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Nonspherical Deltahedra in Low-Energy Dicarbalane Structures Testing the Wade−Mingos Rules: The Regular Icosahedron Is Not Favored for the 12-Vertex Dicarbalane Amr A. A. Attia,† Alexandru Lupan,*,† and R. Bruce King*,‡ †

Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai University, RO-400028 Cluj-Napoca, Romania ‡ Department of Chemistry and Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602, United States S Supporting Information *

ABSTRACT: Theoretical studies on the dicarbalanes C2Aln−2Men (n = 7−14; Me = methyl) predict both carbon atoms to be located at degree 4 vertices of a central C2Aln−2 deltahedron in the lowest energy structures. As a consequence, deltahedra having two degree 4 vertices, two degree 6 vertices, and eight degree 5 vertices rather than the regular icosahedron having exclusively degree 5 vertices are found for the 12-vertex dicarbalane C2Al10Me12. However, the lowest energy C2Aln−2Men (n = 7−11) structures are based on the same most spherical (closo) deltahedra as the corresponding deltahedral boranes. The lowest energy structures for the 13- and 14-vertex systems C2Aln−2Men (n = 13 and 14) are also deltahedra having exactly two degree 4 vertices for the carbon atoms. The six-vertex C2Al4Me6 system is exceptional since bicapped tetrahedral and capped square pyramidal structures with degree 3 vertices for the carbon atoms are energetically preferred over the octahedral structure suggested by the Wade−Mingos rules.

1. INTRODUCTION A characteristic feature of the chemistry of boron cages is the high stability and thus low chemical reactivity of structures based on regular icosahedra. For example, the least reactive borane dianions BnHn2− and isoelectronic dicarbaboranes C2Bn−2Hn are 12-vertex systems with central icosahedra.1,2 In such regular icosahedra, all vertices have degree 5, where the degree of a vertex is defined as the number of edges meeting at that vertex. In a series of theoretical papers spanning more than 20 years, Williams has recognized the special stability of degree 5 boron vertices.3−5 These ideas have also been used to rationalize the stabilities of hydrogen-rich nido6 and arachno7 boranes having general formulas B n H n+4 and B n H n+6 , respectively. The regular icosahedron is the 12-vertex member of the series of most spherical polyhedra found in the polyhedral boranes BnHn2− and the isoelectronic carboranes CBn−1Hn and C2Bn−2Hn (Figure 1). Such species with n vertices have 2n + 2 skeletal electrons, as suggested by the Wade−Mingos rules8−10 as well as topological models of borane cluster bonding.11,12 In these most spherical polyhedra, also designated as closo polyhedra, the vertices are as nearly equivalent as possible so that the surface of a sphere can be tiled using these polyhedra with minimum distortion. The most spherical deltahedra with 6 and 12 vertices are the regular octahedron and icosahedron, with complete equivalence of the vertices having degrees 4 and 5, respectively. The most spherical polyhedra with 7−10 vertices have exclusively degree 4 and 5 vertices. Degree 6 vertices appear in the most spherical polyhedra with 11 and 14 © 2015 American Chemical Society

vertices. However, the most spherical 13-vertex polyhedron, found in C2B11 dicarbaboranes, avoids a degree 6 vertex by having a trapezoidal face rather than all triangular faces.13 All of the other most spherical polyhedra are deltahedra with exclusively triangular faces. The most spherical 12- and 14vertex polyhedra, as well as the most spherical 15- and 16vertex polyhedra not discussed in this Article, are the Frank− Kasper polyhedra.14 Such polyhedra have exclusively degree 5 and 6 vertices with no pair of adjacent degree 6 vertices. A question of interest pertains to the existence of aluminum analogues of the polyhedral boranes. Whereas extensive series of very stable polyhedral boranes exist with terminal hydrogen atoms, i.e., B−H vertices, stable polyhedral alanes are likely to require external alkyl groups owing to the inherent reactivity of Al−H bonds. The only AlnRn2− dianion that has been synthesized is the regular icosahedral Al12iBu122− (iBu = isobutyl) anion, isolated as its dipotassium salt.15 An octahedral Al6tBu6− (tBu = tert-butyl) radical anion has also been synthesized.16 These deltahedral aluminum clusters are dark red, in contrast to the typically colorless deltahedral borane dianions BnHn2−. Schnöckel et al. have also isolated and structurally characterized aluminum halide clusters of stoichiometry Al22X20·12L (L = tetrahydrofuran or tetrahydropyran; X = Cl,17 Br18) that can be described as Al12[AlX2·L]10·2L with a compressed Al12 icosahedron as a central core. Theoretical studies on the complete series of AlnHn2− dianions Received: August 31, 2015 Published: November 6, 2015 11377

DOI: 10.1021/acs.inorgchem.5b02014 Inorg. Chem. 2015, 54, 11377−11384

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Inorganic Chemistry indicate energetic preferences for the same most spherical closo structures found in the BnHn2− dianions discussed above (Figure 1).19

addition, the 14-vertex omnicapped cube is the underlying deltahedron in the (AlMe)8(CCH2Ph)5(CCPh) structure, with aluminum atoms at the corners of the underlying cube, now degree 6 vertices, and carbon atoms forming the degree 4 caps. A clear feature of these experimentally known carbalane structures is the availability of degree 4 vertices for all of the cluster carbon atoms. In an effort to understand the still underdeveloped area of carbalane chemistry, we start with the dicarbalanes C2Aln−2Men (n = 6−14) as analogues of the well-known dicarbaboranes C2Bn−2Hn. Methyl groups rather than hydrogen atoms were used as external groups in these dicarbalanes in order to provide more realistic models for systems likely to be synthesized in the future. Our density functional theory studies on the C2Aln−2Men systems show that the preferred structures for the systems having 7−11 vertices are the same most spherical closo deltahedra found in the deltahedral boranes (Figure 1) with both carbon atoms at degree 4 vertices. However, this tendency for carbon atoms in carbalanes to occupy degree 4 vertices is so strong that the preferred polyhedron for the 12-vertex C2Al10Me12 system is not the regular icosahedron with exclusively degree 5 vertices. Instead, the lowest energy C2Al10Me12 structures have central C2Al10 deltahedra with at least two degree 4 vertices for the carbon atoms balanced by an equal number of degree 6 vertices.

2. THEORETICAL METHODS Full geometry optimizations were carried out on the C2Aln‑2Men systems (n = 6−14) at the B3LYP/6-31G(d) level of theory.21−24 The lowest energy structures were then reoptimized at a higher level, i.e., PBE0/def2-TZVPP,25−27 and these are the structures presented in the Article. A recent benchmarking study on bare aluminum clusters showed that this approach provides the best results.28 The initial structures were chosen by systematic substitution of two aluminum atoms by two carbon atoms in various AlnMen polyhedral frameworks. The large number of different starting structures for the optimizations included 25 structures of the 6-vertex clusters C2Al4Me6, 61 structures of the 7-vertex clusters C2Al5Me7, 117 structures of the 8vertex clusters C2Al6Me8, 129 structures of the 9-vertex clusters C2Al7Me9, 176 structures of the 10-vertex clusters C2Al8Me10, 136 structures of the 11-vertex clusters C2Al9Me11, 114 structures of the 12-vertex clusters C2Al10Me12, 99 structures of the 13-vertex clusters C2Al11Me13, and 51 structures of the 14-vertex clusters C2Al12Me14 (see the Supporting Information). The natures of the stationary points after optimization were checked by calculations of the harmonic vibrational frequencies. If significant imaginary frequencies were found, then the optimizations were continued by following the normal modes corresponding to imaginary frequencies to ensure that genuine minima were obtained. All of the optimized C2Aln−2Men structures discussed in this Article have substantial HOMO−LUMO gaps, ranging from 2.0 to 4.9 eV. All calculations were performed using the Gaussian 09 package29 with the default settings for the SCF cycles and geometry optimization, namely, the fine grid (75 302) for numerically evaluating the integrals, 10−8 hartree for the self-consistent field convergence, maximum force of 0.000450 hartree/bohr, RMS force of 0.000300 hartree/bohr, maximum displacement of 0.001800 bohr, and RMS displacement. The figures of the clusters presented in this Article have been rendered using the Chimera package.30 The C2Aln−2Men (n = 6−14) structures are numbered as Al(n− 2)C2−x, where n is the total number of polyhedral vertices and x is the relative order of the structure on the energy scale (PBE0/def2TZVPP including zero-point corrections). The lowest energy optimized structures discussed in this Article are depicted in Figures 3−11. Only the lowest energy and thus potentially chemically significant structures are considered in detail in this Article. More

Figure 1. Most spherical (closo) polyhedra having from 6 to 14 vertices. All of these polyhedra are known experimentally in derivatives of the dicarbaboranes C2Bn−2Hn.

The dicarbaboranes C2Bn−2Hn (historically called simply carboranes) are neutral species isoelectronic with the BnHn2− dianions.2 No examples of the aluminum analogues of the dicarbaboranes, namely, the dicarbalanes C2Aln−2Hn, have yet been synthesized. However, carbon-richer carbalanes are known, as exemplified by the tetracarbalane (AlEt)7(CCH2Ph)4(μ-H)2.20 The central 11-vertex C4Al7 deltahedron in this structure has four degree 4 vertices for the carbon atoms balanced with three degree 6 vertices for the aluminum atoms (Figure 2). It is thus topologically different from the most spherical 11-vertex closo-borane deltahedron found in B11H112−, C2B9H11, and their derivatives (Figure 1). In

Figure 2. (Left) C4Al7 deltahedron found in (AlEt)7(CCH2Ph)4(μH)2; (Right) C6Al8 omnicapped cube found in (AlMe)8(CCH2Ph)5(CCPh). Degree 4, 5, and 6 vertices are in red, black, and green, respectively. 11378


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Figure 3. Six lowest energy 12-vertex C2Al10Me12 structures.

Table 1. Six Lowest Energy C2Al10Me12 Structuresa

comprehensive lists of structures, including higher energy structures, are given in the Supporting Information.

3. RESULTS 3.1. 12-Vertex C2Al10Me12 System (Figure 3 and Table 1). The only known structures for the 12-vertex dicarbaboranes C2B10R12 (R = H, Me) have central C2B10 regular icosahedra with only degree 5 vertices and represent extremely stable molecules. 2 However, none of the structures of the corresponding 12-vertex dicarbalanes C2Al10Me12 within 30 kcal/mol of the global minimum have a central regular C2Al10 icosahedron with only degree 5 vertices. Instead, the lowenergy C2Al10Me12 structures are based on less symmetrical 12vertex deltahedra with at least two degree 4 vertices for the carbon atoms (Figure 3 and Table 1). Such 12-vertex deltahedra necessarily have the same number of degree 6 vertices as degree 4 vertices. No low-energy C2Al10Me12 structures are found with degree 7 vertices similar to the predicted lowest energy structure for the molybdatricarbaborane CpMoC3B8H11 (Cp = η5-C5H5).31 The two lowest energy C2Al10Me12 structures, Al10C2−1 and Al10C2−2, have two degree 4 vertices for the carbon atoms and correspondingly two degree 6 vertices and are essentially degenerate, lying within 1.3 kcal/mol of each other (Figure 3 and Table 1). In Al10C2−1, the carbon atoms are located at opposite ends of the deltahedron with a C···C distance of 4.685 Å. However, in Al10C2−2, the carbon atoms are separated by a single aluminum atom, leading to a shorter C···C distance of 3.234 Å. Less spherical 12-vertex C2Al10Me12 structures having more than two degree 6 vertices lie at higher energies starting with the oblate (flattened) structure Al10C2−3, with three degree 6 vertices and nonadjacent carbon atoms lying 11.8 kcal/mol above Al10C2−1 (Figure 3 and Table 1). Other structures with three degree 6 vertices are the oblate structure Al10C2−4 and the prolate (elongated) structure Al10C2−6, lying 12.6 and 17.7 kcal/mol, respectively, in energy above Al10C2−1. Both Al10C2−4 and Al10C2−6 have the two carbon atoms at

structure

ΔE

degree 6

(symmetry)

kcal/mol

vertices

0.0 1.3 11.8 12.6 14.8 17.7

2 2 3 3 4 3

Al10C2−1 Al10C2−2 Al10C2−3 Al10C2−4 Al10C2−5 Al10C2−6

(C2) (C2v) (Cs) (Cs) (C2) (Cs)

C−C distance 4.685 3.234 3.844 1.472 3.654 1.474

Å Å Å Å Å Å

a

Distances between adjacent carbon atoms sharing an edge are indicated in italic.

adjacent degree 4 vertices, forming edges of length ∼1.47 Å. The lowest energy C2Al10Me12 structure with four degree 6 vertices as well as four degree 5 and four degree 4 vertices is the oblate structure Al10C2−5, lying 14.8 kcal/mol in energy above Al10C2−1. 3.2. C2Aln−2Men (n = 7−11) Systems (Figures 4−8 and Table 2). The most spherical deltahedra with 7−11 vertices (Figure 1) all have at least two degree 4 vertices for the carbon atoms in the dicarbalanes C2Aln−2Men and are found in the lowest energy structures, consistent with the Wade−Mingos rules (Table 2 and Figures 4−8).8−10 Thus, the two lowest energy 7-vertex C2Al5Me7 structures are both C2v structures with central C2Al5 pentagonal bipyramids and substantial HOMO−LUMO gaps exceeding 4.6 eV (Figure 4). The lower energy of these structures, Al5C2−1, has the two carbon atoms at adjacent equatorial vertices, forming a C−C edge of length 1.484 Å. The next higher energy C2Al5Me7 structure Al5C2−2, lying 7.3 kcal/mol in energy above Al5C2−1, has the two carbon atoms at nonadjacent equatorial vertices. The two lowest energy structures for the 8-vertex C2Al6Me8 system are based on the most spherical 8-vertex deltahedron, namely, the bisdisphenoid (Figure 1), with the carbon atoms again at degree 4 vertices (Figure 5 and Table 2). The lower energy of these two structures, namely, Al6C2−1, has adjacent carbon atoms forming an edge of length 1.488 Å. The next 11379


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Table 2. Lowest Energy C2Aln−2Men (n = 7−11) Structuresa carbon vertices structure (symmetry)

ΔE kcal/mol

degrees

7-vertex systems (up to 19 kcal/mol) Al5C2−1 (C2v) 0.0 4,4 Al5C2−2 (C2v) 7.3 4,4 8-vertex systems (up to 25 kcal/mol) Al6C2−1 (C2v) 0.0 4,4 Al6C2−2 (C2) 6.4 4,4 Al6C2−3 (C2v) 17.8 3,3 9-vertex systems (up to 32 kcal/mol) Al7C2−1 (C2v) 0.0 4,4

Figure 4. Two lowest energy 7-vertex C2Al5Me7 structures.

C2Al6Me8 structure Al6C2−2, lying 6.4 kcal/mol in energy above Al6C2−1, has nonadjacent carbon atoms separated by an aluminum atom. The third C2Al6Me8 structure, namely, Al6C2−3, lying 17.8 kcal/mol in energy above Al6C2−1, is of interest since the central C2Al6 polyhedron is not the most spherical bisdisphenoid but, instead, is a bicapped octahedron with the carbon atoms at the two degree 3 vertices (Figure 5 and Table 2). This deviation from the most spherical 8-vertex deltahedron provides degree 3 vertices for the carbon atoms, thereby allowing tetrahedral carbon coordination to three aluminum atoms and the external methyl group. The lowest energy structure for the 9-vertex C2Al7Me9 Al7C2−1 is the unique structure based on the most spherical 9-vertex deltahedron, namely, the tricapped trigonal prism (Figure 1), with the carbon atoms at two of the three degree 4 vertices (Figure 6 and Table 2). Structure Al7C2−1 is a very favorable structure since it lies 9 kcal/mol in energy below any of the other C2Al7Me9 structures. The next two C2Al7Me9 structures in terms of energy have a central C2Al7 isocloso deltahedron with a single degree 6 vertex surrounded by four degree 4 vertices as well as two degree 5 vertices (Figure 6 and Table 2). In Al7C2−2, lying 9.3 kcal/ mol in energy above Al7C2−1, the carbon atoms are located at adjacent degree 4 vertices, forming a C−C edge of length 1.435 Å. However, in Al7C2−3, lying 13.2 kcal/mol in energy above Al7C2−1, the carbon atoms are located at nonadjacent degree 4 vertices. The next two 9-vertex C2Al7Me9 structures have relatively high energies of ∼25 kcal/mol above Al7C2−1, but they provide examples of unusual 9-vertex deltahedra (Figure 6 and Table 2). Structure Al7C2−4 has a central C2Al7 capped bisdisphenoid, thereby providing a degree 3 vertex for one of the carbon atoms. The rather oblate structure Al7C2−5 has a

C−C distance 1.484 Å 2.728 Å

pentagonal bipyramid pentagonal bipyramid

1.488 Å 2.991 Å 3.587 Å

bisdisphenoid bisdisphenoid bicapped octahedron

3.012 Å

tricapped trigonal prism isocloso 9-vertex deltahedron isocloso 9-vertex deltahedron capped bisdisphenoid 2v6 9-vertex deltahedron

Al7C2−2 (Cs)

9.3

4,4

1.435 Å

Al7C2−3 (C2)

13.2

4,4

3.440 Å

Al7C2−4 (Cs) Al7C2−5 (C2v)

25.3 25.5

3,4 4,4

4.509 Å 2.870 Å

10-vertex systems (up to 34 kcal/mol) Al8C2−1 (D4d) 0.0 4,4

4.108 Å

Al8C2−2 (C1)

1.469 Å

29.8

4,4

polyhedron

Al8C2−3 (C1) 31.9 4,4 11-vertex systems (up to 21 kcal/mol) Al9C2−1 (C2v) 0.0 4,4 Al9C2−2 (Cs) 8.6 4,4

3.522 Å

Al9C2−3 (C2v)

12.1

4,4

1.433 Å

Al9C2−4 (C1)

16.9

4,4

3.114 Å

3.727 Å 4.475 Å

bicapped square antiprism 2v6 10-vertex deltahedron one quadrilateral face 11v closo deltahedron 2v6 11-vertex deltahedron tricapped square antiprism 2v6 11-vertex deltahedron

a

Distances between adjacent carbon atoms sharing an edge are indicated in italic.

central C2Al7 deltahedron with two degree 6 vertices and both carbon atoms at degree 4 vertices. The most spherical 10-vertex deltahedron, namely, the bicapped square antiprism, has exactly two degree 4 vertices (Figure 1). The unique C2Al8Me10 structure Al8C2−1 based on this deltahedron, with the carbon atoms at the degree 4 vertices, is obviously a very favorable structure since it lies ∼30 kcal/mol below the next lowest energy structure (Figure 7 and Table 2). In addition, the HOMO−LUMO gap of 4.6 eV for Al8C2−1 is one of the highest HOMO−LUMO gaps for the structures discussed in this Article. The next higher energy C2Al8Me10 structure Al8C2−2, at a very high energy of 29.8 kcal/mol above Al8C2−1, has a central C2Al8 deltahedron with

Figure 5. Three lowest energy 8-vertex C2Al6Me8 structures. 11380


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Figure 6. Five lowest energy 9-vertex C2Al7Me9 structures.

Figure 7. Three lowest energy 10-vertex C2Al8Me10 structures.

the bicapped tetrahedral structure Al5C2−1 (Figure 9 and Table 3). The octahedral C2Al4Me6 structure Al4C2−3 has the two carbon atoms at adjacent vertices with a predicted C−C distance of 1.509 Å. The two lowest energy C2Al4Me6 structures have central C2Al4 polyhedra with degree 3 vertices for both carbon atoms (Figure 9 and Table 3). Structure Al4C2−1 has a central bicapped tetrahedron similar to that found in the osmium carbonyl Os6(CO)18 with the carbon atoms at the two degree 3 vertices.32 The central C2Al4 polyhedron in Al4C2−2, lying 12.6 kcal/mol in energy above Al4C2−1, is a trapezoidal pyramid with a capped triangular face. This polyhedron has three degree 3 vertices and a trapezoidal basal face. The carbon atoms are located at the two degree 3 vertices in the basal trapezoid. 3.4. C2Aln−2Men (n = 13 and 14) Systems (Figures 10 and 11). The lowest energy structures for the 13- and 14vertex dicarbalanes C2Aln−2Men (n = 13, 14) (Figures 10 and 11) follow the same general principle as that for the 12-vertex C2Al10Me12 system. Thus, the lowest energy structures for both the 13- and 14-vertex systems have central C2Aln−2 deltahedra with exactly two degree 4 vertices for the carbon atoms. This necessarily leads to n − 10 degree 6 vertices in these deltahedra. The major differences between these rather unsymmetrical lowest energy structures are the distances between the two degree 4 vertices corresponding to the C···C distances. For the 13-vertex C2Al11Me13 system (Figure 10), the lowest energy

two degree 6 vertices and adjacent degree 4 vertices for the two carbon atoms, forming a C−C edge of length 1.469 Å. The third C2Al8Me10 structure Al8C2−3, at 31.9 kcal/mol above Al8C2−1, has a central C2Al8 deltahedron with one quadrilateral face and 14 triangular faces. The most spherical 11-vertex deltahedron, like the most spherical 10-vertex deltahedron, has exactly two degree 4 vertices (Figure 1). The lowest energy structure for the 11vertex C2Al9Me11 system Al9C2−1 is the unique structure based on this deltahedron with the carbon atoms at the degree 4 vertices (Figure 8 and Table 1). The higher energy C2B9Me11 structures Al9C2−2 and Al9C2−4, lying 8.6 and 16.9 kcal/mol in energy above Al9C2−1, respectively, have central C2Al9 deltahedra with two degree 6 vertices and thus three degree 4 vertices, including two for the carbon atoms. The central C2Al9 deltahedron in the C2B9Me11 structure Al9C2−3, lying 12.1 kcal/mol in energy above Al9C2−1, can be derived from the bicapped square antiprism, i.e., the most spherical 10-vertex deltahedron, by adding a degree 4 vertex adjacent to one of the original two degree 4 vertices. The carbon atoms in Al9C2−3 are located at these adjacent degree 4 vertices, forming a C−C edge of length 1.433 Å. 3.3. C2Al4Me6 System (Figure 9 and Table 3). The most spherical 6-vertex deltahedron is the regular octahedron, which has exclusively degree 4 vertices. An octahedral C2Al4Me6 structure Al4C2−3 is found, but it is at an energy of 16.7 kcal/mol above the lowest energy C2Al4Me6 isomer, namely, 11381


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Inorganic Chemistry Table 3. Lowest Energy C2Al4Me6 Structures carbon vertices structure (symmetry)

ΔE kcal/mol

degrees

C−C distance

6-vertex systems Al4C2−1 (C2v)

0.0

3,3

3.038 Å

Al4C2−2 (Cs)

12.6

3,3

1.387 Å

Al4C2−3 (C2v)

16.7

4,4

1.509 Å

polyhedron bicapped tetrahedron capped square pyramid octahedron

Figure 10. Two lowest energy 13-vertex C2Al11Me13 structures. Figure 8. Four lowest energy 11-vertex C2Al9Me11 structures.

The regular icosahedron has 12 equivalent degree 5 vertices and thus lacks degree 4 vertices for the carbon atoms in the dicarbalane C2Al10Me12. The tendency for carbon atoms to occupy degree 4 vertices in dicarbalanes is so strong that none of the low-energy C2Al10Me12 structures has central regular C2Al10 icosahedra. Instead, the two lowest energy C2Al10Me12 structures Al10C2−1 and Al10C2−2, lying within 2 kcal/mol in energy, are based on deltahedra having two degree 4 vertices for the carbon atoms balanced by two degree 6 vertices, leaving eight degree 5 vertices. Such 12-vertex deltahedra can be viewed as distortions of the regular icosahedron through diamond-square-diamond processes33 to provide two degree 4 vertices for the carbon atoms. These two deltahedral C2Al10Me12 structures with two degree 4 and two degree 6 vertices lie more than 10 kcal/mol in energy below the next lowest energy C2Al10Me12 structure with a central 12-vertex deltahedron having three degree 4 vertices, three degree 6 vertices, and six degree 5 vertices. The same trend continues for the 13- and 14-vertex systems. The most spherical 13-vertex polyhedron found in R2C2B11H11 carboranes is no longer a deltahedron but, instead, is the socalled hencosahedron with a trapezoidal face with two adjacent degree 4 vertices that could accommodate carbon atoms

structure Al11C2−1 is a prolate (elongated) structure with a C···C separation of 3.738 Å. The next C2Al11Me13 structure Al11C2−2 is an oblate (flattened) structure with a much longer C···C separation of 5.318 Å. The three lowest energy 14vertex C2Al12Me14 structures Al12C2−1, Al12C2−2, and Al12C2−3 have C···C separations of 3.708, 5.033, and 3.223 Å, respectively. The most spherical 14-vertex deltahedron, namely, the bicapped hexagonal antiprism, like the regular icosahedron, has no degree 4 vertices (Figure 1) and thus is not found in a low-energy C2Al12Me14 structure.

4. DISCUSSION Theoretical studies19 on the polyhedral alane dianions AlnHn2− (n = 5−12) indicate that the lowest energy structures are the same most spherical deltahedra (Figure 1) as those found in the corresponding borane dianions BnHn2−. Thus, the AlnHn2− anions follow the Wade−Mingos rules.8−10 We also find the most spherical deltahedra in the lowest energy structures of the dicarbalanes C2Aln−2Men (n = 7−11). In all of the low-energy structures of these dicarbalanes, the carbon atoms are located at the degree 4 vertices.

Figure 9. Three 6-vertex C2Al4Me6 structures. 11382


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Figure 11. Three lowest energy 14-vertex C2Al12Me14 structures.

(Figure 1).34 However, both of the low-energy 13-vertex dicarbalane structures Al11C2−1 and Al11C2−2 have two degree 4 vertices for the carbon atoms balanced by three degree 6 vertices. For the 14-vertex system, as exemplified by R2C2B12H12 carboranes, the most spherical polyhedron is the bicapped hexagonal antiprism having two degree 6 vertices and 12 degree 5 vertices.35 However, the three low-energy 14-vertex dicarbalanes C2Al12Me14 have deltahedral structures with the two degree 4 vertices for the carbon atoms balanced by four degree 6 vertices. These deviations from the most spherical polyhedra in borane chemistry in the dicarbalanes C2Aln−2Men (n = 12−14) do not necessarily represent a violation of the Wade−Mingos rules.8−10 Analyses of the 2n + 2 skeletal electron requirement suggested by the Wade−Mingos rules for the deltahedral borane structures using either graph theory11 or spherical harmonics12 do not depend on the details of the deltahedral topology. Instead, they depend on the approximation of the actual deltahedron by a sphere. Thus, as long as the distances between the center of gravity of the deltahedron to each of the vertices are similar enough for a multicenter core bond using all of the vertex atoms, the Wade−Mingos rules can still apply. The diamond-square-diamond processes33 necessary to convert the most spherical 12- to 14-vertex deltahedra to deltahedra providing two degree 4 vertices for the carbon atoms do not distort the deltahedron enough from sphericity to eliminate the multicenter core bond. Unlike the C2Aln−2Men (n = 7−14) systems, the six-vertex C2Al4Me6 system clearly violates the Wade−Mingos rules. Thus, the octahedral C2Al4Me6 structure Al4C2−3 suggested by the Wade−Mingos rules is a higher energy structure than the bicapped tetrahedral structure Al4C2−1 and the capped square pyramidal structure Al4C2−2 by 16.7 and 4.1 kcal/mol, respectively. Both Al4C2−1 and Al4C2−2 provide degree 3 vertices for both carbon atoms. Such carbon atoms at degree 3 vertices are four-coordinate tetrahedral sp3 systems, implying localized bonding to the three adjacent cage atoms as well as the external group. The Wade−Mingos rules leading to 2n + 2 skeletal electrons for an n-vertex polyhedron imply fully delocalized bonding in the deltahedron, including a multicenter core bond at the polyhedron center. However, degree 3 carbon vertices and their adjacent polyhedral vertices are pockets of localization in the polyhedral structure. Therefore, C2Aln−2Men systems with carbon atoms at degree 3 rather than degree 4 vertices would no longer be expected to follow the Wade− Mingos rules.

deltahedron. For the systems having 7−11 vertices, the lowest energy structures are based on the same most spherical deltahedra as those found in the deltahedral borane dianions BnHn2− or the isoelectronic dicarbaboranes C2Bn−2Men and in accord with the Wade−Mingos rules. However, the deltahedra for the 12−14 vertex C2Aln−2Men (n = 12−14) systems deviate from the most spherical deltahedra in order to provide two degree 4 vertices for the carbon atoms. As a consequence, the regular icosahedron is no longer the preferred polyhedron for the 12-vertex system C2Al10Me12, in contrast to theoretical predictions for Al12H122− and experimental observations for the well-known species B12H122− and C2B10H12. The lowest energy structures for the six-vertex C2Al4Me6 system are bicapped tetrahedral and capped square pyramidal structures with degree 3 vertices for the carbon atoms. The octahedral C2Al4Me6 structure suggested by the Wade−Mingos rules is a higher energy structure, at ∼17 kcal/mol above the lowest energy isomer.

5. SUMMARY The low-energy structures of the dicarbalanes C2Aln−2Men (n = 7−14) all have the two carbon atoms at degree 4 vertices of a

(1) Boron Hydride Chemistry; Muetterties, E. L., Ed.; Academic Press: New York, 1975. (2) Grimes, R. N. Carboranes; Academic Press: New York, 1970. (3) Williams, R. E. Inorg. Chem. 1971, 10, 210.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b02014. Complete Gaussian 09 reference, initial structures, energy ranking data, distance tables for lowest-lying structures, orbital energies, and HOMO−LUMO gaps (PDF) Concatenated xyz file containing all of the optimized structures (XYZ)

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AUTHOR INFORMATION

Corresponding Authors

*(A.L.) E-mail: [email protected]. *(R.B.K.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding from the Romanian Ministry of Education and Research (grant PN-II-RU-TE-2014-4-1197) and the U.S. National Science Foundation (grant CHE-1057466) is gratefully acknowledged.

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