0 ≤ φ < 2π. To determine the birefringence, we\n0\nca⋅lculated a series of candidate Δn values based on the relation\n𝜑𝜑0 +𝑚𝑚⋅2𝜋𝜋\nΔ𝑛𝑛𝑚𝑚 = 𝜆𝜆 (𝑅𝑅3)\nwhere Δn denotes the candidate Δn values 2co𝜋𝜋r𝐿𝐿responding to each assumed integer m,\nm\nL is the crystal thickness and λ is the wavelength. where and L is the crystal thickness.\nOwing to the periodicity of phase retardation, the Equation R3 yields multiple Δn\nsolutions for each measurement. Because the thickness appears in the denominator, the\nsame 2π phase increment results in different Δn intervals for samples of different\nthicknesses.\nBy comparing two such sets of Δn values for three obtained from HGSC crystals\nm\nwith thicknesses of 12.9 μm and 14.0 μm, we identified a pair of values that closely\nagree, thereby uniquely determining the correct integer m and the birefringence (Table\nR13). As a result, the birefringence Δn was determined to be 0.517@1064 nm,\n0.468@1540 nm, and 0.453@3500 nm. It should be noted that due to the challenges in\ngrowing b(x)-plane HGSC crystals, we were only able to perform the dual-thickness\nmeasurement. If more samples were available, a multi-thickness measurement would\nallow for better cross-validation of Δn and m, enhancing the reliability of the results.\nTable R13. Comparison of Δn for different wavelengths derived from two HGSC\nm\nsamples with different thicknesses to determine the integer m.\n1064 nm\nThickness 12.9 μm: Δn = 0.0223 + m×0.0824\nm\nThickness 14.0 μm: Δn = 0.0610 + m×0.0760\nm\nΔn\nm\n(12.9 μm) with m Δn\nm\n(14.0 μm) with m Difference\n0.1047 1 0.1371 1 -0.0324\n0.1871 2 0.2131 2 -0.026\n0.2695 3 0.2891 3 -0.0196\n0.3519 4 0.3652 4 -0.0133\n0.4343 5 0.4412 5 -0.0069\n0.5167 6 0.5172 6 -0.0005 √\n0.5991 7 0.5933 7 0.0058\n0.6814 8 0.6693 8 0.0121\n0.7638 9 0.7453 9 0.0185\n1540 nm\nThickness 12.9 μm: Δn = 0.1087 + m×0.1192\nm\nThickness 14.0 μm: Δn = 0.0287 + m×0.1100\nm\nΔn\nm\n(12.9 μm) with m Δn\nm\n(14.0 μm) with m Difference\n0.1087 0 0.1388 1 -0.0301\n0.2279 1 0.2488 2 -0.0209\n0.3472 2 0.3589 3 -0.0117\n0.4665 3 0.4689 4 -0.0024 √\n0.5857 4 0.5789 5 0.0068\n0.7050 5 0.6890 6 0.0160\n0.8242 6 0.7990 7 0.0252\n0.9435 7 0.9091 8 0.0344\n1.0628 8 1.0191 9 0.0437\n3500 nm\nThickness 12.9 μm: Δn = 0.1801 + m×0.2710\nm\nThickness 14.0 μm: Δn = 0.2041 + m×0.2501\nm\nΔn\nm\n(12.9 μm) with m Δn\nm\n(14.0 μm) with m Difference\n0.1801 0 0.2041 0 -0.0240\n0.4511 1 0.4542 1 -0.0031 √\n0.7222 2 0.7044 2 0.0178\n0.9932 3 0.9545 3 0.0387\n1.2643 4 1.2046 4 0.0597\n1.5353 5 1.4547 5 0.0806\n1.8064 6 1.7048 6 0.1016\n2.0774 7 1.9549 7 0.1225\n2.3485 8 2.2050 8 0.1435\nQuestion 3: The manuscript claimed a space group of P2 /c while the CIF file shows\nrefinement residuals and atomic displacement parameters. Without a direct\nrepresentation of the diffraction pattern, such as precession maps along hk0, h0l, 0kl\nreciprocal planes, one cannot discriminate crystal disorder from the diffraction data\nquality. Especially, the large atomic displacement parameters of Hg and Cl atoms near\nz=0.75 can be a hint of symmetry lowering or disorders while forming domains.\nResponse: We appreciate the reviewer for pointing out this important issue. Upon\nverification, we confirm that due to an oversight during our proofreading process, the\nspace group of the compound HGSC was incorrectly labeled as P2 /c in the original\n1\nmanuscript. The correct space group for this compound should be C2/c. We have made\nthis correction in the revised manuscript and have accordingly updated all affected\nstructural descriptions and tables.\nRegarding the concern about the relatively large refinement residuals and atomic\ndisplacement parameters, especially for Hg and Cl atoms near z ≈ 0.75, we conducted\na comprehensive analysis of the single-crystal diffraction data. The (0kl) precession\nimage reveals elongation of certain reflections along the k-direction (c*), suggesting\nthe presence of domain disorder within the crystal. Additionally, in the high-resolution\nregime, several reflections exhibit smearing along the l-direction, further indicative of\npacking irregularities. Notably, such structural imperfections are absent in the (h0l),\n(hhl), and (hkl) precession images (Fig. R21).\nFig. R21 Single-crystal XRD diffraction pattern of HGSC in reciprocal space. a (0kl)\nplane. b (h0l) plane. c (hhl) plane. d (hk0) plane.\nQuestion 4: The 0 to 30 to 60 to 90 degrees angle dependence of transmitted images\nunder orthogonally polarized light in Fig.4 is a basic property of any birefringent\ncrystal, regardless of how large the birefringence is. At 0 and 90 degrees, polarization\nis parallel to the fast or slow axis of the crystal, thus maintaining the incoming\npolarization. Between 0 and 90 degrees, incoming light is decomposed into the fast and\nslow axes of the crystal and creates an elliptically polarized light which, in part, passes\nthrough the orthogonal polarizer. Such works for any birefringent crystal, but of a\ndifferent amount for different retardances. Therefore, Fig. 4 is a qualitative argument\nthat HGSC is optically anisotropic, and demands quantitative modeling to discriminate\nwhether the optical anisotropy is large or small.\nResponse: We would like to express our sincere gratitude for the valuable suggestions\nfrom the reviewers. To provide more convincing support for the “ultra-large optical\nanisotropy” feature of HGSC, we have included new quantitative birefringence\nmeasurements for the (010) plane HGSC crystals in the revised manuscript. In the\noriginal submission, polarized optical microscopy was not employed for birefringence\nquantification to avoid potential misinterpretation of interference colors. Notably, this\nmaterial exhibits significant thermochromic properties (Fig. R22), transforming into a\ncolorless and transparent state below the critical temperature (140 K), thereby greatly\nenhancing interference color resolution and providing an ideal window for quantitative\nbirefringence measurement.\nFig. R22 Thermochromic process of HGSC from 300 K to 77 K.\nWe successfully synthesized (010) plane HGSC single crystals and conducted\npolarized light microscopy measurements at 90 K, effectively avoiding potential\nmisinterpretation of interference color gradients that may arise from the pale-yellow\ncoloration of HGSC crystals at room temperature. Under an incident wavelength of\n546.1 nm, a bidirectional optical path difference measurement was performed on the\nHGSC crystal with a thickness of 14.030 μm using a rotating compensator. The total\noptical path difference obtained from the experiment was 12224.52 nm (Figs. R23 a-d).\nAccordingly, using the formula: R = ∆n × d, the measured birefringence value is n =\nexp\n0.871@ 546 nm.\nWe have incorporated all the above discussions into the “Results and Discussion”\nsection of the revised manuscript, with marked revisions indicated on page 10.\nFurthermore, Figures R22 and R23 have been added to the revised manuscript.\nFig. R23 a Photograph of HGSC crystal for the measurement of birefringence. b The\nthickness of. c,d The positive and negative rotation of compensatory.\nQuestion 5: The assignment of Raman peaks lacks clarification. The references cited\nalso indicate large uncertainty of which Raman shift [Hg ]2+ and [Hg ]2+ has,\n3 2\ndependent on the exact compound and the physical form. The Ga-Cl and Hg-Se also\nneed clarification, especially Hg-Se, which can hardly be treated as a functional unit\nwith a characteristic vibrational mode.\nResponse: We sincerely appreciate the reviewer’s insightful comment. We fully agree\nthat individual Hg-Hg and Hg-Se bonds cannot be treated as functional units possessing\nwell-defined, molecule-like characteristic vibrational modes, especially given the\ndiversity of Hg-based coordination environments reported in the literature. To address\nthis concern, we performed first-principles simulations of the Raman spectrum of the\nHGSC compound. The calculated results show strong agreement with the experimental\ndata (Fig. R24a), thereby validating the overall reliability of the mode assignments. In\ntotal, nine distinct Raman peaks were observed in the experimental spectra.\nBased on the simulated vibrational eigenmodes, the A peaks at 137.10 cm-1 and\ng\n215.89 cm-1 correspond to lattice motions primarily influenced by the [Hg Se ] unit,\n2 2\nwhereas the A mode at 110.70 cm-1 involves substantial contributions from the [Hg Se ]\ng 3 2\nunits. It is important to emphasize that these assignments do not imply localized\nmolecular-like vibrational modes, but instead reflect collective lattice dynamics\npredominantly governed by the bonding environments within the Hg-based structural\nunits. Regarding the Ga-Cl vibrations, the [GaCl ] tetrahedra are embedded as isolated\n4\nzero-dimensional units. The A modes observed at 360.87 cm-1 and 380.67 cm-1 show\ng\nexcellent agreement with the calculated symmetric stretching vibrations of the [GaCl ]\n4\nunits (Fig. R24b).\nTo address the inquiries about the Raman peaks mentioned in these comments, the\nrelevant discussions have been included in the “Results and Discussion” section of the\nrevised manuscript, with tracked changes indicated on page 8.\nFig. R24 a Comparison between experimental and calculated Raman spectrum (λ =\n622 nm). b The lattice vibration A modes appear at 110.70, 137.10, 215.89, 360.87,\ng\nand 380.67 cm-1. The blue and yellow spheres represent Hg and Se atoms, respectively,\nwhile the cyan and green spheres represent Ga and Cl atoms. The red arrows indicate\nthe direction of atomic displacement.\nQuestion 6: Fig. 3 misses a few figures according to their caption.\nResponse: We sincerely appreciate the valuable feedback provided by the reviewers.\nRegarding the inconsistencies between the image and the textual description in Figure\n3, we have carefully revised both the figure and the corresponding text to ensure\ncomplete alignment. The updated version has been incorporated into the revised\nmanuscript.\nQuestion 7: The authors made unsubstantiated speculations on perspective properties\nin a decisive tone multiple times, such as on the superconductive, thermoelectric, and\noptoelectronic properties of HGSC. The authors should try rationalizing such claims if\nthey want to stick to ambitious speculations.\nResponse: We sincerely appreciate the valuable comments provided by the reviewers.\nWe fully understand your rigorous requirements regarding the speculative assertions in\nthe manuscript concerning the superconductivity, thermoelectricity, and optoelectronic\nproperties of HGSC materials. We acknowledge that some statements in the original\nmanuscript lacked sufficient experimental or theoretical support. To address this, we\nhave conducted a systematic study of the physical properties of HGSC crystal materials,\nincluding a series of experimental verifications such as analysis of thermochromic\nbehavior, low-temperature superconducting transition tests, and evaluations of\nthermoelectric and optoelectronic response characteristics.\nResponse：(i) Thermochromic property: Systematic characterization confirmed that\nthe HGSC crystal exhibits reversible thermochromic behavior (Fig. R25a). Upon\ncooling from 300 K to 77 K, the crystal undergoes a reversible color change from pale-\nyellow to colorless, which remains stable over more than 100 thermal cycles (Fig.\nR25b), demonstrating the robustness and repeatability of the thermochromic behavior.\nFig. R25 a Thermochromic behavior observed upon cooling from 300 K to 77 K. b The\nphenomenon of thermal cycling of HGSC crystals for 100 cycles.\nResponse (ii) Low-temperature superconductivity and thermoelectric properties:\nDuring the preliminary synthesis stage of HGSC compounds, we observed a slight\nrepulsive phenomenon between the sample and the magnet when the sample was\nimmersed in liquid nitrogen (Supplementary Video 1). This prompted initial speculation\nregarding potential low-temperature superconductivity, referencing prior reports on\nHg₃²⁺-based compounds Hg AsF [Ref. 3 Phys. Rev. 45, 4836 (1992)]. However,\n3-δ 6\nduring the revision process, we confirmed through systematic low-temperature\nresistivity measurements (0-300 K) that HGSC did not display superconducting\ntransition characteristics within this temperature range (Fig. R26). These data clearly\nrule out the possibility of superconductivity in HGSC under the measured conditions.\nAdditionally, due to the material's high resistive properties (with a room temperature\nresistivity of 1 MΩ), the realization of its thermoelectric performance is fundamentally\nlimited. Based on the aforementioned experimental evidence, we have removed the\nrelevant speculations regarding superconductivity and thermoelectric performance in\nthe revised manuscript. We thank the reviewers for their in-depth discussion on this\nissue, which helped us avoid potential biases in our conclusions.\nFig. R26 The variation of electrical resistivity of HGSC crystals with temperature.\nResponse: (iii) Photodetection Properties of HGSC: Under standard testing\nconditions, the photovoltaic conversion ability of HGSC crystals was evaluated. Two\ncopper wires (diameter: 50 μm) were symmetrically fixed on the surface of the HGSC\nsingle crystal using highly conductive silver paste to construct a dual-electrode\nmeasurement configuration. The sample was illuminated by a monochromatic light\nsource of 350 nm (incident power density: 7 mW/cm2), and the photocurrent-time (I-t)\nresponse was recorded using a high-precision digital source meter (Keithley 2450) (Fig.\nR27). This result suggests that the HGSC crystals can effectively generate, separate,\nand transport photogenerated carriers under UV irradiation, thus providing\nexperimental evidence for their potential application in UV photodetection.\nFig. R27 The time-dependent photocurrent (I-t) density curve of the of HGSC crystals.\nWe thank the reviewer for the helpful suggestion. The inappropriate speculative\nstatement has been removed from the main text, and the latest experimental results have\nbeen incorporated into both the revised manuscript and the Supplementary (page 6).\nQuestion 8: The fitting of the XPS doesn’t match well with the experimental data at the\nshoulders of the peak. Might explain that.\nResponse: We sincerely appreciate the valuable comments from the reviewers\nregarding the peak fitting of the XPS spectrum. In response, we re-evaluated the\nelectronic structure characteristics associated with different oxidation states of mercury\n(Hg2+, Hg+, and Hg0), particularly considering the lower binding energy of metallic Hg⁰\ndue to its instability. Based on this analysis, we re-optimized the fitting parameters to\nmore accurately reflect the expected binding energy positions. The sentence “ In the\nanalysis of the Hg4f and Hg4f peaks, we observed six distinct peaks and determined\n5/2 7/2\ntheir binding energy positions to be 105.0, 103.9, and 104.3 eV (Hg4f ) and 101.1,\n5/2\n99.8, and 100.4 eV (Hg4f ), respectively.” has been modified to “The high-resolution\n7/2\nspectra revealed six different components, located at 105.3, 104.8, and 103.8 eV for the\nHg 4f levels, and at 101.2, 100.7, and 99.7 eV for the Hg 4f levels.” (Page 7). The\n5/2 7/2\nupdated fitting results, as shown in the revised Figure 2d (Fig. R28), show significantly\nimproved agreement with the experimental spectrum, particularly in the shoulder\nregions of the peaks.\nFig. R28. XPS spectrum of HGSC sample.\nQuestion 9: Quantitative analysis is commonly needed in EDS.\nResponse: We sincerely appreciate the reviewers' comments. Following their\nrecommendations, we conducted a comprehensive compositional analysis of the\ncompound HGSC using various characterization techniques.\nThe elemental ratio obtained from energy-dispersive X-ray spectroscopy (EDS)\nmeasurements was Hg : Ga : Se : Cl = 30.38 : 11.39 : 14.06 : 44.16 (Table R14), which\ndeviates from the expected stoichiometric ratio of Hg : Ga : Se : Cl = 36 : 16 : 16 : 64.\nWe attribute this deviation primarily to the influence of surface roughness, which can\naffect the accuracy of EDS quantitative analysis.\nTo obtain more precise quantitative data, we employed inductively coupled plasma\nmass spectrometry (ICP-MS), which offers higher quantitative accuracy. The results\nfrom the ICP analysis align closely with the crystallographic data, yielding a molar ratio\nof Hg : Ga : Se = 18.19 : 7.82 : 8.27 (Table R15). The following statement has been\nadded to the revised manuscript: “Inductively coupled plasma (ICP) quantitative\nanalysis confirms the molar ratio of Hg: Ga: Se as 18.19:7.82:8.27, consistent with the\nresults from single crystal refinement”. Please note that due to the low sensitivity of the\ninstrument to halogens, it was not possible to determine the chlorine content using\ninductively coupled plasma mass spectrometry (ICP-MS).\nTo address the issues regarding elemental quantitative analysis raised in this\ncommentary, the above discussion has been added to the \"Results and Discussion\"\nsection of the revised manuscript, with revision marks retained, as detailed on pages 5.\nFurthermore, Table R15 has been added as Supplementary Table S4 in the revised\nsupplementary materials.\nTable R14. EDS analyses of HGSC.\nElement Wt% Wt% Sigma Atomic %\nHg 63.72 0.18 30.38\nGa 8.30 0.11 11.39\nSe 11.61 0.12 14.06\nCl 16.37 0.11 44.16\nTotal 100.00 100.00\nTable R15. ICP-MS analyses of HGSC.\nTheoretical Atomic Theoretical\nElement Amount W%\nW % ratio atomic ratio\nHg 61.755 61.49% 60.84 18.19 18\nGa 9.148 9.18% 9.39 7.82 8\nSe 11.012 11.01% 10.64 8.27 8\nCl / / / / 32\nQuestion 10: Aside from the above, the authors might benefit from professional\nassistance in enhancing the clarity of the language.\nResponse: We appreciate the reviewer’s suggestions regarding the clarity of the\nlanguage. We have carefully reviewed the manuscript and sought the assistance of a\nprofessional editor to enhance its readability and ensure it meets the highest linguistic\nstandards. Any modifications will be highlighted in red in the revised version of the\npaper.\nSep 17th, 2025\nPoint-by-Point Responses to the Reviewers’ Comments\nDear reviewers：\nWe sincerely thank the reviewers for the insightful comments concerning our\nmanuscript. These comments are very valuable for further improving the quality of this\nmanuscript. In the revised manuscript, we have tried our best to address all the concerns\nraised by the reviewers, and the point-by-point responses are listed below. The changes\nto the manuscript are indicated in RED in the marked version of the revised manuscript.\nWe hope that the quality of our manuscript has met the publication standard after this\nrevision.\nReviewer # 1 (Remarks to the Author):\nThe authors have revised the manuscript in accordance with the requirements, resulting\nin substantial improvements. While the manuscript is acceptable for publication, three\nfollow-up concerns are noted below:\nQuestion 1: The author computed the Raman spectrum of HGSC; however, the\nfollowing key information is recommended for supplementation: (i) the specific value\nof the scissors operator; (ii) critical parameter settings employed in the calculations\n(e.g., cutoff energy, k-point grid).\nResponse: We would like to express our gratitude for the valuable comments provided\nby the reviewers. In the revised manuscript, we have included the following key\nparameters in the “Computational Methods” section on Page 17: A scissor operator\ncorrection of 0.449 eV was applied in the Raman spectra calculations; the plane-wave\ncutoff energy was set to 800 eV; and a (2 × 2 × 1) Monkhorst-Pack k-point mesh was\nemployed to ensure adequate convergence of the phonon frequencies and electronic\nstructure. These additions improve the transparency and reproducibility of the\ncomputational methodology.\nQuestion 2: The author supplemented the low-temperature test data of HGSC crystals\nin the revised manuscript. To ensure the reproducibility of the experiment, it is\nrecommended that the authors elaborate on the relevant instrument information in the\n“Characterization” section.\nResponse: We sincerely thank the reviewer for this constructive suggestion.\nAccordingly, we have now elaborated on the relevant experimental setup and conditions\nin the “Characterization” section of the revised manuscript (Page 16).\nQuestion 3: There are some formatting errors in the reference section that should be\ncorrected, such as Ref.47, 48, and 49.\nResponse: We thank the reviewer for this comment. The reference format has been\nupdated according to author guideline of Nature Communications.\n47. Chen, C. A., Li, Y., Jin, C. C., Ok, K. M. CsICl : A record-birefringent inorganic\n2\ncrystal with ultrawide infrared transparency enabled by linear interhalogen ICl –\n2\nunits. J Am Chem Soc. 147, 26133-26138 (2025).\n48. Li, G. M., Yang, Z. H., Hou, X. L., Pan, S. L. Chain‐like [S ](x = 2-6) units\nx\nrealizing giant birefringence with transparency in the near‐infrared for\noptoelectronic materials. Angew. Chem. Int. Ed. 135, e202303711 (2023).\n49. Zhang, B., Huang, W. Q., Zhang, J. X., Wu, X. T., Lin, H., Zhu, Q. L. [TeSeS ]2–:\nbirefringence. Angew. Chem. Int. Ed. e202508555 (2025).\nReviewer # 3 (Remarks to the Author):\nWhile the authors have addressed some of the previous feedback in their response,\nseveral critical points still require further clarification and elaboration to strengthen\nthe manuscript:\nQuestion 1: The authors stated that “only (001) plane crystals were available for\nmeasurement.” However, as a single crystal should inherently contain all\ncrystallographic planes, the claim of having “only (001) face” appears potentially\nmisleading. Could the authors clarify whether the absence of the (010) face is\nattributable to the crystal growth habit—where only the (001) face is naturally\nexposed—or whether it stems from technical limitations in cutting or measurement?\nAdditionally, please provide a detailed description of how the (010) face was\nsubsequently prepared or obtained for analysis.\nResponse: We thank the reviewer for the thoughtful comment and acknowledge that\nour original wording was imprecise. In the initial growth batches, most crystals\nnaturally developed with the (001) plane exposed, owing to the intrinsic anisotropic\ngrowth habit of HGSC under the applied conditions. A small number of (010)-oriented\ncrystals were also present, but they were extremely thin and small (typically < 1 mm in\nwidth), and thus unsuitable for optical characterization.\nDuring the revision, we optimized the CVT growth conditions by adjusting the\nprecursor ratio and extending the reaction time in the high-temperature zone, which\nenhanced supersaturation and promoted crystal growth. As a result, several millimeter-\nsized (010) single crystals were obtained, making them suitable for birefringence\ntesting, although the overall yield remained low.\nWe appreciate this valuable suggestion, which helped us to improve both the\naccuracy and completeness of the experimental section. The corresponding details have\nbeen added to the revised manuscript (see Page 15, “Synthesis and crystal growth”).\nQuestion 2: The birefringence (Δn) of HGSC at 546 nm was measured using a polarized\nmicroscopy technique that differs from the method applied at other wavelengths. Please\nelaborate on the rationale for employing a distinct methodology at this specific\nwavelength and provide comprehensive details regarding the experimental setup and\nprocedures. For consistency across wavelengths, could the polarization modulation\nmethod—which was successfully used at 1064 nm, 1540 nm, and 3500 nm—also be\napplied at 546 nm? If not, please justify the methodological divergence.\nResponse: We thank the reviewer for the thoughtful concern regarding the consistency\nof birefringence measurement methods across different wavelengths, and we are\npleased to clarify our rationale in detail.\nThe polarized microscopy technique combined with a compensator is a well-\nestablished and widely adopted standard method for birefringence measurements in the\nvisible region. This method originated in mineralogy and has been extensively used in\nthe optical anisotropy characterization of wide-bandgap crystals [Ref. 1 Optical\nMineralogy. UCL Press: London, (1992). Ref. 2 J. Am. Chem. Soc. 144, 9083-9090\n(2022), Ref. 3 Angew. Chem. Int. Ed. 64, e202422818 (2025)]. The polarizing\nmicroscopy technique typically uses a broadband white light source and assumes\nnegligible birefringence dispersion within the visible range. The 546 nm is commonly\ntaken as a representative wavelength for such measurements. Therefore, the method is\nnot readily applicable to other specific wavelengths. In addition, this approach relies on\nvisual interpretation of interference color patterns after compensator rotation, which\nmay become unreliable for colored or absorbing samples. During the revision process,\nwe discovered that HGSC exhibits a pronounced thermochromic behavior (Fig. 2e,\nPage 7): the crystal becomes optically transparent at around 140 K. This phenomenon\nsignificantly improves the reliability of interference color identification. We believe this\nlow-temperature measurement using the polarized microscope and compensator\nprovides a rigorous and valid determination of Δn at 546 nm. The experimental setup is\nshown in Fig. R1 and the step-by-step procedures are provided in the Supplementary\nInformation (Page 27).\nFig. R1 Experimental setup for birefringence characterization of an HGSC crystal at\n546 nm.\nAs for its applicability at 546 nm, the polarization-modulation method is less\nsuitable for such short wavelengths. As mentioned in the “Birefringence measurement\nof HGSC” section of the previous version of the Supplementary Information, “Second,\nthis method is better suited for long-wavelength birefringence measurements. In such\ncases, the phase retardation (hence the value of m) remains relatively small, which\nsupports a reliable and efficient determination.” At 546 nm, the birefringence of the\nmaterial is larger than in the infrared region, and the shorter wavelength also results in\nsmaller values of Δn and narrower Δn gap between adjacent m values. These factors\nm m\nlead to difficulties in resolving Δn unambiguously. Therefore, this method was only\nm\napplied at longer wavelengths (IR region) in our study.\nWe thank the reviewer once again for prompting this detailed explanation.\nQuestion 3: The principle and mathematical formulation used to determine the integer\nm in the polarization modulation method require clearer explanation. Although the\ntabulated data show that Δn values are similar across multiple m-values for two\ndifferent crystal thicknesses, the practice of selecting m based on minimal difference\nbetween these values may lack robustness. Given that, in principle, birefringence\nshould be independent of sample thickness, what accounts for the slight discrepancies\nobserved? Please propose a more rigorous approach for determining m or Δn to\nenhance the reliability of the results.\nResponse: We sincerely thank the reviewer for raising this critical point regarding the\ndetermination of birefringence (Δn) and phase order (m) in the polarization modulation\nmethod.\nIn this study, we adopted a reliable method for determining the birefringence of\nmicroscale crystals. As is well known, the available techniques for accurately\nmeasuring Δn in micron-sized crystals are extremely limited. Except for The polarized\nmicroscopy technique, there are few established options. Our measurement principle is\nsimilar to that of the polarized microscopy technique, both relying on extracting the\nphase difference (Δφ) between orthogonally polarized components to deduce Δn.\nHowever, our method offers significant advantages in terms of accuracy and\nreproducibility.\nThe polarized microscopy technique typically uses a broadband white light source\nand assumes zero birefringence dispersion within the visible range. In contrast, our\nmethod employs near-monochromatic laser sources (1064, 1540, and 3500 nm),\navoiding any assumption of dispersion. By directly calculating Δn from the known\nsample thickness and measured phase delay, this approach is better suited for precise\nevaluation of birefringence in small crystals during the material screening stage.\nRegarding the reviewer’s concern on how the integer m is determined, we have\nclarified the mathematical model in the revised manuscript. As mentioned in the\nSupplementary Information, the birefringence is calculated using the equation:\n𝜆𝜆(𝜑𝜑0 +2𝜋𝜋𝑚𝑚)\nwhere λ is the laser wavelength, Δφ𝑛𝑛 𝑚𝑚is =the measured phase delay (within 0 ≤ φ < 2π), d\n0 0\n2𝜋𝜋𝜋𝜋\nis the crystal thickness, and m is the integer ambiguity due to 2π periodicity. Since Δn\nm\nis a linear function of m (n = a × m + b), it is difficult to unambiguously determine m\nm\nfrom a single-thickness measurement. One must rely on whether the resulting Δn falls\nwithin a physically reasonable range, which introduces uncertainty. To address this\nissue, we developed a “dual-thickness cross-validation” strategy. Crucially, because the\nthickness appears in the denominator, the same 2π phase increment results in different\nΔn intervals for samples of different thicknesses. We measured two HGSC samples\nm\nwith different thicknesses (12.9 μm and 14.0 μm) under identical conditions, and\ncalculated Δn for each m. By presenting Δn vs. m and locating the intersection of the\nm m\ntwo linear functions, we identified the unique correct values of m and Δn based on the\nprinciple that Δn should be thickness-independent. As shown in Supplementary Table\n15, by comparing two such sets of Δn values obtained from HGSC samples, we\nm\nidentified a pair of values that closely agree, thereby uniquely determining the correct\ninteger m and the birefringence, then the actual results are as follows: At 1064 nm,\nperfect Δn matching occurred at m = 6 for both samples. At 1540 nm, matching occurred\nbetween m = 3 (12.9 μm) and m = 4 (14.0 μm), corresponding to a typical “1-order\noffset”. At 3500 nm, matching was achieved at m = 1. These consistent results across\ndifferent thicknesses and wavelengths demonstrate the validity and robustness of our\nmethod for determining m. The modifications have been incorporated into the\n“Birefringence Measurement of HGSC” section in the Supplementary Information\n(Page 33).\nAs for the reviewer’s concern about small discrepancies in Δn values between\nsamples of different thicknesses, we believe these (typically < 0.01) mainly result from\ncommon experimental uncertainties rather than a violation of the thickness-\nindependence principle. Potential error sources include: thickness measurement error:\nvariation in d directly affects the denominator and hence Δn; elliptical polarization\nmeasuring and fitting error: φ is derived from fitting the polarization ellipse, limited\n0\nby measurement accuracy of the parameters ψ and χ; crystal inhomogeneities: such as\nresidual stress, microcracks, or surface imperfections may cause local optical variations.\nTo further improve the reliability of m determination, we plan the following future\noptimizations:\n(i) Multi-thickness validation: If three or more samples with different thicknesses\nare available, multiple Δn –m curves can be generated. The intersection of these curves\nm\nnot only enhances the uniqueness and reliability of determining the correct m value, but\nalso allows for cross-sample curve fitting, enabling a more accurate and statistically\nrobust evaluation of Δn.\n(ii) Reducing phase accumulation: As shown in the equation, reducing thickness d\nor using longer wavelength λ increases the Δn gap between adjacent m values,\nm\nimproving resolution.\n(iii) Improving measurement precision: Enhancing the accuracy of both ellipticity\nparameters and thickness measurements will reduce cumulative errors and improve\noverall fitting quality.\nIn summary, although determining m has some inherent ambiguity, the dual-\nthickness matching method we applied provides stable and repeatable Δn results under\ncurrent experimental conditions. We fully agree with the reviewer that multiple-\nthickness measurements can further enhance the reliability of the method, and related\nexperiments will be carried out in the future. We will continue improving the\nmeasurement method.\nWe sincerely thank the reviewer again for this valuable suggestion, which helped\nus clarify the principle and reliability of our approach more thoroughly in both the main\ntext and supplementary materials.\nQuestion 4: The method used for crystal orientation (crystallographic axis\ndetermination) prior to birefringence measurements is not clearly described. Please\ninclude a detailed account of the orientation technique employed—such as X-ray\ndiffraction or optical conoscopy—along with corresponding results in the manuscript\nto ensure reproducibility and clarity.\nResponse: As mentioned in the Supplementary Information, “The orientation of the\ncrystallographic axes (a, b, c) was determined by a single-crystal X-ray diffractometer,\nand the approximate angles between the optical principal axes (x, y, z) and the\ncrystallographic axes were established based on theoretical calculations.” Here, we\nprovide a more detailed account of the crystal orientation procedure to improve the\nreproducibility and clarity of the experimental method.\nHGSC Crystals with a natural b-plane (i.e., b x) were first selected for\nbirefringence measurements, and the orientation of the e∥xposed face was determined\nby XRD (Supplementary Fig. 3). Subsequently, the crystallographic axis c of the crystal\nwas determined using a Bruker D8 VENTURE single-crystal X-ray diffractometer, and\nthe crystal orientation was subsequently analyzed with APEX Ⅲ software. As shown\nin Figure R2, the (001) plane (i.e., c-axis) was identified and the direction was\naccordingly marked. To establish a reliable connection between the crystallographic\norientation and the subsequent optical measurements, we next performed first-\nprinciples calculations to estimate the angular relationship between the crystallographic\naxes and the optical principal axes (x, y, z) in the HGSC crystal. The results indicated\nthat the b-axis is parallel to the x-axis, while the a- and c-axes are close to the y- and z-\naxes, respectively, with angular deviations of ~2 ° for a/y and ~5 ° for c/z. These\ntheoretical insights provided a reliable basis for guiding the subsequent optical\nalignment. Then the b-oriented crystal was placed in the laser beam path with normal\nincidence along the b-axis. A linear polarizer was set before the crystal, and an analyzer\nwas positioned behind. By rotating the crystal about the b-axis and monitoring the\npolarization state of the transmitted light, we identified the two orthogonal dielectric\naxes in the y-z plane. When the input polarization was aligned with one of the dielectric\naxes, the output polarization remained unchanged, allowing us to distinguish the optical\ny- and z-axes, where the axis closer to the c-direction was assigned as z.\nFig. R2 The crystal orientation of the HGSC crystal indexed by X-ray single-crystal\ndiffraction.\nThis integrated approach, combining X-ray diffraction, first-principles calculation,\nand polarization-resolved optical testing, enabled us to accurately correlate the\ncrystallographic and optical axes. The relevant experimental details and orientation\ndiagrams have been added to the supplementary information section “Birefringence\nMeasurement of HGSC.” (Page 28).\nWe again thank the reviewer for this helpful suggestion, which prompted us to\nimprove the methodological clarity of this key experimental step.\nQuestion 5: The birefringence measurement of HGSC at 546 nm was conducted at 130\nK, a temperature that is non-standard for such characterization. Please explicitly state\nthe rationale for selecting this temperature and include the measurement temperature\nin the main text to provide appropriate context for the reported values.\nResponse: We thank the reviewer for the constructive comment. The reason for\nperforming the birefringence measurement at 546 nm under cryogenic conditions lies\nin the optical appearance of the HGSC crystal. At room temperature, HGSC exhibits a\nnoticeable yellow coloration, which can significantly hinder the reliable identification\nof interference color orders under polarized light microscopy. Since the compensator\nmethod relies on distinguishing discrete interference fringes or color transitions, any\ninherent color in the crystal may obscure these visual cues and introduce ambiguity in\ndetermining the optical path difference. During our experimental optimization, we\nobserved that HGSC undergoes a distinct thermochromic transition: as detailed in the\nmanuscript (Fig. 2e, Page 7), the crystal becomes fully transparent below approximately\ninterference colors, thereby enabling accurate birefringence estimation. Thus, the\nchoice of 140 K was not due to a specific optical property at that temperature, but rather\nbecause it provides the necessary optical clarity for measurement using this method.\nWe also conducted additional measurements at 90 K to evaluate the temperature\ndependence of birefringence. Both sets of conditions were described in the manuscript\n(Pages 9-10). We thank the reviewer again for this helpful suggestion.\nReviewer #4 (Remarks to the Author):\nRen et al. have responded to my prior concerns point by point by incorporating\nappropriate experiments and analyses and have made the necessary revisions. These\nchanges substantially strengthen the validity of the optical characterization and\nhighlight the uniqueness of their polarization-anisotropic [Hg Se ] clusters. The\n3 2\nrevised manuscript, titled “Unprecedented [Hg Se ]2- cluster drives giant optical\n3 2\nanisotropy and broad infrared transparency,” is suitable for publication in Nature\nCommunications.\nI am pleased that several of my suggestions have led to essential improvements,\nstrengthening the central claim of exceptional experimental birefringence. I encourage\nthe authors to articulate, for a broad audience, the challenges that comprehensive\noptical measurements pose for small crystals, and to highlight their efforts in\ndeveloping and validating alternative approaches.\nResponse: We are deeply grateful to the reviewer for the many insightful comments\nand suggestions, which not only helped us identify areas for improvement but also\ngreatly guided us in revising and refining the manuscript. We fully agree with the\nreviewer’s observation regarding the challenges of comprehensive optical\nmeasurements on small crystals, and we have added corresponding clarifications in the\nrevised text. We sincerely thank the reviewer for the valuable guidance and constructive\nfeedback, which have enabled us to more clearly highlight the significance of this work\nin both the exploration of highly anisotropic materials and the development of reliable\nbirefringence measurement approaches.\nQuestion 1: One element remains, however, for a comprehensive determination of the\nbirefringence Eigenvalues: in the monoclinic C2/c space group (unique axis b), the off-\ndiagonal dielectric tensor component ε_ac is, in general, nonzero. Its magnitude rotates\nthe fast/slow axes away from the crystallographic a and c directions and modifies the\ncorresponding Eigenvalues of the dielectric function. The authors should comment on\nthe necessity of including ε_ac when selecting the measured optical axes and, if possible,\nquantify its magnitude—either experimentally or via theory.\nResponse: (i) Clarification on the ε component in the dielectric tensor\nac\nWe sincerely thank the reviewer for raising this important point. Indeed, in the\ncrystallographic coordinate system (a, b, c) of a monoclinic crystal, the dielectric tensor\ngenerally contains a non-zero off-diagonal component ε . The relative dielectric tensor\nac\ncan be expressed as:\n𝜀𝜀𝑎𝑎𝑎𝑎 0 𝜀𝜀𝑎𝑎𝑎𝑎\n𝜀𝜀 =  0 𝜀𝜀𝑏𝑏𝑏𝑏 0 \nIt should be noted that in monocli\n𝜀𝜀\nn𝑎𝑎i𝑎𝑎c sys\n0\ntems\n𝜀𝜀\n, 𝑎𝑎t𝑎𝑎he b-axis is perpendicular to the a-\nc plane, but the a and c axes are not orthogonal to each other. Therefore, the\ncrystallographic axes (a, b, c) do not constitute an orthogonal coordinate system. While\nthe b-axis typically aligns with one dielectric principal axis, the a and c axes do not\ncoincide with the dielectric principal axes.\nThe dielectric constants are directly related to the refractive indices, but since at\nthe current stage, the refractive indices of this crystal cannot be experimentally\ndetermined, we rely on theoretical calculations. Therefore, we have calculated the\ndielectric tensor using first-principles calculations (CASTEP), yielding the following\nresult:\n3.51569 0.00000 −0.05643\n𝜀𝜀 =  0.00000 3.15764 0.00000 \nThe magnitude of the off-diagonal component is ε = −0.05643.\n−0.05643 0.00000 5.a0c9191\n(ii) Analysis of the effect of the off-diagonal dielectric tensor component on\nbirefringence determination\nIt is worth emphasizing that, in monoclinic systems where the optical principal\naxes are unknown, the dielectric tensor expressed in Cartesian coordinates typically\ntakes the form:\n𝜀𝜀11 0 𝜀𝜀13\n𝜀𝜀 =  0 𝜀𝜀22 0 \nwhere ε\n13\nis the only non-zero off-di\n𝜀𝜀\na1g3onal\n0\nterm 𝜀𝜀3. 3Upon diagonalization, the tensor\nbecomes:\n𝜀𝜀𝑦𝑦 0 0\n𝜀𝜀 = 0 𝜀𝜀𝑥𝑥 0\nthe corresponding principal refractive in 0 dices 0 are: 𝜀𝜀 𝑧𝑧\n, , ,\nThis confirms that the d𝑛𝑛ie 𝑥𝑥 le=ctric𝜀𝜀 𝑥𝑥 tens𝑛𝑛or 𝑦𝑦 c=an b𝜀𝜀e 𝑦𝑦 dia𝑛𝑛g 𝑧𝑧 on=al√iz𝜀𝜀e 𝑧𝑧 d to orthogonal dielectric\nprincipal axes theoretically.\nMoreover, we performed first-principles calculations to determine the polarization\ndirection and further computed the angular relationship between the dielectric principal\naxes (x, y, z) and crystallographic axes (a, b, c). The results indicate that x is parallel to\nb, the y-axis is ~2 ° from the a-axis, and the z-axis is ~5 ° from the c-axis. These results\nshow that y is close to a, and z is close to c.\nIn the experiment, we used a b-surface crystal as the sample and a linearly\npolarized laser propagated along the crystallographic b-axis. By monitoring the\ntransmitted polarization state while rotating the crystal about b, we identified two\northogonal dielectric axes in the a-c plane. The axis closer to the c direction was\nassigned as z, and the orthogonal direction as y. This method allows reliable\nidentification of dielectric principal axes and ensures that the measured birefringence\ncorresponds to the correct eigenvalue difference.\nWe again thank the reviewer for highlighting this subtle yet crucial point.\nOct 17th, 2025\nPoint-by-Point Responses to the Reviewers’ Comments\nDear editor and reviewers：\nWe sincerely thank the reviewers for the insightful comments concerning our\nmanuscript. These comments are very valuable for further improving the quality of this\nmanuscript. In the revised manuscript, we have tried our best to address all the concerns\nraised by the reviewers, and the point-by-point responses are listed below. The changes\nto the manuscript are indicated in RED in the marked version of the revised manuscript.\nWe hope that the quality of our manuscript has met the publication standard after this\nrevision.\nReviewer #4 (Remarks to the Author)\nRen et al. have addressed the challenges of extracting optical properties from small\ncrystals and resolved my earlier concern regarding the off-diagonal dielectric constants\nthrough appropriate theoretical calculations. The revised manuscript is, in my view,\nsuitable for publication in Nature Communications. In addition, I have the following\nsuggestions to further improve the clarity of the article:\nThe authors are encouraged to incorporate a brief discussion of the monoclinic\ndielectric constants in the manuscript and laid out the details in the Supplementary\nInformation. Otherwise, directly stating “According to theoretical analysis, the\nrelationship between the crystallographic axes and the principal dielectric axes of this\ncrystal is a ≈ y, b = x, and c ≈ z” may confuse a general audience.\nSupplementary Table 11: Including a schematic illustration of the optical path with\nthe corresponding optical components, along with a mathematical expression showing\nhow sample phase retardation is compensated by the compensator phase retardation,\nwould help the audience better understand the method.\nSupplementary Table 15: I recommend condensing the table into three figures\nshowing the intersections between two (or more, if available in the final version) Δn-m\nlines, with error bars at each integer m. This would improve readability and error\ninterpretation.\nResponse: (i) We appreciate the valuable comments from the reviewers. To address this\nissue, we have moved the discussion regarding the dielectric constant of the monoclinic\nlattice, which was previously mentioned in our response letter during the second round,\nto the “Birefringence Measurement” section in the supplementary information.\nIt is worth emphasizing that, in monoclinic systems where the optical principal\naxes are unknown, the dielectric tensor expressed in Cartesian coordinates typically\ntakes the form:\n𝜀𝜀11 0 𝜀𝜀13\n𝜀𝜀 =  0 𝜀𝜀22 0 \nwhere ε\n13\nis the only non-zero off-di\n𝜀𝜀\na1g3onal\n0\nterm 𝜀𝜀3. 3Upon diagonalization, the tensor\nbecomes:\n𝜀𝜀𝑦𝑦 0 0\n𝜀𝜀 = 0 𝜀𝜀𝑥𝑥 0\nthe corresponding principal refractive in 0 dices 0 are: 𝜀𝜀 𝑧𝑧\n, , ,\nThis confirms that the d𝑛𝑛ie 𝑥𝑥 le=ctric𝜀𝜀 𝑥𝑥 tens𝑛𝑛or 𝑦𝑦 c=an b𝜀𝜀e 𝑦𝑦 dia𝑛𝑛g 𝑧𝑧 on=al√iz𝜀𝜀e 𝑧𝑧 d to orthogonal dielectric\nprincipal axes theoretically.\nTo establish a reliable connection between the crystallographic orientation and the\nsubsequent optical measurements, we next performed first-principles calculations to\nestimate the angular relationship between the crystallographic axes and the optical\nprincipal axes (x, y, z) in the HGSC crystal. The results indicated that the b-axis is\nparallel to the x-axis, while the a- and c-axes are close to the y- and z-axes, respectively,\nwith angular deviations of ~2 ° for a/y and ~5 ° for c/z. These theoretical insights\nprovided a reliable basis for guiding the subsequent optical alignment.\n(ii) We agree with the reviewer that a clear optical path diagram and corresponding\nmathematical description would greatly improve the clarity of the methodology.\nAccordingly, the following revisions have been made to Supplementary Figure 11:\nSchematic Diagram: A schematic diagram illustrating the working principle of the\npolarized light microscope (Fig. R1) has been added to the Supplementary Information.\nThis diagram clearly labels the positions of the optical components and the direction of\nthe light path.\nFig. R1 Schematic diagram of polarizing microscope.\nMathematical description: Additionally, a mathematical derivation has been included\nbelow the diagram to quantitatively describe the principle of phase compensation. Its\ncore expressions are as follows:\nTo compensate for birefringence in the sample, the compensator introduces an\nadditional birefringence via a thin sheet of calcite which is cut perpendicular to the\noptical axis. The retardance (optical path difference) ΔΓ introduced can be calculated\nvia:\n2 2\n𝑠𝑠𝑠𝑠𝑛𝑛 𝛼𝛼 𝑠𝑠𝑠𝑠𝑛𝑛 𝛼𝛼\n∆𝛤𝛤 = 𝑛𝑛𝑜𝑜𝑑𝑑1− 2 −1− 2 \nwhere d is the thickness of the calcite plate𝑛𝑛, 𝑜𝑜 nₒ and nₑ are t𝑛𝑛h 𝑒𝑒 e refractive indices for the\nordinary and extraordinary rays, and α is the tilt angle. The refractive indices of calcite\nare determined using the Sellmeier equations:\n2 2\n𝐵𝐵 𝜆𝜆 𝐷𝐷 𝜆𝜆\n𝑛𝑛 = 𝐴𝐴+ 2 + 2\nwhere λ is the wavelength (nm). The co𝜆𝜆eff−ic𝐶𝐶ients𝜆𝜆 fo−r t𝐸𝐸he ordinary and extraordinary\nrays are as follows: A = 1.7336, B = 0.9646, C = 1.9433×10−2, D = 1.8283, E = 120\no o o o o\nfor the ordinary ray and A = 1.3586, B = 0.8243, C = 1.0669×10−2, D = 0.1443, E =\ne e e e e\n120 for the extraordinary ray. The retardance values calculated using these formulae are\ntabbed out for different wavelengths and tilt angles on the following pages (Table. R1).\n(This dataset was provided with technical support from ZEISS.)\nTo address the suggestion regarding the operating principle of the compensator in\nthis comment, all the above discussions have been added to Figure 11 of the\nsupplementary information in the revised manuscript with revision traces.\nTable. R1 Retardance lookup sheet.\n(iii) We are deeply grateful to the reviewer for the invaluable suggestions. We concur\nthat presenting the data in a visual format substantially improves the clarity and\nreadability of the charts, rendering the difference analysis more intuitive. The revised\nfigure has been updated in supplementary information Figure 15 (Fig. R2). Once more,\nwe appreciate your contribution to enhancing the quality and lucidity of our manuscript.\nFig. R2 a-c Δnₘ values of HGSC crystals with two different thicknesses (12.9 μm and\n14.0 μm) at laser wavelengths of 1064 nm, 1540 nm, and 3500 nm, respectively. The\nred arrows indicate the m values where the Δnₘ curves intersect, corresponding to the\nunique solution for birefringence. Notably, in panel (b), the m value for the 14.0 μm\nsample is one order higher (m + 1) than that for the 12.9 μm sample. d-f Differences in\nbirefringence |Δn -Δn | as a function of m for the three wavelengths, used to\n12.9 14.0\ndetermine the optimal matching point with minimum deviation."}]

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