Using density functional theory and quantum Monte Carlo calculations, we show that B_{12}H _{
n
} and B_{12}F _{
n
} (n = 0 to 4) quasi-planar structures are energetically more favorable than the corresponding icosahedral clusters. Moreover, we show that the fully planar B_{12}F_{6} cluster is more stable than the three-dimensional counterpart. These results open up the possibility of designing larger boron-based nanostructures starting from quasi-planar or fully planar building blocks.

Background

The icosahedral B_{12}H_{12}^{2-} cluster is the most stable molecule among the number of polyhedral boranes synthesized so far [1]. A large-scale and efficient synthesis of fully fluorinated boron hydrides, e.g., icosahedral B_{12}F_{12}^{2-}, has been also reported [2]. On the other hand, the all-boron C_{3v}-B_{12} cluster is quasi-planar, and it was reported to be one of the most stable all-boron clusters. It was also established by extensive computations that the quasi-planar B_{12} cluster is much lower in energy than the all-boron icosahedral B_{12} cluster. This was reported not only for the neutral clusters [3], but also for the charged ones [4]. It is then interesting to investigate what happens with the relative stability of the two (quasi-planar and three-dimensional (3D)) all-boron structures upon addition of hydrogen or fluorine atoms. This is the purpose of this study.

Quasi-planar and 3D boron clusters with the number of hydrogen atoms smaller than the number of boron atoms have been studied both theoretically [5–11] and experimentally [12–14]. Ohishi et al. [12] reported the formation of B_{12}H_{
n
}^{+} (n = 0 to 12) cationic clusters through ion-molecule reactions of the decaborane ions (B_{10}H_{
n
}^{+}, n = 6 to 14) with diborane molecules (B_{2}H_{6}) in an external quadrupole static attraction ion trap. The mass spectrum analysis revealed that among the B_{12}H_{
n
}^{+} clusters with different hydrogen content n, the B_{12}H_{8}^{+} molecule was the main product. In the same study, using first principle calculations with the Becke 3-parameter Lee-Yang-Parr (B3LYP) hybrid functional and the 6-31G(d) basis set, the authors compared the relative energies of quasi-planar and 3D B_{12}H_{
n
}^{+} clusters with n varying from 0 to 12. According to that study, two-dimensional (2D) clusters with n = 0 to 5 are energetically preferred over the 3D structures, whereas 3D clusters are energetically favored for n ≥ 6. In a more recent combined experimental/theoretical study, Ohishi et al. [14] suggested that quasi-planar B_{12}H_{
n
}^{+} with n = 0 to 3 clusters can be obtained by further removal of H atoms from the decaborane ions. This opens up the possibility of changing the structure of the B_{12}H_{
n
}^{+} cluster by controlling the number of hydrogen atoms in the cluster.

To our knowledge, there are no previous studies on the structure and properties of quasi-planar B_{12}F _{
m
} clusters. However, the structures of two polyboron fluorides, B_{8}F_{12} and B_{10}F_{12}, revealing unusual open structures were recently determined [15].

Methods

The initial search for the most stable structures of the boron hydrides B_{12}H _{
n
} and boron fluorides B_{12}F _{
n
} was done at the B3LYP/6-31G(d) level of theory using the FreeON code [16] with no symmetry restrictions. For clusters with an even number of hydrogen or fluorine atoms (even number of electrons), the computations were performed for the singlet multiplicity only, whereas doublet and quartet multiplicities were considered for clusters with an odd number of hydrogen or fluorine atoms (odd number of electrons). In the later case, structures with lower multiplicity were energetically more favorable. For the charged structures, a similar analysis has been done, and the ground state was found to have the lowest multiplicity. Next, the low-lying isomers of B_{12}H _{
n
} and B_{12}F _{
n
} have been re-optimized using the GAMESS-US code [17] at the B3LYP/6-311++G(d, p) level of theory, and for the resulting structures, the vibrational analysis has been done to identify true local minima. It is important to mention that the 'bare' B_{12} icosahedron undergoes distortions after structural optimization and that its symmetry is S_{2}[3], not I_{h}. However, we will refer to that structure and its derivatives as icosahedral or 3D. The quasi-planar or fully planar clusters, for a change, will be often labeled as 2D structures.

The nucleus independent chemical shift (NICS) values and magnetic susceptibility tensors were calculated using the Gaussian 03 package [18] at the B3LYP/6-311++G(d, p) level of theory. To obtain the NICS values, we have used the gauge-independent atomic orbital method, and the magnetic susceptibility tensors were calculated using the continuous set of gauge transformations method.

The quantum Monte Carlo calculations have been done using the QWalk [19] package in two steps. The first step involved optimizing the trial many-body wave function by doing variational Monte Carlo calculations. The trial wave function was of the Slater-Jastrow form. The Slater determinants were constructed using B3LYP orbitals, generated using the GAMESS-US code with the previously optimized geometries within the B3LYP/6-311++G(d, p) level of theory. For the calculations, we have used Gaussian basis sets with effective core potentials [20]. In the second step, we have done fixed-node diffusion Monte Carlo (DMC) calculations with the previously optimized trial wave functions. In the computations, we have used a time step of 0.005 a.u. The DMC error bars are about 0.1 eV.

Results and discussion

Planar vs icosahedral structures

The procedure for determining the most stable isomers of B_{12}H _{
n
} was very similar to that reported in [12], namely, we started with optimized icosahedral and quasi-planar B_{12} clusters, and for a given n, we have calculated the total energies of all possible clusters that resulted from adding hydrogen atoms to the vertices of the distorted icosahedron or to the outer boron atoms of the quasi-planar structure; 2D clusters with an even n have been considered in our previous work [6], and here, we have extended the investigation to an odd n. The energetically most favorable 2D and 3D B_{12}H _{
n
} structures are shown in Figure 1. The minimum-energy cluster structures of B_{12}F_{
n
}, shown in Figure 2, have been found by replacing the hydrogen atoms of the low-lying B_{12}H _{
n
} isomers by fluorine atoms. Interestingly enough, the resulting structures are similar to those found for B_{12}H_{
n
}. One of the small differences is that the B-F bonds are on average 13% longer than the B-H bonds.

In Figure 3, we have plotted the total energy difference between quasi-planar (or fully planar) and icosahedral B_{12}X _{
n
} (X = H, F) clusters as a function of n, the number of H or F atoms in the cluster. As can be seen from the figure, the quasi-planar clusters with up to four hydrogen atoms are more stable than the corresponding icosahedral structures (a similar result has been recently reported [11] for B_{12}H_{
n
}^{0/-} clusters). The same is true for the fully planar B_{12}F_{6} molecule, which is 0.63 eV lower in energy than the 3D cluster. The 2D and 3D B_{12}F_{5} isomers are almost degenerated in energy. From Figure 3, we can also see that the energy difference, E_{2D} - E_{3D}, increases monotonically with n, with the exception of the two 'minima' for B_{12}F_{4} and B_{12}F_{6}. These two minima may suggest an additional stabilization of the 2D structures over the 3D counterparts due to the presence of aromatic stabilization energy.

Similar results to those presented in Figure 3 were reported for the icosahedral and quasi-planar B_{12}H_{
n
}^{+} structures [12]. However, in their recent work, Ohishi et al. [14] have used the PBE0 functional instead of the B3LYP functional to determine the energies of the B_{12}H_{
n
}^{+} clusters. The authors' choice was motivated by the fact that the B3LYP functional may overestimate the energy difference between 2D and 3D structures. To address this problem, we calculated the energy difference between the 2D and 3D structures of B_{12}, B_{12}H_{6}, and B_{12}F_{6} using the very accurate DMC approach. The DMC E_{2D} - E_{3D} values are -5.13, 0.79, and -0.47 eV for B_{12}, B_{12}H_{6}, and B_{12}F_{6}, respectively, whereas the corresponding B3LYP values are -5.34, 0.41, and -0.63 eV, respectively (see Figure 3). This means that the DMC values are shifted up by a value not larger than about 0.4 eV with respect to the B3LYP values. This, however, does not affect the conclusions that are drawn from Figure 3, since even if we shift up the curves by 0.4 eV, the quasi-planar B_{12}H _{
n
} and B_{12}F _{
n
} (n = 0 to 4), and the fully planar B_{12}F_{6} clusters still remain energetically favorable.

Fully planar clusters: B_{12}H_{6} vs B_{12}F_{6}

As calculated here and also reported in [11], the fully planar B_{12}H_{6} cluster corresponds to a local minimum of energy, whereas the D_{3h}-B_{12}F_{6} structure wins the competition with other 2D and 3D isomers and corresponds to a global minimum of energy. Many properties of the B_{12}H_{6} cluster have been previously described in [6], but for consistency purposes, we have repeated some of those calculations at the B3LYP/6-311++G(d, p) level of theory. The highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gaps of the planar B_{12}H_{6} and B_{12}F_{6} structures are 3.54 and 4.39 eV, respectively, whereas the HOMO-LUMO gaps of the corresponding 3D clusters are the same and equal to 2.73 eV. The B-H and B-F interatomic distances are 1.179 and 1.326 Å in B_{12}H_{6} and B_{12}F_{6}, respectively. For comparison, the computed B-H and B-F bond lengths in borane (BH_{3}) and boron trifluoride (BF_{3}) are 1.190 and 1.318 Å, respectively.

While both 2D structures, B_{12}H_{6} and B_{12}F_{6}, have similar shape and size, they exhibit quite different magnetic properties that are directly related to aromaticity. First, we have computed the anisotropy of magnetic susceptibility. The values for B_{12}H_{6} and B_{12}F_{6} are -208.1 and -125.8 cgs ppm, respectively. The isotropic values of the magnetic susceptibility are -91.9 and -118.2 cgs ppm for B_{12}H_{6} and B_{12}F_{6}, respectively. These results suggest that the induced ring current is stronger for B_{12}H_{6} than for B_{12}F_{6}. Similarly, as reported in [6] for B_{12}H_{6}, the central part of the B_{12}F_{6} molecule has a paratropic current flowing inside the inner B_{3} triangle. The antiaromaticity of the inner triangle is, however, smaller for B_{12}F_{6} than for B_{12}H_{6} since the NICS(0) values are 3.9 and 13.3 ppm, respectively. A global aromatic current is dominant above the B_{12}F_{6} (B_{12}H_{6}) molecule since the NICS values are negative, NICS(1) = -5.5 ppm (-3.6 ppm) and NICS(2) = -4.8 ppm (-5.0 ppm), 1 and 2 Å above and below the center of the cluster.

To examine the influence of charge on the structure and stability of the fully planar clusters, we have also studied charged 2D and 3D B_{12}X_{6} (X = H, F) structures. The lowest energy 2D structures identified for B_{12}H_{6} and B_{12}F_{6} were used as initial structures for the structural optimization at a given charge state. For the 3D structures, we have made a search over all possible configurations of the hydrogen or fluorine atoms. For the lowest energy structures with an even number of electrons, a singlet multiplicity has been assumed, whereas doublet and quartet multiplicities were considered for clusters with an odd number of electrons. In the later case, clusters with lower multiplicity were energetically more favorable. The structures of the 2D and 3D charged clusters are shown in Figure 4. It has been previously reported that the fully planar D_{3h}-B_{12}H_{6} cluster undergoes structural distortions if charged with one electron, although the quasi-planarity is preserved [6]. In general, all the charged 2D B_{12}X_{6} (X = H, F) clusters are quasi-planar rather than fully planar, as can be seen in Figure 4. In Figure 5, we have plotted the energy difference between 2D and 3D [B_{12}X_{6}] ^{
q
} (X = H, F) structures as a function of the cluster charge state q. We have found that the addition of one or two electrons to fully planar B_{12}H_{6} and B_{12}F_{6} clusters (or the removal of one electron from them) makes those structures even less energetically favorable with respect to the corresponding 3D isomers. This is, however, less pronounced for B_{12}H_{6} than for B_{12}F_{6} as shown in Figure 5. Finally, it should be noted that the quasi-planar B_{12}F_{6}^{2+} cluster is much more stable than its 3D isomer. Finally, all structures and energies are provided in Additional file 1.

Conclusions

Our density functional theory and quantum Monte Carlo results show that the B_{12}H _{
n
} and B_{12}F _{
n
} (n = 0 to 4) quasi-planar structures are energetically more favorable than the corresponding icosahedral clusters and that the fully planar B_{12}F_{6} cluster is more stable than the 3D counterpart. We have also shown that negative or positive charge further stabilizes the 3D over the 2D B_{12}X_{6} (X = H, F) clusters (except for B_{12}X_{6}^{2+}, where the opposite is observed). Our findings are potentially useful for designing larger boron-based nanostructures starting from quasi-planar or fully planar building blocks.

Declarations

Acknowledgements

The authors would like to acknowledge the support given by the Robert A. Welch Foundation (grant J-1675), the NSF CREST CRCN center (grant HRD-1137732), and the Texas Southern University High Performance Computing Center (http://hpcc.tsu.edu/; grant PHY-1126251).

Authors’ Affiliations

(1)

Institute of Theoretical Physics, Faculty of Physics, University of Warsaw

(2)

Department of Physics, Texas Southern University

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