Borates are in the spotlight owing to their extended structural diversity and broad applications in the design of photonic and optoelectronic devices1,2,3,4,5, nuclear waste separation and sequestration6,7, commercial glasses8,9, amorphous oxide catalysts10,11, as well as Li-ion12,13 and Mg-ion14,15 rechargeable batteries. Consequently, considerable effort has been put toward the search for borates with distinctive atomic structures and properties16,17,18,19,20,21,22. For borate-based nonlinear optical materials, (i) absorption edge energy, (ii) birefringence, and (iii) nonlinear optical coefficient are the three critical parameters. They are mainly affected by the electronic properties of the microscopic B–O basic units, especially for the alkali and/or alkaline earth metal borates (abbreviated as A-borates)23,24,25,26. Anionic group theory provides the basis for the structure–property relationships27. That is to say, the limits of the optical properties of borates lie in the capability of the microscopic units to produce a large bandgap, polarizability anisotropy, and hyperpolarizability18. Thus, the maximum thresholds of bandgap, birefringence, and nonlinear optical coefficient of A-borates comprising BO3 and BO4 units, are ~150 nm, ~0.07@1064 nm, and ~0.64 pm/V, respectively, assuming all the basic units are in optimally aligned configurations18,27. However, compounds with a higher absorption edge energy (lower wavelength), larger birefringence, and larger nonlinear optical coefficient are required to enable smaller optical devices leading to new uses and increased efficiency in industrial and scientific applications. For example, birefringent materials are essential for modulating polarized light and thus underpin critical optical devices such as beam-splitting polarizers for microscopes and isolators, circulators, and interleavers used in fiber optics for telecommunications18. The realization of strategies to push the current limitations of A-borates is still a grand challenge in the field of optical materials.

Vacant p orbitals in boron cause it to behave as an electron pair acceptor to interact with oxygen nucleophiles and form sp, sp2, and sp3 hybridized chemical bonds to construct linear BO2, triangular BO3, and tetrahedral BO4 basic units. Despite the nearly 300 natural borate minerals, 3000 borate crystals, and extensive glass research on non-crystalline alkali borates, the structure models are almost exclusively built up from the BO3 and BO4 polyhedra. Even, linear BO2 units are rare and have only been observed in apatite-type A10(PO4)6–xy(SiO4)x(BO4)y(BO2) and Gd4(BO2)O5F28,29,30,31. It should be noted that the apatite compounds contain atomic disorder and the two isolated Gd4(BO2)O5F crystals are twinned31. Thus, among the thousands of borates, there is an absence of studies of borate-based optical materials with BO2 units. Structurally, the realization of borates with the BO2 functionality will dramatically enrich the structural diversity of this technologically important family of materials and offer a path toward improving the three critical optical parameters in A-borates. This basis stimulates the exploration of A-borates with BO2 configurations and routes toward a more fundamental understanding of the origin of this unique building unit.

Herein, a mixed metal borate K5Ba2(B10O17)2(BO2) with BO2 units was obtained. Boron solid-state nuclear magnetic resonance (NMR) spectroscopy has long been a core characterization tool in the study of borate crystals32,33 and glasses34,35. The resonances ascribed to BO4 units are observed from 2 to –4 ppm and are narrow due to their small quadrupolar coupling constants of typically <1 MHz while the resonances ascribed to BO3 units appear from 12 to 19 ppm and are broader with quadrupolar coupling constants of 2–3 MHz34,36,37. Differences in the number of non-bridging oxygen atoms within a given local coordination can often be determined through careful analysis of 11B shielding33,38. The 11B spectral characteristics of linear BO2 observed here in K5Ba2(B10O17)2(BO2) do not follow the chemical shift trend but rather are unique in the anisotropy of the shielding interaction. The ordered, diamagnetic structure enables clear identification of BO2 even in the presence of BO3 and BO4 units. The solid-state NMR signatures of the 11 distinct boron sites in K5Ba2(B10O17)2(BO2) are examined from multifield experimental spectra recorded under static and magic-angle spinning conditions through the support of DFT calculations and multiple-quantum magic-angle spinning (MQMAS) experiments.

K5Ba2(B10O17)2(BO2) expands the frontiers of structural diversity and functionality. Structurally, this is the first compound that contains all three borate motifs simultaneously, namely linear BO2, trigonal BO3, and tetrahedral BO4. The BO2 units have the largest polarizability anisotropy, and thus push the maximum thresholds of birefringence in A-borates to 0.18@1064 nm, which is much higher than can be attained in compounds with only BO3 and BO4. Such a large theoretical birefringence from this motif sets a target for the design of optical materials.

Results and discussion

Crystal structure of K5Ba2(B10O17)2(BO2)

Single crystals of K5Ba2(B10O17)2(BO2) were obtained (Supplementary Fig. 1) by a high-temperature solution method in an open system, and a phase-pure polycrystalline sample was obtained by a stoichiometric solid-state reaction. The crystal structure was solved and refined from single-crystal X-ray diffraction (XRD) data. K5Ba2(B10O17)2(BO2) crystallizes in the triclinic crystal system in space group \(P\bar{1}\) (Supplementary Table 1). The crystal structure of K5Ba2(B10O17)2(BO2) features two dimensional (2D) 2[B10O17] double layers with K- and Ba-based polyhedra residing in the tunnels and interlayers (Fig. 1). The asymmetric unit of K5Ba2(B10O17)2(BO2) contains three, one, eleven, and eighteen crystallographically independent K, Ba, B, and O atoms, respectively (Supplementary Table 2). With respect to the B–O polyanionic framework, the B(11) atom resides in linear BO2 coordination, while B(1, 2, 4, 6, 8, 9) atoms sit in trigonal BO3 sites and B(3, 5, 7, 10) have tetrahedral oxygen coordination (Fig. 1a and Supplementary Table 2). The B–O bond lengths in BO2 (1.255 Å) are shorter than the mean values of BO3 (1.385 Å) and BO4 (1.475 Å), respectively, and the O–B–O angle refined to 180°, which conforms to sp hybridization. The geometries of BO3 and BO4 here are consistent with those of reported borates28,29,30,31,39,40,41,42. In K5Ba2(B10O17)2(BO2), three BO3 and two BO4 units form a B5O11 ring, which is further linked to form a B10O21 double ring by sharing a common O(7) atom. The double rings spread out over the ac- plane by sharing terminal O atoms to construct a 2[B5O9] infinite layer, then adjacent layers further link to form a 2[B10O17] double layer (Fig. 1c). The double layers are stacked along the [010] direction in the –AAAA– sequence and held together via electrostatic K–O and Ba–O interactions (Fig. 1b). Topologically, the structural framework of K5Ba2(B10O17)2(BO2) can be simplified to a four-connected unimodal net with the Schläfli symbol {48·62}, in which the B5O11 single ring is considered as the four-connected node. According to the rules proposed by Xue et al.2, the linear BO2 unit can be denoted by the letter ‘L’, and the algebraic description of the title compound can be written as {11:2[2 < 3∆+2 T>]&L}.

Fig. 1: Crystal-structural features of K5Ba2(B10O17)2(BO2).
figure 1

a B–O basic units and their relative positions in the asymmetric unit of K5Ba2(B10O17)2(BO2). K5Ba2[B10O17]2(BO2) presents the first case that contains all the three available basic motifs simultaneously, namely BO2, BO3, and BO4 units. b Crystal structure of K5Ba2(B10O17)2(BO2) viewed along the a-axis. c The 2[B5O9] infinite layer and isolated [BO2] anions in K5Ba2(B10O17)2(BO2). The adjacent 2[B5O9] infinite layers further link to form 2[B10O17] double layers and then stack along the [010] direction in the –AAAA– sequence. The atoms in lavender, green, black, and red are K, Ba, B, and O atoms, respectively. All the bonds of K–O and Ba–O are removed for clarity. The O atoms in isolated [BO2] units are non-bridging atoms.

Solid-state nuclear magnetic resonance spectroscopy of K5Ba2(B10O17)2(BO2)

Multifield 11B static and magic-angle spinning (MAS) NMR spectroscopy, aided by DFT calculations of the chemical shielding and quadrupolar coupling, was used to study the local structure of the boron moieties and to gain fundamental physical insights into the anisotropy of the BO2 site (Fig. 2, Supplementary Fig. 2). The signal centered about 1.5 ppm in the MAS spectra (Fig. 2a, Supplementary Fig. 2a) was ascribed to the four overlapped BO4 resonances predicted from DFT32,34. This low-frequency signal is better resolved at a high magnetic field (Supplementary Fig. 2a) and is resolved into three distinct sites (in 2:1:1 ratio) by multiple-quantum MAS (MQMAS) (Supplementary Figs. 34)43, confirming the expected number of distinct BO4 sites. The low-field (9.4 T) MAS spectrum also revealed a broad shoulder centered about 13 ppm that is characteristic of BO333. This feature is nearly baseline-resolved at high field (16.4 T, Supplementary Fig. 2a). The overall MAS lineshapes could almost be reproduced by performing numerical simulations of the shift and quadrupolar parameters of the four BO4 and six BO3 sites directly from DFT calculations of the K5Ba2(B10O17)2(BO2) XRD structure model. Owing to a large number of adjustable shift and quadrupolar parameters from the ten overlapping distinct BO4 and BO3 crystallographic sites, all parameters except the isotropic shift were fixed to their calculated values and not refined in the fit (see Supplementary Table 5). However, after calculating the overall lineshape from BO3 and BO4 contributions, a small feature resembling a powder ‘horn’ line-shape was unaccounted for, which is most distinct at –12 ppm in the low-field MAS data where it does not overlap with other boron sites. Turning to the static 11B spectra, the BO4 and BO3 regions are poorly resolved due to the presence of broadening from chemical shift anisotropy (CSA), dipolar, and second-order quadrupolar interactions. Meanwhile, small but distinct features were observed at ca. 100 and –110 ppm at low field (Fig. 2b) and ca. 80 ppm at high field (Supplementary Fig. 2b), outside the expected range for known borates. DFT calculations predicted a relatively large CQ for BO2 but within the same range as BO3; the CSA of BO2, on the other hand, was calculated to be an order of magnitude larger than those of BO3 and BO4 units32. Finally, the calculated shift of 9 to 10 ppm is intermediate between BO4 at lower frequency and BO3 at higher frequency. A simulated lineshape based on the DFT-calculated CSA and quadrupolar parameters for the B(11) (BO2) site agrees well with the unexplained features in the MAS and static spectra. Allowing the lineshape to be refined against the experimental data yielded an isotropic shift of 13.5(3) ppm, a reduced CSA (Haeberlen convention, see “Methods”) of –123(3) ppm, a quadrupolar coupling constant of 3.31(5) MHz, asymmetries near zero, and coincident shielding and quadrupolar tensors aligned along O–B–O (Fig. 2, Supplementary Figs. 2, 5, and 6, and Supplementary Table 5). The BO2 site could not be detected in the MQMAS spectrum due to its low intensity and large CQ44,45; however, the BO3 and BO4 resonances gave good agreement with the DFT-calculated NMR parameters for the K5Ba2(B10O17)2(BO2) structure model. That the observed CQ and CSA were smaller than the calculated values for the BO2 B(11) site—and B(11) only—is an indication of partial averaging from dynamics within this isolated (i.e., not covalently bound) moiety46. This observation is in agreement with the diffraction-derived atomic displacement parameters of B(11) and O(17) from BO2 being approximately a factor of three larger than all the other boron and oxygen atoms for which motion is restricted within the polymeric borate framework.

Fig. 2: Experimental and simulated 11B NMR spectra.
figure 2

(Left) The experimental powder lineshape along with the 11 individual boron sites that comprise the summed simulation are shown for a 40 kHz magic-angle spinning and b static conditions. The shift and quadrupolar tensors are given in Supplementary Table 5. (Right) Highlighted spectral features of the linear BO2 motif and its contribution to the overall lineshape of K5Ba2(B10O17)2(BO2).

The intermediate 11B shift of BO2 is unexpected and does not follow the trend of, e.g., 27Al in aluminates wherein the shift increases as the coordination number decreases (δ 27Al: AlO4 > AlO5 > AlO6)47. However, the origin of the intermediate shift can be explained by a change in bonding from B–O single bonds in BO4 and BO3 to B = O double bonds in BO2. Note that O=B=O has a formal negative charge on the boron center. The borate NMR picture is consistent with the isoelectronic carbon analogues: The 13C shift of CO2 (126–129 ppm) is shielded relative to CO3 (163–171 ppm)48,49. Moreover, the large CSA of BO2 (–123(3) ppm) is analogous to the large CSA of CO2 (–210 ppm)50, further highlighting the similarities between these two species. Finally, the shift and quadrupolar asymmetry parameters ηCSA and ηQ, respectively, for the central atom of a linear species with D∞h symmetry should be zero (δXX = δYY; VXX = VYY). An ηCSA of zero was found for solid 13CO2 within experimental uncertainty47. Similarly, ηCSA and ηQ for 11BO2 in K5Ba2(B10O17)2(BO2) were calculated to be slightly above zero due to interactions with neighboring atoms and were observed to be zero or near zero within the experimental uncertainty.

The chemical shift anisotropy is by far the most distinctive NMR feature of the elusive BO2 moiety and thus careful experiments are required to take advantage of this characteristic. Acquiring high signal-to-noise and performing measurements at multiple magnetic field strengths is useful to differentiate line broadening arising from CSA versus the quadrupolar interaction. Furthermore, magic-angle spinning effectively averages CSA but only partially averages second-order quadrupolar interactions, so it is useful to compare static and MAS spectra. The 11B parameters of the BO2 site are consistent between considerations of symmetry and bonding, DFT calculations, and experimental lineshapes across multiple magnetic field strengths and MAS/static conditions. It was possible to identify the linear BO2 unit in K5Ba2(B10O17)2(BO2) although it only comprises 1/21 of the boron atoms in the structure and thus 5 % of the integrated intensity, which is distributed over a broad frequency range. Extending these principles beyond this compound, solid-state 11B NMR, as well as 17O NMR (see Supplementary NMR Discussion in the Supporting Information), may serve as powerful tools for the identification of BO2 groups in borates even in the absence of atomically ordered single crystals.

Spectral analysis and optical properties combining theory and experiment

The infrared spectrum measurement and vibrational modes calculation of K5Ba2(B10O17)2(BO2) were respectively performed to further confirm the coordination environment of the B atoms. Clearly, the experimental infrared spectrum matches well with the first-principles calculations (Supplementary Fig. 7). In the upper range of both spectra (4000–2500 cm–1), no vibrational bands caused by OH or H2O were detectable, which is consistent with the results concluded from the single-crystal data. With respect to the linear BO2 units, the weak peak at 2009 cm–1 and the strong peak at 1938 cm–1 are assigned to the 10B–O and 11B–O stretch in the linear B(11)O2 units29. K5Ba2(B10O17)2(BO2) is stable up to 732 °C (see TG–DSC curves and powder XRD patterns, Supplementary Fig. 8), thus sizable single crystals should be grown below this decomposition temperature. The UV−vis−NIR spectrum measured from a polycrystalline sample shows that the reflectance at 190 nm is still about 16.5% and thus its UV cutoff edge is shorter than 190 nm (corresponding to a bandgap larger than 6.52 eV) (Supplementary Fig. 9). Theoretical analysis indicates that K5Ba2(B10O17)2(BO2) has an indirect bandgap of 6.21 eV when calculated with a hybrid (HSE06) functional. With pure DFT, the predicted band gap is only 4.61 eV (Supplementary Fig. 10). This well-known underestimation mainly results from the approximation of exchange-correlation energy under DFT methods; nevertheless, it is useful for qualitative analysis51. The electronic transitions among the states near the Fermi level were analyzed. The top of the valence band is mainly composed of O 2p orbitals (Supplementary Fig. 11). The O 2p orbitals from the linear BO2 units are located deeper in the valence band than those in the 2[B10O17] framework (Fig. 3a–c). In order to further probe the effect of the BO2 units on the electronic structure, the polarizability anisotropy and HOMO–LUMO bandgap were calculated for BO2, BO3, and BO4 units (Fig. 3d). The HOMO–LUMO bandgap of BO2 is larger than that of BO3 but smaller than that of BO4. More importantly, the polarizability anisotropy of BO2 is even larger than those of BO3 and BO4. Thus, larger birefringence, as determined by polarizability anisotropy, can be expected in BO2-containing borates. Structurally, the linear BO2 units can replace the position of halide ions in the related apatite structure28,29,30,31. In order to probe the structure–function relationship of BO2 units on the optical properties, BO2 was computationally substituted for halide ions (Cl, Br) to generate the isostructural halides of K5Ba2(B10O17)2(BO2), namely, K5Ba2(B10O17)2Cl, and K5Ba2(B10O17)2Br. Results show that the halides exhibit similar bandgaps (Supplementary Figs. 10 and 12) but smaller birefringence (0.057@1064 nm) than K5Ba2(B10O17)2(BO2) (0.062@1064 nm) (Supplementary Fig. 13). The birefringence enhancement from BO2 units is small due to the low density (1/21) of the boron sites in K5Ba2(B10O17)2(BO2). Facing this, we postulate that a borate with a high density of BO2 units would be a suitable candidate for a material with a wide bandgap and large birefringence. To test this hypothesis, a structure comprising solely BO2 units was investigated: K(BO2) (Figs. 3e, 3f). In the phonon spectrum of K(BO2), there are no soft modes at any wave vectors and no imaginary phonon modes are observed (Fig. 3g), demonstrating that K(BO2) is dynamically stable. The bandgap is 6.11 eV (HSE06 hybrid functional) and the birefringence is 0.18@1064 nm. This birefringence is larger than that of any previously reported A-borate, including Ca(BO2)2 (0.124@1064 nm) comprising infinite chains of co-planar BO3 trigonal rings52. The predicted birefringence in K(BO2) would lead to higher-efficiency beam splitting in light polarization processes across telecommunications and scientific instrumentation applications and provide enhanced ability to create and control polarized light. The BO2 functionality could enable optical capabilities, specifically in the rarely accessible deep UV range, and thus sets a target for synthetic crystal growth.

Fig. 3: The structure-property relationships of K5Ba2(B10O17)2(BO2).
figure 3

ac The structure-property relationships of K5Ba2(B10O17)2(BO2). Frontier orbitals (isosurface) and partial density of states of O in K5Ba2(B10O17)2(BO2). d Calculated HOMO–LUMO gap (Eg) and polarizability anisotropy of B(11)O2, B(1)O3, and B(3)O4 units in K5Ba2(B10O17)2(BO2). The polarizability anisotropy of BO2 units is even larger than those of BO3 and BO4 units. Thus, larger birefringence, as determined by polarizability anisotropy, can be expected in BO2-containing borates. e, f Theoretical structural model of K(BO2) with solely BO2 units derived from K5Ba2(B10O17)2(BO2). g The phonon dispersion of K(BO2). There are no soft modes at any wave vectors and none of the imaginary phonon modes are observed, demonstrating that K(BO2) is dynamically stable.

In summary, a mixed metal borate K5Ba2(B10O17)2(BO2) with linear BO2 units was obtained successfully. It also presents as the first compound that contains all three available B–O bonding motifs: BO2, BO3, and BO4 units. Solid-state 11B NMR spectroscopy unequivocally identified the nature of the BO2 units in K5Ba2(B10O17)2(BO2) and the experimental and calculated NMR parameters provide a quantitative basis for the future identification of BO2 units in polycrystalline and non-crystalline samples. The beneficial role of BO2 units on the bandgap and birefringence in K5Ba2(B10O17)2(BO2) were discussed and a model, K(BO2), with only BO2 units was designed and analyzed theoretically to motivate future directions for deep UV optical materials. Aligned BO2 units with the highest polarizability anisotropy push the maximum thresholds of birefringence in A-borates to 0.18@1064 nm—far beyond the birefringence limit with only BO3 and BO4. K5Ba2(B10O17)2(BO2) expands the frontiers of complex borate chemistry while the unique properties and spectroscopic characterization of the elusive BO2 unit motivate and provide a methodology for its broader study.


Synthesis and crystal growth

KNO3 (aladdin, 99.5%), Ba(NO3)2 (aladdin, 99.5%), K2CO3 (aladdin, 99.0%), B2O3 (aladdin, 99.9%) and H3BO3 (aladdin, 99.5%) were used as received. Single crystals of K5Ba2(B10O17)2(BO2) were grown by the high-temperature solution method. A mixture of KNO3 (0.379 g, 3.75 mmol), Ba(NO3)2 (0.392 g, 1.50 mmol), and B2O3 (0.548 g, 7.87 mmol) were loaded into a platinum crucible. The crucible was heated to 830 °C in 10 h, and held at this temperature for 17 h, and then cooled to 30 °C with a rate of 1.5 °C/h. Colorless and block-shaped crystals of K5Ba2(B10O17)2(BO2) were obtained and manually picked out from the crucible for structural characterization (Supplementary Fig. 1). Polycrystalline samples of K5Ba2(B10O17)2(BO2) can also be obtained by the lower temperature through solid-state reaction based on the following reaction: 2.5K2CO3 + 2Ba(NO3)2 + 21H3BO3 → K5Ba2(B10O17)2(BO2) + 2.5CO2↑ + 4NO2↑ + 31.5H2O↑. A mixture of K2CO3, Ba(NO3)2, and H3BO3 was preheated at 200 °C for 10 h. After that, the temperature was gradually raised to 700 °C with several intermediate grindings and mixings, and then held at a selected temperature for 24 h. The powder XRD pattern of the polycrystalline samples matches well with the one calculated from single-crystal XRD analysis except for some preferred orientation peaks.


Single-crystal XRD data were collected on a Bruker SMART APEX II 4 K CCD diffractometer using Mo Kα radiation (λ = 0.71073 Å) at room temperature. Data integration, cell refinement, and absorption corrections were carried out with the program SAINT53. The structure was solved by direct methods and refined on F2 by full-matrix least-squares techniques using the program suite SHELXTL54. Solutions were checked for missed symmetry using PLATON. The crystal data give Rint = 0.0249 and the structure solution parameters are R1 = 0.0340, wR2 = 0.0812, GOF = 1.077. Powder XRD data were collected with a Bruker D2 PHASER diffractometer (Cu Kα radiation with λ = 1.5418 Å, 2θ = 5–70°, scan step width = 0.02 °, and counting time = 1 s/step). During the structure analysis, we found that K(3) and Ba(1) can also be set to share the same sites with the corresponding Ba and K (occupancies of about 97 and 92%, respectively.). But, it has no effect on the B-O framework, thus, in order to build an ideal model to study the BO2 units, a disorder-free model was considered. Thermal gravimetric (TG) and differential scanning calorimetry (DSC) measurements were carried out on a simultaneous NETZSCH STA 449 F3 thermal analyzer instrument in a flowing N2 atmosphere. The sample was placed in Pt crucible, heated from 40 to 900 °C at a rate of 5 °C/min. Infrared spectroscopy was carried out on a Shimadzu IR Affinity-1 Fourier transform infrared spectrometer in the 400–4000 cm–1 range. UV–vis–NIR diffuse-reflectance spectroscopy data in the wavelength range of 190–2600 nm were recorded at room temperature using a powder sample of K5Ba2(B10O17)2(BO2) on a Shimadzu SolidSpec-3700DUV spectrophotometer. 11B MAS NMR experiments were performed in static magnetic fields of 9.4 and 16.4 T on Bruker Avance spectrometers at 11B Larmor frequencies of 128.25 MHz or 224.60 MHz, respectively. MAS 11B NMR spectra were recorded with a single non-selective excitation pulse in a 1.6 mm Phoenix probe (9.4 T) or Bruker 4.0 mm probe (16.4 T) using a short pulse length in an effort to obtain quantitative results under quadrupolar nutation. Static spectra were recorded under similar conditions with the exception of a spin-echo pulse sequence at 16.4 T to mitigate probe background. For the MAS spectra, 2400 scans were acquired with a recycle delay of 40 s (9.4 T) and 16 scans with a recycle delay of 0.25 s (16.4 T); for the static spectra, 768 scans were acquired with a recycle delay of 5 s (9.4 T) and 65,536 scans with a recycle delay of 1 s (16.4 T). The measured T1, integrated over the overlapped resonances from 19 to –2 ppm, was 6.0 s (via saturation recovery) but no difference was seen in the lineshape for recycle delays from 0.02 to 90 s. Experimental details for MQMAS experiments are given in Supplementary Figs. 3 and 4. Lineshape simulations were performed in the Solid Lineshape Analysis program in Bruker TopSpin 3.6.1. 11B isotropic shifts were externally referenced to 0.1 M aqueous H3BO3 at 19.6 ppm37. In this work, the Haeberlen convention is used to describe the chemical shift tensor in terms of the isotropic shift (δiso), chemical shift anisotropy (δCSA, sometimes called the reduced CSA), and shift asymmetry parameter (ηCSA)55,56,57. In terms of the principal components of the shift tensor (δXX, δYY, and δZZ):

$${\delta }_{{iso}}=\frac{{\delta }_{{XX}}+{\delta }_{{YY}}+{\delta }_{{ZZ}}}{{\delta }_{{iso}}}$$
$${\delta }_{{CSA}}={\delta }_{{ZZ}}-{\delta }_{{iso}}$$
$${\eta }_{{CSA}}=\frac{{\delta }_{{YY}}-{\delta }_{{XX}}}{{\delta }_{{ZZ}}-{\delta }_{{iso}}}$$

and the principal components are ordered such that \(\left|{\delta }_{{{ZZ}}}-{\delta }_{{{iso}}}\right|\ge \left|{\delta }_{{{XX}}}-{\delta }_{{{iso}}}\right| > \left|{\delta }_{{{YY}}}-{\delta }_{{{iso}}}\right|\). The quadrupolar tensor is described by the nuclear quadrupolar coupling constant (CQ) and quadrupolar asymmetry parameter (ηQ). In terms of the principal components of the electric field gradient (VXX, VYY, VZZ):

$${\eta }_{Q}=\frac{{V}_{{XX}}-{V}_{{YY}}}{{V}_{{ZZ}}}$$

where e is the electric charge, Q is the nuclear quadrupole moment, and h is Planck’s constant.

Computational methods

The electronic structure as well as optical property calculations were performed by employing CASTEP58, a plane-wave pseudopotential DFT package, with the norm-conserving pseudopotentials (NCP)59. The Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA) was used.60 The plane-wave energy cutoff was set at 850 eV. NMR calculations utilized ultrasoft pseudopotentials generated “on-the-fly” in CASTEP61,62. The all-electron magnetic response was calculated with the gauge-including projector-augmented wave (GIPAW) method63. Self-consistent field (SCF) calculations were performed with a convergence criterion of 1 × 10−6 eV/atom on the total energy. The k-point separation was set as 2π × 0.04 Å–1 in the Brillouin zone corresponding to primitive cell, resulting in Monkhorst–Pack k-point meshes of 4 × 3 × 4. In addition, we also calculated bandgaps adopting the nonlocal exchange functional HSE06 a widely used hybrid functional with relatively high efficiency in calculating bandgaps64. The HSE XC functional energy is calculated as follows,

$${E}_{xc}^{{\rm{HSE}}}=a{E}_{x}^{{\rm{HF}},{\rm{SR}}}(\omega )+(1-a){E}_{x}^{{\rm{PBE}},{\rm{SR}}}(\omega )+{E}_{x}^{{\rm{PBE}},{\rm{LR}}}(\omega )+{E}_{c}^{{\rm{PBE}}}$$

In HSE06, the parameters are suggested as a = 0.25 and ω = 0.11 bohr–1.