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

Domain: physics
Published: 2025-12-12
Editorial outcome: minor_revision

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 material science1–3, 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 challenging5–7. 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 distortions8,9. To date, more than 100,000 perovskite-type compounds have been synthesized and engineered across diverse fields including catalysis, photovoltaics, and luminescence10–12. 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 structures13–17.

In addition to perovskites, anti-perovskites (XBA3; 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 framework19–21. 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 range22–25. In previous research, a series of high-performance anti-perovskite NLO crystals, such as Ae3Q[GeOQ3] (Ae=Ba, Sr; Q=S, Se)26, K3X[B3O6] (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 Na3ClMoO4 30 and Ba3ClFeS4 31) 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 effects33,34. 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 compounds35–37) 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).

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 growth39,40. 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, [Cs2NaCd2Cl3]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, [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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
Crystal structure
[Cs2NaCd2Cl3]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 reveal that [Cs2NaCd2Cl3]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 states42,43.

Structuralevolution 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 [Cs2NaCd2Cl3]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, [NaCdVNa]Cl(MoO4) (where ‘VNa’ 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 [NaCdVNa]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. Due 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, [Cs2NaCd2Cl3]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.

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 [Cs2NaCd2Cl3]Cl2(MoO4)2 are highly reasonable, resulting in a structurally stable framework46,47. 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 [Cs2NaCd2Cl3]Cl2(MoO4)2 undergoes complete melting without decomposition. A high-quality single crystal of [Cs2NaCd2Cl3]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, [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]Cl2(MoO4)2 were subsequently investigated based on the obtained high-quality crystals.

The Ultraviolet-Visible-Infrared (UV–vis-IR) transmission spectrum of [Cs2NaCd2Cl3]Cl2(MoO4)2 was measured using a 1-mm-thick crystal wafer at room temperature. As shown in Fig. 4b, [Cs2NaCd2Cl3]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 crystals49–51. 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 (LiNaMo5O9: 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, [Cs2NaCd2Cl3]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 Ag6Mo2TeO16 (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 [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 LiNaMo5O9 (1.2 GW/cm2), suggesting its strong potential for high-power mid-IR laser applications40,60.

Due to its polar structure, the SHG intensities of [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]Cl2(MoO4)2 across the near- to mid-infrared region. Furthermore, in accordance with the mm2 point group symmetry and Kleinman’s symmetry rules, [Cs2NaCd2Cl3]Cl2(MoO4)2 is predicted to have three independent non-zero components of the second-order NLO susceptibility tensor (dij)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 [Cs2NaCd2Cl3]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 (deff) in mm2 symmetry and the relatively large phase-matching angles imposed by the crystal’s small birefringence64,65.

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 [Cs2NaCd2Cl3]Cl2(MoO4)2 was evaluated through the first-principles calculations. The refractive index difference (nz – ny) exceeds (ny – nx), classifying [Cs2NaCd2Cl3]Cl2(MoO4)2 as a positive biaxial crystal with the birefringence of 0.034 @ 1064 nm (Fig. 4g). These results indicate that [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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)4, each O2- 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, [Cs2NaCd2Cl3]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 O2– 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.

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 [Cs2NaCd2Cl3]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 O2– 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 [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 experimentally74,75. 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 4d 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 the electrostatic and covalent factors, plays a critical role in establishing the NCS lattice and the resulting NLO properties.

To gain further insight into the electronic structure and charge density distributions are localized around the oxygen atoms, a characteristic signature of highly polarized covalent Mo–O bonds76,77. 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 macro-scaled 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, [Cs2NaCa2Cl3]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 [Cs2NaCd2Cl3]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 [Cs2NaCa2Cl3]Cl2(MoO4)2 was also calculated and compared with [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]Cl2(MoO4)2, the Cd 3s and 3p orbitals show weak contribution near the Fermi level, while in [Cs2NaCa2Cl3]Cl2(MoO4)2, the Ca2+ contribution near the Fermi level is negligible. In addition, the Cd 4d orbitals form narrow and deep bands centered around −7 eV, and show weak direct hybridization with the coordinating ion p states (Supplementary Fig. 11c,d). Affected by this, Cd2+ ions and their coordinating ions show obvious asymmetric electron density distribution, while Ca2+ exhibits an 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 [Cs2NaCa2Cl3]Cl2(MoO4)2 and [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 correspond 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 [Cs2NaCd2Cl3]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:

α = (e2/2ℏ) * (1/ε∞ - 1/ε0) * (2m*ωLO/ℏ)1/2 (1)

Here, the longitudinal optical phonon frequency is taken as ωLO = 343 cm−1 (from the Raman peak), while the transverse optical phonon frequency is estimated as ωTO ≈ 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 ε0/ε∞ = (ωLO/ωTO)2. An effective electron mass of m* = 0.7 me, 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, [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 melting recrystallization (Supplementary Fig. 4) indicate that [Cs2NaCd2Cl3]Cl2(MoO4)2 melts congruently. When [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]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 at a rate of 2°C/h. 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 [Cs2NaCd2Cl3]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 well-faceted single crystal of [Cs2NaCd2Cl3]Cl2(MoO4)2 was selected for X-ray diffraction analysis. Data were collected at 273 K using a Bruker SMART APEX III diffractometer equipped with a 4K 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 F287. 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 [Cs2NaCd2Cl3]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 = E/(πr2τp), where E is the laser energy of a single pulse, r is the spot radius, and τp is the pulse width.

The powder SHG measurement
The powder SHG responses for [Cs2NaCd2Cl3]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 [Cs2NaCd2Cl3]Cl2(MoO4)2 were calculated using the bond-valence method and then projected along the [001], [100] and [010] crystallographic directions43,89. 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 employed90,91. 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 Gaussian 09 package at the 6–31G level.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In recent years the Heterocationic doping in compound functional material design is widely accepted to tune or modulate the local symmetry or microscopic interactions in the crystal, which can produce pronounced properties of them. the authors used such a strategy to get a NLO material Cs2NaCd2Cl3(MoO4)2 with a fasinating parameters, with such a advancement, there are still some questions need to be clearified as follows:\n1: Why the Cs and Cd ions are selected, what is the principle for their selection? what is their respective role in the lattice?\n2: the heterocationic ions are choosed for octahedra with ionicity, how their incorporations influence their structure stability ?\n3. the critical dipole unit is MoO4, how to realize the alligning of MoO4 tetrahedra in the lattice ? what effect of Cd2+ play in the MoO4 allignment ? only the acid-base theory can do the explanation of structural distortion in such a compound with Cd2+ (d10) and Mo(VI)O4 ? does d orbitals of them didnot contribute ?\n4. what is the working range of NLO proceses in this materials?\n5. water adsorptions can easily occurred in this halide crystal, how it influence the working situation of this crystal?\n6. as indicated the mid-infrared NLO behavior， the phonon-charge coupling should be involved, can authors supply some discussions on this microscopic interactions in this crystal?\n7. three Cd2+ layer with MoO4 tetrahedra, may modulate and form the periodical charge distribution as the significant NLO behavior, can authors supply some ionic orbital level descriptions to underscores the NLO properties, but not only the mascroscopic or traditional chemistry diagram? for example, with the calculated results from the first principle DFT calculations.", "reviewer_2": "The manuscript entitled \"Rational Design of A Polar Anti-Perovskite Nonlinear Optical Crystal via Heterocationic Strategy\" by Yang, et al., presents a comprehensive study on the synthesis, crystal structure, properties, and theoretical calculations of a polar antiperovskite, Cs2NaCd2Cl3(MoO4)2. The reported material crystallizes in a polar noncentrosymmetric space group, Pba2, exhibiting a wide band gap of 4.37 eV, strong second harmonic generation efficiency (exp. 3.5 × KDP), and an exceptional transparency range of 0.26–5.42 µm and 6.25–10.46 µm. The synthetic strategy, synthesis, crystal structure, and large single crystal growth of the reported material are interesting. Overall, the work has been done carefully, and the manuscript is well-written. Therefore, the manuscript might be a possible publication once the following issues are well-addressed by the authors.\n1. Details of the structures used in the dipole moment and DFT calculations need to be provided. As the O sites are disordered, the exact configurations used in the calculations should be described.\n2. The authors stated that the calculated d15 coefficient at a wavelength of 1064 nm is more than 20 times greater than the d36 of KDP. However, the experimentally measured SHG intensity of the reported material is only approximately 3.5 times that of KDP in the range of 75–106 µm. Could the authors provide a detailed explanation for this discrepancy?\n3. The authors state that the reported material is a promising mid-IR NLO candidate. For mid-IR NLO materials, the SHG efficiencies at around 2 µm and the laser-induced damage threshold (LIDT) are critical factors. Therefore, SHG and LIDT results of the reported material, along with benchmark materials–including AgGaS2 (AGS) and others–at 2090 nm should be provided.\n4. In the ELF diagram, the authors noted that \"As show in Figure 6c, asymmetric electron densities, which exhibit lobe-like shapes, are concentrated around the oxygen atoms, contributing to the SHG-active groups with polar covalent Mo–O bonds.\" Could the authors provide a detailed explanation of the impacts of these asymmetric electron densities on the bonding characteristics of the oxygen atoms?\n5. The SHG measurement using the Kurtz–Perry method mainly depends on wavelength, temperature, and particle size. Consequently, this method is generally employed for preliminary screening of NLO properties. In contrast, the Maker–Fringe method provides reliable NLO coefficients and is essential for the proper characterization of SHG materials, although it requires sufficiently large single crystals. It is notable that the authors successfully grew relatively large crystals (18 × 11 × 5 mm3). In light of this, could the authors consider conducting SHG measurements using the Maker–Fringe method?\n6. According to Table 1, the bond distortion and bond angle variance for MoO4 tetrahedron and [ClCs3NaCd2] octahedron were calculated to be 0.03 and 4.69, and 0.14 and 48.48, respectively. These values indicate that the [ClCs3NaCd2] octahedron exhibits greater structural distortion compared to the MoO4 tetrahedron. However, as shown in the dipole moment calculations presented in Table S7, the dipole moment originating from the MoO tetrahedron is larger than that from the [ClCs3NaCd2] octahedron. Could the authors provide a detailed explanation?", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The rational design of NCS inorganic materials remains a longstanding challenge due to unpredictable solid-state reactions and stringent requirements for symmetry breaking. This manuscript reports a novel heterocationic strategy for the first time, leading to the rational design and successful synthesis of a new polar anti-perovskite molybdate, [Cs2NaCd2Cl]Cl2(MoO4)2. By introducing chemically distinct Cs+ and Cd2+ cations into the anti-perovskite framework, the authors effectively induced strong octahedral distortions, breaking inversion symmetry and aligning [MoO4]2– tetrahedra. As a result, the compound exhibits outstanding NLO properties, including an extremely broad infrared transparency window (0.26–5.42 μm and 6.25–10.46 μm), a wide band gap (4.37 eV), and a large SHG coefficient (d15 = 9.31 pm/V). These findings provide important insights for the design of high-performance mid-infrared NLO materials and offer a new strategy for distortion engineering in anti-perovskite compounds. So, I think this paper is very interesting and informative. I will recommend its acceptance after the following issues are addressed:\n1) In the discussion of structural evolution of the heterocationic framework, the authors introduce a hypothetical intermediate compound, [NaCdVNa]Cl(MoO). Is there any experimental evidence supporting the existence of such an intermediate? Could this structure be realistically synthesized or isolated?\n2) The manuscript lacks discussion on vibrational spectroscopy (e.g., Raman or IR) to confirm the existence of MoO4 units. Have such measurements been carried out?\n3) It is better to index the crystal plane as shown in Figure 4f for the measurement of birefringence experiment. This is important for determine the error between the measured value and real birefringence.\n4) In Figure 4f, the bandgap (Eg) value of KTP is incorrect. Please correct this error.\n5) Are the band-gap and UV cutoff algorithms correct in this paper? How to deal with the big difference between the measured value and the theoretical calculation band gap?\n6) The authors attribute the ordered orientation of [MoO4] tetrahedra along the c-axis to the anchoring effect of more electronegative Cd2+. Could the authors discuss whether similar effects could be achieved with other aliovalent cations (e.g., Zn2+ or Mn2+)?"}, "author_response": "Response: Thanks for reviewer’s kind comments! In our design, Cs+ and Cd2+ were intentionally selected based on the parent compound Na3MoO4Cl, aiming to independently control charge and size effects to break inversion symmetry. The selection principle is as follows:\n1) Cd2+ was chosen as an aliovalent substitution for Na+. Its ionic radius (97 pm) is nearly identical to that of Na+ (95 pm), allowing substitution without introducing significant size-related strain. This charge change perturbs the local electronic environment to promote asymmetric Mo–O coordination.\n2) Cs+ was chosen as a homovalent substitution. It shares the +1 valence with Na+ but has the largest ionic radius among alkali metals (~167 pm). This maximizes size mismatch to induce substantial lattice distortion, which effectively transfers strain to the entire lattice and serving as the main driving force for inducing macroscopic symmetry from centrosymmetric to non-centrosymmetric.\nThus, the two cations play distinct yet complementary roles in the Cs2NaCd2Cl3(MoO4)2 lattice: Cd2+ functions as a structural and electronic modulator, while Cs+ serves as a distortion inducer and symmetry breaker. Together, they satisfy charge balance, stabilize the lattice, and provide a rational route to enhance the non-centrosymmetric structure required for NLO activity.\nAccordingly, the following sentences have been added at Line 38 in the right column of Page 2 in the paper: ‘Here, Cd2+ serves as a structural and electronic modulator, whereas Cs+ acts as a distortion inducer and symmetry breaker.’\nLine 9 in the left column of Page 4 in the paper: ‘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.’\nAnd line 24 in the left column of Page 4 in the paper: ‘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.’\n2: the heterocationic ions are chose for octahedra with ionicity, how their incorporations influence their structure stability?\nResponse: Thanks for reviewer’s kind comments! Although the incorporation of heterocations may introduce lattice distortion and local charge imbalance, the overall structural stability of our crystal is well preserved. Structurally, the heterovalent cations were carefully selected: Cd2+ has an ionic radius comparable to that of Na+, ensuring minimal mismatch in coordination geometry, while Cs+ ions occupy interstitial positions, complementing both spatial and electrostatic requirements within the lattice.\nMoreover, quantitative stability indicators confirm the stability of the structural framework. Despite increased octahedral distortion resulting from heterocationic substitution, the global instability index (GII) remains as low as 0.086 vu and the bond strain index (BSI) is 0.043 vu, both well within the range for structurally stable compounds. These values indicate that all atomic positions in Cs2NaCd2Cl3(MoO4)2 are geometrically reasonable and free from excessive strain.\nIn addition, thermal analyses further substantiate this conclusion: TG/DSC measurements (Supplementary Figure 3) and melting–recrystallization experiments (Supplementary Figure 4) show that the compound undergoes congruent melting without decomposition. The ability to grow a high-quality single crystal of dimensions 18 × 11 × 5 mm3 using the TSSG method with self-flux (Figure 4a), further attests to its excellent thermal and structural stability. These details are discussed on page 4, line 7 in the right column of the paper.\n3. the critical dipole unit is MoO4, how to realize the alligning of MoO4 tetrahedra in the lattice? what effect of Cd2+ play in the MoO4 allignment? only the acid-base theory can do the explanation of structural distortion in such a compound with Cd2+ (d10) and Mo(VI)O4? does d orbitals of them did not contribute?\nResponse: Thank you for the reviewer’s insightful comments and valuable suggestions! The ordered alignment of [MoO4] tetrahedra along the c-axis in our compound is primarily governed by electrostatic interactions, especially the anchoring effect of more electronegative and polarizable Cd2+ ions. Compared with Na+ or Cs+, Cd2+ exhibits stronger Lewis acidity, enabling tighter coordination with oxygen atoms and thereby effectively constraining the spatial orientation of the [MoO4] units. This anchoring behavior is well supported by Mulliken population analysis based on first-principles calculations, as discussed on Page 6, Line 19 in the left column of the “Structure–Property Relationships” section in the manuscript.\nIn addition, according to the reviewer’s suggestion, we have re-analyzed the projected density of state. As shown in Figure 6b of this paper, clear evidence of orbital hybridization is observed among Cd 4d, O 2p, and Mo 4d orbitals, particularly near the valence band maximum. Although Cd2+ (d10) and Mo6+ (d0) are generally considered less active in directional bonding than partially filled d-cations, the presence of such orbital overlap suggests a weak yet non-negligible covalent contribution. This interaction complements the Lewis acid–base theory, indicating that both electrostatic forces and weak orbital hybridization contribute to the directional stabilization of the [MoO4] tetrahedra.\nAccordingly, the following sentences have been revised and added at Line 17 in the left column of Page 7 in the paper: ‘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 4d orbitals, while the bottom of conduction band (CB) consists primarily of O 2p, Mo 4d and Cd 5s orbitals (Figure 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 significant 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.’\n4. what is the working range of NLO processes in this material?\nResponse: Thanks for reviewer’s kind comments and good suggestions! The working range of NLO processes in [Cs2NaCd2Cl]Cl2(MoO4)2 is mainly determined by its optical transparency and phase-matching ability. As shown in the UV–Vis–IR diffuse reflectance spectrum (Figure 4b), the compound exhibits a wide transparent window with high transmittance ~80%, covering 0.26–5.42 μm (UV–Vis to near-IR) and 6.25–10.46 μm (mid- to long-wave IR), with the absorption edges originated from characteristic Mo–O vibrational modes.\nExperimentally, strong SHG signals were observed under fundamental wavelengths of both 1064 nm and 2090 nm, demonstrating efficient NLO activity across the near- to mid-infrared region (up to at least ~2.09 μm). The measurement at 2090 nm was additionally performed in response to Reviewer #2’s third comment, which specifically requested mid-IR SHG and LIDT tests, and the corresponding data and discussion have been incorporated into the revised manuscript. Furthermore, phase-matching behavior was verified at both excitation wavelengths, further demonstrating the broad working range of this NLO material.\nAccordingly, the following sentences have been revised and added at Line 9 in the right column of Page 5 in the paper: ‘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.’\n5. water adsorptions can easily occur in this halide crystal, how it influenced the working situation of this crystal?\nResponse: Thanks for reviewer’s kind comments! Although halide crystals are often susceptible to moisture, [Cs2NaCd2Cl]Cl2(MoO4)2 demonstrates outstanding stability under ambient conditions and does not undergo deliquescence. According to the reviewer’s suggestion, we re-polished the crystal wafers (10× 5 × 1 mm2) previously used for transmittance measurements and intentionally exposed them to air ambient with ~50% relative humidity for one month. As shown in Supplementary Figure 6, no evident deliquescence or surface degradation was observed after prolonged exposure, with only minor dust accumulation observed. Furthermore, the infrared transmission spectra collected before and after long-term air exposure revealed negligible changes, confirming that the crystal is the absence of hygroscopic degradation and maintains its structural integrity and optical performance under natural atmospheric conditions.\nThese results collectively verify the excellent environmental stability of [Cs2NaCd2Cl]Cl2(MoO4)2, ensuring its reliability in practical optical applications.\nAccordingly, the following sentence has been added at Line 40 in the right column of Page 6 in the paper: ‘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 Figure 6).’ And a new Figure as Supplementary Figure 6 has been added in the supporting information.\nSupplementary Figure 6. Ambient stability evaluation of [Cs2NaCd2Cl]Cl2(MoO4)2 crystal before and after exposure to air for one month. (a) Polished crystal before exposure to ambient air. (b) Crystal after exposure to ambient air for one month. (c–d) Corresponding polarized optical microscope images (×40) of a–b. (e) Transparency spectra of [Cs2NaCd2Cl]Cl2(MoO4)2 crystal before and after exposure to air for one month from 4000 to 400 cm–1.\n6. as indicated the mid-infrared NLO behavior, the phonon-charge coupling should be involved, can authors supply some discussions on this microscopic interaction in this crystal?\nResponse: Thanks for reviewer’s kind comments and good suggestions! According to the reviewer’s suggestion, we have calculated the phonon vibrational spectra of [Cs2NaCd2Cl]Cl2(MoO4)2 using the first-principle calculations and measured its Raman spectra. The phonon dispersion spectrum exhibits a distinct “forbidden gap” between 400–800 cm–1, which correlates with the absence of significant peaks in both the IR and Raman spectra. This gap originates from the separation of vibrational modes dominated by heavy atoms (Cs/Cd) at low frequencies and those dominated by light atoms (O) at high frequencies. In particular, the Mo–O stretching vibrations, observed as a strong Raman peak at 918 cm–1, lie above this range, resulting in the decoupling of low- and high-frequency phonons and the consequent formation of the gap. Based on the calculated phonon spectra and experimental spectroscopic results, we performed a preliminary evaluation of the phonon–charge coupling constant and analyzed the phonon–charge coupling and its impact on the NLO properties of the material.\nTo quantify the phonon–charge coupling strength, we calculated the Fröhlich coupling constant (α) using the established formula (Adv. Phys. 3, 325-361 (1954); Phys. Rev. B 99, 085207 (2019)):\nα = (e^2 / 2ℏωLO) * (1/ε∞ - 1/ε0) * sqrt(m* / 2ℏωLO)\nHere, the longitudinal optical phonon frequency is taken as ωLO = 343 cm−1 (from the Raman peak), while the transverse optical phonon frequency is estimated as ωTO ≈ 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 ε0 = ε∞ * (ωLO^2 / ωTO^2). An effective electron mass of m* = 0.7 me, appropriate for a wide-gap semiconductor (Eg = 4.37 eV), yields a Fröhlich coupling constant of α = 0.11. This value indicates weak phonon–charge coupling, slightly stronger than that in GaAs (α ≈ 0.06) but much weaker than in ZnO (α ≈ 0.3). (Phys. Rev. 121, 752-758 (1961); Phys. Rev. B 36, 4821-4830 (1987))\nIn summary, the phonon–charge coupling in [Cs2NaCd2Cl]Cl2(MoO4)2 is relatively weak (α = 0.11), which suggests that its direct contribution to the intrinsic NLO response is limited. Instead, the small coupling constant primarily contributes to the suppression of phonon-assisted absorption and anharmonic scattering, thereby helping to maintain a wide mid-IR transparency window and low optical losses. Meanwhile, the presence of polar optical phonons, as reflected in the finite LO–TO splitting, ensures the existence of macroscopic electric fields required for sustaining second-order nonlinear processes. Therefore, although the weak phonon–charge coupling does not significantly enhance the nonlinear susceptibility, it plays an indirect but important role in enabling efficient SHG performance by preserving high transparency and minimizing dissipation.\nAccordingly, the following paragraph has been added at Line 20 in the left column of Page 8 in the paper: ‘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 (Supplementary Figure 12):\nα = (e^2 / 2ℏωLO) * (1/ε∞ - 1/ε0) * sqrt(m* / 2ℏωLO)\nHere, the longitudinal optical phonon frequency is taken as ωLO = 343 cm−1 (from the Raman peak), while the transverse optical phonon frequency is estimated as ωTO ≈ 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 ε0 = ε∞ * (ωLO^2 / ωTO^2). An effective electron mass of m* = 0.7 me, 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.’ And the new Supplementary Figure 11 has been added in the supporting information.\nSupplementary Figure 12. (a) The phonon vibrational spectrum of [Cs2NaCd2Cl]Cl2(MoO4)2. (b) Raman spectra of [Cs2NaCd2Cl]Cl2(MoO4)2 measured as polished crystal plate.\nWe have also added the references ‘[76] Fröhlich, H. Electrons in lattice fields. Adv. Phys. 3, 325-361 (1954). [77] Lautenschlager, P., Garriga, M., Vina, L. & Cardona, M. Temperature dependence of the dielectric function and interband critical points in silicon. Phys. Rev. B 36, 4821-4830 (1987).’\n7. three Cd2+ layer with MoO4 tetrahedra, may modulate and form the periodical charge distribution as the significant NLO behavior, can authors supply some ionic orbital level descriptions to underscores the NLO properties, but not only the macroscopic or traditional chemistry diagram? for example, with the calculated results from the first principle DFT calculations.\nResponse: Thanks for reviewer’s kind comments and good suggestions! According to the reviewer’s suggestions, we performed electron density difference (EDD) analysis based on the first-principles DFT calculations and examined the frontier molecular orbitals of an isolated MoO4Cd3 unit using the Gaussian 09 package. The corresponding results are shown in Supplementary Figures 10–11. As presented in Supplementary Figure 10a, the three-dimensional EDD isosurface clearly demonstrates pronounced electron accumulation (yellow regions) along Mo–O and Cd–O bonds, indicating strong covalent interactions. In contrast, negligible electron depletion around Cs+ and Na+ is consistent with their ionic character. The planar (001) slice (Supplementary Figure 10b) further reveals alternating regions of charge accumulation and depletion, establishing a periodic polarization along the crystallographic c-axis. This NCS charge redistribution provides the microscopic origin of the observed SHG response.\nFrontier orbital analysis (Supplementary Figure 10) further supports this picture: the calculated highest occupied molecular orbital (HOMO) is predominantly localized on the O 2p orbitals of the MoO4 tetrahedra, whereas the lowest unoccupied molecular orbital (LUMO) is largely delocalized around the Cd2+ centers. This spatial separation suggests a preferential charge transfer pathway from MoO4 to Cd2+ layers. Combining this orbital asymmetry with the EDD-derived charge modulation demonstrate how the periodic arrangement of Cd2+ layers with MoO4 tetrahedra gives rise to non-centrosymmetric polarization, facilitating efficient NLO conversion.\nAccordingly, the following paragraph on ionic orbital level descriptions has been also added at Line 11 in the right column of page 7 of the paper: ‘To 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 Gaussian09. The EDD results (Supplementary Figure 10) 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 Figure 11), 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.’ And two new Figures as Supplementary Figure 10 and Supplementary Figure 11 have been added in the supporting information.\nSupplementary Figure 10. (a) The isosurfaces of EDD analysis of [Cs2NaCd2Cl]Cl2(MoO4)2; (b) the planar EDD map of the (001) slice. The yellow and blue regions represent electron accumulation and depletion, respectively; the red, yellow, cyan, grey, pink and green spheres indicate Cs, Na, Cd, Mo, O, and Cl atoms, respectively; isosurface value: 0.05 e·Å–3.\nSupplementary Figure 11. The frontier molecular Orbitals of [MoO4Cd3] unit. Cyan Mo; red, O; yellow Cd.\nIn addition, the following sentence has also been added in the ‘Methods’ in the paper: ‘The electronic structures of MoO4Cd3 group in molecular level were calculated using the DFT method implemented by the Gaussian09 package at the 6-31G level.’\nReviewer #2 (Remarks to the Author):\nThe manuscript entitled \"Rational Design of a Polar Anti-Perovskite Nonlinear Optical Crystal via Heterocationic Strategy\" by Yang, et al., presents a comprehensive study on the synthesis, crystal structure, properties, and theoretical calculations of a polar antiperovskite, Cs2NaCd2Cl3(MoO4)2. The reported material crystallizes in a polar noncentrosymmetric space group, Pba2, exhibiting a wide band gap of 4.37 eV, strong second harmonic generation efficiency (exp. 3.5 × KDP), and an exceptional transparency range of 0.26–5.42 µm and 6.25–10.46 µm. The synthetic strategy, synthesis, crystal structure, and large single crystal growth of the reported material are interesting. Overall, the work has been done carefully, and the manuscript is well-written. Therefore, the manuscript might be a possible publication once the following issues are well-addressed by the authors.\n1. Details of the structures used in the dipole moment and DFT calculations need to be provided. As the O sites are disordered, the exact configurations used in the calculations should be described.\nResponse: Thanks for reviewer’s kind comments and good suggestions! We apologize for the insufficient clarity in our original description and would like to provide more precise details of the structural models used in our calculations. As shown in Supplementary Table 2, the occupancies of the disordered O atoms at alternative positions are close to 50%. To accurately account for this disorder in the dipole moment analysis of the MoO4 groups, we considered two fully ordered configurations: (O2A, O3, O4A) and (O2, O3A, O4) (Figure 1). These configurations maintain both the geometric integrity of each MoO4 tetrahedron and the crystallographic constraint of mutually exclusive site occupancy. Dipole moments were first calculated for each configuration and then averaged with equal weight (50% each) to yield the final dipole moment values.\nFor the first-principles DFT calculations, a periodic model capable of representing the positional disorder was required. We therefore constructed a 1 × 1 × 2 supercell along the crystallographic c-axis, in which one half of the supercell adopts the (O2A, O3, O4A) arrangement and the other half adopts the (O2, O3A, O4) arrangement. All DFT calculations, including structural optimization and property analyses, were performed on this supercell, which provides a reasonable representation of the disordered structure.\nFigure 1. Detailed information on the occupancy of oxygen atoms\nIn addition, the following clarifications have also been added to the ‘Methods’ of the manuscript:\n‘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. 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.’\nComputation 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.’\n2. The authors stated that the calculated d15 coefficient at a wavelength of 1064 nm is more than 20 times greater than the d36 of KDP. However, the experimentally measured SHG intensity of the reported material is only approximately 3.5 times that of KDP in the range of 75–106 µm. Could the authors provide a detailed explanation for this discrepancy?\nResponse: Thanks for reviewer’s kind comments and insightful question! The SHG intensity measured via the Kurtz-Perry powder method represents an averaged response that is close to the effective nonlinear coefficient (deff), rather than the maximum theoretical value of any individual tensor component in single-crystal measurements. Our compound, [Cs2NaCd2Cl]Cl2(MoO4)2, crystallizes in the mm2 point group, where the effective nonlinear coefficients can be expressed as:\nx-y layer (θ = 90°): deff = d24 cosφ\ny-z layer (φ = 90°): deff = d15 cosθ\nx-z layer: deff = d24 sin2θ + d15 cos2θ\nAs seen from these expressions, deff is strongly correlated with the cosine terms of the phase-matching angles (θ and φ). Since our crystal exhibits relatively small birefringence, the corresponding phase-matching angles are relatively large in the measurement range. (Handbook of Nonlinear Optical Crystals, Springer (1999); J. Appl. Phys. 39, 3798 (1968)) From the above equations, it is evident that deff decreases with increasing phase-matching angle, which explains why the experimentally observed SHG intensity from powder samples is significantly smaller than that the calculated coefficient.\nIn addition, as the reviewer also pointed out in Questions 3 and 5, the SHG response depends on the fundamental wavelength. In nonlinear optical crystals, the phase-matching angle is generally wavelength-dependent due to birefringent dispersion. In many practical cases, below the phase-matching cutoff wavelength, the phase-matching angle decreases with increasing fundamental wavelength. This trend strongly affects the deff, as it determines the projection of the nonlinear susceptibility tensor along the beam propagation direction. (J. Appl. Phys. 39, 3798–3813 (1968); Boyd, Nonlinear Optics, 3rd ed., Academic Press (2008)). In our compound, the small birefringence leads to relatively large phase-matching angles at 1064 nm, thereby reducing the deff. At longer wavelengths (e.g., 2090 nm), the phase-matching angle decreases and correspondingly deff increases, which is consistent with our experimental observations. Specifically, we measured the SHG response of our crystal at 2090 nm and found it to be about 0.2× that of AGS (d36 = 13.4 pm/V), which is higher than the relative value obtained at 1064 nm. This result further confirms our interpretation that the relatively small birefringence and the associated large phase-matching angles are the main factors limiting the powder SHG intensity.\nIn addition, to clarify this point for readers, the following sentence has also have been also added at Line 23 in the right column of page 5 in the paper: ‘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 (deff) in mm2 symmetry and the relatively large phase-matching angles imposed by the crystal’s small birefringence.’\nWe have also added the references ‘[61] Maker, P.D., Terhune, R.W., Nisenoff, M. & Savage, C.M. Effects of Dispersion and Focusing on the Production of Optical Harmonics. Phys. Rev. Lett. 8, 21-22 (1962). [62] Armstrong, J.A., Bloembergen, N., Ducuing, J. & Pershan, P.S. Interactions between Light Waves in a Nonlinear Dielectric. Phys. Rev. 127, 1918-1939 (1962).’\n3. The authors state that the reported material is a promising mid-IR NLO candidate. For mid-IR NLO materials, the SHG efficiencies at around 2 µm and the laser-induced damage threshold (LIDT) are critical factors. Therefore, SHG and LIDT results of the reported material, along with benchmark materials–including AgGaS2 (AGS) and others–at 2090 nm should be provided.\nResponse: Thanks for reviewer’s kind comments and good suggestions! We fully agree that the second harmonic generation (SHG) efficiency at around 2 μm and the laser-induced damage threshold (LIDT) are critical performance indicators for mid-IR NLO materials. Accordingly, we have carried out additional experiments as follows:\nSHG efficiency at 2090 nm: Powder SHG measurements were performed at 2090 nm using AgGaS2 (AGS) as a reference (Figure 4e). The SHG intensities of [Cs2NaCd2Cl]Cl2(MoO4)2 increase with particle size, confirming phase-matching behavior in the mid-IR. For particle sizes of 75–106 µm, the SHG intensity is ~0.2 × that of AGS, slightly exceeding the SHG intensity observed at 1064 nm.\nLIDT measurements: Due to the superior stability and reproducibility of high-power Nd:YAG lasers, LIDT tests were performed at 1064 nm using the single-pulse method. Given that multiphoton absorption thresholds increase with wavelength, the 1064 nm result provides a conservative estimate for 2 µm operation. The measured LIDT of [Cs2NaCd2Cl]Cl2(MoO4)2 is ~0.96 GW/cm2, which exceeds that of AGS by two orders of magnitude (Supplementary Table 6), and is comparable to benchmark oxide-based NLO crystals such as LiNbO3 (0.84 GW/cm2) and LiNa5Mo9O30 (1.2 GW/cm2). This high damage resistance is consistent with its wide bandgap, and underscores its promise for high-power mid-IR applications.\nAccordingly, the following paragraph on the LIDT has been also added at Line 24 in the left column of page 5 of the paper: ‘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 outstanding 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.’\nAnd the following sentences have been revised on the SHG intensities at Line 37 in the left column of page 5 of the paper: ‘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, respectively. As shown in Figure 4d and 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.’ Besides, two new Figures (Figure 4d and Figure 4e) have been revised and added in the paper and one Supplementary Table 6 has been add at supporting information.\nFigure 4e. (a)The SHG intensities versus particle sizes for [Cs2NaCd2Cl]Cl2(MoO4)2 and AGS as a reference; (b) SHG intensities of [Cs2NaCd2Cl]Cl2(MoO4)2 and AGS at a particle size of 75–106 μm;\nSupplementary Table 6. Powder laser-induced damage threshold (LIDT) measurements.\nEnergy (mJ) Spot area (cm2) Pulse width (ns) Repetition frequency (Hz) LIDT (MW/cm2)\n[Cs2NaCd2Cl]Cl2(MoO4)2 432 0.079 6.03 100 961.3\nAgGaS2 13 0.074 17.07 5 9.6\nIn addition, the following paragraph has also been added at the ‘Methods’ in manuscript: ‘Laser-induced damage threshold measurement. The LIDTs of the [Cs2NaCd2Cl]Cl2(MoO4)2 and AGS powder 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/(πr2τp), where E is the laser energy of a single pulse, r is the spot radius, and τp is the pulse width.’\nWe also have added the references ‘[40] Zhang, W. et al. LiNa5Mo9O30: Crystal Growth, Linear, and Nonlinear Optical Properties. Chem. Mater. 28, 4483-4491 (2016). [57] Bass, M. Nd:YAG laser-irradiation-induced damage to LiNbO3 and KDP. IEEE J. Quantum Electron. 7, 350−359 (1971).’\n4. In the ELF diagram, the authors noted that \"As show in Figure 6c, asymmetric electron densities, which exhibit lobe-like shapes, are concentrated around the oxygen atoms, contributing to the SHG-active groups with polar covalent Mo–O bonds.\" Could the authors provide a detailed explanation of the impacts of these asymmetric electron densities on the bonding characteristics of the oxygen atoms?\nResponse: Thanks for reviewer’s kind comments and good suggestions! The asymmetric electron localization observed around oxygen atoms is a signature of highly polarized covalent bonds, specifically within the Mo–O bonds of the [MoO4]2– tetrahedra, which plays a fundamental role in establishing the strong second-order NLO response (Nature 371, 683–686 (1994); Angew. Chem. Int. Ed. 36, 1808–1832 (1997)). This pronounced lobe-like electron density, strongly localized on the oxygen side of the Mo–O bond, indicates a significant shift of the shared electron pair toward the more electronegative oxygen atom. This charge separation generates a strong, localized dipole moment within the bond. The oxygen atoms thus exhibit highly directional lone-pair-like orbitals, which are readily distorted by an external electric field, giving rise to a large microscopic second-order polarizability. Crucially, the directional nature of these Mo–O bonds allows the localized dipoles from multiple polar covalent units to align cooperatively within the NCS lattice. This additive, vectorial alignment of dipoles is the key for breaking macroscopic inversion symmetry, thereby facilitating the observed SHG activity (hyperpolarizability, β) (Acc. Chem. Res. 54, 3322–3333 (2021); Chem. Mater. 33, 6995–7003 (2021)). Thus, the [MoO4]2– tetrahedra, characterized by these asymmetric electron density distributions, act as the essential SHG-functional motifs, resulting in the macroscopic NLO response.\nAccordingly, the following sentences have been added at Line 35 in the left column of page 7 of the paper: ‘As shown in Figure 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. This reflects strong directional covalency and charge separation, resulting in significant 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 non-centrosymmetric lattice leads to additive macroscale dipoles, breaking inversion symmetry and facilitating strong SHG response. Thus, MoO4 groups act as key SHG-active motifs.’\nWe have also added the references ‘[73] Silvi, B. & Savin, A. Classification of chemical bonds based on topological analysis of electron localization functions. Nature 371, 683–686 (1994). [74] Savin, A., Nesper, R., Wengert, S. & Fässler, T.F. ELF: The Electron Localization Function. Angew. Chem. Int. Ed. 36, 1808-1832 (2003).’\n5. The SHG measurement using the Kurtz–Perry method mainly depends on wavelength, temperature, and particle size. Consequently, this method is generally employed for preliminary screening of NLO properties. In contrast, the Maker–Fringe method provides reliable NLO coefficients and is essential for the proper characterization of SHG materials, although it requires sufficiently large single crystals. It is notable that the authors successfully grew relatively large crystals (18 × 11 × 5 mm3). In light of this, could the authors consider conducting SHG measurements using the Maker–Fringe method?\nResponse: Thanks for reviewer’s kind comments and good suggestions! We fully agree the Maker–Fringe measurements for SHG coefficients mentioned is very important for the practical application of a NLO crystal. But obviously, it is difficult for one paper to complete and contain all properties of an entirely new NLO crystal. Currently, although we have successfully grown a centimeter-size [Cs2NaCd2Cl]Cl2(MoO4)2, the crystal growth conditions are still unstable and many growth technological parameters are under investigations. This is primarily due to the volatile nature of multiple halide components (e.g., CsCl, CdCl2, NaCl) in the melt growth process, which leads to fluctuations in stoichiometry and unstable growth environments, often resulting in defects such as screw dislocation and cladding (Figure a and b). Although we have obtained a crystal with relatively high optical quality (Figure c and d), the predominant growth along the (110) plane makes it particularly challenging to achieve sufficient thickness along the a- and b- axes. For Maker–Fringe measurements, not only are large, optically homogeneous single crystals required, but they also must be precisely oriented and cut into specific wafers, followed by refractive index determination based on Minimum Deviation Angle Method to complete Maker–Fringe fitting. We have attempted processing and testing with the currently available crystals, but the limited size and growth quality have prevented us from obtaining optical components that meet the stringent requirements for high-precision measurements. So right now, we have no more new centimeter-size [Cs2NaCd2Cl]Cl2(MoO4)2 single crystals that can be cut into new crystal devices for other measurements. After we get the better crystal growth conditions and can grow the more and larger-size [Cs2NaCd2Cl]Cl2(MoO4)2 crystals, we will certainly do all of these measurements.\nFigure 2. Representative single crystals of [Cs2NaCd2Cl]Cl2(MoO4)2 grown via the top-seeded solution growth method, illustrating variations in crystal habit and quality.\n6. According to Table 1, the bond distortion and bond angle variance for MoO4 tetrahedron and [ClCs3NaCd2] octahedron were calculated to be 0.03 and 4.69, and 0.14 and 48.48, respectively. These values indicate that the [ClCs3NaCd2] octahedron exhibits greater structural distortion compared to the MoO4 tetrahedron. However, as shown in the dipole moment calculations presented in Table S7, the dipole moment originating from the MoO tetrahedron is larger than that from the [ClCs3NaCd2] octahedron. Could the authors provide a detailed explanation?\nResponse: Thanks for reviewer’s kind comments and good suggestions! The bond‐length distortion (D) and bond‐angle variance (σ2) were evaluated using the definitions implemented in VESTA:\nD = 1/n * Σ( |di - d̅| / d̅ ); σ2 = 1/m * Σ(θj - θideal)^2\nwhere d is length of the i-th bond, d̅ is average bond length, and n = number of bonds; and θ is the X–M–X bond angles, θideal is ideal tetrahedral angle, and m = number of bond angles. These purely geometric metrics show that the [ClCs3NaCd2] octahedron is more distorted (D = 0.14, σ2 = 48.48) than the MoO4 tetrahedron (D = 0.03, σ² = 4.69). By contrast, the local dipole moment is an electrostatic quantity, computed as:\nμ⃗ = Σ qi * r⃗i\nwith qi the effective charges (from Bader/Born analyses) and r⃗i the atomic position vectors measured from the polyhedral centroid; the macroscopic polarization then depends on the vector sum and orientation of these local dipoles within the unit cell. With this distinction, the apparent contradiction is resolved. Although [ClCs3NaCd2] exhibits larger geometric distortion, its mixed-cation framework (Cs+, Na+, Cd2+ around a central Cl–) yields a relatively diffuse and compensating charge distribution: individual cation–anion contributions are oriented in different directions and partially cancel, leading to a small net μ⃗. In contrast, MoO4 units possess intrinsically high bond polarity arising from the large electronegativity difference in Mo–O and directional Mo6+(d0)–O 2p hybridization. Thus, each tetrahedron possesses a significant local dipole. More importantly, our calculations show that these MoO4 dipoles are uniformly aligned with a dominant z-component, so they add constructively across the crystal lattice. Consequently, although the [ClCs3NaCd2] octahedron exhibits greater geometric distortion, the strong intrinsic polarity and coherent alignment of the MoO4 units dominate the net dipole moment, consistent with Supplementary Table 8. In short, geometric distortion of [ClCs3NaCd2] octahedron is a secondary factor. The macroscopic polarity is primarily determined by the effective charges and vector alignment of local bond dipoles, i.e. the MoO4 tetrahedra in this compound.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nReviewer #4 (Remarks to the Author):\nThe rational design of NCS inorganic materials remains a longstanding challenge due to unpredictable solid-state reactions and stringent requirements for symmetry breaking. This manuscript reports a novel heterocationic strategy for the first time, leading to the rational design and successful synthesis of a new polar anti-perovskite molybdate, [Cs2NaCd2Cl]Cl2(MoO4)2. By introducing chemically distinct Cs+ and Cd2+ cations into the anti-perovskite framework, the authors effectively induced strong octahedral distortions, breaking inversion symmetry and aligning [MoO4]2– tetrahedra. As a result, the compound exhibits outstanding NLO properties, including an extremely broad infrared transparency window (0.26–5.42 μm and 6.25–10.46 μm), a wide band gap (4.37 eV), and a large SHG coefficient (d15 = 9.31 pm/V). These findings provide important insights for the design of high-performance mid-infrared NLO materials and offer a new strategy for distortion engineering in anti-perovskite compounds. So, I think this paper is very interesting and informative. I will recommend its acceptance after the following issues are addressed:\n1) In the discussion of structural evolution of the heterocationic framework, the authors introduce a hypothetical intermediate compound, [NaCdVNa]Cl(MoO). Is there any experimental evidence supporting the existence of such an intermediate? Could this structure be realistically synthesized or isolated?\nResponse: Thanks for reviewer’s kind comments and good suggestions! This intermediate does not exist experimentally. We would like to clarify that the compound [NaCdVNa]Cl(MoO) is a purely hypothetical structural model solely to conceptually illustrate the effect of Na+ vacancies on the heterocationic framework evolution. As stated in the manuscript (Figure 3c), this intermediate has not been experimentally realized.\nTo further evaluate its plausibility, we attempted synthesis using the stoichiometric ratio corresponding to this composition. However, PXRD analysis of the obtained product (Figure 3) unambiguously revealed the formation of CdMoO4 as the dominant phase, rather than the targeted heterocationic intermediate. This result indicates that the proposed structure is thermodynamically unstable under conventional solid-state conditions, likely due to the high concentration of cationic vacancies. Therefore, while the hypothetical model is conceptually valuable for understanding the structural evolution mechanism, it cannot be realistically synthesized or isolated.\nFigure 3. Experimental PXRD pattern (black) of the product synthesized using the stoichiometric ratio corresponding to the hypothetical compound [NaCdVNa]Cl(MoO), which matches well with the standard pattern of CdMoO4 (red), confirming the formation of CdMoO4 rather than the intended intermediate.\n2) The manuscript lacks discussion on vibrational spectroscopy (e.g., Raman or IR) to confirm the existence of MoO4 units. Have such measurements been carried out?\nResponse: Thanks for reviewer’s kind comments and good suggestions! To confirm the existence of MoO4 units, we assigned the infrared (IR) absorption peaks and performed Raman spectroscopic analyses, and the detailed assignments are shown in Supplementary Figure 7. The IR spectrum exhibits some characteristic absorption peaks around 1800–1600 cm–1, which assign to the stretching vibration of Mo–O bonds in tetrahedral MoO4 groups. Raman spectroscopy further supports this assignment, exhibiting a strong peak at 890–930 cm–1 and a medium-intensity peaks at 800–840 cm–1, corresponding to the stretching vibration modes of Mo-O bonds. Weaker bands observed at 300–400 cm–1 are assigned to O–Mo–O bending modes. These spectral features are fully consistent with reported vibrational signatures of MoO4 units in similar materials, confirming their successful incorporation into the structure.\nAccordingly, the following sentence has been added at Line 39 in the right column of page 4 of the paper: ‘The prominent absorption peaks observed at 1845 cm–1 and 1600 cm–1 in the IR spectrum were unambiguously assigned to the characteristic stretching vibrational modes of Mo−O groups. This assignment was further supported by complementary Raman spectroscopic measurements, which further revealed the corresponding vibrational signatures of MoO4 units (Supplementary Figure 7).’ And the new Figure as Supplementary Figure 7 has been added in the supporting information.\nSupplementary Figure 7. The IR and Raman spectra of [Cs2NaCd2Cl]Cl2(MoO4)2.\n3) It is better to index the crystal plane as shown in Figure 4f for the measurement of birefringence experiment. This is important for determine the error between the measured value and real birefringence.\nResponse: Thanks for reviewer’s kind comments and good suggestions! According to the reviewer’s suggestion, we have indexed the orientations of the crystal that we used for the birefringence measurement using the “Index Crystal Faces” program in Bruker SMART APEX III. Clearly, the crystal wafer we used is the (010) wafer.\nThe following sentence has also been added at Line 34 in the right column of Page 5 in the paper: ‘The [010] crystallographic orientation was determined by analysis conducted using a Bruker SMART APEX III diffractometer (Supplementary Figure 9).’\nIn addition, the following Supplementary Figure 9 has been added in the supporting information.\nSupplementary Figure 9. The crystal plane picture of [Cs2NaCd2Cl]Cl2(MoO4)2 has been indexed using the “Index Crystal Faces” program in Bruker SMART APEX III.\n4) In Figure 4h, the bandgap (Eg) value of KTP is incorrect. Please correct this error.\nResponse: Thanks for reviewer’s kind comments and good suggestions! We have corrected the bandgap (Eg) value of KTP in Figure 4h.\n5) Are the band-gap and UV cutoff algorithms correct in this paper? How to deal with the big difference between the measured value and the theoretical calculation band gap?\nResponse: Thanks for reviewer’s comments! The band-gap and UV cutoff algorithm are correct in this paper. The measured band-gap value of this compound is converted to the absorption according to the following Kubelka-Munk Equation:\nF(R) = K/S = (1−R)2/2R\nwhere R represents the reflectance, K the absorption, and S the scattering. On the absorption curve, elongating the linear part to zero, wide band gap of 4.37 eV can be obtained for the compound. This algorithm is commonly used in many papers, such as Angew. Chem. Int. Ed. 2021, 60, 24131-24136. Chem. Mater. 2020, 32, 6780-6787. J. Am. Chem. Soc. 2022, 144, 21916–21925. Angew. Chem. Int. Ed. 2021, 60, 17415-17418. J. Mater. Chem. C 2022, 10, 3300-3306. While the calculated band-gap was carried out using the Perdew-Burke-Ernzerhof (PBE) function based on density functional theory (DFT). The theoretical band-gaps are smaller than the experimental results, which is due to the discontinuity of the exchange-correlation energy functional leading to the underestimation of the band-gap. In general, the band-gaps calculated by this method is 30% ~ 50% lower than the experimental result. (J. Phys.: Condens. Matter 2022, 34, 404002; J. Chem. Phys. 1999,110, 5029-5036.) However, the band gap we calculate by this method is 3.19 eV, which is within a reasonable range.\nAccordingly, the following sentence has been added at Line 15 in the left column of Page 7 in the text: ‘Typically, this calculation method yields band-gap value that is 30-50% smaller than those obtained experimentally.’\nWe have also added the references ‘[71] Mao, G. et al. DFT-1/2 and shell DFT-1/2 methods: electronic structure calculation for semiconductors at LDA complexity. J. Phys.: Condens. Matter 34, 403001 (2022). [72] Ernzerhof, M. & Scuseria, G.E. Assessment of the Perdew–Burke–Ernzerhof exchange-correlation functional. J. Chem. Phys 110, 5029-5036 (1999).’\n6) The authors attribute the ordered orientation of [MoO4] tetrahedra along the c-axis to the anchoring effect of more electronegative Cd2+. Could the authors discuss whether similar effects could be achieved with other aliovalent cations (e.g., Zn2+ or Mn2+)?\nResponse: Thanks for reviewer’s kind comments and good suggestions! Similar anchoring effects could potentially be achieved using other aliovalent cations. However, the anchoring effects could theoretically be achieved using other aliovalent cations, the selection of substituents must satisfy multiple structural and thermodynamic criteria, including ionic radius compatibility and appropriate melting points. In the case of [Cs2NaCd2Cl]Cl2(MoO4)2, Cs+ and Cd2+ were specifically chosen due to: (i) they relatively low melting points, which facilitate synthesis and (ii) the close match between the ionic radii of Cd2+ (97 pm) and Na+ (95 pm), minimizing lattice strain and enabling effective substitution while maintaining structural stability.\nWe experimentally explored substitutions with Zn2+ and Mn2+ under stoichiometric conditions, but both attempts failed to incorporate these cations into the [Cs2NaCd2Cl]Cl2(MoO4)2 structure, as evidenced by XRD analysis. For Zn2+ substitution ([Cs2NaZn2Cl]Cl2(MoO4)2), the XRD pattern showed dominant phase impurities: Cs2Zn2(MoO4)3 and CsNaMo2O7, while Mn2+ substitution ([Cs2NaMn2Cl]Cl2(MoO4)2) resulted in incomplete reaction with secondary phases: Cs2Mn2(MoO4)3, CsCl(Mo3O10) and MoO3, indicating unsuccessful incorporation. These results align with the theoretical constraints, as the significant mismatch in ionic radii (Zn2+: 74 pm; Mn2+: 80 pm vs. Na+: 95 pm) and higher melting points likely hindered stable cation replacement. We view these results as important guidelines for future studies and plan to systematically investigate this substitution strategy further, with the aim of identifying optimal cationic replacements.\nFigure 4. XRD patterns of attempted substitutions with Zn2+ (a) and Mn2+ (b) for Cd2+ in [Cs2NaCd2Cl]Cl2(MoO4)2."}, {"round": 2, "reviewer_comments": {"reviewer_1": "the authors have replied to the comments from referees， most questions are answered, but not acurately explained the concerns of questions. authors do very focus on the metal ion size on the structure formation and NLO properties, that is important, here i would like to cocus on the metal ion type, if Ca ion replace Cd, the crystal can show the same thing, because Cd and Ca show very close size. I want to know the exact effect of d orbital of Cd in this crystal. Mn and Zn cannot do, i hope to know the NLO behavior of Ca replace of Cd in the crystal, NLO property change or not.", "reviewer_2": "Most of the issues raised by the reviewers have been well addressed by the authors. However, a few minor points still require attention before final acceptance. Detailed comments are provided below:\n1. In the newly added Methods subsection, Dipole Moment Calculation, the authors state:\n“… 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 ….”\nAn appropriate reference should be added.\n2. In Figures 4d and 4e, the x-axis units are incorrect and should be revised to µm.\n3. The stretching vibrations of Mo–O are typically observed as distinct peaks below 1000 cm–1. The peaks reported at 1845 cm–1 and 1600 cm–1 are unlikely to originate from Mo–O stretching and may instead arise from other factors. Could the authors provide further details and measure an additional IR spectrum using a polycrystalline sample?", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "I have received the revised manuscript (NCOMMS-25-51761A) and noted that the authors have made careful and thorough revisions to the original draft. The authors have also provided detailed point-by-point responses to my comments, explaining how each of the suggestions was addressed. I believe the revised version has fully resolved my main concerns. In my opinion, the scientific rigor and readability of the paper have been significantly enhanced through these revisions. I consider the manuscript now meets the publication standards of Nature Communications."}, "author_response": "Response: Thanks for the reviewer’s insightful comments!\nActually, when we initially intended to synthesize the heterocationic perovskite from the maternal template Na3Cl[MoO4], we had already attempted to synthesize its Ca-isostructural analogue. Unfortunately, the Ca-based analogue was not obtained; instead, the reactions consistently yielded a mixture of CaMoO4, CsCl, and Cs2Mo2O7Cl (Supplementary Figure 10a, b), despite attempts with various calcium precursors and flux-assisted solid-state reaction conditions. That may be attributed to the high stability and energetically favorable formation of CaMoO4 in this system.\nAlthough the experimental substitution of Cd with Ca was not achieved, to investigate the specific role of Cd 4d orbitals 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). The structure relaxed smoothly to a local minimum, confirming the geometric stability of the Ca analogue and its suitability for electronic comparison.\nElectronic-structure analysis reveals that the Ca-analogue has slightly larger band gaps (3.36 eV) than [Cs2NaCd2Cl]Cl2(MoO4)2 (3.19 eV) (Supplementary Figure 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) and CaMoO4 (3.96 eV) vs. CdMoO4 (3.31 eV). (J. Appl. Phys. 2008, 103, 073524; Ceram. Int., 2024, 50, 26361-26370; Cryst. Growth Des., 2008, 8, 3595–3601) Furthermore, based on the electronic structures, the NLO coefficients of [Cs2NaCa2Cl]Cl2(MoO4)2 and [Cs2NaCd2Cl]Cl2(MoO4)2 were also calculated (Supplementary Table 9). 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 both compounds are provided in Supplementary Figure 11c-e. The PDOS show that the contributions of Cs, Na, Mo, O and Cl are 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 4d orbitals form narrow and deep bands centered around −7 eV, and show weak direct hybridization with the coordinating ions p states (Supplementary Figure 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 Figure 11e), which indicates the unique polarized characteristic of Cd 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 non-centrosymmetric framework, thereby amplifying the macroscopic second-order susceptibility. This has also been observed in other papers, such as Phys. Rev. B, 2011, 84, 195439; Phys. Rev. B, 2004, 70, 035112; J. Appl. Phys. 2008, 103, 073524. 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)2 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.\nAccordingly, the following sentences have been revised and added at Line 16 in the right column of Page 7 in the paper: ‘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?\nTo 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 Figure 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. 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 Figure 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)2 was also calculated and compared with [Cs2NaCd2Cl]Cl2(MoO4)2 (Supplementary Table 9). 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 Figure 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 4d orbitals form narrow and deep bands centered around −7 eV, and show weak direct hybridization with the coordinating ions p states (Supplementary Figure 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 Figure 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 susceptibility.81 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)2 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.’ And these new Supplementary Figure 10, 11 and Supplementary Table 9 have been added in the supporting information.\nWe have also added the references ‘[78] R. Lacomba-Perales, J. Ruiz-Fuertes, D. Errandonea, D. Martínez-García & Segura, A. Optical absorption of divalent metal tungstates: Correlation between the band-gap energy and the cation ionic radius. J. Appl. Phys. 83, 37002 (2008); [79] Gancheva, M., Iordanova, R., Koseva, I., Avdeev, G. & Ivanov, P. Direct mechanochemical synthesis of CaMoO4 and Dy3+ doped CaMoO4 nanoparticles and their photoluminescent properties. Ceram. Int. 50, 26361-26370 (2024).; [80] Lin Zhou, Wenzhong Wang, Haolan Xu & Sun, S. Template-Free Fabrication of CdMoO4 Hollow Spheres and Their Morphology-Dependent Photocatalytic Property. Cryst. Growth Des. 8, 3595–3601 (2008).; [81] Menéndez-Proupin, E., Gutiérrez, G., Palmero, E. & Peña, J.L. Electronic structure of crystalline binary and ternary Cd−Te−O compounds. Phys. Rev. B 70, 035112 (2004).’\nSupplementary Figure 10. (a) XRD patterns of the stoichiometric reaction mixtures heated at different temperatures (350–550 °C) in air, using CaO as the calcium source for the Ca-target compound. (b) Using different calcium sources (CaCl2, CaCO3, Ca(OH)2 and CaO) for the Ca-target compound at 500 °C.\nSupplementary Figure 11. (a, b) Electronic band structures of [Cs2NaCd2Cl]Cl2(MoO4)2 and [Cs2NaCa2Cl]Cl2(MoO4)2, respectively. (c, d) The total and partial DOS projected on the constitutional atoms of [Cs2NaCd2Cl]Cl2(MoO4)2 and [Cs2NaCa2Cl]Cl2(MoO4)2, respectively. (e) The ELF maps of [Cs2NaCd2Cl]Cl2(MoO4)2 and [Cs2NaCa2Cl]Cl2(MoO4)2 along Cd2+ and Ca2+ cationic slices.\nSupplementary Table 9. Calculated theoretical nonlinear optical coefficients of [Cs2NaCd2Cl]Cl2(MoO4)2 and Ca-substituted model [Cs2NaCa2Cl]Cl2(MoO4)2.\nCompounds d15 d24 d33\n[Cs2NaCd2Cl]Cl2(MoO4)2 9.31 8.58 −8.53\n[Cs2NaCa2Cl]Cl2(MoO4)2 8.77 6.55 −5.52\nReviewer #2 (Remarks to the Author):\nMost of the issues raised by the reviewers have been well addressed by the authors. However, a few minor points still require attention before final acceptance. Detailed comments are provided below:\n1. In the newly added Methods subsection, Dipole Moment Calculation, the authors state:\n“… 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 ….”\nAn appropriate reference should be added.\nResponse: Thanks for reviewer’s kind comments! As suggested, we have added references [90] and [91] to the “Dipole Moment Calculation” subsection in the Methods section. The relevant citation numbers in the manuscript have been updated accordingly.\n[90] Brown, I.D. & Altermatt, D. Bond-valence parameters obtained from a systematic analysis of the Inorganic Crystal Structure Database. Acta Cryst. B B41, 244-247 (1985).\n[91] Maggard, P.A., Nault, T.S., Stern, C.L. & Poeppelmeier, K.R. Alignment of acentric MoO3F3− anions in a polar material: (Ag3MoO3F3)(Ag3MoO3F3)Cl. Journal of Solid State Chemistry 175, 27-33 (2003).\n2. In Figures 4d and 4e, the x-axis units are incorrect and should be revised to µm.\nResponse: Thanks for reviewer’s kind comments! We have revised the x-axis units of Figures 4d and 4e to µm.\n3. The stretching vibrations of Mo–O are typically observed as distinct peaks below 1000 cm–1. The peaks reported at 1845 cm–1 and 1600 cm–1 are unlikely to originate from Mo–O stretching and may instead arise from other factors. Could the authors provide further details and measure an additional IR spectrum using a polycrystalline sample?\nResponse: Thanks for the reviewer’s kind comments and suggestions! According to the reviewer’s suggestion, we have measured the IR spectrum of a polycrystalline sample of [Cs2NaCd2Cl]Cl2(MoO4)2. As shown in Supplementary Figure 7a, a distinct absorption peak is observed at 804 cm–1, which we assign to the fundamental Mo–O stretching vibration of [MoO4]2– unit. This observation is fully consistent with the expected vibrational signature below 1000 cm–1. To further validate this assignment, we performed DFT calculations at the B3LYP level with Gaussian 09 based on a larger cluster model Cs4Na2Cd3Cl6[MoO4]4 representing the local crystal field. The calculated vibrational frequencies show excellent agreement with the experimental IR and Raman spectra (Supplementary Figure 7c, d), unambiguously confirming the origin of the sub-1000 cm–1 feature.\nThe additional peaks at 1651 cm–1 and 1757 cm–1 are present only in the single-crystal IR spectrum (Supplementary Figure 7b) and are not obvious in the polycrystalline measurement. We attribute these features not to fundamental vibrations but to multi-phonon absorption processes, specifically overtones and combination bands arising from the coupling between fundamental Mo–O stretches and lattice phonons. The detection of these weak second-order signals is facilitated by the high optical quality and low scattering loss in the single crystal, whereas they are obscured by dominant scattering effects in the powder sample. (Phys. Rev. B 20, 1747-1749 (1979); Phys. Rev. 158, 433-445 (1967); Angew. Chem. Int. Ed. 64 (2025); Dalton Trans. 46, 2778-2784 (2017); J. Mater. Chem. C 13, 4374-4378 (2025); Chem. Sci. 15, 7104-7110 (2024)) The influence of multi-phonon absorption on the infrared cutoff behavior is a well-established phenomenon in complex oxides, such as LiNbMoO6 (479-5200 nm), and CdTeMoO6 (345-5400 nm) La3ZrGa5O14 (239-7800nm), and the mechanism aligns with the observations in our system.\nAccordingly, the following sentences have been added at Line 38 in the right column of Page 2 in the paper: ‘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. 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 Figure 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 Figure 7c, d).’\nWe have also added the references ‘[49] Bendow, B., Lipson, H.G. & Mitra, S.S. Multiphonon infrared absorption in highly transparent MgF2. Phys. Rev. B 20, 1747-1749 (1979); [50] Lu, D. et al. La3ZrGa5O14: Band‐Inversion Strategy in Topology‐Protected Octahedron for Large Nonlinear Response and Wide Bandgap. Angew. Chem. Int. Ed. 64 (2025); [51] Li, P.-F., Hu, C.-L., Mao, J.-G. & Kong, F. Pb2(SeO3)(SiF6): the first selenite fluorosilicate with a wide bandgap and large birefringence achieved through perfluorinated group modification. Chem. Sci. 15, 7104-7110 (2024).’ Besides, the Supplementary Figure 7 has been revised at supporting information.\nSupplementary Figure 7. (a) The IR spectrum of [Cs2NaCd2Cl]Cl2(MoO4)2 polycrystalline. (b) The IR spectrum of [Cs2NaCd2Cl]Cl2(MoO4)2 single-crystal. (c) The Raman spectra of [Cs2NaCd2Cl]Cl2(MoO4)2. (d) The vibrational frequencies of larger cluster model Cs4Na2Cd3Cl6[MoO4]4 representing the local crystal field.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nReviewer #4 (Remarks to the Author):\nI have received the revised manuscript (NCOMMS-25-51761A) and noted that the authors have made careful and thorough revisions to the original draft. The authors have also provided detailed point-by-point responses to my comments, explaining how each of the suggestions was addressed. I believe the revised version has fully resolved my main concerns. In my opinion, the scientific rigor and readability of the paper have been significantly enhanced through these revisions. I consider the manuscript now meets the publication standards of Nature Communications."}]

Decision letter
Thank you for the opportunity to publish in Nature Communications
Sincerely yours,
Hongwei Yu
Professor of Chemistry
State Key Laboratory of Crystal Materials
Tianjin Key Laboratory of Functional Crystal Materials
Institute of Functional Crystal
Tianjin University of Technology
No. 391 Bin Shui Xi Dao Road
Tianjin 300384, China
Phone: (086)022-60216056
E-mail: yuhw@email.tjut.edu.cn