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 (2c2e) peripheral BB 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 þ 2þ þ 2 showed that pure boron clusters could have double (- and -) aromaticity (B3 ,B4,B5 ,B6 ,B7 ,B7 ,B8,B8 ,B9 , þ þ 2 B10,B11 ,B12, and B13 ), double (- and -) antiaromaticity (B6 ,B15), or conflicting aromaticity (B5 ,-antiaromatic 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 2c2e BB 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 overall 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; chemical 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 (2c2e) 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 2c2e chemical bonds. Sec- imental and theoretical studies have been reported for a number 14 15 16 17 18 ond, MOs, which cannot be localized into 2c 2e bonds, are of boron clusters, B3 and B4 , B5 , B6 , B7 , B8 and B9 , 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 in boron clusters: localized MOs and delocalized MOs. In the been shown, that high symmetric planar boron clusters B3 2 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- 2 (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 2 ron clusters with 4n -electrons such as B6 (4 -electrons) and forming localized 2c2e CC 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,þ2,1,2 ron clusters Bx (x ¼ 2–15). We will consider here þ 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. Also, the use of the Figure 2. The HOMO (1a2 )isa-MO, formed by the out-of- concept of aromaticity in cationic boron clusters was limited pri- plane overlap of 2pz-AOs of the three B atoms. We localized þ 5a,6 0 0 marily by B13 cluster. the remaining set of valence MOs (1e -HOMO-1 and 1a1- HOMO-2) into three 2c2e BB bonds with the occupation numbers (ON) 1.89 |e| using NBO analysis at the B3LYP/6-311 3þ 02 04 Theoretical Methods þ G* level of theory in the putative B3 cation (1a1 1e elec- þ tronic configuration) at the geometry of the B3 cluster. The 3þ 0.96 1.03 In our current study, we used previously determined geometries strong s–p hybridization in the B3 cation (2s 2p )isre- þ 2 þ 2þ 2 þ 2 for B3 ,B3 ,B4,B4 ,B5 ,B5 ,B6 ,B6,B6 ,B7 ,B7 ,B8,B8 , sponsible for bonding character of the lowest three valence MOs þ þ B9 ,B10,B11,B11,B12,B13 ,B14, and B15 clusters, which have [correction to our statement in the ref. 14a, where we stated that been summarized in recent review.63 Their structures are shown the bonding effect from these MOs should be small]. The two

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 253

þ þ 2þ 2 þ 2 Figure 1. Global minimum structures of B3 ,B3 ,B4,B5 ,B5 ,B6 ,B6,B6 ,B7,B7,B8 ,B9,B10, þ þ B11,B11,B12,B13,B14, and B15 as reported in ref. 63.

þ þ electrons in the fully delocalized -HOMO make B3 -aromatic, in B3 , Fig. 2), which is a -molecular orbital, formed by the ra- obeying the 4n þ 2Hu¨ckel rule for n ¼ 0. Three other MOs dial overlap of the 2p-atomic orbitals on boron atoms (Fig. 2). represent 2c2e bonds, even though their ON (1.89 |e|) are The two electrons in the fully delocalized -HOMO make B3 somewhat lower than 2.00 |e| for the classical 2c2e bonds. Its -aromatic. The doubly occupied -HOMO-1 is responsible for 0 0 aromatic character is confirmed by highly symmetric structure -aromaticity in B3 .1a1 HOMO-3 and 1e HOMO-2 can be and highly negative NICS values: NICS(0) ¼66.3 ppm; transformed into three 2c2e BB bonds. Thus, B3 is a dou- NICS(0.5) ¼46.3 ppm, and NICS(1.0) ¼15.9 ppm (Table bly (- and -) aromatic system. Its doubly aromatic character is 1). The small first singlet vertical excitation energy (0.77 eV, confirmed by NICS value: NICS(0) ¼73.6 ppm; NICS(0.5) ¼ Table 1) reflects the presence of low-lying completely bonding 57.9 ppm; and NICS(1.0) ¼28.2 ppm (Table 1). High sym- 0 -aromatic LUMO (2a1). metry and rather high first singlet vertical excitation energy In the triangular ground electronic state of the B3 clus- (2.65 eV at TD-B3LYP/6-311 þ G*, Table 1) also confirm the 14,49,55 0 ter, the extra pair of electrons occupies 2a1 -MO (LUMO doubly-aromatic character of B3 .

Journal of Computational Chemistry DOI 10.1002/jcc 254 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

Table 1. Computed Features of Chemical Bonding in Boron Clusters.a

Number of Number of totally Number of totally First singlet

2c2e delocalized delocalized vertical excitation Eatomiz Cluster BB bonds -MOs -MOs NICS (ppm) energy (eV) (kcal/mol per atom)

þ 1 0 B3 D3h ( A1 )3 0 1 66.3 (0.0) 0.77 56.4 46.3 (0.5) 15.9 (1.0) 1 0 B3 D3h ( A1 )3 1 1 73.6 (0.0) 2.65 88.43 57.9 (0.5) 28.2 (1.0) 1 B4 D2h ( Ag)4 1 135.6 (0.0) 3.08 79.65 24.5 (0.5) 7.7 (1.0) þ 1 0 B5 D5h ( A1 )5 1 1 36.2 (0.0) 2.97 88.04 31.0 (0.5) 18.8 (1.0) 1 B5 C2v ( A1)5 2 1 10.8 (0.0) 1.10 98.10 16.9 (0.5) 14.3 (1.0) 2þ 1 B6 D6h ( A1g)6 1 1 29.6 (0.0) 1.94 70.02 26.2 (0.5) 17.8 (1.0) 0 1 B6 C5v ( A1)5 3 159.1 (0.15) 2.34 89.18 41.9 (0.65) 23.3 (1.15) 2 1 B6 D2h ( Ag)6 2 2 3.6 (0.0) 0.88 93.57 3.4 (0.5) 8.9 (1.0) þ 1 B7 C6v ( A1)6 3 1 42.3 (0.1) 1.99 98.74 36.0 (0.6) 20.4 (1.1) 2 1 0 B8 D7h ( A1 )7 3 3 84.7 (0.0) 1.75 106.31 27.0 (0.5) 24.8 (1.0) 1 B9 D8h ( A1g)8 3 3 28.3 (0.0) 2.79 110.62 23.3 (0.5) 13.7 (1.0) 0 1 B10 C2h ( Ag)9 3 3 17.0 (0.0) 1.95 105.19 15.2 (0.5) 13.3 (1.0) þ 1 0 B11 Cs ( A )103 3 19.6 (0.0) 1.34 108.20 20.0 (0.5) 15.3 (1.0) 1 B11 C2v ( A1)104 3 18.5 (0.0) 1.81 114.27 20.1 (0.5) 17.0 (1.0) 0 1 B12 C3v ( A1)123 3 28.4 (0.04) 2.57 107.24 27.1 (0.54) 19.9 (1.04) þ 1 B13 C2v ( A1)133 3 17.2 (0.0) 2.09 111.21 21.5 (0.5) 20.2 (1.0) 1 B14 C2v ( A1)143 4 14.5 (0.6) 1.50 108.74 19.0 (0.1) 11.9 (0.4) 1 B15 C1 ( A) 15 4 4 11.4 (0.0) 1.02 114.5 12.0 (0.5) 8.8 (1.0)

aAll data at B3LYP/6-311 þ G*.

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 255

2 1 The doubly charged B4 cluster has a square planar D4h A1g 2 4 2 2 2 2 (1a1g 1eu 1b1g 1b2g 2a1g 1a2u ) structure according to Sundholm 8a 2 and coworkers, who used the isoelectronic analogy with Al4 , where extensive search for the global minimum structure has been 1a,1b 1 performed, and the D4h A1g square planar structure was 2 found to be the most stable isomer. The Al4 dianion has been studied extensively and it was shown using a variety of criteria that it is a doubly (-and-) aromatic system [1a and references 2 therein]. Thus, the isoelectronic, isostructural B4 is also doubly (-and-) aromatic system with four 2c2e peripheral BB 8a Figure 2. Molecular orbitals of the Bþ cation. bonds. Sundholm and coworkers also confirmed aromaticity in 3 2 B4 by calculating the ring-current susceptibility, which was found to be 7.4 nAT1. That is only 10% smaller than that for 2– the value of the prototypical aromatic benzene molecule, thus B4 and B4 Clusters confirming the aromatic nature of the dianion. The structure of neutral B4 was carefully studied by Martin 35 1 et al. They predicted the D2h Ag rhombus global minimum + – B5 and B5 Clusters structure (see Fig. 1). Such a distortion from the perfect square comes from the second-order (or ‘‘pseudo’’) Jahn-Teller effect, Kato et al.36 and Rica and Bauschlicher42 reported that in the 35 þ as it was discussed by Martin et al. Because of the nature of global minimum, B5 adopts the C2v structure, which is a slightly the distortion, the barrier for ‘‘squareness’’ is rather small (0.7– distorted planar pentagon. Rica and Bauschlicher stated that the 14a,58 1 0 02 04 04 002 02 0.8 kcal/mol) and even at moderate temperature, the B4 D5h ( A1,1a1 1e1 1e2 1a2 2a1 ) planar pentagon has two imagi- cluster is effectively square. nary frequencies at the B3LYP/cc-pvTZ level of theory and in the 1 2 2 2 2 1 2 2 2 2 2 2 2 2 Molecular orbitals of the D2h ( Ag,1ag 1b1u 1b2u 1b3g global minimum, C2v ( A1,1a1 1b2 2a1 3a1 1b1 2b2 4a1 3b2 ) 2 2 1b3u 2ag ) structure of B4 are shown in Figure 3. The lowest structure BB bonds are only slightly distorted. This distortion is four MOs (HOMO-2 (1b3g), HOMO-3 (1b2u), HOMO-4 (1b1u), due to the second order Jahn-Teller Effect. According to our cal- 1 HOMO-5 (1ag)) can be localized, as it has been shown by NBO culations (CCSD(T)/6-311 þ G*), the global minimum C2v ( A1) analysis, into four classical peripheral 2c2e BB bonds (Ta- structure (Fig. 1) is only 0.365 kcal/mol lower in energy than the 1 0 ble 2). Again, the strong s–p hybridization on the both types second-order saddle point D5h ( A1) planar pentagon structure of boron atoms: 2s0.952p1.14 and 2s1.042p0.83 (from HOMO-2 and after ZPE correction (harmonic frequencies at CCSD(T)/6- 1 0 through HOMO-5) is responsible for the formation of four clas- 311 þ G*), the vibrationally averaged D5h ( A1) structure is 1 sical BB bonds. The remaining two MOs are globally delocal- actually lower in energy than the vibrationally averaged C2v ( A1) ized, and participate in the global bonding in the cluster. NBO structure by 0.010 kcal/mol. Thus, for all practical purposes, we þ analysis in this case shows eight lone pairs with the occupation can consider the B5 cluster as a planar pentagon. þ number 0.5 e (Table 2) on each atom. The HOMO-1 (1b3u)isa The beautiful planar pentagonal structure of B5 can be completely bonding -molecular orbital formed by the out-of- understood from its molecular orbital analysis (Fig. 4). The 0 0 plane overlap of 2pz-AOs on the B atoms. The two electrons NBO analysis showed that HOMO-2 and HOMO-2 (1e2), 0 0 0 populating this MO make the cluster -aromatic. The HOMO HOMO-3 and HOMO-3 (1e1), and HOMO-4 (1a1) can be local- 0 (2ag)ofB4 is a -radial molecular orbital, just like the HOMO ized into five peripheral 2c2e BB bonds. The HOMO (2a1) 0 þ 00 (2a1 )ofB3 , formed by the radial overlap of 2p-AOs. The sys- in B5 is a globally bonding -MO, and HOMO-1 (1a2 )isa tem thus can be characterized as -aromatic, i.e., B4 is a doubly- globally bonding -MO. Thus, they make the cation doubly (- aromatic molecule as it was first recognized by Zhai et al.14a This and -) aromatic. Double aromaticity in conjunction with the conclusion is supported by effective high symmetric (square) presence of five 2c2e BB peripheral bonds is responsible structure, calculated the first singlet vertical excitation energy for the vibrationally averaged highly symmetry D5h structure of 1 1 þ þ (3.08 eV X Ag?A B3g at TD-B3LYP/6-311 þ G*, Table 1) the B5 cluster. Also, double aromaticity in B5 manifests itself and calculated NICS index, which is highly negative at the center in the high first singlet vertical excitation energy (2.97 eV at of the cluster (35.7 ppm), but quickly diminishes and changes TD-B3LYP/6-311 þ G*, Table 1), highly negative values of the sign at 1.0 A above the center (þ7.7 ppm) (Table 1). NICS: NICS(0) ¼36.2 ppm; NICS(0.5) ¼31.0 ppm, and

Figure 3. Molecular orbitals of the B4 cluster.

Journal of Computational Chemistry DOI 10.1002/jcc 256 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

a 1 Table 2. Localized MOs of the D2h, Ag Structure of B4. and it is a first order saddle point. Geometry optimization following the imaginary frequency normal mode led eventually to the global Occupation minimum structure. LMO type number |e| Composition 0 In the global minimum structure, the LUMO þ1(2e1 -MO) 00 is partially occupied instead of LUMO (1e1 ). The singlet B –B 1.977 50.49% B : 2s – 47.79%; 2p – 52.03% 02 04 04 002 02 00 1 2 1 1a1 1e1 1e2 1a2 2a1 1e12 electronic configuration of the D5h 49.51% B2: 2s – 49.93%; 2p – 49.91% structure of B5 should also lead to the first order Jahn-Teller B1–B3 1.977 50.49% B1: 2s – 47.79%; 2p – 52.03% 1 2 2 distortion. Indeed, it was shown that the C2v ( A1,1a1 1b2 49.51% B3: 2s – 49.93%; 2p – 49.91% 2 2 2 2 2 2 2a1 3a1 1b1 2b2 4a1 3b2 ) planar structure is the B5 global mini- B2–B4 1.977 49.51% B2: 2s – 49.93%; 2p – 49.91% þ þ 50.49% B : 2s – 47.79%; 2p – 52.03% mum structure. The LUMO 1inB5 belongs to a partially bond- 4 B –B 1.977 49.51% B : 2s – 49.93%; 2p – 49.91% ing/antibonding -orbital related to the completely bonding - 3 4 3 0 50.49% B4: 2s – 47.79%; 2p – 52.03% HOMO (2a1). These three MOs are a part of the set of 5 MOs Lone pair B1 0.576 B1: 2s – 0.0%; 2p – 99.5% formed by the 2p-radial AOs of B and responsible for global - Lone pair B1 0.529 B1: 2s – 5.1%; 2p – 93.9% bonding. Thus, B5 has four electrons on globally -delocalized Lone pair B4 0.576 B4: 2s – 0.0%; 2p – 99.5% HOMO-1 (4a1) and HOMO (3b2) and two electrons on globally Lone pair B 0.529 B : 2s – 5.1%; 2p – 93.9% 4 4 delocalized HOMO-3 (1b1), which makes B5 asystemwithcon- Lone pair B2 0.503 B2: 2s – 1.6%; 2p – 97.7% flicting aromaticity (-antiaromatic and -aromatic). NBO analy- Lone pair B2 0.424 B2: 2s – 0.0%; 2p – 99.6% sis for the B5þ cation at the geometry of B and with the Lone pair B 0.503 B : 2s – 1.6%; 2p – 97.7% 5 5 3 3 1a 21b 22a 23a 22b 2 electronic configuration shows that there are Lone pair B 0.424 B : 2s – 0.0%; 2p – 99.6% 1 2 1 1 2 3 3 five 2c2e BB peripheral bonds (ON ¼ 1.76–1.91 |e|). The 1 aLMO calculated at B3LYP/6-311 þ G* using the NBO analysis. C2v A1 structure of B5 has been experimentally established in joint photoelectron and ab initio study by Zhai et al.15 Because the geometric structure of -antiaromatic and -aromatic B5 NICS(1.0) ¼18.8 ppm (Table 1) and most importantly it anion has lower symmetry, we believe that antiaromaticity over- þ explains why B5 is a magic cluster in collision induced dissoci- whelms aromaticity in this case and we prefer to call this cluster ation (CID) experiments by Anderson and coworkers.23–31 The ‘‘net antiaromatic’’. The low first singlet vertical excitation energy 1 0 distortion of the D5h ( A1) structure can be explained by the sec- (1.10 eV, at TD-B3LYP/6-311 þ G*) provides us additional sup- ond-order Jahn-Teller effect, because there is nothing in occu- port in our overall assignment of aromaticity in spite of negative pied MOs (Fig. 4) of the pentagon indicating the deviation from values of NICS in this case (Table 1). The B5 anion is a remark- high symmetry. That case is similar to the distortion of the B4 able example showing that if we would limit our chemical bond- cluster into rhombus and like in that case the second-order dis- ing analysis to -electrons only, we will not be able to explain þ tortion makes the B5 potential energy surface very shallow. why -aromatic (2 -electrons) B5 cluster has low C2v symmetry For the B5 cluster, we can predict the global minimum structure and low first singlet vertical excitation energy. þ 00 þ 1 0 if we start with the B5 cluster. The 1e1-LUMO in B5 D5h ( A1 )is a doubly degenerate -MO, which is a partially bonding/antibond- 2+ 2– 00 B6 ,B6, and B6 Clusters ing orbital related to the completely bonding 1a2-HOMO-1 -MO. þ These three MOs are a part of the set of 5 MOs formed by the 2pz- The six-atomic cyclic analog of the B5 cluster should be the 2þ AOs of B and responsible for global -bonding. Occupation of one B6 dication. It has 16 valence electrons and assuming the for- of doubly degenerate LUMO by two electrons should lead to distor- mation of six peripheral 2c2e BB bonds out of HOMO-2 0 tion of the D5h structure to the C2v structure because of the first (1b2u), HOMO-3 and HOMO-3 (1e2g), HOMO-4 and HOMO- 0 order Jan-Teller effect. Our calculations have shown that the result- 4 (1e1u), and HOMO-5 (1a1g), we should have four electrons on 1 2 2 2 2 2 2 2 2 ing C2v ( A1,1a1 1b2 2a1 3a1 1b1 2b2 4a1 1a2 ) structure is 50.6 two completely bonding -HOMO (2a1g) and -HOMO-1 (1a2u) 2þ (B3LYP/6-311 þ G*) kcal/mol higher than the global minimum (Fig. 5a). This makes the B6 dication doubly aromatic. Our cal-

þ Figure 4. Molecular orbitals of the B5 cation.

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 257

1 2þ Figure 5. (a) Molecular orbitals of the D6h ( A1g) structure of the B6 cluster; (b) molecular orbitals 1 1 0 of the C5v ( A1) structure of the B6 cluster; (c) molecular orbitals of the D5h ( A1) structure of the B6 1 2 cluster; (d) molecular orbitals of the D2h ( Ag) structure of the B6 dianion.

Journal of Computational Chemistry DOI 10.1002/jcc 258 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

2þ 1 culations proved that the B6 D6h ( A1g) structure indeed is a MOs. The HOMO-4 is completely bonding, and the HOMO is a 2 minimum at three levels of theory (B3LYP/6-311 þ G*, MP2/6- partially bonding orbital. The B6 dianion has 4 - and 4 - 311 þ G*, and CCSD(T)/6-311 þ G*). electrons on the globally delocalized MOs and six 2c2e pe- 2 In the B6 cluster, in addition to cyclic structures, we observe ripheral BB bonds. Thus, we can assign B6 to doubly (- the emergence of a new type of structure—pentagonal pyramid, and -) antiaromatic systems. The question may arise, why a which now corresponds to the global minimum.16 The planar doubly antiaromatic structure is the global minimum. We 2 pentagonal structure with the boron atom located at the center of believe that this is because the B6 cannot favorably support the five-atomic ring is not a minimum because the cavity inside six delocalized electrons in either -or-subsystems. Electro- of the pentagon is too small to favorably accommodate a boron static field from the screened boron nuclei does not provide atom at the center. However, as we will see in the large boron enough stabilization for six electrons in either -and-subsys- clusters (B8 and B9 ), with the increase of the size of the central tems and that leads to a compromised globally doubly antiaro- cavity, a boron atom can be favorably accommodated at the cen- matic structure. 20 2 ter of the appropriate polygon leading to planar highly symmet- In a recent article, Aihara et al. claimed that the boron B6 ric global minimum structures. The most accurate calculations16 cluster is highly aromatic on the basis of topological resonance 2 reveal that the global minimum structure of B6 is the pyramid energy (TRE). The calculated TRE for B6 in terms of the reso- 1 2 4 2 4 2 4 C5v ( A1,1a1 1e1 2a1 1e2 3a1 2e1 ) structure with the boron nance integral between two bonded boron atoms (| BB|) is 0.549 ˚ atom located 0.94 A above the center of the B5 perfect penta- | BB|. This value expressed in terms of the resonance integral 3 2 2 2 2 2 2 2 2 gon. The triplet C2h Au (1ag 1bu 2ag 2bu 3ag 1au 3bu 4ag between two bonded carbon atoms (| CC|) is 0.478 | CC|. For the 1 1 1 2 2 2 2 2 2 2 2 2 4bu 1bg ) and the singlet C2 A (1a 1b 2b 2a 3a 4a 3b 5a 4b ) reference, the TRE for benzene is 0.273 | CC|. Thus, according 20 2 structures originating from the cyclic B6 geometry upon Jahn- to Aihara et al., the B6 cluster is clearly showing the pres- Teller distortions were found to be 7.2 and 8.2 kcal/mol ence of aromaticity. However, this large resonance energy does 2 (CCSD(T)/6-311 þ G(2df)//B3LYP/6-311 þ G*), respectively, not contradict our assignment of B6 to doubly antiaromatic sys- 16 2 higher in energy. tems. Our assignment of B6 to -antiaromatic system is based To simplify interpretation of molecular orbitals, let us first on the presence of 4 -electrons, its highly distorted (D2h) struc- 1 0 02 04 00 0 2 02 0 perform MO analysis for the D5h ( A1,1a1 1e1 1a2 1e2 2a1 2e14) ture, and paratropic ring currents calculated by Fowler and co- structure (Fig. 5c), in which the central atom is pushed into the workers (see ref. 16 for details). We see this cluster as being 0 0 center. The set of five MOs (HOMO-2 and HOMO-2 (1e2), antiaromatic globally. It does not, however, mean that this clus- 0 0 0 HOMO-4, and HOMO-4 (1e1), and HOMO-5 (1a1)) can be ter cannot have positive resonance energy. In fact, according to 2 localized into five 2c2e BB bonds, so they are responsible our MO analysis, we can consider the -system in B6 as being 00 for the peripheral bonding. The HOMO-3 (1a2 ) is formed by split into two subsystems, with two -electrons localized over 2 2pz-AOs, and it is responsible for the global -bonding. The each of two triangular regions. -MOs of B6 cluster can be 0 0 0 HOMO and HOMO (2e1 ), and HOMO-1 (2a1) are formed from viewed as composed of two aromatic B3 clusters (see ref. 16 2p-radial AOs, and they are responsible for global -bonding in for detailed discussion). Indeed, 1b2g-HOMO and 1b3u-HOMO-4 1 0 the B6 cluster. Thus, B6 in the D5h A1 configuration is a doubly are a pair of bonding and antibonding -MOs in B3 . Thus, -MOs (-and-) aromatic system with 2-and6-electrons. How- do not contribute significantly to chemical bonding between two 2 ever, as we mentioned above, the central cavity in the B5 pen- B3 groups. This allows us to speculate that -MOs in B6 give rise tagon is too small to favorably accommodate the central boron to an island -aromaticity in this cluster. Similar analysis for the 1 atom, and therefore a C5v A1 pyramidal structure corresponds delocalized -MOs reveals that we also have an island -aromatic- 2 2 to the global minimum. While in the pyramidal structure - ity in B6 . Thus, the globally antiaromatic B6 system can be con- and -MOs are mixed, we believe that the bonding picture sidered as having two island aromatic subunits. The island -aro- 1 0 2 20 developed for the D5h A1 structure is still qualitatively valid maticity is responsible for positive TRE in B6 in Aihara et al. 20 and can explain why B6 clusters adopts such a structure. calculations. Indeed, Aihara et al. stated that three (a2–a4) out of 1 When two extra electrons are added, the planar D2h Ag four circuit currents (Fig. 6) are paratropic indicating antiaromatic- 2 2 2 2 2 2 2 2 2 2 (1ag 1b1u 2ag 1b2u 1b3g 1b3u 3ag 2b2u 2b1u 1b2g ) structure be- ity and the a1 circuit current, which is located over triangular 2 7b,16 comes the global minimum for the B6 dianion. This struc- region, is highly diatropic. It overwhelms the antiaromatic con- ture is originating from the D6h hexagon, which underwent the first order Jahn-Teller distortion. Molecular orbital analysis helps 2 us to interpret chemical bonding in B6 (Fig. 5d). Six MOs (HOMO-2 (2b2u), HOMO-5 (1b3g), HOMO-6 (1b2u), HOMO-7 (2ag), HOMO-8 (1b1u), and HOMO-9 (1ag)) can be localized into six 2c2e BB bonds, so they are responsible for the peripheral bonding in this cluster. The remaining four MOs are re- 2 sponsible for the global bonding in the B6 cluster. The HOMO-1 (2b1u) and HOMO-3 (3ag) are -radial MOs, with the HOMO-3 being completely bonding, and the HOMO-1 being 2 partially antibonding. Thus, B6 has two globally delocalized - MOs, which makes this dianion -antiaromatic. The two other 2 delocalized orbitals HOMO-4 (1b3u) and HOMO (1b2g) are - Figure 6. Nonidentical circuits in B6 (adopted from ref. 20).

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 259 tributions from the a2a–4 a1 circuit currents and results in the electron spectroscopic and ab initio study17 suggests that at least 3 1 overall positive TRE. This result clearly supports the presence two isomers C6v A2 and C2v A1 could coexist in the B7 beam 2 22 of island aromaticity in B6 . Alexandrova et al. have shown and contribute to the photoelectron spectra of B7 . 3 that for Li2B6 molecule in the gas phase, the global minimum MO analysis was performed for the planar B7 D6h A2g 1 þ 2 4 4 2 2 2 4 2 3 structure is C2h ( A1) with two Li ions located above and (1a1g 1e1u 1e2g 1b2u 2a1g 1a2u 2e1u 1e1g ) (In ref. 17 the A1 2 1 2 4 4 2 below the B3 triangular areas in B6 . The D2h ( Ag) structure, spectroscopic state was reported for the 1a1 1e1 1e2 2a1 3a1 þ þ 2 2 4 2 containing one Li cation above and another Li below the 1b1 2e1 3e1 electronic configuration of the B7 C6v structure as 2 plane of B6 dianion, was found to be a saddle point on the determined by the Gaussian 03 program. However, the correct 3 potential energy surface. These results confirmed the presence of spectroscopic state for such electronic configuration is A2. Sim- 3 2 4 the -island aromaticity in the globally antiaromatic system. ilarly, in ref. 17 the A1g was reported for the 1a1g 1e1u 1e2g 4 2 2 2 4 2 1b2u 2a1g 1a2u 2e1u 1e1g configuration of the B7 D6h structure, 3 + – while the correct spectroscopic state is A2g) model system. B7 and B7 Clusters NBO analysis showed that the set of low-energy MOs (Fig. 7b): þ 0 The global minimum structure of the cationic B7 cluster is the HOMO-7 (1a1g), HOMO-6 and HOMO-6 (1e1u), HOMO-5 and 1 2 4 4 2 2 2 4 36,39,41,55,60 0 C6v A1 (1a11e11e22a11b13a12e1) pyramid with the HOMO-5 (1e2g) and HOMO-3 (1b2u) can be localized into six central boron atom located 0.72 A˚ above the plane (Fig. 1). To peripheral 2c2e BB bonds, forming the hexagonal frame- simplify the interpretation of the molecular orbitals, we per- work. HOMO-2 (1a2u), which is a completely bonding -MO and 1 2 4 4 2 0 formed MO analysis of the D6h ( A1g,1a1g 1e1u 1e2g 1b2u partially antibonding and partially occupied HOMO, and HOMO 2 2 4 2a1g 1a2u 2e1u ) structure (Fig. 7b), in which the central atom is (1e1g) make this cluster -aromatic with 4 -electrons according pushed into the plane. The set of six MOs (HOMO-3 (1b2u), to the inverse 4n rule for triplet states. HOMO-3 (2a1g) and 0 0 0 HOMO-4 and HOMO-4 (1e2g), HOMO-5, and HOMO-5 (1e1u) HOMO-1 and HOMO-1 (2e1u) are delocalized -MOs, which and HOMO-6 (1a1g)) is responsible for the peripheral bonding make this cluster -aromatic. Thus, the B7 cluster is a doubly (- and can be localized into six 2c2e BB bonds. The HOMO-1 and -) aromatic system with six peripheral BB bonds. (1a2u) is formed by 2pz-AOs and is responsible for the global - 0 bonding. The HOMO and HOMO (2e1u), and HOMO-2 (2a1g) 2– – B8,B8 , and B9 Clusters are formed from 2p-radial AOs and they are responsible for þ þ 1 global -bonding in the B7 cluster. Thus, B7 in the D6h A1g The neutral B8 cluster has a triplet perfect heptagon structure 3 0 02 04 04 02 04 002 04 002 configuration is doubly (- and -) aromatic system with 2- D7h A2 (1a1 1e1 1e2 2a1 1e3 1a2 2e1 1e1 ) in its ground and 6-electrons. However, the central cavity in the B6 hexagon electronic state (correction to the previous statement in ref. 18) is too small to favorably accommodate the central boron atom, as it was established by Zhai et al.18 Another wheel-type struc- 1 1 0 18 1 0 and therefore, the C6v A1 pyramidal structure corresponds to the ture Cs A was identified as a low-lying isomer. The Cs A 1 global minimum. While in the pyramidal C6v A1 structure - and isomer is a Jahn-Teller distorted heptagon, because in the hepta- 00 -MOs are now mixed, we believe that the bonding picture we gon singlet structure only one out of two doubly degenerate 1e1 - 1 developed for the D6h A1g structure is still qualitatively valid. HOMOs is occupied. þ 3 0 The seven-atomic cyclic structure of the B7 cluster is a local In the triplet D7h A2, perfect heptagon structure HOMO-3 3 0 02 04 04 04 002 02 002 0 0 0 0 minimum with the A2 (1a1 1e1 1e2 1e3 1a2 2a1 1e1 )spectro- and HOMO-3 (1e3), HOMO-5 and HOMO-5 (1e2), HOMO-6 0 0 0 scopic state. It has 20 valence electrons and assuming the forma- and HOMO-6 (1e1), and HOMO-7 (1a1) (Fig. 8) can be local- 0 00 tion of seven peripheral 2c2e BB bonds from HOMO-3 and ized into seven 2c2e BB bonds, HOMO and HOMO (1e1 ), 0 0 0 0 00 HOMO-3 (1e3), HOMO-4 and HOMO-4 (1e2), HOMO-5 and and HOMO-2 (1a2 ) (Fig. 8) are formed from 2pz-AOs, and they 0 0 0 HOMO-5 (1e1) and HOMO-6 (1a1), we should have six electrons are responsible for the global -bonding. The HOMO-1 and 0 0 0 for global bonding. Two of them occupy the completely bonding HOMO-1 (2e1 ) and HOMO-4 (2a1) are formed from 2p-radial 0 -HOMO-1 (2a1). Two electrons occupy the completely bonding AOs, and they are responsible for global -bonding in the B8 00 3 0 -HOMO-2 (1a2) and two electrons occupy the partially bonding cluster. Thus, the D7h A2 structure is a doubly (-and-) aro- 00 doubly degenerate -HOMO (1e1) with the triplet coupling. This matic system with 4 -electron (satisfying the inverse rule 4n þ makes the B7 dication doubly aromatic in the cyclic structure. rule for aromaticity for triplet coupled electrons), with 6 -elec- þ 3 0 However, the B7 D7h ( A2) structure is significantly higher (63.4 trons (satisfying the 4n þ 2 rule for aromaticity for singlet kcal/mol at B3LYP/6-311 þ G*) in energy than the global mini- coupled electrons), and with seven 2c–2e peripheral BB 1 0 mum C6v ( A1) structure, because of unsupported dangling elec- bonds. 1 0 tron density at the center of the cycle. Thus, the cyclic structures In the singlet Cs A isomer, one of the doubly degenerate are not favorable anymore beyond six boron atoms. HOMOs is now occupied by a pair of electrons with another 17 According to the Alexandrova et al. calculations, the B7 one being empty, which results in the Jahn-Teller distortion. 3 2 4 4 2 2 2 cluster has a very flat triplet C6v A2 (1a11e11e22a13a11b1 This system is the one with conflicting aromaticity: there are 4 4 2 78 þ 2e13e1) pyramidal structure similar to the B7 structure as the -electron (satisfying 4n rule for antiaromaticity for a singlet 1 2 2 2 2 global minimum. The second lowest C2v A1 (1a11b21b1 2a1 coupled electrons) and 6 -electrons (satisfying 4n þ 2 rule for 2 2 2 2 2 2 2 1a23a12b14a1 2b23b13b2) structure (Fig. 1) was found to be just aromaticity for a singlet coupled electrons), and seven 2c–2e pe- 0.7 kcal/mol higher (RCCSD(T)/6-311þG(2df)//RCCSD(T)/6- ripheral BB bonds. 5 0 2 4 4 311 þ G*) in energy than the global minimum structure. Thus, We also optimized the cyclic B8 (D8h, A1,1a1g 1e1u 1e2g 1e 4 2 2 2 2 2 these two structures are almost degenerate. The combined photo- 3u 2a1g 1b2g 1a2u 2e1u 1e1g ) doubly aromatic structure and found

Journal of Computational Chemistry DOI 10.1002/jcc 260 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

1 þ Figure 7. (a) Molecular orbitals of the C6v ( A1) structure of the B7 cluster; (b) molecular orbitals of 1 þ the D6h ( A1g) structure of the B7 cluster.

þ 2 that it is a local minimum at B3LYP/6-311 þ G*. However, it is Li cation is located above the slightly distorted B8 heptagon about 98 kcal/mol higher in energy than the boron-centered B8 agrees well with the experimentally recorded spectra of the anion. 3 0 1 0 2 (D7h, A2) doubly aromatic global minimum structure. The singlet D7h A1 structure of B8 is a doubly (-and-) Thus, the cyclic structures even being doubly aromatic and aromatic system with 6 -electron, 6 -electrons, and 7 2c2e corresponding to local minima are getting less and less stable peripheral BB bonds. This analysis of chemical bonding for þ 2 18 with the increase of the size of the cluster starting from B7 and B8 was first proposed by Zhai et al. Its double aromaticity is cannot be considered even as low-energy isomers. also confirmed by very high values of NICS: NICS(0) ¼84.7 2 1 0 02 The doubly-charged B8 anion has a planar D7h A1 (1a1 ppm, NICS(0.5) ¼27.0 ppm, and NICS(1.0) ¼24.8 ppm 04 04 02 04 002 04 004 1e1 1e2 2a1 1e3 1a2 2e1 1e1 ) singlet global minimum structure (Table 1). 1 2 (Fig. 1), in which high symmetry is restored again because the dou- The anionic B9 has the perfect planar D8h ( A1g,1a1g 1e1u 00 4 4 4 2 2 2 4 4 bly degenerate 1e1 -HOMO is now occupied by four electrons 1e2g 1e3u 2a1g 1b2g 1a2u 2e1u 1e1g ) wheel-shaped structure as (Fig. 8). While isolated dianion was not studied experimentally, its the global minimum (Fig. 1), which was established in a joint high-symmetric structure was experimentally confirmed in a joint photoelectron and ab initio study by Zhai et al.18 The perfect oc- photoelectron spectroscopy and ab initio calculations of the LiB8 tagon structure of B9 is unprecedented in chemistry and repre- cluster by Alexandrova et al.21 It was shown that calculated photo- sents the first example of octacoordinated atom in a planar envi- electron spectrum of the half-sandwich structure of LiB8 in which ronment.

1 0 2 Figure 8. Molecular orbitals of the D7h ( A1) structure of the B8 dianion.

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 261

1 Figure 9. Molecular orbitals of the D8h ( A1g) structure of the B9 anion.

The remarkable planar octagon structure of B9 can be easily ment is rather small and it allows one to consider the fluxional rationalized on the basis of the presence of double (- and -) CB8 system as one with the effective octacoordination of the þ aromaticity (Fig. 9). central atom. The two other SiB8 and PB8 clusters were found Chemical bonding in B9 is remarkably similar to bonding to have a perfect octagonal structure. We would like to stress 2 pattern in B8 . As before, eight MOs (Fig. 9): HOMO-3 (1b2g), that on the basis of our analysis of chemical bonding in B9 , the 0 0 þ HOMO-5, HOMO-5 (1e3u), HOMO-6, HOMO-6 (1e2g), valence isoelectronic CB8, SiB8, and PB8 species are also -ar- 0 HOMO-7, HOMO-7 (1e1u), and HOMO-8 (1a1g) can be local- omatic with six -electrons and they also have eight 2c2e pe- ized into 8 2c2e BB peripheral bonds. The other valence ripheral BB bonds. - and -aromaticity together with eight MOs are delocalized over the octagon and they are responsible peripheral BB bonds are responsible for the beautiful octago- for global bonding between the central B atom and peripheral B nal structure of these species. 0 atoms. The three -MOs: HOMO, HOMO (1e1g), and HOMO-2 (1a2u) are responsible for -aromaticity and the three -MOs: + – 0 B10,B11, and B11 Clusters HOMO-1, HOMO-1 (2e1u), and HOMO-4 (2a1g) are responsible þ for -aromaticity in B9 . Again, such chemical bonding analysis The B10,B11, and B11 clusters in their global minimum struc- 18 for B9 was first proposed by Zhai et al. The double (-and-) tures (Fig. 1) have a common feature—two boron atoms located þ aromaticity in B9 is supported by high symmetry, high first sin- inside either eight- (B10) or nine- (B11 and B11) membered ring. glet vertical excitation energy (2.79 eV at TD-B3LYP/6-311 þ Therefore, we will consider chemical bonding in these clusters G*), and highly negative NICS values: NICS(0) ¼28.3 ppm, together. ¼ ¼ 19 NICS(0.5) 23.3 ppm, and NICS(1.0) 13.7 ppm The global minimum of B10 according to Zhai et al. and 44 1 (Table 1). Boustani is the C2h, Ag structure (Fig. 1), which is nonplanar In addition to the wheel planar structure of B9 , Minkin and with eight boron atoms forming a planar cycle around two atoms coworkers78,79 reported planar structures and -aromatic charac- at the center, with one of the central atoms located above the þ ter in the CB8, SiB8, and PB8 species. The authors found, how- plane and the other one below the plane. The chemical bonding ever, that in the case of octacoordinated carbon, the central cav- analysis of the global minimum structure of B10 was performed ity is now too big to be stabilized through the accommodation by Zhai et al.19 only for the -system, and it was shown that this of only one carbon nucleus. The D8h structure of CB8 is a sec- structure has 6 -electrons and thus it is -aromatic. We propose ond-order saddle point. The normal mode displacements lead to here an explanation of chemical bonding in B10 including both 1 1 aC2v ( A1) structure, in which the central C-atom is shifted to - and -electrons. As before, let us first flatten the C2h Ag 1 the side. However, the barrier on the intramolecular rearrange- structure into the planar D2h Ag structure for simplicity and per-

1 Figure 10. Molecular orbitals of the D2h ( Ag) structure of the B10 cluster.

Journal of Computational Chemistry DOI 10.1002/jcc 262 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

1 þ Figure 11. (a) Molecular orbitals of the C2v ( A1) structure of the B11 cation; (b) molecular orbitals of 1 the C2v ( A1) structure of the B11 anion. form MO analysis (Fig. 10) for the planar structure. NBO analy- of that this structure is 60.5 kcal/mol higher than the global min- sis shows eight peripheral 2c2e BB bonds (ON ¼ 1.87–1.93 imum structure with two boron nuclei at the center. Also, this |e|) and one 2c2e BB bond (ON ¼ 1.52 |e|) between central structure is the forth order saddle point. Thus, starting from B10 atoms, which could be approximately assigned to HOMO-5 cluster, the structures with one boron atom at the center are not (2b3g), HOMO-7 (2b2u), HOMO-8 (3ag), HOMO-9 (2b1u), low energy isomers anymore. However, the eight-membered ring HOMO-10 (1b3g), HOMO-11 (2ag), HOMO-12 (1b2u), HOMO- is still small to favorably accommodate two boron atoms within 13 (1b1u), and HOMO-14 (1ag). Rather low occupation number the plane. 42 1 0 for the central BB bond shows that we should treat the exis- Ricca and Bauschlicher reported a quasi-planar Cs A þ tence of this bond with caution. Three MOs: HOMO (1b1g), structure for the B11 cluster (Fig. 1). In this case, the nine-atomic HOMO-1 (1b2g), and HOMO-6 (1b3u) are responsible for the ring is again too small to accommodate two boron atoms within þ 1 global -bonding and the remaining three -MOs: HOMO-2 the plane. We plotted MOs of B11 for the ﬂattened C2v A1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (4ag), HOMO-3 (3b1u), and HOMO-4 (3b2u) are responsible for (1a1 2a1 1b2 3a1 2b2 4a1 3b2 5a1 6a1 4b2 1b1 7a1 5b2 8a1 2b1 2 the global -bonding. Thus, B10 is a doubly (- and -) aromatic 1a2 ) structure in Figure 11a. NBO analysis reveals nine 2c2e cluster with eight peripheral 2c2e BB bonds and one peripheral BB bonds (ON ¼ 1.81–1.94 |e|) and one central 2c2e central BB bond. A relatively high ﬁrst singlet vertical BB bond (ON ¼ 1.53 |e|), which could be approximately excitation energy (1.95 eV, at TD-B3LYP/6-311 þ G*) and neg- assigned to 10 lowest canonical MOs (from HOMO-6 to ative NICS values (Table 1) support our description of chemical HOMO-15). Three MOs: HOMO (1a2), HOMO-1 (2b1), and bonding in B10. HOMO-5 (1b1) are responsible for the global -bonding and 1 0 We calculated the B10 (D9h, A1) planar structure with only three MOs: HOMO-2 (8a1), HOMO-3 (5b2), and HOMO-4 (7a1) þ one boron atom at the center of the nine-atomic ring. The radius are responsible for the global -bonding. Thus, the B11 cation is of the external ring was found to be 2.202 A˚ (B3LYP/6-311 þ a doubly aromatic system with 6- and 6-delocalized electrons, G*), and the central cavity is now too big to be stabilized nine 2c–2e BB peripheral bonds and somewhat less pro- through the accommodation of only one boron nucleus; because nounced central BB bond.

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 263

1 The global minimum structure C2v A1 for B11 was reported trons is affiliated with three or four boron atoms. We need to de- by Zhai et al.19 (Fig. 1). Our MO plots for this dianion are velop new software tools for making -bonding analysis in such shown in Figure 11b. NBO analysis reveals nine 2c2e periph- systems more precise. þ eral BB bonds (ON ¼ 1.93–1.96 |e|) and one BBbond The B13 cationic cluster attracted a lot of attention, because between central atoms (ON ¼ 1.56 |e|). We believe that the low- Anderson and coworkers25–31 reported that it has anomalously est 10 canonical MOs (from HOMO-7 to HOMO-16) are approxi- high stability and low reactivity in comparison with other cati- mately responsible for the formation of these 10 BB bonds. onic boron clusters. Initially, this high stability was attributed to þ 28 34 Three MOs: HOMO (2b1), HOMO-2 (1a2), and HOMO-6 (1b1) B13 having a filled icosahedron structure. Kawai and Weare þ are responsible for the global -bonding, making this cluster -ar- have shown that a filled icosahedral of B13 is not even a minimum omatic as it was initially reported by Zhai et al.19 Four MOs: on the potential energy surface using Car-Parrinello ab initio HOMO-1 (6b2), HOMO-3 (8a1), HOMO-4 (7a1), and HOMO-5 molecular dynamics simulations. The global minimum structure of þ 42 (5b2) are approximately responsible for the global -bonding, B13 was established by Ricca and Bauschlicher, who predicted 1 making this system formally -antiaromatic. However, the shape the planar C2v A1 structure (Fig. 1). Three boron atoms can fit of HOMO-1, HOMO-3, HOMO-4, and HOMO-5 hints that the perfectly into the plane of the 10-membered ring. Fowler and globally delocalized electrons may in fact break into four local- Ugalde5a were the first who proposed that exceptional stability þ ized areas (giving rise to island -aromaticity) over the B11 clus- and low reactivity of B13 is related to its aromatic character. On 2 ter, similar to -delocalized electrons in the B6 cluster, where the basis of plotted MOs, Fowler and Ugalde concluded that they are split into two subsystems each localized over three boron three doubly occupied -MOs give six -electrons in a round atoms. At this point, it is hard to point out which atoms in the system, a situation reminiscent of benzene and the Huckel aro- 6 B11 clusters belong to which regions of island -aromaticity. maticity. Aihara evaluated the topological resonance energy (TRE) for -electrons using his graph theory of aromaticity. þ + He found that the TRE of B13 is positive in sign and very large B12 and B13 Clusters in magnitude: TRE ¼ 2.959 | BB|. This number can be com- Zhai et al.19 and Boustani39 reported that the global minimum pared with the aromatic hydrocarbons of the similar size such as 1 þ for B12 is the quasi-planar convex structure C3v A1 (Fig. 1). In the phenalenium (C13H9 ) TRE ¼ 0.410 | BB|, antracene this case, three central boron atoms cannot fit into the plane of (C14H10) TRE ¼ 0.475 | BB|, and phenanthrene (C14H10) TRE þ the nine-membered ring. As before, let us first flatten the C3v ¼ 0.576 | BB|. On the basis of the TRE value, B13 is much more 1 1 0 A1 structure into the planar D3h A1 structure for simplicity aromatic than polycyclic aromatic hydrocarbons of the similar and perform MO analysis for the planar structure. NBO analysis size. shows nine peripheral 2c2e BB bonds (ON ¼ 1.89–1.94 However, like in case of other large boron clusters, the - þ þ |e|). Unlike in B10,B11, and B11,inB12 NBO analysis does not bonding has not been discussed. Molecular orbitals for the B13 show 2c2e BB bonds between three central atoms. Instead, cation are plotted in Figure 13. NBO analysis shows ten 2c2e NBO analysis shows the presence of three ‘‘lone pairs’’ with the BB peripheral bonds (ON ¼ 1.89–1.93 |e|) and three ‘‘lone average occupation number about 1.1 |e| and with the total accu- pairs’’ with the average occupation number about 1.1 |e| and mulation of 3.2 |e| on each of three central atoms. Such unusual total accumulation of 3.2 |e| on each of three central atoms. As accumulation of electron density could be a deficiency of the before, let us assume that we have also three 2c2e BB employed NBO method and a hint that two electrons on every bonds between central atoms, which makes the total number of central boron atoms could be involved into the formation of 2c2e BB bonds thirteen. These 13 bonds take 26 out of 38 three 2c2e BB bonds. Let us assume that indeed, we have valence electrons. Molecular orbital picture (Fig. 13) also shows also three 2c2e BB bonds between central atoms and that the presence of three globally delocalized -MOs: HOMO-6 makes total number of 2c2e BB bonds twelve. These twelve (1b1), HOMO-2 (1a2), and HOMO-1 (2b1), which reveals -aro- bonds take 24 out of 36 valence electrons. Molecular orbital pic- maticity, as it was previously reported by Fowler and Ugalde5a ture (Fig. 12) also shows the presence of three globally delocal- and Aihara.6 We can assign six remaining electrons on HOMO- 00 0 00 ized -MOs: HOMO-5 (1a2 ), HOMO-1 and HOMO-1 (1e ), 4 (9a1), HOMO-3 (6b2), and HOMO (10a1) to globally delocal- which reveals -aromaticity, as it was initially reported by Zhai ized -bonding. Again, we would like to stress that such chemi- et al.19 We have also six electrons on globally delocalized - cal bonding description should be considered at this point as ten- 0 0 0 MOs: HOMO-2 (4a1), HOMO, and HOMO (5e ), which reveals tative. However, if we assume that our description is correct þ -aromaticity. With our previous assumptions, we can assign the then the presence of double aromaticity in B13 can explain high B12 cluster as being doubly (- and -) aromatic with nine first singlet vertical excitation energy (2.09 eV, at TD-B3LYP/6- 2c2e peripheral BB bonds and three 2c2e central BB 311 þ G*), highly negative NICS values (all in Table 1), and bonds. We would like to stress that this description is tentative the most importantly the anomalously high stability and low þ at this point. The presence of double aromaticity in B12 is sup- reactivity of B13 in comparison with other cationic boron clus- ported by high first singlet vertical excitation energy (2.57 eV, ters observed by Anderson and coworkers.25–31 at TD-B3LYP/6-311þG*) and highly negative NICS values (all in Table 1). B14 and B15 Clusters In an alternative explanation of -bonding in B12, one may 19 consider that central boron atoms donate their electrons to form Zhai et al. reported that the global minimum for B14 is the 1 islands of -aromaticity where each pair of delocalized -elec- quasi-planar structure C2v A1 (Fig. 1). We performed MO anal-

Journal of Computational Chemistry DOI 10.1002/jcc 264 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

1 0 Figure 12. Molecular orbitals of the D3h ( A1) structure of the B12 cluster.

1 ysis for the planar D2h ( Ag) structure (Fig. 14). NBO analysis orbital picture (Fig. 14) also shows the presence of four globally shows 10 peripheral 2c2e BB bonds (ON ¼ 1.88–1.95 |e|). delocalized -MOs: HOMO-10 (1b3u), HOMO-4 (1b1g), HOMO- It also shows the presence of three ‘‘lone pairs’’ with the average 3 (1b2g), and HOMO (2b3u), which reveals global -antiaroma- occupation number about 1.1 |e| and the total accumulation of ticity, as it was previously reported by Zhai et al.19 The global 2.9–3.3 |e| on each of four central atoms. Like before, let us -antiaromaticity results in the formation of small areas of - assume that there are four 2c2e BB bonds between central aromaticity (island aromaticity). The remaining six electrons atoms. That makes total number of 2c2e BB bonds 14. occupy globally delocalized -MOs: HOMO-6 (5ag), HOMO-2 These 14 bonds take 28 out of 42 valence electrons. Molecular (5b2u), and HOMO-1 (6ag), which reveals -aromaticity. Thus,

1 þ Figure 13. Molecular orbitals of the C2v ( A1) structure of the B13 cation.

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 265

1 Figure 14. Molecular orbitals of the D2h ( Ag) structure of the B14 cluster.

we can tentatively assign the B14 cluster as having conﬂicting occupation number about 1.1 |e| with the total accumulation of ((-aromatic and -antiaromatic) aromaticity with 10 2c2e pe- 2.9–3.3 |e| on each of four central atoms. Let us assume that we ripheral BB bonds and four 2c2e central BB bonds. have also four 2c2e BB bonds between central atoms. That 19 Zhai et al. reported that the global minimum for B15 is the makes total number of 2c2e BB bonds 15. These 15 bonds 1 quasi-planar structure C1 A (Fig. 1). We performed MO analysis take 30 out of 46 valence electrons. Molecular orbital picture 1 for the planar C2v ( A1) structure (Fig. 15). NBO analysis shows (Fig. 15) shows the presence of four globally delocalized -MOs: 11 peripheral 2c2e BB bonds (ON ¼ 1.88–1.95 |e|). It also HOMO-11 (1b1), HOMO-5 (1a2), HOMO-2 (2b1), and HOMO-1 shows the presence of the three ‘‘lone pairs’’ with the average (3b1), which reveals global -antiaromaticity. We have also eight

1 Figure 15. Molecular orbitals of the C2v ( A1) structure of the B15 anion.

Journal of Computational Chemistry DOI 10.1002/jcc 266 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

electrons on globally delocalized -MOs: HOMO (11a1), HOMO- are talking about global antiaromaticity of boron clusters with 3(8b2), HOMO-4 (10a1), and HOMO-7 (9a1), which reveals 4n -electrons. We, however, agree that a globally -antiaro- 2 global -antiaromaticity. Thus, we can tentatively consider the matic molecule such as B6 could have islands of -aromaticity. B15 cluster as being doubly (-and-) antiaromatic with eleven The island -aromaticity is responsible for the high TRE ¼ 2 2c2e peripheral BB bonds and four 2c2e central BB 0.549 | BB| energy in B6 . However, to explain low symmetry 2 bonds. Again the -and-antiaromaticity result in the formation (D2h instead of D6h)ofB6 , we must consider this cluster as of small areas of aromaticity (island aromaticity). being globally -antiaromatic. The same is true for larger -antiaromatic clusters. Similarly, in B5 , the high TRE ¼ 1.058 | BB| cannot explain the low symmetry (C2v instead of D5h)ofB5 as Overview well as the small first singlet vertical excitation energy. The - electrons must be included in chemical bonding analysis. When þ On the basis of chemical bonding analysis performed for B3 , the global -antiaromaticity in the B5 cluster is recognized, the 2 þ 2þ 2 þ 2 B3 ,B4,B4 ,B5 ,B5 ,B6 ,B6,B6 ,B7 ,B7 ,B8,B8 ,B9 ,B10, low symmetry C2v structure and small first singlet vertical exci- þ þ B11,B11,B12,B13,B14, and B15 clusters in this work, we pro- tation energy have rather simple explanation, which is not possi- pose the next chemical bonding model for planar or quasi-planar ble if only -aromaticity in this cluster is considered. boron clusters: 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 The number of 2c2e peripheral BB bonds in all consid- cycles. Peripheral 2c2e BB bonds are built up from s–p ered here planar or quasi-planar clusters is equal to the num- hybrid atomic orbitals and this enforces the planarity of the ber of peripheral edges. 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. There are globally delocalized -MOs, which make a cluster Otherwise, central atoms come out of the plane of the cycle. þ either globally -aromatic if it has 4n 2 -electrons or Initially, from B3 to B6 the cyclic (perfect or distorted globally antiaromatic if it has 4n -electrons for singlet depending on their aromatic or antiaromatic character) structures coupled electrons. For triplet coupled -electrons, the number correspond to global minima. In the B6 cluster, in addition to of electrons should satisfy the inverse 4n rule for aromaticity. cyclic structures, we observe the emergence of a new type of structure, pentagonal pyramid, which now corresponds to the There are globally delocalized -MOs, which make a cluster global minimum. The planar pentagon structure with the boron either globally -aromatic if it has 4n þ 2 -electrons or glob- atom located at the center of the pentagon is not a minimum ally antiaromatic if it has 4n -electrons for singlet coupled because the cavity inside of the pentagon is too small to favor- þ electrons. For triplet coupled -electrons, the number of elec- ably accommodate a boron atom at the center. The cyclic B7 3 0 trons should satisfy the inverse 4n rule for aromatic systems. D7h ( A2 ) structure is significantly higher (63.4 kcal/mol at B3LYP/6-311 þ G*) in energy than the global minimum C6v 1 0 ( A1 ) structure, because of unsupported dangling electron den- þ This bonding model works well for B3 –B9 clusters. Some sity at the center of the cycle. Thus, the cyclic structures are not þ boron clusters can be doubly (- and -) aromatic: B3 ,B4,B5 , favorable anymore beyond six boron atoms. Starting with B8, þ 2 B7 ,B7 ,B8,B8 , and B9 . Some clusters can be globally doubly the central cavity can favorably accommodate one boron atom at 2 antiaromatic, for example, B6 . Global antiaromaticity can be the center of the appropriate polygon leading to planar highly also described in terms of formation of areas of island aromatic- symmetric global minimum structures. Starting from B10 cluster, 2 ity. In the B6 clusters, the globally delocalized -and - elec- the structures with one boron atom at the center are not low trons can be localized over two areas composed of three boron energy isomers anymore, because it takes more than one boron atoms. Some clusters may have conflicting aromaticity, such as nucleus to make a good fit for the central cavity. B5 (which is -aromatic and -antiaromatic). For larger clusters, We believe that this approach in which we combine 2c2e in addition to peripheral 2c2e BB bonds globally delocal- bonds (or lone pairs) with global (or island) - and -aromatic- ized -MOs, and globally delocalized -MOs, we can introduce ity is a perspective way to characterize chemical bonding in þ one 2c2e central BB bond (B10,B11,B11), three 2c2e large boron clusters and could potentially become a useful tool þ central BB bonds (B12,B13) or four 2c2e central BB in addition to -delocalization and -aromaticity in case of other 2b,2c bonds (B14,B15). The presence of central 2c2e bonds was new planar clusters such as hyparenes, aromatic boron þ 80,81 confirmed up to certain degree only in B10,B11, and B11 clus- wheels with more than one carbon atom at the center, and 82 ters. In other large clusters it was postulated. There is an alterna- fen-shaped BnE2Si (E ¼ CH, BH, or Si, n ¼ 2–5) clusters. tive approach in which electrons located at central boron atoms are thought to participate in global delocalization resulting in formation of several areas with island -aromaticity. Acknowledgments We would also like to point out that there is no conflict between our assignment of planar boron clusters with 4n -elec- Computer time from the Center for High Performance Computing trons to antiaromatic and Aihara et al. assignment of the same at Utah State University is gratefully acknowledged. DYZ wishes molecules to aromatic. The disagreement is purely semantic. We to thank Utah State University for a Presidential Fellowship.

Journal of Computational Chemistry DOI 10.1002/jcc Comprehensive Analysis of Chemical Bonding in Boron Clusters 267

References 18. Zhai, H. J.; Wang. L. S.; Alexandrova, A. N.; Boldyrev, A. I. Angew Chem Int Ed 2003, 42, 6004. 1. (a) Boldyrev, A. I.; Wang, L. S. Chem Rev 2005, 106, 3716; (b) Li, 19. Zhai, H. J.; Kiran, B.; Li, J.; Wang, L. S. Nat Mater 2003, 2, 827. X.; Kuznetsov, A. E.; Zhang, H. F.; Boldyrev, A. I.; Wang, L. S. 20. Aihara, J.; Kanno, H.; Ishida, T. J Am Chem Soc 2005, 127, 13324. Science 2001, 291, 859; (c) Kuznetsov, A. E.; Birch, K. A.; Bol- 21. Alexandrova, A. N.; Zhai, H.-J.; Wang, L. S.; Boldyrev, A. I. Inorg dyrev, A. I.; Li, X.; Zhai, H.-J.; Wang, L. S. Science 2003, 300, Chem 2004, 43, 3588. 622; (d) Boldyrev, A. I.; Kuznetsov, A. E. Inorg Chem 2002, 41, 22. Alexandrova, A. N.; Boldyrev, A. I.; Zhai, H.-J.; Wang, L. S. J Chem 532; (e) Alexandrova, A. N.; Boldyrev, A. I. J Phys Chem A 2003, Phys 2005, 122, 054313. 107, 554. 23. Alexandrova, A. N.; Koyle, E.; Boldyrev, A. I. J Mol Mod 2006, 2. (a) Chen, Z.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; 12, 569. Schleyer, P. v. R. Chem Rev 2005, 106, 3842; (b) Exner, K.; 24. Zhai, H.-J.; Wang, L. S.; Zubarev, D. Yu.; Boldyrev, A. I. J Phys Schleyer, P. v. R. Science 2000, 1937, 290; (c) Wang, Z.-X.; Chem A 2006, 110, 1689. Schleyer, P. v. R. Science 2001, 292, 2465; (d) Tanaka, H.; Neuker- 25. Hanley, L.; Anderson, S. L. J Phys Chem 1987, 91, 5161. mans, S.; Janssens, E.; Silverans, R. E.; Lievens, P. J Am Chem Soc 26. Hanley, L.; Anderson, S. L. J Chem Phys 1988, 89, 2848. 2003, 125, 2863. 27. Hanley, L.; Whitten, J. L.; Anderson, S. L. J Phys Chem 1988, 92, 3. (a) Tsipis, C. A. Coord Chem Rev 2005, 249, 2740; (b) Tsipis, 5803. A. C.; Tsipis, C. A. J Am Chem Soc 2003, 125, 1136. 28. Hintz, P. A.; Ruatta, S. A.; Anderson, S. L. J Chem Phys 1990, 92, 4. (a) Jemmis, E. D.; Jayasree, E. G. Acc Chem Res 2003, 36, 816; (b) 292. Balakrishnarajan, M. M.; Hoffmann, R.; Pancharatna, P. D.; Jemmis, 29. Ruatta, S. A.; Hintz, P. A.; Anderson, S. L. J Chem Phys 1991, 94, E. D. Inorg Chem 2003, 42, 4650; (c) Jemmis, E. D.; Balakrishnara- 2833. jan, M. M.; Pancharatna, P. D. Chem Rev 2002, 102, 93. 30. Hintz, P. A.; Sowa, M. B.; Ruatta, S. A.; Anderson, S. L. J Chem 5. (a) Fowler, J. E.; Ugalde, J. M. J Phys Chem A 2000, 104, 397; (b) Phys 1991, 94, 6446. Mercero, J. M.; Ugalde, J. M. J Am Chem Soc 2004, 126, 3380. 31. Sowa-Resat, M. B.; Smolanoff, J.; Lapiki, A.; Anderson, S. L. J Chem 6. Aihara, J. J Phys Chem A 2001, 105, 5486. Phys 1997, 106, 9511. 7. (a) Fowler, P. W.; Havenith, R. W. A.; Steiner, E. Chem Phys Lett 32. Kawai, R.; Weare, J. H. J Chem Phys 1991, 95, 1151. 2001, 342, 85; (b) Havenith, R. W. A.; Fowler, P. W.; Steiner, E. 33. La Placa, S. J.; Roland, P. A.; Wynne, J. J Chem Phys Lett 1992, Chem—Eur J 2002, 8, 1068; (c) Fowler, P. W.; Havenith, R. W. A.; 190, 163. Steiner, E. Chem Phys Lett 2002, 359, 530; (d) Havenith, R. W. A.; 34. Kawai, R.; Weare, J. H. Chem Phys Lett 1992, 191, 311. van Lenthe, J. H. Chem Phys Lett 2004, 385, 198; (e) Havenith, R. 35. Martin, J. M. L.; Franc¸ois, J. P.; Gijbels, R. Chem Phys Lett 1992, W. A.; De Proft, F.; Fowler, P. W.; Geerling, P. Chem Phys Lett 189, 52. 2005, 407, 391. 36. Kato, H.; Yamashita, K.; Morokuma, K. Chem Phys Lett 1992, 190, 8. (a) Juselius, J.; Straka, M.; Sundholm, D.; J Phys Chem A 2001, 105, 361. 9939; (b) Lin, Y.-C.; Juselius, J.; Sundholm, D.; J Chem Phys 2005, 37. Roland, P. A.; Wynne, J. J. J Chem Phys 1993, 99, 8599. 122, 214308; (c) Lin, Y.-C.; Sundholm, D.; Juselius, J.; Cui, L.-F.; Li, 38. Kato, H.; Yamashita, K.; Morokuma, K. Bull Chem Soc Jpn 1993, X.; Zhai, H.-J.; Wang, L. S. J Phys Chem A 2006, 110, 4244. 66, 3358. 9. Zhan, C.-G.; Zheng, F.; Dixon, D. A. J Am Chem Soc 2002, 124, 39. Boustani, I. Int J Quantum Chem 1994, 52, 1081. 147. 40. Boustani, I. Chem Phys Lett 1995, 233, 273. 10. Santos, J. C.; Andres, J.; Aizman, A.; Fuentealba, P. J Chem Theory 41. Boustani, I. Chem Phys Lett 1995, 240, 135. Comput 2005, 1, 83. 42. Ricca, A.; Bauschlicher, C. W., Jr. Chem Phys 1996, 208, 233. 11. (a) Datta, A.; Pati, S. K.; J Phys Chem A 2004, 108, 9527; (b) 43. Boustani, I. Surf Sci 1997, 370, 355. Datta, A.; Pati, S. K. J Am Chem Soc 2005, 127, 3496; (c) Datta, 44. Boustani, I. Phys Rev B 1997, 53, 16426. A.; Pati, S. K.; J Chem Theory Comput 2005, 1, 824; (d) Datta, A.; 45. Nie, J.; Rao, B. K.; Jena, P. J Chem Phys 1997, 107, 132. John, N. S.; Kulkarni, G. U.; Pati, S. K. J Phys Chem A 2005, 109, 46. Gu, F. L.; Yang, X.; Tang, A.-C.; Jiao, H.; Schleyer, P. V. R. J Com- 11647; (e) Rehaman, A.; Datta, A.; Mallajosyulu, S. S.; Pati, S. K. J put Chem 1998, 19, 203. Comput Theory Comput 2006, 2, 30. 47. Reis, H.; Papadopoulos, M. G.; Boustani, I. Int J Quant Chem 2000, 12. (a) Havenith, R. W. A.; Fowler, P. W.; Steiner, E.; Shetty, S.; Kan- 78, 131. here, D.; Pal, S. Phys Chem Chem Phys 2004, 6, 285; (b) Shetty, S.; 48. Yang, C. L.; Zhu, Z. H. J Mol Struct (Theochem) 2001, 571, 225. Kar, R.; Kanhere, D. G.; Pal, S. J Phys Chem A 2006, 110, 252. 49. Cao, P.-L.; Zhao, W.; Li, B.; Song, X.-B.; Zhou, X.-Y. J Phys: Condens 13. (a) Hu, X. B.; Li, H. R.; Liang, W. C.; Han, S. J.; Chem Phys Lett Matter 2001, 13, 5065. 2004, 397, 180; (b) Hu, X.; Li, H.; Liang, W.; Han, S. Chem Phys 50. Li, Q. S.; Jin, H. W. J Phys Chem A 2002, 106, 7042. Lett 2005, 402, 539; (c) Hu, X. B.; Li, H. R.; Liang, W. C.; Han. S. 51. Ma, J.; Li, Z.; Fan, K.; Zhou, M. Chem Phys Lett 2003, 372, 708. J. New J Chem 2005, 29, 1295; (d) Wang, F.-F.; Li, Z.-R.; Wu, D.; 52. Jin, H. W.; Li, Q. S.; Phys Chem Chem Phys 2003, 5, 1110. Sun, X.-Y.; Chen, W.; Li, Y.; Sun, C.-C. Chem Phys Chem 2006, 7, 53. Li, Q. S.; Jin, Q. J Phys Chem A 2004, 108, 855. 1136. 54. Li, Q. S.; Jin, Q.; Luo, Q.; Tang, A. C.; Yu, J. K.; Zhang, H. X. Int 14. (a) Zhai, H. J.; Wang, L. S.; Alexandrova, A. N.; Boldyrev, A. I.; J Quant Chem 2003, 94, 269. Zakrzewski, V. G. J Phys Chem A 2003, 107, 9319; (b) Kuznetsov, 55. Wyss, M.; Riaplov, E.; Batalov, A.; Maier, J. P.; Weber, T.; Meyer, W.; A. E.; Boldyrev, A. I. Struct Chem 2002, 13, 141. Rosmus, P. J Chem Phys 2003, 119, 9703. 15. Zhai, H. J.; Wang, L. S.; Alexandrova, A. N.; Boldyrev, A. I. J 56. Cias, P.; Araki, M.; Denisov, A.; Maier, J. P. J Chem Phys 2004, Chem Phys 2002, 117, 7917. 121, 6776. 16. Alexandrova, A. N.; Boldyrev. A. I.; Zhai, H. J.; Wang, L. S.; 57. Batalov, A.; Fulara, J.; Shnitko, I.; Maier, J. P. Chem Phys Lett Steiner, E.; Fowler, P. W. J Phys Chem A 2003, 107, 1359. 2005, 404, 315. 17. Alexandrova, A. N.; Boldyrev, A. I.; Zhai, H. J.; Wang, L. S. J Phys 58. Gillery, C.; Linguerri, R.; Rosmus, P.; Maier, J. P. Z. Phys Chem Chem A 2004, 108, 3509. 2005, 219, 467.

Journal of Computational Chemistry DOI 10.1002/jcc 268 Zubarev and Boldyrev • Vol. 28, No. 1 • Journal of Computational Chemistry

59. Li, Q.-S.; Gong, L.-F. J Phys Chem A 2004, 108, 4322. Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, 60. Li, Q.-S.; Gong, L.-F.; Gao, Z.-M. Chem Phys Lett 2004, 390, 220. J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; 61. Li, Q.; Zhao, Y.; Xu, W.; Li, N. Int J Quant Chem 2005, 101, 219. Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Fores- 62. Lau, K. C.; Deshpande, M.; Pandey, R. Int J Quant Chem 2005, man, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cio- 102, 656. slowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; 63. Alexandrova, A. N.; Boldyrev, A. I.; Zhai, H.-J.; Wang, L. S. Coord Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Chem Rev 2006, 250, 2811. Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; John- 64. Reed, A. E.; Curtiss, L. A.; Weinhold, F. Chem Rev 1988, 88, 899. son, B. G.; Chen, W.; Wong, M. W.; Gonzales, C.; Pople, J. A. Gaus- 65. Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and sian 03 (revision A. 1); Gaussian, Inc.: Pittsburgh, PA, 2003. Molecules; Oxford University Press: Oxford, 1989. 74. Schaftenaar, G. MOLDEN3.4; CAOS/CAMM Center: The Netherlands, 66. Becke, A. D. J Chem Phys 1993, 98, 5648. 1998. 67. Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, 75. (a) Chandrasekhar, J.; Jemmis, E. D.; Schleyer, P. V. R. Tetrahedron M. R.; Singh, D. J.; Fiolhais, C. Phys Rev B 1992, 46, 6671. Lett 1979, 39, 3707; (b) Schleyer, P. v. R.; Jiao, H.; Glukhovtsev, 68. Clark, T.; Chandrasekhar, J.; Spitznagel, G.W.; Schleyer, P. v. R. J Comput M. N.; Chandrasekhar, J.; Kraka, E. J Am Chem Soc 1994, 116, Chem 1983, 4, 294. 10129. 69. Frisch, M. J.; Pople, J. A.; Binkley, J. S. J Chem Phys 1984, 80, 76. Martin-Santamaria, S.; Rzepa, H. S. Chem Commun 2000, 1503. 3265. 77. Pra¨sang, C.; Hofmann, M.; Geiseler, G.; Massa, W.; Berndt, A. 70. Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H.; Hommes, Angew Chem Int Ed 2002, 41, 1526; (b) Pra¨sang, C.; Mlodzianow- N. J. R. v. E. J Am Chem Soc 1996, 118, 6317. ska, A.; Sahin, Y.; Hofmann, M.; Geiseler, G.; Massa, W.; Berndt, 71. Bauernshmitt, R.; Alrichs, R. Chem Phys Lett 1996, 256, 454. A. Angew Chem Int Ed 2002, 41, 3380; (c) Mesbah, W.; Pra¨sang, 72. Casida, M. E.; Jamorski, C.; Casida, K. C.; Salahub, D. R. J Chem C.; Hofmann, M.; Geiseler, G.; Massa, W.; Berndt, A. Angew Chem Phys 1998, 108, 4439. Int Ed 2003, 42, 1717; (e) Hofmann, M.; Berndt, A. Heteroatom 73. Frisch, M. J.; Trucks, G. M.; Schlegel, H. B.; Scuseria, G. E.; Robb, Chem 2006, 17, 224. M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, 78. Minkin, V. I.; Minyaev, R. M. Mendeleev Comm 2004, 14, 43. K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Bar- 79. Minyaev, R. M.; Gribanova, T. N.; Starikov, A. G.; Minkin, V. I. one, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, Mendeleev Comm 2001, 11, 213. G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fakuda, R.; 80. Erhardt, S.; Frenking, G.; Chen, Z.; Schleyer, P. v. R. Angew Chem Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Int Ed 2005, 44, 1078. Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, 81. Wu, Y.-B.; Yuan, C.-X.; Yang, P. J Mol Struct (Theochem) (in press). J. B.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; 82. Li, S.-D.; Miao, C.-C.; Guo, J.-C.; Ren, G.-M. J Am Chem Soc Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; 2004, 126, 16227.

Journal of Computational Chemistry DOI 10.1002/jcc