Theoretical Study of closo -Borate Anions [B n H n ] 2 − ( n = 5–12): Bonding, Atomic Charges, and Reactivity Analysis

: This study has focused on the structure, bonding, and reactivity analysis of closo -borate anions [B n H n ] 2 − ( n = 5–12). Several descriptors of B–H interactions have been calculated. It has been found that the values of electron density and total energy at bond critical point are the most useful descriptors for investigation of B–H interactions. Using results from the descriptor analysis, one may conclude that orbital interactions in [B n H n ] 2 − increase with increasing the boron cluster size. Several approaches to estimate atomic charges have been applied. Boron atoms in apical positions have more negative values of atomic charges as compared with atoms from equatorial positions. The mean values of boron and hydrogen atomic charges tend to be more positive with the increasing of boron cluster size. Global and local reactivity descriptors using conceptual density functional theory (DFT) theory have been calculated. Based on this theory, the closo -borate anions [B n H n ] 2 − ( n = 5–9) can be considered strong and moderate electrophiles, while the closo -borate anions [B n H n ] 2 − ( n = 10–12) can be considered marginal electrophiles. Fukui functions for electrophilic attack have been calculated. Fukui functions correlate well with atomic charges of the closo -borate anions. Boron atoms in apical positions have the most positive values of Fukui functions.

The possibility to substitute terminal hydrogen atom with exo-polyhedral groups allows one to obtain various derivatives of closo-borate anions. There are a lot of approaches to functionalize this type of anions [22][23][24][25]. Among these approaches, the most convenient and studied is electrophile-induced nucleophilic substitution (EINS) [26,27].
Currently, many useful methods focused on structure and reactivity analysis have been described. The quantum theory of atoms in molecules (QTAIM) analysis gives clear information about covalent and non-covalent interactions [28][29][30]. Previously, several closo-borate anions have been calculated by QTAIM analysis [31,32]. The main focus in these studies has been directed to B-B interactions and the investigation of the exopolyhedral bonds of the general type B-X, where X = C, O, N [33][34][35]. Conceptual density functional theory (DFT) allows one to obtain information about reactivity using only values of highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) energies [36][37][38][39]. Fukui functions allow one to obtain information about molecular reactivity site [40][41][42]. Some main conceptual DFT descriptors of closo-borate anions [BnHn] 2− (n = 5-12) have been calculated [43], but the most convenient index for estimation reactivity-electrophilicity index-remains undiscussed.
In present work, we have focused on the comprehensive investigation of closo-borate anions [BnHn] 2− (n = 5-12). The study of B-H bonds allows one to obtain information about their stability and possibility to break in substitution reactions. Using several bonding descriptors, we have found main trends in the B-H interactions. Calculation of atomic charges allows one to obtain information about electronic density distribution in molecules. In the present work, several approaches to estimate atomic charges have been used.
For finding main trends in reactivity of the closo-borate anions, conceptual density functional theory (DFT) has been used. As shown previously, the most interesting reaction type for closo-borate-anions is the EINS process. In these reactions, complexes with Lewis acids are formed on the first stage. As electrophile inductor H + , carbocations and AlCl3 can be used [44,45]. For finding most suitable site for electrophilic attack, Fukui functions have been used.

Computational Details
The full geometry optimization procedure for all model structures has been performed at the ωB97X-D3/6-31++G(d,p) level of theory [46,47]. Note that ωB97X-D3 functional provides accurate results for QTAIM analysis, charge density distribution, and conceptual DFT descriptors in a satisfactory level [48][49][50], and we already successfully applied this DFT functional in similar theoretical studies of different boron clusters chemical systems [33][34][35]. Tight SCF (self-consistent field) convergence has been employed during the calculations. All computations have been performed with the help of the ORCA 4.2.1 program package [51]. All the closo-borate anions [BnHn] 2− (n = 5-12) have closed electron shells, thus the spin restricted approximation has been applied for all structures. Symmetry operations have not been applied during the geometry optimization procedures for all structures. All stationary points on the potential energy surfaces have been characterized as minima by eigenvalue analysis of the diagonalized Hessians (no imaginary frequencies). The natural population analysis and natural bond orbital (NBO) calculations has been performed by using NBO7 program package [52]. The topological analysis of the electron density according to quantum theory of atoms in molecules (QTAIM) formalism developed by Bader [53] has been carried out with the Multiwfn program (version 3.7) [54]. The main reactivity indices (electronic chemical potential μ, chemical hardness η, and softness S, global electrophilicity ω) have been calculated using the Equations (1)-(4) [55]: The condensed Fukui functions (fk + ) have been calculated with the help of natural population analysis (NPA), Hirshfeld, and AIM charges using the following equation:

= −
where qk (anion) and qk (neutral) are the charges at atom k on the anion and neutral species, respectively.

B-H Bond
We  As seen in Table S1,

Atomic Charges
We estimated atomic charges of [BnHn] 2− (n = 5-12) using AIM, NBO, and Hirshfeld approaches (Table S3). These methods give quite different results for values of atomic charges. In the case of the AIM method, there is a pronounced difference between boron and hydrogen atomic charges. Boron atoms have positive values of atomic charges, while hydrogen atoms have negative values. In the case of NBO atomic charges, we found the opposite situation: boron atoms have more negative charges than hydrogen atoms. In the case of Hirshfeld method, boron and hydrogen atoms have similar values of atomic charges. The mean values of Hirshfeld atomic charges for boron atoms are shown in Figure 4 (for hydrogen atoms, see Figure S1). However, despite these peculiar properties of each method, the main trends are the same. Initially, we found trends for each type of boron cluster.

Fukui Functions
We calculated Fukui functions for electrophilic attack using the AIM, NBO, and Hirshfeld approaches (Table S4). Cartesian coordinates of all optimized structures are given in the Table S5. Applying AIM charges leads to a lot of negative values of Fukui function, which are meaningless. For closo-borate anions, the more appropriate approaches to estimate Fukui functions are the Hirshfeld and NBO methods.   Table S1: Main descriptors of B-H interactions in closo-borate anions. Table S2: Reactivity descriptors of B-H interactions in closo-borate anions. Table S3. NBO, QTAIM, and Hirshfeld atomic charges of [BnHn] 2− (n = 5 -12). Table S4: Fukui function values based on the NBO, QTAIM, and Hirshfeld atomic charges in [BnHn] 2− (n = 5-12). Table S5: Cartesian atomic coordinates of the calculated optimized equilibrium model structures. Figure