**tight - binding**

tight-binding

(Parte **1** de 5)

Thermochemistry and Electronic Structure of Small Boron Clusters (Bn, 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 UniVersiteit LeuVen, B-3001 LeuVen, Belgium, and Department of Chemistry, The UniVersity of Alabama, Shelby Hall, Box 870336, Tuscaloosa, Alabama 35847-0336

ReceiVed: September 4, 2009; ReVised Manuscript ReceiVed: October 27, 2009

Thermochemical parameters of a set of small-sized neutral (Bn) and anionic (Bn-) boron clusters, with n ) 5-13, were determined using coupled-cluster theory CCSD(T) calculations with the aug-c-pVnZ( n ) D, T, and Q) basis sets extrapolated to the complete basis set limit (CBS) plus addition corrections and/or G3B3 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

between the calculated adiabatic electron affinity and the adiabatic detachment energy for B6 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 3Au, C2h and 1Ag, D2h excited states of B6, which have geometries similar to the 1Ag, D2h state of B6-, 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 characterizedby a number of experimentaltechniques including mass spectrometry.1 Experimental and theoretical studies on the electronic structure,2-5 chemical bonding,6-8 and spectroscopic properties9-15 of bare boron clusters as well as doped boron clusters16-20 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-phaseclusters to determinetheir fundamentalpropertiesand growth patterns.

There are a substantial number of computational studies of boron clusters, and we describe some here. Boustani24 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 experimentalstudies.Wang, Boldyrev,and co-workers25-28 used photoelectronspectroscopy(PES) in combinationwith quantum chemical calculations to investigate the structural and electronic properties of a series of boron cluster anions Bn- (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 suggestedthat the π-electrondelocalization in Bn follows the Huckel model for aromaticity and antiaromaticity as found in cyclic hydrocarbons. In contrast, Aihara and co-workers29 analyzed the structure of boron clusters in terms of topological resonance (TRE) arguments and concluded that their aromaticityis not relatedto the total number of π-electrons, and the Huckel rule can therefore not be applied. Subsequently, Zubarev and Boldyrev6 reexamined the chemical bonding of

Bn 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 charactersfollowingthe Huckel 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 Bn, boron oxide BnOm clusters, and their anions, with n e 4, using high accuracyquantum chemicalmethods.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

Bn and their anions Bn-, 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.

* Corresponding author. E-mail: dadixon@bama.ua.edu (D.A.D.); minh.nguyen@chem.kuleuven.be (M.T.N.).

† Katholieke Universiteit Leuven. ‡ The University of Alabama.

J. Phys. Chem. A X, x, 0 A

10.1021/jp9085848 C: $40.75 X American Chemical Society

Computational Methods

All quantum chemical calculations were carried out using the Gaussian 0331 and Molpro 200632 suites of programs.Enthalpies of formation of the Bn and Bn- clusters were evaluated from the corresponding total atomization energies (TAE).3 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 approach34 were used. The G3B3 (G3/ B3LYP) approachis a compositetechniquein 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 computationally less demanding than a CBS counterpart, we used only the

G3B3 approach for the larger Bn 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-c-pVDZ and aug-c-pVTZbasis sets. The fully unrestrictedformalism(UHF, UMP2) was used for open-shell system calculations done with Gaussian 03. The valence electronic energies were computed using coupled-cluster CCSD(T) theory35 extrapolated to the complete basis set limit (CBS) using the correlation-consistent basis sets.36 The single-pointelectronic energies were calculated by using the restricted coupled-cluster R/UCCSD(T) formalism37-39 in conjunction with the correlation-consistent aug-cpVnZ( n ) D, T, and Q) basis sets at the (U)MP2/aug-c-pVDZ or (U)MP2/aug-c-pVTZoptimizedgeometrieswith the Molpro program. For simplicity, the basis sets are labeled as aVnZ. The CCSD(T) energies were extrapolated to the CBS limit energies using expression 1:40

where x ) 2, 3, and 4 for the aVnZ basis D, T, and Q, respectively (total CCSD(T) electronic energies as a function of basis set are given in Table S1 of the SupportingInformation). 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 (∆ECV) were obtained at the CCSD(T)/c-pwCVTZ level of theory with

Molpro.41 Douglas-Kroll-Hess (DKH) scalar relativistic cor- rections(∆EDKH-SR), which accountfor changesin 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.42-4 ∆EDKH-SR is defined as the difference in the atomization energy between the results obtained from basis sets recontracted for DKH calculations43 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)/c-pVTZ and the CCSD(T)/c-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 Moore45 is used. The total atomization energy (∑D0 or TAE) of a compound is given by eq 2:

By combining our computed ∑D0 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 ∆Hf° values at0Kf or the molecules in the gas phase. In this work, we used the value of ∆Hf°(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 evaluatethe electronaffinitiesand otherenergeticquantities.30,48

The analysis of chemical bonding phenomenon was performed using the electron localization function (ELF)49 supplemented by analyses of topological bifurcation50 and canonical MOs. The ELF is a local measureof the Pauli repulsionbetween electrons owing to the exclusion principle in 3D space. The definition of ELF, η(r), is given by following eq 3:

where DP and Dh 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 correspondingbifurcation value in an antiaromatic system is very low. The density for the ELF analysisof the lowest-energystatein each spin manifold was obtainedat the (U)B3LYP/6-311+G(d) level.The totalELF 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 Bn and Bn - 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 ∑D0 values are summarized in Table S3 of the Supporting Information. The optimized geometries of the lowest-lyingisomersusing 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

formation of the clusters derived using the ∑D0 obtained from bothCBS and G3B3methodsare givenin Table2. The adiabatic electron affinities (EAs) of the neutrals Bn are given in Table

3, and the verticaldetachmentenergies(VDEs)of the Bn- anions computed using the single-point G3B3 and CBS methods are

B J. Phys. Chem. A, Vol. x, No. x, X Tai et al.

given in Table 4. The dependence of the singlet-triplet gaps with respect to the basis set is given in Table 5.

The heats of formation of B2,B 3, and B4 and their anions were determined in our previous work at the CBS level and compared to previous experimentaland theoretical results.30 For purposes of comparison, both the CBS and the G3B3 values for these clusters are also listed in Table 2. We note that the

G3B3 value of 206.0 kcal/mol for ∆Hf at 298Ko fB 2 (3Σg-)i s close to the CBS result of 207.4 kcal/mol, and both are at the high end of the range of the experimental value of 198.3 ( 8.0

respective CBS values of 211.7 and 226.4 kcal/mol.30 For the larger species, the differences between the G3B3 and CBS values vary from 1 to 3 kcal/mol, which reflects the degree of accuracy of the G3B3 results. Different aspects of the geometric and electronic structures of the clusters have been analyzed in detail in numerous previous studies, and thus do not warrant additional comments. We focus the discussion only on the new thermochemical results and analyses of the bonding.

B5. The B5 and B5- clusters have been extensively studied both experimentally and theoretically. Kato et al.2 reported that

B5 has either a C2V (2A2)o ra C2 (2B) ground state. Boustani,4 Li and Jin,56 Boldyrev and co-workers,6,57 and Ricca et al.58 reported that the most stable structure of B5 is a planar C2V 2B2 state. In contrast, Ray et al.58 as well as Niu et al.59 suggested a trigonal bipyramid D3h as the lowest energy structure. Our calculations concur with the prediction that the B5 radical possesses a 2B2 ground state I (Figure 1). The high spin state still has a C2V structure (4B2) and is the second lowest-lying isomer.The quartet-doubletgap is ∼43 kcal/mol,whichis close to the value of 40 kcal/mol estimated by Alexandrova et al.25

Figure 1. Shape of the lowest-energy structures of Bn and Bn- (n ) 5-8) and their ELF localization domains.

Small Boron Clusters (Bn, n ) 5-13) J. Phys. Chem. A, Vol. x, No. x, X C

from GEGA/B3LYP/3-21G calculations. At the B3LYP/

6-31G(d) level, a search of the 2A1 state of B5 led to a nonplanar structure (C2V), which is ∼58 kcal/mol higher in energy than I.

A ∑D0 of 407 kcal/mol for B5 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 ∑D0 estimate differs significantly from the present values of 420.6 (G3B3) and 423.8 (CBS) kcal/mol.

Attaching one electron to the neutral B5 (2B2) does not affect the geometry much, and the closed-shell 1A1 structure is found to be the most stable structure for the B5- anion I (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 C2 3B structure, and the 3B-1A1 gap of B5- 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) calculations.25

The adiabatic electron affinity (EA) of B5 calculated from the heats of formation at 0K of I (2B2) and I (1A1) is 2.48 (G3B3) and 2.29 eV (CBS). As compared to the experimental value of 2.3 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 larger57 (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 C2V structure of the B5 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 electrons into these vacant orbitals to obtain B5 and B5- , respectively, is subject in their low spin states to a Jahn-Teller effect whose stabilization leads to a lower symmetry C2V structure. The high spin triplet state of B5- arising from the

(Parte **1** de 5)