Effect of Electronic and Geometric Shell Closures on the Stability of

May 28, 2010 - Theoretical studies on the electronic structure and stability of neutral and anionic Ti-doped sodium clusters have been carried out to ...
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J. Phys. Chem. C 2010, 114, 10739–10744

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Effect of Electronic and Geometric Shell Closures on the Stability of Neutral and Anionic TiNan (n ) 1-13) Clusters J. Ulises Reveles,*,† Prasenjit Sen,‡ Kalpataru Pradhan,†,‡ Debesh R. Roy,† and Shiv N. Khanna*,† Department of Physics, Virginia Commonwealth UniVersity, 701 West Grace Street, Richmond, Virginia 23284-2000, and Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India ReceiVed: February 26, 2010; ReVised Manuscript ReceiVed: May 14, 2010

Theoretical studies on the electronic structure and stability of neutral and anionic Ti-doped sodium clusters have been carried out to examine the role of electronic and geometric shells on the physical and chemical behavior. Our studies delineate the effect of the filling of atomic electronic states, electronic shells in a twoand three-dimensional confined electronic gas, and the completion of geometrical shells on the energetic stability and chemical reactivity. In particular, a TiNa9 cluster is shown to be marked by a filled 1S2 1P6 super shell, and a half-filled atomic electronic d-level on the Ti atom, with the composite closure leading to a magnetic superatom. Introduction A promising direction in the field of clusters and nanostructures is the possibility of synthesizing materials where clusters serve as the elementary building blocks. As the properties of clusters change with size, composition, shape, and the charged states, cluster assemblies offer the unique prospect of making materials with tunable characteristics.1,2 Identifying stable cluster motifs and understanding the features controlling their stability is the first step toward such an objective. Consequently, there has been great interest in identifying factors that govern the stability of clusters. Early on, experiments on sodium clusters in beams led to the observation of stable clusters (magic numbers) containing 2, 8, 18, 20, 34, 40, ... atoms.3 The stability of these clusters was attributed to electronic shells resulting from bunching of quantum states in a nearly free electron gas confined to a spherical region.4 The magic numbers then correspond to clusters with filled shells. These findings highlighted the importance of electronic contributions to stability. Further research showed that the shape of the confining potential can lead to different grouping of shells, and more recently, filling of shells in a two-dimensional cluster of a transition metal (TM)doped Aun cluster has also been found to result in increased stability.5,6 As the clusters grow larger, the spacing between the electronic shells grows smaller and experiments on sodium clusters showed that in the regime of large sizes, clusters with complete geometric shells were more stable.7 These experiments thus demonstrated the role of geometry on the stability. However, sodium clusters are marked by metallic bonding with no directionality, the geometric effects on enhanced stability thus show up only when there are a few thousand atoms in the cluster. On the other hand, clusters of Al atoms show more pronounced effects of geometry on their stability. Early experiments on the ionization potential of Al clusters concluded that all the observations cannot be explained by shell models alone, and that the structure plays an important role.8 More specifically, * To whom correspondence should be addressed. E-mail: snkhanna@ vcu.edu (S.N.K.) and [email protected] (J.U.R.). † Virginia Commonwealth University. ‡ Harish-Chandra Research Institute.

Al13- has been found to be an exceptionally stable cluster because of both a 40-electron filled shell and a filled geometric shell of an icosahedron.2 Combined, these observations led to the idea that it should be possible to design highly stable species by simultaneously attaining electronic and geometric shell closure. Since electronic and geometric shell closures occur through different sequences, it is difficult to find examples where both effects combine in a coherent fashion; Al13- is one of the rare examples. While clusters with filled electronic shells are marked by enhanced stability, departures from the filled status can result in chemical behaviors reminiscent of reactive atoms. It was almost 20 years ago that Khanna and co-workers showed that while Al13- with 40 electrons has filled electronic shells, an Al13 cluster had a high electron affinity9 and hence could be regarded as a superatom resembling a halogen atom. Since then, its superhalogen behavior has been confirmed both theoretically and experimentally.9 More recently, Al14 with two electrons more than a filled shell of 40 electrons has been shown to behave like an alkaline earth element.10 Na3O and K3O clusters, being one electron more than a filled shell structure, act as superalkalis,11,12 and Al7- with 22 valence electrons having two and four more electrons than the 18 and 20 closed shells acts as a germanium atom with multiple valences.13 In a recent work,14,15 the present authors demonstrated that an interesting situation arises when the confined electron gas has components that have localized as well as diffuse subspaces. Such a situation arises, for example, when alkali metal (AM) clusters are doped with a transition metal atom. The TM atoms have both diffuse s and localized d orbitals whereas the alkali atoms only contribute s-electrons. We showed that in Sc-, Ti-, and V-doped alkali clusters, the valence electronic spectrum has both localized 3d-like and delocalized 1S, 1P, 1D, 2S, ... shells.14,15 Moreover, there can be transfer of charge between the localized and the delocalized orbitals as the states are hybridized. This opens up an exciting possibility where stability can be enhanced by a filled free electron shell in the delocalized pool, while magnetism is supported by a partially filled localized set of 3d orbitals. In fact, we showed that both VCs8 and MnAu24(SH)18 exhibit special stability because of a filled 1S21P6

10.1021/jp101757n  2010 American Chemical Society Published on Web 05/28/2010

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shell structure and a half-filled 3d5 atomic shell, and behave as magnetic superatoms.15 While these findings are interesting, they could not delineate the stability arising from half or completely filled d-shell vis-a`-vis the stability from S and P supershells. This is because in the V-alkali clusters, a V atom goes to a d5 configuration with only two alkali neighbors. This raises the issue if it is possible to have a system where the stability from localized d-states, from superatom S, P, ... shells, and from the geometric shell closure could be separately identified. We show that TiNan clusters offer such a possibility. In this paper we present our findings on neutral and anionic TiNan clusters with n ) 1 to 13, to demonstrate distinct stabilities originating from the closure of two (2D)- and three (3D)-dimensional free electron shells, from half-filled atomic d orbitals, and from geometric shell closure. Indeed, TiNa9 is a unique stable cluster in the TiNan series in which half filling of localized atomic d orbitals stabilizes the cluster. The major results presented here are the following: (i) addition of successive Na atoms with 3s1 valence electrons to a Ti atom (3d2 4s2) first results in a stable TiNa5 due to shell filling in a twodimensional electrons gas, (ii) with addition of further Na atoms, a stable TiNa7 cluster results with a filled 1S21P6 shell, (iii) next, a stable TiNa9 cluster is obtained with a half-filled atomic 3d5 shell and a filled 1S21P6 shell, and (iv) finally, a stable TiNa12 cluster is obtained with a complete icosahedral geometric shell. Our studies delineate the role of delocalized and localized electronic states, and geometric effects on the stability in the same series. These results may lead to new insights into designing superatoms with novel properties. Theoretical Methods The theoretical studies were carried out within the framework of gradient corrected spin polarized density functional theory (DFT). To minimize uncertainties due to choice of basis sets or numerical procedure we studied the clusters using both a real space approach adapted to free clusters and a periodic cell approach where one calculates the electronic structure of a periodic system using plane waves. In the latter approach, the interactions between clusters in different cells can be minimized by choosing a large unit cell. For the real space approach, studies were carried out by expressing molecular orbitals as a linear combination of Gaussian functions centered at the atomic sites. The actual calculations employed the deMon2k code.16 Here, we used the PW86 generalized gradient approximation (GGA) functional,17 and the double ζ with valence polarization (DZVP) basis sets optimized for gradient corrected exchange-correlation functionals.18 The calculation of four-center electron repulsion integrals is avoided through the use of the variational fitting of the Coulomb potential.19,20 Further, the auxiliary density was expanded in primitive Hermite Gaussian functions by using the A2 auxiliary function set. The exchange-correlation potential was calculated by a numerical integration on an adaptive grid21 from the orbital density. To determine the geometry and spin multiplicity of the ground state, the configuration space was sampled by starting from several initial configurations and spin multiplicities and optimizing the geometry employing the quasiNewton Levenberg-Marquardt method. All structures were fully optimized in delocalized redundant coordinates without using symmetry constrains.22 The resulting ground states were further ascertained via a frequency analysis. The benchmarks on the accuracy of the theoretical calculations for 3d transition metal atoms have been reported in a previous study by Calaminici et al.18 In the case of Ti atom, we find a triplet ground state and an ionization potential of 7.04

Reveles et al. eV, in agreement with the experimental ground state multiplicity and a measured ionization energy of 6.83 eV.23 For the Na atom, our studies predict an ionization potential of 5.52 eV, an electron affinity of 0.59 eV, and a lowest excitation energy between 2S and the 2P states of 2.01 eV, in good agreement with the experimental values of 5.14,23 0.55,24 and 2.10 eV,25 respectively. As the present work focuses on magnetic ground states, one has to consider any spin contamination errors arising from the use of unrestricted DFT. To this end, spin contamination errors of the optimized geometries were checked by analyzing the expectation value of the total spin 〈S2〉 after spin projection. In most clusters, 〈S2〉 differed from s(s + 1) by less than 1%. Here s equals one-half times the number of unpaired electrons. The only exception was neutral TiNa12 in which the ground state with a magnetic moment of 2 µB presented a larger spin contamination with respect to its nearly degenerated state of 4 µB (energy difference ) 0.04 eV). Additional restricted open DFT calculations were conducted for this cluster that confirmed a ground state with a magnetic moment of 2 µB for TiNa12. To further eliminate any uncertainty arising from the choice of basis set or the numerical procedure, supplementary calculations were carried out for many of the clusters by using planewave basis sets within a supercell approach as implemented in the VASP code.26–28 Here, an energy cutoff of 500 eV was used, and the cluster was placed in a large cubic box of sides 20 Å in order to reduce its interaction with its images. The potential between the ion cores and the valence electrons was expressed in terms of projector augmented waves (PAW). The exchangecorrelation effects were treated with the PW9129 GGA functional. Brillouin zone integrations were carried out with use of only the Γ-point. Structures were relaxed using the conjugate gradient (CG) method for different fixed spin multiplicities, and without any symmetry constraints. The geometries and molecular orbital charge densities were plotted with the Schakal30 and Molekel31 software, respectively. Results and Discussion Our primary interest is in identifying clusters that are energetically more stable. To identify such TiNan species we examined the gain in energy (∆ENa) when successive Na atoms bind to a single Ti atom:

∆ENa(n) ) E(TiNan-1) + E(Na) - E(TiNan)

(1)

where the E values are the total energies of the respective atoms or clusters. A large energy gain ∆ENa in the formation of a cluster from a stoichiometry with one less Na atom, and a small gain in energy when one extra Na atom binds to the cluster attest for energetic stability, as more energy is gained in forming the cluster and less energy is gained in growing to the next size. In addition to energetic stability we also examined the gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital (HOMO-LUMO gap). While the present paper does not address the chemical activity, large HOMO-LUMO gaps can identify species that will be less chemically active. The calculated ∆ENa values are presented in Figure 1a while the geometries of the ground state structures are shown in Figure 2. ∆ENa for the TiNan series has distinct peaks at n ) 5, 7, 9, and 12. The HOMO-LUMO gap also shows peaks at these sizes, except at n ) 12 (Figure 1c). This indicates that clusters with 5, 7, 9, and 12 Na atoms are particularly stable. As we now show, the underlying mechanisms for their stability are very different.

Stability of Neutral and Anionic TiNan Clusters

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Figure 1. Energy gain (∆ENa) in the addition of Na atoms to neutral (a) and anionic (b) TiNan clusters. Variations of HOMO-LUMO gaps vs the number of Na atoms for the neutral (c) and anionic (d) TiNan series. Variation of the relative energy Erel between the ground state and the nearest higher-energy state of different magnetic moment in the neutral (e) and anionic (f) TiNan clusters.

First, the enhanced stability at n ) 5 is due to a filled shell configuration in the 2D electron gas. As shown in Figure 2, a TiNa5 cluster has a planar structure. For a free-electron system confined to a 2D region (circular, square, or triangular), 6 is found to be a shell filling number with an 1S2 1P4 like configuration.5,6 The Ti atom has an electronic configuration 3d24s2, while each Na contributes one s-electron. In a TiNa5 cluster there are, thus, 2 d and 7 s electrons. In forming the cluster, one of the s electrons is transferred to the empty majority d orbitals of the Ti atom, and the rest lead to a filled 1S2 1P4 electronic supershell. This can be inferred from the one-electron energy level diagram where the analysis of the global shape of the molecular orbitals (MO’s) of TiNa5 is shown in Figure 3. We have also shown the atomic levels in a Na and Ti atom to guide how the individual atomic states combine to form MO’s. It is found that one 1S and two 1P orbitals originating from the overlap of Na sp and Ti s states are occupied in each spin channel. In addition, three d orbitals in the majority spin channel on Ti are occupied. Although the one-electron energy levels

diagram was worked out earlier,14 the shell filling in 2D, and consequent stability of this cluster was not fully appreciated at that time. The enhanced stability at n ) 7 results from a filled shell configuration in a three-dimensional electron gas. Of the 9 s electrons in this cluster, 8 lead to a filled 1S21P6 shell. The remaining one is transferred to the Ti d orbitals for it to have a d3 configuration as seen in the energy level diagram and MO’s charge densities shown in Figure 4a. As both the super shell and the localized atomic 3d states coexist, different grouping of the energy levels can be found. Indeed, in TiNa7 the energy level of a 3d R-spin orbital is lower in energy than that of the highest occupied single p-like orbital of β-spin. Perhaps the most interesting aspect of the TiNan series is an enhanced stability at n ) 9. An analysis of the MO’s of this cluster reveals that all the five majority d-like orbitals on the Ti atom are occupied. In fact, in this series, n ) 9 is the smallest size at which the d shell on the Ti atom becomes half-filled for the first time. A half-filled d shell on the Ti atom, along with a

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Figure 2. Ground state geometries of neutral TiNan with n ) 1 to 13 clusters. The bond lengths are given in angstroms and superscripts indicate spin multiplicity. The Mulliken atomic spin charges for the symmetry inequivalent atoms are marked below them.

Figure 3. One-electron energy levels (in eV) and atomic/molecular orbital charge density isosurfaces in Na (a), TiNa5 (b), and Ti (c). The continuous lines represent occupied levels; the dotted lines correspond to unfilled states. The degeneracy is marked next to each level. The angular characters of the levels with upper-case letters for delocalized 1S, 1P, and 2S shells, and lower-case letters for localized 3d atomic orbitals are also given. The arrows indicate the majority (up) and minority (down) spin states.

filled 1S21P6 shell of delocalized electrons leads to a stable magnetic superatom. The above descriptions are also consistent with the variation of the HOMO-LUMO gap with increasing size shown in Figure 1c. The figure further shows that beyond n ) 9, the gap drops, and there is very little variation after that until n ) 13. This is because from n ) 10 onward, the minority d orbitals on the Ti atom start to fill up. To illustrate this Figure 5a presents the variation of magnetic moment (µ) vs the number of Na atoms as well as the nature of frontier orbitals in the TiNan series. Until TiNa9 the minority 3d spin orbitals are empty and there is a large exchange splitting between occupied majority and

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Figure 4. One-electron energy levels (in eV) and molecular orbital charge density isosurfaces in TiNa7 (a) and TiNa9 (b). See the caption of Figure 3.

unoccupied minority orbitals, resulting in a large HOMO-LUMO gap. However, once the minority spin orbitals start to fill up in TiNa10, the gap is between an occupied and an unoccupied orbital of minority 3d shell, which is small as long as the minority orbitals are not completely filled. Finally, the enhanced stability at n ) 12 is due to a geometric effect. A TiNa12 cluster has an icosahedral structure with the Ti atom at the center as shown in Figure 2. Our calculations show that any other structure at this size has higher energy. In fact, starting from n ) 11 until n ) 13, the ground state structures of TiNan have an icosahedral based motif. Compact icosahedral structure leads to significant stability of a TiNa12 cluster. As seen in Figure 1, this is the only case where a peak in ∆ENa is not accompanied by a peak in the HOMO-LUMO gap, indicating that the origin of this stability is not electronic. Having identified the causes leading to enhanced stability at n ) 5, 7, 9, and 12 in TiNan, we proceeded to check if our understanding of the electronic structure of these clusters is consistent with other physical properties. First, we analyze in detail the variation of the ground state magnetic moment of these clusters with size, as well as the character of the occupied frontier orbitals (Figure 5a). Over most of the size range studied here, the magnetic moment on the clusters is larger than that on an isolated Ti atom. Here we point out that the moments on the clusters at n ) 5, 7, and 9 are 3, 3, and 5 µB, respectively. These are consistent with the electronic structure denoted at each size in Figure 5a. A further confirmation of this comes from a Mulliken population analysis of these clusters. Structures and Mulliken spin charges on symmetry inequivalent atoms of the TiNan clusters are shown in Figure 2. The spin charges on the Ti atom in TiNa5 and TiNa7 are nearly 3, indicating a nearly d3 configuration. In TiNa9, however, the spin moment is only 3.57 µB. This is because the frontier orbitals in this cluster result from a larger overlap of Ti d and Na sp orbitals. Another measure of the spin stability of a cluster is given by the energy difference between the ground state and the nearest higher-energy state of different magnetic moment called Erel. This quantity for the TiNan clusters is shown in Figure 1e. Erel also has peaks at n ) 5, 7, and 9 indicating robustness of these clusters with respect to spin excitations. This behavior is consistent with the fact that all these clusters have filled 1S 1P

Stability of Neutral and Anionic TiNan Clusters

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Figure 6. Ground state geometries of anionic TiNan- with n ) 1 to 13 clusters. See the caption of Figure 2. The antiprism cage isomer of TiNa8- is also shown.

Figure 5. Variation of magnetic moment (µ) vs the number of Na atoms, and nature of frontier orbitals for neutral (a) and anionic (b) TiNan clusters.

shells. The n ) 12 cluster, on the other hand, having a partially filled d shell, has a relatively low Erel. Erel, in fact, has a local dip at this size. This again brings out the fact that enhanced stability of the TiNa12 cluster is due to geometric shell closure. A more direct visualization of the nature of the frontier orbitals can be obtained from isosurface plots of these MO’s and the energy level diagram of the MO’s for TiNa9 is shown in Figure 4b. As we mentioned earlier, one S-type and three P-type delocalized orbitals are filled in both the spin channels. Moreover, there are five predominantly d-type orbitals on the Ti atom that form the nearly degenerate HOMO. Our discussions so far establish that the enhanced stability n ) 5, 7, and 9 in TiNan clusters is electronic in origin, while the one at n ) 12 has geometric shell filling as its cause. Therefore, if the charge state of the cluster was to change, the n ) 12 cluster should still show enhanced stability, while the n ) 5, 7, and 9 clusters would be expected to lose their magic character. To check this possibility, we studied the ∆ENa and HOMO-LUMO gap for anionic TiNan- with n ) 1-13 clusters. As shown in panels b and d of Figure 1 peaks in ∆ENa now occur at n ) 4, 6, and 12. Enhanced stability at n ) 4 is due to a shell filling in a 2D electron gas similar to the case of neutral TiNa5. This is seen from the nature of the occupied frontier orbitals as indicated in Figure 5b, and the calculated ground state moment

of 3 µB is also consistent with this (Figure 6). Enhanced stability at n ) 6 in TiNa6- is due the filling of the 1S and 1P shells of the nearly free electron gas similar to the case of TiNa7. The resulting electronic structure 1S21P6 3d3 is consistent with a moment of 3 µB. The enhanced stability at n ) 12 is due to geometric effects as in the case of neutral clusters. As seen in Figure 6, the TiNa12- cluster also has an icosahedral ground state structure with the Ti atom at the center. An analysis of the calculated HOMO-LUMO gap presented in Figure 1d shows maxima at n ) 4 and 6. These results are in agreement with the discussion for the neutral series, showing that once the minority 3d spin orbitals start to fill up the gap is reduced. To find out the ground state spin stability of the anion clusters, we plot Erel for these clusters as a function of the number of Na atoms in Figure 1f. TiNa4- and Tina6- clusters, having their stability originating from filled shells, have large values of Erel. TiNa12- has a small Erel, again consistent with the facts that it has a partially filled shell, and its stability originates from geometric shell closure. What is surprising is that in the anionic series we do not see any signature of enhanced stability when the Ti atom attains a half-filled 3d shell. This happens at n ) 8, as seen in Figure 5b. But there is no peak in ∆ENa at this size. The reason for this can be understood in terms of the differences in sizes of the neutral Ti and anionic Ti- species. The negative charge is localized at the Ti site, increases its size, and it no longer fits the expected antiprism Na8 cage found previously for VNa8.15 In the case of TiNa8- the ground state corresponds to an open structure with the Ti atom partially covered by the Na atoms (Figure 6). An analysis of the low-energy isomers of TiNa8found a compact antiprism cage lying 0.15 eV higher than the ground state, with 10% longer Na-Na bonds when compared with the ground state (Figure 6). The fact that the TiNa8- ground state is not a compact structure, a requirement for the free electron energy stability, is the cause of the lack of stability in the ∆E, HOMO-LUMO gap, and Erel. A possible way to stabilize a species analogous to TiNa8- will then be to use a larger alkali atom in the TiAM8- (AM ) K, Rb, or Cs) clusters. Supplementary calculations on the ground state of TiK8- indicate an open structure with the Ti atom partially covered by the K atoms and a magnetic moment of 4 µB. In this case, the compact antiprism lies only 0.08 eV higher than the ground state. Further calculations with an even larger alkali atom, namely TiCs8-, did show a compact antiprism ground state with a magnetic

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moment of 4 µB, and a corresponding maximum in the Cs energy gain ∆ECs. As also pointed out in a recent paper, it is possible to control the relative order of the super shells and the localized 3d states by changing the alkali atom, and thus stabilize different magnetic states.32 In summary, the findings in the anionic clusters are consistent with the previous overall picture. Because the enhanced stabilities at n ) 5 and 7 in the neutral series were due to electronic effects, they have moved to n ) 4 and 6 respectively in the anionic series. The enhanced stability at n ) 12 is robust with respect to change in the charge state since it is a result of geometric shell closure. Conclusions It is shown that Ti-doped alkali metal neutral and anionic TiNan clusters can be stabilized by electronic and geometrical effects. Clusters of 2 and 3 dimensions are stabilized by closed shells of nearly free electrons delocalized over the clusters. In addition to these shells the localized atomic 3d orbital defines the magnetic characteristic of the clusters. TiNa9 is shown to correspond to a closed 1S2 1P6 3d5 species being another member of the magnetic superatom family recently discovered by the authors.15 In addition to the electronic effects, the geometrical factors are also shown to explain the stability of the icosahedral species. We believe that our findings would stimulate experiments in such clusters to complement the theoretical findings as one proceeds to the identification of nonmagnetic and magnetic superatoms. Acknowledgment. J.U.R., D.R.R., and S.N.K. gratefully acknowledge support from the U.S. Department of the Army through an ARO MURI grant (Grant No.W911NF-06-1-0280). K.P. and P.S. gratefully acknowledge use of the cluster computing facilities (http://cluster.hri.res.in) at the HRI. References and Notes (1) Claridge, A.; Castleman, A. W., Jr.; Khanna, S. N.; Murray, C. B.; Sen, A.; Weiss, P. S. ACS Nano 2009, 2, 244. (2) Castleman, A. W., Jr.; Khanna, S. N. J. Phys. Chem. C 2009, 113, 2664. (3) Knight, W. D.; Clemenger, K.; de Heer, W. A.; Saunders, W. A.; Chou, M. Y.; Cohen, M. L. Phys. ReV. Lett. 1984, 52, 2141. (4) de Heer, W. A. ReV. Mod. Phys. 1993, 65, 611.

Reveles et al. (5) Neukermans, S.; Janssens, E.; Tanaka, H.; Silverans, R. E.; Lievens, P. Phys. ReV. Lett. 2003, 90, 033401. (6) Janssens, E.; Tanaka, H.; Neukermans, S.; Silverans, R. E.; Lievens, P. New J. Phys. 2003, 5, 46. (7) Martin, T. P.; Bergmann, T.; Golich, H.; Lange, T. Chem. Phys. Lett. 1990, 172, 209. (8) Schriver, K. E.; Persson, J. L.; Honea, E. C.; Whetten, R. L. Phys. ReV. Lett. 1990, 64, 2539. (9) Khanna, S. N.; Jena, P. Phys. ReV. B 1995, 51, 13705. Bergeron, D. E.; Calstleman, A. W., Jr.; Morisato, T.; Khanna, S. N. Science 2004, 304, 84. (10) Bergeron, D. E.; Roach, P. J.; Castleman, A. W., Jr.; Jones, N. O.; Khanna, S. N. Science 2005, 307, 231. (11) Clayborne, P.; Jones, N. O.; Reber, A. C.; Reveles, J. U.; Qian, M. C.; Khanna, S. N. J. Comput. Methods Sci. Eng. 2007, 7, 417. (12) Reber, A. C.; Khanna, S. N.; Castleman, A. W., Jr. J. Am. Chem. Soc. 2007, 129, 10189. (13) Reveles, J. U.; Khanna, S. N.; Roach, P. J.; Castleman, A. W., Jr. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 18405. (14) Pradhan, K.; Sen, P.; Reveles, J. U.; Khanna, S. N. Phys. ReV. B 2008, 77, 045408. (15) Reveles, J. U.; Clayborne, P. A.; Reber, A. C.; Khanna, S. N.; Pradhan, K.; Sen, P.; Pederson, M. R. Nat. Chem. 2009, 1, 310. (16) Ko¨ster, A. M.; Calaminici, P.; Casida, M. E.; Flores-Moreno, R.; Geudtner, G.; Goursot, A.; Heine, T.; Ipatov, A.; Janetzko, F.; , M.; Del Campo, J.; Patchkovskii, S.; Reveles, J. U.; Salahub, D. R.; Vela, A. deMon2k, ver 2.7; The International deMon Developers Community: Cinvestav, Me´xico, 2010; available at http://www.deMon-software.com. (17) Perdew, J. P.; Wang, Y. Phys. ReV. B 1986, 33, 8800. (18) Calaminici, P.; Janetzko, F.; Ko¨ster, A. M.; Mejia-Olvera, R.; Zuniga-Gutierrez, B. J. Chem. Phys. 2007, 126, 044108. (19) Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. J. Chem. Phys. 1979, 71, 4993. (20) Mintmire, J. W.; Dunlap, B. I. Phys. ReV. A 1985, 25, 88. (21) Ko¨ster, A. M.; Flores-Moreno, R.; Reveles, J. U. J. Chem. Phys. 2004, 121, 681. (22) Reveles, J. U.; Ko¨ster, A. M. J. Comput. Chem. 2004, 25, 1109. (23) Ionization potentials of atoms and atomic ions. In CRC Handbook of Chemistry and Physics, 79th ed.; Lide, D. R., Ed.; CRC Press, Inc.: Boca Raton, FL, 1998. (24) Ilin, R. N.; Sakharov, V. I.; Serenkov, I. T. Opt. Spectrosc. (USSR) 1987, 62, 578. (25) Forsberg, P. Phys. Scr. 1991, 44, 446. (26) Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, 558. (27) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (28) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (29) Perdew, J. P. In Electronic Structure of Solids; Ziesche, P., Eschrig, H., Eds.; Akademie: Berlin, Germany, 1991. (30) Keller, E. Chem. Unserer Zeit 1980, 14, 56. (31) Flu¨kiger, P.; Lu¨thi, H. P.; Portmann, S.; Weber, J. Molekel 4.0; Swiss National Supercomputing Centre CSCS, Manno, Switzerland, 2000. (32) Pradhan, K.; Reveles, J. U.; Sen, P.; Khanna, S. N. J. Chem. Phys. 2010, 132, 124302.

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