Ionic versus Metallic Bonding in Al n Na m and Al n Mg m (m 3, n+m 15) Clusters

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1 Ionic versus Metallic Bonding in Al n Na m and Al n Mg m (m 3, n+m 15) Clusters Cameron J. Grover, Arthur C. Reber, and Shiv N. Khanna Department of Physics, Virginia Commonwealth University 701 West Grace Street, Richmond, Virginia 23220, United States *Correspondence to: snkhanna@vcu.edu First principles electronic structure studies on the ground state geometries, stability, and the electronic structure of Al n Na m and Al n Mg m ( m 3, n+m 15) clusters have been carried out to examine the nature of bonding between Na or Mg and Al. Identifying whether the bonding is ionic or metallic in bulk materials is typically straightforward; however, in small clusters where quantum confinement is important, the nature of bonding may become unclear. We have performed a critical analysis of the bonding in these bimetallic clusters using charge analysis, electrical dipole moments, hybridization of the atomic orbitals, the Laplacian of the charge density at the bond critical points, and the change in the bonding energy between neutral and anionic forms of the cluster. For Na n Al m clusters, we find that the Na binding is primarily ionic while the bonding in Al n Mg m is primarily metallic. We find that the Mulliken population of the 3p orbital of Na and Mg can provide a rapid assessment of the nature of bonding. We also find that the Hirshfeld charge and dipole moments are effective indicators, when placed in context. We found that the Laplacian of the charge density at the bond critical points can be misleading in identifying whether the bonding is ionic or metallic in small clusters. 1

2 Materials in nature exhibit various classes of bonding including metallic, ionic, or covalent bonding. 1 4 Metallic bonds in solids are marked by highly delocalized electrons that stabilize the close-packed ions. 5 The metal s finite density of states at the Fermi energy allows the highly delocalized electrons to respond to any electric fields. Metals are therefore good conductors of electricity and this grants them the ability to screen stray electric fields thereby annihilating any dipole moments. 6 8 The lack of a band gap energy and the delocalization of electronic states enables identifying metallicity in the solid state straightforward. In ionic bonding, energy levels of the atomic orbitals are significantly different in energy so that one atomic orbital is fully occupied through an electron precise charge transfer, and the other atomic orbital is empty. Ionic bonding leads to counter ions with significant local electric dipole moments. Ionic solids typically have appreciable band gap energies and are generally poor conductors of electricity. These classifications of bonding types become murky as one reduces the size to clusters containing few to few hundred atoms, 9 14 and in fact several papers have come to differing conclusions as to the nature of bonding in Al n Mg m clusters The quantum confinement leads to formation of electronic shells in clusters of metallic elements. The electronic states in different shells can be assigned orbital character (S, P, D, F..) through examination of their orbital shapes and such an analysis has been useful in classification of clusters into various groups and has led to the conceptual framework of superatoms. Depending on the number of valence electrons, clusters of a given element can have filled or open electronic shells depending on their size and shape For this reason, bimetallic clusters have attracted a great deal of attention due to the ability to tune the properties through size, shape, and doping This has led to attempts to describe the metallicity in clusters by either the size of the gap between the filled and unfilled states, or the existence of dipole moment A 2

3 metal cluster may have a well-defined HOMO-LUMO gap (Highest Occupied Molecular Orbital, Lowest Unoccupied Molecular Orbital) and yet the valence electrons of the cluster can be delocalized over entire sphere of the cluster. 15,16 Furthermore, in bulk materials the long range coulombic interaction stabilizes the charge transfer resulting in an electron precise charge transfer from the cation to the anion; however in smaller particles the effect is smaller so only partial charge transfers typically occur. For example, in a solid the NaCl has a net charge transfer of 1.00 electron from the Na to Cl, however in the NaCl molecule, the charge transfer is only 0.56 e -. This makes identifying whether a bond is ionic or a polar metal bond more difficult. In order to critically examine the nature of bonding in bimetallic clusters we have examined a number of criteria for assessing whether the bonding is ionic or metallic in Al n Mg m and Al n Na m. 1) We have evaluated the Hirshfeld charge as the integrated charge associated with an ionic core that can indicate if the bonding is ionic. 2) We have examined the electrical dipole moment to examine the polarity of the bonds in singly-doped clusters. 3) We have examined the hybridization of the 3s and 3p states in the Na and Mg. The occupation of 3p orbitals is a signature of metallic bonding, especially in Mg ) We have calculated the binding energy of Na and Mg in a neutral and anionic cluster. If the bonding is ionic, the excess electron should significantly weaken the binding. 47,48 5) We have determined if the electronic structure of the cluster is affected by removing the dopant Na or Mg atoms and replacing them with the appropriate number of extra valence electrons. 6) We have evaluated the Laplacian of the charge density at the bond critical points in which a positive Laplacian indicates charge depletion and ionic bonding, while a negative Laplacian indicates charge aggregation and metallic bonding. 49,50 By examining these different criteria in two sets of clusters we may understand which criteria are most useful for analyzing the bonding in bimetallic clusters and nanoparticles. 3

4 The purpose of the present paper is to apply this critical analysis on the nature of bonding to the Al-Na and Al-Mg clusters. Sodium, with a single s-electron in the outer shell (3s 1 ) is an alkali atom with a low ionization potential (5.14 ev) and hence one could expect ionic bonding Magnesium has a filled s-subshell (3s 2 ), has a higher ionization potential (7.65 ev) and the nature of bonding between aluminum clusters and magnesium is not so obvious and while some previous studies have classified it as ionic, others have proposed it to be metallic. 9,10 It has been argued that Al n Mg m clusters may be thought of as Zintl-like ions 10 using integrated charge analysis, while another paper has found that the bonding is ionic using the Laplacian of the charge density at the bond critical points. 13 Our studies focus on the size range Al n Na m (1 n 14, 1 m 3; where n+m 15) and Al n Mg m (1 n 14, 1 m 3; where n+m 15) where the number of Na or Mg atoms is limited from 1-3. Understanding the nature of bonding in bimetallic clusters is highly valuable because the type of bonding effects the tendency for a nanoparticle to form an alloy structure or a phase separated structure. 22,23,57 59 We also note that the distinction between metallic and ionic bonding is useful since long range coulombic interactions have been found to promote self-assembly that could facilitate synthesis of clusterassembled materials II. Methods The electronic structure calculations are carried out within a density functional approach. The exchange correlation effects are included using gradient corrected density functional (PBE) proposed by Perdew et. al. 68 This functional has been previously shown to provide a reasonable description of the electronic structure in pure Al n, Na n, or Mg n clusters. 9,47,69,70 The cluster wave functions are formed from a linear combination of slater type orbitals located at the atomic sites. 4

5 The actual computations are performed using Amsterdam Density Functional (ADF) code. 71 Up to fifteen geometries per Al n Na m or Al n Mg m species were optimized to ensure a diverse sampling of the configurational space, and structure have been taken from the literature. 54,72,73 The T2ZP basis was used for all calculations. During the geometry optimization, the energy convergence criterion was 10-8 Hartree. The molecular orbitals were visualized using ADF viewer and were assigned subshell labels based on the nodes of the wave function. The Mulliken population and Hirshfeld charges were examined. 74,75 A fragmented molecular orbital (FMO) analysis was completed to ascertain the level of charge transfer between the cationic sodium or magnesium and the anionic aluminum motif. 71 III. Results We start by considering the ground state geometries of the clusters. Our investigations covered both the neutral and anionic clusters for the sodium doped and magnesium doped systems. The ground state geometries of the Al n Na m species are shown in Figure 1, and the structures of Al n Mg m are shown in Figure 2. The ground state structures of the corresponding anionic structures are provided in Figure S1 and Figure S2. Note that the Na or Mg sites do not generally bond to each other, with only a few exceptions to this rule being Al 2 Na 2, Al 2 Mg 3, Al 3 Mg 3, Al 4 Mg 3, Al 5 Mg 3, Al 6 Mg 3, Al 9 Mg 2, and Al 10 Mg 3. This is partly because Na-Na (0.74 ev) or Mg-Mg (0.10 ev) bond strengths are weaker than Al-Al (1.18 ev), NaAl (0.90 ev), and Mg-Al (0.53 ev) bond strengths. Additionally, the Na sites tend not to embed itself inside the aluminum motif. In order to quantify this difference, we calculated the average distance to the center of mass for each element type and examined the difference between the two as a function of cluster size and composition, as shown in Figure 3. This indicates how much farther the 5

6 dopant atom is from the center of mass than the aluminum atoms, so a positive value indicates that the dopant is farther from the center of mass, and a negative value indicates that the average dopant is closer to the center of mass. In Fig. 3, we show this trend as a function of the cluster size and composition. In nearly all cases, the dopants were farther away than the aluminum atoms. At sizes larger than seven atoms, a different trend emerges. Here, the sodium sites are farther away from the center of mass, compared to the aluminum than in case of magnesium. One might expect that if the dopant is undergoing metallic bonding with the neighboring aluminum then its distance from the center of mass should be in line with that of aluminum, while if the bonding is ionic, one might expect that the aluminum cluster would form a compact core and the dopant counterion would then show a markedly larger distance from the center of mass. This trend is important when determining if an atom contributes its valence electrons to the aluminum motif or has a more metallic like bonding. The present results suggest that sodium may bind ionically while the nature of the magnesium bonding may be closer to metallic. Also, we note that almost no Na-Na bonds are seen, while a few Mg-Mg bonds are seen. One would expect that ionic bonding would destabilize bonding between the counterions, while metallic bonding might still occur between the dopant atoms. The most straightforward way to gain insight into whether a bond is ionic or not is to analyze the charge density, this can be done by either analyzing the atomic charge densities or the dipole moment. Towards this end, we carried out an analysis of the charge distribution using a Hirshfeld charge analysis that allows a basis set independent determination of the difference between the cluster and unrelaxed atomic charge densities. We further calculated the net electric dipole moment to quantify the charge transfer and the separation of the charges. In an idealized ionic system, one would expect a 100% charge transfer of the valence electron from the cation to 6

7 the anion. However, the charge transfer in clusters is typically smaller than in a bulk system with ionic bonding. Figure 4 shows the Hirshfeld charges per Na or Mg sites and the electric dipole moment for the neutral series with a single dopant. The dipole moment of species with multiple Na or Mg sites are included in Fig. S3. For Na atoms, the charge transfer from sodium to the aluminum generally increases with cluster size with some small oscillations in the Al n Mg 3 clusters. For Mg sites, the trends with size are less evident as there are substantial fluctuations with size. The average net charge on Na is consistently more positive that Mg, with the average charge being e - for Na with n>5, while Mg ranges from e - in this size range. A similar pattern is observed for dipole moment of clusters containing a single Na or Mg atom shown in Fig. 4. For Na based clusters, the dipole moment initially increases (except for dips at n=4,5) to a value of around 6 Debye and then decreases for Al 14. For Mg atoms, the diploe moment remains almost constant between 2 and 12 atoms with a value around 2.5 Debye, then shows a sharp drop at Al 13, and a rise at Al 14. The dipole moments of the multiply doped clusters are shown in Figure S3 and also show that the Al n Na m clusters generally have larger dipole moments than the Al n Mg m clusters, although we note that the position of the dopants plays a significant role in the dipole moment, so the correlation between the ionic nature of the bonding and the dipole moment is tenuous. Three general conclusions can be drawn. 1) The change transfer from Na sites is greater than charge transfer from Mg sites for the same number of Na or Mg atoms. 2) The dipole moment for singly doped Al n Na clusters are almost twice as large as for Al n Mg clusters. 3) Al 13 Mg shows an unusually small dipole moment. 1) and 2) confirm that the sodium sites donate more charge than magnesium, which also causes a larger dipole moment. These findings are consistent with the hypothesis that sodium binds more ionically than magnesium. For a metallic system one would expect a smaller dipole moment as 7

8 the delocalized electrons can respond to the electric field to quench the dipole, while an unbalanced ionic system will have a large dipole moment. One of the striking results from charge density analysis is the unusually low dipole moment of Al 13 Mg and Al 11 Na 2. We first start with Al 11 Na 2 that has almost negligible dipole moment. As Fig. 1 shows, this is not related to the bonding being metallic but to the atomic structure where the two Na sites are located at the opposite sides of the Al 11 motif thereby cancelling dipole moment from each other. This shows that one has to be careful in using the dipole moment of multiply doped species to characterize the bonding type. We next consider Al 13 Mg. Before we discuss this in detail, previous studies on pure aluminum clusters have shown that an Al 13 with 39 valence electrons has an electron affinity close to a Cl atom and an - Al 13 has a closed electronic shell with a large HOMO-LUMO gap exceeding 1.80 ev Clusters with filled electronic shells and large HOMO-LUMO gaps are highly stable and have low polarizability. To examine the unusually low dipole moment, we therefore examined the charge density distribution in the cluster. Figure 5 shows a visualization of the Hirshfeld charge at various sites. Note that the magnesium site has a net positive charge of e -. What is interesting is that this charge is transferred primarily to the neighboring Al sites. With such a small charge transfer, the bonding would be expected to be more metallic. Note that the Al 13 motif undergoes a strong polarization with atoms farther from the Mg site having an positive net charge that balances the positively charge Mg atom. This is very different as compared to Al 13 Na that has a large dipole moment of 6.23 Debye and is ionic with a structure analogous to Al - 13 Na +. Thus the combination of the weakly polar Al-Mg bond and the fact that the cluster has an open electronic shell allowing it to screen the electric field results in the negligible electric dipole. 8

9 We now examine the hybridization between Na or Mg states and aluminum sites to further analyze the nature of bonding. In Figure 6 we show the average Mulliken population for the dopant elements within each cluster for both aluminum-sodium and aluminum-magnesium series. The 3s and 3p orbitals were the orbitals of interest for both Na and Mg. A small Mulliken occupation of the 3s orbital is indicative of the electrons being transferred to the aluminum site, because the 3s orbital contains the valence electrons for both Na and Mg. Figure 6 shows that the Na atoms have a much lower 3s occupation as compared to Mg atoms in clusters of the same size. For sodium, 3s occupation quickly falls with increasing cluster size, while the 3p occupation of Na is negligible. The occupation in the magnesium 3s orbital also decreases while the 3p occupation increases. We observe that while magnesium has two valence electrons, its 3s occupation is reduced by less charge than the sodium. This is consistent with the electric dipole and Hirshfeld charge analysis that also showed greater charge transfer from Na than from Mg. We also note that the occupation of the 3p orbital in Na, Mg and Al is an important marker of metallic bonding. Constructing delocalized orbitals out of only 3s orbitals becomes difficult as the number of electrons in a cluster increases, therefore an increase in the occupation of the 3p orbital is a marker of metallic behavior. Previous studies on pure Mg n clusters have shown that while Mg atom has a filled 3s 2 shell with an empty 3p shell, as the cluster size increases, the p- states become occupied leading to a metallic transition. 43,44,69 We notice a similar progression here where the Mg 3p orbitals are occupied indicative of a metallic transition. In fact despite the very high energy of the atomic 3p orbital, the occupation of the 3p orbital of Mg is approximately the same as the 3s occupation of the Na atoms. This suggests that the 3p occupation may be an alternative tool for quickly checking the nature of bonding in alkali or alkaline earth atoms. 9

10 Another indicator of the ionic or metallic bonding is the change in binding energy of the Na or Mg as an extra electron is added. Ionic bonds are stabilized first by an electron transfer where the cation donates charge to an anion, and then stabilized by a Coulomb interaction between the two charges. If an extra electron is added, it is expected that the bond formed between the anion and cation will be weaker since the LUMO in ionic materials is typically located on the counterion, and the extra electron would result in less charge transfer and less Coulombic stabilization. Consequently, we calculated the change in removal energies between the anion and cation of the same type, i.e. E R.E. = [dopant removal energy - Anion] [dopant removal energy - Neutral] If this quantity is negative, the bond between the dopant (magnesium or sodium) is expected to be stronger in the neutral species than in the anion species, which suggests an ionic bond, while a positive value indicates that the bonding is stronger in the anion than the neutral suggesting the bonding is not ionic. Figure 7 shows this quantity for the sodium and magnesium series. For the sodium series, this quantity was always negative, with a few exceptions that were slightly positive, but always below 0.2 ev. Conversely, the magnesium series did not have a clear trend, with roughly the same number of positive and negative changes in the removal energies reiterating that the bonding of Mg is less ionic than that of Na. One approach to ascertain charge transfer is to compare the electronic structure of the cluster to the electronic structure of the aluminum motif with an extra charge equal to the number of valence electrons transferred from the cation. This is analogous to the Zintl concept where the aluminum cluster may be thought of as a multiply charged polyatomic anion. 79,80 If the electrons were transferred to the aluminum motif, one would expect a similar pattern of cluster orbitals in the Al n Na m and Al -m n cluster for sodium and Al n Mg m and Al -2m n for Mg. One also might expect 10

11 that the atomic structure of the negatively charged aluminum motif would be similar to the bimetallic cluster. We now examine this concept for the case of Al 12 Na -. First, we plot the electronic structure of the optimized geometry of the bimetallic cluster in Figure 8. Next, we plot the electronic structure of the optimized Al 2-12 cluster where the sodium atom is replaced with an extra electron. We then compare the electronic structures with that obtained in a single point calculation with the unoptimized Al -2 12, where the sodium atom is removed and replaced with an extra electron. We have labelled the electronic orbitals by their orbital character based on the confined nearly free electron gas model. 16 If the sodium binds ionically, one would expect the three electronic structures to be similar. Figure 8 shows that the three clusters indeed have very similar electronic structure. This indicates that the Al 12 Na - -2 cluster is isoelectronic with the Al 12 cluster, implying that the principal role of the sodium atom is to donate an electron. This result is fully consistent with the argument that the bonding between Na and Al in the NaAl - 12 cluster is ionic. We next used the Zintl concept to analyze the bonding in the Al 12 Na - 3 cluster. Again, the three plots, shown in Figure 9, represent the electronic structures of the optimized Al 12 Na - 3, the optimized Al -4 12, and the single point calculation of Al where the Al 12 structure comes from the optimized Al 12 Na - 3 cluster, with the sodium atoms removed. Figure 9 shows that the electronic structure of the three clusters are very similar. It is possible that the sodium counterions cause crystal field splitting within the orbitals of the aluminum cluster, however despite that, the order of the orbital character is consistent among the three clusters. The optimized Al cluster has more degeneracy in its F and D shells than the single point calculation and in the Al 12 Na - 3 because it is more symmetrical. We find that while Al 12 Na - is 11

12 isoelectronic with Al 12-2, Al 12 Na 3 - is isoelectronic with Al These results suggest that the main contribution of the sodium atoms in the electronic structure is the donation of the electrons. We now try to analyze the bonding in Al 11 Mg 3- using the Zintl hypothesis. Mg is divalent and hence the corresponding cluster with electrons transferred is Al Intriguingly, this is similar to the In cluster that is a stable Zintl compound. 81 The three plots are shown in Figure 10 that diagrams the electronic structures of the optimized Al 11 Mg cluster, the optimized Al 11 cluster, and the single point calculation of Al -7 11, where the Al 11 structure comes from the Al 11 Mg - 3 cluster. Note that the electronic structures of the three clusters show rearrangements in the electronic states. In the single point calculation for Al -7 11, the crystal field splitting is so extreme that one of the 1F orbitals becomes unoccupied while the 3S orbital becomes filled. - Furthermore, the HOMO-LUMO gap of this cluster is now only 0.08 ev, while the Al 11 Mg 3 cluster has a closed electronic shell with a HOMO-LUMO gap of 1.48 ev. These plots demonstrate that the Al 11 Mg - 3 cluster has a significantly different electronic structure than the Al clusters. This shows that the contribution of the magnesium does not stop at its electrons, differentiating it from the sodium doped clusters. The origin of this important difference between the magnesium doped clusters and the sodium doped clusters is that the magnesium may sometimes bond as it is embedded into the aluminum motif, while the sodium typically attaches to the outside and donates charge. The large difference in the electronic plots for the magnesium -7 doped cluster between the optimized Al 11 and single point calculation Al is due to the magnesium being needed to form the approximately spherical structure of the cluster which allows for the cluster to have a closed electronic shell. Our final approach for analyzing the bonding is to examine the Laplacian at the bond critical points. 49,50,82 A positive value for the Laplacian at the bond critical point is a signature of 12

13 charge depletion, while a negative value is a marker of charger accumulation. Conventional wisdom is that a negative value is a marker of a covalent bond, while a positive value is the marker of an ionic bond. This analysis has been used by Xing et al. to argue that the bonding in AlMg is primarily ionic. 13 For comparison, we have evaluated the Laplacian of the charge density at the bond critical points of several representative clusters in Table 1. We see that the Al-Al bonding in Al - 13 is positive with a value of which should be a marker of ionic bonding. The Al-Al bond is nonpolar due to symmetry, so the fact that this analysis indicates the bond is ionic suggests that this is not an effective method of analyzing the bonding. We do find that the Al-X bonds with both Na and Mg are positive, and that the values are more positive than in the case of Na than Mg. Furthermore, our previous analysis has found that the bonding between Al and Mg is more metallic, while the bonding between Al and Na is best described as ionic. Thus we conclude that the Laplacian of the charge density may not be an effective method for identifying the nature of bonding in small metal clusters. Conclusions The present studies indicate that sodium and magnesium bond differently in bimetallic clusters with aluminum. By critically analyzing the bonding within the two sets of clusters we find that the bonding between Na sites are marked with low Mulliken populations of the 3s and 3p orbitals, high dipole moments, and negative changes in removal energy between the neutral and anionic species all suggesting that the bonding between Na and Al n motifs is largely ionic. The nature of ionic bonding is further reinforced by the fact that the electronic structure of Al n Na m clusters are isoelectronic with the corresponding Zintl Al n -m clusters. In contrast to the Al n Na m 13

14 clusters, the Al n Mg m clusters have a higher 3s and 3p Mulliken population, lower dipole moments, and no clear trend in the change in removal energy between neutral and anionic species. Coupled with the observation that the electronic structure of clusters containing magnesium atoms changes substantially when Mg sites are replaced by extra valence electrons suggests that the bonding of Mg with Al n clusters is best described as metallic. Our studies also show that a method to quickly and accurately distinguish between ionic and metallic bonding within a cluster is to look at the hybridization of the 3s and especially the 3p orbitals since a relatively large 3p occupation is a marker of metallic bonding. The Hirshfeld charge analysis is effective, but the calculated atomic charges are much less than electron precise charge transfer that is often assumed in ionic bonding, so a reference charge transfer is necessary for a thorough analysis. The electric dipole moment is effective in singly doped clusters, but can sometimes lead to ambiguous conclusions in multiply doped clusters. The change in removal energy between a neutral and anionic species and the change in electronic structure in the absence of the cation are although good markers, they require extra calculations and analysis. Lastly, we found that evaluating the Laplacian of the charge density at the bond critical point can be misleading for evaluating the nature of bonding in small metal clusters. We would like to add that there is some indication that the ionic bonding can help in stabilizing the cluster materials as the combination of counter ions can act as structure seekers. In this regards, it is interesting to note that it has indeed been possible to stabilize cluster assembled materials containing aluminum clusters and K atoms

15 Figure 1. Structures of neutral Al n Na m clusters, m=1 3, n+m

16 Figure 2. Structures of neutral Al n Mg m clusters, m=1 3, n+m 15. Figure 3. Average difference in distance from center of mass between Na and Mg and Al in Al n Na m +/0 and Al n Mg m +/0. 16

17 Figure 4. Average Hirshfeld Charge of Al n Na m, and Al n Mg m, and Electric Dipole of Al n Na, and Al n Mg. Figure 5. Hirshfeld Charges on Al 13 Mg. 17

18 Figure 6. 3s and 3p Mulliken population of Na and Mg in Al n Na m and Al n Mg m clusters. Figure 7. Change in Na and Mg removal energy in Al n Na m and Al n Mg m versus Al n Na m and Al n Mg m. 18

19 Figure 8. Comparison of the electronic structure of Al 12 Na, Al 12 2 optimized and Al 12 2 in the structure of Al 12 Na. Figure 9. Electronic structure comparison of Al 12 Na 3 Al 12 4, and Al 12 4 in the structure of Al 12 Na 3. 19

20 Figure 10. Electronic structure comparison of Al 11 Mg 3, Al 11 7 and Al 11 7 in the structure of Al 11 Mg 3. 20

21 Al Al (Interior) Al Al (Exterior) Al X Al K 2 Al 10 Methyl Al 13 Na Al 12 Na Al 12 Mg Al 12 Mg Table 1. The average value of the Laplacian at the bond critical points for interior Al Al bond, Al Al exterior bonds, and Al Na and Al Mg bond in the indicated clusters. 21

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