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Reactivity and Regioselectivity of Aluminum Nanoclusters: Insights from Regional Density Functional Theory David J. Henry,† Pawez Szarek,‡,§ Kosuke Hirai,‡ Kazuhide Ichikawa,‡ Akitomo Tachibana,‡ and Irene Yarovsky*,† †
School of Applied Sciences, RMIT University, Melbourne Victoria 3001, Australia Department of Micro Engineering, Kyoto University, Kyoto 606-8501, Japan § Institute of Physical and Theoretical Chemistry, Wroczaw University of Technology, Wroczaw 50-370, Poland ‡
bS Supporting Information ABSTRACT: The structure and bonding in charged and doped aluminum clusters have been investigated using Regional Density Functional Theory (RDFT). The RDFT method provides a measure of the electronic stress tensor from which it is possible to determine bond indices and electronic chemical potential. We find that the distribution of bond indices on the surface of Al12X clusters provides not only an understanding of the internal bonding and stability but also some insight into the regioselectivity observed in reactions of the clusters.
1. INTRODUCTION The properties of metal clusters can differ significantly from that of bulk materials and are often dependent on not only the composition but also the size of the cluster. As a result, clusterbased materials are increasingly being investigated for their reactivity and catalytic activity which may be enhanced due to their high surface-to-bulk ratio.1-4 Research has recently been directed to the study of aluminum clusters and nanowires for a range of applications including for hydrogen storage,5-7 as nanocatalysts,8 and for optical applications.9 To aid in the design of these materials, it is useful to understand the nature of the bonding in aluminum clusters and to identify features that will lead to improved properties. Experimental studies10-22 of aluminum clusters have identified Al13 as a particularly stable structure which has also been supported by a range of theoretical studies.23-34 This stability has been attributed to the fact that Al13 has 39 valence electrons and therefore is only one electron short of being closed shell in the Jellium model.35 In fact, the Al13- anion and Al12Si clusters, which have 40 valence electrons and are therefore closed shell, are even more stable than Al13,12-16,36-44 potentially making these species suitable for the preparation of novel cluster assembled materials. In comparison, the increased stability makes these species less reactive to many small molecules.45 Recently,45-49 we have been investigating theoretically the interaction of hydrogen with both neutral and charged X-centered aluminum clusters (Al12X, X = Mg, Al, Si) to identify trends in the reactivity of these species. Importantly, our analysis of the reaction energetics for H2 adsorption indicates that these r 2011 American Chemical Society
processes are largely controlled by thermodynamics.45 However, there are strong correlations between the degree of charge transfer in the transition state and the reaction barriers and also between the barriers and cluster distortion energies. We also found47 that dissociation of a H2 molecule between closely spaced 38 valence electron Al12Mg clusters proceeds spontaneously when in a linear arrangement. Doping of the cluster with an electropositive atom (Mg) appears to result in transfer of electron density to the Al cage, which enhances H2 dissociation. However, a more detailed analysis of the bonding and intrinsic properties of Al12X clusters would provide greater insight into the chemistry and regioselectivity of reactions of these species and aid in the design of new materials for specific applications. Many of the early studies of aluminum clusters focused on the evolution of structural and electronic properties with size from diatomics toward that of the bulk metal. For example, Jones50 investigated Aln (n e 10) using Car-Parrinello methods and reported that the variety of structures found was consistent with the metallic nature of the element, implying that even in these small species the bonding is predominantly metallic. Rao and Jena32 note that with increasing size the s-electron content of the valence shell decreases, with a corresponding increase in p content. For clusters with less than five atoms, the Highest Occupied Molecular Orbital (HOMO) is clearly s-like, while for clusters with more than seven atoms it is p-like. This corresponds Received: October 12, 2010 Revised: December 13, 2010 Published: January 7, 2011 1714
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The Journal of Physical Chemistry C with the size range in which cluster structures preferentially become three-dimensional, and there is a sudden increase in the number of bonds and coordination numbers, i.e., the bonding becomes more metallic. Photoelectron studies12b,15 suggest that the 3s and 3p bands begin to overlap somewhere between n = 6 and n = 12. Complementing these experimental results is the theoretical study of Bauschlicher and Pettersson24a that found there is little hybridization of the 3s and 3p orbitals in Al2; however, by the time the cluster size has increased to Al13, there is strong hybridization which allows for the formation of many metal-metal bonds. Similarly, Yang et al.28 report plots of the HOMO-LUMO gap as a function of cluster size, showing that aluminum clusters become more metallic as the cluster size increases, with Al55 and Al147 having values close to that of the bulk metal. However, the HOMO-LUMO gap for Al13 is quite large (∼2.0 eV), suggesting nonmetallic character. McHenry et al.23 also found that the density of states for Al13 was different from the free electron behavior of crystalline aluminum, showing more structure and a particularly high peak at the Fermi energy. The HOMO has π-bonding character which is very delocalized, extending over the first coordination shell. The presence of dopants can potentially have a significant effect on the structure and properties of the cluster. Sun et al.42 analyzed the bonding in pure and doped closed shell aluminum clusters (Al6, Al12Si, Al12C, Al55, and Al54Si) using the electron localization function (ELF). This function provides a measure of the parallel-spin correlation by defining the conditional probability of finding an electron in the neighborhood of another electron with the same spin. It can therefore be used to identify regions of space where electrons are well localized such as in covalent bonds or lone pairs and can clearly distinguish freeelectron behavior. They found that in regions between the core and surface atoms of Al12Si and Al12C the ELF displays jelliumlike behavior with nearly delocalized charge distribution, and therefore, bonding is metallic. However, outside the cluster, the electrons are highly localized. Similarly, Rao and Jena51 studied carbon-doped aluminum clusters (Aln, n = 3, 4, 5, 11, 12, and 13) and found that while the bonding of carbon to aluminum in most of the clusters is primarily covalent, carbon in the neutral Al12C cluster behaves more like a metal atom. However, Seitsonen et al.38b used the atomic pair correlation function to investigate Al12C and found very large fluctuations, which is unusual for covalent or metallic bonds. They indicate that while the isolated Al12C cluster is very stable this stability is due largely to the bonding in the Al cage rather than the contribution from carbon. Ma~ nanes et al.52 investigated the bonding in Al13 and its reactivity toward the H atom. Electron density plots suggested metallic character for the bonding in Al13, and plots of the electron deformation density show how the metallic bonding is established by moving electronic charge from the Al atoms to the interstitial regions. Local Fukui functions and condensed Fukui indexes of the ground state of Al13 were used as indicators of reactivity and to predict the equilibrium location of H in Al13H. Analysis of the Al-H bond in Al13H suggests covalent rather than metallic character. In this work, we apply the Regional DFT53 and electronic stress tensor method to obtain a greater understanding of the bonding and electronic structure in Al12XZ clusters (where X = Mg, Al, or Si and Z = -1, 0, or 1) and how this influences observed trends in cluster reactions.
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2. THEORETICAL PROCEDURES A. Regional Density Functional Theory (RDFT). The RDFT approach53-56 allows one to assign energy density to any point in space according to the associated electronic density. This energy density can be decomposed into: the kinetic energy density, the external potential energy density, and the interelectron potential energy density. The concept of the electronic drop (RD) and electronic atmosphere (RA) regions, separated by an interface surface (S), is used to define the shape of atoms and molecules.54 The kinetic energy density (nT(rB)) is positive in the RD region, and classical movement of electrons is permitted. However, in the electronic atmosphere RA the kinetic energy is negative, and only quantum effects for electrons are possible; consequently, S defines a turning point for an electron. The total force acting on electrons in the system is composed ! ^ r Þ and the tension force ! τ SR^ð B r Þ. For of the Lorentz force L SR ð B a system in a stationary state, the total force at every point in space equals zero, thus the Lorentz force exactly cancels the tension force. S ! ! ^S ^S τ^ R ðrBÞ þ L R ðrBÞ F R ðrBÞ ¼ !
ð1Þ
If one investigates the tension force in the bonding regions of a molecule in a stationary state, one might find a point where (along with the condition FB(rB) = 0) the tension (as well as Lorentz force) itself will vanish, and any force acting on electron density at that point will be zero.54 This Lagrange point (rBLagrange) is a stationary point for the electron density distribution in a molecule56,57 and can be used to characterize interaction between the local atoms. Covalent bond formation is characterized by the concept of the spindle structure, which is a geometrical region where the principal electronic tensile stress is positive along the line of the principal axis that connects a pair of atom or molecule RD's. The trace over the eigenvalues of stress produces the energy density ε(rB) in the nonrelativistic limit of the rigged QED.54 1X S εðrBÞ ¼ τ ðr Þ 2 k RB
ð2Þ
The energy density based bond order index (bε) is defined as56,57 bε ¼
εAB ðrBLagrange Þ
εHH ðrBLagrange Þ
ð3Þ
where εAB(rBLagrange) is the energy density at the Lagrange point of bond of interest (A-B) and εHH(rBLagrange) is the energy density at the Lagrange point of the H-H bond in a H2 molecule. The ratio of energy density to electronic density gives a linear approximation of the local electronic chemical potential μR.56,57 From the electronic chemical potential, we can derive an alternate bond index (bμ) called the chemical potential bond order !-1 εAB ðrBLagrange Þ εHH ðrBLagrange Þ bμ ¼ ð4Þ nAB ðrBLagrange Þ nHH ðrBLagrange Þ where nAB(rBLagrange) and nHH(rBLagrange) denote the electronic density for the A--B and H-H bonds, respectively. 1715
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Table 1. Comparison of Selected Properties Calculated with PBE and PW91 Functionals and the 6-31G(d) and 6-311G(d) Basis Sets PBE/6-31G(d) Al13þa
Al13
PBE/6-311G(d) Al13-
Al13
þa
Al13
PW91/6-31G(d) Al13-
Al13
þa
Al13
PW91/6-311G(d)
Al13-
Al13
þa
Al13
Al13-
bε bond order X-Al b Al-Al b
0.095 0.069
0.082 0.060
0.084 0.061
0.077 0.060
0.100 0.071
0.102 0.072
0.097 0.070
0.080 0.058
0.082 0.060
0.080 0.058
0.098 0.069
0.100 0.070
bμ bond order X-Alb Al-Alb
0.521 0.445
0.528 0.452
0.533 0.456
0.625 0.534
0.634 0.534
0.640 0.539
0.512 0.442
0.521 0.446
0.526 0.451
0.619 0.529
0.628 0.529
0.635 0.534
-0.414 (-0.080) 0.108 (0.075) 0.125 (0.102) -0.306
-0.432 (0.064) 0.032 (-0.004) 0.039 (-0.006) -0.183
-0.462 (0.125) -0.045 (-0.094)
-0.452 (-2.461) 0.034 (0.195) 0.041 (0.215) -0.183
-0.469 -0.427 (-2.778) (0.050) -0.044 0.109 (0.148) (0.067) 0.129 (0.093) -0.035 -0.306
-0.445 (0.182) 0.034 (-0.015) 0.040 (-0.015) -0.183
-0.473 (0.233) -0.044 (-0.103)
-0.035
-0.443 (-2.332) 0.110 (0.256) 0.131 (0.302) -0.306
-0.458 (-2.546) 0.034 (0.202) 0.042 (0.222) -0.184
-0.477 (-3.044) -0.044 (0.170)
-0.031
-0.438 (-2.413) 0.111 (0.264) 0.130 (0.307) -0.305
-0.161 -0.125 -0.132
-0.149 -0.125 -0.130
-0.135 -0.120 -0.127
-0.146 -0.118 -0.126
-0.135 -0.119 -0.124
-0.124 -0.113 -0.120
-0.161 -0.125 -0.134
-0.149 -0.125 -0.131
-0.136 -0.120 -0.127
-0.146 -0.118 -0.126
-0.135 -0.118 -0.124
-0.124 -0.114 -0.120
2.82 6.39 9.71
2.78 6.54 9.91
2.71 6.39 10.23
4.01 6.14 8.79
3.90 6.03 8.59
3.73 5.68 8.51
2.76 6.53 9.90
2.77 6.58 9.94
2.72 6.39 10.21
4.05 6.27 9.04
3.92 6.13 9.10
3.74 5.90 8.92
chargesc Xb Al min Al max chemical potential d atope bridgee hollowe electron density ( 10-3) atop f bridge f hollow f
-0.041
Data are for the Al13þ triplet structure. b Core aluminum (X) and cage aluminum (Al). c ATP charges with corresponding Mulliken charges given in parentheses. d Vertical chemical potential of the cluster. e Chemical potential on the interface surface. f Maximum of electron density at atop, bridge, and hollow sites on the interface surface. a
B. Computational Details. Standard density functional theory calculations were performed using the DMol358,59 and GAUSSIAN0360 computer programs. Geometries for Al12Mg, Al13, Al12Si, and their respective monocations and monoanions were calculated with the PBE61 and PW9162 functionals using the DNP, 6-31G(d), and 6-311G(d) basis sets. Calculations with the 6-31G(d,p) and 6-311G(d,p) basis sets were performed in GAUSSIAN. The respective electronic wave functions were used for the electronic stress tensor and energy density calculations in the RDFT program package.63 Calculations in DMOL3 were performed with the double numerical polarized (DNP) basis set. Each of these basis sets includes a d-type polarization function on heavy atoms and a p-type polarization function on hydrogen. The DNP basis set is comparable in size to the Gaussiantype 6-31G(d,p) basis set. It was previously shown that an allelectron basis set with the addition of d functions is essential for a proper description of high-valence Al atoms.64 Vibrational frequency analysis was performed to characterize all stationary points reported here as true minima. Partial atomic charges were determined using the atomic polar tensors approach of Cioslowski.65 The interface surface (zero isosurface) of the kinetic energy density around the valence electron region has been used to define the molecular surface. Bond order indicies are based on quantities calculated at the stationary point of electron density (Lagrange point) between two atoms, where the forces acting on electrons locally disappear. Bond inidices are weighted by quantities obtained from the H2 molecule, calculated at the same level of theory. An iso-value of 0.022 e/Å3 was used for plots of electron deformation density. Visualizations of structures were done with VMD66 and PyMol.67
3. RESULTS AND DISCUSSION A. Effect of Functionals and Basis Sets. Previously46 we
have shown that the PBE and PW91 functionals perform well against benchmark levels (CCSD and G3(MP2)-RAD) for both aluminum cluster geometries and binding energies. However, the electronic stress tensor and properties derived from it can be sensitive to both choice of functional and basis set. Therefore, we begin this study by first investigating the sensitivity of the electronic stress tensor in these systems to the choice of exchangecorrelation functional and basis set. Table 1 presents selected properties calculated at the PBE/631G(d), PBE/6-311G(d), PW91/6-31G(d), and PW91/6311G(d) levels for comparison. It is clear from Table 1 that there are generally no significant effects with the choice of exchange-correlation functionals used in these calculations, with PBE and PW91 giving very similar results for the same choice of basis set. However, basis set size was found to contribute to larger differences in some properties. In particular, it was found that the smaller 6-31G(d) basis set results in consistently lower chemical potential bond indices (bμ) compared to those obtained with 6-311G(d). Nevertheless, in both cases there is a gradual increase in core-to-surface (X-Al) and surface-to-surface (Al-Al) bμ bond orders with charge (z = þ1 to z = -1). For the calculated energy density bond indices (bε), the changes with basis set are less systematic. We observe that bε bond orders are lower with 6-31G(d) for Al13 and Al13-; however, for Al13þ they are in fact higher. Consequently, while the 6-311G(d) basis set leads to a similar gradual increase in bε bond orders as noted for bμ bond orders, the 6-31G(d) basis set produces the reverse trend. 1716
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The Journal of Physical Chemistry C Mulliken charges obtained with the 6-31G(d) basis set are generally quite small in magnitude for core and surface atoms (Table 1), while the larger 6-311G(d) basis leads to excessive charge on the core atoms of the clusters. In comparison, charges calculated with the APT scheme appear quite reasonable and show much less variation due to the basis set. In particular, all levels of theory predict a gradual increase in the negative charge of the core (X) and decreasing positive charges on the surface atoms, with increase of overall charge (z = þ1 to z = -1). Chemical potential values on the interface surface are generally unaffected by choice of basis set, showing a steady decrease with charge. However, the chemical potentials determined for the atop, bridge, and hollow sites are generally more nega tive with the smaller 6-31G(d) basis set than with 6-311G(d), although similar trends with charge are observed. The sensitivity of electron density to basis set and functional varies for different locations, which may have a bearing on predicted reaction energetics and regioselectivity of the respective sites. For example, the maximum values for atop sites are consistently lower with 6-31G(d), while the reverse is true for bridge and hollow sites. We also note that with the 6-311G(d) basis set there is a decrease in electron density at each of the sites, as the overall charge of the cluster changes from z = þ1 to z = -1. However, this is not observed with the 6-31G(d) basis set, particularly for bridge and hollow sites. This indicates that with the 6-31G(d) basis set there is a tendency to concentrate greater density in the hollow sites. Unlike the other properties discussed so far, the electron density exhibits a variation with choice of functional, particularly for bridge and hollow sites, where PBE predicts slightly lower density than PW91. On the basis of the above findings and the previous assessment against benchmark levels of theory,46 we recommend the PBE/6311G(d) level be used for determination of aluminum cluster properties derived from the electronic stress tensor, in conjunction with charges determined using the APT scheme. B. Structure and Bonding of Al12X (X = Al, Mg, and Si). In this section, we discuss the trends in the structure and bonding of Al12X clusters as a function of the dopant X and charge. We observe that the “magic” 40 electron clusters exhibit compact structures, and generally, the greater the deviation of the total number of valence electrons from 40 the greater the enlargement of the cluster. We also note a correlation between cluster size and the electronegativity of X, with the silicon-centered clusters generally smaller than isoelectronic species with Al or Mg cores. We begin our analysis of the bonding by considering first the electron deformation densities of each of the clusters, which is the difference between the total cluster electron density and the density of the isolated atoms (Figure 1). The deformation densities for all nine clusters exhibit the same general feature of a delocalized framework with electronic charge located in the interstitial regions between the surface atoms, which is consistent with metallic bonding between the atoms and in agreement with earlier studies of Al13, Al13-, and Al12Si.6,24,28,42,52 However, we observe changes in the degree to which this deformation density penetrates the region between the surface atoms and the core atom. For the Al12Mgz clusters, we observe that the deformation density on the 0.022 electrons Å-3 isosurface extends from the surface toward the core, but there is no concentration of charge close to the core atoms. This reflects the electropositivity of the Mg atom, which donates its valence electrons to the Jellium orbitals so that they are more closely aligned with the Al framework than with the Mg core. In
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Figure 1. Electron deformation densities for Al12Xz clusters.
comparison, the Al12Siz clusters exhibit a significant concentration of charge about the electronegative silicon core that is distinct from the framework holding the surface atoms together. This is consistent with the observations of Sun et al.42 for Al12Si, who found that while Si loses electrons the excess charge is distributed in bonds between the Si and the cage Al atoms. The deformation density of the Al13 cluster differs somewhat from those of Al12Mgz and Al12Siz. In particular, the delocalized charge of the surface atoms spreads over the entire cluster and is continuous with the charge network, filling the internal space of the cluster. All Al atoms are donating their valence electrons to the charge framework indicating metallic bonding throughout the cluster, in agreement with the earlier LDA study of Ma~ nanes et al.52 The deformation density of the closed shell Al13- is similar to that of neutral Al13; however, the density in the core region of the cluster contracts somewhat toward the core Al atom. Ma~nanes et al.52 also reported that the deformation density of Al13- shows only very small changes from that of neutral Al13. For the electron-deficient Al13þ, the deformation density consists of two concentric frameworks: the inner one exists between the core Al atom and the cage Al atoms, while the outer framework spreads over the entire cluster with a high concentration between Al atoms. Overall, the deformation electron densities indicate a significant degree of metallic bonding throughout the framework of these clusters. However, the electron deformation density alone is unable to provide a detailed description of the bonding in these clusters. The RDFT approach enables the determination of the electronic tensile stress between pairs of atoms and can indicate if there is localization of electrons in these regions which indicates a degree of covalent character to the bond. Figure 2 displays the structures of the clusters considered in this study with the calculated chemical potential bond indices between adjacent atoms (left) shown as colored lines and the largest eigenvalue of the stress tensor (Figure 2, right) shown in red. We find that for the clusters in this study the “spindle structure” (with corresponding tensile stress region) does not appear at all inside the cluster cages indicating that there is no covalency to the X-Al bonds of these species; i.e., the X-Al 1717
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Figure 2. Chemical potential bond indices and largest eigenvalue of the stress tensor.
Table 2. Bond Lengths and Average Bond Indices (bε, bμ) of Al12Xz Clusters (X = Si, Al, and Mg, z = þ1, 0, and -1) at the PBE/ 6-311G(d) Level bμ indices
bond length bond
a
Al12Xþa
Al12X
Al12X-
Al12Xþa
Al12X
bε indices Al12X-
Al12Xþa
Al12X
Al12X-
X;Si
2.660, 2.687
2.651
2.628, 2.855
0.642
0.655
0.649
0.112
0.117
0.114
Al-Al
2.744 - 2.951
2.788
2.749-2.865
0.555
0.564
0.552
0.074
0.077
0.074
Al-Alb Al-Al
2.641 - 2.967 2.638 - 2.985
2.660, 2.709 2.759 - 2.962
2.671 2.808
0.582 0.539
0.634 0.534
0.640 0.539
0.085 0.074
0.100 0.071
0.102 0.072
Al-Mg
2.717, 2.761
2.709, 2.759
2.688, 2.728
0.623
0.564
0.634
0.069
0.061
0.075
Al-Al
2.867 - 2.907
2.820 - 2.923
2.832 - 2.920
0.492
0.451
0.510
0.063
0.055
0.067
Al13þ data are for the singlet structure. b Alcore-Alsurface bonds
bonds are metallic in character. However, regions of tensile stress can be identified in surface Al-Al bonds, and the character of these bonds differs depending on the dopant atom and cluster charge. These observations are in agreement with the work of Sun et al.42 who used the electron localization function (ELF) to investigate the bonding in Al12Si. They found that between the surface and central atoms the ELF indicates jellium-like behavior with nearly delocalized charge distribution; however, outside the surface atoms the electrons are highly localized. It is clear from Figure 2 that for each series of clusters the level of electron localization (i.e., covalent character) in the Al cage decreases as the charge on the cluster changes from þ1 to -1. Similarly, we note that the level of covalent character also decreases as the core element is changed from Mg to Al to Si. Bonds with higher covalent character should be more amenable to homolytic addition reactions such as H2 chemisorption, and generally we observe that the barriers for H2 addition reflect these trends in bonding character. In the RDFT scheme, bond indices are calculated relative to the bond strength of the H-H bond in the H2 molecule. Table 2 displays bond lengths, average chemical potential bond indices, and average energy density bond indices for the clusters of this study. We find that the average chemical potential bond orders
(bμ) in our Al12X clusters are about half of the value for the H-H bond, while bond strengths measured with the energy density bond order (bε) are about 1 order of magnitude lower than that of the H-H bond in a H2 molecule. Although the absolute values for bond indices determined from the energy density differ from those determined from the chemical potential, the observed trends are generally similar; therefore, we will focus on the bμ values. As noted above, the silicon-centered clusters are generally more compact than isoelectronic species with Al or Mg cores. In conjunction with this, we note that the silicon-doped clusters display the highest average bond indices for X-Al bonds, while the magnesium-doped clusters display the smallest X-Al bond indices for each of the respective charges. It is also clear that the bond indices for surface Al-Al bonds are always smaller than for the corresponding X-Al bonds within the same cluster. This is due largely to the weakening of the metallic bonding between the surface atoms due to electron localization. We find that the chemical potential bond orders are ∼14% smaller for Al-Al bonds than for X-Al bonds, in all Al12Si and Al13 clusters, while the corresponding difference in all Al12Mg clusters is about 20%. In comparison, the average bε bond orders of surface Al-Al bonds were approximately 34%, 28%, and 9% lower than indices associated with bonds involving the central X = Si, Al, and Mg 1718
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The Journal of Physical Chemistry C atom, respectively. These results highlight that while the absolute values for X-Al and Al-Al bond indices are dependent on the charge/total number of valence electrons the relative X-Al and Al-Al bond strengths within a cluster type are largely dependent on the dopant X. The distribution of bμ indices on the surface of Al12X clusters provides not only an understanding of the bonding and stability but also some insight into the regioselectivity observed in reactions of these clusters. The “magic” 40 valence-electron systems (Al13- and Al12Si) have Ih symmetry and are characterized by high average bond order indices, reflecting the exceptional stability of these species. Due to the high symmetry, the bond indices are uniform for all 12 X-Al bonds of each cluster, although bμ is slightly higher for Al12Si than for Al13-. Similarly, the surface Al-Al bond indices are uniform across the surface of the respective clusters, with bμ again slightly higher in Al12Si than in Al13-. In addition to providing a measure of bond strength, the chemical potential bond order also provides a measure of the “electron affinity” of bonds. The bond indices indicate that removal of an electron will be easier from a surface Al-Al bond rather than a core X-Al bond; however, the uniformity of the bonding in these clusters indicates that there is no preferred face for this to occur. Additionally, the relative values of bμ indicate that it is slightly harder to strip an electron from Al12Si than from Al13-, which is in agreement with the calculated ionization potentials for Al12Si and Al13-.46 Our previous study45 of hydrogen interactions with Al12X clusters found that for both Al12Si and Al13- dissociative chemisorption of H2 proceeds via bridge-type transition structures with high energy barriers. Analysis of the kinetic energy density indicates that the surface bonds of Al12Si and Al13exhibit only a low level of covalency (i.e., are predominantly metallic), which hinders neutral addition type reactions such as H2 chemisorption and therefore contributes to the high reaction barriers. The uniformity of the Al-Al bonds of these 40 electron clusters also means that there is no preferred edge/face for adsorption of H2. Consequently, the lowest-energy pathway is achieved via a bridge-type transition structure that requires direct involvement of only two surface Al atoms and therefore primarily leads to disruption of only one surface Al-Al bond. The Al12Mg-, Al13, and Al12Siþ clusters have 39 valence electrons and should therefore be slightly less stable than their 40 electron counterparts. Table 2 and Figure 2 reveal that there is relatively little change in the bond indices for the internal (X-Al) bonds of these clusters, with regard to each other and the 40 electron systems. Consequently, the lowering of stability is largely reflected in changes of the bond indices of the surface Al-Al bonds. The lowest Al-Al surface bond indices in all three clusters are observed between triangles of Al atoms on either side of the clusters indicating these are the regions from which it will be the most easy to remove an electron. These triangular faces are also the regions in which the spin density of the unpaired electron is located (Table S1, Supporting Information). The tensile stress for the bonds comprising these triangles indicates a significant degree of covalency, making these faces ideally suited to accommodate a hollow-type transition state for H2 adsorption, in agreement with our previously observed regioselectivity.45 The bonds between the Al atoms of the central region also exhibit small bμ and significant covalency, which would make them suitable for bridge-type transition states. However, as discussed previously,45 Al13 is the only cluster for which we could identify
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both bridge and hollow transition structures, suggesting that other factors also contribute to the selectivity and barrier height. Interestingly, the remaining surface Al-Al bonds of Al12Mg-, located between the triangle faces and the central region, also exhibit small levels of positive stress. However, the same bonds in Al13 and Al12Siþ exhibit no covalency and have higher bμ bond indices. There are no obvious similarities observed between the two 38 electron systems (triplet Al12Mg and singlet Al13þ). For Al12Mg, the axial pair of X-Al bonds displays a significantly lower bond strength than the remaining X-Al bonds, and the spin density is largely located around the Mg core of the cluster (Table S1, Supporting Information). In comparison, in singlet Al13þ there are significant variations in bμ across the 12 X-Al bonds, and these are generally weaker than those of Al12Mg. The spin density for Al13þ is largely located on the surface of the cluster (Table S1, Supporting Information). In regard to the surface atoms, the Al-Al bond indices for Al12Mg are the lowest for the central region of the cluster and display significant covalency. The characteristics of these bonds are ideal for the low-energy barrier bridge-type transition states reported previously.45 The bμ bond indices are higher for the bonds to the axial Al atoms; however, these also display a degree of covalency. Interestingly, the neutral Al12Mg cluster actually has lower Mg-Al as well as Al-Al bond strengths than those in either of the charged Al12Mg(1 clusters, which might be related to the high spin density on the Mg atom in Al12Mg.68 This is also reflected in the lower barrier for addition of H2 to Al12Mg compared to the charged Al12Mg(1 clusters.45 The surface bond indices for singlet Al13þ vary significantly across the surface of the cluster but are generally quite low, and this is consistent with both the low barrier for addition of H2 and the observation that Al13þ forms an association complex. For the 37 electron cluster (Al12Mgþ) (D3d symmetry), the X-Al bond strengths are divided into two groups of six bonds each. In comparison, the surface of the cluster exhibits a fairly uniform value for bμ across all Al-Al bonds which also exhibit significant covalency. The spin density is largely located around the Mg core with small regions also located in the hollow sites on the surface of the cluster. These characteristics facilitate the low barrier for addition of H2 to Al12Mgþ and the preference for a hollow-type transition structure. For the 41 electron Al12Si(D5d symmetry) cluster, the X-Al bond strengths for the long pair of bonds along the rotational axis are slightly lower than those of the other 10 atoms. This also corresponds with the atoms on which the spin density is located (Table S1, Supporting Information). Similarly, the strongest Al-Al bonds are observed between Al atoms of the central region, and the bond strengths become progressively weaker for bonds closer to the axial Al atoms; however, Al12Si- appears to have almost no covalency in its bonding. This accounts for the observed high barrier for the addition of H2 via a bridge-type transition structure. Figure 3 shows a plot of the chemical potential bond indices versus bond length for X-Al and Al-Al bonds of all the clusters. There is a general correlation between bond strength and bond length with shorter metallic bonds exhibiting higher bond indices and longer more covalent bonds having smaller values.69 This correlation is particularly strong for the metallic bonds between core and surface atoms, except for the Al12Mg cluster where the bμ values fall below the general trend for X-Al bonds. The surface Al-Al bonds of these clusters follow a similar linear trend; however, there is a larger scatter in the points, and once 1719
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Table 3. Electronic States, Atomic Radii of Core Atoms (rX), Chemical Potential around the Core (μX), and Partial Atomic Charges of Core (qX) and Surface (qAl) Atoms μX
rX
qX
A2u
-0.251
1.818
-0.034
þ0.078 to þ0.091
A1g
-0.254
1.811
-0.147
-0.049 to þ0.025
2
A2u
-0.254
1.809
-0.257
-0.031 to -0.093
1
A
-0.398
1.511
-0.438
þ0.110 to þ0.130
A2u 1 Ag
-0.405 -0.408
1.501 1.499
-0.452 -0.469
þ0.034 to þ0.041 -0.044
Al12Siþ
2
A2u
-0.633
1.257
-0.627
þ0.134 to þ 0.137
Al12Si
1
Ag
-0.637
1.255
-0.576
þ0.048
Al12Si-
2
A1g
-0.819
1.148
-0.473
-0.036 to -0.084
2
Al12Mg
3
Al12MgAl13þ Al13 Al13-
2
Al12Mg
Figure 3. Chemical potential bond orders versus bond lengths.
again the values for Al12Mg fall below the general trend. We also note that there is generally a trend between the chemical potential bond indices and the degree of covalency for the individual cluster bonds. The longer more covalent bonds tend to have smaller bond indices, whereas the shorter more metallic bonds tend to have larger bond indices. A key finding of our previous work45 on H2 addition to Al12Xz clusters was the identification of a strong correlation between reaction barrier and the cluster distortion energy in the transition state. The distortion energy is the amount of energy required to change the geometry of the cluster from its equilibrium structure to that of the geometry in the transition structure. Generally higher distortion energies correlate with higher H2 addition barriers and vice versa. The chemical potential bond orders provide a clear explanation of this trend. For example, in the closed shell Al12Si and Al13- clusters the surface Al-Al bonds have high bμ and low covalency, and therefore the energy to distort these bonds, to allow addition of H2, is high and makes a significant contribution to the barrier. In comparison, the 37, 38, and 39 electron clusters have a number of surface Al-Al bonds with low bμ and high covalency, and these bonds require significantly less energy to distort to accommodate the approaching H2 molecule; consequently, there is a smaller contribution to the barrier. The electronic chemical potential is a measure of the effective potential experienced by electrons and reflects the electron affinity of the region. Therefore, it is not surprising that the chemical potential around the core region (μX) of Al12Si is much greater (more negative) than the corresponding regions of Al13 and Al12Mg (Table 3). The diameter of the core regions (rX) was assumed to be confined by the internal interface of the kinetic energy density enclosing the valence electron region and correlates with the atomic radii of the core elements. For all of the clusters, the central atom carries a negative partial charge (Table 3), and the magnitude of this charge increases with the electronegativity of the core atom. The charge on the core is balanced by an even distribution (within each symmetry) of small positive charges on the surface atoms. The electrophilicity of the individual surface atoms is much smaller, and hence the chemical potential across the surface of the cluster is also smaller than for the core. Figure 4 presents the electronic chemical potential mapped on the zero kinetic energy density isosurfaces of each of the clusters. The chemical potential or electron affinity of the cluster surface will have some bearing on the reactivity of the cluster toward different species. Regions of greater μ (more negative) indicate poor shielding and therefore reflect the strong electric potential of the nucleus. Therefore, regions of greater
þ
qAl
Figure 4. Electronic chemical potential for Al12Xz clusters.
electronic chemical potential are electrophilic. In comparison, regions of weaker μ (less negative) are better shielded and in relative terms can be considered nucleophilic. Considering first the neutral clusters, we can see that there are significant regions with greater μ (red and yellow regions) on the surface of the Al12Si cluster reflecting the greater electron affinity of the surface atoms of this species. In comparison, the Al13 and Al12Mg clusters have a greater distribution of regions with weaker μ (blue and green) on their surfaces, indicating a weaker electron affinity. We note that for all three clusters the regions with the weakest electronic chemical potential are located in the bridge and hollow sites, and this corresponds with the delocalized framework of the electron deformation density plots. In general, isoelectronic clusters display similar relative distribution patterns of chemical potential (μ) on the cluster surfaces; however, the absolute magnitudes of the chemical potentials generally differ with charge and dopant X. A number of studies6,48,52 have shown that addition of hydrogen to Al12X clusters actually results in transfer of electron density to the hydrogen atom rather than from the hydrogen to the cluster. Clusters with smaller chemical potential on the surface (Al12Mg and Al13) should have greater potential to transfer density to hydrogen, and this is generally reflected in the higher binding energies for hydrogen to Al12Mg and Al13 1720
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The Journal of Physical Chemistry C compared with Al12Si. Similarly, in our earlier study of H2 chemisorption on Al12X clusters,45 we found that there was a strong correlation between barrier height and the charge on the H atom moving to the atop position in the transition structure. We find that the cluster cations which have the largest chemical potentials in the atop regions are more electrophilic, resulting in less charge transfer to H in the transition states for chemisorption of H2. Consequently, these clusters generally exhibit earlier (more reactant like) transition structures than the corresponding neutral cluster. In comparison, cluster anions are more nucleophilic, leading to greater charge transfer to H, resulting in later (more product like) transition structures. The chemical potential plots can also be used to explain the regioselectivity of other reactions. For example, Khanna and Jena70 report that the preferred location for the potassium atom interacting with Al13 is a hollow site. Similarly, Majumder et al.71 report that the lithium atom also occupies a hollow site on the surface of Al13. Both of these atoms are electropositive and therefore will donate an electron to the cluster. The chemical potential plot of Al13 reveals that the regions of greatest chemical potential are the hollow sites associated with the triangles of atoms on either side of the cluster. These regions have the greatest electron affinity and are therefore the preferred location for K and Li to adsorb. We can see that for the cluster cations there are more regions with greater μ (red and yellow) on the surface of the clusters reflecting the greater electron affinity of the surface of these species. In comparison, the anionic clusters have a greater distribution of regions with weaker μ (blue and green) on their surfaces reflecting a weaker electron affinity. Charkin et al.72 studied Al13L- clusters where L is the ligands H, F, Cl, Br, OH, NH2, CH3, and C6H5 and found that L predominantly occupies an atop position on the cluster. Each of these ligands is electrophillic with regard to the cluster and therefore has a preference for regions with weaker chemical potential. The plot of Al13- indicates that the regions of weakest chemical potential are the atop positions, and along the lines of the metallic framework, however, the atop positions are more accessible. These results indicate that analysis of the kinetic energy density can provide a detailed understanding of the bonding and observed regioselectivity of Al12X clusters in a range of different types of reactions. This information can be used as a predictive tool to identify the reactive sites for different species in different chemical environments.
4. CONCLUSIONS The structure and bonding in charged and doped aluminum clusters have been investigated using Regional Density Functional Theory (RDFT). The RDFT method provides a measure of the electronic stress tensor from which it is possible to determine bond indices and electronic chemical potential. We find that for the clusters in this study the “spindle structure” (with tensile stress region) does not appear at all inside the cluster cages indicating that there is no covalency to the X-Al bonds of these species; i.e., these bonds are metallic. However, regions of tensile stress can be identified in surface Al-Al bonds, and the character of these bonds differs depending on the dopant atom and cluster charge. Overall there is a general correlation between bond strength and bond length with shorter metallic bonds exhibiting higher bond indices, while longer more covalent bonds have smaller bond indices. This correlation is particularly strong for the bonds between core and surface atoms with the exception of Al12Mg.
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The distribution of bμ indices on the surface of Al12X clusters provides not only grounds for understanding of the bonding and stability but also some insight into the regioselectivity observed in reactions of the clusters. Surface bonds that exhibit only a low level of covalency (i.e., are more metallic) hinder neutral addition type reactions, such as H2 adsorption, and therefore this contributes to the high reaction barriers. The uniformity of the Al-Al bonds of the highly symmetric 40 valence electron clusters means that there is no preferred edge/face for adsorption of H2. Consequently, the previously observed preference for a bridge-type transition structure can be accounted for since this requires direct involvement of only two surface Al atoms and, therefore, primarily leads to disruption of only one surface Al-Al bond. In comparison, the lowest Al-Al surface bond indices in all three 39 valence electron clusters (Al12Mg-, Al13, and Al12Siþ) are observed between triangles of Al atoms on either side of the clusters indicating these are the regions from which it will be the most easy to remove an electron. The tensile stress for the bonds comprising these triangles indicates a significant degree of covalency, making these faces ideally suited to accommodate a hollow-type transition state for H2 adsorption, in agreement with our previously observed regioselectivity.45 The bonds between the Al atoms of the central region also exhibit small bμ and significant covalency and would be suitable for bridge-type transition states as previously observed for Al13. The chemical potential bond orders also provide a clear explanation of the strong correlation observed between reaction barrier and the cluster distortion energy in the transition states for H2 addition to Al12Xz clusters. Surface Al-Al bonds with high bμ and low covalency (i.e., are more metallic) require significant energy to distort to facilitate addition of H2, making a significant contribution to the barrier. In comparison, surface Al-Al bonds with low bμ and high covalency require significantly less energy to distort to accommodate the approaching H2 molecule, and consequently, there is a smaller contribution to the barrier. Clusters with smaller electronic chemical potential (μ) on the surface should have greater potential to transfer density to an electrophile, and this is generally reflected in the higher binding energies for hydrogen to Al12Mg and Al13 compared with Al12Si. Similarly, the cluster cations exhibit greater μ on the surface of the clusters reflecting the greater electron affinity of these species. In comparison, the anionic clusters have a greater distribution of small μ on their surface reflecting a weaker electron affinity. In general, isoelectronic clusters display similar relative distribution patterns of chemical potential on the cluster surfaces; however, the absolute magnitudes of the chemical potentials generally differ with charge and dopant X.
’ ASSOCIATED CONTENT
bS
Supporting Information. Table S1 contains spin-density diagrams for the Al12Xz clusters, and Table S2 contains GAUSSIAN archive entries showing PBE/6-311G(d) and PW91/ 6-311G(d) geometries for all species considered in the present work. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 1721
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’ ACKNOWLEDGMENT We gratefully acknowledge the award of an Australian Research Council Discovery Grant (IY and AT) and a Japan Society for the Promotion of Science, Short Term Invitation Fellowship (DJH), to carry out this work. We also gratefully acknowledge allocation of computing time from the Australian National Computational Infrastructure (NCI) facility. ’ REFERENCES (1) Metal Clusters; Moskovits, M., Ed.; Wiley and Sons: New York, 1986. (2) Clusters of Atoms and Molecules: Theory, Experiment and Clusters of Atoms; Haberland, H., Ed.; Springer Series in Chemical Physics; Springer Verlag: New York, 1995; Vol. 52. (3) Clusters and Nanomaterials; Kawazoe, Y., Ohno, K., Kondow, T., Eds.; Springer: New York, 2002. (4) Clusters and Nano-Assemblies; Jena, P., Khanna, S. N., Rao, B. K., Eds.; World Scientific: Singapore, 2005. (5) Ahlrichs, R.; Elliott, S. D. Phys. Chem. Chem. Phys. 1998, 1, 13–21. (6) Goldberg, A.; Yarovsky, I. Phys. Rev. B 2007, 75, 195403. (7) Jung, J.; Han, Y.-K. J. Chem. Phys. 2006, 125, 064306. (8) Wang, L.; Zhao, J.; Zhou, Z.; Zhang, S. B.; Chen, Z. J. Comput. Chem. 2009, 30, 2509–2514. (9) Xie, R.-H.; Bryant, G. W.; Zhao, J.; Kar, T.; Smith, V. H. Phys. Rev. B 2005, 71, 125422. (10) Cox, D. M.; Trevor, D. J.; Whetten, R. L.; Kaldor, A. J. Phys. Chem. 1988, 92, 421–429. (11) Schriver, K.; Persson, J. L.; Honea, E. C.; Whetten, R. L. Phys. Rev. Lett. 1990, 64, 2539–2542. (12) (a) Gantef€or, G.; Meiwes-Broer, K. H.; Lutz, H. O. Phys. Rev. A 1988, 37, 2716–2718. (b) Gantef€or, G.; Eberhardt, W. Chem. Phys. Lett. 1994, 217, 600–604. (13) Taylor, K. J.; Pettiette, C. L.; Craycraft, M. J.; Chesnovsky, O.; Smalley, R. E. Chem. Phys. Lett. 1988, 152, 347–352. (14) (a) Cha, C.-Y.; Gantef€or, G.; Eberhardt, W. J. Chem. Phys. 1994, 100, 995–1010. (b) Gantef€or, G.; Eberhardt, W. Chem. Phys. Lett. 1994, 217, 600–604. (15) Li, X.; Wu, H.; Wang, X.-B.; Wang, L.-S. Phys. Rev. Lett. 1998, 81, 1909–1912. (16) Li, X.; Wang, L.-S. Phys. Rev. B 2002, 65, 153404. (17) De Heer, W. A.; Milani, P.; Ch^atelain, A. Phys. Rev. Lett. 1989, 63, 2834–2936. (18) (a) Jarrold, M. F.; Bower, J. E.; Kraus, J. S. J. Chem. Phys. 1987, 86, 3876–3885. (b) Jarrold, M. F.; Bower, J. E. J. Chem. Phys. 1987, 87, 1610–1619. (c) Ray, U.; Jarrold, M. F.; Bower, J. E.; Kraus, J. S. J. Chem. Phys. 1989, 91, 2912–2921. (19) King, F. L.; Ross, M. M. Chem. Phys. Lett. 1989, 164, 131– 136. (20) Cottancin, E.; Pellarin, M.; Lerme, J.; Baguenard, B.; Palpant, B.; Vialle, J. L.; Broyer, M. J. Chem. Phys. 1997, 107, 757–770. (21) (a) Jarrold, M. F.; Bower, J. E. J. Chem. Phys. 1986, 85, 5373– 5375. (b) Jarrold, M. F.; Bower, J. E. J. Chem. Phys. 1987, 87, 5728–5738. (c) Jarrold, M. F.; Bower, J. E. Chem. Phys. Lett. 1988, 144, 311–316. (d) Jarrold, M. F.; Bower, J. E. J. Am. Chem. Soc. 1988, 110, 70–78. (22) (a) Leuchtner, R. E.; Harms, A. C.; Castleman, A. W. J. Chem. Phys. 1989, 91, 2753–2754. (b) Leuchtner, R. E.; Harms, A. C.; Castleman, A. W. J. Chem. Phys. 1991, 94, 1093–1101. (23) McHenry, M. E.; Eberhart, M. E.; O'Handley, R. C.; Johnson, K. H. Phys. Rev. Lett. 1986, 56, 81–84. (24) (a) Bauschlicher, C. W.; Pettersson, L. G. M. J. Chem. Phys. 1986, 84, 2226–2232. (b) Pettersson, L. G. M.; Bauschlicher, C. W.; Halicioglu, T. J. Chem. Phys. 1987, 87, 2205–2213. (25) (a) Yi, J.-Y.; Oh, D. J.; Bernholc, J.; Car, R. Chem. Phys. Lett. 1990, 174, 461–466. (b) Yi, J.-Y.; Oh, D. J.; Bernholc, J. Phys. Rev. Lett. 1991, 67, 1594–1597.
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The Journal of Physical Chemistry C
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Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford, CT, 2004. (61) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3868. (62) Perdew, J. P.; Wang, Y. Phys. Rev. B 1996, 45, 13244–13249. (63) Doi, K.; Szarek, P.; Nakamura, K.; Senami, M.; Tachibana, A. Molecular Regional DFT program package, Ver. 2; Kyoto, Tachibana Lab.: Kyoto University, 2007. (64) Fowler, J. E.; Ugalde, J. M. Phys. Rev. A 1998, 58, 383–388. (65) Cioslowski, J. J. Am. Chem. Soc. 1989, 111, 8333–8336. (66) Humphrey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33. (67) DeLano, W. L. The PyMOL Molecular Graphics System; DeLano Scientific LLC: San Carlos, CA, USA. (68) The Mulliken spin density of Al12Mg is about twice the value observed for the charged clusters. (69) There are a few Al-Al bonds for which Lagrange points were not found (in Al13þ, Al12Mgþ), and the corresponding bond lengths were longer than 3 Å, which is around 10% longer than the average Al-Al bond distance in those clusters. (70) Khanna, S. N.; Jena, P. Chem. Phys. Lett. 1994, 219, 479–483. (71) Majumder, C.; Das, G. P.; Kulshrestha, S. K.; Shah, V.; Kanhere, D. G. Chem. Phys. Lett. 1996, 261, 515–520. (72) Charkin, O. P.; Kochnev, V. K.; Klimenko, N. M. Russ. J. Inorg. Chem. 2006, 51, 1925–1936.
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dx.doi.org/10.1021/jp109804y |J. Phys. Chem. C 2011, 115, 1714–1723
