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 D4dsquare antiprism for Be@Ge8−, a
Trang 1Endohedral Beryllium Atoms in Germanium Clusters with Eight and Fewer Vertices: How Small Can a Cluster Be and Still Encapsulate a Central Atom?
M M Ut ̧ㆠand 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 D4dsquare antiprism for Be@Ge8−, a D2dbisdisphenoid
for Be@Ge8−, an ideal Ohcube for Be@Ge8, and a C2vdistorted cube for Be@Ge82+ The
Be-centered cubic structures predicted for Be@Ge8and Be@Ge82+differ from the previously predicted
lowest energy structures for the isoelectronic Ge8−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 Genpolyhedron 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 C2vcapped trigonal prismatic structure for Be@Ge7−, a centered D5h pentagonal bipyramidal structure for Be@Ge7, a centered D3h trigonal prismatic structure for Be@Ge6−, and a centered octahedral structure for Be@Ge6 Cluster buildup reactions of the type Be@Genz+ Ge2→ Be@Gen+2z(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
1 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 D4dbicapped
square antiprism encapsulates a group 12 metal atom in the
anionic indium cluster Zn@In108− found in the intermetallic1
K8In10Zn In addition the same D4dbicapped square antiprism
is found in the lead clusters M@Pb102− in
[K(2,2,2-crypt)]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).5The pentagonal prism has
been found as the host polyhedron for an encapsulated iron or
cobalt atom in the clusters M@Ge103−(M = Fe,6Co7)
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-electronfilled d10s2shell 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 post-transition metal atom corresponds to the number of electrons
Received: March 1, 2012
Revised: May 7, 2012
Published: May 7, 2012
pubs.acs.org/JPCA
Trang 2in excess of a filled d10 shell Thus an interstitial Zn atom
donates two skeletal electrons Interstitial Ni, Pd, and Pt atoms
have exactly afilled d shell and thus have no “excess” electrons
available for the polyhedral skeleton They are thus zero skeletal
electron donors.12An interstitial Co atom, such as found in the
experimentally known7Co@Ge103−, has only 9 electrons in its
3d shell and thus takes an electron from the skeleton tofill 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.8−11Note, however, that the Wade−Mingos rules predict
deltahedral structures having the “ideal” 2n + 2 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
10-vertex clusters containing an interstitial atom The known
clusters2,3,7M@Pb102−(M = Ni, Pd, Pt) and Co@Ge103−have
the 22 skeletal electrons (= 2n + 2 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 D5hpentagonal 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
4,4-bicapped 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
have a D4dbicapped 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 D5hpentagonal 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 size-unselective synthetic methods, such as melting metals together
to give intermetallics such as K8ZnIn10and K10NiIn10, suggests that the sizes of the cavities in 10-vertex clusters of many post-transition 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.16The usual 10-vertex Ge10polyhedra (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 1 Four 10-vertex polyhedra found in bare post-transition
element clusters encapsulating an interstitial atom.
Figure 2 Some eight-vertex polyhedra relevant to this research.
| J Phys Chem A 2012, 116, 5227−5234 5228
Trang 3makes 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 B8H8− is the D2d bisdisphenoid, which is
obtained from the cube by drawing six diagonals so that the
vertex degrees alternate between 4 and 5 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 structure19for the empty germanium
cluster Ge8−
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 = 7 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 + 1 vertices
2 THEORETICAL METHODS
Geometry optimizations were carried out at the hybrid DFT
B3LYP level20−23with 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 programs24was 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-consistentfield 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
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 frequen-cies,25 and these values are indicated in the corresponding figures
The optimized structures found for the Be@Genzderivatives 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
3 RESULTS
3.1 Eight-Vertex Be@Ge8z Structures Five Be@Ge8− structures were found within 30 kcal/mol of the global minimum The lowest energy structure for the Be@Ge8− system, namely, 8−22−1, is the D4d square antiprism (Figure 6) This is in accord with the Wade−Mingos rules,8−11which predict a structure with two nontriangular faces for an eight-vertex arachno system with 22 skeletal electrons (= 2n + 6 skeletal electrons for n = 8) In the next two Be@Ge8− structures, namely, the C2singlet 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 3 Starting structures for the optimization of the eight-vertex
structures Be@Ge8 (z = −4, −2, 0, +2).
Figure 4 Starting structures for the optimization of the seven-vertex structures Be@Ge7 (z = −4, −2, 0, +2).
Figure 5 Starting structures for the optimization of the six-vertex structures Be@Ge6 (z = −4, −2, 0, +2).
| J Phys Chem A 2012, 116, 5227−5234 5229
Trang 4form 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@Ge8− structure 8−22−3T retains the
original D3hsymmetry whereas the singlet structure 8−22−2 is
distorted to C2symmetry by bending the original C3axis 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@Ge8− 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@Ge8−
structure 8−22−5T lies 27.4 kcal/mol above 8−22−1
Four Be@Ge8−structures were found within 30 kcal/mol of
the D2d bisdisphenoid global minimum 8−20−1 (Figure 7)
The triplet D4d square antiprism Be@Ge8− structure
8−20−2T lies only 0.4 kcal/mol above this global minimum
The next Be@Ge8− structure 8−20−3, lying 7.7 kcal/mol
above the 8−20−1 global minimum, is a C3vstructure with only
seven of the eight Ge atoms in 8−20−3 within bonding
distance (<2.8 Å) of the central Be atom These Ge atoms form
a capped octahedron The highest energy structure, namely,
8−20−4 at 17.0 kcal/mol above 8−20−1, is an example of an
open structure with the Be atom on the surface rather than in
the center The Ge8 unit in 8−20−4 has split into two Ge4
tetrahedra connected by a Ge−Ge bond The surface Be atom
is connected to six of the eight Ge atoms Structure 8−20−4 is
a good example of the migration of the Be atom to the surface
when the cavity of the germanium polyhedron is too small to
accommodate an interstitial atom
Four structures were also found for the neutral Be@Ge8
within 30 kcal/mol of the global minimum 8−18−1 (Figure 8)
This global minimum is a perfect cube having ideal Oh point
group symmetry This cubic Be@Ge8 structure lies
13.2 kcal/mol in energy below the next lowest energy Be@
Ge8structure, namely, the C2vbicapped trigonal prismatic 8−
18−2 The Wade−Mingos rules8−11 suggest a bisdisphenoid
structure for the Be@Ge8 system with 18 skeletal electrons
Such a structure 8−18−3 is found but at an energy 17.7 kcal/
mol above 8−18−1 Furthermore, structure 8−18−3 has an
imaginary vibrational frequency at 32 i cm−1 Following the
corresponding normal mode leads to the C2vbicapped trigonal prismatic structure 8−18−2 A still higher energy D4d square antiprismatic Be@Ge8structure 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 C2vsymmetry
having 2 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
Ohto 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 Ge7polyhedron 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@Ge7−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 6 Five optimized Be@Ge8−structures.
Figure 7 Four optimized Be@Ge8−structures.
Figure 8 Four optimized Be@Ge8structures.
Figure 9 Three optimized Be@Ge82+ structures.
Figure 10 Five optimized Be@Ge7−structures.
| J Phys Chem A 2012, 116, 5227−5234 5230
Trang 5a C2vcapped trigonal prism However, the three“vertical” edges
of the underlying trigonal prism have elongated to∼3.3 Å for
one of the edges and∼3.9 Å for the other two edges
In addition to these two low-energy Be@Ge7− structures,
three other Be@Ge7− structures were found at energies
between 30 and 33 kcal/mol above the global minimum
7−20−1 (Figure 10) Structure 7−20−3 has a central Be atom
surrounded by a germanium cube missing one vertex In fact
Be@Ge7−structure 7−20−3 can be derived from the neutral
Be@Ge8 cube global minimum 8−18−1 (Figure 8) by
removing a Ge vertex as Ge4+ The other two Be@Ge7−
structures within 35 kcal/mol of 8−18−1, namely, 7−20−4
and 7−20−5, have the Be atom in the center of a seven-vertex
polyhedron In 7−20−4 the Ge7polyhedron is derived from a
C3vcapped octahedron by lengthening the three edges of the
capped face (Ge5−Ge6, Ge6−Ge7, and Ge7−Ge5 in Figure
10) to a nonbonding 3.81 Å This apparently increases the
internal volume of the capped octahedron enough to
accommodate the Be atom, albeit in a relatively high energy
structure The other Ge7 polyhedron in 7−20−5 is a C2v
capped trigonal prism in which two edges of the capped face
(Ge2−Ge6 and Ge3−Ge7 in Figure 10) are lengthened to a
nonbonding 3.62 Å Structure 7−20−5 has an imaginary
vibrational frequency at 32 i cm−1 Following the corresponding
normal mode leads to 7−20−4
Only two structures were found for the dianion Be@Ge7−
within 30 kcal/mol of the global minimum 7−18−1 (Figure
11) In this global minimum the Be atom has migrated to the
surface to form an unsymmetrical BeGe7 eight-vertex
polyhedron The peculiar shape of this polyhedron allows the
Be atom to lie within 3.0 Å of all seven Ge atoms In fact, all but
two of these seven Be−Ge distances are less than 2.3 Å in this
highly unsymmetrical structure The higher energy Be@Ge7−
structure 7−18−2, lying 23.2 kcal/mol above 7−18−1, has the
Be atom at the center of a Ge7 capped trigonal prism of C2v
symmetry The longest edges of the Ge7capped trigonal prism
in 7−18−2 are ∼3.08 Å for two of the edges of the underlying
trigonal prism (Ge2−Ge3 and Ge6−Ge7 in Figure 11) so that
this seven-vertex polyhedron does not need to expand
significantly to accommodate the interstitial Be atom All
seven Ge atoms in 7−18−2 are clearly within bonding distance
of the central Be atom with Be−Ge distances ranging from 2.18
to 2.29 Å
Only two structures are found for the neutral Be@Ge7within
30 kcal/mol of the global minimum 7−16−1 (Figure 12) In
this global minimum the Be atom has migrated to the surface
but remains within bonding distance (<2.62 Å) of all seven Ge
atoms The other Be@Ge7 structure is the beautiful D5h
pentagonal bipyramidal structure 7−16−2, lying 11.8 kcal/
mol above 7−16−1 Structure 7−16−2 follows the Wade−
Mingos rules8−11 since the pentagonal bipyramid is the most
spherical seven-vertex deltahedron expected for a seven-vertex cluster with 16 skeletal electrons (= 2n + 2 for n = 7) Four optimized structures were found for the dication Be@Ge72+within 30 kcal/mol of the global minimum 7−14−1 (Figure 13) In this global minimum the Be atom has migrated
to the surface to form an irregular eight-vertex empty BeGe7 polyhedron Six of the vertex atoms of this BeGe7polyhedron, namely, the Be atom and Ge2,3,5,6,7 (Figure 13), form an octahedral cavity
The lowest energy Be@Ge72+ structure 7−14−1 lies more than 23 kcal/mol below the next lowest energy Be@Ge72+
structure 7−14−2 (Figure 13) This suggests that structure 7−14−1, with the Be atom on the surface, is a highly favored structure relative to structures where the Be atom remains in the center of the polyhedron In structure 7−14−2 the Be atom remains in the center of the Ge7unit, which has broken open giving a large “hole” (Ge2−Ge4−Ge3−Ge7−Ge6 in Figure 13) The Be@Ge72+ structure 7−14−4, lying 26.0 kcal/mol above 7−14−1, also has the Be atom lying inside an opened
Ge7polyhedron
The lowest energy Be@Ge72+ structure with the Be atom inside a true Ge7polyhedron is the C3vstructure 7−14−3, lying 25.1 kcal/mol above the global minimum 7−14−1 (Figure 13)
In 7−14−3 the outer Ge7polyhedron is a C3v trigonal prism with a symmetrical cap on one of the triangular faces In this structure the central Be atom lies within bonding distance (∼2.17 Å) of the six Ge atoms of the underlying trigonal prism However, the Be atom lies a nonbonding distance of 3.43 Å from the seventh Ge atom capping the triangular face to form a tetrahedral cavity
3.3 Six-Vertex Be@Ge6z Structures A six-vertex Ge6 polyhedron appears to be too small to accommodate an interstitial Be atom without opening up or the Be atom migrating to the polyhedral surface Thus none of the lowest energy Be@Ge6z(z =−4, −2, 0, +2) structures consist of an intact six-vertex Ge6 polyhedron with the Be atom inside However, some interesting higher energy Be@Ge6zstructures are found with the Be atom inside an intact Ge6 polyhedron The numbers of Be@Ge6z structures found competitive in energy with the lowest energy structures were much more
Figure 11 Two optimized Be@Ge7−structures.
Figure 12 Two optimized Be@Ge7structures.
Figure 13 Four optimized Be@Ge72+ structures.
| J Phys Chem A 2012, 116, 5227−5234 5231
Trang 6limited than for the larger Gen polyhedra Thus only two
Be@Ge6z structures are discussed for each value of z
The lowest energy Be@Ge6− structure 6−18−1 is a C2v
structure with a BeGe4trigonal bipyramid bridged across the
two axial vertices (Ge4 and Ge7 in Figure 14) with the two
remaining vertices Ge5 and Ge6 A much higher energy
Be@Ge6− structure 6−18−2 is found lying 39.8 kcal/mol in
energy above 6−18−1 in which a Be atom is located in the
center of a D3hGe6trigonal prism The 18 skeletal electrons in
this six-vertex structure correspond to two-electron two-center
bonds along each of the nine edges of the trigonal prism if the
Wade−Mingos rules8−11are used for skeletal electron counting
The lowest energy Be@Ge6− structure 6−16−1 is an
unusual empty C5v BeGe6pentagonal bipyramid in which the
Be is one of the two vertices on the C5 axis (Figure 15) A
triplet Be@Ge6−structure 6−16−2T is found at the extremely
high energy of 70.4 kcal/mol above 6−16−1 The triplet
Be@Ge6− structure 6−16−2T is similar to the singlet
Be@Ge6− structure 6−18−2 in having an interstitial Be
atom inside a Ge6trigonal prism
The lowest energy neutral Be@Ge6 structure 6−14−1 is
again an open structure in which the Be atom has migrated to
the surface of a distorted pentagonal bipyramid to become an
axial vertex (Figure 16) In addition, a regular Oh octahedral
Be@Ge6structure 6−14−2 is found 12.2 kcal/mol in energy
above 6−14−1 This structure conforms to the Wade−Mingos
rules8−11since the 14 skeletal electrons (= 2n + 2 for n = 6) is
the correct number for the regular octahedron
The dication Be@Ge62+ has no low-energy structures with
the Be atom at the center of a closed polyhedron In the lowest
energy Be@Ge62+structure 6−12−1 the Be atom is one of the
degree 3 capping vertices of a BeGe6 tricapped tetrahedron (Figure 17) Close in energy to 6−12−1 is structure 6−12−2,
lying only 1.6 kcal/mol above 6−12−1 In structure 6−12−2 the Be atom has also migrated to the surface to form a BeGe6 distorted pentagonal bipyramid in which the Be atom is a vertex
of the equatorial pentagon
3.4 Thermochemistry The synthesis of numerous 10-vertex clusters containing an encapsulated atom suggests that ten-vertex clusters are a particularly favorable size to encapsulate a central atom To test this idea with Be@Genz
clusters the enthalpies (ΔH) of reactions of the type Be@Genz
+ Ge2 → Be@Gen+2z (n = 6, 8) were investigated using the lowest energy structures (Table 1) To calculate these
enthalpies the absolute energy of the lowest energy Ge2 structure (triplet) was used The absolute energies for the Be@Ge10zstructures were taken from the previous work.16The enthalpies listed in Table 1 are at 298.150 K and 1.000 atm The important point from the data listed in Table 1 is that all
of the cluster buildup reactions Be@Genz+ Ge2→ Be@Gen+2z
(n = 6, 8) are highly exothermic This suggests that the six to eight vertex Genz clusters discussed in this paper are less than the optimum size to contain the interstitial Be atom This is consistent with the lack of experimental reports of stable main group element clusters with eight or fewer vertices containing interstitial atoms
4 DISCUSSION
The lowest energy eight-vertex Be@Ge8z(z =−4, −2, 0, +2) structures all have the Be atom inside an eight-vertex Ge8 polyhedron indicating that suitably chosen Ge8polyhedra can accommodate a Be atom without distortion For the lowest energy Be@Ge8− structure (Figure 6) 8−22−1 the Ge8
polyhedron is the D4d square antiprism predicted by the Wade−Mingos rules8−11 for an arachno system having 2n + 6 skeletal electrons (= 22 for n = 8) The same square antiprism
is the polyhedron predicted for the lowest-energy structure19of
Figure 14 Two optimized Be@Ge6−structures.
Figure 15 Two optimized Be@Ge6−structures.
Figure 16 Two optimized Be@Ge6structures.
Figure 17 Two optimized Be@Ge62+ structures.
Table 1 Thermochemistry of the Buildup of the Be@Genz(n
= 6, 8, 10;z = −4, −2, 0, +2) Clusters Using the Lowest Energy Structuresa
Be@Ge6−+ Ge2→ Be@Ge 8 − −165.4 (−163.9) Be@Ge6−+ Ge2→ Be@Ge 8 − −112.1 (−111.8) Be@Ge6+ Ge2→ Be@Ge 8 −148.7 (−145.0) Be@Ge62+ + Ge2→ Be@Ge 82+ −135.2 (−133.2) Be@Ge8−+ Ge2→ Be@Ge 10 4− −171.3 (−170.8) Be@Ge8−+ Ge2→ Be@Ge 10 2− −154.5 (−153.3) Be@Ge8+ Ge2→ Be@Ge 10 −156.9 (−156.6) Be@Ge82+ + Ge2→ Be@Ge 10 −192.0 (−190.5)
a The ΔH values with zero point corrections are given in parentheses.
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Trang 7the isoelectronic Ge8− indicating that the presence of the
interstitial Be atom has relatively little effect on the preferred
polyhedron However, the dimensions of the empty Ge8−
square antiprism change significantly when a Be2+ dication is
inserted in the center to give Be@Ge8− The eight edges
connecting the two square faces are 2.83 Å in empty Ge8−but
contract to 2.68 Å in thefilled Be@Ge8 − However, the eight
edges of the two square faces of Ge8−change relatively little
when a Be2+ dication inserted into the center to give Be@
Ge8−, contracting only slightly from 2.79 Å in the empty Ge8−
to 2.77 Å in Be@Ge8− Thus the Ge8− square antiprism
becomes more oblate (i.e.,“flattened”) when a Be2+dication is
inserted into the center The contraction rather than expansion
in size when a Be2+ dication is inserted into a square
antiprismatic Ge8−cluster may relate to the reduction of the
negative charge in the original Ge8− by adding the Be2+
dication
The lowest energy Be@Ge8− structure is a D2d
bisdi-sphenoid structure (Figure 7) An analogous D2dbisdisphenoid
structure for the isoelectronic empty Ge8− lies only
2.8 kcal/mol above the global minimum The global minimum
of Ge8− is a more open structure analogous to that found
experimentally in the borane B8H12 A similar B8H12structure is
not found even in the higher energy Be@Ge8−structures
The lowest energy structures for the centered eight-vertex
electron poorer clusters Be@Ge8and Be@Ge82+ have the Be
atom in the center of a Ge8cube (Figures 8 and 9) For neutral
Be@Ge8 this cube is an ideal cube having the maximum Oh
symmetry with all 12 edges essentially equivalent in length
However, for the dication Be@Ge82+ the cube is distorted to
C2vsymmetry with two of the edges much longer (3.08 Å) than
the other ten edges (2.53 Å) These cubic Be@Ge8 and
Be@Ge82+ structures are different from the lowest energy
structures found for the isoelectronic Ge8 − and Ge8,
respectively Thus for Ge8− the lowest energy structure is
the Td tetracapped tetrahedron The D2d bisdisphenoid
predicted by the Wade−Mingos rules8−11 lies only 2.8 kcal/
mol above the Tdstructure The low energy structures for the
neutral Ge8 are somewhat open structures of relatively low
symmetry No cubic structures were found for either Ge8−or
Ge8 within the energy range investigated.19 Thus the size
demands of the interstitial Be atom in Be@Ge8and Be@Ge82+
make the outer Ge8 polyhedron the maximum volume cube
rather than the polyhedra with smaller internal volumes found
in the low-energy structures of the isoelectronic empty Ge8−
and Ge8clusters This appears to be an analogous phenomenon
to the experimental observation7of a D5hpentagonal prismatic
structure for the trianion Co@Ge103− rather than the D4d
bicapped square antiprism expected from the Wade−Mingos
rules However, in all four Be@Ge8z systems (z =−4, −2, 0,
+2) there is an eight-vertex Ge8polyhedron with an internal
volume large enough to accommodate the interstitial Be atom
without significant distortion in the lowest energy structure
It appears to be much more difficult for a seven-vertex Ge7
polyhedron to provide enough volume for an interstitial Be
atom without opening up Thus the two lowest energy
structures of the Be@Ge7− tetraanion, namely, 7−20−1 and
7−20−2 (Figure 10), have the Be atom inside open Ge7
networks rather than a closed Ge7 polyhedron Three much
higher energy Be@Ge7− structures, namely, 7−20−3,
7−20−4, and 7−20−5 (Figure 10), all lying at least 30 kcal/
mol above the 7−20−1 global minimum, have obvious origins
from a cube, a capped octahedron, and a capped trigonal prism,
respectively In 7−20−3 the Ge7network surrounding the Be atom is clearly a cube missing a vertex The Ge7 polyhedra surrounding the interstitial Be atoms in 7−20−4 and 7−20−5 are clearly opened up by stretching some of the original polyhedral edges beyond bonding distances
The lowest energy Be@Ge7−structure 7−18−1 (Figure 11) has the Be atom on the surface rather than in the center of the
Ge7unit The capped trigonal prismatic structure 7−18−2 with
a truly interstitial Be atom lies 23.2 kcal/mol above 7−18−1 This is a good example where a relatively small polyhedral cluster with an interstitial atom is less favorable energetically than an isomeric empty cluster with the Be atom on the surface The dianion Ge7− has 16 (= 2n + 2 for n = 7) skeletal electrons and thus is expected by the Wade−Mingos rules8−11
to have a structure based on the most spherical deltahedron For a seven-vertex system, the most symmetrical deltahedron is
a pentagonal bipyramid Indeed for Ge7− the pentagonal bipyramid is found to lie more than 20 kcal/mol in energy below the next lowest energy structure.26 A similar centered pentagonal bipyramidal structure 7−16−2 (Figure 12) was found for the isoelectronic neutral Be@Ge7 However, this structure lies 11.8 kcal/mol above the global minimum
7−16−1 in which the Be atom has migrated to the surface forming a low-symmetry BeGe7 eight-vertex polyhedron Inserting Be2+into the center of a Ge7−pentagonal bipyramid has relatively little effect on the edge-lengths of the equatorial pentagon, that is, the eq-eq edges, but increases significantly the edges connecting equatorial pentagon vertices with the axial vertices, that is, the eq-ax edges Thus insertion of Be2+into a
Ge7− pentagonal bipyramid increases the eq-ax edges from 2.83 Å to 2.90 Å and the nonbonding ax···ax distance from 3.58
Å to 4.04 Å but leaves the eq-eq edges at 2.58 Å Thus, the
Ge7−pentagonal bipyramid becomes more prolate (elongated) upon inserting Be2+in the center
The lowest energy structure of the Be@Ge72+ dication by more than 23 kcal/mol, namely, 7−14−1 (Figure 13), is a relatively unsymmetrical structure with the Be atom on the surface rather than in the center The lowest energy Be@Ge72+ structure with the Be atom in the center of a polyhedron is
7−14−3 lying ∼25 kcal/mol above the global minimum 7−
14−1 In 7−14−3 the Be atom is in the center of a C3vcapped trigonal prism The lowest energy structure previously found for the empty neutral cluster Ge7 is an oblate pentagonal bipyramid that has beenflattened to the extent that the ax-ax distance of 2.82 Å is only slightly longer than the essentially equivalent eq-ax and eq-eq edges of 2.68 Å A similar structure was not found for the isoelectronic Be@Ge72+
The lowest energy structures for the centered six-vertex clusters Be@Ge6z (z = −4, −2, 0, +2) continue the trend of having the Be atom inside an open Ge6network or the Be atom
on the surface of a seven-vertex BeGe6 polyhedron Some Be@Ge6z cluster structures with the Be atom inside a closed
Ge6 polyhedron are found, but all such structures lie significantly in energy above the corresponding global minimum Thus for the tetraanion the Be@Ge6− structure
6−18−2 with a Be atom inside a Ge6trigonal prism lies ∼40 kcal/mol above the lowest energy Be@Ge6−structure 6−18−
1, which is an open structure with an exposed Be atom (Figure 14) The 18 skeletal electrons in the trigonal prismatic Be@
Ge6−structure correspond to two-electron two-center bonds along each of the nine edges of the underlying trigonal prism The lowest energy Be@Ge6− structure 6−16−1 is an interesting BeGe6pentagonal bipyramid of C5vsymmetry with
| J Phys Chem A 2012, 116, 5227−5234 5233
Trang 8the Be atom at the surface as one of the axial vertices (Figure
15) The 16 skeletal electrons in 6−16−1 (= 2n + 2 for n = 7)
correspond to the number required by the Wade−Mingos
rules8−11for the most spherical polyhedral structure, which is a
pentagonal bipyramid for a seven-vertex system The lowest
energy centered polyhedral structure Be@Ge6− is a triplet
centered trigonal prism 6−16−2T lying a whopping ∼70 kcal/
mol above 6−16−1
The lowest energy structure of the neutral Be@Ge6, namely,
6−14−1 (Figure 16), is a distorted version of 6−16−1 (Figure
15) with a distorted BeGe6pentagonal bipyramid having the Be
atom on the surface as one of the axial vertices The Oh
octahedral Be@Ge6structure 6−14−2 with the Be atom in the
center of a regular Ge6octahedron lies 12.2 kcal/mol above the
lowest energy structure 6−14−1 The corresponding empty
structure of the isoelectronic Ge6−was found to be the global
minimum, lying ∼11 kcal/mol below the next lowest energy
Ge6−structure.26The lengths of the 12 octahedral edges in the
empty Ge6− cluster of 2.69 Å increase to 2.87 Å in the
isoelectronic Be@Ge6 cluster 6−14−2, apparently to
accom-modate the interstitial Be atom
Two low-energy structures, namely, 6−12−1 and 6−12−2,
were found for the dication Be@Ge62+ (Figure 17) Both of
these structures have their Be atoms at the surface of a
seven-vertex BeGe6polyhedron rather than the Be atom in the center
of a Ge6polyhedron
5 SUMMARY
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@Ge8−, a D2dbisdisphenoid for Be@Ge8−, an ideal Oh
cube for Be@Ge8, and a C2vdistorted cube for Be@Ge82+ The
Be-centered cubic structures predicted for Be@Ge8 and
Be@Ge82+ differ from the previously predicted lowest energy
structures for the isoelectronic Ge8−and Ge8related to the fact
that the cube has the largest internal volume for a closed
eight-vertex polyhedron
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 Genpolyhedron 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 C2vcapped trigonal prismatic structure
for Be@Ge7−, a centered D5hpentagonal bipyramidal structure
for Be@Ge7, a centered D3h trigonal prismatic structure for
Be@Ge6−, 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 relates to the failure to date to observe
experimentally six- and eight-vertex bare post-transition
element clusters with interstitial atom
*S Supporting Information
Table S1, Interatomic Distances in Be@Ge8zStructures; Table
S2, Interatomic Distances in Be@Ge7z Structures; Table S3,
Interatomic Distances in Be@Ge6z Structures; Table S4,
Optimized Coordinates of the Be@Genz Clusters (n = 8, 7,
6); Table S5, Details of the Thermochemistry Calculations in
Table 1 for the reactions Be@Genz + Ge2 → Be@Gen+2z; Complete Gaussian03 Reference (Reference 24) This material
is available free of charge via the Internet at http://pubs.acs.org
Corresponding Author
*E-mail: rbking@chem.uga.edu
Notes The authors declare no competingfinancial interest
This work was supported by CNCSIS-UEFISCSU, project number PNII-RU 465/2010, in Romania and by the National Science Foundation under Grant CHE-1057466 in the U.S.A
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