Density functional study of 8- and 11-vertex polyhedral borane structures: comparison with bare germanium clusters

Density functional theory (DFT) at the hybrid B3LYP level has been applied to the polyhedral boranes B(n)H(n)(z) (n = 8 and 11, z = -2, -4, and -6) for comparison with isoelectronic germanium clusters Ge(n)(z). The energy differences between the global minima and other higher energy borane structures are much larger relative to the case of the corresponding bare germanium clusters. Furthermore, for both B(8)H(8)(2-) and B(11)H(11)(2-), the lowest energy computed structures are the corresponding experimentally observed most spherical deltahedra predicted by the Wade-Mingos rules, namely the D(2)(d) bisdisphenoid and the C(2)(v) edge-coalesced icosahedron, respectively. Only in the case of B(8)H(8)(2-) is there a second structure close (+2.6 kcal/mol) to the D(2)(d) bisdisphenoid global minimum, namely the C(2)(v) bicapped trigonal prism corresponding to the "square" intermediate in a single diamond-square-diamond process that can lead to the experimentally observed room temperature fluxionality of B(8)H(8)(2-). Stable borane structures with 3-fold symmetry (e.g., D(3)(h), C(3)(v), etc.) are not found for boranes with 8- and 11-vertices, in contrast to the corresponding germanium clusters where stable structures derived from the D(3)(d) bicapped octahedron and D(3)(h) pentacapped trigonal prism are found for the 8- and 11-vertex systems, respectively. The lowest energy structures found for the electron-rich boranes B(8)H(8)(4-) and B(11)H(11)(4-) are nido polyhedra derived from a closo deltahedron by removal of a relatively high degree vertex, as predicted by the Wade-Mingos rules. They relate to isoelectronic species found experimentally, e.g., B(8)H(12) and R(4)C(4)B(4)H(4) for B(8)H(8)(4-) and C(2)B(9)H(11)(2-) for B(11)H(11)(4-). Three structures were found for B(11)H(11)(6-) with arachno type geometry having two open faces in accord with the Wade-Mingos rules.     

Cyclopentadienyl ruthenium, rhodium, and iridium vertices in metallaboranes: geometry and chemical bonding

Most cyclopentadienylmetallaboranes containing the vertex units CpM (M = Co, Rh, Ir; Cp = eta(5)-cyclopentadienyl ring, mainly eta(5)-Me(5)C(5)) and CpRu donating two and one skeletal electrons, respectively, have structures closely related to binary boranes or borane anions. Smaller clusters of this type, such as metallaborane analogues of arachno-B(4)H(10) (e.g., (CpIr)(2)B(2)H(8)), nido-B(5)H(9) (e.g., (CpRh)(2)B(3)H(7) and (CpRu)(2)B(3)H(9)), arachno-B(5)H(11) (e.g., CpIrB(4)H(10)), B(6)H(6)(2)(-) (e.g., (CpCo)(4)B(2)H(4)), nido-B(6)H(10) (e.g., CpIrB(5)H(9) and (CpRu)(2)B(4)H(10)), and arachno-B(6)H(12) (e.g., (CpIr)(2)B(4)H(10)), have the same skeletal electron counts as those of the corresponding boranes. However, such clusters with eight or more vertices, such as metallaborane analogues of B(8)H(8)(2)(-) (e.g., (CpCo)(4)B(4)H(4)), arachno-B(8)H(14) (e.g., (CpRu)(2)B(6)H(12)), and nido-B(10)H(14) (e.g., (CpRu)(2)B(8)H(12)), have two skeletal electrons less than those of the corresponding metal-free boranes, analogous to the skeletal electron counts of isocloso boranes relative to those of metal-free deltahedral boranes. Some metallaboranes have structures not analogous to metal-free boranes but instead analogous to metal carbonyl clusters such as 3-capped square pyramidal (CpRu)(2)B(4)H(8) and (CpRu)(3)B(3)H(8) analogous to H(2)Os(6)(CO)(16) and capped octahedral (CpRh)(3)B(4)H(4) analogous to Os(7)(CO)(21). In the metallaborane structures closely related to metal-free boranes, the favored degrees of BH and CpM vertices appear to be 5 and 6, respectively.     

Some examples of unusual skeletal bonding topologies in metallaboranes containing two or three early transition metal vertices

The metallaboranes (CpM)(2)B(n)H(n+4) (M = Cr, Mo, W; n = 4, 5; Cp = eta(5)-C(5)H(5), eta(5)-C(5)Me(5)), (CpW)(2)B(7)H(9), (CpRe)(2)B(7)H(7), and (CpW)(3)B(8)H(9) have the 2v or 2v + 2 skeletal electrons for closo or isocloso deltahedra (v = number of polyhedral vertices) if the early transition metal vertices are assumed to contribute four or more internal orbitals rather than the usual three internal orbitals for BH vertices. The polyhedra for the metallaboranes (CpM)(2)B(n)H(n+4) (M = Cr, Mo, W; n = 4, 5) are derived from (n + 1)-gonal bipyramids by removal of an equatorial vertex. The deltahedra for the larger metallaboranes (CpW)(2)B(7)H(9), (CpRe)(2)B(7)H(7), and (CpW)(3)B(8)H(9) are derived from the corresponding B(n)H(n)(2)(-) deltahedra (n = 9 and 11 in these cases) by sufficient diamond-square-diamond processes to provide vertices of degrees > or = 6 for each of the CpM vertices. Reasonable skeletal bonding topologies in accord with the availability of skeletal electrons and orbitals consist of surface 2c-2e and 3c-2e bonds supplemented by metal-metal bonding through the center of the polyhedron.     

Density functional theory study of twelve-atom germanium clusters: conflict between the Wade-Mingos rules and optimum vertex degrees

Density functional theory (DFT) at the hybrid B3LYP level has been applied to Ge(12)(z) bare germanium clusters (z = -6, -4, -2, 0, +2, +4, +6) starting from 11 initial configurations. The Wade-Mingos rules are seen to have limited value in rationalizing the results since they frequently require vertex degrees higher than the optimum vertex degree of 4 for germanium. Thus the expected I(h) regular icosahedron is no longer the global minimum for Ge(12)(2-) although it remains a low energy structure for Ge(12)(2-) lying only 5.6 kcal mol(-1) above a bicapped arachno structure conforming to the Wade-Mingos rules. The three lowest energy structures for Ge(12)(4-) within 11 kcal mol(-1) are a prolate (elongated) polyhedron with six quadrilateral faces and eight triangular faces, the dual of the bisdisphenoid with four trapezoidal and four pentagonal faces, and a polyhedron with two quadrilateral and 16 triangular faces related but not identical to the polyhedron found in the known tetracarbon carboranes R(4)C(4)B(8)H(8). The lowest energy structures for the neutral Ge(12) are seen to be distorted versions of the icosahedron and the bicapped 10-vertex arachno lowest energy structures for Ge(12)(2-). The low energy structures for the even more hypoelectronic Ge(12)(2+) and Ge(12)(4+) are even more unusual including a hexacapped octahedron, a tetracapped square antiprism, and a double cube for Ge(12)(2+) and a C(2v) structure with a central unique degree 6 vertex for Ge(12)(4+).     

A unifying electron-counting rule for macropolyhedral boranes, metallaboranes, and metallocenes.

A generally applicable electron-counting rule-the mno rule-that integrates macropolyhedral boranes, metallaboranes, and metallocenes and any combination thereof is presented. According to this rule, m + n + o number of electron pairs are necessary for a macropolyhedral system to be stable. Here, m is the number of polyhedra, n is the number of vertices, and o is the number of single-vertex-sharing condensations. For nido and arachno arrangements, one and two additional pairs of electrons are required. Wade's n + 1 rule is a special case of the mno rule, where m = 1 and o = 0. B20H16, for example has m = 2 and n = 20, leading to 22 electron pairs. Ferrocene, with two nido polyhedral fragments, has m = 2, n = 11, and o = 1, making the total 2 + 11 + 1 + 2 = 16. The generality of the mno rule is demonstrated by applying it to a variety of known macropolyhedral boranes and heteroboranes. We also enumerate the various pathways for condensation by taking icosahedral B12 as the model. The origin of the mno rule is explored by using fragment molecular orbitals. This clearly shows that the number of skeletal bonding molecular orbitals of two polyhedral fragments remains unaltered during exohedral interactions. This is true even when a single vertex is shared, provided the common vertex is large enough to avoid nonbonding interactions of adjacent vertices on either side. But the presence of more than one common vertex results in the sharing of surface orbitals thereby, reducing the electronic requirements.     

Ab initio studies on the electronic structure and properties of aluminum hydrides that are analogues of boron hydrides

Although the boron hydrides are well-known in the literature, the aluminum hydride chemistry is limited to very few systems such as AlH(3), its dimer, and its polymeric form. In view of the recent experimental studies on the possible existence of the aluminum hydrides, herein, we have undertaken a systematic study on the electronic structure and properties of these aluminum hydrides. Under this, we have studied different classes of hydrides, viz., closo (Al(n)H(n+2)), nido (Al(n)H(n+4)), and arachno (Al(n)H(n+6)), similar to the boranes. All the aluminum hydrides are found to have exceptionally large highest-occupied molecular orbital-lowest-unoccupied molecular orbital gaps, low electron affinities, large ionization potentials and also large enthalpy and free energy of atomization. In addition, most of the structures are also found to have high symmetries. These exceptional properties can be indicative of the pronounced stability, and hence, it is expected that other aluminum hydride complexes can indeed be observed experimentally.     

Structure and bonding in cyclic isomers of B2AlH(n)(m) (n = 3-6, m = -2 to +1): a comparative study with B3H(n)(m), BAl2H(n)(m) and Al3H(n)(m)

The structure, bonding and energetics of B(2)AlH(n)(m) (n = 3-6, m = -2 to +1) are compared with corresponding homocyclic boron, aluminum analogues and BAl(2)H(n)(m) using density functional theory (DFT). Divalent to hexacoordinated boron and aluminum atoms are found in these species. The geometrical and bonding pattern in B(2)AlH(4)(-) is similar to that for B(2)SiH(4). Species with lone pairs on the divalent boron and aluminum atoms are found to be minima on the potential energy surface of B(2)AlH(3)(2-). A dramatic structural diversity is observed in going from B(3)H(n)(m) to B(2)AlH(n)(m), BAl(2)H(n)(m) and Al(3)H(n)(m) and this is attributable to the preference of lower coordination on aluminum, higher coordination on boron and the higher multicenter bonding capability of boron. The most stable structures of B(3)H(6)(+), B(2)AlH(5) and BAl(2)H(4)(-) and the trihydrogen bridged structure of Al(3)H(3)(2-) show an isostructural relationship, indicating the isolobal analogy between trivalent boron and divalent aluminum anion.     

Structural relationships among two vertex sharing macropolyhedral boranes

Various two vertex sharing macropolyhedral boranes were computed at the B3LYP/6-311 + G**//B3LYP/6-31G* level of theory to determine the preferred fragments for the thermodynamically most stable isomers. These are nido-10 and arachno-9 vertex fragments for neutral macropolyhedral boranes. The thermodynamically most stable isomers of the nido:nido-, arachno:nido- and arachno:arachno-macropolyhedral borane classes are structurally related to each other by the successive removal of one open face vertex as in the case of simple polyhedral boranes. For these classes, the stabilities of the thermodynamically most stable macropolyhedra relative to isomeric simple polyhedra follow similar trends with respect to the number of skeletal electrons.     

Quantum Chemistry Study on the Geometrical Structures of Small Centipedo－boranes

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Macropolyhedral boron-containing cluster chemistry. The reversible disassembly and reassembly of the hexagonal pyramidal {B7} feature in the [S2B18H19]- anion

The [S(2)B(18)H(19)](-) anion 1, from syn-B(18)H(22) 2 with NaH and elemental sulfur, has an unusual arachno-type eleven-vertex {SB(10)} subcluster that has an open hexagonal pyramidal {B(7)} structural feature. This is conjoined, with two boron atoms in common, to a second {SB(10)} subcluster of conventional nido eleven-vertex geometry. Protonation of 1 forms neutral [S(2)B(18)H(20)] 4. Subsequent deprotonation of 4 yields the fluxional [S(2)B(18)H(19)](-) anion 5, which is isomeric with 1. Neutral 4and anion 5 do not have the {B(7)} hexagonal pyramidal feature. Neutral 4 consists of conventional nido eleven-vertex {SB(10)} and arachno ten-vertex {SB(9)} subclusters conjoined with a single spiro boron atom in common. Anion 5 is closely related to 4, but with an additional inter-boron intercluster link. Anion 5 spontaneously reverts to anion 1 over a few hours at room temperature, remarkable in that the open {B(7)} hexagonal pyramid is regenerated. DFT B3LYP/6-31G* calculations suggest definitive structures for 4 and 5 that are substantiated by agreement between observed NMR delta((11)B) values and boron nuclear shielding as calculated by the GIAO approach on the DFT-calculated structures. Extension of this approach additionally defines transition states and intermediates for the fluxionality of 5, and also for the reassembly of the starting anion 1, together with its {B(7)} feature, from fluxional 5. The fluxionality of 5 involves the inter-subcluster transfer of a {BH} unit. The reassembly of 1 from 5 involves a DSD rearrangement and two successive hydrogen-atom hops. Confidence in the application of this method to these large macropolyhedral assemblies is afforded in the first instance by good agreement between delta((11)B)(OBS) and delta((11)B)(CALC) for the structurally characterised original anion 1, the only species amongst these to be crystallographically established.     

Can the bis(diboranyl) structure of B(4)H(10) be observed? The story continues

Pathways for the conversion of the unknown bis(diboranyl) isomer of tetraborane(10) (B(4)H(10)) to the known arachno isomer have been determined for the first time with the use of an electron correlation ab initio quantum chemical method and without the use of constraints in determination of the transition structures. Two isomers of tetraborane(10), one new, with a pentacoordinated boron atom have been found on the theoretical potential energy surface. Several other pathways for molecular rearrangement of tetraborane(10) have also been characterized. The theoretical method was MP2 theory with the 6-31G(d,p) basis set. The most likely pathway for the conversion of the bis(diboranyl) isomer of tetraborane(10) to the arachno isomer is a concerted pathway with two pentacoordinated intermediates. The highest energy transition state for this pathway lies 27.7 kcal/mol above the bis(diboranyl) isomer. At the same level of the theory, the bis(diboranyl) isomer lies 9.2 kcal/mol above the known arachno isomer. The two isomers with a pentacoordinated boron atom lie 12.5 and 13.1 kcal/mol above the arachno isomer.     

Structural paradigms in macropolyhedral boranes

Macropolyhedral borane clusters are concave polyhedra constituting fused convex simple polyhedra. They are formally obtained by condensation of simple polyhedral boranes under elimination of between one and four BH(3) or isoelectronic units. The number of eliminated vertexes from simple polyhedra equals the number of shared vertexes in macropolyhedral boranes. For each of the eight classes with general formulae ranging from B(n)H(n-4) to B(n)H(n+10), more than one structure type is possible, differing in the number of shared vertexes and in the types of the two combined cluster fragments. However, only one type of "potential structures" is represented by experimentally known examples and is found to be favored by theoretical calculations. A sophisticated system exists among the favored macropolyhedral borane structures. For each class of macropolyhedral boranes, the number of skeletal electron pairs is directly related to the general formula, the number of shared vertexes and the type of fused cluster fragments. In order to predict the distribution of vertexes among the fused fragments, we propose the concept of preferred fragments. Preferred fragments are those usually present in the thermodynamically most stable structure of a given class of macropolyhedral boranes and are also frequently observed in the experimentally known structures. This allows us to completely predict the cluster framework of the thermodynamically most stable macropolyhedral borane isomers.     

Redox energetics of hypercloso boron hydrides B(n)H(n) (n = 6-13) and B12X12 (X = F, Cl, OH, and CH3)

The reduction potentials (E°(Red) versus SHE) of hypercloso boron hydrides B(n)H(n) (n = 6-13) and B(12)X(12) (X = F, Cl, OH, and CH(3)) in water have been computed using the Conductor-like Polarizable Continuum Model (CPCM) and the Solvation Model Density (SMD) method for solvation modeling. The B3LYP/aug-cc-pvtz and M06-2X/aug-cc-pvtz as well as G4 level of theory were applied to determine the free energies of the first and second electron attachment (ΔG(E.A.)) to boron clusters. The solvation free energies (ΔG(solv)) greatly depend on the choice of the cavity set (UAKS, Pauling, or SMD) while the dependence on the choice of exchange/correlation functional is modest. The SMD cavity set gives the largest ΔΔG(solv) for B(n)H(n)(0/-) and B(n)H(n)(-/2-) while the UAKS cavity set gives the smallest ΔΔG(solv) value. The E°(Red) of B(n)H(n)(-/2-) (n = 6-12) with the G4/M06-2X(Pauling) (energy/solvation(cavity)) combination agrees within 0.2 V of experimental values. The experimental oxidative stability (E(1/2)) of B(n)X(n)(2-) (X = F, Cl, OH, and CH(3)) is usually located between the values predicted using the B3LYP and M06-2X functionals. The disproportionation free energies (ΔG(dpro)) of 2B(n)H(n)(-) → B(n)H(n) + B(n)H(n)(2-) reveal that the stabilities of B(n)H(n)(-) (n = 6-13) to disproportionation decrease in the order B(8)H(8)(-) > B(9)H(9)(-) > B(11)H(11)(-) > B(10)H(10)(-). The spin densities in B(12)X(12)(-) (X = F, Cl, OH, and CH(3)) tend to delocalize on the boron atoms rather than on the exterior functional groups. The partitioning of ΔG(solv)(B(n)H(n)(2-)) over spheres allows a rationalization of the nonlinear correlation between ΔG(E.A.) and E°(Red) for B(6)H(6)(-/2-), B(11)H(11)(-/2-), and B(13)H(13)(-/2-).     

Elongation of planar boron clusters by hydrogenation: boron analogues of polyenes

Dihydrogenated boron clusters, H(2)B(n)(-) (n = 7-12), were produced and characterized using photoelectron spectroscopy and computational chemistry to have ladderlike structures terminated by a hydrogen atom on each end. The two rows of boron atoms in the dihydrides are bonded by delocalized three-, four-, or five-center σ and π bonds. The π bonding patterns in these boron nanoladders bear similarities to those in conjugated alkenes: H(2)B(7)(-), H(2)B(8), and H(2)B(9)(-), each with two π bonds, are similar to butadiene, while H(2)B(10)(2-), H(2)B(11)(-), and H(2)B(12), each with three π bonds, are analogous to 1,3,5-hexatriene. The boron cluster dihydrides can thus be considered as polyene analogues, or "polyboroenes". Long polyboroenes with conjugated π bonds (analogous to polyacetylenes), which may form a new class of molecular wires, should exist.     

Electronic structures and thermochemical properties of the small silicon-doped boron clusters B(n)Si (n=1-7) and their anions

We perform a systematic investigation on small silicon-doped boron clusters B(n)Si (n=1-7) in both neutral and anionic states using density functional (DFT) and coupled-cluster (CCSD(T)) theories. The global minima of these B(n)Si(0/-) clusters are characterized together with their growth mechanisms. The planar structures are dominant for small B(n)Si clusters with n≤5. The B(6)Si molecule represents a geometrical transition with a quasi-planar geometry, and the first 3D global minimum is found for the B(7)Si cluster. The small neutral B(n)Si clusters can be formed by substituting the single boron atom of B(n+1) by silicon. The Si atom prefers the external position of the skeleton and tends to form bonds with its two neighboring B atoms. The larger B(7)Si cluster is constructed by doping Si-atoms on the symmetry axis of the B(n) host, which leads to the bonding of the silicon to the ring boron atoms through a number of hyper-coordination. Calculations of the thermochemical properties of B(n)Si(0/-) clusters, such as binding energies (BE), heats of formation at 0 K (ΔH(f)(0)) and 298 K (ΔH(f)([298])), adiabatic (ADE) and vertical (VDE) detachment energies, and dissociation energies (D(e)), are performed using the high accuracy G4 and complete basis-set extrapolation (CCSD(T)/CBS) approaches. The differences of heats of formation (at 0 K) between the G4 and CBS approaches for the B(n)Si clusters vary in the range of 0.0-4.6 kcal mol(-1). The largest difference between two approaches for ADE values is 0.15 eV. Our theoretical predictions also indicate that the species B(2)Si, B(4)Si, B(3)Si(-) and B(7)Si(-) are systems with enhanced stability, exhibiting each a double (σ and π) aromaticity. B(5)Si(-) and B(6)Si are doubly antiaromatic (σ and π) with lower stability.     

The prevalence of isocloso deltahedra in low-energy hypoelectronic metalladicarbaboranes with a single metal vertex: manganese and rhenium derivatives

Theoretical studies on the hypoelectronic metalladicarbaboranes CpMC(2)B(n-3)H(n-1) (M = Mn, Re; n = 9, 10, 11) having 2n skeletal electrons indicate that true isocloso MC(2)B(n-3) deltahedra are highly energetically favored in which the metal atom occupies the single degree 6 vertex. This contrasts with the previously studied isoelectronic diferradicarbaboranes Cp(2)Fe(2)C(2)B(n-3)H(n-1) for which the isocloso structure is clearly favored only for the 10-vertex system. For the 12-vertex hypoelectronic manganadicarbaborane CpMnC(2)B(9)H(11) with 2n (= 24) skeletal electrons the lowest energy structures have central MnC(2)B(9) icosahedra. However, for the corresponding rhenadicarbaborane CpReC(2)B(9)H(11) the lowest energy structures have central non-icosahedral ReC(2)B(9) deltahedra with two degree 6 vertices, one of which is occupied by the rhenium atom. The low-energy structures for the metalladicarbaboranes studied in this work relate to the preferences of transition metal atoms for degree 6 vertices but those of boron and carbon for degree 5 and 4 vertices, respectively.     

Flattening the b(6)h(6)(2-) octahedron Ab initio prediction of a new family of planar all-boron aromatic molecules

The structural chemistry of boron is dominated by 3D structures (polyhedra), while in carbon structural chemistry the planar aromatic structures are more abundant. In this Communication we present results of ab initio calculations showing that the polyhedral boranes can be flattened into planar aromatic structures similar to their carbon analogues. We predicted that a B6H62- octahedron (in Li2B6H6), a B5H52- trigonal bipyramid (in Li2B5H5), a B7H72- pentagonal bipyramid (in Li2B7H7), and a B10H84- bioctahedron with a joint edge (in Li4B10H8) can be reduced to a planar aromatic B6H66- hexagon (in Li6B6H6), to a planar pentagon B5H56- (in Li6B5H5), to a planar heptagon B7H76- (in Li6B7H7), and to a naphthalene-like B10H810- (in Li10B10H8). Ab initio prediction of these new planar aromatic boranes shows that a large new family of planar aromatic all-boron molecules is possible.     

Tridurylboranes extended by three arylethynyl groups as a new family of boron-based pi-electron systems

A series of tris(phenylethynylduryl)boranes (R-C(6)H(4)-C&tbd1;C-duryl)(3)B with various substituents R have been prepared as air-stable solids owing to the steric protection of the boron atom by the three bulky duryl groups. These compounds show unique photophysical properties due to the p(pi)-pi conjugation through the p-orbital on the boron atom. In particular, a push-pull type derivative with R = NMe(2) exhibits a significant solvatochromism of fluorescence from blue to orange colors.     

Analysis of why boron avoids sp2 hybridization and classical structures in the BnHn+2 series

We performed global minimum searches for the B(n) H(n+2) (n=2-5) series and found that classical structures composed of 2c-2e B-H and B-B bonds become progressively less stable along the series. Relative energies increase from 2.9?kcal?mol(-1) in B(2) H(4) to 62.3?kcal?mol(-1) in B(5) H(7). We believe this occurs because boron atoms in the studied molecules are trying to avoid sp(2) hybridization and trigonal structure at the boron atoms, as in that case one 2p-AO is empty, which is highly unfavorable. This affinity of boron to have some electron density on all 2p-AOs and avoiding having one 2p-AO empty is a main reason why classical structures are not the most stable configurations and why multicenter bonding is so important for the studied boron-hydride clusters as well as for pure boron clusters and boron compounds in general.     

Endohedral beryllium atoms in germanium clusters with eight and fewer vertices: how small can a cluster be and still encapsulate a central atom?

Structures of the beryllium-centered germanium clusters Be@Ge(n)(z) (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@Ge(8)(z) clusters (z = -4, -2, 0, +2) all have the Be atom at the center of a closed polyhedron, namely, a D(4d) square antiprism for Be@Ge(8)(4-), a D(2d) bisdisphenoid for Be@Ge(8)(2-), an ideal O(h) cube for Be@Ge(8), and a C(2v) distorted cube for Be@Ge(8)(2+). The Be-centered cubic structures predicted for Be@Ge(8) and Be@Ge(8)(2+) differ from the previously predicted lowest energy structures for the isoelectronic Ge(8)(2-) and Ge(8). 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@Ge(n)(z) (n = 7, 6; z = -4, -2, 0, +2) no longer have the Be atom at the center of a closed Ge(n) polyhedron. Instead, either the Ge(n) 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 Ge(n) (n = 7, 6) polyhedron. Examples of such structures are a centered C(2v) capped trigonal prismatic structure for Be@Ge(7)(2-), a centered D(5h) pentagonal bipyramidal structure for Be@Ge(7), a centered D(3h) trigonal prismatic structure for Be@Ge(6)(4-), and a centered octahedral structure for Be@Ge(6). Cluster buildup reactions of the type Be@Ge(n)(z) + Ge(2) → Be@Ge(n+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.