The growth mechanism of nanoclusters can be studied by their geometric structures. Three low-lying energy isomers of each size AuMg_n (n = 2–12) nanoclusters are presented in Figure 1 and Figure 2. Under each structure, the “i” in AuMg_n-i is their energy order, with “1” indicating the lowest and “2” the second-lowest ones. The energy difference (eV) between the AuMgn-i and AuMg_n-1 at each size can also be found. In addition to the symmetry, and the electronic structure information is also shown in Figures 1, 2. All information about the lowest energy state of AuMg_n (n = 2–12) nanoclusters is summarized in Table 1, where the results of the frequency calculations show the lowest and highest vibrational frequencies satisfying the requirements that the results of the frequency calculations cannot contain any imaginary frequency. As can be seen from Figure 1, isomers AuMg₂-1 (D_∞h symmetry with ²Σ_g electronic state) and AuMg₂-2 (C_2v symmetry with ²Σ_g electronic state) have a similar linear structure, the difference being that the Au atom is in the center of the former and on the side of the latter. Isomers AuMg₂-3 (C_2v symmetry with ²B₂ electronic state) show a 2D planar isosceles triangle structure. Relative to the lowest energy state AuMg₂-1 isomer energy, AuMg₂-2 and AuMg₂-3 isomers have 0.24 and 0.39 eV higher energy than it, respectively. The isomer AuMg₃-1 (D_3h symmetry with ²Aʹ₁ electronic state) has an equilateral triangular geometry, while the isomer AuMg₃-3 (C_2v symmetry with ⁶A₂ electronic state) has an isosceles triangular structure, and interestingly the Au atoms are located at the center of their triangular structures, respectively. The structure of the isomer AuMg₃-2 (C_2v symmetry with ²A₁ electronic state) is a planar combination of Au-Mg-Mg isosceles triangle and Mg-Mg-Mg isosceles triangle. Calculations show nanoclusters of 2 Mg atoms doped with one Au atom, where the second and third lowest isomers are 0.03 and 4.20 eV higher than the lowest energy isomer, respectively. The structures of the isomers AuMg₄-1 (C_3v symmetry with ²A₁ electronic state) and AuMg₄-2 (C_2v symmetry with ²A₁ electronic state) can be considered as formed based on the tetrahedral (pyramid-like) structure of Au-Mg-Mg-Mg adsorbing an Mg atom in different directions. The isomer AuMg₄-3 (C_2v symmetry with ²B_3g electronic state) shows a rectangular structure in which the 4 Mg atoms are at the vertices while Au atom is at the geometric center. For the isomer AuMg₄-1, AuMg₄-2 and AuMg₄-3 are higher in energy by 0.01 and 0.49 eV, respectively. The structure of the isomer AuMg₅-1 (C_2v symmetry with ²A₁ electronic state) is based on the formation of AuMg₄-2 by adsorbing an Mg atom on its top. The isomers AuMg₄-3 and AuMg₄-1 form the isomers AuMg₅-2 (C _s symmetry with ²Aʹ electronic state) and AuMg₅-3 (C₃ symmetry with ²A electronic state) after pulling up the Au atom into the interior of the polyhedral while adsorbing an Mg atom on their tops. The second- and third-lowest energy isomers of the AuMg₅ nanocluster are 0.08 and 0.16 eV higher than that of the lowest-energy isomer. The isomers AuMg₆-1 (C₂ symmetry with ²B electronic state) and AuMg₆-2 (C_2h symmetry with ²A_g electronic state) have extremely close energies and structures, which exhibit rotational symmetry with Au atom. The isomer AuMg₆-3 (²B_2g electronic state), which is higher 0.04 eV in energy than AuMg₆-1, possesses a high symmetry (D_2h ) octahedron in which the Au atom is located at its center. The structures of the isomers AuMg₇-1 (C₁ symmetry with ²A electronic state), AuMg₇-2 (C _s symmetry with ²Aʹ electronic state) and AuMg₇-3 (C _s symmetry with ²Aʹ electronic state) are all grown based on the diversity of Au-Mg-Mg-Mg tetrahedra-like geometries. In addition, the energy shift of the isomers AuMg₇-2 and AuMg₇-3 relative to the lowest energy state are 0.03 and 0.10 eV, respectively.

As Figure 2 displayed, the medium-sized AuMg_n (8–12) nanoclusters exhibit a diversity of structures. The structures of the isomers AuMg₈-1 (C_2v symmetry with ²A₁ electronic state) and AuMg₈-3 (C₁ symmetry with ²A electronic state) are generated based on the deformation of AuMg₇-3 after adsorption of an Mg atom, while the structure of the isomer AuMg₈-2 (C _s symmetry with ²Aʹ electronic state) can be obtained from the deformation of AuMg₇-2 by adsorption of an Mg atom. The second and third lowest energy isomers of AuMg₈ nanoclusters are higher in energy than the ground state by 0.01 and 0.17 eV. The isomers AuMg₉-1 and AuMg₉-2 have the same symmetry (C_4v ), electronic structure (²A₁), energy and “fascinating cage-like” structures. The isomer AuMg₉-3 (²Aʹ electronic state), which is 0.08 eV higher in energy than AuMg₉-1, has a cage-like structure with C _s symmetry. The structures of the isomers AuMg₁₀-1 (C_2v symmetry with ²A₁ electronic state), AuMg₁₀-2 (C₁ symmetry with ²A electronic state) and AuMg₁₀-3 (C₁ symmetry with ²A electronic state) are generated based on the deformation of AuMg₉-1 by adsorption of Mg atoms in different directions. The second and third lowest energy isomers of AuMg₁₀ nanoclusters are higher in energy than the first lowest energy by 0.46 and 0.84 eV, respectively. Interestingly, the isomers AuMg₁₁-1 (C _s symmetry with ²Aʹ electronic state) and AuMg₁₁-2 (C₁ symmetry with ²A electronic state) are easily obtained by the deformation of AuMg₁₀-2 by adsorption of an Mg atom. On the other hand, AuMg₁₁-3 (C₁ symmetry with ²A electronic state) is formed by the deformation of AuMg10-1 after the adsorption of an Mg atom. AuMg₁₁-2 and AuMg₁₁-3 have higher energies than AuMg₁₁-1 at 0.03 and 0.39 eV. The isomer AuMg₁₂-1 (C _s symmetry with ²Aʹ electronic state) is generated by the adsorption of an Mg atom by AuMg₁₁-1. The isomer AuMg₁₂-2 (C_2v symmetry with ²A₁ electronic state), on the other hand, exhibits a deformed tubular-like structure, while the isomer AuMg₁₂-3 (C _s symmetry with ²Aʹ electronic state) has a pyramid-like structure. Furthermore, compared to the energy of AuMg₁₂-1, AuMg₁₂-2 and AuMg_12-3 are 0.05 and 0.40 eV higher, respectively. Because the lowest energy state isomers of nanoclusters often require more comprehensive studies to explore their various physical and chemical properties, the atomic coordinates of the AuMg_n-1 (n = 2–20) nanoclusters are shown in Supplementary Table S1 in the Supplemental Material.

In conclusion, based on the small size of AuMg_n-1 or smaller, AuMg_n (n = 2–12) nanoclusters can usually be formed by adsorption of Mg atoms in different directions, and the interesting point is that the direction of adsorption does not have a fixed pattern. Such result is consistent with many existed Mg-based nanoclusters reported (Li et al., 2017; Zeng et al., 2020, 2021; Zhu et al., 2020). However, despite the many similarities, the structures of gold-doped Mg nanoclusters have unique properties compared to other Mg-based nanoclusters studies. For example, the structure of AuMg₃ nanoclusters is 2D planar, while the lowest energy heterostructures of Be (Zeng et al., 2020), Si (Zhu et al., 2020), C, Ge, Sn (Zeng et al., 2021), Zn-doped (Li et al., 2017) Mg nanoclusters of corresponding sizes are all ortho-tetrahedral in shape. Interestingly, although the ground-state structures of AuMg₉ and BeMg₉ (Zeng et al., 2020) look similar, the significant difference between them is that the Au atom locates on the surface of AuMg₉ while the Be atom is absorbed into the inside of BeMg₉.

Since clusters exhibit different physical and chemical properties at different sizes, their relative stability is well worth studying. The relative stability of cluster can be calculated through the following three quantities, that is the binding energy per atom (E _b in eV), the second-order energy difference (Δ₂ E in eV) and the HOMO-LUMO energy gap (E _gap in eV). Equations 1–3 display the above three energies for AuMg_n-1 (n = 2–20) nanoclusters in Figure 1. E b ( AuMg n ) = [ n E ( Mg ) + E ( Au ) − E ( AuMg n ) ] / ( n + 1 ) (1) Δ 2 E ( AuMg n ) = E ( AuMg n + 1 ) + E ( AuMg n - 1 ) − 2 E ( AuMg n ) (2) E gap ( AuMg n ) = E LUMO ( AuMg n ) − E HOMO ( AuMg n ) (3)

E (Au) and E (Mg) denote the energies of free Au and Mg atoms, E (AuMg_n) means the energy of the corresponding nanocluster. The lowest unoccupied molecular orbital (LUMO) and highest occupied molecular orbital (HOMO) energies are E _LUMO and E _HOMO.

The theoretically calculated values of these quantities for AuMg_n-1 (n = 2–20) nanoclusters are presented in Table 1 and their curves with size are showed in Figure 3. As Figure 3A displayed, overall, the E _b curve becomes larger as the size of the nanoclusters increases, implying that the atoms within the AuMg_n nanoclusters bind more stably as the Au atoms are doped. Locally, the maximum value of E _b appears at AuMg₁₀ (1.90 eV), indicating that this nanocluster has the robust stability. Secondly, a small local peak (1.84 eV) appears at AuMg₄, indicating that it is slightly more stable than its neighbors. Figure 3B shows the curve of the Δ₂E, which can be detected experimentally by mass spectrometry. Interestingly, as in the case of the local peaks of the E _b curve, AuMg₄ and AuMg₁₀ have local maximum Δ₂E values of 0.29 and 0.28 eV, respectively. This conclusion suggests that they are both the most stable and have a high probability of being observed in mass spectrometry experiments. The thermodynamic stability of nanoclusters can be characterized by the value of their E _gap. Since AuMg_n nanoclusters are open-shell systems, they have both α and β-electrons, and the E _gap of α and β-electrons are illustrated in Figure 3C and Figure 3D. For α-electrons E _gap curve of AuMg_n nanoclusters, the local peaks appear at n = 5, 9 and 11, while AuMg₁₀ has the largest local β-electron E _gap, indicating that the thermal stability of these clusters is relatively high.

In conclusion, AuMg₄ and AuMg₁₀ nanoclusters show the robust stability. However, it is noteworthy that the AuMg₉ nanocluster is always the second largest value in both E_b and Δ₂E curves, although they are not the maximum values. Therefore, combined with the E _gap maximum for α-electrons, AuMg₉ nanocluster also has considerable robust stability and be worthy studying.

The natural charge population (NCP) results from the NBO calculations can reveal the charge transfer properties in the nanoclusters. The NCP values of Au atoms in Table 1 are in the range of [−2.99, −0.74] e, indicating that Au atoms play the role of electron receiver in all nanoclusters. The curve of NCP on Au atom with the size is ploted in Figure 4. The AuMg₉ nanocluster appears to be very special, which is probably originated from its high symmetry structure. Supplementary Table S2 in the Supplementary Material shows the NCP values on the Mg atoms. Except for 4 Mg atoms in AuMg₉ with a charge of −0.05 e and 1 Mg atom in AuMg₁₂ with an NCP value of −0.01 e, all other Mg atoms have positive NCP values, distributed from 0.65 to 0.01 e, suggesting that they are losing electrons. In other words, Mg atoms are electron donors in the AnMg_n nanoclusters. The charge transfer property depends on the electronegativity of the atom, the greater the electronegativity, the easier it is to get electrons. The electronegativity value of Mg atom is 1.31, while that of Au atom is 2.54, so the charge transfer is mostly from Mg atom to Au atom.

The ELF values of atomic bonding regions and their 2D maps are useful tools for analyzing the chemical bonding properties of nanoclusters. ELF is a value greater than 0 and less than 1, which characterizes the degree of electron localization and thus can determine the bonding properties. A region with ELF > 0.5 implies high electron localization and covalent bonding in the bonding region, while a region with ELF < 0.5 has low electron localization and non-covalent bonding in the bonding region. Figure 5 and Supplementary Figure S1 in the Supplementary Material display the 2D distribution of the EFL value for the ground state AuMg_n (n = 2–12) nanoclusters. The calculations show that Mg-Mg in AuMg_n nanoclusters is covalently bonded, while Au-Mg bond is non-covalent. The relatively large value of ELF around Au atom and low in bonding region indicates that the valence layer of Au atom is solidified around it, so it does not form covalent bonds with Mg. Furthermore, considering that the Au atom always gains electrons and the Mg atom around it loses electrons to be positively charged, it can be concluded that Au-Mg is ionic bonding. Another noteworthy point is that the critical size for Mg-Mg bonding is AuMg₄. ELF distribution map shows that Mg-Mg does not covalently bond in AuMg₂ and AuMg₃ nanoclusters.

Static polarizabilities α (∞) and hyperpolarizabilities β (∞) for the ground state AuMg_n (n = 2–20) nanoclusters were calculated to analyze their nonlinear optical property. Specifically, the coupled-perturbed Kohn-Sham (CPKS) method (Jensen, 2017) was adopted for the AuMg_n nanoclusters to compute the polarizabilities and hyperpolarizabilities in the zero-frequency limit (λ→∞). The results of α(∞) and β(∞) calculations are presented in Supplementary Table S3 in the Supplementary Material and are shown in Figure 6. From Figure 6A, it can be seen that the polarization anisotropy α_aniso (∞) and isotropy α_iso (∞) of AuMg_n nanoclusters do not change consistently with the size. The α_iso (∞) shows an overall upward trend, except for AuMg₉, while the α_aniso (∞) oscillates with increasing size. In addition, α_xx (∞), α_yy (∞), α_zz (∞) of each nanocluster are also shown in Figure 6A. Due to the diversity of the nanoclusters structures, these quantities display irregular oscillations in different directions. However, AuMg₉ nanocluster with high structural symmetry exhibits synchronous local minimum anisotropic and isotropic polarization, suggesting that it has special nonlinear optical properties compared to other nanoclusters. In order to study the polarization of AuMg₉ more intuitively, the unit sphere representation of its polarization tensor is plotted in Figure 6C. One can find the anisotropic polarization of AuMg₉, more specifically, the small polarization rate in the x-y plane and the maximum polarization rate in the z-direction (i.e., the direction of the line connecting the leftmost Au and the rightmost Mg in the figure).

Figure 6B exhibits the static hyperpolarizability β (∞) of the AuMg_n nanoclusters and their projection values in the dipole moment direction β_prj (∞). Since β_prj (∞) can be measured by the electric field-induced second harmonic generation experiment (EFISH), it serves as a guide for experiments. Specifically, β increases from AuMg₂ to a maximum value of AuMg₅, then gradually decreases to a minimum value of AuMg₉, and increases again afterward. Interestingly, the β and β_prj of AuMg₉ and the nanoclusters smaller than it are exactly equal, indicating that its β is isotropic with the dipole moment. However, starting from AuMg₁₀ nanocluster, the two curves are reversed, forming a mirror-symmetric trend. Figure 6D shows the unit sphere representation of static hyperpolarizability tensor β of AuMg₉, it is found that β is also anisotropic, with a maximum in the z-direction, and changes in the x-y plane as the Mg atoms surround it. In conclusion, relative to other nanoclusters, AuMg9 exhibits distinctive nonlinear optical properties.

For the ground state AuMg_n nanoclusters, the infrared and Raman spectra with weighted average of the Boltzmann distribution were calculated for guidance experiments. The motivation for considering the Boltzmann distribution is due to the difficulty of observing only the ground state nanoclusters in experiments, especially in the gas-phase nanoclusters. Figure 7 display the IR and Raman spectra of the weighted average of the Boltzmann distribution at room temperature. The Boltzmann distribution probabilities of each isomer at different temperatures were also calculated by the relevant equations in the Supplementary Material. The small 3D plots in each map are the corresponding weighted average spectra at 100 k, 300 k and 1000 k temperatures. As shown in Figures 7–9, the strongest absorption peaks of IR spectra are distributed in the 40–350 cm⁻¹ frequency band, while the strongest peaks of Raman spectra are distributed in a narrower band of 20–220 cm⁻¹. However, the most intense peaks of both IR and Raman weighted average spectra appear around 200 cm⁻¹ as the size increases. In addition, as can be seen from the small 3D plots in each figure and Supplementary Figure S2 in the Supplementary Material, the location of the most intense peak of the weighted average spectrum does not shift as the temperature increases, but the intensity changes.

Specifically, for the weighted average IR spectra at room temperature, the nanoclusters of AuMg_n (n = 4–12) are easily distinguished as separate strong bands in the 200 cm⁻¹ regions of the spectra, except for the AuMg₂ and AuMg₃ nanoclusters. This result is in good agreement with the results of infrared spectroscopy of pure Mg nanoclusters studied by Belyaev et al. (Belyaev et al., 2016). For the Raman weighted average spectrum at room temperature, although the strongest Raman activity peaks of AuMg₄, AuMg₅ and AuMg₆ nanoclusters appear in the low-frequency band (20–50 cm⁻¹), they still have many strong peaks in the 200 cm⁻¹ regions. Therefore, for all the Raman spectra of the AuMg_n (n = 2–12) nanoclusters, the 200 cm⁻¹ regions can be more easily distinguished as individual strong bands. In conclusion, it was computationally shown that the formation of AuMg_n (n = 2–12) nanoclusters at room temperature is possible to identify these nanoclusters by IR and Raman spectroscopy.

In order to provide more data support for future possible experiments, the average bond distance and average nearest neighbor distance were calculated. As shown in Figure 10, the average nearest neighbor distance and bond distance for Au-Mg and Mg-Mg in the ground state of AuMg_n (n = 2–12) clusters display some interesting conclusions. Figures 10A,B show that, overall, the nearest neighbor distance for Au-Mg becomes larger as the cluster size increases (except for n = 8 and 12), while Mg-Mg is overall decreasing. The average nearest neighbor distance of Au-Mg is 2.68 Å, and that of Mg-Mg is 3.44 Å. Figure 10C gives the average bond distance of Au-Mg with cluster size dependence similar to the nearest neighbor distance of Au-Mg, i.e., increasing overall. However, Figure 10D demonstrates that the average Mg-Mg bond distance decreases and then increases with cluster size. Another interesting conclusion is that the local turning points of the average bond and nearest-neighbor distances for Mg-Mg and the average bond distance curves for Au-Mg occur at AuMg₉, suggesting that they influence the local stability of the clusters.