ARTICLE pubs.acs.org/JPCA
Endohedral Beryllium Atoms in Ten-Vertex Germanium Clusters: Effect of a Small Interstitial Atom on the Cluster Geometry R. B. King,*,† I. Silaghi-Dumitrescu,‡,§ and M. M. Ut-a‡ † ‡
Department of Chemistry, University of Georgia, Athens, Georgia 30602, United States Faculty of Chemistry and Chemical Engineering, Babes--Bolyai University, Cluj-Napoca, Roumania
bS Supporting Information ABSTRACT: Ten-vertex clusters are unusually versatile because polyhedra with 3-, 4-, and 5-fold symmetry are possible and are found in experimentally known structures. Such clusters therefore provide useful probes for subtle effects on cluster structure such as changing the electron count or introducing an interstitial atom. In this connection, DFT shows that one of the smallest possible interstitial atoms, namely beryllium, has relatively little effect on the structures of Be@Ge10z (z = þ2, 0, 2, 4) clusters. Thus the same C3v and D4d polyhedra are found as the lowest energy structures for the isoelectronic pairs Be@Ge102þ/Ge10 and Be@Ge10/Ge102. Even for the more complicated potential energy surfaces of the Be@Ge102/Ge104 and Be@Ge104/Ge106 systems, the lowest energy structures are remarkably similar. Thus the same C2v structures are the global minima for both Be@Ge102 and Ge104. Similarly, the same slipped pentagonal prism structures are the global minima for both Be@Ge104 and Ge106.
1. INTRODUCTION Four different ten-vertex bare post-transition element polyhedra have been shown to host interstitial guest atoms in M@E10 structures that have been synthesized and characterized structurally by X-ray diffraction (Figure 1). Thus a D4d bicapped square antiprism is found to encapsulate a group 12 metal atom in the anionic indium cluster Zn@In108 found in the intermetallic1 K8In10Zn as well as the lead clusters M@Pb102 in [K(2,2,2crypt)]2[M@Pb10] (M = Ni, Pd, Pt).2,3 However, in the M@In1010 clusters found in the K10In10M intermetallics (M = Ni, Pd, Pt), isoelectronic with Zn@In108, the encapsulating polyhedron is a C3v tetracapped trigonal prism.4 The pentagonal antiprism is the host polyhedron for an interstitial palladium atom in the cationic bismuth cluster Pd@Bi104þ in Bi14PdBr16 (= [Pd@Bi10][BiBr4]4).5 The pentagonal prism has been found as the host polyhedron for an iron or cobalt atom in the clusters M@Ge103 (M = Fe,6 Co7). These pentagonal prismatic clusters are particularly unusual since the pentagonal prism has no triangular faces at all but only rectangular and pentagonal faces. This is in stark contrast to the fundamental deltahedra of borane chemistry in which all of the faces are triangles. Furthermore, recent theoretical studies8,9 suggest that in some cases the preferred cluster polyhedron for enclosing a central metal atom is simply the polyhedron with the largest internal volume rather than a polyhedron meeting the electronic requirements of the WadeMingos rules.1013 For the tenvertex systems, this polyhedron is the pentagonal prism with the minimum number of edges and hence no triangular faces. r 2011 American Chemical Society
The number of electrons contributed by the endohedral atoms to the skeletal bonding of polyhedral metal clusters is of interest, particularly in connection with the applicability of the WadeMingos rules.7,1013 This is particularly crucial for the ten-vertex clusters where four different host polyhedral structures (Figure 1) have been found, as noted above. In addition, this is also critical for clusters containing interstitial transition metals exhibiting multiple oxidation states, as exemplified by iron and cobalt in the recently reported and structurally characterized centered pentagonal prismatic clusters M@Ge103 (M = Fe,6 Co7). In order to assess the role of an interstitial atom in determining cluster shape, the ten-vertex clusters Be@Ge10z (z = þ2, 0, 2, and 4) have been investigated by DFT. Beryllium was chosen as the central atom for the following reasons: (1) Beryllium has a single oxidation state, namely þ2. Therefore, the number of electrons donated by an interstitial beryllium is expected to be unambiguous, namely two electrons. (2) Beryllium has the smallest size of any feasible metal. Therefore, an interstitial beryllium atom is likely to fit inside a Ge10 cluster polyhedron with minimal distortion. Preliminary studies suggest that incorporation of alkaline earth metals larger than beryllium, namely magnesium Received: November 8, 2010 Revised: February 24, 2011 Published: March 16, 2011 2847
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Figure 2. Two low-energy Be@Ge102þ structures.
Figure 1. Four ten-vertex polyhedra found as host polyhedra for interstitial metal atoms. For clarity, the edges of capping vertices are shown in green.
and particularly calcium, lead to significant distortion and sometimes complete rupture of the host polyhedron.
2. THEORETICAL METHODS Geometry optimizations were carried out at the hybrid DFT B3LYP level1417 with the 6-31G(d) (valence) double-ζ quality basis functions extended by adding one set of polarization (d) functions for both the interstitial and germanium atoms. The Gaussian 03 package of programs18 was used in which the fine grid (75,302) is the default for numerically evaluating the integrals and the tight (108) hartree stands as default for the self-consistent field convergence. Computations were carried out using six initial geometries including ten-vertex polyhedra with 3-fold, 4-fold, and 5-fold symmetry including the four polyhedra in Figure 1 as well as a prolate C3v structure and the D4h bicapped cube. The symmetries were maintained during the initial geometry optimization processes. Symmetry breaking using modes defined by imaginary vibrational frequencies was then used to determine optimized structures with minimum energies. Vibrational analyses show that all of the final optimized structures discussed in this paper are genuine minima at the B3LYP/6-31G(d) level without any significant imaginary frequencies (Nimag = 0). In a few cases the calculations ended with acceptable small imaginary frequencies19 and these values are indicated in the corresponding figures. The optimized structures found for the Be@Ge10z derivatives are labeled by the number of skeletal electrons and the order of the principal rotation axis of the final structure. Distorted structures derived from an initial structure with 5-fold symmetry are starred. The clusters have been optimized at both singlet and triplet spin multiplicities. The triplet optimized structures are indicated by T. Thus, the structure of singlet neutral Be@Ge10 obtained from the bicapped square antiprism of D4d symmetry is labeled 224, whereas the triplet neutral Be@Ge10 obtained from the tetracapped trigonal prism of C3v symmetry is labeled 223T (Figure 3). Additional details of all of the optimized structures, including all interatomic distances, the initial geometries leading to a given optimized structure, and structures with
energies too high to be of possible chemical relevance are provided in the Supporting Information. In assigning polyhedra to the optimized structures, the GeGe distances less than ∼3.2 Å were normally considered as polyhedral edges; significant exceptions are noted in the text. Similarly, BeGe distances less than ∼2.8 Å are considered bonding distances; most such BeGe bonding distances were less than ∼2.5 Å except for some of the less regular polyhedra. Only structures within 40 kcal/mol of the global minima are discussed in the text; some higher energy structures are included in the Supporting Information.
3. RESULTS 3.1. Dication. Be@Ge102þ (Figure 2). The lowest energy
structure for the 20 skeletal electron Be@Ge102þ system is the C3v structure 203 (Figure 2) similar to the predicted lowest energy structure of Ni@Ge10 and the experimental structure4 of the anion Ni@In1010 found in the intermetallic K10NiIn10. These nickel-centered ten-vertex clusters can be considered as 20 skeletal electron systems like Be@Ge102þ because the interstitial nickel atom functions as a zero electron donor because of its filled d10 shell. The C3v Be@Ge102þ structure 203 has three quadrilateral faces meeting at the unique vertex on the C3 axis as well as 10 triangular faces. A higher energy D4d bicapped square antiprismatic structure 204 is also found for Be@Ge102þ at only 6.0 kcal/mol above the global minimum 203. Previous computational studies have also shown that similar C3v and D4d clusters at this particular skeletal electron count are very close in energy. Thus, for the Ni@Ge10 clusters, the difference in energy between the C3v global minimum and the structure of D4d symmetry is 4.6 kcal/mol, computed at the same level of theory used in this study.20 However, for the isoelectronic Zn@Ge102þ clusters discussed in the same article, the D4d cluster is the global minimum in accord with the experimental structure,1 closely followed by the C3v cluster, at 1.0 kcal/mol. Another computational study places the D4d Ni@Ge10 as global minimum, followed by the C3v structure at 1.8 kcal/mol,8 but using a different basis set. The differences in energy for the two Ni@Ge10 structures depending on the level of theory are discussed there. However, for the neutral bare Ge10 clusters, the C3v structure is a global minimum, whereas a triplet D4d structure lies at 35.6 kcal/mol above, computed at the same level of theory as the one used here.21 Apparently the encapsulation of a beryllium ion leads to a change in the spin state of the D4d clusters, significantly lowering the gap in energy between the C3v and the D4d structures. 3.2. Neutral Be@Ge10 (Figure 3). The lowest energy structure for the 22 skeletal electron neutral Be@Ge10 species is the D4d 2848
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Figure 3. Two low-energy Be@Ge10 structures.
Figure 4. Six low-energy Be@Ge102 structures.
bicapped square antiprism 224 (Figure 3). The next higher energy structure found for Be@Ge10 is a triplet C3v structure (223T) closely related to the lowest energy singlet structure 203 found for the dication Be@Ge102þ (Figure 2). The triplet spin state for the neutral Be@Ge10 arises in this case by addition of two electrons into a doubly degenerate LUMO of the corresponding dication Be@Ge102þ. A singlet spin state of this C3v structure would result in a JahnTeller distortion thereby lowering its symmetry. However, the Be@Ge10 structure 223T lies 16.1 kcal/mol above the global minimum 224 indicating that the latter Be@Ge10 structure is highly favorable in accord with the expectation of the WadeMingos rules.1013 The D4d bicapped square antiprism found in 224 is the most spherical ten-vertex deltahedron and is also the polyhedron found in the very stable ten-vertex polyhedral borane dianion B10H102. Our previous computational studies predicted the D4d structure to be also the global minimum for the isoelectronic empty bare Ge102 clusters.21 Such a bicapped square antiprismatic Ge102 cluster has been recently found experimentally as a monodentate ligand in the manganese carbonyl trianion [Ge10Mn(CO)4]3-.22 Similarly, our previous calculations on the 22 skeletal electron germanium clusters doped with nickel and zinc show the D4d structure as the global minimum.8,20 3.3. Dianion Be@Ge102 (Figure 4). The potential energy surface for the 24-skeletal electron system Be@Ge102 appears to be significantly more complicated than for the Be@Ge102þ and Be@Ge10 energy surfaces since six structures are found within 20 kcal/mol of the global minimum (Figure 4). The global
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minimum structure 242 for Be@Ge102 is a C2v structure with 12 triangular faces and two rectangular faces sharing an edge. The next two Be@Ge102 structures are both based on C3v polyhedra but have rather different topologies (Figure 4). The C3v singlet Be@Ge102 structure 243 at 2.1 kcal/mol above 242 is similar to the Be@Ge10 structure 223T and the Be@Ge102þ structure 203. This structure has three quadrilateral faces meeting at the unique vertex on the C3 axis as well as 10 triangular faces. All ten germanium atoms in the cluster are within bonding distance (i. e., < 2.5 Å) of the central beryllium atom so the 243 C3v polyhedron is an essentially spherical polyhedron. The higher energy C3v triplet Be@Ge102 structure 243T is a prolate (elongated) structure, also with three quadrilateral and ten triangular faces. This structure is derived by elongating the underlying trigonal prism so that the original vertical edges are too long to represent direct bonds. The unique vertex in 243T located on the C3 axis then forms a tetrahedral cavity. Only 9 of the 10 germanium atoms in 243T are within bonding distance (i. e., < 2.4 Å) of the central beryllium atom. Thus, the unique vertex in 243T located on the C3 axis is ∼3.62 Å away from the beryllium atom, obviously a nonbonding distance. The three remaining Be@Ge102 structures within ∼20 kcal/mol of the global minimum, namely the various 245* structures, represent various distortions of a pentagonal prismatic structure (Figure 4). These three structures have similar relative energies, namely 18 to 20 kcal/mol above the Be@Ge102 global minimum 242. A general feature of these structures is the acentric location of the central beryllium toward one of the pentagonal faces of the original pentagonal antiprism thereby forming a distorted pentagonal pyramidal cavity. A related hexagonal pyramidal cavity is found in the solid state beryllium boride carbide BeB2C2.23 In all three of these 245* structures the beryllium atom is within bonding distance ( 3.0 Å) of the central beryllium atom. These differences have little effect on the relative energies of these structures since the three 245* structures lie within 2 kcal/mol of each other. 3.4. Tetraanion Be@Ge104. The potential energy surface of the tetraanion Be@Ge104 is even more complicated than that of the dianion Be@Ge102. Thus eight structures of the 26-skeletal electron tetraanion Be@Ge104 lie within 18 kcal/mol of the global minimum 265* (Figure 5). These Be@Ge104 structures represent a variety of different structure types. The Be@Ge104 global minimum 265* has a structure closely related to the 245* structures of the dianion Be@Ge102 discussed above (Figure 5). The beryllium atom in 265* is located acentrically inside a distorted Ge10 pentagonal prism so that the beryllium atom is within bonding distance of all five germanium atoms of one of the original pentagonal faces. In addition, the beryllium atom is within bonding distance of two of the remaining germanium atoms (7 and 8 in Figure 5) so that the beryllium atom is within bonding distance of a total of seven germanium atoms. The next Be@Ge104 structure, namely the C2v structure 262, lies 3.6 kcal/mol above the 265* global minimum (Figure 5). In this structure, the germanium atoms have split into two separate Ge5 units. The upper Ge5 unit (Figure 5) forms a distorted BeGe5 octahedral cluster with the beryllium atom. In 2849
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starting structure but has the beryllium inside a Ge8 cube that is part of the Ge10 system. The remaining two germanium atoms cap one of the faces of the cube. The beryllium atom is within bonding distance (i. e., < 2.8 Å) of the eight germanium atoms forming the cube as well as one of the capping germanium atoms. The next Be@Ge104 structure, namely 265 lying 15.7 kcal/mol above the global minimum 265*, is of interest since it is the regular D5d pentagonal antiprism expected by the Wade-Mingos rules1013 for this system with 26 skeletal electrons. Thus structure 265 is an arachno structure derived from a Be@Ge12 regular icosahedron by removal of an antipodal pair of vertices. All of the ten BeGe distances in 265 are equivalent bonding distances of 2.518 Å. The next Be@Ge104 structure 262, lying 16.8 kcal/mol above 265*, is a C2v polyhedron with eight triangular and four quadrilateral faces. The beryllium atom in 262 is within bonding distance (i. e., < 2.8 Å) of all ten germanium atoms. The final structure among the eight Be@Ge104 structures within 18 kcal/mol of the global minimum is a relatively open Cs structure 261 lying 17.2 kcal/mol above 265*. The beryllium atom is within bonding distance (i. e., < 2.7 Å) of eight of the ten germanium atoms in irregular cubic coordination.
Figure 5. Eight low-energy Be@Ge104 structures.
addition, the beryllium atom is within bonding distance of two of the germanium atoms of the other Ge5 unit. The total of 26 skeletal electrons for the Be@Ge104 structure 262 can be partitioned into 14 electrons for the upper octahedral BeGe5 unit corresponding to the 2n þ 2 = 14 skeletal electrons (n = 6) expected from the WadeMingos rules.1013 The remaining 12 skeletal electrons of 262 can be allocated to the lower Ge5 trigonal bipyramidal unit, again in accord with the WadeMingos rules. The next Be@Ge104 structure 263, lying 10.2 kcal/mol above the global minimum 265*, is a singlet prolate C3v polyhedral structure (Figure 5). This structure is very similar to the triplet Be@Ge102 structure 243T (Figure 4). In the C3v Be@Ge104 structure 263 the HOMO appears to be a full doubly degenerate orbital whereas in the similar C3v Be@Ge102 structure 243T this doubly degenerate orbital is only half full, thereby leading to the triplet spin multiplicity. As in 243T the beryllium atom in 263 is within bonding distance of nine of the ten germanium atoms. The next Be@Ge104 structure 261, at 10.4 kcal/mol above 265*, is an irregular structure with the beryllium atom in the middle of a bent pentagonal face (atoms 2, 7, 4, 6, and 10 in Figure 5) with BeGe distances in the range 2.14 to 2.28 Å. The central beryllium forms longer bonds (2.75 and 2.91 Å) with two of the remaining germanium atoms thereby accounting for this highly distorted structure. The Be@Ge104 structure 265* of C1 symmetry lying 15.6 kcal/mol above the global minimum 265* of Cs symmetry is interesting since it arises from a pentagonal prism
4. DISCUSSION Comparison of the structures of the beryllium-centered tenvertex germanium clusters Be@Ge10z with those predicted in a previous study21 of the isoelectronic empty clusters Ge10z2 using similar theoretical methods indicates that the interstitial beryllium atom is small enough not to affect the preferred lowest energy structures. Thus, for the dication Be@Ge102þ with 20 skeletal electrons (Figure 2) the lowest energy structure is the spherical C3v structure 203 followed by a D4d bicapped square antiprismatic structure 204 at ∼6 kcal/mol above this global minimum. A similar C3v structure is the global minimum for the isoelectronic empty neutral cluster Ge10. However, for the empty Ge10 cluster the D4d bicapped square antiprism is predicted to lie ∼29 kcal/mol above this global minimum. The neutral Be@Ge10 cluster is a 22 skeletal electron system, which is isoelectronic with Ge102 as well as the polyhedral borane dianion B10H102, known experimentally24 to have the D4d bicapped square antiprismatic structure (Figure 1) predicted by the Wade-Mingos rules.1013 An experimental Ge102- cluster of D4d symmetry has been recently characterized in the complex [Ge10Mn(CO)4]3.22 This structure is predicted theoretically to be the lowest energy structure for both Be@Ge10 and Ge102. In both cases a C3v structure is predicted to lie ∼16 kcal/mol above the D4d global minimum. The patterns of the three lowest energy structures for the isoelectronic 24 skeletal electron systems Be@Ge102 and Ge104 are rather similar. In both cases the lowest energy structures, for example, 242 for Be@Ge102 (Figure 4), are C2v structures with 12 triangular faces and two rectangular faces sharing an edge. Also, for both Be@Ge102 and Ge104 the spherical C3v structures, for example, 243 for Be@Ge104 (Figure 4), lie ∼2 kcal/mol above the C2v global minima. The third lowest energy structures for both Be@Ge102 and Ge104 are prolate triplet spin state C3v structures, for example, 243T for Be@Ge104 (Figure 4). For Ge104 this prolate C3v structure lies ∼8 kcal/mol above the C2v global minimum, whereas for Be@Ge102 the corresponding structure lies somewhat higher at ∼14 kcal/mol above the corresponding global minimum. 2850
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The Journal of Physical Chemistry A This energy difference may relate to the difficulty of fitting even the relatively small beryllium atom inside the nine-vertex polyhedral cavity of the prolate C3v structure of 243T. The spherical tenvertex C3v polyhedron of 243 (Figure 4) has a single cavity formed by a Ge10 polyhedron using all ten germanium vertices. However, the prolate C3v polyhedron of 243T has its internal volume partitioned into two cavities, namely a cavity inside a ninevertex polyhedron encompassing all of the germanium vertices except the unique vertex on the C3 axis and a second much smaller tetrahedral cavity including this unique germanium vertex. For the 26-skeletal electron Be@Ge104 and Ge106 clusters the lowest energy structures are polyhedra conveniently described as “slipped pentagonal prisms” (e.g., structure 265* for Be@Ge104 in Figure 5) rather than the arachno D5d pentagonal antiprismatic structures predicted by the WadeMingos rules.1013 The arachno D5d pentagonal antiprismatic structures (e.g., structure 265 for Be@Ge104 in Figure 5) are also found but at the higher energies of ∼17 kcal/mol for both systems. Intermediate in energy between these two structures for both systems are singlet prolate C3v structures at 10.2 kcal/mol for Be@Ge104 (structure 263 in Figure 5) and 5.9 kcal/mol for Ge106above the corresponding slipped pentagonal prismatic global minima. Two additional Be@Ge104 structures were found lying in energy between the global minimum 265* and the arachno pentagonal antiprismatic structure 265 (Figure 5). These structures reflect clearly the effect of the interstitial beryllium atom. In structure 262 the Ge10 polyhedron is split into two separate Ge5 units. One of these Ge5 units combines with the beryllium to form a BeGe5 distorted octahedron. The other Ge5 unit remains as a trigonal bipyramid with the beryllium atom bridging an edge. The 26 skeletal electrons in the Be@Ge104 structure 262 can be partitioned between these two polyhedra in accord with the Wade-Mingos rules.1013 Thus, 14 of these skeletal electrons can be allocated to the octahedral BeGe5 octahedral cavity in accord with the requirement of 2n þ 2 skeletal electrons for a closed most-spherical deltahedron—here n = 6. The remaining 12 skeletal electrons can be allocated to the Ge5 trigonal bipyramid in a similar manner. The other low -energy Be@Ge104 structure 261 (Figure 5) not found for the isoelectronic Ge106 is a highly unsymmetrical structure with the beryllium atom in nearly the center of a warped pentagonal face. This polyhedron would appear to be very unlikely in the absence of such an interstitial atom to hold together a pentagonal face. The binding energies of the beryllium atom to the bare germanium clusters have also been computed for all charges considered in this article. The results show that the incorporation of the beryllium cation is highly exothermic, especially for the negatively charged clusters: Ge10 þ Be2þ f Be@Ge10 2þ þ 345:9 kcal=mol Ge10 2- þ Be2þ f Be@Ge10 þ 715:6 kcal=mol Ge10 4- þ Be2þ f Be@Ge10 2- þ 1061:5 kcal=mol Ge10 6- þ Be2þ f Be@Ge10 4- þ 1385:0 kcal=mol The energies of the corresponding global minima for both bare germanium clusters and clusters doped with beryllium have
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been considered. Therefore, we can conclude that such beryllium doped clusters might prove to be more stable than the corresponding bare ones, since the presence of the interstitial beryllium atom stabilizes the cluster without causing significant distortion.
5. SUMMARY Ten-vertex clusters are unusually versatile since polyhedra with 3-, 4-, and 5-fold symmetry are found in their structures (Figure 1). They therefore provide a useful probe for subtle effects on cluster structure such as changing the electron count or introducing an interstitial atom. The theoretical studies reported in this paper show that the smallest possible interstitial atom, namely beryllium, stabilizes the cluster with relatively little effect on the preferred cluster structures. Thus the same C3v and D4d polyhedra are found as the lowest energy structures for the isoelectronic pairs Be@Ge102þ/Ge10 and Be@Ge10/Ge102. Even for the more complicated potential energy surfaces of the Be@Ge102/Ge104 and Be@Ge104/Ge106 systems, the lowest energy structures are remarkably similar. Thus, the same C2v structures are the global minima for both Be@Ge102 and Ge104. Similarly, the same “slipped pentagonal prism” structures are the global minima for both Be@Ge104 and Ge106. ’ ASSOCIATED CONTENT
bS
Supporting Information. Tables of distances; complete Gaussian reference (ref 18). This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Notes §
Deceased December 25, 2009.
’ ACKNOWLEDGMENT This work was supported by CNCSIS-UEFISCSU, project number PNII-RU 465/2010, in Romania and by the National Science Foundation under Grant CHE-0716718 in the U.S.A. ’ REFERENCES (1) Sevov, S. C.; Corbett, J. C. Inorg. Chem. 1993, 32, 1059. (2) Esenturk, E. N.; Fettinger, J.; Eichhorn, B. Chem. Commun. 2005, 247. (3) Esenturk, E. N.; Fettinger, J.; Eichhorn, B. J. Am. Chem. Soc. 2006, 128, 9178. (4) Henning, R. W.; Corbett, J. D. Inorg. Chem. 1999, 38, 3883. (5) Ruck, M.; Dubenskyy, V.; S€ohnel, T. Angew. Chem., Int. Ed. 2003, 45, 2978. (6) Zhou, B.; Denning, M. S.; Kays, D. L.; Goicoechea, J. M. J. Am. Chem. Soc. 2009, 132, 2802. (7) Wang, J.-Q.; Stegmaier, S.; F€assler, T. F. Angew Chem., Int. Ed. 2009, 48, 1998. (8) King, R. B.; Silaghi-Dumitrescu, I.; Ut-a, M. M. J. Phys. Chem. A 2009, 113, 527. (9) King, R. B.; Silaghi-Dumitrescu, I.; Ut-a, M. M. Inorg. Chem. 2009, 48, 8508. (10) Wade, K. Chem. Commun. 1971, 792. (11) Wade, K. Adv. Inorg. Chem. Radiochem. 1976, 18, 1. 2851
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