Prediction of Large Second Harmonic Generation in the Metal-Oxide/Organic Hybrid Compound CuMoO 3 (p2c)

: Noncentrosymmetric hybrid framework (HF) materials are an important system in dis-covering new practical second-order nonlinear optical materials. We calculated the second harmonic generation (SHG) response of a noncentrosymmetric (NCS) organic–inorganic HF compound, CuMoO 3 (p2c) (p2c = pyrazine-2-carboxylate) to ﬁnd that it exhibits the largest SHG response among all known NCS HF materials with one-dimensional helical chains. Further atom response theory analysis revealed that the metal atoms Cu and Mo contribute much more strongly than do nonmetal atoms in determining the strength of the SHG response, which is a novel example in nonlinear optical materials known to date.


Introduction
Nonlinear optical (NLO) crystals play a vital role in modern laser technologies and sciences due to their ability to convert the frequency of an incident laser beam through the second harmonic generation (SHG) process [1][2][3][4][5]. A noncentrosymmetric (NCS) arrangement of atoms is a prerequisite for the generation of non-zero second-order NLO properties in bulk materials. The search for new NCS structures with excellent SHG properties remains a hot scientific challenge. Although there exist a number of extensively studied and commercially available inorganic NLO crystals, there has been tremendous interest in finding new NLO materials in other systems such as organic molecular crystals, inorganic-organic hybrid nanocomposites, self-assembled chromophoric superlattices and inorganic-organic hybrid framework materials (including both nanoporous metal-organic frameworks (MOFs) and dense inorganic-organic frameworks (IOFs)) [6][7][8][9][10]. Hybrid framework materials are constructed by employing modular synthetic procedures with metal ions (i.e., Zn 2+ , Cd 2+ , Mn 2+ , Ag + , Cu + ) or inorganic nodes (i.e., metal-oxide, metalfluoride, metal-chalcogenides and metal-halides) that are covalently and/or coordinately bonded to various organic linkers. Because of the advantages over conventional inorganic materials (e.g., structural tunability at a molecular level, chemical stability, and ease of synthesis on a large bulk scale), NCS hybrid framework materials are expected to form a new class of potentiality practical NLO materials [11][12][13].
Numerous hybrid framework materials have been reported to be SHG-active. According to the dimensionality of the coordination networks, they can be roughly classified Vienna ab-initio simulation package (VASP) [27][28][29] with the projector augmented wave (PAW) method [30]. The generalized gradient approximation (GGA) within the Perdew-Burke-Ernzerhof (PBE)-type exchange-correlation potentials [31] was used throughout this work. The employed PAW-PBE pseudopotentials [32] with 11 (3d 10 4s 1 ), 14 (4s 2 4p 6 4d 5 5s 1 ), 6 (2s 2 2p 4 ), 5 (2s 2 2p 3 ), 4 (2s 2 2p 2 ) and 1 (1s 1 ) valence electrons for Cu, Mo, O, N, C, and H were used to describe the electron−ion interactions, respectively. The plane wave cutoff energy for the expansion of wave functions was set at 600 eV and the tetrahedron method with Blöchl corrections was used for integrations. The numerical integrations in the Brillouin zones were performed by utilizing 7 × 7 × 4 Monkhorst-Pack k-point mesh, which showed an excellent convergence of the energy differences (0.005 eV) and stress tensors (0.001 eV/Å). The quasi-Newton algorithm as implemented in the VASP code was used in all structural relaxations. In this work, both the cell volume and the atomic positions were all allowed to relax to minimize the internal forces. The optimized lattice parameters, a = 7.791 Å and c = 11.229 Å, are slightly overestimated with respect to the experimental, which were measured at room temperature as usual with the PBE functional. Details of the optimized structure and the agreement with the experimental values are given in Table S1 of the Supporting Information.
In our calculations for the linear and the nonlinear optical properties, we employed the sum over states (SOS) method [33][34][35] using the code we developed [24] based on the calculated electronic structures from the VASP optical module. The SOS formalism for second-order susceptibility was derived by Aversa and Sipe [33] and later modified by Rashkeev et al. [34,36] and Sharma et al. [35,37]. To gain insight into the origin of SHG response, the contributions A τ of the individual atoms τ to a specific component, e.g., the largest, of the total SHG response tensor, were determined by performing atom response theory (ART) analysis [24,38] for the CuMoO 3 (p2c) structure. Note that, the chirality may lead to higher-order interactions beyond the electric dipole approximation, for which one needs to use model studies instead of the first-principles method [39].
Partial response functional (PRF) method. The contribution of a certain occupied energy region between E B and valence band maximum (VBM), ijk , l = 1, 2, 3, 4, 5, 6 is determined by considering only those excitations from all occupied states between E B and VBM to all the unoccupied states of the conduction bands (CBs) and the contribution, δζ V (E B ), of specific occupied states of energy E B to each d il by the excitations from that energy to all unoccupied states of the CBs.
Similarly, the contribution, ζ C (E B ), of a certain unoccupied region between the conduction band minimum (CBM) and E B to each d il is determined by the excitations from all occupied states of the VBs only to all unoccupied states between CBM and E B , and the contribution, δζ C (E B ), of specific unoccupied states of energy E B to each d il by the excitations from all occupied states of the VBs only to that energy.
Atom response theory (ART) analysis. To evaluate the individual atom contributions to the SHG components, d il , it is computationally more convenient to express the corresponding PRFs in terms of the band index I B , ζ(I B ), where the band index I B runs from 1 to N tot (i.e., the total number of band orbitals) with increasing energy, E B , from E min to E max . Here, ζ V (I B ) and ζ C (I B ) are denoted as VB ζ j and CB ζ j , respectively, with I B replaced by a subscript j. Suppose that a specific atom τ has L atomic orbitals with a coefficient VB C → k j Lτ in the valence band j at a wave vector → k . The total contribution VB A τ of an atom τ makes to the SHG coefficient from all the VB bands j is written as where Ω is the unit cell volume, VB ζ j is the corresponding PRFs in terms of the band index j. Similarly, the total contribution CB A τ of an atom τ makes to the SHG coefficient from all the CB bands j is written as in which we assumed that the atom has L atomic orbitals with coefficient CB C → k j Lτ in the conduction band j at a wave vector → k . To calculate the actual contribution of each constituent atom in a unit cell to the total SHG response, one needs to consider the signs of VB ζ j and CB ζ j .
The total contribution, A τ , each individual atom makes to the SHG response from both the VBs and the CBs (i.e., from all the bands) is given by where the factor of 1/2 is applied to remove the double-counting of each excitation.

Structure
The Cu/Mo-oxide CuMoO 3 (p2c) crystallizes in the NCS space group P3 2 ( Figure 1 and Table S1 (Figure 1a). Besides, this zigzag chain propagates along the 3 2 screw axis, that is, the MoO 5 N octahedron could duplicate itself by rotating 120 • within the ab plane and gliding 2c/3 along the c axis. In addition, the Cu atoms form a distorted CuO 3 N tetrahedron with the surrounding three O atoms (O2, O3, O5) and one N2 atom with the Cu-O and Cu-N distances of~2.03 Å and~1.99 Å, respectively. Each CuO 3 N tetrahedra is corner shared with two neighbouring MoO 5 N octahedra through the O3 and O5 atoms, forming the inorganic CuMoO 3 helical chains. Further, the N1 and O1 atoms from one p2c ligand simultaneously coordinate with the Mo atom, while the O2 and N2 atoms from two different p2c ligands connect the Cu atoms ( Figure 1b). Thus, the 3D inorganic-organic hybrid network is constructed through the bridging p2c ligands. Note that, the Mo-O1 and the Cu-O2 distances (2.16 and 2.08 Å) connecting to the p2c ligands are slightly longer than the other Mo-O and Cu-O bonding lengths (1.86 and 2.01 Å) within each MoO 5 N octahedra and CuO 3 N tetrahedra, respectively. This fact reflects that the metal/ligand interaction is relatively weak compared with the interaction within the inorganic framework.

Electronic Structures
Our electronic calculations reveal that CuMoO3(p2c) has a small indirect bandga (Eg PBE ) of 0.743 eV, which is smaller than the experimentally measured (Eg exp = 1.32 eV). I computing optical properties, this deficiency of the DFT [26] is often corrected empiricall by employing the scissor operation [40] in which the conduction bands (CBs) are shifte in energy to have the experimental bandgap [41,42].

Electronic Structures
Our electronic calculations reveal that CuMoO 3 (p2c) has a small indirect bandgap (E g PBE ) of 0.743 eV, which is smaller than the experimentally measured (E g exp = 1.32 eV). In computing optical properties, this deficiency of the DFT [26] is often corrected empirically by employing the scissor operation [40] in which the conduction bands (CBs) are shifted in energy to have the experimental bandgap [41,42].

Electronic Structures
Our electronic calculations reveal that CuMoO3(p2c) has a small indirect bandgap (Eg PBE ) of 0.743 eV, which is smaller than the experimentally measured (Eg exp = 1.32 eV). In computing optical properties, this deficiency of the DFT [26] is often corrected empirically by employing the scissor operation [40] in which the conduction bands (CBs) are shifted in energy to have the experimental bandgap [41,42].    The crystal orbital Hamilton population (COHP) [43,44] analysis (Figure 2b) shows that the frontier orbital states (−4.4 eV to E F ) are described mainly by nonbonding states made up of Cu-3d and O-2p orbitals with a small mixture of weak Cu-O/N d-p antibonding states ( Figure S2). Generally, these filled nonbonding and antibonding states near E F are highly polarizable and hence are important for the optical properties. Besides, some O-2p and N-2p states make weak bonding interaction with Mo-4d and Cu-3d states as well as C-2p orbitals around −9.0 to −4.4 eV, leading to a dispersive orbitals feature. Strong Cu-O and Mo-O s-p bonding interaction can be found at around −18 eV. The covalent character of the C-O/N bonds is much stronger than that of the Mo-O/N and the Cu-O/N bonds (Figure 2b).

Optical Properties
Calculations of the refractive indices (n o and n e ) as a function of wavelength ( Figure 3) reveal that CuMoO 3 (p2c) can meet the Type-I phase matching condition at λ = 1.88 µm, a value in the IR range. Besides, the calculated birefringence value ∆n for CuMoO 3 (p2c) is 0.21 at 1910 nm ( Figure S5). Such a large birefringence reflects a strong optical anisotropy of CuMoO 3 (p2c). The value of ∆n in a uniaxial optical material is the difference between n o and n e , ∆n = |n o − n e |. As n o > n e in CuMoO 3 (p2c), it is a negative uniaxial crystal.
Symmetry 2022, 14, x FOR PEER REVIEW 6 The crystal orbital Hamilton population (COHP) [43,44] analysis (Figure 2b) sh that the frontier orbital states (−4.4 eV to EF) are described mainly by nonbonding s made up of Cu-3d and O-2p orbitals with a small mixture of weak Cu-O/N d-p antib ing states ( Figure S2). Generally, these filled nonbonding and antibonding states ne are highly polarizable and hence are important for the optical properties. Besides, O-2p and N-2p states make weak bonding interaction with Mo-4d and Cu-3d stat well as C-2p orbitals around −9.0 to −4.4 eV, leading to a dispersive orbitals feature. St Cu-O and Mo-O s-p bonding interaction can be found at around −18 eV. The cov character of the C-O/N bonds is much stronger than that of the Mo-O/N and the Cu bonds (Figure 2b).

Optical Properties
Calculations of the refractive indices (no and ne) as a function of wavelength (F 3) reveal that CuMoO3(p2c) can meet the Type-I phase matching condition at = 1.88 a value in the IR range. Besides, the calculated birefringence value Δn for CuMoO3(p 0.21 at 1910 nm ( Figure S5). Such a large birefringence reflects a strong optical anisot of CuMoO3(p2c). The value of Δn in a uniaxial optical material is the difference betw no and ne, Δn = |no − ne|. As no > ne in CuMoO3(p2c), it is a negative uniaxial crystal. Due to the point group of 3, the SHG tensor of CuMoO3(p2c) has 13 non-zero ponents, in which 5 of them are independent, i.e., d11 = −d12 = −d25, d22 = −d21 = −d16, d31 = d24 = d15, d14 = −d25 and d33, as presented in Table S2. As the Kleinman symmetry, i.e. −d25 = 0, is not followed in CuMoO3(p2c), it was not enforced in calculating the NLO p erties in this work. The effective , an average SHG coefficient over all possible o tations of the powder crystals, is estimated from the formula derived by Kurtz [47] and obtain maximum value of ~168.7 pm/V at = 0.0°and φ = 45.0°at ω = 1910 nm, which is l than the powder . Using the deff and the refractive indices, one can get a much l 2058.7 (pm/V) 2 than that 150 (pm/V) 2 , of LiNbO 3 , which suggests a great potential value of CuMoO 3 (p2c)-though the large crystal has not yet been obtained. We also calculated the SHG responses for several NCS hybrid framework compounds with helical chains, as presented in Table S2, which also lists the available experimental SHG values. Clearly, CuMoO 3 (p2c) has the highest d p e f f value among all reported NCS hybrid framework materials with helical chains. This remarkably strong SHG response is larger than those found in the majority of hybrid framework materials reported to date.  [53], and δ-Ga 2 Se 3 (I 2ω powder of 2.3 × AGS) [54]. These facts indicate that CuMoO 3 (p2c) is a promising IR hybrid NLO crystal material.

Atom Response Theory Analyses
We investigate the origin of the SHG responses further by employing the ART analysis [24]. Shown in Figure 2c is the partial response functionals (PRFs), ζ V (E B ) and ζ C (E B ) as well as their derivatives ( Figures S6 and S7),  (Table S3), and in the order Mo >> Cu > N > C > O> H in the CB contributions (Table S3). These findings show that the SHG of CuMoO 3 (p2c) is governed largely by the occupied states of Cu 3d, Mo 4p, and O 2p, and by the unoccupied states of Mo 4d, Cu 3d, and N-2p. The metal atoms Cu and Mo contribute much more strongly than do the nonmetal atoms in determining the strength of the SHG response in CuMoO 3 (p2c), which is quite special among the NLO materials known to date. Table 1. Contributions of the atoms to the largest components of the SHG tensors d 11 of CuMoO 3 (p2c). N A refers to the number of the atom type (on the same Wyckoff site) in a unit cell; A τ , N A A τ , VB A τ , CB A τ refer to the contributions (in %) from a single atom, the total atoms of the same type, all VBs for a single atom, and all CBs for a single atom, respectively. d τ denotes the actual value of the contribution (in pm/V) to the SHG for a single atom. According to the individual atomic contribution to the SHG response, the contribution of an atomic group can be calculated by summing the contributions of the center atom and those of its ligands. In this work, we partition the contribution of an anion (O 2− and N 3− ) equally to all the atomic groups it belongs to. The contribution of an atomic group can be calculated by summing the contributions of the center atom and those of its coordinated atoms [55]. Considering the coordination number for each O and N atom (2 and 3, respectively, Figure S8), the group CuO 3 N can be rewritten as CuO2 1/2 O3 1/2 O5 1/2 N2 1/3 . Therefore, the group contribution of CuO 3 N to SHG response is calculated as follows, The group with anion coordination numbers and the total group contributions for the largest SHG component d 11 of CuMoO 3 (p2c) are given in Figure 4, which shows that the metal-centered group [CuO 3 N] and [MoO 5 N] contribute much more strongly to the SHG response than does the organic p2c ligand. That is, the total contribution of the inorganic part is~79.1%, which far surpasses that of the organic part (i.e.,~20.9%). Our results reflect that the inorganic part contributes dominantly to the SHG response, while the organic part is important in the stabilization of the crystal structure of CuMoO 3 (p2c). It is worth mentioning that we did not separately calculate the hyperpolarizability tensor β ijk by cutting out the ligand or groups from the structure of the compound. Such an approach will unavoidably lead to uncontrolled errors; thus, they are not used in our calculations.  According to the individual atomic contribution to the SHG response, the contribu tion of an atomic group can be calculated by summing the contributions of the center atom and those of its ligands. In this work, we partition the contribution of an anion (O 2− an N 3− ) equally to all the atomic groups it belongs to. The contribution of an atomic grou can be calculated by summing the contributions of the center atom and those of its coor dinated atoms [55]. Considering the coordination number for each O and N atom (2 and 3 respectively, Figure S8), the group CuO3N can be rewritten as CuO21/2O31/2O51/2N21/ Therefore, the group contribution of CuO3N to SHG response is calculated as follows, The group with anion coordination numbers and the total group contributions fo the largest SHG component d11 of CuMoO3(p2c) are given in Figure 4, which shows tha the metal-centered group [CuO3N] and [MoO5N] contribute much more strongly to th SHG response than does the organic p2c ligand. That is, the total contribution of the inor ganic part is ~79.1%, which far surpasses that of the organic part (i.e., ~20.9%). Our result reflect that the inorganic part contributes dominantly to the SHG response, while the or ganic part is important in the stabilization of the crystal structure of CuMoO3(p2c). It i worth mentioning that we did not separately calculate the hyperpolarizability tensor by cutting out the ligand or groups from the structure of the compound. Such an approac will unavoidably lead to uncontrolled errors; thus, they are not used in our calculations

Conclusions
Although a number of NCS HF materials with 3D and 2D frameworks have been reported with large SHG response, only low SHG intensities have been measured for HF materials with 1D helical chains. Our first-principles calculations predict that the recently synthesized CuMoO 3 (p2c) exhibits the largest SHG response among all the NCS HF materials with 1D helical chains. Its static effective SHG response, 30.6 pm/V, is about 3.12 times greater than that of commercial AGS, and 92.7 times greater than that of KDP. This value also exceeds those of most NCS hybrid framework materials reported so far. Our ART analysis shows that the SHG of CuMoO 3 (p2c) is determined largely by the occupied states composed of Cu 3d, Mo 4p, and O 2p, and by the unoccupied states composed of Mo 4d, Cu 3d, and N 2p. The metal atoms Cu and Mo contribute much more strongly than do the nonmetal atoms in determining the strength of the SHG response in CuMoO 3 (p2c). The latter is quite special in the NLO materials known to date. Our work based on the quantitative calculations at the electronic and the atomic level reveals the importance of the contribution from metal atoms and the metal-centered inorganic groups for the SHG response of CuMoO 3 (p2c).

Supplementary Materials:
The following are available online at https://www.mdpi.com/article/10 .3390/sym14040824/s1, Figure S1: Calculated band structure for P3 2 -CuMoO 3 (p2c). Figure S2: HOMO (M 1 ) and LUMO (A 1 ) for CuMoO 3 (p2c). Figure S3: Calculated partial DOS for P3 2 -CuMoO 3 (p2c). Figure S4: Calculated partial DOS of O and N atoms at independent crystallographic positions. Figure S5: Frequency-dependent refractive indices n (left) and birefringence ∆n (right) of CuMoO 3 (p2c). Figure Figure S8: Coordination environment for each inequivalent O and N atom of CuMoO3(p2c). Table S1: The optimized crystal structure data for P3 2 -CuMoO 3 (p2c). Table S2: Calculated SHG tensors d il and d p e f f for CuMoO 3 (p2c) and several NCS HF compounds with helical chains. For the compounds with no available experimental band gap (E g ), the calculated E g based on HSE06 with mixing parameter α = 0.3 is applied. The available experimental measured SHG responses are also presented. In this work, Kleinman symmetry is not enforced in calculating the NLO properties. Table S3: Contributions of the individual atoms to the SHG component d 11 of CuMoO 3 (p2c). W A refers to the number of the same type of atoms (on the same Wyckoff site) in a unit cell. A τ is the contribution (in %) from a single atom τ, and C A from all atoms of the same type. VB A τ , is the contribution (in %) the VBs, and CB A τ from the CBs. The contributions from the s, p, and d states of the atom τ to of VB A τ and CB A τ are also shown [56,57].