Characterization and Application of Nanomaterials (2020) Volume 3 Issue 2 doi:10.24294/can.v3i2.761 73 Original Research Article Relative stability of planar clusters B11, B12, and B13 in neutral- and charged-states Levan Chkhartishvili Department of Engineering Physics, Georgian Technical University, Tbilisi, Georgia; E-mail: chkharti2003@yahoo.com ABSTRACT Theoretically, within the diatomic model, the relative stability of most abundant boron clusters B11, B12, and B13 with planar structures in neutral, positive and negative charged-states is studied. According to the specific (per at- om) binding energy criterion, B12 + (6.49 eV) is found to be the most stable boron cluster, while B11 – + B13 + (5.83 eV) neutral pair is expected to present the preferable ablation channel for boron-rich solids. Obtained results would be ap- plicable in production of boron-clusters-based nanostructured coating materials with super-properties such as lightness, hardness, conductivity, chemical inertness, neutron-absorption, etc., making them especially effective for protection against cracking, wear, corrosion, neutron- and electromagnetic-radiations, etc. Keywords: Cluster; Charge State; Specific Binding Energy; Diatomic Model; Relative Stability; Clusters-based Coating Material; Boron ARTICLE INFO Article history: Received 12 June 2020 Received in revised form 4 July 2020 Accepted 7 July 2020 Available online 20 July 2020 COPYRIGHT Copyright © 2020 Levan Chkhartishvili. doi: 10.24294/can.v3i2.761 EnPress Publisher LLC. This work is li- censed under the Creative Commons Attribu- tion-NonCommercial 4.0 International Li- cense (CC BY-NC 4.0). http://creativecommons.org/licenses/by/4.0/ 1. Introduction Boron-clusters-based nanostructured coating materials due to their super-properties such as lightness, hardness, conductivity, chem- ical inertness, neutron-absorption, etc. can serve for effective protec- tion against cracking, wear, corrosion, neutron- and electromagnet- ic-radiations, etc. [1,2] . This makes boron clusters Bn, n=1,2,3,…, inter- esting to be investigated in details. Summarizing experimental and theoretical data available in the literature on all-boron nanomaterials (e.g. see our overviews on sub- ject [3-5] ), one can conclude that, according to the specific (per consti- tuting atom) binding energy criterion, ultra-small, with , bo- ron clusters prefer (quasi)planar structures, while at higher , they should be wrapped first into cylinders and then into spheres form- ing boron nanotubes or boron fullerenes, respectively. In (quasi)planar clusters, increases with because of increasing in mean coordina- tion number of constituent atoms. Within the initial approximation, specific binding energy of boron clusters is expected to be almost sat- urated around . But, when the polarity deal into the bonding — commonly characteristic of structures of identical atoms with differ- ently coordinated atomic sites — is taken into account, there is ex- pected a weak maximum instead. In general features, the experimental mass-spectra fits these theoretical findings. However, formation prob- ability peak is well-pronounced for species B11, B12, and B13. Naturally, 20n n  n 10n http://creativecommons.org/licenses/by/4.0/ 74 such a behavior is related not only to the ener- gy-factor, but mainly to the process kinetics. Usual- ly, boron clusters are formed by the ablation of el- emental boron or boron-rich solid materials, main structural motifs of which are slightly deformed regular icosahedra of boron atoms B12 [6] . Present paper aims to theoretically study the relative stability of most abundant boron clusters B11, B12, and B13 with (quasi)planar structures in different charged-states. 2. Method of calculations In clusters of identical atoms, binding energy per atom, i.e., specific binding energy, serves for the key factor determining their relative stabilities and, consequently, concentrations of clusters with dif- ferent numbers of atoms in the products of ablation of corresponding solid materials. Based on various ab initio methods specif- ic binding energy and some other important physi- cal characteristics of boron clusters were numeri- cally calculated by Boustani et al. (summarizing of these studies see in the study of Boustani [7] ) and some other teams (see above cited reviews and ref- erences therein). But, more vividly these species can be descried based on the diatomic model. It should be noted that such model-based approach provides quite good quantitative results as well. The application of the Fermi’s old diatomic model [8] to the multi-atomic structures is based on the property of interatomic bonding to be saturated. In the initial, i.e., pair interactions, approximation the binding energy of a structure simply equals to the sum of energies of interactions between neigh- boring atoms. Based on the pair interactions ap- proximation, microscopic theory of expansion and its generalization to the periodical structures allow correct estimation of the thermal expansion coeffi- cient for number of crystalline substances [9] . De- spite its simplicity, the diatomic model is still suc- cessfully used to calculate anharmonic effects in solids [10] . An analogous approach was used by us to explain isotopic effects of thermal expansion and melting in all-boron lattices [11-15] . Suppose that the index i = 1,…, n numbers the atoms constituting cluster of n atoms, and Ci are their coordination numbers, respectively. In the ini- tial approximation, binding energies between each pair of adjacent atoms E0 are equal, and it turns out that:     ni i iC n E 1 0 2  (1) Here, the factor ½ is introduced to correct the double sum which includes every pair of neighbor- ing atoms twice. We used approximated formula (1) to estimate ground-state binding energy of all-boron nanostructures, mainly, boron nanotubes [16-18] . However, clusters are finite structures of atoms and, consequently, coordination numbers of sites at center are higher than that at periphery. This leads to the redistribution of the outer valence shell elec- trons and, as a result, differences between binding energies of neighboring atomic pairs. Note the re- port [19] on high-pressure experiments and ab initio evolutionary crystal structure predictions that ex- plore the structural stability of boron under pressure and reveal a partially ionic high-pressure boron phase. Our formulation of the first approximation to the diatomic model theory takes into account cor- responding polarity of the bonds. In case of identi- cal constituent atoms, it is obvious to assume that these electrons between the atoms are divided pro- portionally to their coordination numbers. This im- plies that the atoms develop non-zero effective stat- ic atomic charges with charge numbers Zi deter- mined from the relation:     nj j j ii C nC V nZ 1 1 (2) respectively, Here V is the total number of the outer valence shell electrons in the atoms constituent cluster. Thus, the binding energy per atom in the first approximation, i.e., including correction related to the interatomic bonds polarity, is: 75        ni i Ck k kii ii i i ZZECE n 1 110 )( 2 1  (3) Where the index Ki = 1,…,Ci numbers the nearest neighboring atoms of i -atoms, respectively; d e E 0 2 1 4  (4) is the energy-dimension parameter (e and are elemental charge and electric constant, respectively); and d is the initial bonds length in the equilibrium. In our previous studies [20-23] , relations (2), (3), (4) are used to estimate effective static atomic charges, dipole moment, and specific binding ener- gy in the boron planar clusters. Further improvement of this approximation means determination of the equilibrium bond length in the first approximation as well. It can be done by the minimization of the system potentials energy including bond length-dependent vibrational and electrostatic interaction energies [24] . Within the frame of the diatomic approach, further refinement of the clusters’ binding energy and other ground-state parameters can be achieved by abandoning the requirement for equality of all the bond lengths. In another study of mine [25] , a general theoretical frame for further studying is provided. Finally, we have got a set of linear equa- tions determining not precisely planar, but qua- si-planar equilibrium cluster configurations. Finally, all the above described theory-levels of the diatomic approach are summarized in the mini-review [26] . By definition, cluster binding energy is the difference between sum of energies of isolated atomic particles and energy of their bounded struc- ture. Energy of a neutral atom exceeds that of posi- tive ion by the ionization potential (IP), while is less by the electron affinity (EA) that of negative ion. Let Q=0, ±1,…,±n be the ionic charge num- ber of the cluster, then, n QEQ Q  (5) should be the specific binding energy correction related to its charge state if EQ is IP or EA depend- ing on the sign of Q. Taking into account this cor- rection, in present work, specific binding energy of planar boron clusters are estimated from the rela- tion:              QEZZECE n Q ni i Ck k kii ii i i1 110 )( 2 11  (6) 3. Species studied and results Because B12 icosahedral clusters are the main structural units of structural modifications of boron and boron-rich compounds, here we studied stabil- ity of planar boron clusters B11, B12 and B13 with almost the same number of constituent atoms both in neutral- and charged-states. For obtaining numerical results, one needs in- put parameters such as E0 and E1 (or d) characteris- tic of pair interaction between two boron atoms. They can be found e.g. from the quasi-classical B– B potential [27] : E0 ≈ 2.80 eV, E1 ≈ 8.09eV; and d ≈ 1.78 Å. Note that this potential was successfully applied to explaining of the above mentioned iso- topic effects of boron atoms also in geometric mod- els for boron nanostructures [28] . As for IP and EA of boron atom, they equal to E+1 ≈ 8.30 eV and E–1 ≈ 0.28 eV, respectively [29] . Outer valence shell of the isolated B-atom contains a single 2p-electron. Consequently, their total num- ber V=10 in B11 + , V=11 in B11 0 and B12 + , V=12 in B11 – , B12 0 and B13 + , V=13 in B12 – and B13 0 , and V=14 in B13 – , respectively. Obtained results of calcula- tions confirmed the expectation that cluster-isomers with symmetrical shapes and without holes (vacant sites) in their structure, i.e., with the maximal num- ber of interatomic bonds, should be most stable. Correspondingly, in the tables below, we present structures and characteristics only for ground-state isomers. Table 3-1 shows that among the neutral clus- ters studied, the atomic charges are maximal in B11, apparently because of its asymmetric structure. On contrary, symmetric clusters B12 and B13 reveal higher polarity of bonding in their charged-states. 0 76 Table 3-1. Static charge number of atomic sites in most abundant boron clusters Ground-state isomer Cluster Coordination number Number of sites Static charge number B11 + 2 1 + 11/21 3 4 + 2/7 4 4 + 1/21 6 2 – 3/7 B11 0 2 1 + 10/21 3 4 + 3/14 4 4 – 1/21 6 2 – 4/7 B11 – 2 1 + 3/7 3 4 + 1/7 4 4 – 1/7 6 2 – 5/7 B12 + 3 6 + 5/16 4 3 + 1/12 6 3 – 3/8 B12 0 3 6 + 1/4 4 3 0 6 3 – 1/2 B12 – 3 6 + 3/16 4 3 – 1/12 6 3 – 5/8 B13 + 3 6 + 4/13 4 4 + 1/13 6 3 – 5/13 B13 0 3 6 + 1/4 4 4 0 6 3 – 1/2 B13 – 3 6 + 5/26 4 4 – 1/13 6 3 – 8/13 Table 3-2. Specific binding energy of most abundant boron clusters Clusters Binding energy per atom, eV B12 + 6.49 B13 + 6.34 B11 + 6.21 B13 0 5.60 B12 0 5.47 B11 0 5.34 B13 – 5.25 B11 – 5.24 B12 – 5.10 From the Table 3-2, one can see that positively charged clusters are certainly more stable than neu- tral clusters, while lasts are more stable than nega- tively charged ones. Table 3-3. Specific binding energy of neutral pairs of most abundant boron clusters Pairs of clusters Binding energy per atom, eV B11 – + B13 + 5.83 B12 + + B12 – 5.79 B11 + + B13 – 5.69 B11 0 + B13 0 5.48 B12 0 + B12 0 5.47 According to the date presented in the Table 3-3, preferably channels of ablation of icosahe- dral boron-rich materials are related to the for- mation of neutral pairs of differently charged clus- ters. 4. Discussion and conclusions Before making conclusions based on results 77 obtained by us, we need to discuss shortly the theo- retical data available in the literature on B12 cluster isomers stability. As is known, boron atoms are unique in their strong preference for forming icosahedral structural units. In the icosahedral cage B12, each atom has 5 other atoms as nearest neighbors. Such B12 units are unstable by themselves, but distorted B12 icosahedra do form a stable framework for boron and many bo- ron compounds. In particular, the α-rhombohedral crystalline boron may be considered an almost per- fect cubic close packing of B12 icosahedra. In gen- eral, the persistence of the icosahedral structure is explained by the fact that the chemical valence of boron is not completely saturated under 5-fold coordination and that there exist outwardly di- rected bonds which serve to link icosahedra. The Xα-method of the scattered-wave self- consistent-field (SCF–Xα–SW) with muffin-tin-pot- ential and local exchange was applied [30] to deter- mine quantitatively electronic structure and binding energy of icosahedral B12. Then the binding energy was used to estimate the cohesive energy of the B12 units in the α-rhombohedral structure. From this, an effective B12–B12 interaction potential in the solid was constructed. The binding energy of 35.6 eV was obtained at an equilibrium bond distance of 1.96 Å, which is within 10 % of the bond distance of 1.77 Å found in B12 icosahedra in bulk boron. The electronic structure of B12 corresponded to an open-shell configuration so that such an isolated cluster would be chemically unstable. Geometries and electronic structures of the B12 cluster have been investigated using a Car–Parrine- llo ab initio molecular dynamics simulation [31] . The icosahedral structure was found to be locally sta- ble, but with a few dangling bonds. On annealing or melting, this structure rearranges to a more open geometry. The new structure has a significantly lo- wer energy despite a lower coordination. But, bonds are stronger and there are no dangling bonds. The structure and stability of small boron clusters were investigated employing density func- tional theory (DFT) [32] . The search for minima was performed using gradient methods at the local spin density (LSD) level. Most of the final structures prefer planar or quasi-planar forms and can be con- sidered to be fragments of planar or spherical sur- faces. A group of spherical boron clusters may exist. However, their energies are generally higher than those of the convex or quasi-planar clusters. This also means that clusters of real bulk, sections of the boron lattice, have less stable configurations. They try to close the open spheres with a small number of atoms. In particular, the distorted icosa- hedral B12 cluster with closed structure has ener- gy by 2.01–3.28 eV higher than that of the convex B12 structure. Based on ab initio quantum-chemical methods, accurate calculations on small boron clusters were carried out to determine their electronic and geo- metric structures [33] . The geometry optimization with a linear search of local minima on the poten- tial-energy surface (PES) was performed using an- alytical gradients in the framework of the restricted Hartree-Fock SCF approach. Most of the final structures of the boron clusters are composed of two fundamental units: either of hexagonal or of pen- tagonal pyramids. The resulting quasi-planar and convex structures can be considered as fragments of planar surfaces and as segments of nanotubes or hollow spheres, respectively. In particular, the most stable isomers of B12 clusters are 2 planar and 1 convex clusters. In contrast to the convex or the quasi-planar clusters, the structures of the cage-clusters are rather similar to those of the well-known - and -rhombohedral boron crystals, or to those of the boron hydrides. The energies of the cage-clusters on average are between 2 and 5 eV higher than those of the convex or the quasi-planar clusters. Taking into account that interesting features of elemental boron and boron compounds are the oc- currence of highly symmetric icosahedral clusters and rich chemistry of boron also dominated by cage-structures, in the study of Zhai et al. [34] , the authors reported experimental and theoretical evi- dences that small boron clusters prefer planar structures and exhibit aromaticity and antiaromatic- ity according to the Huckel rules, akin to planar 78 hydrocarbons. The electronic and geometric structures, in- cluding binding energies, of small and neutral boron clusters have been investigated using DFT [35] . Line- ar, planar, convex, quasi-planar, open-cage and cage structures have been found. None of the lowest en- ergy structures and their isomers has an inner atom; i.e., all the atoms are positioned at the surface. Within size range under the consideration, the pla- nar and quasi-planar (convex) structures have the lowest energies. In particular, 11 different structures of B12 cluster were investigated. Their binding en- ergies are ranging from 4.037 to 4.599 eV/atom. The first lowest energy isomer is a convex structure containing three dovetailed hexagonal pyramids. Only ninth and tenth isomers are cages, slightly distorted icosahedral structures without the central atom. Among the several aromatic boron clusters, B12 and B13 + are unique. They show three distinct sets of sextets, resulting in extraordinary kinetic stability. A novel way to analyze them was pro- posed in the study of Kiran et al. [36] , in which the cluster is partitioned as inner and outer rings. The molecular orbital analysis, based on this fragmenta- tion, reveals that the delocalized valence electrons in B12 and B13 + clusters can be trifurcated leading to triple aromaticity, which is unique to these clusters. Recently, Bhattacharyya et al. [37] conducted a comprehensive numerical study of the ground-state structures of isomers of B12 cluster. Geometry opti- mization was performed at a level of theory em- ploying the extended basis sets. Once the geometry of a given isomer was optimized, its ground state energy was calculated more accurately at the level of theory employing even larger basis set. Thus, computed values of binding energies of various isomers are expected to be quite accurate. Geometry optimization revealed 10 distinct isomers. The vi- brational frequency analysis performed on the 3 lowest energy isomers showed them to be stable. While, in boron-rich solids, icosahedron is the basic structural unit, in the isolated form, it was demon- strated to be unstable. The disc-like lowest-energy structure of B12 cluster can be seen as a conse- quence of the Jahn–Teller distortion of its icosahe- dral isomer. Computed binding energies per atom are 4.60 and 4.31 eV for quasi-planar and icosahe- dral isomers with C3v and C2h point group symme- tries, respectively. This work, based upon ab initio geometry optimization, verifies early results and predicts the B12 icosahedron to be higher in energy as compared to the lowest energy quasi-planar B12 cluster. This clearly illustrates the tendency of the icosahedral structure towards distortion to a lower symmetry one, consistent with the Jahn–Teller the- orem. Thus, neutral B12 icosahedral clusters consti- tuting boron-rich materials, on ablation have to be converted into (quasi) planar disc-like isomer with higher specific binding energy. The charged cluster B13 + also should have higher stability. Consequently, the formation of B11, B12 and B13 clusters in differ- ent charged-states is expected. Our calculations performed within the diatom- ic model using quasi-classical B–B interatomic po- tential, lead to following hierarchies in the for- mation probabilities of single boron clusters in var- ious charged states and their neutral pairs: B12 + > B13 + > B11 + > B13 0 > B12 0 > B11 0 > B13 – > B11 – > B12 – and B11 – + B13 + > B12 + + B12 – > B11 + + B13 – > B11 0 + B13 0 > B12 0 + B12 0 , respectively. The obtained results would be helpful in con- trolling the synthesis of nanoboron materials with specific engineering properties. References 1. Becker R, Chkhartishvili L, Martin P. Boron, the new graphene? Vacuum Technology & Coating 2015; 16 (4): 38–44. 2. Becker R, Chkhartishvili L, Martin P. Tribological applications for boron. Vacuum Technology & Coating 2015; 16 (10): 36–41. 3. Chkhartishvili L. Micro- and nano-structured boron. In: Perkins GL (editor). Boron. Compounds, pro- duction and application. New York: Nova Science Publishers; 2011.p. 221–294. 4. Chkhartishvili L. Nanoboron (An overview). Nano Studies 2011; 3: 227–314. 5. Chkhartishvili L. All-boron nanostructures. In: Kharisov B I, Kharissova O V, Ortiz–Mendez U (editors). CRC concise encyclopedia of nanotech- nology. Boca Raton: CRC Press; 2016. p. 53–69. 79 6. Albert B, Hillebrecht H. Boron: Elementary chal- lenge for experimenters and theoreticians. Angewa- ndte Chemie International Edition 2009; 48(46): 8640–8668. 7. Boustani I. Towards novel boron nanostructural materials. In: Springborg M (editor). Chemical Mo- delling: Applications and theory. Cambridge: Royal Society of Chemistry; 2011. p. 1–44. 8. Fermi E. Molecules, crystals, and quantum statistics. New York, Amsterdam: W. A. Benjamin INC; 1966. 9. Novikova SI. Thermal Expansion of Solids. Mos- cow: Nauka; 1974. 10. Slutsker AI, Gilyarov VL, Luk’yanenko AS. Energy features of an adiabatically loaded anharmonic os- cillator. Physics of the Solid State 2006; 48(10): 1947–1953. 11. Chkhartishvili L, Gabunia D, Tsagareishvili O, et al. Estimation of isotopic composition effect on sub- stance melting temperature. Bulletin of the Geor- gian National Academy of Sciences 2004; 170(3): 530–532. 12. Chkhartishvili LS, Gabunia DL, Tsagareishvili OA. Estimation of the isotopic effect on the melting pa- rameters of boron. Inorganic Materials 2007; 43(6): 594–596. 13. Chkhartishvili LS, Gabunia DL, Tsagareishvili OA. Effect of the isotopic composition on the lattice pa- rameter of boron. Powder Metallurgy and Metal Ceramics 2008; 47(9-10): 616–621. 14. Gabunia D, Tsagareishvili O, Chkhartishvili L, et al. Isotopic composition dependences of lattice con- stant and thermal expansion of -rhombohedral bo- ron. Journal of Physics: Conference Series 2009; 176(012022): 1–10. 15. Chkhartishvili L, Tsagareishvili O, Gabunia D. Iso- topic expansion of boron. Journal of Metallurgical Engineering 2014; 3 (3): 97–103. 16. Chkhartishvili L. On quasi-classical estimations of boron nanotubes ground-state parameters. Jour- nal of Physics: Conference Series 2009; 176(1): 1– 9. 17. Chkhartishvili L. Molar binding energy of the boron nanosystems. In: Konuk A, Kurama H, Ak H, et al. (editors). Proceedings of the 4th international boron symposium. Ankara: Osmangazi University; 2009. p.153–160. 18. Chkhartishvili L. Nanotubular boron: Ground-state estimates. In: Chikoidze E, Tchelidze T (editors). New developments in materials science. New York: Nova Science Publishers; 2011. p. 67–80. 19. Oganov AR, Chen J, Gatti C, et al. Ionic high- pressure form of elemental boron. Nature 2009; 457(7251): 863–867. 20. Chkhartishvili L, Becker R. Effective atomic charg- es and dipole moment of small boron clusters. Pro- ceedings of the ICANM 2015. Ottawa: IAEMM; 2015. p. 130–147. 21. Becker R, Chkhartishvili L. Dipole moment of qua- si-planar boron clusters. Nano Studies 2015; 11: 29– 48. 22. Chkhartishvili L, Becker R, Avci R. Relative stabil- ity of boron quasi-planar clusters. In: Darsavelidze G, Guldamashvili A, Chedia R, et al. (editors). Pro- ceedings of the international conference ―Advanced Materials & Technologies‖. Tbilisi: Universal; 2015. p. 42–46. 23. Chkhartishvili L. Small elemental clusters in pair interaction approximation. Proceedings of the ICANM 2016. Montreal: IAEMM 2016. p. 128– 132. 24. Chkhartishvili L. Planar clusters of identical atoms in equilibrium: 1. Diatomic model approach. Amer- ican Journal of Nano Research & Applications 2017; 5(3-1): 1–4. 25. Chkhartishvili L. Quasi-planar elemental clusters in pair interactions approximation. Open Physics 2016; 14(1): 617–620. 26. Chkhartishvili L. Boron quasi-planar clusters. In: Pogrebnjak A D (editor). A mini-review on diatomic approach. Proceedings of the IEEE 7th international conference on nanomaterials: Applications & prop- erties (NAP—2017), Part 4, Track: Nanomaterials for electronics, spintronics and photonics; Sumy: Sumy State University; 2017. p. 1–5. 27. Chkhartishvili L, Lezhava D, Tsagareishvili O. Qua- si-classical determination of electronic energies and vibration frequencies in boron compounds. Journal of Solid State Chemistry 2000; 154(1): 148–152. 28. Chkhartishvili L, Mamisashvili N, Maisuradze N. Single-parameter model for multi-walled geometry of nanotubular boron. Solid State Sciences 2015; 47: 61–67. 29. Hayes WM (editor-in-chief). Handbook of Chemis- try and Physics (94th Ed.). Boca Raton: CRC Press; 2013. p. 10–147 & 10–197. 30. Bambakidis G, Wagner RP. Electronic structure and binding energy of the icosahedral boron cluster B12. Journal of Physics and Chemistry of Solids 1981; 42(11): 1023–1025. 31. Kawai R, Weare JH. Instability of the B12 icosahe- dral cluster: Rearrangement to a lower energy structure. The Journal of Chemical Physics 1991; 95(2): 1151–1159. 32. Boustani I. Structure and stability of small boron clusters. A density functional theoretical study. Chemical Physics Letters 1995; 240(1-3): 135–140. 33. Boustani I. Systematic ab initio investigation of ba- re boron clusters: Determination of the geometry and electronic structures of Bn (n = 2–14). Physical Review B 1997; 55(24): 16426–16438. 34. Zhai H, Kiran B, Li J, et al. Hydrocarbon analogues of boron clusters — planarity, aromaticity and anti- aromaticity. Nature Materials 2003; 2(12): 827–833. 35. Atis M, Ozdogan C, Guvenc ZB. Structure and en- ergetic of Bn (n = 2–12) clusters: Electronic struc- ture calculations. International Journal of Quantum 80 Chemistry 2007; 107(3): 729–744. 36. Kiran B, Kumar GG, Nguyen MT, et al. Origin of the unusual stability of B12 and B13+ clusters. Inor- ganic Chemistry 2009; 48(21): 9965–9967. 37. Bhattacharyya P, Boustani I, Shukla A. First princi- ples electronic structure study of B12 isomers: Jahn– Teller distortion flattens the icosahedron into a disc. arXiv:1802.01072v1 [physics.atm-clus] 4 Feb 2018; 1–32.