The B₂ Structural Motif as a Tool for Modulating Ring Currents in Monocyclic Li Clusters

Abstract

Magnetically induced current densities, calculated at the M06-2X/def2-TZVP level using the diamagnetic-zero version of the continuous transformation of origin of current density (CTOCD-DZ) method, were employed to study the aromaticity in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$. It was found that the ${\mathrm{Li}}_3/{\mathrm{Li}}_4$ rings in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ remarkably resemble the monocyclic ${\mathrm{Li}}_3{}^{+}$ and ${\mathrm{Li}}_4{}^{2+}$ clusters. Unlike the parent ${\mathrm{Li}}_3{}^{+}$ and ${\mathrm{Li}}_4{}^{2+}$ systems that sustain negligibly weak global current density circulation, the ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ clusters exhibit a strong diatropic current density. The present work demonstrates how structural modifications introduced by the ${\mathrm{B}}_2$ unit can be used for modulating the current density in cyclic Li-based clusters.

1. Introduction

Cyclic electron delocalization is among the most intriguing phenomena in chemistry. If present in a molecule, cyclic electron delocalization determines rather unique structural, magnetic, thermodynamic, and reactivity-based properties [1,2,3]. All these features coming due to cyclic electron delocalization are usually referred to as aromaticity. Understanding, defining, and evaluating the magnetic aspects of molecular aromaticity have been in focus of chemists for almost a whole century [4,5]. Already, London [6] recognized that an aromatic molecule sustains an induced ring current as a response to the applied external magnetic field perpendicular to the molecular plane. Nowadays, it is widely accepted that the specific behavior of molecules in the presence of the external magnetic field can be used for qualitative and quantitative assessment of aromaticity [4,5,7,8]. The magnetically induced current density is the most important concept in the evaluation of the magnetic aspect of aromaticity. Based on the current densities, one can measure both global and local aromaticity [8]. In addition, other indices of magnetic aromaticity, including the nucleus-independent chemical shift (NICS) [9], as one of the most employed aromaticity indices, can be obtained from the induced current density [4,10]. On the other hand, there is no unique way to reconstruct current density maps from NICS values [11].

One of the most important challenges in the computation of magnetically induced current densities and related molecular magnetic response properties is how to treat the so-called gauge origin problem [8]. The gauge origin problem refers to the dependence of the calculated magnetic properties on the translation of the origin of the coordinate system. The gauge origin problem can be generally eliminated either by employing a basis set with an explicit magnetic field dependence or by an explicit treatment of the gauge origin dependence [8]. The first group of methods are based on perturbation-dependent basis functions, such as gauge-including atomic orbitals (GIAO) [6]. Calculation of the induced current density by means of the GIAO can be performed by means of the gauge-including magnetically induced current (GIMIC) method [8,12]. Within the second approach, the most employed methods are based on the continuous transformations of origin of current density (CTOCD) technique [13,14]. In this approach, the gauge origin problem is resolved by using a different gauge origin for each point for which the current density is calculated. It should be noted that this method was originally introduced as the so-called continuous set of gauge transformation (CSGT) method and that the name CTOCD was suggested somewhat later [15,16,17]. Over almost a half of a century, Riccardo Zanasi contributed to the development of the CTOCD method and other approaches for calculation of the magnetic response properties, as well as in the software implementation of these theories [18,19]. In the present work, the diamagnetic zero (DZ) version of the CTOCD method was employed. In the CTOCD-DZ, also known as the ipsocentric method [20,21], the current density at each point is calculated by choosing itself as the origin of the vector potential. More details on the CTOCD method can be found in reviews [4,16] and elsewhere [22].

There are many cases showing that the NICS and current density results do not always agree [23,24,25,26,27]. One of the most-known examples is the monocyclic ${\mathrm{Li}}_3{}^{+}$ cluster. ${\mathrm{Li}}_3{}^{+}$ has two valence electrons, and according to the Hückel 4n + 2 rule, this system should be aromatic. Large negative NICS values wrongly predicted the aromatic character of ${\mathrm{Li}}_3{}^{+}$ [23] because this system does not sustain global ring currents, but rather local circulations around Li atoms. In what follows, the magnetically induced current densities in two recently characterized complexes [28], ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$, are examined and compared to those in the corresponding reference ${\mathrm{Li}}_3{}^{+}$ and ${\mathrm{Li}}_4{}^{2+}$ systems. To the best of the authors’ knowledge, the target ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ clusters have not so far been experimentally observed. However, a closely related species, ${\mathrm{Na}}_3{\mathrm{Al}}_2{}^{-}$, which is isoelectronic to ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$, has been recently experimentally characterized [29]. In what follows, it is shown how the presence of the ${\mathrm{B}}_2$ unit can drastically change magnetic response properties of the parent ${\mathrm{Li}}_3/{\mathrm{Li}}_4$ ring.

2. Computational Methods

The molecular structures were fully optimized at the M06-2X/def2-TZVP level of theory by means of the Gaussian 09 program [30]. Frequency calculations confirmed that the obtained structures have no imaginary frequencies. The NBO analysis [31] was performed at the same level as implemented in Gaussian 09 program. Magnetically induced current densities were calculated at the M06-2X/def2-TZVP level using the CTOCD-DZ approach. These calculations were performed using our in-house Fortran code, which requires the data from the Gaussian 09 wfx-file (obtained with the keyword NMR = CSGT). The current densities were calculated for a grid of points belonging to the plane formed by Li atoms. The so-obtained current density maps were visualized using the Paraview program [32], utilizing the following convention: counterclockwise/clockwise circulations represent diatropic/paratropic current densities. The bond current strengths were obtained through the disc-based quadrature scheme [33], which employs numerical integration, as devised by Elhay and Kautsky [34]. The integration disc perpendicularly bisects the given bond, and the disc radius is the average of the covalent radii of the two bonded atoms.

The multicenter delocalization index (MCI) [3], which quantifies electron delocalization in a given ring, can be calculated using different partition schemes [35,36,37]. In this work, the natural atomic orbital (NAO) density matrices obtained from NBO analysis were used to calculate the MCI. In order to eliminate the ring-size dependence of the MCI, the nth root of the original index, denoted by ${\mathrm{MCI}}^{1/\mathrm{n}}$, was employed [3]. These calculations were performed using our in-house FORTRAN program. The NICS values were calculated at the M06-2X/def2-TZVP level of theory through the gauge-including atomic orbital (GIAO) method [38,39]. The dissected canonical molecular orbital NICS (CMO-NICS) was calculated using the NBO 7.0 program [40].

3. Results and Discussion

The optimized structures of ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$, having ${\mathrm{C}}_{2\mathrm{v}}$ and ${\mathrm{D}}_{4\mathrm{h}}$ symmetries, respectively, are presented in Figure 1. These structures can be described as inverse sandwich complexes, consisting of the ${\mathrm{Li}}_3/{\mathrm{Li}}_4$ ring sandwiched between two boron atoms. Both studied complexes are characterized by a short B−B distance: 1.53 and 1.52 Å in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$, respectively. These B−B distances are shorter than those reported in the ${\mathrm{Al}}_3{\mathrm{B}}_2{}^{-}$ cluster, for which the respective B−B bond was characterized as triple [41]. It is worth noting that, according to the NBO analysis, the B−B bond in studied clusters is a triple bond: 1σ + 2π bonds.

It has previously been shown that the ${\mathrm{Li}}_3{}^{+}$ cluster exhibits the ${\mathrm{D}}_{3\mathrm{h}}$ structure. It was found that the presence of the ${\mathrm{B}}_2$ unit in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ elongates the Li−Li distances (being equal to 3.30 and 3.65 Å) with respect to that in monocyclic ${ \mathrm{Li}}_3{}^{+}$ cluster (3.05 Å) optimized at the same level of theory. The analogous planar ${\mathrm{D}}_{4\mathrm{h}}$ geometry of the bare ${\mathrm{Li}}_4{}^{2+}$ complex does not correspond to minima on the potential energy surfaces. The global minimum of ${\mathrm{Li}}_4{}^{2+}$ was found to have tetrahedral ${\mathrm{T}}_{\mathrm{d}}$ structure [42]. The strain in the ${\mathrm{Li}}_4$ ring of the ${\mathrm{Li}}_4{\mathrm{B}}_2$ complex is reduced in comparison to the ${\mathrm{Li}}_3$ ring in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$. Along these lines, it was observed that Li−Li distance in ${\mathrm{Li}}_4{\mathrm{B}}_2$ is 2.89 Å, which is significantly shorter than that in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$.

Natural atomic charges of the studied systems are shown in Figure 2. As can be seen, B atoms in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ carry negative charges of −1.18 and −1.27, respectively. This reveals a significant charge transfer from Li atoms to the central ${\mathrm{B}}_2$ unit. The obtained charges on B atoms are approximately equal to −1, and ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ can be viewed as charge-transfer complexes: ${\left[\mathrm{B}\right]}^{-}{{[\mathrm{Li}}_3]}^{+}{\left[\mathrm{B}\right]}^{-}$ and ${\left[\mathrm{B}\right]}^{-}{{[\mathrm{Li}}_4]}^{2+}{\left[\mathrm{B}\right]}^{-}$. The calculated dissociation energies (

Table 1. Dissociation energies (DE, in kcal/mol) at the M06-2X/def2-TZVP level of theory.

Dissociation Path	DE
${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-} \to { \mathrm{Li}}_3{}^{+}{+\mathrm{B}}_2{}^{2-}$	343.24
${\mathrm{Li}}_4{\mathrm{B}}_2 \to { \mathrm{Li}}_4{}^{2+}{+\mathrm{B}}_2{}^{2-}$	569.61
${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-} \to { \mathrm{Li}}_3{+\mathrm{B}+\mathrm{B}}^{-}$	233.26
${\mathrm{Li}}_4{\mathrm{B}}_2 \to { \mathrm{Li}}_4{}^{+}{+\mathrm{B}+\mathrm{B}}^{-}$	358.96
${\mathrm{B}}_2{}^{2-} \to { 2\mathrm{B}}^{-}$	−18.92

) showed a strong stabilizing electrostatic interaction between the ${\mathrm{Li}}_3{}^{+}/{\mathrm{Li}}_4{}^{2+}$ and ${\mathrm{B}}_2{}^{2-}$ fragments in the studied clusters.

The obtained structural and electronic features of ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ indicate that the ${\mathrm{Li}}_3$ ring in this complex bears a significant resemblance to the monocyclic ${\mathrm{Li}}_3{}^{+}$ cluster. In what follows, the nature of magnetically induced current densities in the ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ complex is compared with that in the monocyclic ${\mathrm{Li}}_3{}^{+}$ cluster. As mentioned above, the global minimum of ${\mathrm{Li}}_4{}^{2+}$ has a tetrahedral structure. Therefore, the current density induced in ${\mathrm{Li}}_4{\mathrm{B}}_2$ will be contrasted with the monocyclic ${\mathrm{Li}}_4{}^{2+}$ cluster whose structure was extracted from the optimized geometry of the ${\mathrm{Li}}_4{\mathrm{B}}_2$ complex.

The calculated NICS and ${\mathrm{MCI}}^{1/\mathrm{n}}$ values of the studied systems are presented in

Table 2. ${\mathrm{MCI}}^{1/\mathrm{n}}$, ${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$, and CMO-${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$ values of the Li-atom rings in the studied systems. The results obtained with the ma-def2-TZVP basis set are given in the brackets.

Molecule	${\mathbf{MCI}}^{1/\mathbf{n}}$	${\mathbf{NICS}\left(0\right)}_{\mathbf{ZZ}}$	$\mathsf{\sigma }-{\mathbf{NICS}\left(0\right)}_{\mathbf{ZZ}}$	$\mathsf{\pi }-{\mathbf{NICS}\left(0\right)}_{\mathbf{ZZ}}$
${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$	0.4214 (0.4012)	−54.62 (−53.58)	−45.48	−6.2
${\mathrm{Li}}_4{\mathrm{B}}_2$	0.3176 (0.3081)	−46.21 (−46.50)	−36.17	−6.64
${\mathrm{Li}}_3{}^{+}$	0.7631 (0.7631)	−9.00 (−9.06)	−11.47	-
${\mathrm{Li}}_4{}^{2+}$	0.6575 (0.6575)	−9.59 (−9.73)	−11.08	-

. In order to verify these values, the diffuse functions were introduced through the ma-def2-TZVP basis set. It was revealed that the addition of the diffuse functions invokes only negligible changes in the aromaticity index values. The reference ${\mathrm{Li}}_3{}^{+}$ and ${\mathrm{Li}}_4{}^{2+}$ systems are characterized by large and negative ${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$ values, in agreement with the previous work [23]. The CMO σ-${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$ showed the ${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$ values in these two systems are practically fully determined by the HOMO level contribution. It should be noted that the NICS(0) in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ is calculated in, or very close, to the B–B bond critical point [43]. Therefore, the ${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$ values cannot be used to compare the aromaticity of the target complexes and their reference ${\mathrm{Li}}_3{}^{+}{/\mathrm{Li}}_4{}^{2+}$ systems. The ${\mathrm{MCI}}^{1/\mathrm{n}}$ values predict a high extent of electron delocalization in the reference ${\mathrm{Li}}_3{}^{+}$ and ${\mathrm{Li}}_4{}^{2+}$ systems. Based on the ${\mathrm{MCI}}^{1/\mathrm{n}}$ values, the introduction of the ${\mathrm{B}}_2$ fragment into the ${\mathrm{Li}}_3/{\mathrm{Li}}_4$ rings causes a reduction in the intensity of cyclic electron delocalization. This result goes in line with the calculated NBO-based Wiberg bond indices, which also suggest notably weaker Li–Li bonding interactions in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ than in the reference bare Li-clusters.

Current density maps of the studied ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ complexes, as well as those of the reference ${\mathrm{Li}}_3{}^{+}$ and ${\mathrm{Li}}_4{}^{2+}$ systems, are shown in Figure 3 and Figure 4. In agreement with the previous study [23], it can be seen that the ${\mathrm{Li}}_3{}^{+}$ cluster shows no significant ring currents, but only local circulations around Li atoms. The same was found for the ${\mathrm{Li}}_4{}^{2+}$ structure (Figure 4b). On the other hand, the ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ systems sustain rather strong diatropic ring currents. The most striking feature of the ring currents in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$/${\mathrm{Li}}_4{\mathrm{B}}_2$ is that the strongest circulations occur inside the ${\mathrm{Li}}_3/{\mathrm{Li}}_4$ rings and around the B–B bonds. The obtained current densities in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$/${\mathrm{Li}}_4{\mathrm{B}}_2$ resembles the current densities of π-electrons in the isolated ${\mathrm{B}}_2{}^{2-}$. However, the current density maps from Figure 3 and Figure 4 clearly show that the Li atoms are substantially involved in the circulation occurring in the ring center. In addition, the isolated ${\mathrm{B}}_2{}^{2-}$ unit was found to be unstable relative to the dissociation product 2${\mathrm{B}}^{-}$ (Table 1), which also confirms a strong stabilization of the ${\mathrm{B}}_2{}^{2-}$ fragment thorough the interaction with Li atoms in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$. The obtained current densities completely oppose the conclusions derived from the MCI values.

The nature of the current densities induced in the studied complexes was further analyzed by examining structures in which the B–B distances were simultaneously elongated by Δd value, while the Li-nuclei positions were kept frozen. The current density maps of the so-obtained geometries are displayed in Figure 5. As can be seen in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$, increasing the B–B distances leads to a continuous weakening of the current density induced inside the Li-atom rings.

Table 3. Natural atomic charges of B atoms, ${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$, and ${\mathrm{MCI}}^{1/\mathrm{n}}$ values of the structures in which the B–B distances were simultaneously elongated by Δd value, while the Li-nuclei positions were kept unchanged.

Δd	Charge of B		${\mathbf{NICS}\left(0\right)}_{\mathbf{ZZ}}$		${\mathbf{MCI}}^{1/\mathbf{n}}$
	${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$	${\mathrm{Li}}_4{\mathrm{B}}_2$	${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$	${\mathrm{Li}}_4{\mathrm{B}}_2$	${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$	${\mathrm{Li}}_4{\mathrm{B}}_2$
0.0	−1.175	−1.265	−54.62	−46.21	0.4213	0.3178
0.3	−1.179	−1.265	−49.89	−46.03	0.2581	0.3146
0.6	−1.161	−1.148	−40.86	−36.03	0.1613	0.1495
0.9	−1.089	−0.947	−31.87	−28.46	0.2728	0.2141
1.2	−0.886	−0.717	−25.67	−24.02	0.2571	0.1968
1.5	−0.740	−0.551	−22.21	−22.17	0.2444	0.1821
1.8	−0.621	−0.414	−20.16	−21.45	0.2315	0.1899
2.1	−0.513	−0.293	−19.01	−21.55	0.2237	0.2258

contains natural atomic charges of B atoms, as well as aromaticity indices of Li-atom rings obtained for the structures with the modified B–B distances. As can be seen, the charge of B atoms changes from around −1, at the equilibrium distance, to approximately −1/2 at the dissociation limit. In other words, the ${\mathrm{B}}_2$ unit changes the charge from −2 to −1. While the ${\mathrm{NICS}\left(0\right)}_{\mathrm{ZZ}}$ values show the same trend as the current densities and NBO charges, the ${\mathrm{MCI}}^{1/\mathrm{n}}$ magnitudes do not change regularly with increasing distance between B atoms in the studied complexes. The ${\mathrm{MCI}}^{1/\mathrm{n}}$ also confirms that the B–B bond elongation reduces the intensity of cyclic delocalization in the two clusters. This implies that both ${\mathrm{MCI}}^{1/\mathrm{n}}$ and current density give the same prediction on the aromatic character of the ${\mathrm{Li}}_3{/\mathrm{Li}}_4{}^{+}$ systems obtained by the B–B bond elongation in the examined ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ clusters. The energetic data from Table 1 show that the dissociation of the studied complexes in ${\mathrm{Li}}_3{/\mathrm{Li}}_4{}^{+}$ and ${\mathrm{B}}_2{}^{-}$ fragments is thermodynamically preferred over the dissociation in ${\mathrm{Li}}_3{}^{+}{/\mathrm{Li}}_4{}^{2+}$ and ${\mathrm{B}}_2{}^{2-}$ fragments.

Current density maps are mainly used as a qualitative tool, and for quantitative purposes, the bond current strengths can be utilized.

Table 4. Bond current strengths (in nA T⁻¹) for all symmetry-unique bonds.

Bond	${\mathbf{Li}}_3{}^{+}$	${\mathbf{Li}}_4{}^{2+}$	${\mathbf{Li}}_3{\mathbf{B}}_2{}^{-}$	${\mathbf{Li}}_4{\mathbf{B}}_2$
Li1–Li2	2.28	1.59	8.81	8.11
Li2–Li3			8.57

presents bond current strengths for the studied molecules. The ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ complexes sustain strong diatropic current densities, with the respective bond current strengths of 8.81/8.57 and 8.11 nA T⁻¹. For comparison, the C–C bond current strength for benzene at the same level of theory is 4.75 nA T⁻¹. The obtained results show that the magnetic properties of the ${\mathrm{Li}}_3{}^{+}{/\mathrm{Li}}_4{}^{2+}$ systems are substantially modulated with the introduction of the ${\mathrm{B}}_2{}^{2-}$ unit.

Within the ipsocentric method, total current density can be partitioned into molecular orbital contributions. The HOMO electrons in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ contribute to a rather weak circulation that occurs in the very center of the ${\mathrm{Li}}_3/{\mathrm{Li}}_4$ ring (Figure 3d and Figure 4d). On the other hand, the quasi-degenerate HOMO-1 ${(2\mathrm{b}}_1{+5\mathrm{a}}_1)$ level in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and the doubly degenerate HOMO-1 ${(2\mathrm{e}}_{\mathrm{u}})$ level in ${\mathrm{Li}}_4{\mathrm{B}}_2$ give the most important contribution to the total current density (Figure 3e and Figure 4e).

Relevant subsections of the orbital interaction diagrams of the ${\mathrm{Li}}_3{}^{+}{/\mathrm{Li}}_4{}^{2+}$ and ${\mathrm{B}}_2{}^{2-}$ fragment valence orbitals are shown in Figure 6 and Figure 7. The HOMO (${6\mathrm{a}}_1$) in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ is formed by mixing HOMO of ${\mathrm{Li}}_3{}^{+}$ and LUMO of the ${\mathrm{B}}_2{}^{2-}$ fragment. The quasi-degenerate HOMO-1 level, consisting of ${2\mathrm{b}}_1{ \mathrm{and} 5\mathrm{a}}_1$ orbitals, comes as a result of the combination of LUMO of the ${\mathrm{Li}}_3{}^{+}$ fragment and the degenerate HOMO-1 level of the ${\mathrm{B}}_2{}^{2-}$ fragment. The same bonding situation was found in ${\mathrm{Li}}_4{\mathrm{B}}_2$ (Figure 7).

The CTOCD-DZ approach provides a unique interpretative capability, as the orbital current density contributions can be related to the virtual transitions between occupied and virtual orbitals. The virtual transitions must obey symmetry-based selection rules [20,21]. If the product of symmetries of the occupied and virtual orbitals contains the in-plane translation/rotation, then the given virtual excitation contributes to diatropic/paratropic current density. As already shown [23], the HOMO current density rather weakly contributes to the global currents in ${\mathrm{Li}}_3{}^{+}$ because the effect of the translational transition from HOMO ${(2\mathrm{a}}_1{}^{\prime })$ to LUMO $\left(2{\mathrm{e}}^{\prime }\right)$ is annulled by the rotational transition to the virtual orbital ${1\mathrm{a}}_2{}^{\prime }$ (Figure 6). In ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$, the HOMO electrons give small contribution to the global circulation. This is because the translational transition from HOMO $\left({6\mathrm{a}}_1\right)$ to LUMO $\left({3\mathrm{b}}_1\right)$ is accompanied by the rotational transition to the ${5\mathrm{b}}_1$ orbital. There are translational transitions from the quasi-doubly degenerate HOMO-1 ${(2\mathrm{b}}_1{+5\mathrm{a}}_1)$ level to the quasi-doubly degenerate virtual orbital level ${4\mathrm{b}}_1{+9\mathrm{a}}_1$. These transitions are of the most importance for the global circulation in the ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ cluster.

A very similar relation between the found magnetic properties and electronic structure can be found in the ${\mathrm{Li}}_4{\mathrm{B}}_2$ system. Figure 7 demonstrates that the HOMO electrons in ${\mathrm{Li}}_4{}^{2+}$ induce weak currents because of the two supposed transitions: translational and rotational from HOMO ${(4\mathrm{a}}_{1\mathrm{g}}$) to LUMO ${(3\mathrm{e}}_{\mathrm{u}})$. This way, the HOMO current density contribution is quite small. On the other hand, the translational transition from the doubly degenerate HOMO-1 ${(2\mathrm{e}}_{\mathrm{u}})$ level to the virtual ${1\mathrm{b}}_{2\mathrm{g}}$ level gives rise to the strong currents induced by the HOMO-1 electrons.

4. Conclusions

The present paper shows how the ${\mathrm{B}}_2$ unit can be used for the rational structural modifications of cyclic Li-based clusters, which can cause substantial changes in their aromatic characteristics. The magnetically induced current densities, as the most reliable magnetic criterion, revealed the aromatic character of ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ clusters. The orbital analysis showed that in both ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$, the main contribution to the global currents comes from the doubly degenerate HOMO-1 level, which can be deduced from the combination of the LUMO of the ${\mathrm{Li}}_3{}^{+}{/\mathrm{Li}}_4{}^{2+}$ fragment and the doubly degenerate π-MO level of the ${\mathrm{B}}_2{}^{2-}$ fragment. Quite surprisingly, the MCI values predict a low level of electron delocalization in ${\mathrm{Li}}_3{\mathrm{B}}_2{}^{-}$ and ${\mathrm{Li}}_4{\mathrm{B}}_2$ clusters and, on the other hand, intensive electron delocalization in the reference ${\mathrm{Li}}_3{}^{+}{/\mathrm{Li}}_4{}^{2+}$ systems. It is difficult to understand such a difference between the current-density- and electron-delocalization-based aromaticity predictions, and its rationalization will remain a task for the future.