Origin of Large Second-Harmonic Generation in Nonpolar Molybdenum Tellurite Compounds

Abstract

Molybdenum tellurite compounds have attracted increasing interest as promising nonlinear optical (NLO) materials, yet their microscopic second-harmonic generation (SHG) mechanisms remain unclear. In this work, the electronic structures and SHG responses of ATeMoO₆ (ATM, A = Mg, Cd, Zn) are systematically investigated using first-principles calculations combined with atom response theory. The results show that the SHG responses are mainly governed by the occupied nonbonding O 2p states and the unoccupied Mo 4d and Te 5p states. Our atom response theory analysis reveals that a strong synergistic effect between stereochemically active lone pairs (SCALPs) on Te atoms and nonbonding O 2p states critically enhances the SHG response in ZnTM and MgTM. In contrast, the relative weaker Te SCALPs in CdTM fail to provide a comparable contribution, leading to its lower SHG performance. The structure group analysis reveals that MoO₄ units dominate the SHG response, while TeO₄ units provide secondary contributions. Moreover, group dipole moments are found to be insufficient to explain the SHG behavior. These findings provide microscopic insights into SHG origins and offer guidance for NLO material design.

1. Introduction

Nonlinear optical (NLO) crystals are essential for frequency conversion in the mid-infrared (mid-IR) region, which is of great technological importance for applications such as spectroscopy, biomedical imaging, and environmental monitoring [1,2,3]. To date, only a few of mid-infrared NLO crystals have achieved commercial application. For instance, ZnGeP₂ and AgGaS₂, despite their large second-harmonic generation (SHG) coefficients, commonly suffer from inherent drawbacks such as low laser-induced damage thresholds or poor thermal stability. Moreover, the growth of large, high-quality single crystals continues to pose significant challenges in terms of scalability, yield control, defect regulation, and cost management, factors that severely challenge their industrial adoption [4,5,6,7,8,9,10,11,12]. As a result, numerous promising candidates remain confined to laboratory research due to persistent synthesis and processing difficulties [13,14]. Consequently, the discovery of new high-performance mid-IR NLO crystals that can meet the demands of modern industry remains a critical scientific challenge to be overcome.

In recent years, the exploration of NLO crystals has attracted considerable attention. Among these, quaternary molybdenum tellurite systems [15,16,17,18,19] have gradually emerged as a research hotspot due to their excellent SHG performance. These systems contain cations susceptible to second-order Jahn–Teller (SOJT) distortion [1,20,21], specifically octahedrally coordinated d⁰ transition metals (such as Nb⁵⁺, V⁵⁺, Mo⁶⁺, and W⁶⁺) and cations with stereoactive lone pairs (such as I⁵⁺, Te⁴⁺, Se⁴⁺, and Sb³⁺). Both types of cations typically reside in asymmetric coordination environments. In 2006, Eun et al. [22] reported the noncentrosymmetric monoclinic compound Na₂Te₃Mo₃O₁₆ (space group: I2), which exhibits an exceptionally large SHG response of approximately 500 × α-SiO₂. Subsequently, Zhou et al. [23] reported Ag₂Te₃Mo₃O₁₆, which also crystallizes in the I2 space group and shows an SHG intensity of about 8 × KDP. In addition, Yu et al. [15] reported BaTeMo₂O₉ with the C2 space group, whose effective nonlinear coefficient reaches 10.3 pm·V⁻¹. Feng et al. [17] synthesized Cs₂TeMo₃O₁₂ single crystals belonging to the hexagonal P6₃ space group, for which the NLO coefficients d₃₂ and d₃₃ were measured to be approximately 6.8 and 6.5 pm·V⁻¹, respectively, indicating favorable nonlinear optical performance. Wu et al. [24] further reported novel mid-infrared NLO crystals Ca₃(TeO₃)₂(MO₄) (M = Mo, W). Both compounds crystallize in the polar monoclinic space group P2₁, with SHG responses reaching 3.5 and 2.8 times that of KDP for the Mo- and W-based compounds, respectively, demonstrating promising potential for mid-infrared applications. Overall, the reported polar molybdenum tellurites generally exhibit strong SHG responses.

In particular, the nonpolar molybdenum tellurite family ATM (A = Mg, Cd, Zn; TM = TeMoO₆) has emerged as a promising mid-IR NLO system, exhibiting unexpectedly large SHG responses. These materials provide an excellent platform for investigating the microscopic origin of SHG beyond conventional macroscopic polarity considerations [25,26,27,28,29]. As early as 1980, Forzatti et al. [30] were among the first to synthesize MgTM and related compounds using solid-state reactions and high-temperature solution (flux) methods. In 2012, Zhang et al. [25] reported the growth of MgTM single crystals with SHG responses up to 1.5 × KTP. Based on local dipole moment calculations, they suggested that the strong SHG response of MgTM mainly originates from three NLO-active structural units, namely TeO₄, MoO₄, and MgO₆ polyhedra. Later, Zhao et al. [26] synthesized CdTM single crystals and reported a large SHG response (2 × KTP) and found that the TeO₄ and MoO₄ polyhedra contribute more significantly to the SHG response than the CdO₄ unit. Subsequently, Li et al. [28] successfully grew large CdTM single crystals using an improved top-seeded solution growth method. The powder SHG intensity was reported to be approximately twice that of KTP, with a nonlinear optical coefficient d₃₆ of 8.5 pm·V⁻¹. Additionally, they also pointed out that the strong SHG response originates from the cooperative contributions of three asymmetric structural units. In the same period, Zhao et al. [27] reported the successful growth of ZnTM single crystals with an SHG efficiency of approximately 10.5 times that of KDP, which was associated with the high polarizability of three asymmetric structural units. Although their fundamental structural units have been highlighted in prior studies, the microscopic origin of the SHG effect in the ATM family, as well as the specific structural motifs that dominate the overall NLO response, remain unresolved.

Theoretically, Jiang et al. [31] investigated the NLO properties of these ATM compounds using systematic first-principles calculations combined with an atomic-cutting analysis. The calculated SHG coefficients d₁₄ of MgTM, CdTM, and ZnTM are 14.04, 11.75, and 10.61 pm·V⁻¹, respectively, showing that nonpolar molybdenum tellurite crystals can still exhibit strong SHG responses, following the trend MgTM > CdTM > ZnTM. They further suggested that the SHG intensity does not depend on the intrinsic static dipole moment of the crystal structure but mainly arises from the favorable molecular-level superposition of MoO₄ tetrahedra. Moreover, Cammarata et al. [32] combined experimental measurement and first-principles calculations to elucidate the origin of the SHG response in nonpolar ATM compounds. Their calculations yielded effective nonlinear coefficients $d_{eff}^p$ of 9.61, 16.96, and 12.65 pm·V⁻¹ for Mg, Cd, and Zn compounds, respectively, indicating the trend CdTM > ZnTM > MgTM. They further demonstrated that the SHG response is primarily determined by the degree of orbital hybridization in the A–O bonds, rather than by the intrinsic structural dipole moment or the magnitude of noncentrosymmetric distortion. Despite previous investigations, the physical origin of the NLO response in these crystals remains debated, and systematic theoretical studies elucidating the underlying structure–property correlations are lacking. Consequently, a detailed analysis of the SHG origin at atomic and orbital levels is essential to understand the observed NLO performance.

In this study, we employed first-principles calculations combined with partial response functional (PRF) and atom response theory (ART) [33,34] to investigate the electronic structures and the origin of the SHG response in the molybdenum tellurite series. Our results indicate that the MoO₄ groups contribute substantially more to the SHG response of the ATM compounds than the TeO₄ groups, suggesting that Mo-centered polyhedra are the primary structural units responsible for the strong nonlinear optical activity in these materials. The strength of stereochemically active lone pairs (SCALPs) on Te atoms is identified as an essential factor modulating the SHG response in these compounds as an effective and positive synergy with O 2p states amplifies the response in ZnTM/MgTM, whereas relative weak Te SCALPs in CdTM lead to a weaker effect. We further found that dipole moments alone are insufficient to account for the strong SHG responses in these materials.

2. Results and Discussion

2.1. Structural Optimization

The relaxed structures of molybdenum tellurite materials MgTM, CdTM, and ZnTM, are shown in Figure 1a,b and Figure S1a. The optimized crystallographic parameters are summarized in Tables S1–S3. A comparison between the calculated structural parameters and the corresponding experimental data shows good agreement, indicating that the optimized structures reliably reproduce the experimental crystal structures. MgTM and ZnTM exhibit similar layered crystal structure with the same orthorhombic space group of P2₁2₁2. Each unit cell contains three distinct coordination polyhedral in which TeO₄ tetrahedra formed by Te coordinated with O1 and O2 (TeO1₂O2₂), MoO₄ tetrahedra formed by Mo coordinated with O2 and O3 (MoO2₂O3₂), and MgO₆/ZnO₆ octahedra formed by Mg/Zn coordinated with O1 and O2 (MgO1₄O2₂/ZnO1₄O2₂). In contrast, CdTM crystallizes in a different layered structure with a tetrahedra space group of $P\overline{4}$2₁m, as shown in Figure 1b. Each unit cell likewise contains three distinct structural groups, TeO₄ tetrahedra formed by Te atoms coordinated with O1 and O2 atoms, MoO₄ tetrahedra formed by Mo atoms coordinated with O2 and O3 atoms, and CdO₄ tetrahedral groups formed by Cd atoms coordinated with O1 atoms.

It is important to note that the topology of the A-site metal atom layer in CdTM differs significantly from that in ZnTM and MgTM. In orthorhombic CdTM, the Cd atoms on the (001) plane form a centered rectangular (or face-centered square) lattice (Figure S2), whereas Zn and Mg atoms in their respective compounds adopt a simple square lattice. This structural distinction directly results in a much shorter Cd–Cd distance (~3.78 Å) compared to the Zn–Zn distance (~5.31 Å), which induces the change of atom coordination environment and the related chemical interactions. The Mo–O bond lengths remain highly consistent across all three compounds, with Mo–O2 ranging from 1.873 to 1.885 Å and Mo–O3 from 1.722 to 1.725 Å. In contrast, the Te–O bonds show notable variation. While Te–O1 stays nearly constant (1.915–1.935 Å), Te–O2 elongates significantly in CdTM (2.169 Å) and contracts in ZnTM (2.111 Å), indicating greater distortion of the TeO₄ tetrahedron in CdTM (Table S4). Therefore, although all three compounds share the same stoichiometry, CdTM exhibits distinct crystal structure and coordination environments around the A-site metal compared to the other two compounds.

2.2. Electronic Structures

Based on GGA+U calculations performed using VASP, the indirect band gaps (E_g^PBE) of MgTM, CdTM, and ZnTM are 3.12, 3.22, and 3.10 eV, respectively (

Table 1. Calculated values of E_g^PBE, E_g^HSE, static SHG tensors d_ij, static effective SHG response $d_{eff}^p$, and birefringence at 1064 nm compared with available experimental data and literature references.

Compound	MgTM	CdTM	ZnTM
Space group	P21212	$P\overline{4}$21m	P21212
EgPBE (eV)	3.12 a; 3.09 b	3.22 a; 3.17 b	3.10 a; 3.07 b
EgHSE (eV)	4.39 c	4.51 c	4.33 c
EgEXP (eV)	3.12 [25]	3.59 [26]	3.54 [27]
dij (pm/V)	d14 = 15.64 a; d25 = 16.19 a; d36 = 16.05 a d14 = 13.79 b; d25 = 13.42 b; d36 = 15.56 b d14 = 7.72 c; d25 = 7.93 c; d36 = 8.36 c d14 = 14.04 [31]; d14 = 6.57 [32]	d14 = d25 = 12.04 a; d36 = 12.65 a d14 = d25 = 10.40 b; d36 = 12.50 b d14 = d25 = 7.27 c; d36 = 8.00 c d14 = 11.75 [31]; d14 = 14.19 [32]	d14 = 12.61 a; d25 = 12.65 a; d36 = 13.27 a d14 = 11.29 b; d25 = 11.05 b; d36 = 13.28 b d14 = 8.13 c; d25 = 8.16 c; d36 = 8.88 c d14 = 10.61 [31]; d14 = 8.64 [32]
$d_{eff}^p$ (pm/V)	13.49 a; 12.05 b; 6.76 c; 9.61 [32]	10.34 a; 9.38 b; 6.35 c; 16.96 [32]	10.86 a; 10.03 b; 7.09 c; 12.65 [32]
EXP. SHG (pm/V)	1.5 × KTP [25]; 250$\times \mathrm{\alpha }$-SiO2 [32]	2 × KTP [26,28]; 300 $\times \mathrm{\alpha }$-SiO2 [32] d36 = 8.5 [28];	10.5 × KDP [27]; 250$\times \mathrm{\alpha }$-SiO2 [32]
Δn at 1064 nm	0.29 a; 0.26 [31]	0.22 a; 0.16 [31]	0.26 a; 0.23 [31]

^a Band gaps are calculated using VASP, and d_ij and Δn are calculated based on the SOS method using VASP + ARTATOP with scissor S^EXP. ^b Band gaps are calculated using ABINIT, and d_ij is calculated based on the SOS method using ABINIT with scissor S^EXP. ^c HSE06 band gaps are calculated using VASP, and d_ij is calculated based on the SOS method using VASP + ARTATOP with scissor S^HSE.

), which are relatively smaller than the corresponding experimental values ($E_g^{\mathrm{E}\mathrm{X}\mathrm{P}}$) of 3.12, 3.59, and 3.54 eV. In addition, hybrid functional HSE06 calculations were carried out, yielding band gaps (E_g^HSE) of 4.39, 4.51, and 4.33 eV for MgTM, CdTM, and ZnTM, respectively, all of which are larger than the experimental band gaps. Therefore, both the experimentally measured band gap (E_g^EXP) and the HSE06 band gap (E_g^HSE) were used as lower and upper bounds in the scissor operation (S^EXP and S^HSE) for the optical property calculations to improve the result reliability. At all computational levels examined, CdTM consistently exhibits the largest band gap, in agreement with experimental observations.

To elucidate the electronic states involved in the optical excitations responsible for the SHG response, we examined the electronic band structures, PDOS, and COHP of the three molybdenum tellurite materials (Figure 1 and Figure S1). In all three compounds, the valence bands (VBs) near the Fermi level (from ~−2.3 eV to E_F) are predominantly composed of nonbonding O 2p states. Near the VBM, these states exhibit significant Te–O antibonding character (~−1.1 eV to E_F), which reflects the presence of SCALP electrons on Te atoms, as supported by the -pCOHP analysis (Figure 1f,i and Figure S1d). This conclusion is further corroborated by the partial charge density (PCD) for the VBM and the top VB states (Figure 1a,b and Figure S1a), as well as by the electron localization function (ELF) profiles (Figure S3). Both analyses reveal a highly asymmetric electron density lobe localized on the Te atom along the c axis, confirming the existence of SCALPs in ZnTM and MgTM. However, a distinct contrast is observed in CdTM as its PCD at the k-point of the VBM shows essentially no such localized electron cloud on Te (Figure S4b), indicating a markedly weaker contribution from Te SCALPs (Figure 1d). The diminished NLO accumulation at VBM contribution in CdTM likely originates from a comparatively weaker Te–O interaction. This is evidenced by CdTM having the lowest integrated COHP (-ICOHP) value (4.53 eV, Table S4) among the series, which correlates with a reduced Te–O antibonding interaction near E_F. This electronic structure difference is also reflected in the fat band representations (Figure 1c,d and Figure S1b). In CdTM, the Te-derived states are primarily concentrated along F–K of the Brillouin zone (BZ). In contrast, for ZnTM and MgTM, the Te contributions are distributed more uniformly across the entire BZ.

Below this region, between approximately −5.8 eV and −2.3 eV, strong Mo–O and Te–O covalent bonding interactions are observed. The conduction bands (CBs), extending approximately from E_g to 7.5 eV, are mainly composed of O 2p and Mo 4d states, with a smaller contribution from Te 5p orbitals. These CB features correspond predominantly to Mo–O and Te–O antibonding states, with the Mo 4d contribution being notably larger than that of Te 5p. This suggests that the key electronic transitions involved in the optical excitations originate mainly from O–Mo interactions. The principal distinction among the three compounds lies in the position and role of the A-site metal orbitals and their interaction with oxygen. In MgTM, Mg states are negligible near the top of the VBs, consistent with a more ionic character in the Mg–O bonding. In contrast, strong Cd 4d and Zn 3d states appear around −6.0 eV and −4.5 eV, respectively, indicating pronounced covalent bonding character in the Cd–O and Zn–O interactions. We also found that the Cd–O interaction exhibits the highest -ICOHP value (0.95 eV) among the three compounds, indicating the strongest A–O bonding. This is likely influenced by the comparatively narrow Cd–Cd distance within the (001) plane (Figure S2). These differences in the A-site electronic structure are reflected in the variations of bond lengths, polyhedral distortion, and ultimately in the linear and nonlinear optical properties of the three tellurites.

2.3. SHG Responses and ART Analysis

The point groups of MgTM and ZnTM are 222, whereas CdTM crystallizes in the $\overline{4}2m$ point group. Consequently, three nonzero SHG tensor components, i.e., d₁₄, d₂₅, and d₃₆, are obtained, as listed in Table 1. In all three compounds, Kleinman symmetry (d₁₄ = d₂₅ = d₃₆) is not strictly obeyed. For this reason, Kleinman symmetry constraints were not enforced in the calculation of nonlinear optical properties throughout this study. Based on the experimental band gaps (E_g^EXP), the effective SHG coefficients $d_{\mathrm{e}\mathrm{f}\mathrm{f}}^p$ (averaged over all possible crystal orientations according to the Kurtz–Perry powder method) of MgTM, CdTM, and ZnTM were calculated to be 13.49, 10.34, and 10.86 pm·V⁻¹ (Table 1). For cross-validation, SHG coefficients were further evaluated using the ABINIT code with the same experimental band gaps, yielding values of 12.05, 9.38, and 10.03 pm·V⁻¹. The results indicate that the SHG responses obtained using the sum-over-states (SOS) method are of the same order of magnitude as those reported by Jiang et al. [24]. When the HSE-calculated band gaps are employed, the corresponding values become 6.76, 6.35, and 7.09 pm·V⁻¹ for MgTM, CdTM, and ZnTM, respectively, which align well in magnitude with the results reported by Cammarata et al. [25]. Overall, the computed SHG coefficients show reasonable agreement with available experimental data, though variations exist among different studies (Table 1). These discrepancies can be attributed to differences in computational methods, band gap correction schemes, and experimental measurement conditions.

To quantitatively clarify the origin of the SHG responses in the ATMs, we employed ART combined with the PRF method. By analyzing the energy-dependent PRFs for d₃₆ and d₁₄ components, the dominant atomic orbitals and structural units responsible for the SHG can be identified (Figure 1g,j and Figure S1e). For all three ATMs, the SHG responses show a highly consistent microscopic origin. Taking the d₃₆ component of CdTM as an example, the function (Figure 1j) increases significantly as E_B decreases from E_F to −1.5 eV, indicating that the occupied nonbonding O 2p states near the VBM contribute strongly to the SHG response. In contrast, it decreases noticeably between −2.3 eV and −5.8 eV, where Mo–O and Te–O bonding interactions dominate, suggesting that these bonding states provide a negative contribution to the SHG response. Meanwhile, the sharp variation of from the CBM up to ~7.5 eV reflects the essential role of unoccupied Mo 4d and Te 5p antibonding states. Similar PRF profiles are observed for the d₃₆ component in the other two compounds, confirming a general similar electronic-response pattern across the series. Notably, in all three systems, PRF varies only weakly in energy regions associated with A-site cation orbitals, indicating that the direct contributions of these cations to the SHG response are limited. Instead, A-site substitution primarily modifies the crystal field and polyhedral distortion, thereby indirectly tuning the SHG response intensity. Therefore, the SHG responses in these ATMs are determined predominantly by occupied nonbonding O 2p states and unoccupied Mo 4d and Te 5p states, with the A-site cations acting only as secondary tuning factors.

Based on ART, the atomic-scale origin of the dominant SHG coefficient in the ATMs can be quantitatively resolved (

Table 2. Contributions of individual atoms to the d₃₆ SHG component in the ATM compounds. W_A refers to the number of the same type of atoms (i.e., on the same Wyckoff site) in a unit cell. The total atomic contribution A_τ represents each atom’s net contribution to the SHG response from both the VBs and CBs, i.e., from all bands. The factor of 1/2 is applied in the definition to avoid double-counting of each excitation process. C_A represent that from all atoms of the same type. ^VBA_τ is the contribution (in %) of the VBs, and ^CBA_τ is that from the CBs. The contributions from the s, p, and d states of the atom τ to ^VBA_τ and ^CBA_τ are also shown. The ART analysis for the d₁₄ component can be found in Table S5.

Compound	Atom	WA	Aτ	CA	VBAτ	CBAτ	VBsAτ	VBpAτ	VBdAτ	CBsAτ	CBpAτ	CBdAτ
MgTM	O1	4	6.0	23.9	4.4	1.5	0.5	4.0	0.0	0.3	1.3	0.0
	O2	4	5.1	20.6	3.5	1.7	0.6	2.9	0.0	0.2	1.5	0.0
	O3	4	5.1	20.5	3.6	1.5	1.0	2.6	0.0	0.1	1.4	0.0
	Mg	2	0.9	1.8	0.0	1.0	0.0	−0.1	0.0	0.1	0.4	0.4
	Te	2	8.9	17.7	2.1	6.7	1.0	1.0	0.1	0.4	4.4	1.9
	Mo	2	7.7	15.5	1.2	6.5	0.0	1.3	−0.1	0.2	1.1	5.2
CdTM	O1	4	4.9	19.6	3.4	1.5	0.4	3.0	0.0	0.3	1.2	0.0
	O2	4	5.3	21.2	3.6	1.7	0.6	3.0	0.0	0.1	1.6	0.0
	O3	4	5.2	20.7	3.6	1.6	1.3	2.3	0.0	0.1	1.5	0.0
	Cd	2	2.2	4.5	0.7	1.5	0.0	0.1	0.6	1.5	−0.2	0.3
	Te	2	7.2	14.3	1.4	5.7	0.6	0.7	0.1	0.2	4.1	1.4
	Mo	2	9.8	19.7	1.5	8.3	0.1	1.7	−0.3	0.1	0.9	7.4
ZnTM	O1	4	5.5	22.0	3.9	1.6	0.4	3.5	0.0	0.3	1.3	0.0
	O2	4	5.0	19.9	3.4	1.6	0.5	2.8	0.0	0.2	1.5	0.0
	O3	4	4.8	19.0	3.2	1.6	1.1	2.1	0.0	0.1	1.5	0.0
	Zn	2	2.8	5.6	1.0	1.8	0.0	0.1	0.8	1.4	0.1	0.3
	Te	2	7.7	15.4	1.9	5.8	0.9	0.8	0.1	0.2	4.2	1.4
	Mo	2	9.1	18.1	1.6	7.5	0.0	1.6	−0.1	0.1	0.9	6.4

and Table S5). Here, we take CdTM as a representative example for detailed analysis. In CdTM, the Mo atoms exhibit the highest individual atom contribution (A_τ ~9.8%), followed by the Te atom (~7.2%). The individual atom contributions of O and Cd are about 5.1% and 2.2%, respectively. Examining the orbital contributions, the O contributions are concentrated in the VB and are predominantly of O 2p character (^VB_pA_τ ~2.8%), consistent with the role of nonbonding O 2p states as the initial states in the SHG process. The VB contribution of Mo in CdTM (~1.7%) arises mainly from Mo 4p orbitals, whereas its CB contribution is dominated almost entirely by Mo 4d states (~7.4%), indicating that the unoccupied Mo 4d states play an important role in the SHG process. The atom also contributes notably to the CB, largely from Te 5p orbitals (~4.1%), though its overall contribution remains smaller than that of Mo. The same trend holds for the other two compounds. When weighted by the number of each atom in the unit cell (W_A), the total fractional contributions (C_A) of Mo, O, Te, and Cd are 19.6%, 61.3%, 14.3%, and 4.4%, showing that the Mo, O, and Te atoms constitute the primary source of the SHG response in CdTM. The A-site metals (Mg, Cd, and Zn) provide only minor direct contributions (C_A < 6%), indicating that they act mainly as structural modulators, rather than as primary NLO-active centers.

However, the d₁₄ component in CdTM presents a distinct behavior. Its ${\zeta }_V\left(E_B\right)$ exhibits a steady increase as E_B decreases from E_F to −3.1 eV. In contrast, a pronounced peak is observed in the top VBs of ZnTM and MgTM, characterized by a sharp rise at the VBM followed by a sudden decrease around 0.69 eV. The PRF analysis indicates that the synergetic interaction between the Te atom SCALPs and the nonbonding O 2p states in orthorhombic ZnTM and MgTM contributes significantly and positively to the SHG response. In CdTM, however, the SCALPs are relatively weak (PCHG, Figure 1b), failing to induce a similarly drastically contribution in the top VBs. As shown in Table S5, the individual atom contribution of Te atom in VBs (^VBA_τ) is only 1.6%, which is lower than that of ZnTM (2.8%). This may account for the observed overall decrease in SHG performance in CdTM. It should be noted that, despite the different PRF features of the d₁₄ component at the top VBs (caused by Te SCALPs in CdTM vs. ZnTM/MgTM), the overall dominant contributors to SHG remain consistent: the occupied nonbonding O 2p states and the unoccupied Mo 4d/Te 5p states. This pattern is similar to that observed for the d₃₆ component.

According to the individual atomic contributions to the SHG response, the contribution of a given atomic group can be obtained by summing the contributions from the central atom and its coordinated ligands [35]. For example, the group contribution of the MoO₄ (denoted as [MoO2_2/3O3_2/1]) unit to the SHG response in MgTM and ZnTM is calculated as the total contribution of the Mo atom together with those of the surrounding O atoms belonging to this polyhedron:

$$\begin{array}{c}{\chi }_{\left[{\mathrm{M}\mathrm{o}\mathrm{O}2}_{2/3}{\mathrm{O}3}_{2/1}\right]}^{\left(2\right)}={\chi }_{\mathrm{M}\mathrm{o}}^{\left(2\right)}+2\times \left({\displaystyle \frac{{\chi }_{\mathrm{O}2}^{\left(2\right)}}{3}}\right)+2\times \left({\displaystyle \frac{{\chi }_{\mathrm{O}3}^{\left(2\right)}}{1}}\right)\end{array}$$

(1)

For CdTM, the structural unit is defined as [MoO2_2/2O3_2/1] according to its O coordination environment [35]. The total group contribution for the d₃₆ SHG component for ATMs is presented in Figure 2. In terms of individual group contributions (S_G), the MoO₄ groups exhibit the largest values, ranging from 21.3% to 25.5%, which are notably higher than those of the TeO₄ groups (14.7–16.3%) and the A-centered polyhedra (8.7–13.5%). When considering the total group contribution, the MoO₄ units account for 42.6–51.0% of the overall response, confirming their dominant role. The TeO₄ groups also provide a substantial and non-negligible contribution, underscoring their importance in the overall SHG activity of these materials.

We further calculated the dipole moments of individual structural groups (D_G) for the three compounds and examined their correlation with the corresponding SHG contributions. Note that the three ATMs are intrinsically nonpolar; thus, the net dipole moment of the crystal must vanish. As summarized in Figure 2, the single TeO₄ groups exhibit the largest dipole moments (18.8–20.7 Debye), followed by the MoO₄ groups (0.6–1.9 Debye), whereas the metal–oxygen (A–O) groups are nonpolar with zero dipole moments. However, this trend is in sharp contrast to the group SHG contribution analysis, as the group dipole moments and their SHG contributions demonstrate that a large group dipole moment does not necessarily imply a dominant role in generating a strong SHG response. Similar results can be found in our previous analysis on nonpolar KBBF [35]. It was demonstrated that the SHG response does not depend on either the dipole moment of the atomic groups or the static polarization of a crystal.

In our previous work, we have discovered a linear relationship between $d_{eff}^p$ and the ${\alpha }_{\mathrm{s}\mathrm{u}\mathrm{m}}/\left(N{E}_g\right)$ [36,37,38,39] in several families, where ${\alpha }_{\mathrm{s}\mathrm{u}\mathrm{m}}$ is the sum of the polarizabilities of all ions in the primitive unit cell of a compound [40] (Table S7), $N$ is the total number of atoms in the unit cell, and E_g is the band gap. However, this linear relationship meets challenges in the three ATM compounds. MgTM and ZnTM share the same crystal structure and a similar feature of PRF. If we use a line to express the relation of $d_{eff}^p$ and the ${\alpha }_{\mathrm{s}\mathrm{u}\mathrm{m}}/\left(N{E}_g\right)$ for MgTM and ZnTM, we naturally get one line (Figure 3). It is clear that the data point for CdTM deviates significantly from this line. This feature holds true regardless of whether the Hubbard U correction is applied or whether E_g^EXP or E_g^HSE band gaps are used. Given that CdTM crystallizes in a different space group and exhibit a distinct PRF feature for d₁₄ component, its deviation from the line formed by MgTM and ZnTM is understandable. Such deviation from one family to the other has been previously noted in other systems such as borates [36] and Pb₂B₅O₉X [38]. Additionally, we found that the direct lines for different structural families often exhibit distinct slopes and intercepts. Such differences may manifest the atomic structure and electronic structure effects. This is certainly a topic worthy of in-depth investigation.

3. Methods

The structures and electronic properties of three ATM compounds were calculated using the Vienna ab initio Simulation Package (VASP.5.4.4) [41,42,43] based on density functional theory (DFT). The exchange–correlation interactions were treated using the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA) [44]. The valence electron wave functions were described using the projector augmented-wave (PAW) method [45]. The valence electron configurations of Mg, Cd, Zn, Te, Mo, and O were taken as 2p⁶3s², 4d¹⁰5s², 3d¹⁰4s², 5s²5p⁴, 4s²4p⁶5s¹4d⁵, and 2s²2p⁴, respectively [46]. A plane wave energy cutoff of 600 eV was employed. The Brillouin-zone integrations were carried out with a Γ-centered k-point mesh automatically generated using the KSPACING tag set to 0.12 Å⁻¹. Specifically, MgTM and ZnTM compounds were calculated using an 11 × 10 × 6 k-point mesh, while CdTM was calculated with a 10 × 10 × 6 k-point mesh. Structural optimizations were performed until the residual forces on each atom were less than 0.001 eV Å⁻¹ and the total energy was converged to within 0.01 meV. Considering that molybdenum tellurite materials contain transition-metal d electrons, the GGA+U approach [47] was employed in our calculations. Based on systematic tests (details provided in Table S8), a U value of 2 eV was applied only to the Mo 4d states.

The second-order NLO properties were calculated using the SOS [48,49,50] method using the ARTAROP.1.6.3 code, which we recently developed [33,34], based on the electronic structures obtained from the VASP optical modules, and SOS methods in ABINIT.9.6.2 code [51,52,53,54]. Since DFT typically underestimates the band gap compared to experimental values, both the experimentally measured band gap (E_g^EXP) and the HSE06 band gap (E_g^HSE) were used as lower and upper bounds in the scissor operation (S^EXP and S^HSE) for the optical property calculations. The SOS formalism for the second-order susceptibility was originally derived by Aversa and Sipe [48] and subsequently modified by Rashkeev et al. [49,50] and Sharma et al. [55,56]. To evaluate the individual atomic and orbital contributions to the SHG tensor components, ART analysis based on the PRF was employed (details provided in Supplementary Materials S1 and S2) [33].

4. Conclusions

In summary, based on first-principles calculations combined with the ART, we systematically investigated the electronic structures and the origins of the NLO properties of nonpolar molybdenum tellurite compounds, MgTM, CdTM, and ZnTM. Using quantitative analyses at both the electronic and atomic levels, we reveal that the SHG responses in ATM compounds are mainly governed by the occupied nonbonding O 2p states and the unoccupied Mo 4d and Te 5p states. ART analysis attributes the stronger SHG response in ZnTM and MgTM to an effective and positive synergistic interplay between Te SCALPs and nonbonding O 2p states. However, this mechanism is weakened in CdTM due to diminished Te SCALP activity at VBM, resulting in a lower overall SHG response. From a structural group perspective, the MoO₄ units contribute approximately 42.6–51.0% of the total SHG response and thus play a dominant role, while the TeO₄ groups also provide significant and non-negligible contributions. In addition, we find that there is no direct correspondence between the magnitudes of group dipole moments and their SHG contributions. Overall, this work provides a quantitative and microscopic understanding of the physical origins of SHG responses in the ATM family, offering valuable insights into the rational design and development of advanced NLO materials.