ORIGINAL PAPER

Empty versus filled polyhedra: 11 vertex bare germanium clusters

Matei-Maria Uţă & Robert Bruce King

Received: 29 January 2014 /Accepted: 24 February 2014 /Published online: 28 March 2014# Springer-Verlag Berlin Heidelberg 2014

Abstract The structures and energetics of centered 10-vertexGe@Ge10

z (z=−4, −2, 0, +2, +4) clusters have been investi-gated by density functional theory (DFT) for comparison withthe previously studied isomeric empty 11-vertex Ge11

z clus-ters. For the cationic species (z=+2, +4) such centeredGe@Ge10

z structures are shown to be energetically competi-tive (within ∼1 kcal mol−1) to the lowest energy isomericempty Ge11

z structures. These Ge@Ge10z structures can be

derived from the lowest energy empty 10-vertex Ge10z−4

structures by inserting a Ge4+ ion in the center. The outer10-vertex polyhedron in the lowest energy Ge@Ge10

2+

dication structure is the most spherical D4d bicapped squareantiprism, which is also the lowest energy structure of theempty Ge10

2− dianion, as expected from the Wade-Mingosskeletal electron counting rules. For the tetracationic Ge11

4+/Ge@Ge10

4+ system the lowest energy centered Ge@Ge104+

structure can be obtained by inserting a Ge4+ ion in the centerof a C3v deltahedral empty Ge10 cluster. Centered 10-vertexpolyhedral Ge@Ge10

z structures were also found for the neu-tral (z=0) and dianionic (z=−2) systems but at significantlyhigher energies than the lowest energy isomeric empty Ge11

z

structures.

Keywords Cluster polyhedra . Density functional theory .

Germanium . Interstitial atoms .Wade-Mingos rules

Introduction

During the past decadewe have used density functional theoryto investigate the effects of electron count on the preferredgeometries for the homoatomic germanium clustersGen

z of various sizes (n=5 to 12 and 14) and charges(z=−6, −4, −2, 0, +2, +4) [1–7]. However, only emptyGen polyhedra were considered as possible structuresfor these species. The discovery in recent years of avariety of bare post-transition element clusters contain-ing interstitial atoms suggests a reinvestigation of thesesystems also considering alternative homoatomic Gen

z

structures based on an outer n–1 vertex Gen-1 polyhe-dron with the remaining germanium atom in the center,i.e., a Ge@Gen–1

z structure.Experimental information suggests that diverse 10-

vertex polyhedra of germanium and neighboring post-transition elements are particularly favorable for hostinginterstitial atoms (Fig. 1). Thus a D4d bicapped squareantiprism is found to encapsulate a group 12 metal atomin the anionic indium cluster Zn@In10

8− found in theintermetallic [8] K8In10Zn. A similar D4d bicappedsquare antiprism also encapsulates group 10 metal atomsin the lead clusters M@Pb10

2− in [K(2,2,2-crypt)]2[M@Pb10](M=Ni, Pd, Pt) [9, 10]. However, in the M@In10

10− clustersfound in the K10In10M intermetallics (M=Ni, Pd, Pt),isoelectronic with Zn@In10

8−, the encapsulating polyhe-dron is a C3v tetracapped trigonal prism [11]. The pen-tagonal antiprism is the host polyhedron for an intersti-tial palladium atom in the bismuth cluster tetracationPd@Bi10

4+ in Bi14PdBr16 (=[Pd@Bi10][BiBr4]4) [12].The pentagonal prism has been found as the host poly-hedron for an iron or cobalt atom in the clustersM@Ge10

3− (M=Fe [13], Co [14]).The ability of diverse 10-vertex polyhedra to encap-

sulate an 11th atom suggests the possibility of Ge11

Electronic supplementary material The online version of this article(doi:10.1007/s00894-014-2193-9) contains supplementary material,which is available to authorized users.

M.<M. UţăFaculty of Chemistry and Chemical Engineering, Babeş-BolyaiUniversity, Cluj-Napoca, Romania

R. B. King (*)Department of Chemistry, University of Georgia, Athens, GA 30602,USAe-mail: rbking@chem.uga.edu

J Mol Model (2014) 20:2193DOI 10.1007/s00894-014-2193-9

structures consisting of an outer Ge10 polyhedron en-capsulating an 11th germanium atom, i.e., a Ge@Ge10

z.Although the neutral Ge11 cluster is certainly unstablewith respect to polymerization to bulk germanium met-al, photoionization studies of Gen show significant masspeaks for n=6, 7, 10, 11 [15, 16]. Accordingly, we haveused density functional theory to examine the preferredstructures and relative energies of such Ge@Ge10

z struc-tures using as outer Ge10 polyhedra the empty 10-vertexpolyhedra from our previous study of 10-vertex systems[5]. The density functional theory methods used here arewell established having previously been used by ourgroup to study similar bare germanium clusters rangingfrom 5 to 12 and 14 [1–7]. Since the density functionaltheory methods used in the present work are the sameas those previously used to investigate the empty 11-vertex Ge11

z polyhedral structures [3, 17], a direct com-parison of the relative energies of isomeric empty Ge11

z

and filled Ge@Ge10z structures can be made.

Theoretical methods

Geometry optimizations were carried out at the hybridDFT B3LYP level [18–21] with the 6-31G(d) (valence)double-zeta quality basis functions extended by addingone set of polarization (d) functions for the germaniumatoms. The Gaussian 03 package of programs [22] wasused in which the fine grid (75,302) is the default fornumerically evaluating the integrals and the tight (10−8)hartree stands as default for the self-consistent fieldconvergence. Computations were carried out using asinitial geometries the four polyhedra in Fig. 1 with anadditional germanium atom in the center. The symme-tries were maintained during the initial geometry opti-mization processes. Symmetry breaking using modes

defined by imaginary vibrational frequencies was thenused to determine optimized structures with minimumenergies. Vibrational analyses show that all of the finaloptimized structures discussed in this paper are genuineminima at the B3LYP/6-31G(d) level without any sig-nificant imaginary frequencies (Nimag=0). In a few casesthe calculations ended with acceptable small imaginaryfrequencies [23] and these values are indicated in thecorresponding figures.

The new optimized structures found for the centeredGe@Ge10

z derivatives are labeled as Ge@Ge10-x-y,where x is the number of skeletal electrons and y ordersthe structures according to their relative energies. Indetermining the numbers of skeletal electrons x theinterstitial germanium atom is assumed to be a donorof four skeletal electrons in contrast to the germaniumvertices of the outer polyhedron, which are assumed tobe donors of two skeletal electrons each as suggestedby the Wade-Mingos rules [24–27]. Thus the lowestenergy structure of the dication Ge@Ge10

2+ is labeledGe@Ge10-22-1. The previously reported isomeric emp-ty Ge11

z structures [17] are designated Ge11-xA-ywhere x again is the number of Wadean skeletal elec-trons and y designates the relative energy ordering con-sidering only the empty structures in the previous paper.The designation T is added to the few triplet structures.Additional details of all of the optimized structures, in-cluding all interatomic distances, the initial geometriesleading to a given optimized structure, and structureswith energies too high to be of possible chemical inter-est are provided in the Supporting information. Inassigning polyhedra to the optimized structures, theGe–Ge distances less than ∼3.2 Å were normally con-sidered as polyhedral edges, based on experimental re-sults as well as on our previous experience with themethods employed. In the figures the centered Ge@Ge10

z

structures are enclosed in rectangular boxes to distinguishthem from the isomeric empty Ge11

z structures.Note that the 11 atom centered structures of the type

Ge@Ge10z have two more Wadean skeletal electrons than

the isomeric empty Ge11z structures. This arises because all

four valence electrons of the interstitial germanium atom areassumed to be skeletal electrons whereas each of the polyhe-dral vertex germanium atoms are assumed to be donors ofonly two skeletal electrons.

Results and discussion

The Ge11z/Ge@Ge10

z (z=−4, −2, 0) systems

The only centered Ge@Ge104− tetraanion structures

found in this work have energies lying more than

C3v tetracappedtrigonal prism

D4d bicappedsquare antiprism

D5d pentagonalantiprism

D5h pentagonalprism

Fig. 1 The four 10-vertex polyhedra found as host polyhedra for interstitialmetal atoms. For clarity the edges of capping vertices are shown in green

2193, Page 2 of 5 J Mol Model (2014) 20:2193

50 kcal mol−1 above the lowest energy Ge114− struc-

tures. They thus do not appear to be chemically signif-icant and are not considered in this paper. For thedianions Ge11

2−/Ge@Ge102− a D5d centered pentagonal

antiprismatic structure Ge@Ge10-26-1 was found but atan energy 32.3 kcal mol−1 above that of the lowestenergy empty Ge11

2− structure Ge11-24A-1 (Fig. 2).This structure can be derived from the pentagonalantiprismatic Ge10

6− by insertion of an interstitial Ge4+

cation. The 26 skeletal electrons of the outer Ge106−

pentagonal antiprism in Ge@Ge10-26-1 are consistentwith the Wade-Mingos rules [24–27] for a polyhedronwith two non-triangular faces and relates to the generation of apentagonal antiprism by removal of an antipodal pair ofvertices from an icosahedron. For the neutral Ge11/Ge@Ge10system, the lowest energy Ge@Ge10 structureGe@Ge10-24-1 is a centered C3v polyhedral structure lying 16.5 kcal mol−1

above the lowest energy empty Ge11 structure (Fig. 3).Thus in each of the three Ge11

z/Ge@Ge10z (z=−4, −2,

0) systems the lowest energy centered Ge@Ge10z struc-

ture lies at a significantly higher energy than the lowestenergy isomeric empty Ge11

z structure.

The dications Ge112+/Ge@Ge10

2+

The dicationic Ge112+/Ge@Ge10

2+ is the only system forwhich a centered Ge@Ge10

2+ structure, namelyGe@Ge10-22-1 (Fig. 4), has been found of lower ener-gy than any of the isomeric empty Ge11

2+ structures,albeit by only ∼1 kcal mol−1. The Ge@Ge10

2+ structureGe@Ge10-22-1 can be generated by inserting a Ge4+

cation into the center of a Ge102− bicapped square

antiprism. Note that the bicapped square antiprismaticGe10

2− structure (structure 22–1 in reference [5]) is thelowest energy Ge10

2− structure [5] and is based on themost spherical 10-vertex deltahedron in accord with theWade-Mingos rules [24–27] for a 2n+2 skeletal electronsystem. These special features of the Ge@Ge10-22-1structure are required to provide the single example of

a centered Ge@Ge10z structure being of lower energy

than any of the isomeric empty Ge11z structures.

The next higher energy centered Ge@Ge102+ structure

is a triplet structure Ge@Ge10-22-2T, found to lie10.5 kcal mol−1 in energy above Ge@Ge10-22-1(Fig. 4). Four isomeric empty Ge11

2+ structures, allfound in the previous DFT study [17], are predicted tohave energies lying between those of Ge@Ge10-22-1and Ge@Ge10-22-2T, namely Ge11-20-A1, Ge11-20-A2, Ge11-20-A3, and Ge11-20-A4 at 1.2, 4.0, 4.9,and 6.9 kcal mol−1, respectively, above Ge@Ge10-22-1. All four of these empty Ge11

2+ structures have littleor no symmetry and are not based on obvious familiar11-vertex polyhedra. The centered triplet Ge@Ge10

2+

structure Ge@Ge10-22-2T can be obtained by insertinga Ge4+ ion inside a triplet C3v Ge10

2− polyhedron. Thecorresponding empty triplet C3v Ge10

2− polyhedron (22-2T in reference [5]) is found in the previous DFT studyto lie 16.7 kcal mol−1 above the bicapped squareantiprism global minimum.

The tetracations Ge114+/Ge@Ge10

4+

Three centered tetracationic Ge@Ge104+ structures are

found within 10 kcal mol−1 of the global minimum,namely the empty Ge11

4+ structure Ge11-18A-1 basedon the 11-vertex polyhedron found in the E11

7− (E=Ga,In, Tl) structural units in the K8E11 intermetallics [28,29]. In addition to these three centered Ge@Ge10

4+

structures, four isomeric empty Ge114+ structures, in-

cluding the Ge11-18A-1 global minimum, were foundin the previous DFT study [17] to lie in this energyrange (Fig. 5).

The lowest energy centered Ge@Ge104+ structure

Ge@Ge10-20-1 lies only 0.6 kcal mol−1 in energyabove Ge11-18A-1 (Fig. 5). This centered Ge@Ge10

4+

structure Ge@Ge10-20-1 can be obtained by insertingan interstitial Ge4+ cation into the center of a C3v

polyhedron. This C3v polyhedron is derived from thetetracapped trigonal prism by lengthening threesymmetry-related edges (Ge3-Ge4, Ge4-Ge7, and Ge7-

Fig. 2 A comparison of the lowest energy centered Ge@Ge102− structure

with the lowest energy isomeric empty Ge112− structure

Fig. 3 A comparison of the lowest energy centered Ge@Ge10 structurewith the lowest energy isomeric empty Ge11 structure

J Mol Model (2014) 20:2193 Page 3 of 5, 2193

Ge3 in Fig. 5) to ∼3.88 Å, thereby expanding thecentral cavity to accommodate the interstitial germaniumatom. The tetracapped trigonal prism was found to bethe lowest energy neutral Ge10 structure in the previousDFT study [5].

The next centered Ge@Ge104+ structure Ge@Ge10-20-2,

lying 2.0 kcal mol−1 in energy above Ge11-18A-1, is derivedfrom the bicapped square antiprism and retains its D4d sym-metry. However, the eight equivalent square edges of the

underlying square antiprism are lengthened to ∼3.28 Å, there-by expanding the volume to accommodate the interstitialgermanium atom.

The remaining Ge@Ge104+ structure Ge@Ge10-20-3

lies 8.3 kcal mol−1 in energy above Ge11-18A-1. Theouter Ge10 polyhedron in Ge@Ge10-20-3 has no sym-metry and no readily apparent relationship to any of thelow-energy Ge10 polyhedra found in the previous DFTstudy [5].

Fig. 4 A comparison of the twolowest energy centeredGe@Ge10

2+ structures with theisomeric empty Ge11

2+ structuresof similar energies

Ge11-18A-1 (D3h)0.0 kcal/mol

Ge@Ge10-20-1 (C3v)0.6 kcal/mol

Ge11-18A-2 (Cs)2.3 kcal/mol

Ge@Ge10-20-2 (D4d)2.0 kcal/mol

Ge11-18A-3 (Cs)7.4 kcal/mol

Ge11-18A-4 (C2v)8.2 kcal/mol

Ge@Ge10-20-3 (C1)8.3 kcal/mol

Fig. 5 A comparison of the threelowest energy centeredGe@Ge10

4+ structures with theisomeric empty Ge11

4+ structuresof similar energies

2193, Page 4 of 5 J Mol Model (2014) 20:2193

Conclusions

The DFT studies in this paper show that centered Ge@Gen–1z

structures can be energetically competitive with isomericempty Gen

z structures, at least for systems with 11 germaniumatoms (n=11). Thus the dicationic Ge11

2+/Ge@Ge102+ and

tetracationic Ge114+/Ge@Ge10

4+ systems have centeredGe@Ge10

z structures of comparable energies (within ∼1 kcalmol−1) to the lowest energy isomeric empty Ge11

z structures.Such Ge@Ge10

z structures can be derived from the lowestenergy empty Ge10

z–4 structures by inserting a Ge4+ ion in thecenter. This might explain the fact that stable interstitial clus-ters have been found in this study only for cationic species.Thus, the outer 10-vertex polyhedron in the lowest energyGe@Ge10

2+ structure, lying ∼1.2 kcal mol−1 in energy belowthe lowest energy isomeric empty Ge11

2+ structure, is the mostspherical D4d bicapped square antiprism. This deltahedron isalso the lowest energy structure of the empty Ge10

2−, asexpected from the Wade-Mingos skeletal electron countingrules. For the tetracationic Ge11

4+/Ge@Ge104+ system the

lowest energy centered Ge@Ge104+ structure, lying only

∼0.6 kcal mol−1 above the lowest energy isomeric Ge114+

structure, is based on an outer C3v deltahedron, which is thelowest energy structure of the empty neutral Ge10 cluster.

Centered 10-vertex polyhedral Ge@Ge10z structures were

also found for the neutral (z=0) and dianionic (z=−2)systems but at significantly higher energies than thelowest energy isomeric Ge11

z structures. The lowest en-ergy centered neutral Ge@Ge10 structure, lying 16.5 kcalmol−1 above the lowest energy isomeric Ge11 structure,has an outer 10-vertex C3v polyhedron derived from thetetracapped trigonal prism similar to that found in thelowest energy centered tetracation Ge@Ge10

4+. The low-est energy centered dianion Ge@Ge10

2− is an evenhigher energy structure lying 32.3 kcal mol−1 above thelowest energy isomeric Ge11

2− structure. This Ge@Ge102−

structure can be generated by inserting a Ge4+ ion insidethe pentagonal antiprism predicted by the Wade-Mingosrules for a 10-vertex polyhedron with 26 skeletalelectrons.

Acknowledgments This work was supported by Consiliul Naţional alCercetării Ştiinţifice din Învăţământul Superior-Unitatea Executivăpentru Finanţarea Învăţământului Superior şi a Cercetării Ştiinţifice

Universitare (CNCSIS-UEFISCSU), project number PCCE-129 P4, inRomania and by the National Science Foundation under Grant CHE-1057466 in the U. S. A.

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