Article pubs.acs.org/JPCA Endohedral Beryllium Atoms in Germanium Clusters with Eight and Fewer Vertices: How Small Can a Cluster Be and Still Encapsulate a Central Atom? M M Uţa†̆ and R B King*,‡ † Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai University, Cluj-Napoca, Romania Department of Chemistry, University of Georgia, Athens, Georgia 30602, United States ‡ S Supporting Information * ABSTRACT: Structures of the beryllium-centered germanium clusters Be@Genz (n = 8, 7, 6; z = −4, −2, 0, +2) have been investigated by density functional theory to provide some insight regarding the smallest metal cluster that can encapsulate an interstitial atom The lowest energy structures of the eight-vertex Be@Ge8z clusters (z = −4, −2, 0, +2) all have the Be atom at the center of a closed polyhedron, namely, a D4d square antiprism for Be@Ge84−, a D2d bisdisphenoid for Be@Ge82−, an ideal Oh cube for Be@Ge8, and a C2v distorted cube for Be@Ge82+ The Becentered cubic structures predicted for Be@Ge8 and Be@Ge82+ differ from the previously predicted lowest energy structures for the isoelectronic Ge82− and Ge8 This appears to be related to the larger internal volume of the cube relative to other closed eight-vertex polyhedra The lowest energy structures for the smaller seven- and six-vertex clusters Be@Genz (n = 7, 6; z = −4, −2, 0, +2) no longer have the Be atom at the center of a closed Gen polyhedron Instead, either the Gen polyhedron has opened up to provide a larger volume for the Be atom or the Be atom has migrated to the surface of the polyhedron However, higher energy structures are found in which the Be atom is located at the center of a Gen (n = 7, 6) polyhedron Examples of such structures are a centered C2v capped trigonal prismatic structure for Be@Ge72−, a centered D5h pentagonal bipyramidal structure for Be@Ge7, a centered D3h trigonal prismatic structure for Be@Ge64−, and a centered octahedral structure for Be@Ge6 Cluster buildup reactions of the type Be@Genz + Ge2 → Be@Gen+2 z (n = 6, 8; z = −4, −2, 0, +2) are all predicted to be highly exothermic This suggests that interstitial clusters having an endohedral atom inside a bare post transition element polyhedron with eight or fewer vertices are less than the optimum size This is consistent with the experimental observation of several types of 10-vertex polyhedral bare post transition element clusters with interstitial atoms but the failure to observe such clusters with external polyhedra having eight or fewer vertices INTRODUCTION In recent years a variety of species have been synthesized having structures in which a single atom is encapsulated in the center of a bare post-transition element cluster Ten-vertex polyhedra appear to be particularly suitable host polyhedra for bare post-transition metal clusters to encapsulate such an interstitial atom, since four different 10-vertex polyhedra have been found experimentally in post-transition element clusters containing an interstitial atom (Figure 1) Thus a D4d bicapped square antiprism encapsulates a group 12 metal atom in the anionic indium cluster Zn@In108− found in the intermetallic1 K8In10Zn In addition the same D4d bicapped square antiprism is found in 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@Bi104+ in Bi14PdBr16 (= [Pd@Bi10][BiBr4]4).5 The pentagonal prism has been found as the host polyhedron for an encapsulated iron or cobalt atom in the clusters M@Ge103− (M = Fe,6 Co7) © 2012 American Chemical Society Despite the variety of different 10-vertex polyhedra encapsulating interstitial atoms, there are both experimental and theoretical indications of limitations in the size of the interstitial atom that can be encapsulated in 10-vertex polyhedra This can be seen in attempts to use the Wade− Mingos rules,8−11 which are well established for polyhedral borane chemistry, to rationalize the shapes of the 10-vertex polyhedra encapsulating interstitial atoms To apply the Wade− Mingos rules to bare post-transition element clusters, the number of skeletal electrons contributed by each polyhedron vertex atom is taken to correspond to the number of electrons in excess of a 12-electron filled d10s2 shell Thus the group 13 vertex atoms Ga, In, and Tl each donate a single skeletal electron, the group 14 vertex atoms Ge, Sn, and Pb each donate two skeletal electrons, and the group 15 vertex atoms As, Sb, and Bi each donate three skeletal electrons The number of skeletal electrons donated by an interstitial transition or posttransition metal atom corresponds to the number of electrons Received: March 1, 2012 Revised: May 7, 2012 Published: May 7, 2012 5227 dx.doi.org/10.1021/jp302052u | J Phys Chem A 2012, 116, 5227−5234 The Journal of Physical Chemistry A Article have a D4d bicapped square antiprismatic structure with the D5h pentagonal prismatic isomer lying ∼6 kcal/mol above this structure However, for the M@Ge102− clusters containing the larger Pd and Pt atoms, the D5h pentagonal prismatic structure appears to be the lowest energy structure All of the experimental and theoretical studies discussed above suggest that 10-vertex polyhedra can have large enough cavities to accommodate interstitial atoms In fact, the formation of 10-vertex clusters with interstitial atoms by sizeunselective synthetic methods, such as melting metals together to give intermetallics such as K8ZnIn10 and K10NiIn10, suggests that the sizes of the cavities in 10-vertex clusters of many posttransition elements are particularly favorable to accommodate interstitial atoms Bare post-transition element clusters having fewer than 10 vertex atoms and containing interstitial atoms have not been observed as products from such unselective syntheses Are such clusters with fewer than 10 vertex atoms viable? The research discussed in this paper explores the possibility of post-transition element clusters having fewer than ten vertex atoms and containing interstitial atoms Systems of the type Be@Genz (n = 8, 7, 6) were chosen for this work for the following reasons: (1) Beryllium is the smallest atom likely to be inserted into a cluster of this type Use of a beryllium atom as an interstitial atom thus minimizes the size of the polyhedral cavity required to contain an interstitial atom In addition, beryllium has a single oxidation state, namely, +2 This reduces the ambiguity in the electron bookkeeping in such clusters to test the applicability of the Wade−Mingos rules.8−11 (2) Germanium was used as the cluster atom since it or its neighbors in the Periodic Table are the cluster atoms in many of the experimentally known clusters (3) The Be@Ge10z (z = −4, −2, 0, +2) system has already been studied theoretically in some detail.16 The usual 10vertex Ge10 polyhedra (e.g., Figure 1) were shown to be large enough to contain an interstitial beryllium atom with relatively little distortion from the corresponding empty Ge10z−2 structures with the same numbers of skeletal electrons This study was initiated with the eight-vertex systems Be@Ge8z (z = −4, −2, 0, +2) Among possible eight-vertex polyhedra four deserve special mention (Figure 2) The cube has the highest symmetry (Oh) and the minimum number of edges for an eight-vertex closed polyhedron The latter feature Figure Four 10-vertex polyhedra found in bare post-transition element clusters encapsulating an interstitial atom in excess of a filled d10 shell Thus an interstitial Zn atom donates two skeletal electrons Interstitial Ni, Pd, and Pt atoms have exactly a filled d shell and thus have no “excess” electrons available for the polyhedral skeleton They are thus zero skeletal electron donors.12 An interstitial Co atom, such as found in the experimentally known7 Co@Ge103−, has only electrons in its 3d shell and thus takes an electron from the skeleton to fill its 3d shell Therefore, such an interstitial Co atom is a −1 skeletal electron donor, that is, an acceptor of a single skeletal electron With these considerations in mind, the structures of the known bare post-transition element clusters having interstitial atoms can be evaluated in the context of the Wade−Mingos rules.8−11 Note, however, that the Wade−Mingos rules predict deltahedral structures having the “ideal” 2n + skeletal electrons for a particularly stable three-dimensional aromatic polyhedral cluster Such deltahedral structures, in which all faces are triangles, have the maximum numbers of edges for a given number of vertices This leads to smaller cavities for interstitial atoms than in polyhedra having the same numbers of vertices but fewer edges relative to the number of vertices Thus the preferred polyhedra in bare post-transition element clusters to encapsulate atoms are likely to be delicate balances between polyhedra having all or almost all triangular faces predicted by the Wade−Mingos rules and more open polyhedra having the minimum number of edges for a closed polyhedron This effect is illustrated by the experimentally known 10vertex clusters containing an interstitial atom The known clusters2,3,7 M@Pb102− (M = Ni, Pd, Pt) and Co@Ge103− have the 22 skeletal electrons (= 2n + for n = 10) suggested by the Wade−Mingos rules8−11 to have a deltahedral structure For 10-vertex systems this deltahedron is a 4,4-bicapped square antiprism, ideally with D4d symmetry (Figure 1), and found experimentally13,14 in stable polyhedral boranes such as B10H102− Similarly, the M@Pb102− (M = Ni, Pd, Pt) clusters also have this 4,4-bicapped square antiprismatic geometry.2,3 However, the Co@Ge103− cluster has D5h pentagonal prismatic geometry rather than D4d 4,4-bicapped square antiprismatic geometry.7 For the cobalt derivative, a Ge10 pentagonal prism with the minimum number of edges for a 10-vertex deltahedron provides a larger cavity for the cobalt atom than a Ge10 4,4bicapped square antiprism Previous theoretical studies15 predict additional examples of preferred pentagonal prismatic structures for 10-vertex clusters with 22 skeletal electrons containing interstitial atoms Thus Ni@Ge102− is predicted to Figure Some eight-vertex polyhedra relevant to this research 5228 dx.doi.org/10.1021/jp302052u | J Phys Chem A 2012, 116, 5227−5234 The Journal of Physical Chemistry A Article makes it the most suitable eight-vertex polyhedron to contain an interstitial atom This is reflected in the theoretical results reported in this paper, where the cube was found to be the lowest energy structure for Be@Ge8z (z = 0, +2) The square antiprism (Figure 2) is obtained by twisting one of the square faces of the cube 45° relative to the other square face while maintaining the two square faces parallel and D4d symmetry The eight-vertex deltahedron predicted by the Wade− Mingos rules8−11 and found experimentally in polyhedral boranes17 such as B8H82− is the D2d bisdisphenoid, which is obtained from the cube by drawing six diagonals so that the vertex degrees alternate between and The structure is then distorted to make the diagonals and original cube edges more nearly equal in length while maintaining D2d symmetry Another high symmetry eight-vertex deltahedron is the tetracapped tetrahedron of Td symmetry The tetracapped tetrahedron is neither found nor predicted to be found18 in eight-vertex polyhedral borane chemistry However, it is the predicted lowest energy structure19 for the empty germanium cluster Ge82− We have also evaluated the possibility of stable Be@Genz structures (z = −4, −2, 0, +2) having fewer than eight germanium atoms, that is, n = and 6, and still containing a truly interstitial Be atom A few examples of such closed polyhedral Be@Genz (n < 8) structures are found However, in most cases and in the lowest energy structures, such clusters are found to be too small to accommodate the interstitial Be atom As a result, the external Gen (n = 7, 6) cluster either opens up, breaks into two pieces, or the Be atom migrates to the surface to give an empty polyhedron with n + vertices Figure Starting structures for the optimization of the seven-vertex structures Be@Ge7z (z = −4, −2, 0, +2) Figure Starting structures for the optimization of the six-vertex structures Be@Ge6z (z = −4, −2, 0, +2) 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 frequencies,25 and these values are indicated in the corresponding figures The optimized structures found for the Be@Genz derivatives are labeled by the numbers of vertices, the numbers of skeletal electrons, and their relative energies Triplet structures are designated by T Thus the lowest energy neutral Be@Ge8 structure, which is a singlet, is labeled 8−18−1 Additional details of all of the optimized structures, including all interatomic distances, their optimized coordinates, 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 Ge−Ge distances less than ∼3.2 Å were normally considered as polyhedral edges; significant exceptions are noted in the text Similarly Be−Ge distances less than ∼2.8 Å are considered bonding distances; most such Be−Ge 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 THEORETICAL METHODS Geometry optimizations were carried out at the hybrid DFT B3LYP level20−23 with the 6-31G(d) (valence) double-ζ quality basis functions extended by adding one set of polarization (d) functions for both the beryllium and the germanium atoms The Gaussian 03 package of programs24 was used in which the fine grid (75,302) is the default for numerically evaluating the integrals and the tight (10−8) hartree stands as default for the self-consistent field convergence The starting structures for the Be@Ge8z, Be@Ge7z, and Be@Ge6z (z = −4, −2, 0, +2) optimizations are depicted in Figures 3, 4, and 5, respectively Singlet and triplet spin states were investigated The symmetries of the starting structures were maintained during RESULTS 3.1 Eight-Vertex Be@Ge8z Structures Five Be@Ge84− structures were found within 30 kcal/mol of the global minimum The lowest energy structure for the Be@Ge84− system, namely, 8−22−1, is the D4d square antiprism (Figure 6) This is in accord with the Wade−Mingos rules,8−11 which predict a structure with two nontriangular faces for an eightvertex arachno system with 22 skeletal electrons (= 2n + skeletal electrons for n = 8) In the next two Be@Ge84− structures, namely, the C2 singlet 8−22−2 and the D3h triplet 8−22−3T at 11.6 and 21.6 kcal/mol above 8−22−1, respectively, the Ge8 unit has split into two Ge4 units that Figure Starting structures for the optimization of the eight-vertex structures Be@Ge8z (z = −4, −2, 0, +2) 5229 dx.doi.org/10.1021/jp302052u | J Phys Chem A 2012, 116, 5227−5234 The Journal of Physical Chemistry A Article Figure Four optimized Be@Ge8 structures Figure Five optimized Be@Ge84− structures corresponding normal mode leads to the C2v bicapped trigonal prismatic structure 8−18−2 A still higher energy D4d square antiprismatic Be@Ge8 structure 8−18−4 is found at 22.3 kcal/ mol above 8−18−1 Structure 8−18−4, like structure 8−18−3, has an imaginary vibrational frequency at 35 i cm−1 The situation is completely analogous to structure 8−18−3, since following the normal mode in 8−18−4 corresponding to this imaginary vibrational frequency leads to structure 8−18−2 Three structures were found for the dication Be@Ge82+ within 30 kcal/mol of the global minimum 8−16−1 (Figure 9) This global minimum is a distorted cube of C2v symmetry form two Ge4Be trigonal bipyramid cavities sharing the beryllium atom This is an example of the splitting of a polyhedron that is too small to accommodate the interstitial atom The triplet Be@Ge84− structure 8−22−3T retains the original D3h symmetry whereas the singlet structure 8−22−2 is distorted to C2 symmetry by bending the original C3 axis This distortion of 8−22−2 is undoubtedly a consequence of the Jahn−Teller effect Also structure 8−22−3T has a small imaginary vibrational frequency of 10i cm−1 Following the corresponding normal mode leads to 8−22−2 The next higher Be@Ge84− structure 8−22−4, lying 22.2 kcal/mol above 8−22−1, is derived from a cube by distortion from Oh to D3d symmetry Finally, the D2d bisdisphenoid Be@Ge84− structure 8−22−5T lies 27.4 kcal/mol above 8−22−1 Four Be@Ge82− structures were found within 30 kcal/mol of the D2d bisdisphenoid global minimum 8−20−1 (Figure 7) Figure Three optimized Be@Ge82+ structures having long 3.08 Å edges and 10 short 2.53 Å edges Only slightly higher in energy at 1.7 kcal/mol above 8−16−1 is structure 8−16−2 in which the Be atom has migrated to the surface of the polyhedron The remaining Be@Ge82+ structure 8−16−3T is a triplet structure, lying 11.6 kcal/mol above 8−16−1 In structure 8−16−3T the underlying topology is a cube but four edges are elongated to reduce the symmetry from Oh to D4h 3.2 Seven-Vertex Be@Ge7z Structures All four of the lowest energy Be@Ge8z (z = −4, −2, 0, +2) structures consist of an intact Ge8 polyhedron with an interstitial Be atom Therefore, Be@Ge7z (z = −4, −2, 0, +2) structures were optimized to see whether a Be atom could fit into an intact seven vertex Ge7 polyhedron in a stable structure Our results suggest that it is much more difficult to encapsulate a Be atom into a seven-vertex Ge7 polyhedron than an eight-vertex Ge8 polyhedron without the polyhedron opening up to provide enough volume for the Be atom Thus for the tetraanion Be@Ge74− the two lowest energy structures 7−20−1 and 7−20−2 are within 0.4 kcal/mol of each other (Figure 10) Both of these low-energy structures are structures in which the Ge7 polyhedron has opened up enough to accommodate the Be atom Structure 7−20−2 is derived from Figure Four optimized Be@Ge82− structures The triplet D 4d square antiprism Be@Ge 2− structure 8−20−2T lies only 0.4 kcal/mol above this global minimum The next Be@Ge82− structure 8−20−3, lying 7.7 kcal/mol above the 8−20−1 global minimum, is a C3v structure with only seven of the eight Ge atoms in 8−20−3 within bonding distance (