Abstract
In this work, we report electronic structure calculations aimed at computing the linear optical absorption spectrum and static dipole polarizablity of a newly proposed boronbased planar aromatic compound borozene (B_{12}H_{6}). For the purpose, we use the semiempirical INDO model Hamiltonian, accompanied by largescale correlation calculations using the multireference singlesdoubles configurationinteraction (MRSDCI) approach. We present detailed predictions about the energetics, polarization properties, and the nature of manyparticle states contributing to various peaks in the linear absorption spectrum. Our results can be used to characterize this material in future optical absorption experiments. We also argue that one can deduce the aromaticity of the cluster from the optical absorption and static polarizability results.
Keywords:
Borozene; Optical properties; Aromaticity; Semiempirical modelsIntroduction
Next to carbonbased chemistry, boron chemistry is surely one of the most active areas of research [1]. Boron exhibits a variety of polymorphisms which involve threecenter twoelectron (3c–2e) bonds [2], planar aromatic structures [3], nanotubes [4], nanoribbons [5], and fullerenelike cages [6]. From the point of view of nanotechnology, aromatic boron materials are of particular interest, because delocalized electron clouds in these materials will lead to large linear and nonlinear optical susceptibilities, just like corresponding carbonbased structures. In order to achieve aromaticity, one needs to have a planar structure giving rise to πorbitals. Indeed, aromaticity in boronbased clusters has been the subject of many recent investigations. Zhai et al. [7] in a recent joint theoretical and experiment study probed the issue of planarity and aromaticity of boron clusters. In another theoryexperiment study, Zhai et al. [8] reported the discovery of planar boron clusters B_{8} and B_{9} clusters with “molecular wheel”like structures and exhibiting both σ and π aromaticity. Aihara and coworkers [9] presented a theoretical investigation of aromaticity into several planar and quasiplanar boron clusters using the concept of “topological resonance energy”. Johansson [10] demonstrated theoretically the existence of strong magnetically induced ring currents in B_{20} and other toroidal clusters of boron, as an evidence of underlying aromaticity. Rincon et al. [11] theoretically demonstrated the σ aromaticity in several planar boron clusters performing an analysis of the electronlocalization function (ELF). Similarly, Wu et al. [12] using an ab initio approach investigated the aromaticity of planar boroncarbon complex C_{6}B_{12}^{−2}. Another possible approach to achieve planarity, and, therefore, aromaticity in boron clusters is by hydrogenation of boron clusters. For example Alexandrova et al. [13] have demonstrated by means of theoretical calculations, that quasiplanar B_{7}^{−} cluster, upon hydrogenation, acquires a planar structure for B_{7}H_{2}^{−}. In a recent theoretical work, Szwacki et al. [14] proposed a novel planar aromatic cluster which can be obtained by hydrogenation of quasiplanar B_{12} cluster. They argued that the cluster is aromatic by performing a detailed analysis of its molecular orbitals and by examining magnetic properties such as nucleusindependent chemical shift (NICS), and anisotropy of magnetic susceptibility (AMS). Having thus demonstrated the similarities between the aromaticities of this cluster and benzene (including the number of πelectrons), they called it the boron analogue of benzene and named it borozene [14]. Encouraged by the work of Szwacki et al. [14], Forte et al. [15] performed ab initio calculations to predict larger aromatic compounds which can be constructed using borozene as formula unit. It is a wellknown fact that the carbonbased aromatic materials including molecules such as benzene, napthalene, anthracene, and longer πconjugated polymers exhibit intense linear and nonlinear optical response [16]. Therefore, it is of considerable interest to explore similar properties of aromatic materials consisting of elements other than carbon. With this aim in mind, here we present a systematic theoretical study, based upon a largescale configurationinteraction (CI) methodology, of linear optical response of borozene. We believe our results will be useful in optical characterization experiments on this material, as also in exploring its possible application in nanooptics. Furthermore, we also perform calculations of static polarizability of this cluster, with the aim of exploring the signatures of aromaticity in its dielectric response. Aromaticiy is an intuitive concept which essentially implies electron delocalization. Therefore, the consequences of this delocalization of electrons will be manifest in various properties of materials, including their magnetic as well as dielectric response. With the electrons delocalized along the plane of the molecule for conjugated systems as benzene, one would expect its “inplane” dielectric response to be significantly more than its “perpendicular” response. Thus, this anisotropy in the static polarizability tensor α_{ij} can also be viewed as a signature of the aromaticity, in such systems. Indeed, this anisotropy in α has been verified theoretically by other authors in the past calculations on conjugated molecules [1719] and has been used to formulate an “aromaticity scale” for such systems [17].
Remainder of the paper is organized as follows. In “Introduction” section, we briefly describe the theoretical methodology employed for the present calculations and also discuss the geometry of the molecule. This is followed by the presentation and discussion of our results in “Results and Discussion”. Finally, in “Conclusions” section, we present our conclusions.
Theoretical Methodology
The basic structure of borozene is presented in Fig. 1, and it corresponds to point symmetry group D_{3h}[14]. Before proceeding with our calculations, for the purpose of verification, we decided to perform the geometry optimization on our own using the B3LYPbased hybrid DFT approach employing the 6311++g(d) basis set as implemented in GAUSSIAN03 program [20]. The optimized structure thus obtained belonged to the D_{3h} point group symmetry with uniform value of B–H bond length 1.18 Å , and four distinct values of B–B bond lengths 1.63, 1.66, 1.81, and 1.86 Å. We, however, are interested in exploring an even more symmetric structure of borozene, so as to make the comparison with benzene more transparent. Therefore, we separately optimized a highly symmetric structure of borozene (Sborozene, henceforth) with identical B–B bond lengths, in addition to the B–H ones. For this purpose, as well as for the linear absorption spectrum calculations, we adopted an INDO model Hamiltonian [21] based approach implemented recently in a computer program developed by us [22]. INDO model is an effective valenceelectron approach, employing a Slatertype minimal basis set, and some semiempirical parameters [21]. The geometry optimization using our computer program, performed at the HartreeFock (HF) level (INDOHF, henceforth), yielded bond lengths 1.65 Å for the B–B bond and 1.18 Å for the B–H bond. These values agree perfectly with 1.18 Å for the B–H bond length reported both by Szwacki et al. [14] and Forte et al. [15], and the average value of B–B bond length 1.649 Å reported by Forte et al. [15]. The total energy of the geometryoptimized Sborozene was higher than that of the borozene with distinct B–B bond lengths by ≈0.8 eV at the INDOHF level.
Figure 1. (Color online) Structure of Sborozene considered in this work, assumed to lying in the xyplane. Optimized bond lengths at INDOHF level are B–B = 1.65 Å and B–H = 1.18 Å. Yellow dots indicate boron atoms and violet ones hydrogen atoms
Next, largescale correlation calculations were performed on the ground and the excited states of Sborozene, using the multireference singlesdoubles configurationinteraction (MRSDCI) approach as implemented in the MELD package [23] but employing the one and twoelectron matrix elements of the INDO Hamiltonian supplied by our program [22]. Thus, these correlated calculations were entirely within the INDO model and will be called INDOMRSDCI calculations, henceforth. During these calculations, point symmetry group C_{2v} was utilized, as against the D_{3h} symmetry group, because the MELD [23] package is restricted to D_{2h} and its subgroups. Therefore, we classify the manyelectron states of the system in terms of the irreducible representations (irreps) of C_{2v} group. The optical absorption spectra were calculated under the electricdipole approximation employing the Lorenzian line shape. For both the ground and the excited states, these calculations were performed in an iterative manner, until the optical absorption spectra computed from them converged. We have extensively used this approach in our earlier calculations on conjugated polymers [2427] as well as on B_{12} icosahedral and quasiplanar clusters reported recently [28].
Results and Discussion
Symmetry group C_{2v} consists of four irreps labeled a_{1},a_{2},b_{1}, and b_{2}. Assuming that the Cartesian xyplane lies in the plane of the molecule (cf. Fig. 1), the symmetry adapted electronic structure of the HF ground state obtained from our code is 1a_{1}^{2}2a_{1}^{2}1b_{1}^{2}3a_{1}^{2}2b_{1}^{2}4a_{1}^{2}1b_{2}^{2}3b_{1}^{2}5a_{1}^{2}4b_{1}^{2}6a_{1}^{2}7a_{1}^{2}8a_{1}^{2}5b_{1}^{2}2b_{2}^{2}1a_{2}^{2}9a_{1}^{2}6b_{1}^{2}7b_{1}^{2}10a_{1}^{2}8b_{1}^{2}, leading to ^{1}A_{1} manyparticle configuration. Some of the lowest unoccupied orbitals in the ascending order of energy are 3b_{2}, 2a_{2}, 4b_{2}, 11a_{1}, 9b_{1} etc. Of all the orbitals, the ones belonging to a_{2} and b_{2} irreps are π orbitals and a_{1} and b_{1} are of σ type. Thus, we note that the highest occupied molecular orbital (HOMO) is a σ orbital, while the lowest unoccupied molecular orbital (LUMO) is a π orbital. For the sake of visualization, some of the orbitals are given in Fig. 2.
Figure 2. (Color online) Molecular orbitals (iso plots) of Sborozene from HOMO−1 to LUMO+1, obtained from the INDOHF calculations
While our LUMO orbital matches perfectly with that reported by Szwacki et al. [14], however, we have a disagreement on the nature of the HOMO orbital which was reported to be of πtype by them [14], but we obtain it to be of σtype within the INDO model. In order to rule out the possibility that this disagreement could due to the use of the INDO model, or due to the Sborozene geometry, we studied the nature of the HOMO obtained using the 6311++g(d) basis set both at RHF and DFTB3LYP levels of ab initio theory [20] at various geometries. In all such DFT calculations, it turned out to be a σtype orbital, but for the RHF calculations, the nature of HOMO was highly geometry sensitive with some geometries yielding it to be σtype, while others of πtype. This, in our opinion, is a significant difference between the electronic structures of benzene (whose HOMO and LUMO both are π type) and borozene. Thus, borozene, in our opinion is an example of both σ and π conjugation because, close to the Fermi level, occupied orbitals are mainly of σ type, while the unoccupied ones (LUMOLUMO + 1, LUMO + 2) are of π type. Before presenting the results of our calculations of linear optical absorption spectrum of Sborozene, we would like to discuss the influence of geometry on the optical properties of borozene, considering the fact that our calculations were performed on a highly symmetric conformer Sborozene. For the purpose, we performed the optical absorption calculations at the singlesCI (SCI) calculations both on the lowest energy structure of borozene with unequal B–B bond lengths and Sborozene. We found that qualitatively, at the SCI level, spectra were very similar for both the geometries, except that for Sborozene it was slightly redshifted compared to borozene. The first peak for borozene in this SCI calculation was obtained at 2.6 eV in perfect agreement with excitation energy of the first excited reported by Szwacki et al. [14], while that for Sborozene was located at 2.4 eV.
Next, we present and discuss the results of our INDOMRSDCIbased optical absorption calculations on Sborozene. Comparisons are also made with the absorption spectra of quasiplanar B_{12} bare cluster, and also with that of benzene. As the ground state of the system is ^{1}A_{1}, as per electricdipole selection rules, it can make linear optical transitions to excited states belonging to symmetry manifolds ^{1}A_{1}, ^{1}B_{1}, and ^{1}B_{2} through photons polarized along the yaxis, xaxis, and zaxis, respectively. Thus, transitions to ^{1}A_{1} and ^{1}B_{1} are through photons polarized in the plane of the molecule (xyplane in the present case), while those to ^{1}B_{2} will be through perpendicularly polarized photons. The INDOMRSDCI calculations presented here involved construction and diagonalization of very large CI matrices running into millions of configurations, because simultaneously several states were targeted. For example, for the ^{1}A_{1} symmetry manifold, the total number of configurations was more than 2.1 millions, while those for ^{1}B_{1} and ^{1}B_{2} manifolds were in excess of 2.5, and 3 millions, respectively. Because of the largescale nature of these calculations, we believe that the results account for electron correlation effects properly.
In Fig. 3, we present the combined optical absorption spectrum of Sborozene, obtained from our INDOMRSDCI calculations. As is obvious from the figures that the first absorption feature is a small peak located at 2.25 eV, corresponding to a zpolarized transition into a B_{2} state. The manyparticle wave function of this peak consists mainly of two single excitations H→L + 1 and H − 1→L(H≡HOMO;L≡LUMO), both of which are of σ→π^{*} type. After this feature at 2.25 eV, the next peak follows after a large gap of ≈5 eV and is due to two closely placed states at 7.38 eV (B_{1}type) and 7.55 eV (A_{1}type). Thus, this intense peak can be reached through photons which are polarized in the molecular plane, but not through the zpolarized ones. The manyparticle wave functions of both these states are dominated by singly excited configurations, with the main configurations being H→L + 3〉 for the B_{1} state, and H − 1→L + 3〉 for the A_{1} state. Thus, both these states correspond to σ→σ^{*} transitions.
Figure 3. Linear optical absorption spectrum of Sborozene, computed using the INDOMRSDCI approach. Important peaks, along with their polarization characteristics, are labeled. A line width of 0.1 eV was used to compute the spectrum
Peak III of the spectrum, which is a weaker feature located at 8.44 eV, corresponds to a B_{2} state and can be reached through a zpolarized photon. Manyparticle wave function of this state is a mixture of several singly excited configurations which include H − 4→L〉, H − 3→L + 1〉, and H→L + 8〉, all of which are σ→π^{*} type excitations. Finally, feature IV near 9.1 eV draws its oscillator strength from the transition of the ground state to three closely spaced excited states of symmetries A_{1},B_{1}, and B_{2} located at 9.03 9.01, and 9.19 eV, respectively. The peak has mixed polarization features, out of which x and ypolarized transitions are more intense when compared to the zpolarized one. The manyparticle wave function of the A_{1} state is dominated by configurations H − 1→L + 5〉 and H→L + 4〉, while that of B_{1} state consists mainly of H − 1→L + 4〉 and H→L + 5〉, all of which are of σ→σ^{*} type. The dominant configurations in the wave function of the B_{2} state are H→L + 8〉 and H − 1→L + 7〉, which are of σ→π^{*} type. The absorption spectrum of Sborozene consists of several very intense features beyond 10 eV as well, but we are not discussing them here, given the fact that the ionization potential of this material is estimated close to 9 eV [14].
As Szwacki et al. [14] called borozene the boron analogue of benzene, it is interesting to compare the optical absorption spectra of the two materials. It is a wellknown fact that the first dipole allowed transition in benzene is from the ^{1}A_{1g} ground state to the ^{1}E_{1u} excited state with the peak around 6.99 eV, through photons polarized in the plane of the molecule (xyplane in our case) [29]. This compares very well with peak II (cf. Fig. 3) of Sborozene which is also caused by photons polarized in the the xyplane and is located around 7.5 eV. The only difference being that for the the ^{1}E_{1u} state in benzene corresponds to a π→π^{*} transition, while the peak II of borozene corresponds to σ→σ^{*} transitions. This supports the hypothesis that borozene is both π and σ conjugated [14]. The difference between the optics of the two materials is the presence of low energy feature (peak I) in Sborozene corresponding to a zpolarized transition, whose counterpart in benzene does not exist.
Recently, we studied the linear optical absorption in quasiplaner B_{12} cluster which has a convex shape with symmetry group C_{3v}[28] and can be seen as a precursor of borozene [14]. Therefore, a comparison of the optical properties of quasiplanar B_{12} and borozene can help us understand the influence of hydrogen passivation on the optical properties of B_{12}. In Fig. 4, the linear optical absorption spectra of the two clusters are presented, and it is obvious that the first few peaks of Sborozene occur at lower energies when compared to the quasiplanar B_{12}. Moreover, the H→L transition is symmetry forbidden in Sborozene, which is not the case for quasiplanar B_{12}, and indeed contributes to the wave function of the first peak in its spectrum. Thus, the main contribution of hydrogen passivation on the optical properties is that it makes the structure completely planar (as against quasiplanar) leading to conjugation and higher symmetry. Lower energy gap of Sborozene is most certainly a consequence of conjugated nature of electrons in the system.
Figure 4. A comparison of the linear optical absorption spectra between Sborozene (solid line) quasiplaner B_{12}(broken line). The spectrum of Sborozene is significantly redshifted when compared to that of B_{12}. A line width of 0.1 eV was used to compute the spectrum
In order to explore the aromaticity of borozene, Szwacki et al. [14] computed and discussed its NICS plots and anisotropy of magnetic susceptibility (AMS). With a similar aim, we computed the three diagonal Cartesian components of the static polarizability tensor, α_{xx}, α_{yy}, and α_{zz} of borozene both at the INDOHF and INDOSDCI levels. For the purpose, we adopted a finitefield approach and computed the second derivative of the total energy of the system numerically, with respect to the three Cartesian components of the external electric field, whose values were taken to be 0.005 atomic units. The numerical values obtained in our calculations correspond to valenceelectron contribution to the polarizability because of the nature of the INDO Hamiltonian. However, our aim here is not to obtain exact numerical values of the polarizability tensor, rather to compare the values of its various components. It is obvious from the results of our calculations presented in Table 1 that: (a) inclusion of correlation effects does not change the values of α_{ii} very much, and (b) the values of inplane components (α_{xx} and α_{yy}) are significantly larger than the perpendicular component α_{zz}. Ab initio calculations performed on conjugated molecules like benzene, [17,19] anthracene [18], and ethane [19] exhibit similar anisotropy in the components of the polarizability tensor. And, indeed, based upon the magnitude of this “inplane” vs. “perpendicular” anisotropy, Lazzeretti and Tossell [17] argued for an “aromaticity scale” to characterize πconjugated systems. Using the same logic, the same anisotropy found in our calculations clearly implies the aromaticity in borozene, but as a consequence of both π and σ conjugation in the plane of the molecule. This fact is also obvious from the optical absorption spectrum of the material in which intensities of the zpolarized peaks are much smaller than those of x and ypolarized ones.
Table 1. Values of the components of static dipole polarizability tensor of borozene computed using the INDOHF and INDOSDCI methods (in atomic units)
Conclusions
In conclusion, a correlated study on electronic structure and optical properties of a recently proposed boronhydrogen cluster, borozene, was presented using the semiempirical MRSDCIINDO method. Our calculations of the linear optical absorption spectrum can be used to characterize this substance in future optical absorption experiments. We also calculated the diagonal components of the static polarizability tensor α_{ij} of the cluster and found significant larger values for the components along the plane of the molecule, when compared to the one perpendicular to it. This anisotropy in the components of the polarizability tensor implies that the electrons are quite delocalized along the plane of the molecule leading to a considerably larger response to an electric field directed along the plane, when compared to that perpendicular to the plane. Therefore, we argue that this anisotropy in the static dielectric polarizability of the cluster is an evidence of its aromaticity, [17] caused both by σ and π conjugation along the plane of the molecule.
Acknowledgments
We gratefully acknowledge useful private communications with Dr. C. J. Tymczak in which he clarified the optimized geometry obtained in their calculations on borozene [14].
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References

Davidson MG, Hughes AK, Marder TB, Wade K: Contemporary Boron Chemistry. Edited by . Royal Society of Chemistry, Cambridge (UK); 1999.

Alexandrova AN, Boldyrev AI, HuaJin Zhai, Wang LS:
Coord.Chem. Rev.. 2006, 250:2811.
COI number [1:CAS:528:DC%2BD28XoslSjsro%3D]
Publisher Full Text 
Ciuparu D, Klie FR, Zhu Y, Pfefferle LJ:
Phys. Chem. B. 2004, 108:3967.
COI number [1:CAS:528:DC%2BD2cXhvFWltbs%3D]
Publisher Full Text 
Xu TT, Zheng J, Wu N, Nicholls WA, Roth RJ, Dikin AD, Ruoff RS:
Nano Lett.. 2004, 4:963.
COI number [1:CAS:528:DC%2BD2cXis1Citrw%3D]; Bibcode number [2004NanoL...4..963X]
Publisher Full Text 
Szwacki NG, Sadrzadeh A, Yakobson BI:
Phys. Rev. Lett.. 2007, 98:166804.
Bibcode number [2007PhRvL..98p6804G]
PubMed Abstract  Publisher Full Text 
Zhai HJ, Kiran B, Li J, Wang LS:
Nat. Mat.. 2003, 2:827.
COI number [1:CAS:528:DC%2BD3sXptlemtrY%3D]
Publisher Full Text 
Zhai HJ, Alexandrova AN, Birch KA, Boldyrev AI, Wang LS:
Angew. Chem. Int. Ed.. 2003, 42:6004.
COI number [1:CAS:528:DC%2BD2cXltlWi]
Publisher Full Text 
Aihara JI, Kanno H, Ishida T:
J. Am. Chem. Soc.. 2005, 127:13324.
COI number [1:CAS:528:DC%2BD2MXpsVGhtrg%3D]
PubMed Abstract  Publisher Full Text 
J. Phys. Chem. C. 2009, 113:524.
COI number [1:CAS:528:DC%2BD1cXhsFais77N]
Publisher Full Text 
Rincon L, Almeida R, Alverellos JE, GarciaAldea D, Hasmy A, Gonzalez C:
Dalton Trans.. 2009, 17:3328. PubMed Abstract  Publisher Full Text

J. Mol. Struct. Theochem. 2006, 765:35.
COI number [1:CAS:528:DC%2BD28XlvFGgs74%3D]
Publisher Full Text 
Alexandrova AN, Koyle E, Boldyrev AI:
J. Mol. Model.. 2006, 12:569.
COI number [1:CAS:528:DC%2BD28XosVKjt7w%3D]
PubMed Abstract  Publisher Full Text 
Szwacki NG, Weber V, Tymczak CJ:
Nano. Res. Lett.. 2009, 4:1085. Publisher Full Text

Forte G, Magna ALa, Deretzis I, Pucci Bcondmat arXiv:0908.1153

Barford W: Electronic and Optical Properties of Conjugated Polymers. Clarendon Press, Oxford; 2005.

J. Mol. Struct. (Theochem). 1991, 236:403. Publisher Full Text

Chem. Phys. Lett.. 1992, 188:604.
COI number [1:CAS:528:DyaK38XhsF2gsr4%3D]; Bibcode number [1992CPL...188..604P]
Publisher Full Text 
Hinchliffe A, Soscn H.JM, Struct JMol
1993, 300:1.

Frisch MJ, et al.: Gaussian 03, Revision C.02. Gaussian, Inc., Wallingford, CT; 2004.

Pople JA, Beveridge DL, Dobosh PA:
J. Chem. Phys.. 1967, 47:2026.
COI number [1:CAS:528:DyaF2sXltVaqtL4%3D]; Bibcode number [1967JChPh..47.2026P]
Publisher Full Text 
Comp. Phys. Comm.. 2009, 180:724.
COI number [1:CAS:528:DC%2BD1MXjtF2htrc%3D]; Bibcode number [2009CoPhC.180..724S]
Publisher Full Text 
We used modules sortin, cistar, rtsim and tmom of MELD, a molecular electronic structure program from University of Indiana with contributions from E. R. Davidson, L. McMurchie, S. Elbert, and S. Langhoff

J. Chem. Phys.. 2009, 131:014302.
Bibcode number [2009JChPh.131a4302S]
PubMed Abstract  Publisher Full Text 
Phys. Rev. B. 2007, 75:155208.
Bibcode number [2007PhRvB..75o5208S]
Publisher Full Text 
Phys. Rev. B. 2005, 71:165204.
Bibcode number [2005PhRvB..71p5204S]
Publisher Full Text 
Phys. Rev. B. 2002, 65:125204.
Bibcode number [2002PhRvB..65l5204S]
Publisher Full Text 
Z. Physik.. 1973, 263:83.
COI number [1:CAS:528:DyaE3sXlsVGmtrw%3D]; Bibcode number [1973ZPhy..263...83K]
Publisher Full Text