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Thermochemistry and Electronic Structure of Small Boron Clusters (B n , n ) 5 - 13) and
Their Anions
Truong Ba Tai, †^ Daniel J. Grant, ‡^ Minh Tho Nguyen,* ,†,‡^ and David A. Dixon* ,‡
Department of Chemistry, and Mathematical Modeling and Computational Science Center (LMCC), Katholieke Uni V ersiteit Leu V en, B-3001 Leu V en, Belgium, and Department of Chemistry, The Uni V ersity of Alabama, Shelby Hall, Box 870336, Tuscaloosa, Alabama 35847-
Recei V ed: September 4, 2009; Re V ised Manuscript Recei V ed: October 27, 2009
Thermochemical parameters of a set of small-sized neutral (B n ) and anionic (B n - ) boron clusters, with n ) 5 - 13, were determined using coupled-cluster theory CCSD(T) calculations with the aug-cc-pV n Z ( n ) D, T, and Q) basis sets extrapolated to the complete basis set limit (CBS) plus addition corrections and/or G3B calculations. Enthalpies of formation, adiabatic electron affinities (EA), vertical (VDE), and adiabatic (ADE) detachment energies were evaluated. Our calculated EAs are in good agreement with recent experiments (values in eV): B 5 (CBS, 2.29; G3B3, 2.48; exptl., 2.33 ( 0.02), B 6 (CBS, 2.59; G3B3, 3.23; exptl., 3.01 ( 0.04), B 7 (CBS, 2.62; G3B3, 2.67; exptl., 2.55 ( 0.05), B 8 (CBS, 3.02; G3B3, 3.11; exptl., 3.02 ( 0.02), B 9 (G3B3, 3.03; exptl., 3.39 ( 0.06), B 10 (G3B3, 2.85; exptl., 2.88 ( 0.09), B 11 (G3B4, 3.48;, exptl., 3.43 ( 0.01), B 12 (G3B3, 2.33; exptl., 2.21 ( 0.04), and B 13 (G3B3, 3.62; exptl., 3.78 ( 0.02). The difference between the calculated adiabatic electron affinity and the adiabatic detachment energy for B 6 is due to the fact that the geometry of the anion is not that of the ground-state neutral. The calculated adiabatic detachment energies to the 3 Au, C 2 h and 1 Ag, D 2 h excited states of B 6 , which have geometries similar to the 1 Ag, D 2 h state of B 6 - , are 2.93 and 3.06 eV, in excellent agreement with experiment. The VDEs were also well reproduced by the calculations. Partitioning of the electron localization functions into π and σ components allows probing of the partial and local delocalization in global nonaromatic systems. The larger clusters appear to exhibit multiple aromaticity. The binding energies per atom vary in a parallel manner for both neutral and anionic series and approach the experimental value for the heat of atomization of B. The resonance energies and the normalized resonance energies are convenient indices to quantify the stabilization of a cluster of elements.
Introduction
Boron clusters have a wide range of properties and have been characterized by a number of experimental techniques including mass spectrometry.^1 Experimental and theoretical studies on the electronic structure,^2 -^5 chemical bonding,^6 -^8 and spectroscopic properties 9 -^15 of bare boron clusters as well as doped boron clusters^16 -^20 have been reported. Similar to carbon-based materials, boron nanotubes (and fullerenes, if synthesized) have been considered as potential materials for hydrogen storage. 21 Recent findings on novel properties of boron-based nanotubular materials22,23^ are stimulating further studies on the small-sized gas-phase clusters to determine their fundamental properties and growth patterns. There are a substantial number of computational studies of boron clusters, and we describe some here. Boustani^24 predicted that, for the small boron clusters, the quasi-planar and planar structures are more stable than the three-dimensional structures, and this result was confirmed by subsequent theoretical and experimental studies. Wang, Boldyrev, and co-workers^25 -^28 used photoelectron spectroscopy (PES) in combination with quantum chemical calculations to investigate the structural and electronic properties of a series of boron cluster anions B n -^ ( n ) 3 - 20) and proposed that the planarity and quasi-planarity of boron clusters is due to a delocalization of the π-electrons in 2D
structures. They also suggested that the π-electron delocalization in B n follows the Hu¨ckel model for aromaticity and antiaroma- ticity as found in cyclic hydrocarbons. In contrast, Aihara and co-workers 29 analyzed the structure of boron clusters in terms of topological resonance (TRE) arguments and concluded that their aromaticity is not related to the total number of π-electrons, and the Hu¨ckel rule can therefore not be applied. Subsequently, Zubarev and Boldyrev 6 reexamined the chemical bonding of B n clusters using natural bond orbitals (NBO), canonical molecular orbitals (MOs), and nuclear independent chemical shift (NICS) and predicted that the globally delocalized π or σ MOs in boron clusters exhibit both aromatic and antiromatic characters following the Hu¨ckel rule. These authors also showed that the presence of islands of π-aromaticity in a globally π-antiaromatic molecule results in higher structural stability. We recently predicted the heats of formation of the global energy minimum structures of a series of small boron B n , boron oxide B n O m clusters, and their anions, with n e 4, using high accuracy quantum chemical methods.^30 In addition, we analyzed their electronic structure in terms of the topology of electron localization function (ELF) and molecular orbitals. We now extend the calculations and analyses to the larger boron clusters B n and their anions B n - , with n ) 5 - 13. In view of the lack of reliable thermochemical parameters, we first predict their heats of formation. Subsequently, we analyze the bonding of the clusters, in particular the questions related to their electronic structure.
J. Phys. Chem. A XXXX, xxx, 000 A
10.1021/jp9085848 CCC: $40.75 XXXX American Chemical Society
Computational Methods
All quantum chemical calculations were carried out using the Gaussian 03^31 and Molpro 2006^32 suites of programs. Enthalpies of formation of the B n and B n -^ clusters were evaluated from the corresponding total atomization energies (TAE).^33 Two sets of calculations were performed. For n ) 5 - 9, the complete basis set (CBS) approach previously used for the series of n ) 2 - 430 and the G3B3 approach^34 were used. The G3B3 (G3/ B3LYP) approach is a composite technique in which a sequence of ab initio density functional theory and molecular orbital calculations is performed to obtain the total energy of a given molecular species. Because a G3B3 calculation is computation- ally less demanding than a CBS counterpart, we used only the G3B3 approach for the larger B n clusters with n ) 10 - 13. We briefly describe the CBS approach. Geometry parameters were fully optimized at the second-order perturbation theory (MP2) level with the correlation consistent aug-cc-pVDZ and aug-cc-pVTZ basis sets. The fully unrestricted formalism (UHF, UMP2) was used for open-shell system calculations done with Gaussian 03. The valence electronic energies were computed using coupled-cluster CCSD(T) theory^35 extrapolated to the complete basis set limit (CBS) using the correlation-consistent basis sets.^36 The single-point electronic energies were calculated by using the restricted coupled-cluster R/UCCSD(T) formal- ism^37 -^39 in conjunction with the correlation-consistent aug-cc- pV n Z ( n ) D, T, and Q) basis sets at the (U)MP2/aug-cc-pVDZ or (U)MP2/aug-cc-pVTZ optimized geometries with the Molpro program. For simplicity, the basis sets are labeled as aV n Z. The CCSD(T) energies were extrapolated to the CBS limit energies using expression 1: 40
where x ) 2, 3, and 4 for the aV n Z basis D, T, and Q, respectively (total CCSD(T) electronic energies as a function of basis set are given in Table S1 of the Supporting Information). The zero-point energies (ZPE) were calculated from harmonic vibrational frequencies at the MP2/aVDZ level and are given in Table S2. Additional smaller corrections were included in the TAE calculations. Core-valence corrections (∆ E CV ) were obtained at the CCSD(T)/cc-pwCVTZ level of theory with Molpro.^41 Douglas-Kroll-Hess (DKH) scalar relativistic cor- rections (∆ E DKH-SR), which account for changes in the relativistic contributions to the total energies of the molecule and the constituent atoms, were calculated using the spin-free, one- electron DKH Hamiltonian with Molpro.^42 -^44 ∆ E DKH-SR is defined as the difference in the atomization energy between the results obtained from basis sets recontracted for DKH calcula- tions^43 and the atomization energy obtained with the normal valence basis set of the same quality. The DKH calculations were obtained as the differences of the results from the CCSD(T)/cc-pVTZ and the CCSD(T)/cc-pVTZ-DK levels of theory. Finally, a spin-orbit (SO) correction of 0.03 kcal/mol for the B atom obtained from the excitation energies of Moore^45 is used. The total atomization energy (∑ D 0 or TAE) of a compound is given by eq 2:
By combining our computed ∑ D 0 values from either the CBS or the G3B3 calculations, with the known heat of formation at 0 K for the element B, we can derive ∆ H f° values at 0 K for the molecules in the gas phase. In this work, we used the value of ∆ H f°(B) ) 135.1 ( 0.2 kcal/mol,^46 and the rationale for this selection was discussed in our previous work.^30 We obtain heats of formation at 298 K by following the procedures outlined by Curtiss et al. 47 We use the calculated heats of formation at 0 K to evaluate the electron affinities and other energetic quantities.30, The analysis of chemical bonding phenomenon was per- formed using the electron localization function (ELF)^49 supple- mented by analyses of topological bifurcation^50 and canonical MOs. The ELF is a local measure of the Pauli repulsion between electrons owing to the exclusion principle in 3D space. The definition of ELF, η( r ), is given by following eq 3:
where D P and D h are the local kinetic energy density due to the Pauli exclusion principle and the Thomas-Fermi kinetic energy density, respectively, and F is the electron density. These quantities can be evaluated using either Hartree-Fock or Koln-Sham orbitals. The total ELF can then be partitioned in terms of separate ELFσ and ELFπ components. The latter can be used as indices describing the aromaticity of cyclic mol- ecules.^51 A π and σ aromatic ring possesses a high bifurcation value of ELFπ and ELFσ, whereas the corresponding bifurcation value in an antiaromatic system is very low. The density for the ELF analysis of the lowest-energy state in each spin manifold was obtained at the (U)B3LYP/6-311+G(d) level. The total ELF was mapped out using the TOPMOD software,^52 whereas the ELFπ and ELFσ were constructed using the DGrid-4.2 soft- ware.^53 All isosurfaces of the ELF, ELFπ, and ELFσ have been plotted using the Gopenmol software. 54
Results and Discussion The shapes of the equilibrium structures of the B n and B n - clusters are shown in Figures 1 ( n ) 5 - 8) and 2 ( n ) 9 - 13). These include for each cluster the global energy minimum and selected lower-lying isomers. To simplify the presentation of data, the ELF isosurfaces with one bifurcation value are also displayed in these figures. The total G3B3 energies as well as the corresponding ∑ D 0 values are summarized in Table S3 of the Supporting Information. The optimized geometries of the lowest-lying isomers using the B3LYP/6-31G(d) method (within the G3B3 approach) are listed in Table S4. The different components obtained in the CBS protocol to predict the total atomization energies (∑ D 0 ) and the ∑ D 0 of B n and B n - , with n ) 5 - 9 (except for B 9 ), are given in Table 1. The heats of formation of the clusters derived using the ∑ D 0 obtained from both CBS and G3B3 methods are given in Table 2. The adiabatic electron affinities (EAs) of the neutrals B n are given in Table 3, and the vertical detachment energies (VDEs) of the B n -^ anions computed using the single-point G3B3 and CBS methods are
E ( x ) ) A CBS + B exp[-( x - 1)] + C exp[-( x - 1)^2 ] (1)
∑ D 0 )^ ∆ E elec(CBS)^ +^ ∆ E CV +^ ∆ E DKH-SR +^ ∆ E SO -
∆ E ZPE (2)
η( r ) ) 1 1 + ( D P / D h )^2
D P )
2 ∑ i ) 1
N |∇ψ i | 2 -
|∇F| 2
F
D h )
(3π^2 )2/3F5/
F ) ∑
i ) 1
N |ψ i ( r )| 2
B J. Phys. Chem. A, Vol. xxx, No. xx, XXXX Tai et al.
from GEGA/B3LYP/3-21G calculations. At the B3LYP/ 6-31G(d) level, a search of the 2 A 1 state of B 5 led to a nonplanar structure ( C 2 V), which is ∼58 kcal/mol higher in energy than I. A ∑ D 0 of 407 kcal/mol for B 5 I was reported at the CCSD(T)/ 6-311+G(2df) + ZPE level,^25 but no corresponding value for the heat of formation was derived.^25 The latter ∑ D 0 estimate differs significantly from the present values of 420.6 (G3B3) and 423.8 (CBS) kcal/mol. Attaching one electron to the neutral B 5 (^2 B 2 ) does not affect the geometry much, and the closed-shell 1 A1 structure is found
to be the most stable structure for the B 5 -^ anion II (Figure 1), in agreement with the available theoretical results. The lowest- lying triplet state is distorted by an out-of-plane motion to form a C 2 3 B structure, and the 3 B-^1 A 1 gap of B 5 -^ is ∼10 kcal/mol (G3B3 and CBS, Table 5), which is larger than that of 5.3 kcal/ mol previously obtained by UB3LYP/6-311+G(d) calcula- tions. 25 The adiabatic electron affinity (EA) of B 5 calculated from the heats of formation at 0K of I (^2 B2 ) and II ( 1 A 1 ) is 2. (G3B3) and 2.29 eV (CBS). As compared to the experimental value of 2.33 eV,^57 the G3B3 value is ∼0.15 eV too large, and the CBS result is in good agreement as would be expected (Table 3). The difference between the G3B3 calculated VDE of 2.64 eV and the experimental result of 2.40 ( 0.02 eV is slightly larger 57 (Table 4). The nature of chemical bonding and aromaticity of boron clusters have extensively been studied to explain their planarity and high stability. We add to this effort a topological analysis of the ELF, in combination with MO interactions. The global minimum C 2 V structure of the B 5 cluster in both the neutral and the anion states can be understood by considering the geo- metrical distortions from the higher symmetry geometry of the corresponding B 5 +^ cation following electron attachment. The LUMO of B 5 +^ ( D 5 h , 1 A 1 ′) is degenerate. Addition of one or two electrons into these vacant orbitals to obtain B 5 and B 5 - , respectively, is subject in their low spin states to a Jahn-Teller effect whose stabilization leads to a lower symmetry C 2 V structure. The high spin triplet state of B 5 -^ arising from the (e^2 ) occupancy, 3 A 1 ′ ( D 5 h ), is calculated to be ∼87 kcal/mol higher in energy than the singlet II at the B3LYP/6-311+G(d) level. Analysis of the ELF of B 5 and B 5 -^ shows the existence of a trisynaptic basin B2B3B4 in each state (see Figure 1 for atom
Figure 2. Shape of the lowest-energy structures B n and B n -^ ( n ) 9 - 13) and their ELF localization domains.
TABLE 1: CCSD(T)/CBS Total Atomization Energies ( ∑ D 0 ) TAE in kcal/mol) for the Neutral B n and Anionic B n - Boron Clusters ( n ) 5 - 9) and the Different Components a molecule ∆CBS b^ ∆ E ZPE c^ ∆ E CV d^ ∆ E SR e^ ∆ E SO f^ ∑ D 0 (0 K ) B 5 ( 2 B 2 , C 2 V, I ) 429.67 9.91 4.55 - 0.32 - 0.15 423. B 5 -^ ( 1 A 1 , C 2 V, II ) 482.09 9.56 4.61 - 0.34 - 0.15 476. B 5 -^ ( 3 B, C 2 , II-t ) 472.19 11.31 4.80 - 0.35 - 0.15 465. B 6 ( 1 A 1 , C 5 V, III ) 538.89 13.44 6.46 - 0.44 - 0.18 531. B 6 ( 3 A (^) u , C 2 h , IV-t ) 530.11 11.98 5.82 - 0.41 - 0.18 523. B 6 ( 1 A (^) g , D 2 h , IV-d ) 526.64 11.35 5.69 - 0.38 - 0.18 520. B 6 ( 1 A, C 2 , IV-s ) 527.03 11.90 5.48 - 0.39 - 0.18 520. B 6 -^ ( 2 B (^) 2g , D 2 h , V ) 597.20 11.84 6.24 - 0.42 - 0.18 590. B 6 -^ ( 2 B (^) 1u , D 2 h , V-u ) 585.42 15.78 6.42 - 0.43 - 0.18 575. B 6 -^ ( 2 B (^) u , C 2 h , VI ) 592.09 13.55 6.54 - 0.44 - 0.18 584. B 6 -^ ( 2 A (^) u , C 2 h , V-c) 573.27 12.37 5.84 - 0.46 - 0.18 566. B 6 -^ ( 2 A′′, C (^) s , V-s ) 582.14 12.78 6.56 - 0.47 - 0.18 575. B 7 ( 2 B 2 , C 2 V, VII ) 673.37 16.64 7.61 - 0.53 - 0.21 663. B 7 -^ ( 1 A 1 , C 2 V, IX ) 732.94 15.79 7.58 - 0.56 - 0.21 723. B 7 -^ ( 1 A 1 , C 2 V, IX-a ) 723.92 15.68 7.34 - 0.54 - 0.21 714. B 7 -^ ( 3 A 1 , C 6 V, X ) 733.20 17.56 7.85 - 0.58 - 0.21 722. B 8 ( 3 A 2 ′, D 7 h , XI ) 809.68 19.13 8.40 - 0.59 - 0.24 798. B 8 -^ ( 2 B 1 , C 2 V, XII ) 878.20 18.32 8.79 - 0.61 - 0.24 867. B 9 -^ ( 1 A (^) 1g , D 8 h , XV ) 985.98 19.58 9.47 - 0.64 - 0.27 974. B 9 -^ ( 3 A′, C (^) s , XV-t ) 953.96 19.77 9.75 - 0.72 - 0.27 942. a (^) Atomic asymptotes calculated with the R/UCCSD(T) method. b (^) Extrapolated by using eq 1 with the aVDZ, aVTZ, and aVQZ basis sets. c^ Zero-point energies taken as 0.5, the sum of the MP harmonic frequencies. d^ Core-valence corrections obtained with the cc-pwCVTZ basis sets at the optimized CCSD(T) or MP geometries. e^ Scalar relativistic correction based on a CCSD(T)-DK/ VTZ-DK calculation and is expressed relative to the CCSD(T) result without the DK correction. f^ Correction due to the incorrect treatment of the atomic asymptotes as an average of spin multiplets. Values based on C. Moore’s tables, ref 45.
D J. Phys. Chem. A, Vol. xxx, No. xx, XXXX Tai et al.
labeling), which contains 2.8 e for B 5 and 2.7 e for B 5 -^ located at the center of the five-membered ring. This indicates the presence of a three-center-two-electron (3c-2e) bond in both states, which is likely to be the main factor for stability of these planar structures. Previous reports in the literature disagreed with each other about the actual number of delocalized π and σ electrons of B 5. Li et al.^56 suggested that there are three and four delocalized π electrons for B 5 and B 5 - , respectively, and that they are aromatic. By using a different approach, Aihara et al.^29 predicted that there are only two delocalized π electrons in each of the two states and that they are both aromatic. From a MO analysis, Zubarev and Boldyrev^6 suggested that in B 5 -^ there is a conflicting electron distribution with σ-antiaromaticity and π-aromaticity. The canonical MOs of B 5 and B 5 -^ reveal that HOMO-3 is a completely bonding orbital containing globally delocalized π
electrons, whereas HOMO-1 is a globally bonding σ orbital. These delocalized π and σ orbitals can be considered as the origin for the double aromaticity of B 5 and B 5 -. The SOMO of B 5 (HOMO of B 5 - ) is a partially bonding σ orbital with two triangular wings that make an island of σ electron delocalization within the B1-B2-B3 and B3-B4-B5 domains, and thus contributes to the stabilization of the planar structure. The electron populations of the (B1B2), (B1B3), (B2B3B4), (B3B5), and (B4B5) basins in B 5 are close to each other and amount to 2.4, 2.7, 2.8, 2.7, and 2.4 e, respectively. Such a distribution
TABLE 2: Calculated Heats of Formation (∆ H f at 0 and 298 K, kcal/mol) of the Neutral B n and Anionic B n -^ Boron Clusters ( n ) 5 - 13) Using CCSD(T)/CBS and G3B Approaches
∆ H f (0 K) ∆ H f (298 K) structure (state) label symmetry G3B3 CBS G3B3 CBS B 2 ( 3 Σg-) D ∞ h 204.5 205.9 206.0 207.4 a B 2 -^ ( 4 Σ-g ) D ∞ h 159.1 160.9 160.6 162.4 a B 3 ( 2 A 1 ′) D 3 h 209.9 210.1 211.5 211. B 3 -^ ( 1 A 1 ′) D 3 h 142.7 143.7 144.3 145. B 4 ( 1 A (^) g ) D 2 h 226.7 224.6 228.5 226. B 4 -^ ( 2 B (^) 1u ) D 2 h 188.2 185.8 189.8 187. B 5 ( 2 B 2 ) I C 2 V 254.9 251.7 256.8 253. B 5 -^ ( 1 A 1 ) II C 2 V 197.8 198.9 199.6 200. B 5 -^ ( 3 B) II-t C 2 210.3 211. B 6 ( 1 A 1 ) III C 5 V 285.2 279.3 286.9 280. B 6 ( 3 A (^) u ) IV-t C 2 h 291.4 287.2 293.1 289. B 6 ( 1 A (^) g ) IV-d D 2 h 290.2 291. B 6 ( 1 A) IV-s C 2 290.6 292. B 6 -^ ( 2 B (^) 2g ) V D 2 h 210.8 219.6 213.0 220. B 6 -^ ( 2 B (^) 1u ) V-u D 2 h 235.1 236. B 6 -^ ( 2 B (^) u ) VI C 2 h 234.9 226.1 237.1 228. B 6 -^ ( 2 A (^) u ) V-c C 2 h 244.5 246. B 6 -^ ( 2 A′′) V-s C (^) s 235.3 236. B 7 ( 2 B 2 ) VII C 2 V 286.5 282.1 288.3 284. B 7 ( 2 B 2 ) VIII C 2 V 313.1 315. B 7 -^ ( 1 A 1 ) IX C 2 V 223.8 221.7 225.6 223. B 7 -^ ( 1 A 1 ) IX-a C 2 V 230.9 232. B 7 -^ ( 3 A 1 ) X C 6 V 224.9 223.0 226.4 224. B 8 ( 3 A 2 ′) XI D 7 h 285.3 282.7 287.5 284. B 8 -^ ( 2 B 1 ) XII C 2 V 213.5 213.0 216.0 215. B 9 ( 2 A 1 ) XIII C 2 V 311.9 315. B 9 ( 2 B (^) 1g ) XIV C 2 V 323.2 327. B 9 -^ ( 1 A (^) 1g ) XV D 8 h 242.0 240.9 245.3 244. B 9 -^ ( 3 A′) XV-t C (^) s 273.0 275. B 10 ( 1 A (^) g ) XVI C 2 h 312.7 314. B 10 -^ ( 2 A′′) XVII C (^) s 247.0 249. B 11 ( 2 B 2 ) XVIII C 2 V 333.1 336. B 11 ( 2 B 2 ) XIX C 2 V 343.7 346. B 11 -^ ( 1 A 1 ) XX C 2 V 252.9 255. B 11 -^ ( 1 A 1 ) XXI C 2 V 253.2 255. B 12 ( 1 A 1 ) XXII C 3 V 333.3 335. B 12 -^ ( 2 A′) XXIII C (^) s 297.6 282. B 13 ( 2 A 2 ) XXIV C 2 V 369.6 373. B 13 ( 2 B 1 ) XXV C 2 V 369.2 372. B 13 ( 2 B (^) 3u ) XXVI D 2 h 374.2 376. B 13 -^ ( 3 B 2 ) XXVII C 2 V 308.5 311. B 13 -^ ( 3 B 2 ) XXVIII C 2 V 309.5 312. B 13 -^ ( 1 A (^) g ) XXIX D 2 h 285.7 288. a (^) Experimental values of 198.3 ( 8.0 (B 2 ) and 168.5 ( 9.2 (B 2 - ) kcal/mol from ref 55.
TABLE 3: Adiabatic Electronic Affinities (EA, eV) of Boron Clusters (B n , n ) 5 - 13) Calculated Using the G3B3 and CCSD(T) Methods a neutral (state) anion (state) G3B3 CCSD(T) exptl. b B 2 ( 3 Σg-) B 2 -^ ( 4 Σ-g ) 1.97 1.95 1. B 3 ( 2 A 1 ′) B 3 -^ ( 1 A 1 ′) 2.91 2.88 2.82 ( 0. B 4 ( 1 A (^) g ) B 4 -^ ( 2 B (^) 1u ) 1.67 1.68 1.60 ( 0. B 5 ( 2 B 2 ) B 5 -^ ( 1 A 1 ) 2.48 2.29 2.33 ( 0. B 6 ( 1 A 1 ) B 6 -^ ( 2 B (^) 2g ) 3.23 (3.50) c^ 2.59 (2.93) c^ 3.01 ( 0. B 7 ( 2 B 2 ) B 7 -^ ( 3 A 1 ) 2.67 2.56 2.55 ( 0. B 7 ( 2 B 2 ) B 7 -^ ( 1 A 1 ) 2.72 2.62 2.55 ( 0. B 8 ( 3 A 2 ′) B 8 -^ ( 2 B 1 ) 3.11 3.02 3.02 ( 0. B 9 ( 2 A 1 ) B 9 -^ ( 1 A (^) 1g ) 3.03 (3.53) c^ 3.39 ( 0. B 10 ( 1 A (^) g ) B 10 -^ ( 2 A′′) 2.85 2.88 ( 0. B 11 ( 2 B 2 ) B 11 -^ ( 1 A 1 ) 3.48 3.43 ( 0. B 12 ( 1 A 1 ) B 12 -^ ( 2 A′) 2.33 2.21 ( 0. B 13 ( 2 A 2 ) B 13 -^ ( 1 A (^) g ) 3.62 (3.81) c^ 3.78 ( 0. a (^) Evaluated from calculated enthalpies of formation at 0 K. b (^) Experimental values from ref 25 correspond to either EAs or ADEs. c^ ADEs are given in parentheses; see text.
TABLE 4: Vertical Detachment Energies (VDE, eV) of Boron Cluster Anions (B n - , n ) 5 - 13) Calculated Using the G3B3 and CCSD(T) Methods anion (state) neutral (state) G3B3 a^ CCSD(T) b^ exptl. c B 5 -^ ( 1 A 1 ) B 5 ( 2 B 2 ) 2.64 2.47 2.40 ( 0. B 6 -^ ( 2 B (^) 2g ) B 6 ( 3 B (^) 3u ) 3.74 3. B 6 -^ ( 2 B (^) 2g ) B 6 ( 1 A (^) g ) 3.60 3.17 3.01 ( 0. B 7 -^ ( 1 A 1 ) B 7 ( 2 B 2 ) 2.93 2.87 2.85 ( 0. B 7 -^ ( 3 A 2 ) B 7 ( 2 B 2 ) 2.86 2. B 8 -^ ( 2 B 1 ) B 8 ( 3 A 2 ′) 3.12 2.99 3.02 ( 0. B 8 -^ ( 2 B 1 ) B 8 ( 1 A 1 ′) 3.56 3.44 3.35 ( 0. B 9 -^ ( 1 A (^) 1g ) B 9 ( 2 E (^) 1g ) 3.63 3.45 b^ 3.46 ( 0. B 10 -^ (^2 A′′) B 10 ( 1 A′) 3.22 3.06 ( 0. B 11 -^ (^1 A 1 ) B 11 ( 2 B 2 ) 3.63 3.426 ( 0. B 12 -^ (^2 A′) B 12 ( 1 A′) 2.41 2.26 ( 0. B 13 -^ (^1 A (^) g ) B 13 ( 2 B (^) 3u ) 3.98 3.78 ( 0. a (^) G3B3 calculations of the neutrals using the geometries of the corresponding anions. b^ CCSD(T)/CBS values except for B 9 , which was calculated at the CCSD(T)/aVTZ level. c^ Experimental values from ref 25.
TABLE 5: Calculated Singlet - Triplet Gaps (kcal/mol at 0 K) as a Function of the Basis Set at the CCSD(T) Level
reaction aVDZ aVTZ aVQZ
CBS (DTQ) B 5 - ( II ,^1 A 1 ) f B 5 - ( II - t ,^3 B) 10.5 9.9 9.9 9.
B 6 ( III ,^1 A 1 ) f B 6 ( IV ,^3 Bu ) 3.3 6.5 7.9 8.
B 7 - ( IX ,^1 A 1 ) f B 7 - ( X ,^3 B 1 ) - 0.1^ - 0.3^ - 0.3^ - 0.
B 8 (^1 A′) f B 8 ( XI ,^3 A 2 ′) - 10.6 - 10.
B 9 - ( XV ,^1 A1g) f B 9 - (^3 A′) 32.9 32.8 32.4 32.
Small Boron Clusters (B n , n ) 5 - 13) J. Phys. Chem. A, Vol. xxx, No. xx, XXXX E
( C 2 , 1 A) is located, but the energy difference between both D 2 h and C 2 structures is small (0.4 kcal/mol, Table 1). After including all of the corrections shown in Table 1, IV-d becomes 0.4 kcal/mol lower in energy than IV-s. This energy difference increases to 1.0 kcal/mol at 298 K in favor of IV-d (Table 2). In view of the flat potential energy surface, we can conclude that the singlet neutral structure formed following electron detachment from B 6 -^ V essentially conserves the D 2 h point group. The adiabatic energy separation between IV-d and V is 3.06 eV (Figure 3) in agreement with experiment. Geometry optimization starting from the vertical 3 B3u point leads to a D 2 h stationary point with one imaginary frequency. Relaxation of the geometry along the imaginary vibrational mode of this structure (b3g mode of 143i cm-^1 at B3LYP/aVDZ) yields the equilibrium structure IV-t ( C 2 h , 3 Au) described above. The energy gain upon relaxation is 7.1 kcal/mol, and this triplet state is lower than the D 2 h state for B 6 , although neither is the global minimum. The IV-t - V energy difference is 2.93 eV, again in agreement with the experimental ADE. Because of the similarity between their vertical D 2 h and equilibrium D 2 h / C 2 h structures, both transitions of B 6 r B 6 -^ giving the singlet and triplet neutral could be observed by the PES experiment, and the CBS ADE values of 3.06 eV ( IV-d r V ) and 2.93 eV ( IV-t r V ) are in excellent agreement with the experimental estimate of 3.01 ( 0.02 eV. 63 Li et al.^61 derived an AEA(B 6 ) of ∼2.0 eV from DFT (B3LYP and B3PW91) calculations of V for B 6 and a 2 A 1 ( C 2 V) state for B 6 -. Using B3LYP/6-311+G(3df) calculations, Ma et al.^62 reported a value of 2.60 eV for this quantity considering IV-t for B 6 and V for B 6 -^ as the lowest-energy structures. The latter B3LYP value can be compared to the ADE of 2.53 eV (CCSD(T)/CBS) given above. The difference is mainly due to the different assignments of the ground state. Although there is now a consensus on the global minimum structure of B 6 , the appropriate description of its chemical bonding remains a matter of debate. On the basis of the NICS indices, Ma et al.^62 predicted that both B 6 and B 6 -^ are nonaromatic. By using current density maps, Alexandrova et al. 63 stated that B 6 -^ V (^2 B2g ) is doubly antiaromatic. Aihara et al.^29 argued that the B 6 clusters are aromatic in the neutral, anion, and dianion states. More recently, Zubarev and Bolderev^6 considered that B 6 in its planar configuration (^1 A1′, D 5 h ) is doubly aromatic. The dianion B 62 -^ (^2 B2g , D 2 h ) planar configu- ration is also predicted to be doubly antiaromatic, but exhibits some islands for σ and π aromaticity. Our MO and ELF analyses support the view that neutral B (^6) is doubly aromatic, and that B 6 -^ is partially aromatic. From
the total ELF map for B 6 III in Figure 1, the five basins (B1B2B6), (B1B5B6), (B2B4B6), (B3B5B6), and (B3B4B6) are characterized as trisynaptic with each containing ∼3.5 e. This shows the existence of 3c-2e bonds between two boron atoms in the five-membered ring to the central boron. These populations are equally partitioned into σ and π electrons and delocalized over the whole structure. To simplify the interpreta- tion, we now consider a higher symmetry structure ( D 5 h , 1 A 1 ′) for the neutral cluster in which the central atom is pushed into the center in the molecular plane. The MO diagrams for the neutral D 5 h form show that there are six bonding σ electrons and two bonding π electrons distributed among the three bonding σ orbitals (the degenerate HOMO and HOMO-1), and one bonding π orbital (HOMO-3), respectively. The number of σ (6) and π (2) electrons formally satisfy the conventional Hu¨ckel rule of aromaticity, and such an electron partition makes B 6 ( D 5 h , 1 A 1 ′) doubly aromatic. This result can be applied to the global minimum structure III ( C 5 V). The ELFσ of B 6 ( C 5 V) is plotted separately to identify the values of bifurcations. The high value of 0.89 for ELFσ suggests that III also has σ aromatic character, and the neutral B 6 is a doubly σ and π aromatic system. To simplify the analysis for the open shell B 6 -^ anion, we first consider the closed shell B 62 -^ dianion. From the MO diagrams of the global minimum B 62 -^ ( D 2 h ) with a valence orbital configuration of 1 A1g: 1ag^2 1b1u^2 2ag^2 1b2u^2 1b3g^2 3ag^2 2b2u^2 2b1u- (^2) 1b2g (^2) , the dianion can easily be identified with two global bonding orbitals (HOMO for π and HOMO-3 for σ) and two
TABLE 6: Binding Energies ( D e) and Average Binding Energies ( E b) of the B n and B n -^ Clusters (eV) a
n E b (B n ) b^ E b (B n - ) c^ D e (B n ) d^ D e (B n - ) e 2 1.42 2.41 2.79 4. 3 2.82 3.80 5.68 6. 4 3.40 3.82 5.23 3. 5 3.65 4.14 4.68 5. 6 3.80 4.33 4.58 5. 7 4.08 4.47 5.82 5. 8 4.31 4.70 5.83 6. 9 4.36 4.69 4.71 4. 10 4.50 4.79 5.82 5. 11 4.54 4.86 4.97 5. 12 4.65 4.85 5.85 3. 13 4.63 4.91 4.30 6. a (^) Values obtained from the G3B3 heats of formation at 0 K. b (^) Using eq 5a. c (^) Using eq 5b. d (^) Using eq 4a. e (^) Using eq 4b. Figure 4. Size dependence of the binding energies ( D e in kcal/mol) of B n and B n -^ clusters. Values obtained from G3B3 heats of formation at 0 K.
Figure 5. Size dependence of the average energy per atom ( E b in eV) of B n (black line) and B n -^ (violet line) clusters. Values obtained from G3B3 heats of formation at 0 K.
Small Boron Clusters (B n , n ) 5 - 13) J. Phys. Chem. A, Vol. xxx, No. xx, XXXX G
partially bonding orbitals (HOMO-4 for π and HOMO-1 for σ). The HOMO and HOMO- 3 exhibit bonding electron delocalization over the entire cluster, leading to global (σ and π) aromaticity, whereas the remaining electrons form islands. The total ELF reveals that six disynaptic basins are formed from six B ring atoms, in which each of the two (B1B2) and (B3B4) basins contains ∼2.5e, and each of the remaining basins contains ∼3.4e (see Figure 1 for atom numbering). Such a distribution exhibits a high electron concentration within each of the two three-membered (B 3 - ) components and thus supports further the viewpoint of the partial electron delocalization of the dianion. When analyzing the ELFσ and ELFπ separately, a very high bifurcation value of 0.91 for ELFσ is found. This is even higher than the bifurcation value of 0.79 found for Al 42 -^ that was demonstrated to be a multiply aromatic system. 48,51^ Thus, we can assign B 62 -^ as a σ aromatic system. For the ELFπ component, there are two distinct regions; although the first bifurcation value of 0.6 for the separation of two reducible basins is low, the second value is very high. In addition, the basins cannot be separated further; even the ELFπ isosurface attains a high value of 0.99. This observation is consistent with the view that the B 62 -^ is globally non π aromatic, but contains islands of π aromaticity.
For the B 6 -^ radical anion, which is formed upon removal of an electron from the dianion, the unpaired electron is of π character in the 2 B2g V , and, as a consequence, the global character of the bonding in the cluster is not changed. B 7. Our calculations predict that the neutral B 7 ground state exhibits a nearly planar hexagon capped by one element at the center VII ( C 2 V, Figure 1) with a valence orbital configuration of 2 B 2 : 1a 12 1b 22 1b 12 2a 12 1a 22 3a 12 2b 12 4a 12 2b 22 3b 12 3b 21 , which agrees well with previous theoretical results.2,8,15,24,64,65^ The other C 2 V alternative VIII (^2 B2 ) structure is predicted to be ∼27 kcal/mol higher in energy.
The lowest-lying structure of B 7 -^ is not as well established. From B3LYP, B3PW91, and MP2 calculations, Li et al. 65 predicted that the anion cluster has a singlet ground state IX ( C 2 V, 1 A 1 : 1a 12 1b 22 1b 12 2a 12 1a 22 3a 12 2b 12 4a 12 2b 22 3b 12 3b 22 ). On the contrary, Alexandrova et al.^64 reported that B 7 -^ has the high spin pyramidal X (^3 A 1 , C 6 V) as its most stable isomer. However, at the RCCSD(T)/6-311+G(2df) level, the triplet X structure is only 0.7 kcal/mol lower in energy than the singlet IX. G3B calculations predict that the energy difference between both states is small and that the low spin state is favored by 1.1 kcal/ mol. Similarly, the CCSD(T) calculations predict a singlet-triplet separation gap of 0.3 kcal/mol at the CBS limit, but in an opposite direction in favor of the high spin state (Table 5), and it is only upon inclusion of the additional corrections that the lower spin state becomes more stable by 1.3 kcal/mol (Table 2). We thus conclude that B 7 -^ has the low symmetry, low spin ground state X ( C 2 V, 1 A 1 ) but that there is a very close lying (^3) A1 , C 6 V state.
The VDE derived from X ( 3 A1 ) is calculated to be 2.86 and 2.92 eV at the G3B3 and CCSD(T) levels, respectively, and the respective G3B3 and CCSD(T) VDEs from B 7 -^ IX are 2. and 2.87 eV. Both are in excellent agreement with the experimental value of 2.85 eV.^64 The G3B3 and CBS adiabatic electron affinities EA(B 7 ) are 2.72 and 2.62 eV, respectively (Table 3), and both compare favorably with the experimental value of 2.55 ( 0.05 eV. 64
We found that the planar high symmetry triplet ( D 6 h , 3 A1g : 1a (^) 1g^2 1e1u^4 1e2g^4 1a2u^2 2a1g^2 1b2u^2 2e (^) 1u^4 1e1g^2 ) is, although not a local minimum, a low-energy form being only ∼1 kcal/mol higher in energy than IX at the G3B3 level. The UB3LYP/6-31G(d)
geometry of this 3 A (^) 1g state is characterized by all equal BB distances of 1.626 Å, including those connecting the central atom to the peripheral atoms. The lower symmetry forms IX and X can be considered as small distortions from the D 6 h (^3 A1g) structure. The MOs of these different structures of B 7 -^ have been shown to be similar.^64 To simplify the discussion, we only describe the ELFs of the D 6 h structure. The topology of the ELF supports the view that the planar B 7 -^ anion cluster is a π aromatic system. The ELF plots show bifurcation values of ELFσ and ELFπ of 0.92 and 0.91, respectively, making planar B 7 -^ a doubly aromatic molecule. For the singlet B 7 -^ IX , the bifurcation value of 0.94 for ELFσ suggests σ aromaticity. The value of 0.81 for its ELFπ is rather low with respect to the corresponding value of 0.91 for benzene, so it suggests a weak π-aromaticity. The total ELF plots emphasize the existence of four disynaptic basins (B1B2), (B1B6), (B3B4), and (B3B5) between the peripheral atoms, in which each basin contains ∼2.6 e. Each of the two trisynaptic basins (B2B7B5) and (B4B7B6) contains ∼4.4 e, which again indicates the presence of three-center bonds in the singlet state IX ( C 2 V, 1 A 1 ). These three-center bonds and σ aromatic character are consistent with a high stability for the 1 A 1 state, irrespective of the fact that the system is only weakly π-aromatic. B 8. Kato et al.^2 predicted a singlet structure ( C 2 V, 1 A 1 ) for the neutral B 8 cluster, whereas Ray et al.^3 predicted a square antiprism form. Reis et al.^66 reported a high symmetry high spin geometry ( D 7 h , 3 B2′) structure. Many subsequent theoretical studies16,67-^70 using various levels of theory predicted a different ground state. Our calculations concur with the view that the most stable B 8 structure features a heptagon XI ( D 7 h ), but with a high spin 3 A′ 2 ground state. The alternative low spin C 2 V (^1 A 1 ) structure that results from a Jahn-Teller distortion of the unstable D 7 h singlet state is 8.6 kcal/mol higher in energy than XI. For the B 8 -^ radical anion, the planar arrangement XII ( C 2 V, (^2) B 1 ) arising from a slight in-plane distortion of the heptagonal neutral B 8 is predicted to be the most stable isomer.^70 Because the B 8 -^ anion has a doublet ground state, detachment of one electron from it can lead to either a singlet or a triplet neutral. The VDE evaluated from structure XII (^2 B1 ) is 3.12 (G3B3) and 2.99 (CCSD(T)/CBS) eV to form a vertical triplet neutral ( 3 A 2 ′), and 3.56 (G3B3) and 3.44 (CCSD(T)/CBS) eV to produce a vertical singlet neutral (^1 A 1 ′) (Table 4). The CCSD(T)/ CBS values are in excellent agreement with the experimental VDE values of 3.02 ( 0.02 and 3.35 ( 0.02 eV.^70 An adiabatic EA of 3.11 and 3.02 eV is predicted at the G3B3 and CCSD(T) levels between the 3 A′ 2 state of B 8 and the 2 B1 state of B 8 -. For this system, the experimental adiabatic EA(B 8 ) was assigned to be the same as the VDE(B 8 - ).^70 The fact that there is only a very small difference (0.01 eV) between the calculated ADE and VDE of B 8 -^ is consistent with the seven-member rings characterizing both states changing only slightly on electron addition. In the MO wave function of B 8 , there are four π electrons distributed over three global bonding π orbitals (HOMO(e) and HOMO-2) and six σ electrons over three bonding σ orbitals (HOMO-1(e) and HOMO-4). The topological features of the total ELF shown in Figure 1 are associated with the high bifurcation values of 0.94 and 0.91 for the ELFπ and ELFσ components, respectively, and show the doubly σ and π aromaticity of B 8. B 9. The identity of the most stable structure of the neutral nine boron cluster has not been well established yet.^70 Boustani^4 predicted two lower-lying isomers including a nonplanar Cs and
H J. Phys. Chem. A, Vol. xxx, No. xx, XXXX Tai et al.
XIII ( C 2 V, 2 A 1 ). Kato et al. 2 predicted earlier that XIII is the global minimum isomer. Our G3B3 results predict that in the low spin manifold, the low symmetry Jahn-Teller distorted structure XIII corresponds to the most stable isomer, which is 11.3 kcal/mol lower in energy than the higher symmetry XIV.
For the anion B 9 - , we find the high symmetry planar structure XV ( D 8 h , 1 A1g ) to be the global minimum, in agreement with a previous prediction. 16 The next highest energy isomer is a heptagonal bipyramid structure ( D 7 h , 1 A 1 ), which is 10.6 kcal/ mol higher in energy. In both neutral and anionic forms, the high spin states are much less stable and were not further considered. The VDE of the anion XV is generated by removing one electron from its HOMO producing a vertical 2 E (^) 1g state. The G3B3 and CCSD(T)/aVTZ values of 3.63 and 3.45 eV
respectively are consistent with the PES result of 3.46 ( 0. eV.^70 The global adiabatic EA can be defined as the energy difference between ground-state anion XV and ground-state neutral XIII and is 3.03 eV. The experimental ADE is 3.39 ( 0.06 eV^15 in excellent agreement with our G3B3 value of 3. eV for the energy difference between XV and XIV , which have the same molecular shape. Similar to the electronic configuration of B 8 , the MO picture of B9-^ shows three bonding π orbitals (HOMO(e) and HO- MO-2) and three σ bonding orbitals (HOMO-1(e) and HOMO-4). Thus, six π electrons and six σ electrons are distributed within the high symmetrical skeleton of B 9 -^ XV , consistent with its aromaticity based on the ELF analysis below. The ELF maps for B 9 ( D 7 h , 2 A 1 ) and B 9 -^ ( D 8 h , 1 A1g ) show the presence of seven distinct disynaptic basins between the peripheral atoms of the heptagonal bipyramid ( D 7 h , 2 A1 ), in which each basin contains ∼2.7e. This perfect electron delo- calization in the high symmetry form ( D 7 h , 2 A1 ) is consistent with high structural stability. For the planar anion ( D 8 h , 1 A1g ), a similar landscape emerges. Each of the eight peripheral disynaptic basins contains ∼3.0e. This more complete electron delocalization inherently stabilizes the structure. The bifurcation value of 0.97 for ELFπ is very high and clearly indicates a strong π aromaticity. The value of ELFσ of 0.87 is higher than that of benzene. These results show that B 9 -^ XV possesses double aromaticity consistent with the MO picture. B 10. There is consensus as to the global minimum structures of both B 10 and B 10 -^ (Figure 2). Apart from the prediction made by Boustani 4 that B 10 has the convex C 2 V structure, its global minimum has been established to be a nearly planar C 2 h structure XVI ( 1 A (^) g ).2,16,71^ The most stable structure for B 10 -^ is the planar Cs form XVII ( 2 A′′). 16 Our calculations concur with these findings. We also considered for B 10 two additional high symmetry geometries, a perfect plane ( D 9 h , 1 A 1 ′) and an octagonal bipyramid ( D 8 h , 1 A (^) 1g ) structure. These were however found to be much higher in energy than the global minimum. The VDE evaluated by detaching one electron from the HOMO of the C 2 h (^1 Ag ) structure is 3.22 eV at the G3B3 level, in reasonable agreement with the experimental result of 3. ( 0.03 eV,^26 and consistent with the differences found for the other B n clusters. The adiabatic EA of 2.85 eV obtained from the relevant G3B3 heats of formation is consistent with the experimental value of 2.88 ( 0.09 eV. 26 Again, the shape of MOs and the topology of ELF were constructed for a higher symmetry structure of B10, D 2 h (^1 Ag ), in which two out-of-plane inner atoms are placed in the ring plane. There are three bonding π orbitals (HOMO, HOMO-1, and HOMO-4) and three bonding σ orbitals (HOMO-2, HOMO-3, and HOMO-4), which contain six π and six σ electrons, respectively, and thus satisfy the Hu¨ckel rule. The bifurcation value of ELFπ is 0.76, close to the value of 0. previously obtained for naphthalene by Santos et al.^51 The ELFπ values for the polycyclic species are smaller than that of benzene (0.91). The value of ELFσ is very high, 0.95. Although distortion from this ideal D 2 h structure yielding the minimum C 2 h is expected to reduce the electron delocalization, a doubly (σ and π) aromatic character can be assigned to the neutral cluster B 10. B 11. Kato et al. 2 reported a C 2 V (^2 A1 ) structure to be the most stable structure for B 11 , whereas Boustani^4 reported a quasi- planar Cs (^2 A′′) structure containing a shallow hexagonal unit connected to a heptagonal pyramid. Zhai et al.^16 found a similarly shaped structure but with C 2 V symmetry and a 2 B 2 state ( XVIII , Figure 2). According to our calculations, there is only marginal difference in energy (∼0.3 kcal/mol) between a C (^) s
TABLE 8: Calculated Resonance Energies (RE) and Normalized Resonance Energies (NRE) of the B n and B n - Clusters ( n ) 5 - 13) as a Function of the Number of BB Bonds ( m ) a
cluster B n
number of BB bonds RE (B n ) b^ RE (B n - ) c^ NRE (B n ) NRE (B n - ) 3 3 44.2 50.7 14.7 16. 4 5 61.7 39.6 15.4 9. 5 7 67.8 64.2 13.6 12. 6 9 71.8 85.5 11.9 14. 10 21.4 35.1 3.6 5. 7 11 104.8 106.8 15.0 15. 12 54.4 56.4 7.8 8. 8 13 140.3 151.4 17.5 18. 14 89.9 101.0 11.2 12. 9 15 148.0 157.2 16.4 17. 16 97.6 106.8 10.8 11. 10 17 181.5 186.5 18.2 18. 18 131.1 136.1 13.1 13. 11 19 195.4 214.9 17.8 19. 20 145.0 164.5 13.2 15. 21 94.6 114.1 8.6 10. 12 21 229.5 222.5 19.1 18. 22 179.1 172.1 14.9 14. 24 78.3 71.3 6.5 5. 13 23 227.9 250.7 17.5 19. 24 177.5 200.3 13.7 15. 26 76.7 101.5 5.9 7. a (^) Values given in kcal/mol. b (^) Using eq 6a with the TAE(B (^2)
(^1 Σg+)) ) 50.4 kcal/mol. c^ Using eq 6b with the TAE(B 2 -^ ( 4 Σg-)) ) 111.1 kcal/mol.
Figure 6. Size dependence of the resonance energies (RE in kcal/ mol) of B n and B n -^ clusters. Values obtained from G3B3 heats of formation at 0 K.
J J. Phys. Chem. A, Vol. xxx, No. xx, XXXX Tai et al.
and a C 2 V structure of the type XVIII , which both exist on a flat potential energy surface. We conclude that the C 2 V structure XVIII (^2 B 2 ) is the ground state of B 11. The alternative structure XIX is ∼10 kcal/mol higher in energy.
Similar structures are obtained for the anionic cluster B 11 -. However, for the anion, both C 2 V structures XX and XXI have the same energy with the former being only 0.3 kcal/mol lower at 0 K, but 0.1 kcal/mol higher at 298 K than the latter. The adiabatic EA and anion VDE calculated by detaching one electron from the HOMO of B 11 -^ XX ( C 2 V, 2 B 2 ) at the G3B level are 3.48 and 3.63 eV, respectively. The VDE agrees reasonably well with the PES result of 3.426 ( 0.010 eV.^70 The experimental work^70 suggested that the EA and VDE quantities are identical.
The three bonding π orbitals (HOMO, HOMO-2, and HOMO-6) and four bonding σ orbitals (HOMO-1, HOMO-3, HOMO-4, and HOMO-5) in B 11 are responsible for its σ antiaromatic and π aromaticity, respectively. 6 These authors postulated that the globally delocalized electrons can break into four different areas leading to σ aromatic islands. Our ELF analysis reveals that the first bifurcation value of ELFσ of 0. is for isosurfaces separated into four isolated basins, whereas the second value of 0.91 is for separating it completely. The ELFπ value of 0.98 is high and shows that B 11 -^ has global π aromaticity. B 12. The geometry and electronic structure of B 12 and its anion were extensively investigated by different authors.2,4,15,16,29,72, Our G3B3 calculations confirm previous prediction that the structure XXII ( C 3 V, 1 A (^) g ), consisting of three hexagonal pyramids as subunits, constitutes the most stable isomer. The cluster is slightly distorted out-of-the-plane but is rather fluxional as the fully planar D 3 h structure lies only 3.7 kcal/mol higher in energy.
Following attachment of one electron to the doubly degenerate LUMO of B 12 , the C 3 V structure undergoes a Jahn-Teller distortion, giving rise to a three-dimensional Cs structure XXIII ( 2 A′) for the anion B 12 -^ (Figure 2). G3B3 calculations yield the values of 2.33 and 2.41 eV for EA(B 12 ) and VDE(B 12 - ), respectively, in good agreement with the PES results of 2.21 ( 0.04 and 2.26 ( 0.04 eV. 26
Earlier theoretical studies have focused on π-electron delo- calization as an important component for B 12 where six π-electrons occupy orbitals similar to that of benzene. However, as discussed above, B 10 and B 11 -^ also have benzene-like 6 π-electrons, but their HOMO-LUMO gaps are found to be much smaller than that of B 12. Thus, simple π-electrons cannot account for the differences. Zubarev and Boldyrev^6 argued that the three central B atoms donate electrons to form islands of σ-electrons where each pair of delocalized electrons is affiliated with three or four B atoms. As described in previous studies,2,4,15,16,29,74,75^ the BB bond distances in the B 9 subunit of B 12 (1.55-1.61 Å) are similar to that of B 2 (1.59 Å), and all are shorter than those in the B 3 subunit (1.65 Å). The inter-ring B-B distances formed between the atoms 2, 5, and 7 of the inner B 3 subset with the atoms of the outer B 9 ring (cf., structure XXII in Figure 2 for atom numbering) are longer, 1.68-1.81 Å. The distances suggest that in structure XXII there are 24 BB interactions that could be considered as some type of bond formed by the 36 valence electrons of B 12. These 36 electrons are distributed as follows: (i) 18 σ-electrons are used to make the nine BB bonds of the outer B 9 cycle; (ii) 6 σ-electrons to make the three bonds of the inner B 3 cycle; (iii) 6 π-electrons are delocalized over the entire cluster; and (iv) the remaining 6 electrons are delocalized
between the B 9 and B 3 rings. Thus, beside the B 9 skeleton, there is a 6σ- (^6) π- (^6) σdelo trifurcation of electrons. Aihara et al.^29 predicted that the π-delocalization in planar boron clusters reaches a maximum point at B 12. An important result from this analysis is that, in addition to the π-electrons, there are MOs between B 9 and B 3 subunits that contribute to a stabilization of the B 12 cluster. All of the subsets can formally be considered as aromatic (4 n + 2) units. The above description based on orbital interactions can also be obtained from a topological analysis of the ELFπ and ELFσ components. The isosurface for ELFσ for B 12 is separated into three reducible basins having each a bifurcation value of 0.87. The corresponding σ electrons are delocalized over the five- member ring. A higher value of 0.92 is found by separating completely the isosurface into irreducible basins, which leads to an assignment of σ aromaticity for the system. A bifurcation value of 0.88 is also found for ELFπ. We noted an important distribution in that electrons are delocalized within each of the five-membered rings for both ELFπ and ELFσ isosurfaces, which ultimately results in the multiple aromaticity of B 12. B 13. The structure, stability, and spectroscopic properties of B 13 clusters have attracted much attention due to the high stability of the B 13 +^ cation.^76 Two stable structures for B 13 have been proposed: the quasi-planar XXIV ( C 2 V, 2 A 1 ) and the convex XXV ( C 2 V, 2 B 1 ) in which three hexagonal subunits are present.^4 A similar quasi-planar structure ( Cs , 2 A′) was also located,^16 but it is less stable than the two structures given above. We explored further the possible geometries for B 13 and confirmed that the structures XXIV and XXV are essentially degenerate in energy and are the most stable isomers. XXV is 0.4-0. kcal/mol lower than XIV. The next lower-lying isomer is the high symmetry XXVI ( D 2 h , 2 B3u ), at 5.0 kcal/mol higher in energy. Thus, B 13 is characterized by significant structural fluctionality. Attachment of one electron to the SOMOs of the different B 13 isomers does not affect much their geometry (cf., Figure 2). However, a change in the energy ordering does occur. The high symmetry structure XXIX in its low spin state ( D 2 h , 1 Ag ) becomes the lowest-lying form, and the two other structures XXVII and XXVIII have high spin, and are ∼ 23 - 24 kcal/ mol higher in energy (Table 2). In their previous study, Zhai et al.^16 reported that a Cs structure is the most stable, but we cannot locate such a structure. The G3B3 value for the VDE(B 13 - ) of 3.98 eV is larger than the experimental value of 3.78 eV,^16 consistent with the calculations of the VDE for other B n clusters at this level. The adiabatic EA between B 13 XXIV and B 13 -^ XXIX is 3.62 eV. The adiabatic detachment energy of the anion from B 13 -^ XXIX to B 13 XXVI is 3.81 eV. The latter value agrees well with the experimental result of 3.78 ( 0.02 eV. 16 Zhai et al. 16 stated that as the B 13 -^ anion possesses eight electrons, it is π antiaromatic according to the Hu¨ ckel rule. In contrast, Aihara et al. 29 suggested that the B 13 -^ anion is actually aromatic with a large positive resonance energy. Our ELF plots concur rather with the former view. The bifurcation value of ELFπ is found to be 0.65, which is low and makes B 13 -^ a globally π antiaromatic species. However, partial π electron delocalization can be identified in the anion over the three- and four-membered rings. The value of ELFσ of 0.89 is high enough to suggest that the B 13 -^ anion has σ aromaticity. We thus assign B 13 -^ as being globally σ aromatic with islands of π aromaticity. Binding Energies. To probe further the thermodynamic stability of the boron clusters, we now examine the binding
Small Boron Clusters (B n , n ) 5 - 13) J. Phys. Chem. A, Vol. xxx, No. xx, XXXX K
Use of a topological analysis of the electron localization function, in particular a partition of the global electronic density into σ and π components, provides additional insights into the aromaticity of boron clusters. The global aromaticity of these clusters obeys the conventional Hu¨ckel rule, and, in some cases, the analysis emphasizes the role of partially delocalized electrons in a nonaromatic environment. The larger clusters appear to have multiple aromaticity. 48 The evolution of the average binding energies (energy per atom) is nearly parallel for both neutral and anionic series, and they tend to approach the asymptote of the heat of atomization of the elemental boron in the solid state. The resonance energies, as defined from the TAEs, and in particular the normalized resonance energies are convenient indices to quantify the stabilization of a cluster by electron delocalization. Finally, the evolution of the geometry and resonance energy (through the parameter m ) suggests that the boron cluster grows in such a way that each additional B atom brings about two (for small clusters) or three (for larger clusters) new BB bonds, and forms planar or nearly planar rings with high symmetry. There is a preference for low spin electronic state.
Acknowledgment. Funding was provided in part by the Department of Energy, Office of Energy Efficiency and Renew- able Energy under the Hydrogen Storage Grand Challenge, Solicitation No. DE-PS36-03GO93013. This work was done as part of the Chemical Hydrogen Storage Center. D.A.D. is indebted to the Robert Ramsay Endowment of the University of Alabama. M.T.N. thanks the K.U. Leuven Research Council for support (GOA, IDO, and IUAP programs) and Dr. B. Kiran for valuable discussion on B12. T.B.T. thanks the Arenberg Doctoral School for a scholarship.
Supporting Information Available: Tables containing the Cartesian coordinates for optimized geometries, CCSD(T)/aug- cc-pV n Z total energies, MP2/aVDZ vibrational frequencies, and G3B3 total energies and TAEs. This material is available free of charge via the Internet at http://pubs.acs.org.
References and Notes
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