Geometrical structures of FMgn (n = 2–20) clustersFigure 1 shows the geometries of the lowest energy states of the FMgn (n = 2–20) clusters and the point group symmetry of the corresponding electronic states. All values of the lowest vibration frequency in Table 1 are positive, confirming that these isomers are not excited states. The three lowest energy isomers at each size and the energy difference with the lowest one are displayed in Figures S1 and S2 in the Supplementary Material. One can find that except for FMg2, FMg3, and FMg4, which are planar structures, all other FMgn (n = 5–20) exhibit a three-dimensional geometry, and F atoms are adsorbed by Mgn at the periphery of the structure.Figure 1Structures, points group symmetry and electronic state of the lowest energy isomers of FMgn (n = 2–20) clusters.Table 1 Symmetry, electronic state, the lowest vibrational frequency, average bonding energy (Eb), the second order difference energy (∆2E), HOMO–LUMO energy gap (Egap) for α- and β-electrons for optimized lowest energy isomers of FMgn (n = 2–20) clusters.Both FMg2 and FMg3 clusters have C2v symmetry, where the former is an isosceles triangle and the latter is a plane quadrilateral composed of double isosceles. FMg4 is an irregular planar pentagon with Cs symmetry. The structures of FMg5, FMg6, FMg9, FMg11, FMg13, FMg14 and FMg20 contain a tetrahedral unit of FMg3 that adsorbs several atoms in different directions. FMg7, FMg8, FMg10, FMg12, and FMg15–19 exhibit relatively complex structures, with the commonality of F atoms being relatively far from the Mgn host and only two Mg atoms surrounding F atom. In addition, some other interesting conclusions can be drawn. (i). The structure of FMg6–20 clusters exhibits low symmetry (C1), indicating that they tend to disrupt some high-symmetry structures and achieve relatively lower energy to improve stability. (ii). all F atoms in FMgn clusters are pushed to the outermost edge by the Mgn host. (iii). One Mg atom from FMg15–20 clusters is trapped in an irregular magnesium “cage”. Compared with some existing studies on magnesium-based clusters, there are some significant differences in the ground state structure of F-doped magnesium clusters. For example, studies on Si-doped26 and Pd-doped53 magnesium clusters have shown that Si and Pd atoms are trapped by Mgn clusters. The structures of Ga-doped43,54,55, Au-doped56, and Nb-doped57 magnesium clusters have distinct basic structural units of seeds or frameworks, such as tetrahedra and pentahedra, but F-doped magnesium clusters do not exhibit this pattern. In summary, the diversity of ground state structures of FMgn clusters is rare, and their growth mechanisms have their unique features. The atomic coordinates of all FMgn (n = 2–20) clusters of the radical isomers are shown in Table S1 of the Supplementary Material. The two or three Mg atoms that surround the doping result of F atoms are the only ones that directly affect Mgn; the doping result always “floats” on the exterior of the Mgn host. It is intriguing, nevertheless, because FMgn’s structure makes it difficult for a Mg atom to adsorb from the FMgn-1 cluster in other directions. This has been repeatedly verified in other findings on materials containing magnesium.The relative stabilities analysisSystematic investigation is necessary because the size dependency of atomic cluster characteristics dictates that their relative stability differs. Here, we have estimated the three relative stability parameters (Eb, Δ2E, and Egap) of the ground state isomers of FMgn (n = 2–20) clusters. The following is the formula used to calculate these energies.$$ E_{b} ({\text{FMg}}_{n} ) = [E({\text{F}}) + nE({\text{Mg}}) – E({\text{FMg}}_{n} )]/(n + 1) $$
(1)
$$ \Delta_{2} E({\text{FMg}}_{n} ) = E({\text{FMg}}_{n + 1} ) + E({\text{FMg}}_{n – 1} ) – 2E({\text{FMg}}_{n} ) $$
(2)
$$ E_{gap} ({\text{FMg}}_{n} ) = E_{{{\text{LUMO}}}} ({\text{FMg}}_{n} ) – E_{{{\text{HOMO}}}} ({\text{FMg}}_{n} ) $$
(3)
In the righthand of Eqs. (1)–(2), E is the energies of the corresponding atoms (F and Mg) and clusters. In Eq. (3), EHOMO and ELUMO are the energies of the highest occupied molecular orbital and lowest unoccupied molecular orbital.All calculations are presented in Table 1 and plotted in Fig. 2. As shown in Fig. 2a, the Eb curve of FMgn (n = 2–20) clusters overall decreases in size, The maximum and minimum Eb values of FMgn appear at n = 2 and n = 17, respectively, indicating that FMg2 has the highest stability, while FMg17 has the lowest stability. From a local perspective, FMg10 (0.78 eV) and FMg18 (0.66 eV) have higher Eb values than their neighbors, therefore their local stability is higher. The Δ2E in Eq. (2) defines the local stability of clusters, and from Fig. 2b, it can be seen that the Δ2E curve of FMgn clusters oscillates with size. FMg17 and FMg18 have local minimum values (− 1.43 eV) and maximum values (1.25 eV) of Δ2E, respectively, indicating that FMg18 has the highest local relative stability. FMg4, FMg8, and FMg10 also have larger Δ2E values than their neighboring clusters, indicating that their relative stability is also prominent. Egap can reflect the strength of the ability of clusters to participate in chemical reactions to a certain extent, and at the same time, the larger Egap value indicates that it is more difficult for electrons to transition from occupied orbitals to null orbitals, and therefore the relative stability of clusters is high. FMgn cluster, as an open-shell system, does not have equal numbers of α and β electrons, and their Egap values and curves are shown in Table 1 and Fig. 2c, respectively. Calculations show that the Egap for both α and β electrons as a whole decreases with increasing cluster size. The maximum for Egap-α occurs at the FMg2 cluster, and local maxima are found at FMg9, FMg12 and FMg18, suggesting that they are relatively chemically stable when dominated by α electrons. For Egap-β, on the other hand, the FMg3 cluster has a maximum and the FMg8, FMg13 and FMg17 clusters possess localized maxima, implying that these clusters are chemically more stable when β electrons predominantly contribute. Based on the aforementioned theoretical research, the FMg18 cluster has strong overall stability and may be the “magic” cluster. As such, further experimental observations should concentrate on this cluster.Figure 2(a) Average bonding energy Eb, (b) second order energy difference Δ2E, (c) HOMO–LUMO energy gap Egap, for α and β electrons of the lowest energy isomers of FMgn (n = 2–20) clusters.Charge distribution and electron configuration analysisIt is well known that the free-standing states of the magnesium and fluorine atoms have the electronic configurations 1S22S22P5 and 1S22S22P63S2, respectively. When fluorine atoms interact with magnesium atoms to produce FMgn clusters, it would be very interesting to learn more about the distribution of each atom’s charge population, how the charges are transferred, and how their patterns of change with Mgn size are. By performing natural bonding orbital (NBO) calculations on the FMgn ground state clusters, natural charge population (NPA) and natural electronic configuration (NEC) are obtained and listed in Tables S2 and S3 of the Supplementary Material. From Table S2 and Fig. 3a, it can be seen that in all the FMgn clusters, the F atoms always gain electrons (− 0.93 to − 0.90e), while most of the Mg atoms lose electrons (0.46 to 0.02e), and a few Mg atoms also gain electrons (− 2.11 to − 0.01e). One can find that, in Fig. 3a, the number of electrons gained by all F atoms fluctuates very little, indicating that it is relatively stable in its interaction with Mg atom, which is consistent with the results obtained in Fig. 1 only two or three F–Mg bonds are always presented in FMgn (n = 2–20). Since the electronegativity of fluorine atom (3.98) is greater than that of magnesium (1.31), F always acts as an electron receiver in FMgn clusters, while most of Mg plays the role of electron giver. Figure 3a also reveals an interesting conclusion that there is one Mg atom from each of the FMg15–20 clusters that always gets more electrons than the F atom, and this atom is the one that is imprisoned inside the cluster in Fig. 1.Figure 3(a) NCP on F and Mg atoms, (b) NEC on F atoms, (c) NEC on Mg atoms for all the lowest energy isomers of FMgn (n = 2–20) clusters.The FMgn (n = 2–20) clusters’ electronic configurations are shown in Table S3 and Fig. 3b,c. The interaction of F and Mg atoms to form clusters alters their electronic configurations with respect to the free-standing state electronic configuration. Specifically, the 2p and 3p shells of F and Mg acquire electrons, while the 2S and 3S shells of F and Mg atoms lose some electrons. The data in Table S3 reveals a fact that all the 3S orbitals of magnesium atoms in the FMgn cluster lose charge, with a portion of − 0.93 e ~ − 0.90 e transferring to the 2p orbitals of fluorine atom. The rest, regardless of whether they interact directly with fluorine atoms or not, mostly occupy the 3p empty orbitals of magnesium atoms, with a small amount occupying higher empty orbitals. In order to recognize the effect of charge transfer in depth, the total density of states (TDOS) and partial density of states (PDOS) of F and Mg atoms are calculated. As an example, four of the FMgn ground state clusters are shown in Fig. 4 for the smallest size (n = 2), the largest size (n = 20) and two intermediate sizes (n = 10, 15), and the TDOS and PDOS of the remaining clusters are displayed one by one in Figure S2 of the Supplementary Material. According to the calculations, all F atoms always occupy the molecular orbitals with the lowest energy, followed by Mg atoms that are bonded to F. Mg atoms that are not connected to F, on the other hand, occupy molecular orbitals with the highest energy. This outcome perfectly matches the NEC analysis.Figure 4Total density of state (TDOS) and partial density of state (PDOS) for the lowest energy isomers clusters (a) FMg2, (b) FMg10, (c) FMg15, (d) FMg20.Theoretical calculations on Infrared and Raman spectroscopyIn experiments, infrared and Raman spectroscopy can be useful in guiding the determination of cluster structures. The strongest IR and Raman peaks for each cluster are marked with specific labels in Figs. 5 and 6, which display the results of same level frequency and Raman calculations for all ground state isomers of FMgn. The locations of the other strong peaks are listed in Table S4 of the Supplementary Material. Specifically, the strongest absorption IR peak and Raman scattering peak of FMg2 cluster appear at 442 cm−1 and 75 cm−1, respectively. The strongest IR and Raman peaks of FMg3 can be found at 147 cm−1 and 196 cm−1. The strongest IR peaks of FMg4, FMg5 and FMg6 clusters are located at 134 cm−1, 79 cm−1 and 337 cm−1, respectively, while their strongest Raman peaks appear at 155 cm−1, 152 cm−1 and 150 cm−1. The strongest IR peaks of FMg7, FMg8, FMg9 and FMg10 clusters can be detected at 465 cm−1, 435 cm−1, 319 cm−1 and 471 cm−1, and accordingly, their strongest Raman peaks are located at 172 cm−1, 150 cm−1, 91 cm−1, and 164 cm−1, respectively. As the cluster size increases, the strongest IR peaks of FMg11, FMg12, FMg13, FMg14, and FMg15 clusters appear in a wider range, which are located at 345 cm−1, 444 cm−1, 191 cm−1, 229 cm−1, and 465 cm−1, respectively, but the distributions of the strongest peaks of Raman are relatively concentrated, which can be found at 181 cm−1, 168 cm−1, 180 cm−1, 176 cm−1, and 162 cm−1, respectively. Calculations show that the strongest IR peaks of FMg16, FMg17, FMg18, FMg19, and FMg20 clusters still appear in relatively high-frequency bands, and they can be theoretically found at 431 cm−1, 426 cm−1, 481 cm−1, 477 cm−1, and 349 cm−1, respectively, but interestingly, the distribution bands of the Raman’s strongest peaks appear toward smaller bands, and they are located at 132 cm−1, 144 cm−1, 159 cm−1, 151 cm−1, and 139 cm−1, respectively.Figure 5Theoretical calculations on IR spectra for the lowest energy isomers of FMgn (n = 2–20) clusters.Figure 6Theoretical calculations on Raman spectra for the lowest energy isomers of FMgn (n = 2–20) clusters.The last two subplots in Figs. 5 and 6 also show the strongest IR and Raman peaks versus cluster size curves, as well as the frequency distribution of the occurrence of the first six peaks. Overall, the strongest IR peaks of the FMgn (n = 2–20) ground state isomers appear in the band 481 cm−1 ~ 79 cm−1, while that of the Raman spectra appear in the lower band range of 196 cm−1 ~ 75 cm−1. Additionally, one can find that the frequency intervals between multiple strong peaks tend to be small for Raman spectra, but relatively large for IR spectra. These results suggest that, except for clusters with a single Raman peak, it is more difficult to verify the structure of clusters using Raman spectroscopy than their IR spectra. Based on the Raman spectroscopic data for the failure of FMgn clusters, and in order to provide more guiding data for the experiments, this work further calculates the first 50 excited states for all clusters and predicts their UV–Vis spectra. The UV–Vis spectra of four clusters, FMg2, FMg10, FMg15, and FMg20, are displayed in Fig. 7, and the results for the rest clusters are enumerated in Figure S3 of the Supplementary Material. The four excitation leaps that contribute the most to the strongest peak of UV–Vis, as well as the wavelength of the strongest peak, are shown directly in each figure. Specifically, the strongest UV–Vis peaks of the FMg2–6 clusters all appeared in the mid-UV region at 312 nm, 328 nm, 345 nm, 348 nm, and 373 nm, respectively. The strongest UV–Vis peaks of FMg7–11 clusters are located in the visible region, and their wavelengths can be detected at 503 nm, 603 nm, 583 nm, 602 nm and 631 nm, respectively. In contrast, the strongest UV–Vis peaks of FMg12–20 clusters are located in the near-infrared region, and one can find their wavelengths at 821 nm, 708 nm, 915 nm, 850 nm, 864 nm, 933 nm, 1063 nm, 919 nm, and 896 nm, respectively. With the above IR, Raman and UV–Vis spectroscopic data, sufficient guarantees can be provided for experimentally probing FMgn (n = 2–20) clusters.Figure 7UV–Vis of the lowest energy isomers of (a) FMg2, FWHM = 0.667 and oscillator strength > 0.2300. (b) FMg10, FWHM = 0.100 and oscillator strength > 0.0079. (c) FMg15, FWHM = 0.100 and oscillator strength > 0.0017. (d) FMg20, FWHM = 0.100 and oscillator strength > 0.0010.Topological analysis of chemical bondsFinally, we do a thorough investigation of FMgn (n = 2–20) clusters’ chemical bonding characteristics. By computing the topological characteristics of the F–Mg and Mg–Mg bond critical points (BCPs) in each cluster, including variables like the Laplacian of electron density (Δρ), electron localization function (ELF), and interaction region indicator (IRI), it is possible to identify the kind and strength of the chemical bonds within the clusters. Table S5 of the Supplementary Material shows the properties of BCPs for all FMgn clusters. The BCPs property diagrams for the four clusters FMg2, FMg10, FMg15, and FMg20 are shown in Fig. 8, and the corresponding diagrams for the other clusters are displayed one by one in Figure S4 of the Supplementary Material. Specifically in terms of the components in each figure, one can find the NCP coloring based on Table S2, the number of BCPs, and IRI in 2D and 3D are presented in Fig. 8 and Figure S4. The results show that in the NCP coloring diagrams, all F atoms appear white because they gained electrons, and most of the Mg atoms lost electrons and appear blue. However, the FMg14–20 clusters show particularly interesting colors, with distinct white or red Mg atoms appearing inside the Mgn hosts due to the gain of electrons by these Mg atoms, which is in agreement with previous findings in the NCP analysis section. BCPs number calculations show that the number of F and Mg atoms bonded in all FMgn clusters is 2 or 3, which is consistent with the geometry of Fig. 1. IRI is a useful parameter for measuring atomic interactions, and the red and yellow regions in IRI-2D in the figure are the bonding regions, and the blue regions in IRI-3D are the regions with high bonding strengths. The IRI interaction maps of FMgn clusters confirm that F–Mg and Mg–Mg interactions are prevalent in the clusters and that the Mg–Mg interactions are stronger. Specifically, as shown in Table S5, all the FMgn clusters have positive Δρ for F–Mg with ELF values less than 0.5, whereas most of the Mg–Mg have negative Δρ and correspondingly ELF greater than 0.5. This suggests that the F–Mg is bonded as a noncovalent bond, and most of the Mg–Mg is covalently bonded. Considering the NCP results, we can further determine that the chemical bonding of F–Mg is ionic. As for the Mg atoms in the FMg15–20 clusters that have gained many electrons, for example, the 19 Mg proto in FMg20 cluster has an NCP of − 2.1 e. Since it is surrounded by as many as eight Mg atoms to form an Mg–Mg chemical bond, their ELF values and IRI indices reveal that they are still covalently bonded. It is therefore reasonable to conclude that these Mg–Mg are polarized covalent chemical bonds. Corresponding studies in other clusters are shown in detail in the Supplementary Material, and the conclusions are the same as for the FMg20 cluster.Figure 8Natural charge population atomic coloring, bond critical points (BCPs), interaction region indicator (IRI) 2D and 3D for F–Mg and Mg–Mg interaction of (a) FMg2, (b) FMg10, (c) FMg15, (d) FMg20 clusters.
Structural and spectral properties of Gas-phase FMgn (n = 2–20) clusters based on DFT
