Density functional theory-based study on the structural, electronic and spectral properties of gas-phase PbMg n − (n = 2–12) clusters

Gas-phase PbMg n− (n = 2–12) cluster structures were globally searched on their potential energy surfaces by means of the CALYPSO prediction software. Structural optimization and calculations of properties such as relative energy and electronic structure were then carried out by density functional theory for each size of low energy isomer. The structural, relative stability, natural charge population, natural electronic configuration and distribution of the strongest peaks of the infrared and Raman spectra of the low energy isomers of PbMg n− (n = 2–12) clusters were systematically investigated in the present work. It was shown that the PbMg7− cluster ground state isomer exhibits the highest stability, for which special electronic excitation and chemical bonding analyses were performed. It is reasonable to believe that this work enriches the structural, spectroscopic and other data of magnesium-based clusters and provides some theoretical basis for possible future experimental syntheses.


Introduction
The study of sub-nanometre-sized clusters of a few to several tens of Mg and Mg-based doped atoms has attracted a great

Methods
Since the atomic clusters are based on the physical background of atomic sub-nanometre-sized atomic aggregation, they must be calculated by quantum mechanical methods.The initial structures of the PdMg n − (n = 2-12) clusters were searched on the potential energy surface using the CALYPSO quantum chemistry prediction software [51].CALYPSO is a powerful crystal structure prediction program that has been successfully applied for nearly a decade to predict the structure of clusters [52][53][54], two-dimensional layers [55] and three-dimensional crystals [56].CALYPSO is based on a particle swarm optimization (PSO) algorithm that performs global searching on potential energy surfaces for a given chemical composition and external conditions such as pressure to obtain low energy isomers' structures.Specifically, an extensive global minimum search was performed using the CALYPSO code for anionic PbMg n clusters in the range 2 ≤ n ≤ 12. First, 50 initial structures were randomly generated under symmetry conditions, and in each subsequent generation of optimization, 80% of the new structures were generated based on a PSO algorithm that selects the previous generation of structures with high fitness, while the remaining 20% were randomly generated.As the number of generations increases, the exclusion of similar structures is achieved through characterization matrices.Overall, isomers of PbMg n − were generated with 20 structures per generation for a total of 50 generations; therefore, 1000 structures were obtained at each size at B3LYP functional [57] and low-level 6-31G basis set [58] using the Gaussian 09 code [59].Second, a higher level of structural optimization was performed for the 50 lowest energy isomers of the 1000 structures generated by CALYPSO, with the 6-311G(d) basis set [58] for the Mg atom and the lanl2dz pseudopotential basis set [60] for the Pb atom.The choice of this calculation level is based on successful reports of Mg-based clusters in recent years [44,47,50,[61][62][63].Based on the total number of electrons for each size of PbMg n − , four spin multiplicities of 2, 4, 6 and 8 were considered for the optimization of each isomer.In addition, the structural optimization was accompanied by vibration frequency calculations to ensure that the resulting structure is a local minimum energy state on its potential energy surface.The charge transfer properties, such as natural charge population (NCP) and natural electron configuration (NEC), were obtained by a natural bond orbital (NBO) method [64].The electronic excited state analysis of the most stable cluster was studied by time-dependent density functional theory (DFT) calculations of the 50 excited states [65].The chemical bonding analysis was performed by electron localization function (ELF) [66] calculation of the ELF values for the Pb-Mg and Mg-Mg topological critical points.Multiwfn software [67,68] was used in this research work for ELF, UV-Vis, hole-electron analysis and distribution maps of electronically excited states, as well as for density of states (DOS) for PbMg 7 − .

The structural growth of PbMg n − (n = 2-12) clusters
Figure 1 shows a total of 33 isomers named n-i, where n is the number of Mg atoms and i = 1, 2, 3, corresponding to the ground, second and third lowest energy states, respectively.As shown in table 1, the lowest vibrational frequencies of the isomers are positive, indicating that they meet the requirement that there can be no imaginary frequencies in the frequency calculations, i.e. all optimized isomers are not excited states but local lowest energy states on the potential energy surface.In addition, each isomer of the PbMg n − (n = 2-12) cluster is shown with its point group symmetry, electronic states and their energy differences from the ground state.In table S1 of the Supplementary Material, we also show the atomic coordinates of the radical isomers of each cluster for the reader's use.The three lowest energy isomers of the PbMg 2 − cluster are the isosceles triangular shapes 2-1 (C 2v , 4 A 1 ) and 2-3 (C 2v , 2 A 1 ), and the linear shape 2-2 (D ∞h , 4 Σ g ).The ground state isomer has 0.06 and 0.12 eV lower energy than the isomers 2-2 and 2-3, respectively.The ground state isomer 3-1 (C 3v , 4 A 1 ) and the third lowest energy isomer 3-3 (C S , 2 A′) of the PbMg 3 − cluster possess a tetrahedral structure, while 3-2 (D 3h , 4 A 2 ″) exhibits a planar geometry with its Pb atom located at the centre of the triangle.In addition, the isomers 3-3 and 3-2 have 0.12 and 0.10 eV higher energy than their ground state isomers, respectively.For the PbMg 4 − cluster, the second lowest energy isomer 4-2 (C 2h , 2 B u ) still maintains a planar structure, where the Pb atom is located at the centre of a rectangular structure formed by four Mg atoms.The isomers 4-1 (C 3v , 4 A 1 ) and 4-3 (C S , 4 A″), on the other hand, are based on the growth of the tetrahedral structure 3-1, with one Mg atom adsorbed in different directions.The energy of the ground state isomer 4-1 is 0.03 and 0.08 eV lower than that of the isomers 4-2 and 4-3.For the PbMg 5 − cluster, calculations reveal that isomers 5-1 (C S , 2 A″), 5-2 (C S , 2 A′) and 5-3 (C S , 4 A″) are obtained by adsorption of two Mg atoms in different orientations by isomer 3-1 or based on adsorption of one Mg atom in different orientations by isomer 4-1 while slightly changing the original structure.A similar situation occurs in the formation of the three lowest energy isomeric structures of PbMg 6 − .Based on the tetrahedral structure of isomer 3-1 adsorbing three Mg atoms at different positions or fine-tuning its own structure based on the structure of isomer 5-1 while adsorbing one Mg atom, the geometrical structures of isomers 6-1 (C S , 2 A′), 6-2 (C S , 4 A″) and 6-3 (C 5v , 4 A 1 ) can be searched.In addition, the second lowest and third lowest energy isomers of PbMg 5 − and PbMg 6 − can be found at energies above their ground state isomers of 0.16, 0.10, 0.01 and 0.12 eV, respectively.Starting from the PbMg 7 − cluster, a new structural growth pattern emerges.The isomer 7-1 (C S , 2 A′) is a staggered stack of two planar rectangles, while the isomers 7-2 (C S , 2 A′) and 7-3 (C S , 2 A′) are structures formed by the adsorption of three Mg atoms based on a pentahedral structure formed by five atoms.The energies of isomers 7-2 and 7-3 are 0.31 and 0.35 eV higher than their ground states, respectively.The basic structures of isomers 8-1 (C 1 , 2 A), 8-2 (C 1 , 2 A), 9-1 (C S , 2 A′), 9-2 (C S , 2 A″), 10-1 (C S , 2 A′), 10-3 (C 1 , 2 A), 11-1 (C S , 2 A″), 11-3 (C S , 2 A″), 12-1 (C 1 , 2 A) and 12-2 (C S , 2 A″) are almost identical and can be seen as adsorption of another atom at the apex of the structure of isomer 7-1.This basic unit structure is presented in electronic supplementary material, figure S1, but it is worth noting that the position of the Pb atom is often not fixed in the clusters.The structures of isomers 8-3 (C 1 , 2 A), 9-3 (C S , 2 A″), 10-2 (C S , 4 A′), 11-2 royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 240814 (C S , 2 A′) and 12-3 (C S , 2 A′), on the other hand, show an irregular cage-like geometry.Furthermore, the calculations show that the second lowest energy and third lowest energy isomers of the PbMg n − (n = 8-12) cluster have energies 0.11, 0.31, 0.19, 0.25, 0.26, 0.45, 0.15, 0.32, 0.16 and 0.25 eV higher than their ground state counterparts.
In fact, the growth of small-sized Mg-based clusters based on tetrahedral unit and tower-like unit structures has been reported in the study of Be-, Ga-, Ge-, Si-, Na-and other atomic or ion-doped magnesium clusters.This suggests that the geometric structure of small-and medium-sized PbMg n − (n = 2-12) clusters does not vary much with respect to these alkali metals.

The stability properties
As a ground state, the lowest energy isomers of PbMg n − of different sizes in figure 1 are worthy of further investigation.The bonding energy per atom (E b ), the second-order energy difference (∆ 2 E) and the highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) energy gap (E gap ) are calculated at the same level as B3LYP/6-311G(d) to study the relative stability of these ground state isomers.Formulas for these three energies, which measure the relative stability of clusters, are shown in equations (3.1)-(3.3)below: where E (eV) represents the energy of the object in the parentheses on its right side.The energy difference between the HOMO and the LUMO is E gap .Table 1 shows the results for these energies, and their size-dependent curves are plotted in figure 2. As shown in table 1 and figure 2a, the E b values of each size isomer are in the range 0.44-0.53eV.The E b curves show oscillations with increasing cluster size, with PbMg 6 − (0.44 eV) having the smallest E b value and PbMg 8 − (0.53 eV) the largest, followed by PbMg 7 − (0.52 eV).This implies that PbMg 7 − and PbMg 8 − are more stable relative to other clusters.According to equation (3.2), the Δ 2 E value of a cluster is an important parameter to measure its relative stability compared with its neighbour, and the Δ 2 E curve in figure 2b

Natural charge population and natural electron configuration analysis
NBO calculations on all the ground state isomers of PbMg n − yielded their NCP and NEC, both of which are useful for gaining insights into the properties of the electronic structure of clusters.In order to better discuss the charge transfer and electronic configuration properties, the PbMg

Infrared and Raman spectra
The structure of the clusters determines their spectra, so spectral predictions can provide data to guide possible future experiments.As shown in figures 5 and 6, we have theoretically predicted and plotted the infrared (IR) and Raman spectra of the ground state isomers of all PbMg n − (n = 1-12) clusters.Overall, the IR and Raman spectral peaks of PbMg n − (n = 1-12) clusters are distributed in the 10-220 cm −1 frequency band.It is found that for PbMg 1 − the only IR and Raman peaks are at 146 cm −1 .PbMg 2 − has two IR peaks and one Raman peak, with both the strongest IR peak and the Raman peak occurring at the very low frequency of 22 cm −1 .The strongest IR and Raman peaks of PbMg 3 − both appear at 38 cm −1 .PbMg 4 − possesses two distinct IR peaks and three Raman peaks, with the strongest IR and Raman peaks located at 145 and 11 cm −1 , respectively.Calculations show that PbMg 5 − has at least six detectable IR and Raman peaks distributed between 10 and 200 cm −1 , with the strongest IR and Raman peaks occurring at 95and 145 cm −1 , respectively.The strongest IR and Raman peaks of PbMg 6 − can be found at 168 and 101 cm −1 .The strongest IR and Raman peaks of PbMg 7 − are located at 180 and 192 cm −1 , respectively.The strongest IR and Raman peaks of PbMg 8 − can be detected at 213 and 199 cm −1 .The strongest IR and Raman peaks of PbMg 9 − are both neatly located at 207 cm −1 .The strongest IR and Raman peaks of PbMg 10 − are detected at 175 and157 cm −1 .The strongest IR and Raman peaks of PbMg 11 − are located at 209 and 166 cm −1 .Finally, the strongest IR and Raman peaks of PbMg 12 − are found at 218 and 171 cm −1 .royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 240814 In conclusion, as shown in figures 5 and 6, although the theoretically calculated values of the strongest IR and Raman peaks of each cluster are very clear, the IR or Raman spectra exhibit a multi-peak nature as the size of the clusters increases, which can cause some difficulties in detecting cluster size directly through spectroscopic experiments.However, the theoretical data of these spectra can provide reference for the corresponding experimental spectra.

3.5.
Further studies on the PbMg 7 − cluster ground state isomer 3.5.1.Ultraviolet-visible spectrum and excited state analysis PbMg 7 − is worthy of further investigation as a cluster with relative overall excellent stability.Although its IR and Raman spectra have been calculated in the previous section, this section continues with a special look at its ultraviolet-visible (UV-Vis) spectrum.UV-Vis spectroscopy is another useful experimental tool for understanding the structure of matter and is therefore always used in the study of clusters.Figure 7a shows the UV-Vis absorption spectral curve of the ground state isomer of PbMg 7 − , and the vertical short straight lines within the curve indicate the oscillator strength (right-side y-axis).There are two strong peaks with close intensity at 638 and 676 nm, whose oscillator strength reaches 0.025 and 0.035, respectively.Excited state analysis shows that the main contributors to these two strong peaks are from the S0 → S28, S0 → S30, S0 → S32 and S0 → S36 transitions.In order to further investigate the properties of these four excited states, a hole-electron-based study of the excited states  royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 240814 was carried out.Specifically, electronic supplementary material, table S4 shows the analysis of holes and electrons for the four most dominant excited states in the strongest peaks of the UV-Vis spectrum, including the distance between the centre of mass of the holes and electrons (D), the overall average distribution of electrons and holes (H), the overlap between the electron-hole distributions (S r ), the degree of separation between the holes and the electrons (t), the hole-domain departure indices (HDI) and the electron-domain departure indices (EDI).The transition density matrix (TDM) distribution of the excited state is plotted in figure 7b, while electronic supplementary material, figure S2 shows the hole-electron distribution of the excited state, the hole-electron overlap function and the heat map of the overlap function and TDM distribution.As shown in electronic supplementary material, table S4, the t-parameter is less than zero and the S r -parameter is close to 1.0, indicating that there is no significant separation of holes and electrons during the electronic excitation of the four excited states to S28, S30, S32 and S36, while the D-parameter, HDI and EDI indices are relatively small, suggesting that the dominant peaks of the UV-Vis absorption spectrum of the PbMg 7 − cluster ground state isomer are globally excited (GE).In addition, the TDM distribution in figure 7b and the hole-electron distribution and S r distribution plots in electronic supplementary material, figure S2a,b graphically verify the above electronic excitation properties.To further demonstrate the contribution of each atom to the electronic excitation, electronic supplementary material, figure S2c,d shows the hole-electron distribution as well as the TDM by means of heat maps.An obvious conclusion is that the Pb atom contributes much more to electronic excitation than do holes, and the asymmetry of the diagonal part of the thermogram in the TDM heat map confirms that the cluster is GE.

Chemical bonding analysis
Since the nature of the chemical bonding of metal clusters may vary with size, we performed bonding analyses by ELF for the PbMg    [47,62].These findings confirm that even when metal atoms are doped into small-sized magnesium clusters, their chemical bonding has no metal bonding properties.

Total density of states and partial density of states
Atomic clusters, as isolated systems, have discrete energy levels, so in principle, DOS analysis is not possible, but if the discrete energy levels are artificially broadened into curves, DOS diagrams can be a useful tool for analysing the nature of the electronic structure of clusters.Here, we have performed total density of states (TDOS) and partial density of states (PDOS) calculations for the ground state isomer of the PbMg 7 − cluster and plotted them graphically by means of Multiwfn, as shown in figure 9.
In figure 9a, we show the TDOS and PDOS of the α and β electrons of PbMg 7 − , and the PDOS is defined by the different atomic contributions, where Mg-(Pb) represents the Mg atoms that are bonded to the Pb atom, and the rest of the Mg atoms are not bonded to Pb. Figure 9b, on the other hand, combines the contributions of the two electrons, along with the corresponding molecular orbitals (MOs; each shown by a discrete vertical line) energy levels on display.It was shown that Pb atoms contribute most to the two MOs with the lowest energies, the MOs with slightly higher energies are mainly contributed to by the three Mg atoms that are not bonded to Pb atoms, followed by the four Mg atoms bonded to Pb, and the main contributors to the MOs with higher energies, close to the HOMOs, are the Mg-(Pb) and to a lesser extent Mg, a property that holds for all the unoccupied molecular obitals (UMOs).Combining the discussions in §3.3 about NCP and NEC, we can clearly see that the lowest energy MOs of the PbMg 7 − cluster are occupied by the 6p valence electrons of the Pb atoms and the 3s valence electrons of Mg-(Pb), and the electron occupancy of the MOs with higher energies is populated by the corresponding valence electrons of Mg-(Pb) and those of Mg.In other words, the strong p-d hybridization is responsible for the stability of PbMg 7 − .TDOS and PDOS analyses of the other clusters PbMg n − (n = 2-6, 8-12) lead to similar conclusions as for the PbMg 7 − isomers, as described in electronic supplementary material, figure S3.

Conclusion
Using the CALYPSO cluster structure search software, this work presents a systematic study of gas-phase anionic Pb-doped Mg n (n = 2-12) clusters.The geometrical structural features of the three lowest energy isomers of different-size clusters are investigated, and the relative stability of the size dependence is calculated through several characteristic energies.It is shown that the PbMg7 − cluster ground state isomer has the highest integrated stability, and its excited state properties and chemical bonding properties are investigated in §3.5.In addition, the electronic structure of the cluster ground state isomers of various sizes is investigated through NCP and natural electronic configuration, while theoretical calculations predict where the strongest peaks of their IR and Raman spectra would appear.
shows that PbMg 6 − (−0.67 eV) and PbMg 7 − (0.43 eV) have the smallest and the largest values, respectively, indicating that PbMg 7 − possesses the highest local stability.E gap in equation (3.3) can be used to measure the chemical stability.Since the PbMg n − cluster is an open-shell system,

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Isomer n-i structures for the three lowest energy clusters; n is the size of PbMg n − (n = 2-12) clusters, i (i = 1-3) is the energy minimum label for each size cluster.The energies presented are the energy differences (eV) of the ground state isomer of their corresponding sizes.
the Pb-NEC in the ground-state PbMg n -(n = 1-12) clusters Distribution values of the Pb-NEC in the ground-state PbMg n -(n = 1-12) clusters

7 -Figure 9 .
Figure 9.Total density of states (TDOS) and partial density of states (PDOS) analysis for the lowest energy state isomer of PbMg 7 − .

Table 1 .
The bonding energy per atom: E b , second-order difference energy; ∆ 2 E, HOMO-LUMO energy gap (E gap ); the lowest vibrational frequency; Pb atomic natural charge population (NCP-Pb) and natural electron configuration (NEC-Pb) in the lowest energy state of PbMg n − (n = 2-12) clusters.
4royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 240814 it does not have the same number of α and β electrons, as shown in table 1, as well as figure2c,d, both α-E gap and β-E gap show oscillatory behaviours, with a local maximum of the α-E gap value for PbMg 7 − (1.66 eV) and local maxima of β-E gap occurring at PbMg 8 − (1.86 eV) and PbMg 9 − (1.88 eV), which implies that their chemical stability is relatively high.Overall, the above calculations indicate a relatively high stability of the ground state isomer of the PbMg 7 − cluster.Since this higher stability isomer will be studied in more depth later, its side view and atomic numbering are shown in electronic supplementary material, figure S1.
1 − cluster ground state isomer is added to the study in this section.The NCP and NEC calculations for Pb atoms are shown in table 1, while those for all Mg atoms are shown in electronic supplementary material, tables S2 and S3, and figures 3 and 4 graphically plot them.Figure3ashows the colour distribution of NCP values for Pb and Mg atoms, where blue indicates the gain of electrons while red is the loss of electrons.Obviously, it shows that all 12 Pb atoms in PbMg n −(n = 1 -12) clusters always gain electrons in the range of −1.53e to −0.74e (e stands for electrons).Figure3a,b indicates that 41 out of 78 Mg atoms in PbMg n − (n = 1-12) clusters gain electrons in the range 0.02e to 0.29e, and the remaining 37 Mg atoms lose electrons in the range −0.58e to −0.03e.The electron transfer properties of Mg atoms are non-uniform because the object of our study is negatively charged.In similar studies reported for other, neutral Mg-based clusters, the Mg atom always plays the role of losing electrons.Because the electronegativity of the Mg atom (1.31) is less than that of the Pb atom (2.33), it seems reasonable for Pb 7 − cluster ground state isomer.The ELF values for Pb-Mg and Mg-Mg [46]ing sites were calculated and are plotted graphically in electronic supplementary material, tableS5and figure8.Calculations show that all four Pb-Mg ELF values are <0.5, while all eight Mg-Mg ELF values are >0.5, indicating that the Pb-Mg bond in the PbMg 7 − cluster ground state isomer is noncovalent, while the Mg-Mg bond is covalent.Combined with the NCP calculations in table 1 and electronic supplementary material, tableS2, we can further conclude that Pb1-Mg4, Pb1-Mg5, Pb1-Mg6 and Pb1-Mg8 are ionic bonds.This result agrees with previously reported chemical bonding analyses of Mg-based clusters, such as PdMg n[46], AuMg n [49] and BeMg n