Heterocationic strategy to design a nonlinear optical crystal with a polar anti-perovskite structure

Abstract

The rational design of noncentrosymmetric (NCS) inorganic materials remains a longstanding challenge due to unpredictable solid-state reactions and stringent requirements on symmetry breaking. Although perovskites have been widely used as structural templates for NCS materials, their anti-perovskite counterparts remain unexplored. Here, by incorporating Cs⁺ and Cd²⁺ ions into a centrosymmetric Na₃Cl[MoO₄] template, we introduce a heterocationic strategy tailored for anion-centered anti-perovskites that, modifying the conventional BX₆ octahedra, yields the polar molybdate Cs₂NaCd₂Cl₃(MoO₄)₂. The heterocationic sublattice induces pronounced octahedral distortion, breaks inversion symmetry, and aligns the [MoO₄] tetrahedra, resulting in a wide band gap of 4.37 eV, a broad transparency window (0.26–5.42 μm and 6.25–10.46 μm), an SHG coefficient twenty times larger than KH₂PO₄ (d₁₅ = 9.31 pm/V) and a congruent-melting dynamics, enabling single-crystal growth. This work unveils the potential of heterocationic engineering in anion-centered frameworks demonstrating the design of NCS and polar functional materials with excellent nonlinear optical properties.

Introduction

Rationally designing and synthesizing inorganic functional materials is of vital importance for solid-state chemistry and materials science^1,2,3, particularly for compounds exhibiting noncentrosymmetric (NCS) and polar structures. These structures are vital for symmetry-dependent properties such as piezoelectricity, ferroelectricity, pyroelectricity, and second-harmonic generation (SHG)⁴. However, the inherently unpredictable nature of solid-state synthesis and the stringent requirements for symmetry-breaking make the discovery of new NCS compounds both rare and challenging^5,6,7. Perovskites (ABX₃; A, B = cations; X = anion) have long served as exemplary templates for achieving NCS structures due to their inherent structural versatility and the ability to accommodate adjustable distortions^8,9. To date, more than 100,000 perovskite-type compounds have been synthesized and engineered across diverse fields including catalysis, photovoltaics, and luminescence^10,11,12. Within these structures, three primary types of structural distortions have been identified, including tilting of the octahedra, displacement of the A-site ions, and distortion of the octahedra. These distortions not only enhance the structural diversity of perovskites but also offer valuable routes for the rational design of their structures^{13,14,15,16,17}.

In addition to perovskites, anti-perovskites (X₃BA; A, B = anions; X = cation), which are ionic position-exchange derivatives of perovskites, have commonly emerged as promising material templates, because they can offer comparable structural flexibility and tunability to perovskites¹⁸. Especially because the functional anions or anionic groups of anti-perovskites can be anchored at the A-site, where their distortion and orientation can be finely tuned by the surrounding cationic framework^19,20,21. These features are particularly important for designing polar and NCS nonlinear optical (NLO) crystals, which are capable of frequency conversion to extend the laser output range^22,23,24,25. In previous research, a series of high-performance anti-perovskite NLO crystals, such as Ae₃Q[GeOQ₃] (Ae=Ba, Sr; Q=S, Se)²⁶, K₃X[B₆O₁₀] (X=Cl, Br)^27,28, and Sr₃[CO₃][SnOSe₃]²⁹ have been rationally designed within anti-perovskite templates. Despite these advances, these reported anti-perovskite NLO materials primarily rely on structural distortions induced by octahedral tilting (Type-I, such as Na₃ClMoO₄³⁰ and Ba₃ClFeS₄³¹) or displacement of the A-site ions (Type-II, such as Ba₃[TO₃][SnOQ₃])³². The third mode, octahedral distortion (Type-III), remains unexplored in anti-perovskites, presenting a critical knowledge gap.

As is well-known, the octahedral distortion in perovskites can typically be achieved by introducing cations with strong Jahn–Teller effects^33,34. This strategy, however, is not applicable in anti-perovskites, where the octahedral centers are occupied by anions instead of cations. In this research, we propose a “heterocationic strategy” (analogous to the “heteroanionic” approach in cation-centered compounds^35,36,37) that can be employed in anion-centered anti-perovskites to induce obvious octahedral distortion based on the difference of size and electronegativity of various coordinated cations (Fig. 1).

Fig. 1: Three different types of distortions in perovskites and anti-perovskites.

The crystal symmetry is described in brackets below the chemical composition of each structure.

Motivated by these ideas, we chose anti-perovskite molybdate Na₃Cl(MoO₄) as template to design a new polar and NCS infrared NLO crystal with the following considering: i) the [MoO₄] tetrahedron is intrinsically prone to distortion under variations in its local coordination environment, offering a structural basis for symmetry breaking³⁸; ii) molybdate-based compounds can often exhibit broad optical transparency in the mid-infrared (mid-IR) region, moderate birefringence, strong SHG effect, and facile crystal growth^39,40. Starting with centrosymmetric a⁰a⁰c^–-tilted anti-perovskite molybdate Na₃Cl(MoO₄), we creatively utilize the “heterocationic strategy”, that is, introducing larger Cs⁺ and aliovalent Cd²⁺ cations to partly substitute Na⁺ cations in Na₃Cl(MoO₄) to construct a strongly distorted B-site octahedral distortion for designing NCS and polar anti-perovskite molybdate, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. Here, Cd²⁺ serves as a structural and electronic modulator, whereas Cs⁺ acts as a distortion inducer and symmetry breaker. In its structure, largely distorted heterocationic lattice breaks local symmetry of [MoO₄] tetrahedra and constrains their alignment within the octahedral cavity, leading to net polarization along the c-axis. As a result, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ exhibits excellent NLO performances, including: i) a wide band gap of 4.37 eV; ii) strong SHG coefficients (d₁₅ = 9.31 pm/V), over 20 times that of KH₂PO₄ (KDP, d₃₆ = 0.39 pm/V); and iii) an exceptional transparency range of 0.26–5.42 μm and 6.25–10.46 μm, which is unprecedented among oxide-based molybdates. Additionally, it exhibits congruent melting behavior, enabling the growth of high-quality single crystals up to 18 mm using top-seeded solution growth (TSSG). These properties make [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ promising mid-IR NLO crystal. To gain deeper insight into the origin of these properties, we performed Mulliken population analyses, dipole moment calculations, and the first-principles calculations, which collectively elucidate the structure-composition-property relationships. This work highlights the heterocationic strategy as a highly effective approach for the rational design and synthesis of high-performance functional crystals in anion-centered compounds.

Results and discussion

Crystal structure

[Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ was synthesized via the conventional solid-state reaction using stoichiometric amounts of raw reagents (see Synthesis and Crystal Growth for details). Energy-dispersive spectroscopy (EDS) confirmed the presence of Cs, Na, Cd, Cl, Mo and O with atomic ratios consistent with the expected composition (Supplementary Fig. 1). Phase purity was verified by powder X-ray diffraction (PXRD) (Supplementary Fig. 2).

Single-crystal XRD analysis reveals that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ crystallizes in the NCS and polar orthorhombic space group, Pba2 (No. 32) (Supplementary Table 1–4). The asymmetric unit comprises one crystallographic-independent Cs, one Na, one Cd, one Mo, two Cl, and four O atom(s). The Mo⁶⁺ cations adopt a tetrahedral coordination with four O atoms, forming [MoO₄] groups with the Mo–O distances ranging from 1.698(3) to 1.847(3) Å (Fig. 2a). The Cl^– anions exhibit two different coordination environments: the [Cl(1)Cs₃NaCd₂] and [Cl(2)Cs₄Cd₂] octahedra. In the former, Cl(1) is coordinated to three Cs⁺, one Na⁺ and two Cd²⁺ cations with the Cl(1)–Cs distances ranging from 3.578(7)–3.761(6) Å, Cl(1)–Na distance being 3.007(6 Å, and Cl(1)–Cd distances ranging from 2.629(8)–2.729(8) Å (Fig. 2b). In the latter, Cl(2) is bonded to four Cs⁺ and two Cd²⁺ cations, with Cl(2)–Cs distances ranging from 3.578(7)–3.878(2) and Cl(2)–Cd distances of 2.593(2) Å (Fig. 2c). These heterocationic octahedra are interconnected via edge-, corner-, and face-sharing to form a highly distorted three-dimension (3D) anti-perovskite-like framework, in which the aligned [MoO₄] tetrahedra are embedded in the interstitial voids (Fig. 2d). Notably, there exist two topological types of octahedra with different connectivity diagram (CDm)⁴¹ in the 2:1 ratio. Bond-valence calculations yield values of 0.90 (Cs⁺), 0.92 (Na⁺), 1.98 (Cd²⁺), 6.14 (Mo⁶⁺), 1.94 ~ 2.08 (O²⁻), and 1.06 (Cl^-), consistent with the expected oxidation states^42,43.

Fig. 2: Crystal structural features of [Cs₂NaCd₂Cl]Cl₂(MoO₄).

a–c Basic building units of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂: [MoO₄], [Cl(1)Cs₃NaCd₂] and [Cl(2)Cs₄Cd₂]. d 3D structure of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. The structural connectivity diagrams of the octahedra forming the anti-perovskite-like framework are shown on the right side; these disordered oxygen atoms and bonds have been omitted for clarity.

Structural evolution of the heterocationic framework

In this work, we elucidate how the heterocationic strategy influences structural symmetry by examining the progressive evolution of heterocationic octahedral frameworks. The transition from the centrosymmetric (CS) parent compound Na₃Cl(MoO₄) (P4/nmm), to the polar and NCS [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ (Pba2) can be divided into two simple routes: (i) aliovalent cationic substitution and (ii) vacancy and charge compensation. For clarity, a hypothetical intermediate, [NaCdV_Na]Cl(MoO₄) (where ‘V_Na’ denotes a cationic vacancy of Na⁺ cation), is introduced to describe the structural evolution.

In Na₃Cl(MoO₄), the X-site of the anti-perovskite framework is uniformly occupied by homogeneous Na⁺ cations (Fig. 3a), resulting in a minimal distortion (distortion index (D) = 0.019) of [ClNa₆] octahedra (Fig. 3b)⁴⁴. Under such conditions, configurational entropy dominates. Influenced by the shape of the [MoO₄] tetrahedron itself, the [ClNa₆] octahedra are arranged in antiparallel exhibiting a⁰a⁰c^– tilt system, resulting in a CS and nonpolar point group 4/mmm. Partial substitution of four Na⁺ cations in [ClNa₆] octahedra by two Cd²⁺ cations introduces heterocations and creates two cationic vacancies, forming the intermediate [NaCdV_Na]Cl(MoO₄) (Fig. 3c). Cd²⁺ was selected as an aliovalent substituent for Na⁺ owing to its close ionic size and coordination compatibility, which perturbs the local electronic environment and promotes asymmetric Mo–O coordination. The stronger Lewis acidity of Cd²⁺ and the presence of vacancies induce large local distortions, increasing the octahedral distortion index to D = 0.098 (Fig. 3d). Consequently, the framework undergoes cooperative tilting of adjacent heterocationic octahedra along the c-axis (tilt system a⁰b^–c⁺), lowering the symmetry to a polar mm2 point group and further guiding the ordered orientation of the A-site [MoO₄] groups along the c-axis. To stabilize the vacancy-disrupted framework and achieve charge balance, the [Cs₂Cl]_∞ chains are introduced, effectively compensating for one Na⁺ cation and the two associated vacancies. Owing to its considerably larger ionic radius, Cs⁺ occupies large coordination sites, thereby inducing lattice distortion through elongated Cs–O bonds and lowering the overall symmetry. This leads to the formation of the final compound, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ (Fig. 3e). Due to lower Lewis acid strength and the large ionic radius of Cs⁺ (1.69 Å) relative to Na⁺ (0.95 Å), the [ClCs₃NaCd₂] octahedra exhibit an even larger distortion (D = 0.142), transitioning from a simple corner-sharing network to a more complex structure incorporating corner-, edge- and face-sharing connections (Fig. 3f). Moreover, the smaller [MoO₄] tetrahedra, relative to the oversized and distorted anti-perovskite voids, impose additional lattice strain, driving further octahedral tilting and the complete disruption of the original four-fold axis. The synergistic interactions among multi-cations Cs⁺, Cd²⁺, and Na⁺ not only preserve the overall anti-perovskite-like framework but also promote polar alignment of the A-site [MoO₄] groups. Clearly, the heterocationic approach obviously increases octahedral distortion, and simultaneously achieves a symmetry-breaking effect similar to that of heteroanionic groups in cation-centered materials⁴⁵.

Fig. 3: Transition from parent Na₃Cl(MoO₄) (CS) to polar [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂.

Global structures and (b, d, f) local modifications to the octahedron through the transition from (a) the parent Na₃Cl(MoO₄) with its (b) [ClNa₆] octahedron, to (c) the quasi-anti-perovskite [NaCdV_Na]Cl(MoO₄) and (d) its [ClNa₂Cd₂V_Na2] octahedron, finally to (e) the anti-perovskite [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ and (f) its [Cs₄Na₂Cd₄Cl₃] dimer. The distortion index (D) of different octahedra is shown for each structure.

Performance characterization

Despite the progressive increase in octahedral distortion induced by the heterocationic approach, the low global instability index (GII) of 0.086 vu and bond strain index (BSI) of 0.043 vu confirm that all atomic positions in [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ are highly reasonable, resulting in a structurally stable framework^46,47. Consequently, the compound exhibits congruent melting behavior. Thermogravimetric and differential scanning calorimetry (TG/DSC) analyses (Supplementary Fig. 3), combined with melting recrystallization experiments (Supplementary Fig. 4) verify that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ undergoes complete melting without decomposition. A high-quality single crystal of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂, measuring 18 × 11 × 5 mm³, was successfully grown using the TSSG method with self-flux (Fig. 4a). The obtained crystal exhibited well-faceted growth, with a morphology consistent with predictions based on the Bravais-Friedel-Donnay-Harker (BFDH) model implemented in Mercury program (Supplementary Fig. 5)⁴⁸. Furthermore, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ exhibits high stability under ambient conditions with no observed deliquescence, and only minor surface dust accumulation was noted after one month of air exposure (Supplementary Fig. 6). Herein, the linear and NLO properties of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ were subsequently investigated based on the obtained high-quality crystals.

Fig. 4: Spectroscopic and optical property measurements of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂.

a Photograph of a [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ single-crystal sample. b Transmission spectrum of the [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ single crystal, from near infrared to ultraviolet range. c Comparison of the transparency windows of molybdates (red) and other oxide-based NLO materials, including molybdates (orange), niobate (blue), titanates (cyan), borates(purple). Powder SHG measurements at (d) 1064 nm and (e) 2090 nm. The SHG intensity is plotted in arbitrary units (arb. units) to facilitate a direct comparison of the relative intensity. f Demonstration of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ birefringence and cross-polarized images showing interference colors. g Calculated refractive index components of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ at 1064 nm, showing its birefringence. h Performance benchmarking of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ (red) compared to KTP (green) and KDP (blue).

The Ultraviolet-Visible-Infrared (UV–vis-IR) transmission spectrum of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ was measured using a 1-mm-thick crystal wafer at room temperature. As shown in Fig. 4b, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ displays a broad transparent region spanning from 0.26 to 5.42 μm and 6.25 to 10.46 μm, with an average transmittance of approximately 80% within these ranges. The wide absorption peaks observed between 5.42 to 6.25 μm is attributed to the potential multi-phonon absorption, a common characteristic of infrared nonlinear optical crystals^49,50,51. The fundamental Mo–O stretching vibration of the [MoO₄]^2– groups is confirmed at 804 cm^–1 by the IR spectrum of a polycrystalline sample (Supplementary Fig. 7a). This assignment was further supported by complementary Raman spectroscopic measurement and the calculated vibrational frequencies, which further revealed the corresponding vibrational signatures of MoO₄ units (Supplementary Fig. 7c, d). The IR cutoff edge was considerably red-shifted as compared to those of other molybdate compounds (LiNa₅Mo₉O₃₀: 5.26 μm⁴⁰, MoO₂Cl₂: 4.54 μm⁵², LiVMoO₆: 4.95 μm⁵³) and was comparable to those of AgGaS₂ (AGS) (13 μm)⁵⁴ and ZnGeP₂ (13 μm)⁵⁵. To the best of our knowledge, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ is the first molybdate with an IR absorption edge exceeding 10 μm, and its transparent region is one of the widest among all oxide-based NLO materials (Fig. 4c and Supplementary Table 5).

To further evaluate its optical band gap, the UV–vis transmission spectrum was converted to absorption using the Kubelka-Munk function⁵⁶, yielding an optical band gap of 4.37 eV (Supplementary Fig. 8). This value is notably larger than those of commercially available IR NLO crystals and other molybdate-based materials, such as AgGaSe₂ (1.83 eV)⁵⁷, ZnGeP₂ (1.97 eV)⁵⁵, LaBrMoO₄ (3.31 eV)⁵⁸, and Ag₂Mo₃Te₃O₁₆ (2.85 eV)⁵⁹.

Generally, the large band gap implies that the title material may have a high laser-induced damage threshold (LIDT) value. Therefore, the powder LIDTs of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ and AGS at the particle-size range of 75−106 μm were evaluated using nanosecond pulsed lasers at 1064 nm (Supplementary Table 6). The results show that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ exhibits a remarkably high LIDT value, approximately 100 times higher than that of AGS. This strong damage resistance is comparable to other oxide-based NLO materials, such as LiNbO₃ (0.84 GW/cm²) and LiNa₅Mo₉O₃₀ (1.2 GW/cm²), suggesting its strong potential for high-power mid-IR laser applications^40,60.

Due to its polar structure, the SHG intensities of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ were evaluated using Kurtz and Perry method under 1064 nm and 2090 nm laser irradiation, with KDP and AGS serving as the reference materials, respectively⁶¹. As shown in Fig. 4d, 4e, the SHG intensity increases with particle size, indicating its phase-matching (PM) behavior across the near- to mid-infrared region. Notably, for particle sizes in the range of 75–106 μm, the SHG intensity of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ is approximately 3.5 times higher than that of KDP at 1064 nm. Under identical particle-size conditions at 2090 nm, its SHG intensity is about 0.2 times that of AGS, slightly exceeding the SHG intensity observed at 1064 nm. Strong SHG responses were observed at both 1064 nm and 2090 nm fundamental wavelengths, confirming the efficient NLO activity of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ across the near- to mid-infrared region. Furthermore, in accordance with the mm2 point group symmetry and Kleinman’s symmetry rules, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ is predicted to have three independent non-zero components of the second-order NLO susceptibility tensor (d_ij)⁶². At a wavelength of 1064 nm, the calculated SHG coefficients are d₁₅ = 9.31 pm/V, d₂₄ = 8.58 pm/V, and d₃₃ = −8.53 pm/V. Notably, d₁₅ is 20 times stronger than the d₃₆ coefficient of KDP (0.39 pm/V)⁶³, which strongly suggests that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ is a potential NLO crystal. It should be noted that the apparent discrepancy between the large calculated d₁₅ value and the moderate powder SHG intensity originates from the strong orientation dependence of the effective nonlinear coefficient (d_eff) in mm2 symmetry and the relatively large phase-matching angles imposed by the crystal’s small birefringence^64,65.

In addition, its birefringence was measured using a cross-polarizing microscope, revealing a first-order yellow interference color under cross-polarized light (Fig. 4f). Based on the Michel−Levy chart, the retardation (R values) was determined to be 280 nm. The [010] crystallographic orientation was determined by analysis conducted using a Bruker SMART APEX III diffractometer (Supplementary Fig. 9). As the crystal thickness used for the measurement is 7.6 μm, the birefringence could be calculated as 0.036, which is sufficient to support PM behavior⁶⁶. Additionally, the calculated birefringence (Δn) of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ was evaluated through the first-principles calculations. The refractive index difference (n_z– n_y) exceeds (n_y– n_x), classifying [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ as a positive biaxial crystal with the birefringence of 0.034 @ 1064 nm (Fig. 4g). These results indicate that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ is a promising NLO crystal candidate for dual-wavelength applications (Fig. 4h)^67,68.

Structure-property relationships

According to Chen’s anionic group theory, the macroscopic nonlinearity of a NLO crystal arises from the geometrical superposition of the microscopic second-order susceptibility of its NLO-active anionic groups⁶⁹. Therefore, analyzing the distortion and aligned orientation of the [MoO₄] group is crucial to understanding the origin of the SHG response. To elucidate this relationship, we examined the coordination environment of A-site anions or anionic groups across three representative frameworks: the ideal cubic anti-perovskite X₃BA, the CS Na₃Cl(MoO₄) and the NCS [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. In the ideal anti-perovskite X₃BA, the A-site anions are coordinated by twelve equivalent X-site cations, forming a regular [AX₁₂] polyhedron with six short and six long A–X bonds, reflecting minimal distortion (Fig. 5a). In [Na₃Cl](MoO₄), each O^2- anion of the [MoO₄] tetrahedron bonds to one Mo⁶⁺ and three Na⁺ cations, yielding a nearly ideal [(MoO₄)Na₁₂] coordination environment (Fig. 5b), thereby suppressing the distortion of [MoO₄] group (Table 1)⁴⁴. In contrast, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ exhibits a reduced coordination number (from 12 to 9), forming a distinctly asymmetric [(MoO₄)Cs₄Na₂Cd₃] configuration (Fig. 5c). The incorporation of multiple cations imposes diverse bonding interactions on the O^2– anions, promoting the distortion of the [MoO₄] tetrahedra, as reflected in the increased bond length distortion parameters (Table 1). These findings highlight the pivotal role of heterocationic engineering in tailoring the coordination asymmetry and enhancing acentric distortion of A-site group in anti-perovskite-related materials.

Fig. 5: Analysis of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ structure-property relations.

Coordination environment of the A-site anion or anionic groups (white) in (a) the ideal anti-perovskite X₃BA, (b) Na₃Cl(MoO₄) and (c) [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. d, e Orientation of the [MoO₄] tetrahedron in the unit cell and (f, g) its spatial arrangement in the lattice, for (d, f) Na₃Cl(MoO₄) and (e, g) [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. Elements on the lattice are labeled by colored circles as shown in the legend. The crystal reference frame of [MoO₄] (a–c) is indicated by the colored axis. The thick red arrows indicate the orientation of the dipole moment of these [MoO₄] groups (f, g).

Table 1 Structural information of anti-perovskites

Beyond the local distortion, the heterocationic superlattice strongly influences the global spatial orientation of A-site groups. In Na₃Cl(MoO₄), where the octahedra framework are compositionally homogeneous, the [MoO₄] tetrahedra exhibit a diagonally reversed orientation (Fig. 5d). Driven by the principle of local electroneutrality and in conjunction with the a⁰a⁰c^– tilt system, adjacent [MoO₄] tetrahedra are arranged in antiparallel, which results in a CS structure (Fig. 5f). However, the introduction of Cs⁺ and Cd²⁺ cations in [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ alters this behavior. The presence of three Cd²⁺ cations in the equatorial plane exerts strong Coulombic attraction, ‘pinning’ three O²⁻ atoms within the same plane (Fig. 5e), thereby promoting a coherent orientation of the [MoO₄] groups. Simultaneously, the tilt system transitions from a⁰a⁰c⁻ to a⁰b⁺c⁻, favoring a net polar alignment of the [MoO₄] tetrahedra along the [001] direction (Fig. 5g). These show that heterocationic approach is an efficient strategy to achieve “symmetry-breaking” and “structure-directing” of the A-site anion group in anti-perovskites.

To better indicate that, Mulliken population (MP) analysis based on first-principles calculations was performed (Supplementary Table 7)⁷⁰ and the dipole moments of individual structural units within the unit cell were evaluated using the Debye equation (Supplementary Table 8)⁷¹. The MP results reveal that the longer Na–O and Cs–O bonds are characterized by relatively low overlap populations (−0.05 ~ 0.14 and −0.1 ~ −0.04, respectively), indicating their predominantly ionic nature. In contrast, the Cd–O bonds exhibit comparatively higher overlap populations (0.21–0.25), reflecting enhanced covalency and stronger Cd²⁺ and O^2– interactions. This disparity in bonding strengths leads to an electrostatically asymmetric environment around the [MoO₄] tetrahedron. Notably, one triangular face of [MoO₄] unit is preferentially stabilized through stronger bonding of three equatorially positioned Cd²⁺ cations (Fig. 5e), effectively anchoring the tetrahedron within the cavity and promoting ordered alignment. To study the structural orientation, the dipole moments of [MoO₄] units for a unit cell were projected along the [001], [100] and [010] crystallographic directions. As summarized in Supplementary Table 8, the dipole projections along [100] and [010] are both negligible (0.00 Debye), whereas a substantial net dipole of –10.77 Debye is observed along [001], confirming dominant polarization along the c-axis. These findings reinforce that the structural polarity in [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ originates primarily from the cooperative alignment of dipolar units, most notably the [MoO₄] groups, within the asymmetric multi-cationic octahedral framework.

To further investigate the optical properties, the electronic structures of title compound was studied using the first-principles calculations⁷². The results show that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ has an indirect band gap of 3.19 eV (Fig. 6a). The calculated band gap is obviously smaller than the experimental one, which can be attributed to the discontinuity of exchange-correlation energy⁷³. Typically, this calculation method yields band-gap value that is 30–50% smaller than those obtained experimentally^74,75. The total and partial density of states (DOS) analysis indicates that the top of valence band (VB) is dominated by O 2p, Cl 2p and Cd 4 d orbitals, while the bottom of conduction band (CB) consists primarily of O 2p, Mo 4d and Cd 5s orbitals. There is a clear hybridization behavior between these orbitals (Fig. 6b). Furthermore, the element-projected DOS indicates evident hybridization among Cd 4d, Mo 4d, and O 2p orbitals, especially near the top of the valence band. Although Cd²⁺ (d¹⁰) and Mo⁶⁺ (d⁰) are typically considered electronically inert in directional bonding, the observed orbital overlap implies the presence of weak yet vital covalent interactions. These interactions enhance the local polarization around the [MoO₄] tetrahedra and promote their alignment along the c-axis. Therefore, the structural orientation, dictated by both electrostatic and covalent factors, plays a critical role in establishing the NCS lattice and the resulting NLO properties.

Fig. 6: First-principles analysis of structure-property relations of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂.

a Electronic band structure of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. b The total (gray) and partial (colored) density of states, projected on the constitutional atoms of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂, as indicated in the panels of the plot. c Maps of the electron localization function of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ along (001) slices. The isosurface value is set to 0.005 eV Å⁻³ and the label of the color scale represents the normalized ELF. d Calculated infrared (IR) absorption spectra of [MoO₄] tetrahedron and [MoO₄Cd₃] group. The peaks correspond to the vibrational resonances obtained from Gaussian calculations for the [MoO₄] tetrahedron and and [MoO₄Cd₃] groups.

To gain further insight into the electronic structure and charge distribution, the electron localization function (ELF) was employed. As shown in Fig. 6c, pronounced lobe-shaped and asymmetric electron density distributions are localized around the oxygen atoms, a characteristic signature of highly polarized covalent Mo–O bonds^76,77. This reflects strong directional covalency and charge separation, resulting in strong local dipole moments. The directional lone-pair-like orbitals on oxygen are highly polarizable under external fields, enhancing microscopic hyperpolarizability. Furthermore, the cooperative alignment of these Mo–O units in the NCS lattice leads to additive macroscale dipoles, breaking inversion symmetry and facilitating strong SHG response. Thus, MoO₄ groups act as key SHG-active motifs. In contrast, the electron density around Cs⁺ and Na⁺ ions is more symmetrically distributed, characteristic of ionic bonding with negligible contribution to the SHG response. However, it is worth considering whether Cd²⁺ ions with the unique d¹⁰ orbital will have an impact on the NLO properties?

To investigate the specific role of Cd²⁺ ions in the NLO properties, we theoretically constructed a fully Ca-substituted model, [Cs₂NaCa₂Cl]Cl₂(MoO₄)₂, based on the Cd-structure. This model was subjected to first-principles geometry optimization using a high cutoff energy (860 eV) and a strict convergence criterion (2.0 × 10^–6 eV/atom). Although the experimental substitution of Cd with Ca was not achieved due to the high stability and energetically favorable formation of CaMoO₄ in this system (Supplementary Fig. 10a, b), this Ca-substituted model relaxed smoothly to a local minimum, confirming the geometric stability of the Ca analogue and its suitability for electronic comparison (Supplementary Table 9). Electronic-structure analysis reveals that the Ca-analogue has slightly larger band gaps (3.36 eV) than [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ (3.19 eV) (Supplementary Fig. 11a, b). This can be attributed to the higher electropositivity of Ca²⁺ and its weaker orbital contribution near the band edges. This trend is consistent with experimental observations in other Ca–Cd isostructural systems, such as CaWO₄ (4.94 eV) vs. CdWO₄ (4.15 eV)⁷⁸ and CaMoO₄ (3.96 eV)⁷⁹ vs. CdMoO₄ (3.31 eV)⁸⁰. Furthermore, based on the electronic structures, the NLO coefficients of [Cs₂NaCa₂Cl]Cl₂(MoO₄) was also calculated and compared with [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ (Supplementary Table 10). The results clearly show that the Cd-based compound exhibits stronger NLO coefficients than Ca-analogue, confirming that Cd²⁺ cations indeed contribute positively—though not dominantly—to the NLO properties of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. To visually illustrate the contribution of Cd²⁺ cations, the PDOS and ELF for each atom in both compounds is provided in Supplementary Fig. 11c–e. The PDOS show that the contributions of Cs, Na, Mo, O and Cl were almost identical, with the main difference arising from the metal cations Ca²⁺ and Cd²⁺. In [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂, the Cd 3s and 3p orbitals show weak contribution near the Fermi level, while in [Cs₂NaCa₂Cl]Cl₂(MoO₄)₂, the Ca²⁺ contribution near the Fermi level is negligible. In addition, the Cd 4 d orbitals form narrow and deep bands centered around −7 eV, and show weak direct hybridization with the coordinating ions p states (Supplementary Fig. 11c, d). Affected by this, Cd²⁺ ions and their coordinating ions show obvious asymmetric electron density distribution, while Ca²⁺ exhibits a nearly spherically symmetric electron distribution (Supplementary Fig. 11e), which indicates the unique polarized characteristic of Cd²⁺ ions. Thus, while the filled Cd 4d¹⁰ shell contributes relatively little directly to frontier orbital mixing, it enhances the overall polarizability of the lattice and stabilizes the NCS framework, thereby amplifying the macroscopic second-order susceptibility⁸¹. Based on the above analyses, it is clear that the distinct electronic configurations of metal cation have led to fundamentally different structural and NLO behaviors of [Cs₂NaCa₂Cl]Cl₂(MoO₄) and [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂, although Ca²⁺ and Cd²⁺ are close in size. The presence of the polarizable Cd²⁺ (4d¹⁰) center is also indispensable for maintaining both the polar structure and the large NLO response.

To further elucidate the orbital-level origin of the NLO properties, we conducted electron density difference (EDD) analysis via first-principles DFT calculations and also examined the frontier molecular orbitals of the MoO₄Cd₃ group using Gaussian 09⁸². The EDD results (Supplementary Fig. 12) reveal distinct charge redistribution, characterized by strong accumulation along Mo–O and Cd–O bonds and negligible depletion around Cs⁺/Na⁺, which favor the generation of periodic polarization along the c-axis. This spatial modulation correlates with the frontier orbital asymmetry (Supplementary Fig. 13), where the calculated highest occupied molecular orbital (HOMO) is localized on MoO₄ (O 2p orbitals) and the lowest unoccupied molecular orbital (LUMO) extends toward Cd²⁺ centers, indicating pronounced characteristics within the MoO₄Cd₃ groups. Such an orbital arrangement promotes intragroup charge transfer and reinforces the periodic polarization within the Cd-Mo-O layers, providing a direct orbital-level explanation for the observed strong NLO response.

Additionally, we calculated IR vibrational frequency by the Gaussian 09 program on isolated [MoO₄] tetrahedra and coordinated [Cd₃MoO₄] units to elucidate the origin of the unusually broad transparency range up to 10 μm in [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. As shown in Fig. 6d, the IR vibrational modes of isolated [MoO₄] tetrahedra are mainly located in the 8–12 μm range, overlapping with the key atmospheric transmission window and thus limiting mid-IR transparency. In contrast, the main vibrational absorption of the [Cd₃MoO₄] unit exhibits a markedly redshift, with its strongest phonon modes appearing beyond 12 μm. This indicates that the incorporation of heavy, high-electronegativity Cd²⁺ ions into the anionic framework effectively redshifts the local vibrational frequencies via octahedral distortion and lattice softening, thus expanding the phonon-limited infrared cutoff wavelength from the typical ~5 μm seen in conventional molybdates to as far as 12 μm in this heterocationic system. Notably, a relatively strong absorption is still observed in the 5.42–6.25 μm region (Fig. 4b), which corresponds well with the antisymmetric stretching modes of Mo–O bonds in partially retained or weakly distorted [MoO₄] tetrahedra. Together, these results demonstrate that the heterocationic strategy not only introduces distorted anion-centered polyhedra to promote noncentrosymmetry, but also softens lattice dynamics and shifts phonon modes to lower energies, enabling an extended mid-infrared transmission window, which is crucial for IR NLO applications.

To further elucidate the role of phonon-charge coupling in the NLO response of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂, we computed the phonon vibrational spectra using the first principles calculations and evaluated the electron-phonon coupling strength using the Fröhlich polaron model (1) (Supplementary Fig. 14)^83,84:

$$\alpha=\frac{{e}^{2}}{2{\hslash }{\omega }_{{{{\rm{LO}}}}}}\left(\frac{1}{{\varepsilon }_{\infty }}-\frac{1}{{\varepsilon }_{0}}\right)\sqrt{\frac{{m}^{*}}{2{\hslash }{\omega }_{{{{\rm{LO}}}}}}}$$

(1)

Here, the longitudinal optical phonon frequency is taken as ω_LO = 343 cm⁻¹ (from the Raman peak), while the transverse optical phonon frequency is estimated as ω_TO ≈ 285 cm⁻¹ (from the phonon spectrum edge). The high-frequency dielectric constant ε_∞= 4.0 is derived from the refractive index (n ≈ 2.0 at 1064 nm), and the static dielectric constant ε₀ = 5.7 is obtained using the Lyddane–Sachs–Teller (LST) relation $\frac{{\varepsilon }_{0}}{{\varepsilon }_{\infty }}=\frac{{\omega }_{{{{\rm{LO}}}}}^{2}}{{\omega }_{{{{\rm{TO}}}}}^{2}}$). An effective electron mass of m* = 0.7 m_e, appropriate for a wide-gap semiconductor (Eg = 4.37 eV), yields a Fröhlich coupling constant of α = 0.11. This value indicates weak electron–phonon coupling, which limits its direct impact on nonlinear susceptibility yet effectively suppresses phonon-assisted absorption and scattering. Consequently, it maintains a wide mid-IR transparency window and low losses.

In summary, based on anion-centered anti-perovskite, we innovatively proposed a novel “heterocationic” strategy, which exploits the size and electronegativity differences of multi-cations to construct NCS structures and optimize the functional properties of the material. Selecting anti-perovskite Na₃Cl(MoO₄) as a template and exploiting the driving forces resulting from multi-cations to optimize the functional group distortion and alignment orientation in anti-perovskites, we obtained a new NCS and polar anti-perovskite molybdate, [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂. It exhibits the widest transparent region covering 0.26–5.42 μm and 6.25–10.46 μm among all molybdates and excellent NLO properties including a large SHG coefficient (d₁₅ = 9.31 pm/V), wide band gap (4.37 eV), moderate birefringence (Δn = 0.034 @1064 nm) and congruently melting behavior, indicating that it is a potential mid-IR NLO candidate. This work not only establishes the viability of heterocationic approach for symmetry breaking in anion-centered systems, but also underscores the broader potential of anti-perovskites as a promising yet underexplored platform for high-performance functional materials. Our findings provide a versatile design paradigm for future discoveries of NCS crystals through targeted distortion engineering in anion-centered architectures.

Methods

Synthesis and crystal growth

Polycrystalline [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ was synthesized using a traditional solid-state reaction method by mixing the raw materials CsCl (Aladdin Chemistry Co., Ltd., 99.9%), NaCl (Tianjin Fu Chen Chemical Co., Ltd., 99%), CdO (Tianjin Fu Chen Chemical Co., Ltd., 99%), and MoO₃ (Aladdin Chemistry Co., Ltd., 99.5%). A mixture of 0.3367 (2.0 mmol) of CsCl, 0.0584 g (1 mmol) of NaCl, 0.2568 g (2 mmol) of CdO, and 0.2879 g (2 mmol) of MoO₃ was thoroughly ground, placed into a platinum crucible, and heated in air at 450 °C for 48 h with intermittent regrinding to ensure homogeneity and complete reaction. Thermogravimetric and differential scanning calorimetry (TG/DSC) measurements (Supplementary Fig. 3) and combined with meltingrecrystallization (Supplementary Fig. 4) indicate that [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ melts congruently. When [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ was heated to 540 °C, the polycrystalline powder sample completely melted. After the melt was slowly cooled to room temperature, the resulting powder X-ray diffraction (PXRD) patterns matched those of the initial polycrystalline sample.

Single crystals of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ were grown via top-seeded solution growth (TSSG) method. A precursor mixture consisting of the polycrystalline product and additional flux components in a molar ratio of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂: CsCl: NaCl: CdO: MoO₃ = 3:1:2:2 was finely ground and placed in a platinum crucible. Next, the mixture was heated to 550 °C for 24 h to achieve a homogeneous melt. Nucleation was induced on a platinum wire as the temperature was decreased to 470 °C at 2 °C/h. Some primary seed crystals were observed on platinum wires. To determine the saturation temperature, the melt was cooled in 3 °C steps until microcrystals formed, then reheated stepwise by 1 °C increments, establishing a saturation point at 476 °C. A large [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ crystal was grown at 476 °C at a cooling rate of 0.1 °C/day, followed by slow cooling to room temperature.

Structural characterization

A transparent, well-faceted single crystal of [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ was selected for single-crystal X-ray diffraction (SCXRD) analysis. Data were collected at 273 K using a Bruker SMART APEX II diffractometer equipped with a 4 K CCD detector and Mo Kα radiation (λ = 0.71073 Å). And the collected data were processed using the SAINT software suite⁸⁵, and the structure was solved and refined by the SHELXTL program to obtain the final crystal structure data⁸⁶. Subsequently, rational anisotropic thermal parameters for all atoms were obtained by the anisotropic refinement and extinction correction using the full-matrix least-squares techniques on F_o²⁸⁷. Anisotropic displacement parameters were refined for all non-hydrogen atoms. A PLATON program symmetry check confirmed that no additional symmetry elements were present⁸⁸. Crystallographic data for the structures reported in this paper have been deposited with the Cambridge Crystallographic Data Centre. CCDC number 2463604 contain the supplementary crystallographic data for this paper.

Physical property measurements

Elemental analysis and spatial elemental distribution were acquired using energy-dispersive X-ray spectroscopy (EDS) attached to a Quanta FEG 250 field-emission scanning electron microscope (FE-SEM, FEI). The experimentally determined elemental composition was consistent with the theoretical stoichiometry (Supplementary Fig. 1). Phase purity was confirmed by powder X-ray diffraction (PXRD) patterns collected at room temperature on a Rigaku SmartLab 3 kW diffractometer (Supplementary Fig. 2), confirming the phase purity. Optical transparency in the UV–vis–IR region (190–2500 nm) was measured at room temperature with a Hitachi UV–vis–IR spectrophotometer. Infrared spectra were recorded on a Nicolet iS50 FTIR spectrometer equipped with an ATR module in the 400–4000 cm⁻¹ range.

Laser-induced damage threshold measurement

The LIDTs of the [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ and AGS powders at the particle-size range of 75 − 106 μm were evaluated with 1064 nm laser using the single-pulse method. The measurements were performed by gradually increasing the laser power until a damaged spot was observed under a microscope. The damage thresholds were derived from the equation I_(threshold) = E/(πr²τ_p ), where E is the laser energy of a single pulse, r is the spot radius, and τ_p is the pulse width.

The powder SHG measurement

The powder SHG responses for [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ were measured using a modified Kurtz–Perry system with laser radiation at an optical wavelength of 1064 nm and 2090 nm⁶¹. Powder samples were sieved into narrow particle-size ranges (54–75, 75–100, 100–125, 125–150, 150–180 and 180–250), and KH₂PO₄ and AgGaS₂ powders sieved to the same size ranges served as the references. The birefringence values were determined using a polarizing microscope with plate-shaped crystals and calculated via the equation R = Δn × d, where R, Δn, and d are retardation, birefringence, and thickness of the crystal, respectively.

Dipole moment calculation

To elucidate the orientation of structural units, the total dipole moments of [MoO₄], [Cl(1)Cs₃NaCd₂] and [Cl(2)Cs₄Cd₂] units for a unit cell in [Cs₂NaCd₂Cl]Cl₂(MoO₄)₂ were calculated using the bond-valence method and then projected along the [001], [100] and [010] crystallographic directions^43,89. Since the occupancy of disordered oxygen atoms in different positions is close to 50%, the three disordered oxygen atoms were divided into two fully ordered configurations: (O2A, O3, O4A) and (O2, O3A, O4). Dipole moments were calculated separately for both models, then averaged according to the 50% occupancy ratio.

Computation methods

To account for the positional disorder of oxygen atoms, a 1 × 1 × 2 supercell was constructed along the c-axis, where one half adopts the (O2A, O3, O4A) configuration and the other half the (O2, O3A, O4) configuration. All density functional theory (DFT) calculations, including geometry optimization and property analyses, were carried out directly on this supercell to represent the disordered structure. First-principles calculations of electronic structure and optical properties were performed within the framework of plane-wave pseudopotential DFT using the CASTEP code⁷². The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional was used, along with norm-conserving pseudopotentials (NCP) were employed^90,91. The valence electron configurations included Cs (5p⁶6s¹), Na (3s¹), Cd (4d¹⁰5s²), Mo (4d⁵5s¹), O (2s²2p⁴), and Cl (3s²3p⁵). A plane-wave cutoff energy of 810 eV and a Monkhorst–Pack k-point grid of 3 × 2 × 3 were used for all simulations⁹². The electronic structures of MoO₄Cd₃ group at molecular level were calculated using the DFT method implemented by the Gaussian09 package at the 6–31 G level.