Multiple Valence Superatoms

Multiple valence superatoms

J. U. Reveles*, S. N. Khanna*, P. J. Roach†, and A. W. Castleman, Jr.†‡

*Department of Physics, Virginia Commonwealth University, Richmond, VA 23284; and †Departments of Chemistry and Physics, Pennsylvania State University, University Park, PA 16802

Contributed by A. W. Castleman, Jr., October 6, 2006 (sent for review July 25, 2006)

We recently demonstrated that, in gas phase clusters containing by considering Al14 in a ϩ 2 valence state (3). The electronic shell aluminum and iodine atoms, an Al13 cluster behaves like a halogen structure, outlined above, does become modified (16) for com- atom, whereas an Al14 cluster exhibits properties analogous to an pound clusters as the combination of atoms with different alkaline earth atom. These observations, together with our find- electronegativities can rearrange their geometry and influence ؊ ings that Al13 is inert like a rare gas atom, have reinforced the idea the electronic shells. In particular, the shell closings at 18 and 20 that chosen clusters can exhibit chemical behaviors reminiscent of are sensitive to the nature of the compound cluster. atoms in the periodic table, offering the exciting prospect of a new Although the above developments bring out a close analogy dimension of the periodic table formed by cluster elements, called between atoms and superatoms, one of the most important superatoms. As the behavior of clusters can be controlled by size features characterizing atoms is that numerous elements in the and composition, the superatoms offer the potential to create periodic table exhibit multiple valence states. For example, unique compounds with tailored properties. In this article, we ,؊ carbon exhibits both divalent and tetravalent characteristics and provide evidence of an additional class of superatoms, namely Al7 , strongly binds with O or Si atoms forming CO and SiC, both of that exhibit multiple valences, like some of the elements in the which are highly stable. Are there superatoms that share this periodic table, and hence have the potential to form stable com- commonality to atoms? pounds when combined with other atoms. These findings support In this article we present the results of a synergistic effort that the contention that there should be no limitation in finding combines first-principles’ theoretical calculations and the reac- clusters, which mimic virtually all members of the periodic table. tive stability of selected clusters to demonstrate an additional Ϫ member of the superatom family, namely Al7 . What is truly 3d periodic table ͉ cluster ͉ jellium remarkable is that unlike previous members, this superatom exhibits multiple valence states, enabling it to form stable he formation of materials with properties different from compound clusters when combined with diverse atoms. In Tthose of the constituent atoms is a known phenomenon in particular, we first demonstrate (8) the exceptional stability of Ϫ Ϫ nature. For example, the formation of NaCl molecules/solid with Al7C through the production of AlnC clusters in reactions of characteristics different from its constituent elements, Na and aluminum clusters with benzene, and subsequent reactions of the Cl, is a classic example. One of the objectives of the research on clusters with oxygen to identify the stable species. The mass Ϫ

superatoms (1–3) is to explore if one can carry out, what nature spectra show the Al7C peak to be even more pronounced than CHEMISTRY Ϫ does, in a more facile and controlled manner. Developing an Al13 , another member of the superatom family, which we have understanding of the factors governing the chemical behavior of previously identified, and studied extensively (1–3). Theoretical clusters (4–10) and demonstrating that this knowledge can be considerations indicate that a superatom concept enables one to used to design stable building blocks for new materials is critical understand the electronic origin of the exceptional stability often for translating this concept into practice. For metal clusters, a observed for these species. The same considerations also predict Ϫ Ϫ Ϫ ϩ simple electronic shell model called jellium (11) is routinely used stability of other species such as Al7O ,Al7S ,Al7I2 , and Al7 . Ϫ Ϫ Ϫ to describe the global features of the electronic structure. In this Unlike Al7C , however, the stability of Al7O ,Al7S , and Ϫ model, the nuclei together with the innermost electrons form a Al7I2 cannot be ascertained by O2 etching experiments. Ϫ Ϫ positive-charged background, whereupon the valence electrons The Al7O and Al7S clusters are not stable toward the forma- Ϫ Ϫ coming from individual atoms are then subjected to this poten- tion of AlO2 or Al2O2;Al7I2 is unstable toward the formation tial. For pure metal clusters, within a spherical jellium back- of Al2O2. Consequently, their stability has to be verified by a ground, this approach results in a shell structure where the reactant less harsh than O2. Experimental mass spectra, how- electrons are arranged in electronic shells 1s2,1p6,1d10,2s2,1f14, ϩ ever, do show the stability of Al7 in oxygen-etching experi- 6 2 2 6 2 6 2 10 2p ...compared with 1s ,2s,2p,3s,3p,4s,3d ...in ments. The present investigations provide further extensive individual atoms. Similar shell structure is also obtained for support to the general nature of the superatom concept and show square well and harmonic forms of background potential (12), Ϫ Al7 to be a member of the multivalent superatom family. indicating that the shells derived within a jellium picture represent generic features of electronic states in a confined free Results and Discussion electron gas. Clusters containing 2, 8, 18, 20, 34, 40...electrons In brief, the clusters are produced by vaporizing a rotating and correspond to filled electronic shells and exhibit enhanced translating aluminum rod in a helium atmosphere. The produc- stability as seen via abundances in mass spectra of simple metal tion of mixed clusters is then accomplished by mixing the clusters, higher ionization potential, lower electron affinity, and ablation species with a small percentage of the carrier gas chemical inertness seen in reactivity experiments. In this respect, Ϫ containing the desired precursor. For example, AlnC clusters an Al13 cluster with 39 valence electrons and an electronic structure of 1s2,1p6,1d10,2s2,1f14,2p5 lacks a single electron as do halogen atoms, which, upon addition of a single electron, Author contributions: J.U.R., S.N.K., P.J.R., and A.W.C. designed research, performed acquire a filled shell status (13). Indeed, previous studies (14, 15) research, contributed new reagents/analytic tools, analyzed data, and wrote the paper. have shown that Al13 has an electron affinity comparable to The authors declare no conﬂict of interest. halogen atoms, indicating a chemical behavior reminiscent of Abbreviations: HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied halogen atoms. In a similar vein, we had recently shown that in molecular orbital; BE, binding energy; NRLMOL, Naval Research Laboratory Molecular Orbital Library. cluster compounds with iodine, an Al14 cluster exhibits behavior analogous to alkaline earth atoms (3). We had shown that ‡To whom correspondence should be addressed. E-mail: [email protected]. Ϫ Al14I3 is a stable species and that its stability can be reconciled © 2006 by The National Academy of Sciences of the USA

www.pnas.org͞cgi͞doi͞10.1073͞pnas.0608781103 PNAS ͉ December 5, 2006 ͉ vol. 103 ͉ no. 49 ͉ 18405–18410 Downloaded by guest on October 1, 2021 The reacted species are mass-analyzed, thus spectra contain the stable species generated in the original mass distribution and the stable products generated in the oxidation and etching reactions. In this way, the peaks in the mass spectra of the reacted species identify the stable species. Ϫ Fig. 1 shows the mass spectra of AlnC clusters obtained by reacting the aluminum plasma with benzene and subsequently Ϫ exposing the generated AlnC clusters to Ͼ200 standard cubic Ϫ centimeters per minute of oxygen. Fig. 1 shows that Al7C is the only cluster containing one C atom (at small sizes) that survives Ϫ in the mass spectra. The peak at Al7C is conspicuously large and Ϫ even more prominent than Al13 , known to be resistant to oxygen because of its closed shell structure. It was suggested Ϫ earlier (17) that the stability of Al7C could be reconciled within a jellium model framework where the cluster could be looked upon as a compound jellium formed out of an Al6, with a closed shell of 18 electrons and AlCϪ with a closed shell of 8 electrons. Such a picture does have difficulties. The electron affinity of AlC (1.1 eV) is less than that of Al6 (2.5 eV). Our calculations show that, whereas it takes only 3.76 eV to remove an Al atom from Ϫ Ϫ Al7C , it will take 9.09 eV to break it into Al6 and AlC . The Ϫ C atom in Al7C is located inside the Al7 cage bonded to all of the Al sites, whereas one would expect it to bind to a single Al atom if it were a combination of Al and AlCϪ. Fig. 1. Benzene is introduced in the source aluminum plasma to produce 6 Ϫ F Ϫ We begin by demonstrating that the true reason for the Aln ( ) and AlnC ( ). The clusters are subsequently reacted with molecular Ϫ Ϫ stability of Al7C lies in the multiplet nature of Al7 . Fig. 2 oxygen at thermal energy. Ϫ shows the geometry deduced for Al7C . Note that the cluster features an endohedral C atom as obtained in earlier studies. To ⌬ are produced by adding benzene. To identify the stable species, examine the stability, we calculated the gain in energy, E, the generated clusters are subsequently exposed to increasing Ϫ Ϫ ⌬E ϭ E͑Al Ϫ C ) ϩ E͑Al͒ Ϫ E͑Al C ͒ [1] amounts of etching gas introduced into the flow reactor in which n 1 n the clusters are transported. In this way, the reactive anions are as a function of size as successive Al atoms are added to the Ϫ oxidized and often etched into smaller more stable fragments. preceding size AlnϪ1C . Here E is the total energy of the cluster.

Ϫ Ϫ Fig. 2. Structure and energetics of aluminum compound clusters. (a)Al7C -optimized geometry. (b) Energy gained by adding an Al atom to Aln-1C species Ϫ Ϫ Ϫ and HOMO–LUMO gap for the AlnC clusters. (c) Electron charge density of the HOMO in Al7C clusters. (d)Al7O -optimized geometry. (e) Energy gained by Ϫ Ϫ Ϫ adding an Al atom to Aln-1O species and HOMO–LUMO gap for the AlnO clusters. (f) Electron charge density of the HOMO in Al7O .

18406 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0608781103 Reveles et al. Downloaded by guest on October 1, 2021 Ϫ Ϫ Ϫ Fig. 3. One-electron levels in Al7 ,C,Al7C , O, and Al7O . The continuous thin lines represent single occupied levels, the continuous thicker lines represent CHEMISTRY double occupied levels, the dotted thin lines correspond to single unﬁlled states, the dotted thicker lines correspond to double unﬁlled states, and the arrows indicate the majority (up) and minority (down) spin states.

The results are shown in Fig. 2b. A peak in ⌬E is a signature of of two Al atoms requires 6.78 eV of energy. The corresponding stability as it implies a larger gain in energy in forming the stable studies based on the Naval Research Laboratory Molecular species from the preceding size and lower energy in breaking the Orbital Library (NRLMOL) show that the energy required to higher size to form the stable species. Note that there is a peak remove two Al atoms is 7.04 eV, whereas the BE of Al2O2 toward Ϫ Ϫ at Al7C indicating its preferred formation from the growth of breaking into two Al atoms and an O2 is 6.92 eV. Al7C is Ϫ Ϫ Al6C and from the fragmentation of Al8C . Fig. 2b also shows therefore stable toward these fragmentation pathways and hence the gap between the highest occupied molecular orbital is not etched by O2. The same considerations also explain the ϩ ϩ (HOMO) and the lowest unoccupied molecular orbital observation of Al7 in etching of the pure Aln cations by (LUMO). In metallic clusters, the gap is generally indicative of oxygen. In this case, whereas it takes 4.32 eV to remove an Al Ϫ Ϫ stability and chemical inertness. Note that Al7C has the highest atom, it takes 7.22 eV to remove two Al atoms. Why is Al7C so Ϫ HOMO–LUMO gap. stable? Can Al7 combine with other atoms to create stable Ϫ We now discuss the stability of Al7C and other aluminum- species? based clusters toward etching by O2 more quantitatively. The To answer the above questions, we outline the mechanism by Ϫ possible fragmentation pathways involve the formation of AlO2, which the Al7 cluster combines with other atoms to form stable Ϫ AlO2 ,orAl2O2. Note that because the clusters are first compounds. An Al atom has a valence electronic configuration 2 1 thermalized, then exposed to molecular O2, the OOO bond is of 3s 3p , and it has been proposed that Aln clusters undergo a protected by a barrier and remains intact in the formation of transition (18) from monovalent to trivalent starting around n ϭ Ϫ AlO2,AlO2 ,orAl2O2. Our calculations show that the binding 6. The trivalent character in Al7 is further established (19) by the Ϫ ϩ energy (BE) of AlO2 and AlO2 with OOO bond intact is 2.83 mass spectra of the reacted Aln cations that exhibit a sharp peak ϩ and 5.72, eV respectively, whereas the energy required for at Al7 , indicative of a shell closing at 20 electrons within a breaking an Al2O2 into two Al atoms and an O2 molecule is 6.75 jellium picture of the confined free electron gas. If instead, it was eV. These calculations show that those aluminum-based clusters, monovalent in character, it would display prominence as an Ϫ where it takes (i) Ͻ2.83 eV to remove a single Al, (ii) Ͻ5.72 eV eight-electron system, which it does not (13). An Al7 ,onthe to remove an AlϪ,or(iii) Ͻ6.75 eV to remove two Al atoms, will other hand, has 22 valence electrons. Within the jellium picture, 2 6 10 2 be energetically unstable toward etching by O2. In this way, the the cluster has an electronic configuration of 1s ,1p,1d ,2s, stability of clusters can be tested, as less stable clusters are 1f2, and Fig. 3 shows the one-electron levels in the free cluster. oxidized and often etched into smaller more stable fragments. In The molecular orbitals have been labeled by using the jellium Ϫ the case of Al7C , the removal of an Al atom requires 3.76 eV, classification (the levels roughly correspond to the overall shape and the removal of an AlϪ requires 6.40 eV, whereas the removal of the molecular orbitals). A carbon atom has a valence structure

Reveles et al. PNAS ͉ December 5, 2006 ͉ vol. 103 ͉ no. 49 ͉ 18407 Downloaded by guest on October 1, 2021 of 2s2,2p2 with four vacancies in the p-shell, and Fig. 3 shows the Ϫ one-electron levels in carbon. To discuss the formation of Al7C , Ϫ let us start with the one-electron levels in Al7C also shown in Fig. 3. The 2s filled state of carbon is far below the manifold of Ϫ Ϫ Al7 and becomes the 1s state of Al7C . Consequently, the 1s Ϫ and 2s states of pure Al7 become the 2s and 3s states of the combined system and are pushed up in energies (the resulting 3s is high in energy and is not shown). The partially filled p-states Ϫ of carbon are around the same energy as 1p states of Al7 and form the bonding and antibonding combinations. The bonding Ϫ combination leads to the 1p state of Al7C , whereas the antibonding combination leads to the 2p state that characterizes the Ϫ group of highest occupied molecular states in Al7C . Because a carbon atom has four unfilled p-states, the occupation of the p-states can be looked on as the transition of four electrons from Ϫ the 1f and the 2s states of Al7 to these unfilled states. The 1f unfilled states are pushed up in energy, and Fig. 3 shows the Ϫ revised level structure in Al7C . Note that the p-states of C form Ϫ bonding orbitals with Al7 states; hence it is not a charge transfer but a population of these bonding states. A better description is Ϫ the transition of Al7 toward a ϩ 4 valence state, which is similar to the transition of Al14 toward a ϩ 2 state reported previously (3). Obviously such a transition cannot be identified through Mulliken or other charge analysis, as recently undertaken by some authors (20). Fig. 2c shows the electron charge density in the HOMO orbital. As the bonding states are deep in energy, the occupation of bonding states stabilizes the cluster whereas the movement of unoccupied states opens a large HOMO–LUMO gap of 1.69 eV consistent with the stability and inertness of Ϫ Al7C . A similar model would also account for the previously known (21, 22) enhanced stability of neutral Al7N because Al7 has 21 electrons, and the N states can be filled by occupation of three bonding orbitals. The key issue is whether such simple electronic counting rules and shell filling can be used to describe the stability of other members of the second row of the periodic table. Because the confined free electron gas exhibits shell closing at 18 and 20 Ϫ electrons, the above arguments would suggest that Al7M clusters should exhibit enhanced stability for cases where the M atom would require two or four electrons to fill the deep p-bonding Ϫ orbitals. This criterion would indicate that Al7O should also be quite stable, and Fig. 3 shows the levels in this cluster and an O atom (2s state of oxygen is deep in energy and is not shown). The Ϫ bonding then proceeds as for Al7C , with the only difference Ϫ being that only 1f electrons of Al7 are transferred to bonding Ϫ states between Al7 and O. To examine quantitatively the Ϫ stability, we investigated the AlnO clusters containing five to eight Al atoms and computed ⌬E and the HOMO–LUMO gap Ϫ (shown in Fig. 2 d and e). Note that ⌬E peaks at Al7O , indicating enhanced stability. Fig. 2f also shows the HOMO Ϫ orbital. Although Al7O is stable, it is not resistant to etching by O2 because it takes 6.10 eV to remove two Al atoms. It is now interesting to inquire as to whether the above considerations could be extended to other elements. In partic- Ϫ ular, could one use the above model to understand the bonding Fig. 4. Energetics of the Al7M clusters. (a)BEofAl7M clusters, M is an atom of Al Ϫ to other elements in the second, third, and fourth row of of the second, third, and fourth row of the periodic table. (b) HOMO–LUMO 7 Ϫ Ϫ the periodic table. To this end, we have calculated the BE of all gap for the Al7M clusters. (c) BE and HOMO–LUMO gap of the Al7Im clusters. Ϫ of the atoms in the second, third, and fourth rows with Al7 using the equation Ϫ until the forces dropped below a threshold value of 3 ϫ 10 4 a.u. ϭ ͑ Ϫ͒ ϩ ͑ ͒ Ϫ ͑ Ϫ͒ Several spin multiplicities were tried to locate the ground state. BE E Al E M E Al7M . [2] 7 Three different types of geometries were obtained. For B, C, and Ϫ Ϫ Here E(Al7 ), E(M), and E(Al7M ) are the total energies of the N we obtain an endohedral structure (Fig. 2a). For Li, Be, O, F, Ϫ Ϫ Al7 ,M,andAl7M species, respectively. For cases where M is Na, Mg, Al, P, S, Cl, Cu, Zn, Ga, Ge, As, Se, and Br we obtain a halogen atom, the BE was calculated by breaking into Al7 and an external geometry (Fig. 2d), and for Si we obtained a capped Ϫ M fragments. In these calculations, the atoms were initially geometry where the Al7 opens up to accommodate the addi- located at different positions around the Al7 motif. The geom- tional atom. etry was optimized by moving atoms in the direction of forces In Fig. 4a, we show the variation of BE for all of the anionic

18408 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0608781103 Reveles et al. Downloaded by guest on October 1, 2021 clusters. In each case, we have marked the total number of Ϫ Ϫ valence electrons. Note that Al7C and Al7O are not only stable with respect to addition of Al atoms, as discussed earlier, they also have the highest BE across the entire second row of the periodic table. What is quite amazing is that the same trend continues for the anions of the third-row and fourth-row elements. S in the third row and Se in the fourth row exhibit larger gain in BE than previous sizes, again indicative of their stability. In addition to the BE, we examined the variation of the HOMO–LUMO gap (Fig. 4b). Significantly, the HOMO– LUMO gap follows the stability trend, indicating a chemical inertness. For Si and Ge that correspond to a shell filling at 18, the BE only exhibits a shoulder whereas the HOMO–LUMO gap exhibit minor peaks. This behavior could be caused by the highly directional nature of the bonds from these atoms. To further confirm this extensive nature of the multivalence concept and Ϫ reiterate the theoretical picture, AlnS clusters were generated by reacting Aln clusters with SO2. The mass spectra of the Ϫ resulting clusters contained Al7S clusters, including other sulfur-containing clusters as well. When the clusters were ex- Ϫ posed to O2, however, all of the clusters, including Al7S , were Ϫ Ϫ Ϫ etched away. The etching of Al7S is understandable, noting that Fig. 5. Mass spectra of Aln clusters (A) reacted with I2 (B). the energy required to remove two Al atoms is only 5.78 eV whereas energy gain in forming Al2O2 is 6.75 eV. Further, it will Ϫ take 5.73 eV to remove an Al atom, making the formation for Al2O2 a resonant reaction. Corresponding studies based on the Ϫ AlO2 a resonant reaction. The corresponding studies based on NRLMOL indicate the formation of Al2O2 to be exothermic by the NRLMOL, however, indicate that it will take 5.68 eV to 0.03 eV. remove an AlϪ atom, whereas the gain in energy in forming To summarize, we have demonstrated that bonding patterns of Ϫ Ϫ AlO2 is 6.07 eV, opening this channel for fragmentation. In Al7 can be understood by regarding it as a superatom that Ϫ either case, Al7S is unstable toward oxidation. We further exhibits multiple valence. The valence of 2 and 4 would make it Ϫ Ϫ calculated the electron affinity of the Al7M clusters for M analogous to C or Si. The stability of Al7C on the one hand and Ϫ Ϫ Ϫ covering all of the third-row and fourth-row atoms. Al7S and that predicted for Al7O and Al7S on the other hand could then Ϫ Al7Se had the highest electron affinity, attesting to their closed be correlated with the stability of SiC and CO, respectively. In electronic shell character. the past we have shown that Al13 and Al14 could be regarded as

A critical test of the multiple valence would be to combine superhalogen and alkaline earth superatoms. Importantly, un- CHEMISTRY Ϫ Ϫ Ϫ Al7 with atoms that have only one vacancy in the p-level (3) and like Al13 and Al14,Al7 exhibits multiple valence, thus adding a high electron affinity. The formation of the bonding states an additional dimension to the chemistry of superatoms. would then remove one electron at a time whereupon the effect of the 2f and 2s depopulation could be mapped out with the Materials/Methods number of atoms. To carry out this important test, we considered Experimental Methods. Ϫ Ϫ Al7C species were created by ablating an anionic Al7Im clusters. An I atom has an electron affinity of aluminum rod in a flow of 8,000 standard cubic centimeters per 3.06 eV that is considerably higher than the measured electron minute of high-purity helium. Benzene vapor was added restric- affinity (15) of 2.43 for Al . One would therefore expect each Ϫ 7 tively to the source to maximize the population of AlnC species. additional I atom to combine with one electron from the Al7 The clusters and helium carrier gas merged into a laminar flow core. In Fig. 4c we show the variation of the BE and HOMO– regime where they became thermalized. The clusters were then LUMO gap as successive I atoms are added to the cluster. Note Ϫ reacted with molecular oxygen, and products were extracted, that Al7I2 leading to a core of 20 electrons is particularly mass-analyzed by a quadrupole mass filter, and detected by a prominent compared with other sizes, again verifying the shell Ϫ channel electron multiplier. The Al7I2 species were created by closures discussed above. More importantly, the corresponding ablating an aluminum rod in a flow of 8,000 standard cubic HOMO–LUMO gap shows the same trend. The chemical fea- centimeters per minute of high-purity helium. Molecular iodine tures together with the BE again point to the valence status. To Ϫ was added externally to the source by flowing a small amount of examine whether the stability of Al7I2 is borne out by experi- helium through a sublimation cell containing molecular iodine. ments, the aluminum clusters anions were reacted with iodine gas. Fig. 5 shows the mass spectra of the bare and reacted species. Theoretical Methods. The theoretical studies were carried out by Note that the mass spectra indeed exhibits a peak at Al I Ϫ, 7 2 using first-principles electronic structure calculations performed confirming the theoretical prediction. We note that the varia- within a density functional formalism (24) that incorporates tions in BE and HOMO–LUMO gap in Fig. 4c indicate Al I Ϫ 7 4 exchange and correlation using the generalized gradient approx- to be more stable than neighboring sizes, consistent with a shell imation functional proposed by Perdew, Burke, and Ernzerhof closure at 18 electrons; however, it is not seen in Fig. 5. The (25). In particular, Gaussian basis sets were used to construct the reason is that it is energetically unstable toward the reaction atomic wave function, whereas the cluster wave function was Ϫ ϩ 3 Ϫ ϩ formed from a linear combination of atomic orbitals. All cal- Al7I4 I2 Al6I4 AlI2 [3] culations were performed with deMon2k software (26). Here, an Ϫ Ϫ by 0.31 eV. Although Al7I and Al7I3 are less stable than auxiliary basis set was used for the variational fitting of the Ϫ Al7I4 , as discussed in a previous paper (23), their observation Coulomb potential (27, 28). The numerical integration of the is linked to the enhanced stability of the corresponding neutral exchange-correlation energy and potential were performed on Ϫ species. Al7I2 is, however, unstable toward etching by O2 as it an adaptive grid (29). The minimum structures were fully takes 6.76 eV to remove two Al atoms, making the formation of optimized in delocalized internal coordinates without con-

Reveles et al. PNAS ͉ December 5, 2006 ͉ vol. 103 ͉ no. 49 ͉ 18409 Downloaded by guest on October 1, 2021 straints using the rational function optimization method and the This work was supported by U.S. Air Force Office of Scientific Research Grant F49620-01-1-0328 (to A.W.C. and P.J.R.) for the Broyden, Fletcher, Goldfarb, and Shanno update (30). The Ϫ Ϫ double zeta valence polarized and the GEN-A2 auxiliary basis experimental work on Al7C and Al7In , U.S. Air Force Office of sets were used (31). To eliminate any uncertainty from the basis Scientific Research Grant FA9550-05-1-0186 (to J.U.R. and S.N.K.) for the theoretical work on these systems, and U.S. Department of set or the numerical procedure, in selected cases we carried out Energy Grant DE-FG02-02ER46009 for the systematic studies on Ϫ supplementary calculations by using the NRLMOL developed Al7X clusters for X covering the second, third, and fourth rows of the by Pederson and coworkers (32–34) using the same density periodic table. S.N.K. thanks Virginia Commonwealth University for functional. For details, see refs. 3 and 32–34. providing a study/research leave.

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