B3Al4+: A Three-Dimensional Molecular Reuleaux Triangle

We systematically explore the potential energy surface of the B3Al4+ combination of atoms. The putative global minimum corresponds to a structure formed by an Al4 square facing a B3 triangle. Interestingly, the dynamical behavior can be described as a Reuleaux molecular triangle since it involves the rotation of the B3 triangle at the top of the Al4 square. The molecular dynamics simulations, corroborating with the very small rotational barriers of the B3 triangle, show its nearly free rotation on the Al4 ring, confirming the fluxional character of the cluster. Moreover, while the chemical bonding analysis suggests that the multicenter interaction between the two fragments determines its fluxionality, the magnetic response analysis reveals this cluster as a true and fully three-dimensional aromatic system.


Introduction
In 2010, Wang and coworkers detected an anionic cluster of nineteen boron atoms in the gas phase through a photoelectron spectroscopy experiment [1]. The structure has a planar pentagonal inner B 6 core encircled by a B 13 ring (Figure 1a). A couple of years later, Merino and coworkers found that the inner B 6 fragment rotates almost freely with respect to the periphery [2], so they labeled this type of system as a Wankel-type rotor [2][3][4][5][6][7][8]. Over time, other boron clusters were also reported to exhibit similar behavior, including B 13 + as the most prominent case [9,10]. In fact, the dynamic behavior of B 13 + was experimentally verified by Asmis et al. in 2016 using cryogenic ion vibrational spectroscopy [11,12]. The main reason for this fascinating fluxionality is attributed to the multicenter bonds, often found in boron clusters [13]. Moreover, this property is not exclusive to only planar forms; compasses [14,15], drums [16], stirrers, sphere-shaped clusters [17], and other doped boron architectures have similar dynamic behavior [18][19][20][21]. Other related examples are the global minima of B 7 M 2 and B 8 M 2 (M = Zn, Cd, Hg), which can be described as an M 2 dimer spinning freely on a boron wheel resembling a magnetic stirrer placed on a baseplate [22]. The reader interested in more details on fluxionality in boron clusters is referred to [12].
Inspired by these fluxional boron systems, we analyze the dynamical properties of the global minimum of B 3 Al 4 + (Figure 1b). This cluster has an Al 4 square facing a B 3 triangle. Although this system was recently reported by Wen et al. [23], they overlooked its fluxional properties. Moreover, upon further exploration of the nature of the bonding and aromaticity, we find that this system has all the characteristics to define it as a true and fully three-dimensional aromatic system.

Computational Details
The PES was systematically explored using the Coalescence Kick (CK) program and a modified genetic algorithm implemented in GLOMOS [24,25]. Initial screening in singlet and triplet states was performed for both programs at the PBE0/LAN2DZ level [26,27]. In the range of 30 kcal mol −1 above the putative global minimum, the lowest isomers were minimized and characterized at the PBE0/def2-TZVP level [28]. Final energies were computed at the CCSD(T) [29]/def2-TZVP level, including the zero-point energy correction (ZPE) at the PBE0/def2-TZVP level. Thus, the energy discussion is based on the CCSD(T)/def2-TZVP//PBE0/def2-TZVP results. Natural bond orbital analysis was conducted at the PBE0 level to obtain natural atomic charges (using the NBO6 program [30]) and Wiberg bond indices (WBI) [31]. The Born-Oppenheimer molecular dynamics [32] (BOMD) were carried out at the PBE0/6-31G(d) level to ascertain the fluxional behavior. The aromatic character was evaluated by computing the magnetic response to a uniform external magnetic field. This is achieved by calculating the magnetically induced current density [33][34][35][36] (J ind ) and the induced magnetic field [37][38][39] (B ind ) using the GIMIC [33][34][35][36] and Aromagnetic [40] programs, respectively. Since magnetic properties computed with the BHandHLYP [41] functional yield results in good agreement with those obtained by CCSD(T) computations [42], J ind and B ind were calculated at the BHandHLYP/def2-TZVP level using gauge-including atomic orbitals (GIAOs) [43,44]. All these calculations were performed with Gaussian 16 [45]. Furthermore, the adaptive natural density partitioning [46] (AdNDP) analysis was also carried out to understand the nature of the interactions using the Multiwfn program [47].

Discussion
The putative global minimum of B 3 Al 4 + is a singlet with C s symmetry (1), in which an Al 4 square interacts with a B 3 triangle, forming a polyhedron (see Figure 2). The B-B distances range from 1.60-1.63 Å, creating an isosceles triangle with a B-B bond aligned with an Al-Al bond. On the other hand, the Al 4 ring is a trapezoid or quasi-square, with Al-Al bond distances lying between 2.60-2.77 Å. The distance between the two fragments is 1.73 Å. The energetically closest isomer is only 5.1 kcal mol −1 above the global minimum and consists of an Al 4 tetrahedron. The third isomer differs from the putative global minimum by exchanging a B atom for an Al atom and is 8.1 kcal/mol higher in energy. Note that all isomers in Figure S1 have a B 3 unit. As shown in Figure S2, the lowest vibrational frequency of 1 is 69 cm −1 , which corresponds to the rotation of the B 3 fragment. Following this smooth rotation mode, we identify the transition state (TS) belonging to the C s point group. The structural variations between the TS and the global minimum are negligible. The rotation barrier is only 0.1 kcal/mol (including the zero-point energy correction), which means that the B 3 ring can undergo almost free rotation on top of the Al 4 ring. Figure 3 shows the structural evolution of B 3 Al 4 + with the B 3 triangle rotating clockwise. Starting from the global minimum, fragment B 3 must rotate 15 • to reach the maximum of the rotation barrier. A new minimum is achieved by rotating B 3 another 15 • , and a complete rotation lap is obtained by repeating this process twelve times. BOMD simulations at 300 K for 25 ps and starting from the global minimum structure show that the B 3 triangle rotates with respect to the Al 4 ring, just like a three-dimensional (3D) molecular Reuleaux triangle. The average distance between the B 3 triangle and the Al 4 quasi-square remains constant (see Movie 1 in the supplementary material). A chemical bonding analysis is crucial to understanding the dynamical behavior of B 3 Al 4 + . The natural atomic charges and WBIs for 1 and TS are shown in Figure S3. The charge distribution is such that the B 3 fragment has a charge of −2.0 |e| and the Al 4 moiety of 3.0|e|. Instead, the WBI values show that the B-B bonds (WBI around 1.1) have a higher covalent character than the Al-Al bonds (WBI between 0.32 and 0.65), but even more fascinating is that the WBI values for the B-Al bonds are of the same magnitude as the latter (0.49 and 0.62), indicating that there is significant orbital overlap (an electron delocalization) between both fragments. We also employed AdNDP, an extension of the NBO analysis. AdNDP analysis recovers not only Lewis bond elements (lone pairs and 2c-2e bonds) but also multicenter bonds (nc-2e, n ≥ 3). For B 3 Al 4 + , the analysis identifies three 3c-2e σ-bonds with occupation numbers (ON) of 1.99-1.83 |e| and one π-bond (ON = 1.71 |e|) located mainly throughout the B 3 fragment. This electron distribution is like that reported for the aromatic cyclopropenyl cation (C 3 H 3 + ). On the other hand, the Al 4 ring has three 4c-2e σ-bonds with ON of 1.99-1.88 |e|. The B 3 triangle and the Al 4 trapezoid interact via three 7c-2e bonds (Figure 4). Thus, the absence of 2c-2e bonds between the two fragments favors free rotation [48]. The bonding pattern for the TS is similar to that of the global minimum, as shown in Figure S4. The orbital compositions of canonical molecular orbitals (CMOs) for 1 and TS are listed in Tables S1 and S2, which support the bonding pattern provided by AdNDP.
As shown in Figure S5, the B 3 ring has six σand two π-electrons, i.e., in principle, it satisfies Hückel's aromaticity rule for both the σand π-clouds. Now, while the Al 4 ring has six electrons, which we could classify as σ, the region between the two rings is also connected via six electrons, so both the Al 4 fragment and the area between the rings satisfy the Hückel's rule. The reality is that the orbital separation is complicated, and all the bonds in B 3 Al 4 + have a multicentric character, i.e., they are fully delocalized in the small cage. Aromaticity of B 3 Al 4 + is confirmed by its magnetic response to an external magnetic field perpendicular to the Al 4 ring. The B ind analysis reveals strong shielding values (<−50 ppm) of the z-component of B ind (B ind z ) along the region between the Al 4 and B 3 rings. This can be explained by the presence of an entirely diatropic current density in this region (Figure 5a). Thus, the interaction of both rings creates a ring current in the region between the two rings instead of being two separate ring currents, implying that B 3 Al 4 + has true three-dimensional aromaticity (Figure 5b). Integration of J ind in a plane that intersects the Al-Al and B-B bonds yields a ring-current strength (J ind ) of 33.3 nA/T. Changes in the current density can be obtained by calculating the spatial derivative of the ring-current strength along the vertical z-axis (dJ ind /dz) [49]. This indicates that there are no paratropic contributions along the z-profile. Additionally, the ring-current profile shows that the strongest current density flux occurs just below the boron triangle, confirming the three-dimensional delocalization along the z-axis ( Figure S6 in Supporting Information). Thus, B 3 Al 4 + is a true 3D aromatic fluxional cluster. The external magnetic field is oriented parallel to the z-axis, perpendicular to the Al 4 plane.

Summary
After systematically exploring the potential energy surface of clusters with formula B 3 Al 4 + , we confirm that the putative global minimum corresponds to a structure formed by an Al 4 quasi-square facing a B 3 triangle. Most interestingly, the global minimum of B 3 Al 4 + exhibits fluxionality. However, its dynamic behavior involves the rotation of the B 3 triangle on the Al 4 ring, embodying a molecular Reuleaux triangle. Corroborated by BOMD, a complete rotation cycle includes twelve repeating steps comprising the global minimum and a transition state separated by only 0.1 kcal/mol, indicating that the rotation between both fragments is virtually free. Furthermore, the AdNDP analysis suggests that the interaction between the B and Al fragments is via three 7c-2e bonds, favoring such dynamical behavior. Regarding aromaticity, the analysis based on the magnetic response reveals a strong shielding along the z-axis due to an entirely diatropic current density flowing between fragments leading to a three-dimensional double (σ + π)-aromaticity. So, we find a peculiar three-dimensional fluxional 3D aromatic cluster that resembles a molecular Reuleaux triangle.