BAl₄Mg^−/0/+: Global Minima with a Planar Tetracoordinate or Hypercoordinate Boron Atom

Abstract

We have explored the chemical space of BAl${}_4$Mg${}^{-/0/+}$ for the first time and theoretically characterized several isomers with interesting bonding patterns. We have used chemical intuition and a cluster building method based on the tabu-search algorithm implemented in the Python program for aggregation and reaction (PyAR) to obtain the maximum number of possible stationary points. The global minimum geometries for the anion (1a) and cation (1c) contain a planar tetracoordinate boron (ptB) atom, whereas the global minimum geometry for the neutral (1n) exhibits a planar pentacoordinate boron (ppB) atom. The low-lying isomers of the anion (2a) and cation (3c) also contain a ppB atom. The low-lying isomer of the neutral (2n) exhibits a ptB atom. Ab initio molecular dynamics simulations carried out at 298 K for 2000 fs suggest that all isomers are kinetically stable, except the cation 3c. Simulations carried out at low temperatures (100 and 200 K) for 2000 fs predict that even 3c is kinetically stable, which contains a ppB atom. Various bonding analyses (NBO, AdNDP, AIM, etc.) are carried out for these six different geometries of BAl${}_4$Mg${}^{-/0/+}$ to understand the bonding patterns. Based on these results, we conclude that ptB/ppB scenarios are prevalent in these systems. Compared to the carbon counter-part, CAl${}_4$Mg${}^{-}$, here the anion (BAl${}_4$Mg${}^{-}$) obeys the 18 valence electron rule, as B has one electron fewer than C. However, the neutral and cation species break the rule with 17 and 16 valence electrons, respectively. The electron affinity (EA) of BAl${}_4$Mg is slightly higher (2.15 eV) than the electron affinity of CAl${}_4$Mg (2.05 eV). Based on the EA value, it is believed that these molecules can be identified in the gas phase. All the ptB/ppB isomers exhibit $\pi$/$\sigma$ double aromaticity. Energy decomposition analysis predicts that the interaction between BAl${}_4^{-/0/+}$ and Mg is ionic in all these six systems.

1. Introduction

From the time that the concept of planar tetracoordinate carbon (ptC) emerged [1,2], it was extended not only to its group elements (Si, Ge, Sn, etc.) [3,4,5,6,7,8,9,10] but also to other elements such as B [11,12,13,14,15,16,17], Al [18,19], N [20,21,22], P [23], O [24], and lately even to the F atom [25]. There are two main reasons why experimentalists [5,6,26,27,28,29,30,31,32,33] and theoreticians [2,34,35,36,37,38,39,40,41,42,43] have put a great deal of effort into studying these special class of molecules: (i) ptC is a fundamental deviation from the conventional ideas of tetrahedral tetracoordinate carbon [44,45]; (ii) no two structural isomers behave in the same way chemically. Thus, making this new class of molecules enhances our existing understanding about chemical bonding, and one could potentially make new materials. Schleyer and co-workers computationally identified the real local minima in lithium-substituted cyclopropane and cyclopropene molecules for the first time in 1976 [34]. In the last five decades, a large array of molecules containing ptC atoms that are global and local minima were computationally identified [10,14,15,46,47,48,49,50,51,52,53,54,55,56,57,58], and some were experimentally detected [5,26,27,28,29,30,31,32,33,59]. Lately, the idea of ptC has been extended to planar hypercoordinate carbon (phC; Penta [60,61,62,63,64,65,66] and Hexa [67,68,69,70,71] coordination) and also to other elements such as B [72,73,74,75] or N [76] considering their potential applications in material science [77,78].

The idea of planar tetracoordinate boron (ptB) arrived as a byproduct of stabilizing the ptC itself. Hoffmann and co-workers, in their seminal paper [2], outlined that essentially two things are required for the electronic stabilization of the ptC atom: (1) appropriate substituents that would act as a $\sigma$-donor/$\pi$-acceptor simultaneously—that is, $\sigma$-donors would facile an electron transfer to the electron-deficient ptC atom and $\pi$-acceptors would delocalize the ${\mathrm{p}}_z$ or $\pi$-type lone pair; and (2) embedding the ptC atom into a (4n + 2)$\pi$ electron system. Considering the $\sigma$-donating and $\pi$-accepting nature of boron atoms, molecules with ptC atoms were built in the past, where boron was used as a ligating atom to the ptC atom [79,80]. Moreover, phC, phSi, and phGe molecules were also proposed computationally using boron as ligands [67,81,82]. In the strange case of CB${}_4$, the most stable form contains a ptB atom with a tricoordinate carbon and the ptC isomer lies 1 kcal mol${}^{-1}$ above the ptB isomer, calculated using coupled-cluster methods. By altering the charge (CB${}_4^{+}$), the original objective of stabilizing the ptC atom was achieved. Notably, the ptC isomer in CB${}_4^{+}$ was detected through the mass spectrometry of boron carbide by Becker and Dietze [59].

Based on our earlier theoretical work on CAl${}_4$Mg${}^{-/0}$ [56], we were curious to see how replacing C with B would affect the planar tetracoordination around four Al and one Mg atoms. Thus, in this work, we analyzed the various structural isomers of BAl${}_4$Mg${}^{-/0/+}$ computationally using density functional theory (DFT). For the accurate evaluation of relative energies, composite method CBS-QB3 was used for the low-lying isomers. To our surprise, the global minimum geometries of the anion (1a) and cation (1c) contained a ptB atom (see Figure 1), whereas the global minimum of the neutral (1n) contained a planar pentacoordinate boron (ppB) atom. The bonding and kinetic stabilities of these three isomers along with three other key low-lying isomers 2a; ppB, 2n; ptB, and 3c; ppB were studied in detail.

2. Computational Details

Several trial geometries of BAl${}_4$Mg${}^{-/0/+}$ are generated by chemical intuition and the cluster building procedure implemented in the Python program for aggregation and reaction (PyAR) [83,84]. Modeling by intuition was conducted, targeting for ptB and ppB based on similar reported molecules. The automated cluster building was done as follows: first, a diatomic molecule was generated from two randomly chosen atoms from B, Al, and Mg. To achieve the optimized geometry of these diatomic molecules, another randomly chosen atom was added following the procedure described in [84] to generate several (N) estimated geometries. All these geometries were optimized, unique minima were chosen, and the further addition of random atoms was continued until the target chemical formula was reached. We performed 10 different runs with N = 16 orientations for the BAl${}_4$Mg${}^{-/0/+}$ systems. The trial geometries were optimized using the ORCA program [85] interfaced with PyAR [83,84]. The initial geometry optimizations were carried out using PBE [86] functional with the def2-SVP [87] basis set including Grimme’s empirical dispersion corrections (D3) [88] with Becke–Johnson (BJ) damping [89,90] and resolution of identity (RI) approximation. After we filtered all geometries generated from 10 different runs, unique geometries were selected for further analysis. Some geometries were reached from both intuitive and stochastic procedures. Overall, for BAl${}_4$Mg${}^{-/0/+}$, we identified 33 stationary points, each on their singlet, doublet, and singlet potential energy surfaces, respectively.

The geometries of all BAl${}_4$Mg${}^{-/0/+}$ isomers reported here are optimized further using DFT with the (U)$\omega$B97XD hybrid functional [91] and the 6-311++G(2d,2p) basis set [92,93]. Harmonic vibrational frequencies are calculated for each stationary point to confirm whether it is a minimum, transition state, or higher-order saddle-point. The number of imaginary frequencies (NImag) obtained for each stationary point is indicated underneath the geometries (see Figure 2, Figure 3 and Figure 4). To obtain accurate relative energies, calculations are also conducted using the composite method, CBS-QB3 [94], for the first nine low-lying isomers (eight for cations) that lie below 20 kcal mol${}^{-1}$. All molecules were found to have no instabilities from the wavefunction stability analysis [95] at the (U)$\omega$B97XD/6-311++G(2d,2p) level. Triplet and quartet geometry optimization and frequency calculations were also conducted for the low-lying isomers. For brevity, triplet and quartet geometries are given in the supporting information. We carried out ab initio molecular dynamics (AIMD) simulations using the atom-centered density matrix propagation (ADMP) [96] method. These simulations were performed to check the kinetic stability of six different BAl${}_4$Mg${}^{-/0/+}$ isomers that contain ptB (1a, 2n, and 1c) or ppB atoms (2a, 1n, and 3c).

Chemical bonds in the global minima were analyzed using canonical molecular orbitals (CMOs), adaptive natural density partitioning (AdNDP) [97,98], and natural bond order (NBO) approaches [99]. Natural atomic charges (q) and Wiberg bond indices (WBIs) [100] from the NBO analyses were calculated at the (U)$\omega$B97XD/6-311++G(2d,2p) level. Nucleus-independent chemical shift (NICS) [101] values were calculated to gauge the $\pi$/$\sigma$ dual aromaticity in both global and local minima. All the above calculations were carried out with the Gaussian suite of programs [102]. An energy decomposition analysis (EDA) was performed using the Q-Chem program [103] to check the interaction between two fragments. A topological analysis of the electron localization function (ELF) and Laplacian of electron density was carried out for both the neutral and anionic global minima with the Multiwfn program [104] using the wave function file generated by the Gaussian program [102]. For brevity, optimized geometries of high-energy isomers, Cartesian coordinates of all isomers, total energies, zero-point vibrational energies (ZPVEs), net dipole moment, relative energies without and with ZPVE correction for all isomers, and kinetic stability plots of a few isomers are given in the supporting information.

3. Results and Discussion

3.1. Thermal Stability

Optimized geometries of the global minima (top row) and local minima (bottom row) of BAl${}_4$Mg${}^{-/0/+}$ containing either ptB or ppB atoms are shown in Figure 1. Bond lengths (in Å) and Wiberg bond indices (in green color) are shown for each isomer to justify the ptB or ppB scenario in Figure 1.

The low-lying isomers of BAl${}_4$Mg${}^{-/0/+}$ are shown in Figure 2, Figure 3 and Figure 4, respectively. ZPVE-corrected relative energies obtained at the $\omega$B97XD/6-311++G(2d,2p) (for anions and cations) and U$\omega$B97XD/6-311++G(2d,2p) (for neutrals) levels are given for each geometry. The same energies obtained at the CBS-QB3 level are given in parentheses. It is noted here that for a few isomers (2a, 8a, 6c, and 8c), we could not calculate the relative energies using the composite method CBS-QB3. As the latter method carries out geometry optimization using the B3LYP functional [105,106,107] without any empirical dispersion corrections, in some cases, the geometry either transforms to a low-lying isomer or global minimum itself. Thus, the relative energy values are not given for geometries 2a, 8a, 6c, and 8c at the CBS-QB3 level. In these cases, the relative energy values given in parentheses were calculated at the CCSD(T)/6-311++G(2d,2p)//$\omega$B97XD/6-311++G(2d,2p) level. In this work, we rely on the results obtained using the $\omega$B97XD functional as it incorporates empirical dispersion corrections. For BAl${}_4$Mg${}^{-/+}$, singlet isomers with a ptB atom (1a and 1c) were found to be the global minima. The second low-lying isomer for the anion (2a) lies only 0.48 kcal mol${}^{-1}$ above 1a at the $\omega$B97XD/6-311++G(2d,2p) level (see Figure 2). At CCSD(T)/6-311++G(2d,2p)//$\omega$B97XD/6-311++G(2d,2p) level, this gap increases to 0.92 kcal mol${}^{-1}$. For the cation, the second low-lying isomer (2c) contains a tetrahedral tetracoordinate boron atom, whereas the third isomer (3c) makes a clear case for the ppB atom (see Figure 4). Furthermore, 2c and 3c lie 1.99 and 3.89 kcal mol${}^{-1}$ above 1c at the $\omega$B97XD/6-311++G(2d,2p) level. At CBS-QB3 level, this trend is reversed and 3c is more stable by 3.63 kcal mol${}^{-1}$ compared to 2c. For BAl${}_4$Mg neutral, the doublet isomer with ppB atom becomes the global minimum (1n; see Figure 3) and the second low-lying isomer (2n) that contains a ptB atom lies 0.98 kcal mol${}^{-1}$ above 1n at the $\omega$B97XD/6-311++G(2d,2p) level.

3.2. Kinetic Stability

To explore the kinetic stability of ptB and ppB isomers in BAl${}_4$Mg${}^{-/0/+}$, ab initio molecular dynamics simulations were carried out using the ADMP [96] approach as implemented in the Gaussian 16 program [102]. Six different isomers (see Figure 1) were considered for these simulations at a temperature of 298 K and 1 atm pressure for 2000 fs of time. The time evolution of the total energy (in a.u) plots for 1a (anion; ptB) and 3c (cation; ppB) are shown in Figure 5 and Figure 6, respectively. For brevity, similar plots of four other isomers (2a, 1n, 2n, and 1c) are shown in the supporting information. To show the alteration of the structure over the 2000 fs of time, we added snapshots at 400 fs intervals. These plots show balanced oscillations in energy and steadiness in geometries. This represents the kinetic stability of the minimum geometries. However, the trend of energy for the ppB of BAl${}_4$Mg${}^{+}$ (3c) is different from the others. The energy fluctuation is comparatively higher, and geometries (Figure 6) are not uniform. Here, at the end of the trajectory, it tends to form a 3D-like structure with lower energy. After seeing this trend, we also carried out additional simulations at 100 and 200 K for 3c alone, which shows a balanced oscillation in energy and steadiness for this geometry. The time evolution of the total energy plot for 3c at low temperatures is given in the supporting information for brevity. Overall, these results indirectly indicate that low-temperature measurements are necessary to trap the cation 3c that exhibits a ppB atom.

Furthermore, we analyzed the bond length of B1–Mg2 (Figure 7) for all the six geometries over the period of 2000 fs with the time interval of 250 fs. The bond length ranges from 3.063–3.657Å, 3.013–3.727Å, and 3.608–3.889Å for the ptB of BAl${}_4$Mg${}^{-}$ (1a), BAl${}_4$Mg (2n), and BAl${}_4$Mg${}^{+}$ (1c), respectively. Though the cationic ptB system in BAl${}_4$Mg${}^{+}$ contains 16 valence electrons, 1c showed the maximum steadiness of energy as well as minimum change in bond length over the period of 2000 fs at 298 K compared to the other two ptB systems. The anionic ptB BAl${}_4$Mg${}^{-}$ (1a; 18 v. es) shows a slightly lower scale of bond length variation (0.594Å) as compared to the neutral ptB BAl${}_4$Mg (2n; 17v. es), which shows a variation of 0.714Å in bond lengths. Similarly, ppB BAl${}_4$Mg${}^{+}$ (3c) shows a maximum uniformity of the B1–Mg2 bond length (ranging from 2.139–2.315Å) compared to the other two ppB BAl${}_4$Mg${}^{-/0}$ systems (2a and 1n). The bond length ranges for 2a and 1n are 2.523–3.366Å and 2.402–0.684Å, respectively. The latter two ppB systems cover the bond length variation from the ppB to the ptB region, and the fluctuations are quite high compared to the rest of the other four isomers. Furthermore, 2a (18 v. es) depicts a lower range of bond length variation (0.843Å) compared to 1n (17 v. es); i.e., 1.282Å.

In brief, the observation from the ADMP calculations suggests that all the geometries show steadiness in total energy (in a.u) over the period of 2000 fs. Maximum steadiness is perceived for ptB BAl${}_4$Mg${}^{+}$ (1c), whereas ppB BAl${}_4$Mg${}^{+}$ (3c) shows minimum steadiness in terms of total energy. Both the cationic form of ptB and ppB BAl${}_4$Mg${}^{+}$ demonstrate a smooth and minor change of the B1–Mg2 bond length. However, only the trajectory of ppB BAl${}_4$Mg${}^{+}$ produced a comparatively different geometry (3D like) from the planar tetracoordinate (2D) system. Nevertheless, simulations carried out at low temperatures (100 and 200 K) suggest that 3c retains planarity.

3.3. Natural Bond Orbital Analysis

We employed an NBO scheme for the distribution of natural charge over the BAl${}_4$Mg${}^{-/0/+}$ systems and analyzed WBI values for the boron atom coordination. The summary of the natural population analysis (NPA) charges for ptB (1a, 2n, and 1c) and ppB (2a, 1n, and 3c) systems are given in

Table 1. The natural charges (q, |e|) of BAl${}_4$Mg${}^{-/0/+}$ containing ptB and ppB systems and their corresponding valence electronic configuration.

Type	Species	q${}_{\mathit{B}}$	q${}_{\mathbf{Mg}}$	q${}_{\mathbf{Al}3}$	q${}_{\mathbf{Al}4}$	q${}_{\mathbf{Al}5}$	q${}_{\mathbf{Al}6}$	Valence Electronic Configuration of B
ptB	BAl${}_4$Mg${}^{-}$ (1a)	−2.84	0.69	0.18	0.18	0.38	0.38	2s${}^{1.37}$ 2p${}_x^{1.31}$ 2p${}_y^{1.52}$ 2p${}_z^{1.56}$
	BAl${}_4$Mg (2n)	−2.79	0.90	0.25	0.25	0.69	0.69	2s${}^{1.42}$ 2p${}_x^{1.37}$ 2p${}_y^{1.53}$ 2p${}_z^{1.39}$
	BAl${}_4$Mg${}^{+}$ (1c)	−2.74	1.13	0.33	0.33	0.97	0.97	2s${}^{1.44}$ 2p${}_x^{1.35}$ 2p${}_y^{1.56}$ 2p${}_z^{1.32}$
ppB	BAl${}_4$Mg${}^{-}$ (2a)	−2.80	0.83	0.16	0.16	0.32	0.32	2s${}^{1.36}$ 2p${}_x^{1.26}$ 2p${}_y^{1.52}$ 2p${}_z^{1.60}$
	BAl${}_4$Mg (1n)	−2.60	1.12	0.27	0.27	0.47	0.47	2s${}^{1.40}$ 2p${}_x^{1.39}$ 2p${}_y^{1.52}$ 2p${}_z^{1.29}$
	BAl${}_4$Mg${}^{+}$ (3c)	−2.57	1.47	0.44	0.44	0.61	0.61	2s${}^{1.43}$ 2p${}_x^{1.39}$ 2p${}_y^{1.56}$ 2p${}_z^{1.14}$

. This shows that a notable amount of charge transfer takes place from the peripheral aluminum atoms to the central boron atom. The natural charges on the central boron atom vary from −2.74 to −2.84 for ptB systems and −2.57 to −2.80 $\left|e\right|$ in ppB systems. The anions in both ptB and ppB show the highest negative charge on the boron atom. Moving from anion to neutral, the negative charge (absolute value) decreases in both the systems. As a result, an opposite trend is observed in the peripheral atoms; i.e., the positive charge increases. This also indicates that the charge transfer from the surrounding atoms to the boron atom decreases. The negative charges on the boron atom suggest that boron acts as a $\sigma$-acceptor in all the systems. From the valence electronic configuration of the boron given in Table 1, the 2p${}_x$ orbital population is significantly lower than that of the 2p${}_y$ and 2p${}_z$ orbital in the anion. The planner systems lie in the $yz$ plane, and thus 2p${}_x$ orbital is perpendicular with respect to the molecular plane. The perpendicular boron 2p${}_x$ orbital participated in $\pi$-back bonding in the anionic system, whereas this is less probable in neutral and cationic species. The WBI${}_{total}$ is calculated for both ptB and ppB systems by taking the sum of four B–Al and one B–Mg values (see

Table 2. Wiberg bond indices for BAl${}_4$Mg${}^{-/0/+}$ containing ptB and ppB systems.

Type	Species	B1-Mg2	B1-Al3	B1-Al4	B-Al5	B1-Al6	WBI${}_{\mathbf{total}}$ on B
ptB	BAl${}_4$Mg${}^{-}$ (1a)	0.074	0.907	0.907	0.651	0.651	3.190
	BAl${}_4$Mg (2n)	0.064	0.889	0.889	0.614	0.614	3.064
	BAl${}_4$Mg${}^{+}$ (1c)	0.081	0.937	0.937	0.650	0.650	3.255
ppB	BAl${}_4$Mg${}^{-}$ (2a)	0.045	0.842	0.842	0.726	0.726	3.180
	BAl${}_4$Mg (1n)	0.119	0.769	0.769	0.683	0.683	3.023
	BAl${}_4$Mg${}^{+}$ (3c)	0.355	0.843	0.843	0.644	0.644	3.328

). The WBI values of B–Al (3,4) are in the range 0.89 to 0.94 in case of ptB and 0.77 to 0.84 in ppB systems which indicate covalent bonding with the central boron atom. However, B–Al (5,6) WBI values are somewhat smaller than B–Al (3,4), suggesting a lower covalency along that bond. The B–Mg WBI values are comparatively lower than those of the B–Al bonds. This implies a partial covalent character of this bond. The B–Mg WBI value increases from anion to cation in both ptB and ppB systems.

3.4. Molecular Orbital Analysis

Key molecular orbitals of BAl${}_4$Mg${}^{-/0/+}$ containing the ptB atom are given in Figure 8, whereas the same orbitals containing the ppB atom are given in Figure 9. Energies of the MOs obtained at the (U)$\omega$B97XD/6-311++G(2d,2p) level are given in eV units. We further decomposed the MO composition with the natural atomic orbital method. For brevity, compositions and major contributions are given in the supporting information (see Table S7). The highest occupied molecular orbital (HOMO) for anionic and neutral ppB species is mainly composed of the B 2${\mathrm{p}}_z$ orbital and Al (5,6) 3${\mathrm{p}}_z$ orbital. All the ${\mathrm{p}}_z$ orbitals are in a bonding interaction. In contrast, the Mg 3s orbital contributes significantly in the case of anionic and neutral ppB BAl${}_4$Mg. As the cationic species has one electron fewer than the neutral species, its HOMO resembles the HOMO–1 of the neutral and anionic species. It is composed of a B 2${\mathrm{p}}_y$ orbital along with an Al (5,6) 3s orbital for ptB structure. For the ppB cationic case, besides the B 2${\mathrm{p}}_z$ orbital, Al (3,4) 3s and 3${\mathrm{p}}_z$ orbitals participate in HOMO. HOMO–3 of the anionic and neutral system and HOMO–1 of the cationic species are of $\pi$-type MO, which are formed from B 2${\mathrm{p}}_x$ and Al 3${\mathrm{p}}_x$ orbital. A significant contribution is made by the B 2${\mathrm{p}}_x$ orbital. The HOMO–LUMO energy gaps ($\Delta E_{H-L}$) are 4.59, 1.17, and 4.71 eV for anionic, neutral, and cationic ptB BAl${}_4$Mg, respectively, and 0.43, 1.46, and 4.59 eV for anionic, neutral, and cationic ppB BAl${}_4$Mg species, respectively. In the neutral BAl${}_4$Mg, the HOMO has one unpaired electron; i.e., singly occupied MO (SOMO). Moving from anionic to cationic species, $\Delta E_{H-L}$ increases, indicating that cationic BAl${}_4$Mg is more stable than its anionic form in both ptB and ppB cases. This is also evident from the lower HOMO energy compared to HOMO–1 of anionic and neutral BAl${}_4$Mg. However, the Mg atomic orbital contribution is present only in HOMO–2 for all the ptB BAl${}_4$Mg species. In the HOMO–2 of ptB BAl${}_4$Mg${}^{+}$, major contributions come from B 2${\mathrm{p}}_z$, Mg 3s, and Al (3,4) 3${\mathrm{p}}_y$, 3${\mathrm{p}}_z$ orbitals. In contrast to anionic and neutral ptB BAl${}_4$Mg, no significant contribution was found from boron. A similar type of composition is also found for the ppB BAl${}_4$Mg HOMO–2 orbital. In the HOMO of both anionic and neutral ppB BAl${}_4$Mg, there is a significant contribution from the Mg 3s orbital. So, there is a B–Mg coordination present for these species, which is absent in the ptB case. In the LUMO of cationic ppB BAl${}_4$Mg, an orbital contribution from Mg 3s is found besides the B 2${\mathrm{p}}_z$ orbital. This result suggests that, besides making bonds with four aluminum atoms, the central boron also has a fifth coordination with the Mg in the ppB species. As the neutral BAl${}_4$Mg has one unpaired electron, we also calculated the spin density (see Figure 10) for the ptB BAl${}_4$Mg. The unpaired electron is delocalized on four Al atoms and with the boron 2${\mathrm{p}}_z$ orbital. Al (3, 4) have the highest Mulliken spin density.

3.5. Adaptive Natural Density Partitioning (AdNDP) Analysis

We performed AdNDP analysis [97,98] as implemented in Multiwfn [104] for both ptB and ppB geometries of BAl${}_4$Mg${}^{-/0/+}$ to find the bonding scenarios in these systems. After NBO analysis, we mainly focused on the delocalized nc–2e bonds, where n starts from 3 to 6 (total number of atoms). All nc–2e bonding orbitals obtained for 1a, 2n, and 1c that show the ptB atom are given in Figure 11. For the anion and cation, there are two 3c–2e and one 6c–2e $\sigma$ bonds with an occupation number (ON) of 1.99 $\left|e\right|$ each and one 5c–2e $\pi$ bond with an ON of 1.99 $\left|e\right|$. For neutral, the only difference that we observed was that the alpha orbital shows a maximum ON of 1.00 $\left|e\right|$ with a 6c–2e $\sigma$ bond, whereas in beta, it is a 5c–2e $\sigma$ bond. The 5c–2e $\pi$ bond remains the same in both the cases. Overall, for all ptB cases, four multi-center two electron bonds (and thus eight electrons in total) that show the highest ONs support the idea of 2$\pi$/6$\sigma$ double aromaticity. Likewise, for ppB geometries (2a, 1n, and 3c), we performed AdNDP analysis, and the nc–2e bonding orbitals are shown in Figure 12. For the anion, there are two 5c–2e $\sigma$ bonds with an ON of 1.99 $\left|e\right|$ each, one 6c–2e $\sigma$ bond with an ON of 2.00, and one 5c–2e $\pi$ bond with an ON of 1.98 $\left|e\right|$. For the cation, one can clearly see that the delocalization is increased in the system as there are two 6c–2e $\sigma$ bonds with an ON of 2.00 $\left|e\right|$ each, and one 5c–2e $\sigma$ bond and one 5c–2e $\pi$ bond both with an ON of 1.99 $\left|e\right|$ each. For neutral, we see similar trends in both alpha and beta cases. Once again, even for ppB systems, 2$\pi$/6$\sigma$ double aromaticity is maintained.

3.6. NICS Analysis

In all these six isomers of BAl${}_4$Mg${}^{-/0/+}$, the $\pi$/$\sigma$-dual aromaticity can also be independently confirmed through NICS values. These values computed at 0 (on the ring) and 1Å (above the ring) for ptB and ppB isomers at the (U)$\omega$B97XD/6-311++G(2d,2p) level are shown in Figure 13. The pink dots represent the position of the ghost atoms. These positions are approximately chosen from the ring critical points obtained through AIM analysis. All the NICS values obtained are negative in all the cases. This indicates that both $\sigma$- (NICS(0)) and $\pi$-aromaticity (NICS(1)) are present in all these six different isomers, which supports the conclusion reached from the AdNDP bonding patterns.

3.7. AIM Analysis

We calculated various electron density descriptors at (3, –1) bond critical points for six different BAl${}_4$Mg species. These results are summarized in

Table 3. Electron density descriptors (in a.u.) at the (3, –1) bond critical points (BCP) and ring critical point (RCP) obtained from the (U)$\omega$B97XD/6-311++G(2d,2p) level for BAl${}_4$Mg${}^{-/0/+}$ (1a, 2n and 1c) with ptB atom. The topological parameters such as Lagrangian kinetic energy G(r_c), potential energy density V(r_c), energy density E(r_c) or H(r_c), −G(r_c )/V(r_c ), and G(r_c)/$\rho$(r_c) at the critical points are also given.

System	BCP & RCP	$\mathit{\rho }$(rc)	∇2$\mathit{\rho }$(rc)	G(rc)	V(rc)	H(rc)	ELF	−G(rc)/V(rc)	G(rc)/$\mathit{\rho }$(rc)
1a	B1-Al5	0.0611	0.1090	0.0505	−0.0739	−0.0234	0.2250	0.6838	0.8266
	B1-Al4	0.0705	0.1370	0.0628	−0.0912	−0.0284	0.2320	0.6884	0.8909
	B1-Al6	0.0611	0.1090	0.0505	−0.0739	−0.0234	0.2250	0.6838	0.8266
	RCP	0.0213	0.0135	0.0075	−0.0115	−0.0041	0.2850	0.6460	0.3492
	Al3-Mg2	0.0243	0.0123	0.0088	−0.0144	−0.0057	0.3090	0.6065	0.3603
	B1-Al3	0.0705	0.1370	0.0628	−0.0912	−0.0284	0.2320	0.6884	0.8909
	Al4-Mg2	0.0243	0.0123	0.0088	−0.0144	−0.0057	0.3090	0.6065	0.3603
2n	B1-Al6	0.0611	0.1090	0.0505	−0.0738	−0.0233	0.2260	0.6839	0.8254
	B1-Al4	0.0688	0.1250	0.0590	−0.0867	−0.0277	0.2400	0.6805	0.8583
	B1-Al3	0.0688	0.1250	0.0590	−0.0867	−0.0277	0.2400	0.6805	0.8583
	B1-Al5	0.0611	0.1090	0.0505	−0.0738	−0.0233	0.2260	0.6839	0.8254
	RCP	0.0203	0.0104	0.0056	−0.0086	−0.0030	0.3840	0.6502	0.2762
	Al4-Mg2	0.0273	0.0068	0.0088	−0.0159	−0.0071	0.3960	0.5536	0.3215
	Al3-Mg2	0.0273	0.0068	0.0088	−0.0159	−0.0071	0.3960	0.5536	0.3215
1c	B1-Al4	0.0677	0.1040	0.0541	−0.0821	−0.0281	0.2630	0.6584	0.7985
	B1-Al3	0.0677	0.1040	0.0541	−0.0821	−0.0281	0.2630	0.6584	0.7985
	B1-Al6	0.0633	0.1090	0.0521	−0.0770	−0.0249	0.2350	0.6771	0.8231
	B1-Al5	0.0633	0.1090	0.0521	−0.0770	−0.0249	0.2350	0.6771	0.8231
	RCP	0.0197	0.0186	0.0069	−0.0092	−0.0023	0.2600	0.7533	0.3527
	Al4-Mg2	0.0289	0.0076	0.0095	−0.0172	−0.0077	0.4010	0.5550	0.3303
	Al3-Mg2	0.0289	0.0076	0.0095	−0.0172	−0.0077	0.4010	0.5550	0.3303
2a	B1-Al3	0.0689	0.1130	0.0570	−0.0857	−0.0287	0.2540	0.6651	0.8276
	B1-Al4	0.0689	0.1130	0.0570	−0.0857	−0.0287	0.2540	0.6651	0.8276
	B1-Al5	0.0625	0.1130	0.0526	−0.0769	−0.0243	0.2240	0.6841	0.8417
	B1-Al6	0.0625	0.1130	0.0526	−0.0769	−0.0243	0.2240	0.6841	0.8417
	B1-Mg2	0.0296	0.0557	0.0189	−0.0239	−0.0050	0.1550	0.7918	0.6395
1n	B1-Al4	0.0653	0.1100	0.0538	−0.0802	−0.0264	0.2430	0.6708	0.8231
	B1-Al3	0.0653	0.1100	0.0538	−0.0802	−0.0264	0.2430	0.6708	0.8231
	B1-Mg2	0.0336	0.0765	0.0248	−0.0305	−0.0057	0.1460	0.8137	0.7381
	B1-Al6	0.0585	0.0997	0.0472	−0.0695	−0.0223	0.2250	0.6791	0.8078
	B1-Al5	0.0585	0.0997	0.0472	−0.0695	−0.0223	0.2250	0.6791	0.8078
3c	B1-Al5	0.0541	0.0513	0.0343	−0.0558	−0.0215	0.2960	0.6149	0.6339
	B1-Al6	0.0541	0.0513	0.0343	−0.0558	−0.0215	0.2960	0.6149	0.6339
	B1-Mg2	0.0346	0.0739	0.0247	−0.0309	−0.0062	0.1540	0.7985	0.7135
	B1-Al4	0.0649	0.1240	0.0564	−0.0817	−0.0254	0.2220	0.6895	0.8681
	B1-Al3	0.0649	0.1240	0.0564	−0.0817	−0.0254	0.2220	0.6895	0.8681
	RCP	0.0378	0.0184	0.0137	−0.0229	−0.0091	0.4410	0.6006	0.3636
	Al6-Al5	0.0428	−0.0327	0.0061	−0.0204	−0.0143	0.8580	0.2999	0.1429

. The contour diagram of ∇${}^2$$\rho$(r) along with BCP paths and the corresponding ELF plots are shown in Figure 14 and Figure 15. The existence of bonds between the central boron atom and four surrounding Al atoms is confirmed in both ptB (1a, 2n, and 1c) and ppB (2a, 1n, and 3c) systems. However, additional Mg atom coordination is found in the case of all the ppB species. Although the electron density at B–Mg BCP is lower compared to other BCPs, the low values of electron density ($\rho$(r${}_c$)) at the BCPs indicate closed-shell type bonding. This is also evident from the positive ∇${}^2$$\rho$(r${}_c$), which indicates lower electron density at the BCPs. The total energy density H(r${}_c$) = G(r${}_c$)+V(r${}_c$) at the corresponding BCPs is negative for all the cases, suggesting a partial covalent character. The G(r${}_c$)/V(r${}_c$) values are within 0.5 and 1, which indicate the absence of a non-covalent interaction and partial covalent character. This is also supported by the G(r${}_c$)/$\rho$(r${}_c$) values, which are less than 1. The ELF plots show the extent of electron sharing among the central B to the peripheral atoms and electron density delocalization within the BAl${}_4$Mg system. In the ppB system, there is electron delocalization between B and Mg besides B and Al, which is absent in the case of ptB BAl${}_4$Mg species, which supports our previous analyses.

3.8. ALMO-EDA

We obtained an energy decomposition analysis (EDA) based on absolutely localized molecular orbitals (ALMO-EDA) calculations using the QCHEM program [103]. The total interaction energy is composed of four components as shown in Equation (1)

$$\begin{array}{cc}\hfill \Delta E_{int}& =E_{molecule}-\sum _Z^{frags}{E}_Z\hfill \\ \hfill & =\Delta E_{PREP}+\Delta E_{FRZ}+\Delta E_{POL}+\Delta E_{CT}\hfill \end{array}$$

(1)

Two fragments named X(BAl${}_4^{-/0/+}$) and Y(Mg) were considered to check the intramolecular interactions for all the six different structures (both ptB and ppB). The sharing of interaction energy ($\Delta E_{int}$) terms such as the preparation energy ($\Delta E_{PREP}$), frozen energy ($\Delta E_{FRZ}$), polarization energy ($\Delta E_{POL}$), and charge transfer ($\Delta E_{CT}$) are given in Figure 16. Herein, the $\Delta E_{FRZ}$ increases for ppB BAl${}_4$Mg${}^{-/0/+}$ compared to ptB BAl${}_4$Mg${}^{-/0/+}$. This term shows the energy change from the interaction of two fragments without the spin-coupling (SC), polarization (POL), or charge transfer (CT). It includes electrostatics, Pauli repulsion, exchange-correlation as well as dispersion corrections. The smaller value of $\Delta E_{PREP}$ shows that the geometry distortion and orbital rehybridization of each fragment to the original structure are very small in all cases. Two terms $\Delta E_{POL}$ and $\Delta E_{CT}$ demonstrate that polar, charge-shift, and ionic-type interactions are present in all the six cases. Similar observations for ptC systems are reported elsewhere [108]. Thus, our calculations for ptB/ppB systems support the notion that there is charge transfer from Y(Mg) to X(BAl${}_4^{-/0/+}$), as with ptC systems. Therefore, we can formulate our global and local minima of ptB and ppB as [X]${}^{-}$[Y]${}^{+}$.

4. Conclusions

Various isomers of BAl${}_4$Mg${}^{-/0/+}$ are theoretically identified for the first time using a tabu-search algorithm and chemical intuition. Three isomers (1a, 2n, and 1c) containing ptB atom and three isomers (2a, 1n, 3c) containing ppB atom are studied in detail. The global minima for the anion and cation contain a ptB atom with 18 and 16 valence electrons, respectively. The global minimum for the neutral exhibits a ppB atom with 17 valence electrons. The low-lying isomers of the anion (2a) and cation (3c) with 18 and 16 valence electrons, respectively, also exhibit a ppB atom. Ab initio MD simulations carried out at 298 K indicate that all isomers are kinetically stable except 3c. At this temperature, the cation geometry (3c) breaks apart from its original structure. However, low-temperature simulations carried out at 100 and 200 K suggest that 3c retains planarity and thus remains kinetically stable. Therefore, detecting 3c through gas phase experiments is viable at low temperatures. The electron affinity value of BAl${}_4$Mg is 2.15 eV, and therefore it is possible to detect 1n (ppB) through 1a (ptB). Energy decomposition analysis carried out for all these six systems indicates that the interaction between fragments BAl${}_4^{-/0/+}$ and Mg is ionic.