Comprehensive Analysis of Chemical Bonding in Boron Clusters DMITRY YU. ZUBAREV, ALEXANDER I. BOLDYREV Department of Chemistry and Biochemistry, Utah State University, 0300 Old Main Hill, Logan, UT 84322-0300 Received 27 March 2006; Revised 14 May 2006; Accepted 20 June 2006 DOI 10.1002/jcc.20518 Published online 16 November 2006 in Wiley InterScience (www.interscience.wiley.com). Abstract: We present a comprehensive analysis of chemical bonding in pure boron clusters. It is now established in joint experimental and theoretical studies that pure boron clusters are planar or quasi-planar at least up to twenty atoms. Their planarity or quasi-planarity was usually discussed in terms of -delocalization or -aromaticity. In the current article, we demonstrated that one cannot ignore -electrons and that the presence of two-center two-electron (2cÀÀ2e) peripheral BÀÀB bonds together with the globally delocalized -electrons must be taken into consideration when the shape of pure boron cluster is discussed. The global aromaticity (or global antiaromaticity) can be assigned on the basis of the 4n þ 2 (or 4n) electron counting rule for either -or-electrons in the planar structures. We À þ 2þ þ À 2À À showed that pure boron clusters could have double (- and -) aromaticity (B3 ,B4,B5 ,B6 ,B7 ,B7 ,B8,B8 ,B9 , þ þ 2À À B10,B11 ,B12, and B13 ), double (- and -) antiaromaticity (B6 ,B15), or conflicting aromaticity (B5 ,-antiaro- matic and -aromatic and B14, -aromatic and -antiaromatic). Appropriate geometric fit is also an essential factor, which determines the shape of the most stable structures. In all the boron clusters considered here, the peripheral atoms form planar cycles. Peripheral 2cÀÀ2e BÀÀB bonds are built up from s to p hybrid atomic orbitals and this enforces the planarity of the cycle. If the given number of central atoms (1, 2, 3, or 4) can perfectly fit the central cavity then the overall structure is planar. Otherwise, central atoms come out of the plane of the cycle and the over- all structure is quasi-planar. q 2006 Wiley Periodicals, Inc. J Comput Chem 28: 251–268, 2007 Key words: boron clusters; aromaticity; double aromaticity; double antiaromaticity; conflicting aromaticity; chemi- cal bonding Introduction using the above-mentioned chemical bonding models, which makes them irreplaceable in chemistry. Homoatomic and heteroatomic clusters today represent the final However, we do not have similar chemical bonding models frontier for developing unified chemical bonding theory. allowing us to use the ‘‘paper and pencil’’ approach for predicting In organic chemistry, a vast majority of molecules can be isomers of homoatomic and heteroatomic clusters. The brute-force represented by a single Lewis structure (classical molecules) techniques based on genetic algorithm, molecular dynamics, hop- with either single or multiple two-center two-electron (2cÀÀ2e) ping models can help us to find global minima of small clusters bonds and with an appropriate number of lone pairs on electron- as well as their low-lying isomers, but they quickly run into com- rich atoms. Chemical bonding in many inorganic molecules also putational problems in larger systems. Without robust chemical can be represented by a single Lewis structure, but generally in bonding models capable of predicting global minimum structures inorganic chemistry such a description encounters significant and stable isomers of clusters, our progress in understanding clus- problems. If, however, a single Lewis structure description is ter structure and in rational design of molecular/cluster-level elec- not sufficient, then the resonance of Lewis structures is used. tronic and mechanical devices is seriously limited. The last approach is particularly important for description of the chemical bonding in aromatic compounds. The major advantage of the above-mentioned chemical bonding model is that we can Correspondence to: A. I. Boldyrev; e-mail: [email protected] predict possible isomers of classical molecules using just paper Contract/grant sponsor: The Petroleum Research Fund, American Chemi- and pencil and with high certainty we can also predict which cal Society; contract/grant number: ACS-PRF No. 38242-AC6 isomer could be the most stable one for a given stoichiometry. Contract/grant sponsor: National Science Foundation; contract/grant We can also draw a possible mechanism of chemical reaction numbers: CHE-0404937, CTS-0321170 q 2006 Wiley Periodicals, Inc. 252 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry There was some progress in recent years in developing chem- in Figure 1. Chemical bonding analysis was preformed using the ical bonding models for clusters (1–13 and references therein). natural bond analysis (NBO)64 (at the B3LYP/6-311þG* level Boron clusters are the best understood clusters of the main of theory65–69), pictures of Hatree-Fock canonical MOs (RHF/6- group elements. Today we are capable of explaining and predict- 311þG*), and nuclear independent chemical shift (NICS)70 ing their geometric structures and other molecular and spectro- (B3LYP/6-311 þ G*). We also calculated first singlet vertical scopic properties, because of recent advances in developing excitation energies at TD-B3LYP/6-311 þ G*71,72 as a probe of chemical bonding model for these systems.5a,6,7b,14–24 aromaticity or antiaromaticity. All calculations have been per- Pioneering works on pure boron cations have been done by formed using Gaussian 03 program.73 MO pictures were made Anderson and coworkers.25–31 These authors produced boron using the Molden 3.4 program.74 Results of our calculations are cluster cations in molecular beams using laser vaporization and summarized in Table1. studied their chemical reactivity and fragmentation properties. They initially postulated the three-dimensional structures for bo- Chemical Bonding Analysis ron clusters. Consequent quantum chemical calculations32–62 have shown that boron clusters prefer planar or quasi-planar In our chemical bonding analysis, we adopt the following ap- structures. However, these computational predictions were not proach. First, we use the NBO analysis to determine which ca- verified experimentally. In a series of recent articles, joint exper- nonical MOs can be localized into 2cÀÀ2e chemical bonds. Sec- imental and theoretical studies have been reported for a number ÀÀ À À 14 À 15 À 16 À 17 À À 18 ond, MOs, which cannot be localized into 2c 2e bonds, are of boron clusters, B3 and B4 , B5 , B6 , B7 , B8 and B9 , À À 19 identified as -delocalized or -delocalized. Third, the -aroma- B10 –B15 , and their neutrals. The structures of these clusters ticity (or -antiaromaticity) and the -aromaticity (or -antiaro- have been studied computationally and verified through compari- maticity) is assigned to a cluster on the basis of counting of the sons of experimental and theoretical photoelectron spectra. delocalized electrons according to the 4n þ 2 rule for aromaticity These studies have confirmed the two-dimensional or quasi two- (the singlet coupling electrons) and the 4n rule for antiaromaticity dimensional structures of all these clusters. Pure boron clusters (the singlet coupling electrons). For the triplet coupling of elec- have been recently reviewed.63 trons, we use the inverse 4n counting rules for aromaticity. Thus Planarity of boron clusters have been primarily discussed in in our analysis, we mix two ways of describing chemical bonding terms of -delocalization43,44 and -aromaticity.5a,6,7b,14–24 It has À in boron clusters: localized MOs and delocalized MOs. In the been shown, that high symmetric planar boron clusters B3 ÀÀ 2À À result, chemical bonding is expressed in terms of 2c 2e bonds (D3h), B4 (effectively D4h), B8 (D7h), B9 (D8h) have either 2 and lone pairs as well as multiple aromaticity, multiple antiaroma- À 2À À (B3 and B4)or6(B8 and B9 ) -electrons similar to prototypi- ticity or conflicting (-aromaticity and -antiaromaticity, or - þ cal hydrocarbons C3H3 (D3h, with 2 -electrons) or C6H6 (D6h, antiaromaticity and -aromaticity) aromaticity. Such a mixed þ with 6 -electrons), Thus, they formally satisfy the 4n 2 analysis is not new in chemistry. It is constantly used in organic Huckel rule and could be considered as -aromatic clusters. Bo- chemistry. For example, in benzene, -electrons are treated as 2À ron clusters with 4n -electrons such as B6 (4 -electrons) and forming localized 2cÀÀ2e CÀÀC bonds, while -electrons are B14 (8 -electrons) can be considered as -antiaromatic. Aihara 20 treated as completely delocalized over six carbon atoms. and coworkers recently performed analysis of aromaticity of The concept of ‘‘double aromaticity’’ was initially introduced boron clusters B (x ¼ 3–15) in terms of topological resonance x in 1979 by Schleyer and coworkers for explanation of chemical energy (TRE) and concluded that all boron clusters are highly bonding in 3,5-didehydrophenyl cation.75 Double aromaticity -aromatic including systems with 4n -electrons. and antiaromaticity in small carbon rings was discussed by To resolve the controversy about aromaticity or antiaromatic- Martin-Santamaria and Rzepa.76 ( þ )-Double aromaticity and ity of closed shell boron clusters with 4n electrons and also to (, )-mixed aromaticity have been used by Berndt and coworkers include -electrons into discussion in the current work, we pres- for explaining chemical bonding in planar boron compounds.77 ent a comprehensive chemical bonding analysis in the pure bo- 0,þ1,þ2,À1,À2 ron clusters Bx (x ¼ 2–15). We will consider here þ À B3 and B3 Clusters only clusters with even number of electrons. For neutral and ani- þ onic boron clusters with 3–9 atoms, the -aromaticity has been The B3 cluster is a perfect triangle in the closed shell ground 14–18,21–24 1 0 02 04 002 35,36,39,42,49 previously considered ; however, the influence of - spectroscopic state (D3h, A1,1a1 1e 1a2 ) (Fig. 1). electrons on geometric structure of boron clusters with 10–15 Molecular orbital picture of four valence MOs is presented in 5a,6,19,20 00 atoms have not been discussed yet.