#
Theoretical Study of closo-Borate Anions [B_{n}H_{n}]^{2−} (n = 5–12): Bonding, Atomic Charges, and Reactivity Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{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.

## 1. Introduction

_{n}H

_{n}]

^{2−}(n = 5–12) have been extensively studied theoretically and experimentally [1,2,3,4]. These cluster species have highly symmetrical structure and unusual chemical bonding, which cannot be described by classical Lewis structure [5,6,7]. closo-borate anions and their derivatives have a lot of useful features such as high stability, low toxicity, and magnetic properties [8,9,10,11]. The main application of closo-borate anions is boron neutron capture therapy of cancer [12,13,14]. In addition, these anions have a lot of potential applications in photochemistry, medicine, and electrochemistry [15,16,17].

_{n}H

_{n}]

^{2−}(n = 5–12) are convenient building blocks for formatting inorganic and bioinorganic systems with desired properties [18,19,20,21]. 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].

_{n}H

_{n}]

^{2−}(n = 5–12) have been calculated [43], but the most convenient index for estimation reactivity—electrophilicity index—remains undiscussed.

_{n}H

_{n}]

^{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.

^{+}, carbocations and AlCl

_{3}can be used [44,45]. For finding most suitable site for electrophilic attack, Fukui functions have been used.

## 2. Materials and Methods

#### Computational Details

_{n}H

_{n}]

^{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]:

**f**) have been calculated with the help of natural population analysis (NPA), Hirshfeld, and AIM charges using the following equation:

_{k}^{+}**(anion) and q**

_{k}**(neutral) are the charges at atom k on the anion and neutral species, respectively.**

_{k}## 3. Results and Discussion

#### 3.1. B–H Bond

_{n}H

_{n}]

^{2−}(n = 5–12). Optimized structures of the closo-borate anions are shown in Figure 1. Several common descriptors have been used. First, the B-H bond lengths were inspected. We found that the lengths of B–H bonds lie in a tight interval between 1.21–1.22 Ǻ, and this observation correlates well with previous theoretical and experimental studies [31,56,57]. The Wiberg bond index is another useful and convenient descriptor for investigation bond phenomena. The Wiberg bond indices of B–H bonds in closo-borate anions fall in the range 0.93–0.96. The B–H bonding interactions in the closo-borate anions [B

_{n}H

_{n}]

^{2−}(n = 5–12) were theoretically investigated using the QTAIM approach, and main local and integral topological properties of the electron density involved in these interactions were computed. For main topological parameters of electron density for B–H interactions, see Supporting Information, Table S1.

^{2}p(r) for all types of closo-borate anions are negative. The most negative values have the [B

_{12}H

_{12}]

^{2−}anion. All B–H bcps are characterized by negative values of total energy. B–H bcp between B(10) and H atom from [B

_{11}H

_{11}]

^{2−}have the most negative value of total energy. We found a good correlation between p(r) at bcp and value of total energy (Figure 3). The values of delocalization index for B–H interactions are not correlated with other descriptors and thus not useful for investigation of given types of bonds. The values of the main topological parameters for B–H interaction are typical for shared interactions.

_{n}H

_{n}]

^{2−}(n = 5–12) have been elaborated. We found that using the values of the B–H bond length and Wiberg bond index does not allow one to obtain information about general trends of the B–H interaction phenomena, because these descriptors are not sensitive to the nature of closo-borate polyhedra and lie in tight intervals. Furthermore, the delocalization indices for B–H interactions have no correlation with other descriptors, so they are not convenient for investigation of the B–H bond phenomena. The values of electron density and total energy at bcp are the most useful for investigation of B–H interactions. These two types of descriptors correlate well with the nature of the boron cluster. Summarizing all results from descriptor analysis, one may conclude that orbital interactions in [B

_{n}H

_{n}]

^{2−}increase with increasing boron cluster size.

#### 3.2. Atomic Charges

_{n}H

_{n}]

^{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. For symmetrical clusters ([B

_{6}H

_{6}]

^{2−}and [B

_{12}H

_{12}]

^{2−}), all values of boron atomic charges are equal. In case of [B

_{5}H

_{5}]

^{2−}and [B

_{10}H

_{10}]

^{2−}anions, the most negative values of atomic charges are localized on apical boron atoms. In the case of [B

_{7}H

_{7}]

^{2−}anion, equatorial boron atoms have more negative value than apical boron atoms. For [B

_{8}H

_{8}]

^{2−}anion, boron atoms in positions B5–B8 have the most negative values. The boron atom in apical position and atoms B(8) and B(9) have the most negative value of atomic charges in [B

_{9}H

_{9}]

^{2−}. For [B

_{11}H

_{11}]

^{2−}, B(8) atom in apical position and atom B7 have most negative value of atomic charges. The values of atomic charges for hydrogen atoms are equal for all types of the clusters. We have calculated the mean values of atomic charges for each types of atoms for all closo-borate anions. Using mean values of atomic charges is simple approach to find general trends. We have found that with the increasing of boron cluster size, the mean values of boron and hydrogen atomic charges become more positive.

_{5}H

_{5}]

^{2−}anion has the most negative values of mean B and H atomic charges, and the [B

_{12}H

_{12}]

^{2−}anion has the most positive values of mean B and H atomic charges.

#### 3.3. Reactivity Analysis

_{n}H

_{n}]

^{2−}(n = 5–12) are shown in Table S2.

_{12}H

_{12}]

^{2−}anion has the most positive value of the HOMO-LUMO gap. The value of chemical potential μ has reverse trend, whereas the [B

_{5}H

_{5}]

^{2−}anion has the most positive value. The [B

_{12}H

_{12}]

^{2−}anion has the most positive value of chemical hardness

**η**. Electrophilicity ω index value is decreasing with increasing the boron cluster size (Figure 5).

**ω**> 0.80 eV moderate,

**ω**< 0.80 eV marginal), the closo-borate anions [B

_{n}H

_{n}]

^{2−}(n = 5–9) can be considered strong and moderate electrophiles, and the closo-borate anions [B

_{n}H

_{n}]

^{2−}(n = 10–12) can be considered marginal electrophiles.

#### 3.4. Fukui Functions

_{5}H

_{5}]

^{2−}and [B

_{10}H

_{10}]

^{2−}anions, boron atoms in apical positions have the most positive values of Fukui functions. For symmetrical cluster ([B

_{6}H

_{6}]

^{2−}, [B

_{12}H

_{12}]

^{2−}), all boron atoms are equal, but the values of Fukui functions are different. In the case of [B

_{7}H

_{7}]

^{2−}, two equatorial boron atoms have the most positive values of Fukui functions. For [B

_{8}H

_{8}]

^{2−}anion, boron atoms B5–B8 have the most positive values of Fukui functions. The boron atom in apical position and atoms B8 and B9 have the most negative values of atomic charges in [B

_{9}H

_{9}]

^{2−}. For [B

_{11}H

_{11}]

^{2−}, atom in apical position B(8) and atom B(7) have the most positive values of Fukui functions. Thus, the values of Fukui function have similar trends with atomic charges. Atoms with the most negative values of atomic charges have the most values of Fukui functions.

_{n}H

_{n}]

^{2−}anions (n = 5–12). The best way to estimate this descriptor is using the NBO and Hirshfeld approaches. Fukui functions correlate well with atomic charges of the closo-borate anions. In general, boron atoms in apical positions have the most positive values of Fukui functions.

## 4. Conclusions

_{n}H

_{n}]

^{2−}(n = 5–12) has been carried out. 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. These two types of descriptors correlate well with the nature of the boron cluster. Summarizing all 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. In general, 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 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. The best way to estimate this descriptor is using the NBO and Hirshfeld approaches. Fukui functions correlate well with atomic charges of the closo-borate anions. In general, boron atoms in apical positions have the most positive values of Fukui functions.

## Supplementary Materials

_{n}H

_{n}]

^{2−}(n = 5–12). Table S4: Fukui function values based on the NBO, QTAIM, and Hirshfeld atomic charges in [B

_{n}H

_{n}]

^{2−}(n = 5–12). Table S5: Cartesian atomic coordinates of the calculated optimized equilibrium model structures. Figure S1: Mean values of Hirshfeld charges for hydrogen atoms for the closo-borate anions of general form [B

_{n}H

_{n}]

^{2−}(n = 5–12).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIM | atoms in molecules |

Bcp | bond critical points |

DFT | density functional theory |

EINS | electrophile-induced nucleophilic substitution |

HOMO | highest occupied molecular orbital |

LUMO | lowest unoccupied molecular orbital |

NBO | natural bond orbitals |

NPA | natural population analysis |

QTAIM | quantum theory of atoms in molecules |

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**Figure 1.**Optimized structures of the closo-borate anions of general form [B

_{n}H

_{n}]

^{2−}(n = 5–12).

**Figure 2.**Values of electron density p(r) at bond critical points for the closo-borate anions of general formula [B

_{n}H

_{n}]

^{2−}(n = 5–12).

**Figure 3.**Correlation between values of total energy and electron density p(r) at B–H bond critical points.

**Figure 4.**Mean values of Hirshfeld charges for boron atoms for the closo-borate anions of general form [B

_{n}H

_{n}]

^{2−}(n = 5–12).

**Figure 5.**Values of electrophilicity index, ω for the closo-borate anions of general form [B

_{n}H

_{n}]

^{2−}(n = 5–12).

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**MDPI and ACS Style**

Klyukin, I.N.; Vlasova, Y.S.; Novikov, A.S.; Zhdanov, A.P.; Zhizhin, K.Y.; Kuznetsov, N.T.
Theoretical Study of *closo*-Borate Anions [B_{n}H_{n}]^{2−} (*n* = 5–12): Bonding, Atomic Charges, and Reactivity Analysis. *Symmetry* **2021**, *13*, 464.
https://doi.org/10.3390/sym13030464

**AMA Style**

Klyukin IN, Vlasova YS, Novikov AS, Zhdanov AP, Zhizhin KY, Kuznetsov NT.
Theoretical Study of *closo*-Borate Anions [B_{n}H_{n}]^{2−} (*n* = 5–12): Bonding, Atomic Charges, and Reactivity Analysis. *Symmetry*. 2021; 13(3):464.
https://doi.org/10.3390/sym13030464

**Chicago/Turabian Style**

Klyukin, Ilya N., Yulia S. Vlasova, Alexander S. Novikov, Andrey P. Zhdanov, Konstantin Y. Zhizhin, and Nikolay T. Kuznetsov.
2021. "Theoretical Study of *closo*-Borate Anions [B_{n}H_{n}]^{2−} (*n* = 5–12): Bonding, Atomic Charges, and Reactivity Analysis" *Symmetry* 13, no. 3: 464.
https://doi.org/10.3390/sym13030464