Stability and bonding of the multiply coordinated bimetallic boron cycles: B₈M₂²⁻, B₇NM₂ and B₆C₂M₂ with M = Sc and Ti

Abstract

A theoretical investigation of the geometry, stability and aromaticity of boron clusters doped by two Sc and Ti atoms was carried out using DFT calculations. The Sc and Ti atoms form the bimetallic boron cycles not only with the eight-membered ring B₈²⁻ but also with the isoelectronic species B₇N and B₆C₂ rings yielding the planar B₈²⁻, B₇N and B₆C₂ rings sandwiched to two transition metals. The N and C atoms prefer to form eight-membered hetero-rings via the classical 2e–2c bonding rather than occupy high coordination positions. Both C atoms avoid binding each other. The thermodynamic stability of all bimetallic boron cycles is induced by the stabilizing overlap between the bonding and anti-bonding MOs of a metallic dimer M₂ with levels of an eight-membered ring. These stabilizing interactions also release two sets of delocalized σ and π MOs, which obey the (4N + 2) electron count. Such a double σ and π aromaticity feature is clearly supported by the magnetic ring current flows.

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1. Introduction

The diversity in geometrical structures of small boron clusters is bewildering. While it is well known that the B₁₂ icosahedral three-dimensional (3D) cage is a structural unit in many boron polyforms,¹ this structure is found to be much less stable than its planar isomer.² Previous studies on a large number of pure boron clusters show that the bare boron clusters basically have four different classes of structures involving the planar (or quasi-planar), the tubular (double ring, triple ring and multiple ring…), the bowl shaped, and the fullerene-like and cage forms. Most of the boron clusters B_n having sizes smaller than n < 20 atoms are planar or quasi-planar,³ whereas the lowest-lying structures of the even sizes B_2m in the range of 2m = 20–26 have a double ring form. The latter is the simplest tubular structure, which consists of two B_m strings often connected in an anti-prism fashion.⁴ The triple ring form is observed in B₂₇ and B₄₂ which result from a superposition of three B₉ and B₁₄ strings in an anti-prism motif, respectively.^5,6 Regarding the cationic state, 3D structure is already found for the size of B₁₇⁺ (ref. 7) whereas the planar and quasi-planar forms remain the global minimum structures for anionic boron clusters up to the size of B₂₇⁻.^5,8 The B₃₀, B₃₂ and B₃₆ clusters present the bowl shape which contains a large hole, and their electron distribution satisfies the disk-aromaticity.^6,9,10 A fullerene motif of structure is established in the case of B₃₈ and B₄₀ clusters.^11–13 Recently, the B₄₄ cluster has been shown to have a cage-like structure containing two nonagonal holes. Such a feature is unprecedented in elemental clusters.¹⁴

Another class of boron clusters, represented firstly by the anions B₈²⁻ and B₉⁻, has planar cyclic geometry in which a B atom occupies the central position of the B₇ and B₈ rings, respectively.¹⁵ It is remarkable that a doping of one transition metal M gives rise to the M@B_n planar cyclic structures in which the M atom is surrounded by the B₈, B₉ and B₁₀ rings.⁸ The stability of the planar metallic boron cycles can be rationalized in terms of strong orbital interactions where the transition metal uses its d-AOs to form two delocalized bonding patterns. The latter satisfy the classical (4N + 2) electrons count in both series of delocalized π and σ MOs therefore they are classified as doubly π and σ aromatic. Stimulated by the finding that the B@B₅H₅⁺ cluster is a planar cycle,¹⁶ in which one B atom occupies the central position of the B₅H₅⁺ pentagonal cycle, a number of doped M@B_nH_n cyclic forms have been established.¹⁷ The planarity of these systems are rationalized by the aromaticity with 10 π electrons. Although several boron rings centered by other main group elements were previously investigated, none of them was found to be a global minimum on the corresponding potential energy surface.^18,19 The central atom needs to form delocalized π and σ bonds with the B_n ring, but when a main group element having a larger electronegativity than B, it favors connection to other atoms through simpler 2 center–2 electrons (2c–2e) bonds. As for typical examples, carbon–boron mixed cluster reveals many B@B_xC_y boron hetero-cycles where C atoms participate to ring not in central position.^20–23 The electronic partition analysis shows that C atoms provide to the delocalized π and σ MOs and subsequently stabilize boron cycles.

For the boron-based clusters multiply doped by transition metal atoms, some organometallic compounds were observed. In the series of bimetallic complexes B_nX_nRe₂, the boron ring is uniformly coordinated with two Re atoms. Of the derivatives B₅X₅ and B₆X₆, the boron units were found to be perfectly planar pentagon and hexagon motifs, respectively.^24,25 The planar B₆ ring was found to play as a key structural motif in the solid compound Ti₇Rh₄Ir₂B₈,²⁶ in which the B₆ unit is sandwiched by two Ti atoms. Recently, we reported on a novel motif of structure in which two metals are coordinated vertically and oppositely with the planar boron ring.¹⁸ In these bimetallic boron cycles having Fe and Co as dopants, the global energy minima of the resulting B₇Co₂, B₇Fe₂ and B₇CoFe clusters contain each a perfectly planar B₇ cycle, which is sandwiched with two metallic atoms vertically placed along the C₇ axis.²⁷ The high thermodynamic stability of these bimetallic boron cycles arises from a combination of two stabilizing effects. On the one hand, the M₂ dimeric metals introduce their electrons to fill up the empty levels of the B₇ string, and on the other hand, both anti-bonding and bonding MOs of M₂ are significantly stabilized through orbital interactions with the cyclic B₇ counterpart.

Although bimetallic boron clusters have been identified in the seven-membered boron rings with Fe and Co as dopants, the derivatives containing larger boron rings such as eight-membered ring are not known yet. As mentioned above, the main group elements such as the C and N atoms tend to prefer formation of classical 2c–2e bonds, in part due to their higher electronegativity. As a prediction, these atoms possibly connect with boron atoms to reveal hetero-cycles and subsequently, in the presentation of metals, the bimetallic hetero-boron cycles could be established. In this context, we set out to search for new members of the family of bimetallic boron cycles containing eight-membered rings using quantum chemical computations. Our extensive theoretical search on the geometry and electronic structure of several B₈M₂ species reveals that the M = Sc and Ti atoms are quite suitable for this purpose. For the eight-membered rings, beside the neutral B₈ and its dianionic B₈²⁻ ring, simple substitutions generating the B₇N ring (with substitution of B²⁻ by N) and the B₆C₂ ring (with replacement of two B⁻ anions by two C atoms) are also considered. The latter systems allow us to analyze the behavior of the series of isoelectronic systems. The electronic structure and chemical bonding feature of this class of bimetallic boron cycles will be examined using the electron localization indicator. Their aromatic character is further probed using the magnetic response of the electron density as described by the ring current maps.

2. Computational methods

The potential energy surfaces of the B₈M₂, B₈M₂²⁻, B₇NM₂ and B₆C₂M₂ clusters, with M = Sc and Ti, are explored using the stochastic genetic algorithms.²⁸ In addition, the guess structures are also manually built by adding two metal atoms to the framework of the pure B_n species. Initial B₇NM₂ and B₆C₂M₂ structures obtained following substitution of a single B atom by a N, or two B atoms by two C are also generated on the basis of the B₈M₂²⁻ counterparts.

Geometric optimizations and harmonic vibrational analyses are carried out using density functional theory (DFT) with the hybrid TPSSh,²⁹ which was previously shown to give better results on relative energies for boron clusters than those obtained with other available functionals, as compared to the high accuracy MO coupled-cluster method (CCSD(T)).³⁰ The PBE1PBE³¹ and BPW91 (ref. 32) functionals are found to excellently predict geometry and spin state of Fe(C₆H₆)_n systems, therefore these functional are use in the current investigation where the B₆C₂ and B₇N containing structures are considered.³³ In order to evaluate the reliability of three functionals, the bond length, vibrational frequency, dissociation energy and electron affinity (EA) of the dimers Sc₂ and Ti₂ are calculated and compared to available experimental values.^34–40 As shown in Table 1, in comparison to available experimental values, the TPSSh and PBE1PBE in combining with the cc-pVTZ basis set produce better results than the BPW91/cc-pVTZ calculation.

Table 1 Bond length (d(M₂), angstrom), vibrational frequency (Freq., cm⁻¹), dissociation energy (D_o, eV) and electron affinity (EA, eV) of Sc₂ and Ti₂ obtained by TPSSh, PBE1PBE and BPW91 calculations and compared to experiment (Exptl)

	Exptl			TPSSh				PBE1PBE				BPW91
	d(M2)	Freq.	Do	d(M2)	Freq.	EA	Do	d(M2)	Freq.	EA	Do	d(M2)	Freq.	EA	Do
a CASSCF calculations taken from ref. 33.b Ref. 36 and 37.c Ref. 35.d Ref. 38.e Ref. 39.f Ref. 40.
Sc2	2.51^a	238.9^c	1.65^e	2.6	258.3	0.68	1.1	2.6	267.3	0.94	0.76	2.6	241.3	2.9	−0.5
Ti2	1.94^b	407.9^d	1.54^f	1.90	502.3	0.35	2.0	1.95	536.8	1.0	1.2	1.90	470.7	0.7	2.2

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The initial structures are first optimized using the small 3-21G(d) basis set.^41,42 The optimized lower-energy isomers, being within a range of 50 kcal mol⁻¹ on relative energies, are subsequently reoptimized using the same TPSSh functional, but in conjunction with the larger cc-pVTZ basis set.^43,44 Electronic structure theory computations are performed using the Gaussian 09 suite of program.⁴⁵

The magnetic response is calculated using the CTOCD-DZ method⁴³ implemented in the SYSMO program,^46,47 which is connected to the GAMESS-UK package.⁴⁸ The ring current approach^49,50 is calculated using the B3LYP/6-311G(d) level. In the current density map, the contour and shading show the modulus of induced current density and narrows display its projection on plotting plane. As for a convention, anticlockwise or clockwise circulations correspond to diatropic or paratropic current, respectively. The diatropic current flow indicates an aromatic character, whereas the paratropic current flow suggests an anti-aromatic character.

3. Results and discussion

3.1 Geometries of the B₈M₂, B₈M₂²⁻, B₇NM₂ and B₆C₂M₂ clusters

Only the eight-membered boron cycles are considered in this investigation. As for a convention, the structure is noted as M2.X.n, in which M2 = Sc2 and Ti2 stands for two metal dopants, X = B8, C2 and N according to the pure B₈ cycle and the substituted derivatives by two and one B atoms by two C and one N atoms, respectively, and finally n = 1, 2, 3… denotes the isomers with increasing energy ordering. When relative energies are mentioned, the values are consistently referred to the corresponding most stable isomer n = 1. In the dianionic B₈M₂²⁻, the isomers are referred to as M2.B8.da.n.

Fig. 1 displays the shape of the energetically lower-lying isomers of both series B₈M₂ and B₈M₂²⁻ with M = Sc and Ti obtained using the TPSSh/cc-pVTZ level. Geometry of B₆C₂M₂ and B₇NM₂ are displayed in Fig. 2. As for a comparison, their relative energy obtained by using three different TPSSh, PBE1PBE and BPW91 functionals in various spin states are listed in Table 1.

Fig. 1 Energetically lower-lying isomers of (a) B₈Sc₂, (b) B₈Sc₂²⁻, (c) B₈Ti₂ and (d) B₈Ti₂²⁻ clusters. Geometry optimizations and relative energies (kcal mol⁻¹ with ZPE corrections) were obtained using the TPSSh/cc-pVTZ computations.

Fig. 2 Lower-lying isomers of (a) B₆C₂Sc₂ and B₆C₂Ti₂ clusters, and (b) B₇NSc₂ and B₇NTi₂ clusters. Geometry optimizations and energies (kcal mol⁻¹ with ZPE corrections) were performed using the TPSSh/cc-pVTZ computations.

For the neutral B₈Sc₂ cluster, the triplet spin state Sc2.B8.1 is confirmed as the ground state by TPSSh, PBE1PBE and BPW91 calculations whereas the quintet multiplicity is highly unstable. The structural characteristic of Sc2.B8.1 shows that two Sc atoms are vertically coordinated to the planar B₈ string. All DFT calculations demonstrate that the next isomer Sc2.B8.2, in three spin states, is extremely unstable. In the following section, the structural motif in which two metal atoms are vertically sandwiched in the opposite sides of a planar eight-membered ring is referred to as a ‘bimetallic planar cyclic cluster’. Following attachment of two electrons, the bimetallic cyclic structure Sc2.B8.da.1 is also found to be the lowest energy isomer. TPSSh/cc-pVTZ calculations point out that the quintet multiplicity is the ground state of Sc2.B8.da.1, whereas both PBE1PBE/cc-pVTZ and BPW91/cc-pVTZ computations predict the triplet spin state as the ground state. The fish-like structure Sc2.B8.da.2 is predicted to be highly unstable by all DFT calculations (Table 1).

The boron cyclic form is not favored in the neutral B₈Ti₂. The fish-like structure Ti2.B8.1 (C_s ¹A′) in which both metal atoms are coordinated to a B₆ string and two B atoms capped at two different sites of the same B–B edge, is the ground state of B₈Ti₂. The Ti2.B8.2 bimetallic cycle (D_8h ¹A_1g) is much less stable as indicated by TPSSh, PBE1PBE and BPW91 calculations (Table 1). However, the bimetallic boron cycle Ti2.B8.da.1 turns out to be again the lowest-energy structure for the dianion B₈Ti₂²⁻, being ∼10 kcal mol⁻¹ below the corresponding fish-like structure (Table 1). It should be stressed that both Sc and Ti dopants give rise to the bimetallic cyclic boron structures with high symmetry in the dianion state.

Similar to the B₈Ti₂²⁻ and B₈Sc₂²⁻ dianions, both isoelectronic neutral B₆C₂Sc₂ and B₆C₂Ti₂, where the two B⁻ centers of B₈ are replaced by two C atoms, remain stable in the bimetallic cyclic motif with the emergence of a B₆C₂ planar heterocycle. In the case of B₆C₂Sc₂, the TPSSh, PBE1PBE and BPW91 calculations show that the singlet Sc2.C2.1 and Sc2.C2.2 isomers are degenerate with an energy gap of <1 kcal mol⁻¹, and as a consequence they can be regarded as competing ground state for the neutral mixed B₆C₂Sc₂. Within these structures, two Sc atoms are coordinated vertically but oppositely to the B₆C₂ perfect planar ring.

Similar structure is observed also in the case of Ti-dopant. As shown in Fig. 2a, a hetero-boron cycle structure, Ti2.C2.1, is identified as the ground state of B₆C₂Ti₂ with high symmetry and low spin multiplicity (singlet state) by all functionals used. Some other bimetallic boron cycles are observed on the potential energy surface of B₆C₂Ti₂, but they are significantly less stable than Ti2.C2.1. It is remarkable that the bimetallic boron cycles B₆C₂M₂ with M = Sc and Ti containing C–C connection are highly unstable, which results from the fact that in the formation of bimetallic carbon–boron cycles with two C atoms, the C dopants avoid to meet and to bind to each other.

Substitution of a single B site of B₈Sc₂²⁻ and B₈Ti₂²⁻ by an N atom also releases a bimetallic cyclic structure, Sc2.N.1 and Ti2.N.1 again with a planar B₇N hetero-ring. The fish-like structures Sc2.N.2 and Ti2.N.2 are found at ∼10–14 kcal mol⁻¹ above the corresponding ground state, predicted by TPSSh, PBE1PBE and BPW91 calculations, respectively. The next isomers Sc2.N.3 and Ti2.N.3 are highly unstable in both singlet, triplet and quintet states.

Overall, the dianion B₈M₂²⁻ with M = Sc and Ti and their isoelectronic neutral B₆C₂M₂ and B₇NM₂ clusters provide us with the new members of the class of bimetallic boron cycles. Of these species, it is clear that Sc2.C2.1, Sc2.N.1, Ti2.C2.1 and Ti2.N.1 present the bimetallic hetero-boron cycles in which two metals are coordinated to B₆C₂ and B₇N hetero-ring. The most interesting result here is that both N and C atoms participate together with boron atoms in the formation of the eight-membered hetero-ring rather than to be located in highly coordinated position as the two metals.

3.2 Energetic properties

In order to explore the stability of bimetallic boron cycles, the dissociation energy (DE) where parent structure dissociates giving a metallic dimer plus the corresponding ring, is calculated using three functionals and given in Table 3. Additionally, the binding energy (BE, Table 4) are also performed to quantify their stability. The DE values of the channel in which B₈M₂ and B₈M₂²⁻ structures release the global minimum B₈ and B₈²⁻ plus M₂ dimers, are significantly lower (Table 2) and illustrates that B₈M₂ and B₈M₂²⁻ favor the dissociation channel giving B@B₇ and B@B₇²⁻. The DE value of the B₈M₂²⁻ dianion is somewhat lower than that of the neutral. The connection between M₂ and B₈²⁻ dianionic ring becomes thus weaker than that between M₂ and B₈ neutral ring. This is supported by binding energy calculations in which BE value of neutral B₈M₂ is greater than the corresponding B₈M₂²⁻. Similarly, in comparison to B₈M₂ neutral, DE values of B₇NM₂ and B₆C₂M₂ cluster with M = Ti and Sc are smaller, therefore the interactions between M₂ and B₇N and B₆C₂ rings are weaker than that of in the B₈M₂ neutral. Although the substitution of B atom by N and C atoms in B₈²⁻ ring results in many new BN, CB bonds which are stronger than the BB bond in B₈²⁻ ring, the interactions of two M atoms with the rings either in B₇NM₂ and B₆C₂M₂ or in B₈M₂ and B₈M₂²⁻ cycles are of the same order of magnitude.

Table 2 The relative energy (kcal mol⁻¹) of B₈M₂, B₈M₂²⁻, B₆C₂M₂ and B₇NM₂ clusters with M = Sc and Ti at various spins states using TPSSh, PBE1PBE and BPW91 functionals combining with cc-pVTZ basis set

Structure	TPSSh			PBE1PBE			BPW91
	Singlet	Triplet	Quintet	Singlet	Triplet	Quintet	Singlet	Triplet	Quintet
Sc2.B8.1	3.4	0.0	48.7	4.0	0.0	48.0	4.0	0.0	44.3
Sc2.B8.2	22.9	47.5	68.7	22.3	46.0	66.3	22.2	38.1	60.6
Ti2.B8.1	0.0	15.5	41.8	0.0	21.8	41.8	0.0	15.4	42.0
Ti2.B8.2	6.1	6.4	11.8	10.1	9.5	11.4	5.7	7.7	15.9
Ti2.B8.3	14.9	18.8	42.6	15.0	17.6	38.3	12.5	17.7	43.2
Sc2.B8.da.1	6.8	15.2	0.0	12.3	0.0	16.8	15.0	0.0	4.2
Sc2.B8.da.2	25.1	20.3	31.3	38.7	27.3	33.1	33.4	26.1	33.6
Ti2.B8.da.1	0.0	3.5	0.5	0.5	0.0	0.0	0.0	15.9	6.9
Ti2.B8.da.2	10.3	9.0	17.8	8.3	1.1	27.0	9.4	6.7	18.5
Ti2.B8.da.3	28.9	20.4	23.4	26.7	16.2	22.4	27.0	19.5	23.4
Sc2.C2.1	0.0	15.7	49.7	0.0	16.0	45.7	0.0	14.5	47.3
Sc2.C2.2	0.6	20.2	45.2	0.5	20.6	45.7	0.7	18.7	43.4
Sc2.C2.3	6.5	22.6	47.0	6.6	22.3	47.5	6.1	19.5	45.3
Ti2.C2.1	0.0	6.8	11.0	0.0	4.0	6.0	0.0	1.5	14.1
Ti2.C2.2	6.6	4.2	17.5	6.8	1.5	12.6	6.2	6.0	16.5
Ti2.C2.3	12.1	6.1	16.4	12.0	12.1	11.7	10.2	7.4	18.8
Sc2.N.1	0.0	10.1	34.3	0.0	8.2	35.6	0.0	8.9	32.0
Sc2.N.2	13.1	18.4	46.7	11.3	17.7	36.3	11.9	17.1	35.7
Sc2.N.3	22.5	41.0	71.9	20.8	39.4	74.3	21.1	40.3	68.1
Ti2.N.1	0.0	10.6	19.0	0.0	14.3	14.1	0.0	15.1	26.2
Ti2.N.2	11.6	14.6	38.6	10.5	10.7	27.8	14.6	17.5	41.7
Ti2.N.3	18.7	40.0	43.1	17.8	34.7	36.5	21.7	41.1	51.0

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Table 3 The dissociation energy of bimetallic cycles obtained by TPSSh, BPW91 and PBE1PBE calculations

	TPSSh	BPW91	BPE1PBE
B8Sc2 → B@B7 + Sc2	230.8	289.7	224.6
B8Sc2^2− → B@B7^2− + Sc2	174.8	216.0	171.5
B8Ti2 → B@B7 + Ti2	218.9	306.9	215.2
B8Ti2^2− → B@B7^2− + Ti2	165.0	232.9	156.9
B8Sc2 → B8 + Sc2	328.0	312.0	333.5
B8Sc2^2− → B8^2− + Sc2	332.1	271.2	288.3
B8Ti2 → B8 + Ti2	327.4	406.6	334.9
B8Ti2^2− → B8^2− + Ti2	286.5	299.7	289.8
B7NSc2 → B7N + Sc2	286.8	271.1	289.0
B7NTi2 → B7N + Ti2	259.9	257.9	261.1
B6C2Sc2 → B6C2 + Sc2	305.5	289.7	309.5
B6C2Ti2 → B6C2 + Ti2	279.4	272.5	282.2

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Table 4 The binding energy (D₀, kcal mol⁻¹) of bimetallic cycles obtained. Using three different functionals with the cc-pVTZ basis set

	TPSSh	BPW91	PBE1PBE
B8Sc2	106.8	110.1	109.3
B8Sc2^2−	104.8	108.8	107.8
B8Ti2	107.7	111.4	109.3
B8Ti2^2−	105.9	110.0	107.3
B7NSc2	115.5	114.9	118.0
B7NTi2	117.0	117.0	118.3
B6C2Ti2	126.0	130.0	127.6
B6C2Sc2	125.5	129.2	128.4

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It should be noted that although the B₈Ti₂ bimetallic cycle is not the global minimum, this structure has a greater DE value than B₈Ti₂²⁻ cycle which is determined as the ground state. In combination with calculations of the binding energy, these results show that the interaction of M₂ dimer with B₈, B₈²⁻, B₇N and B₆C₂ rings which is one main contributor to the stability of bimetallic boron cycles, is quantitatively comparable. The delocalization or aromaticity is more important in stability, particular for di-anions B₈M₂²⁻, B₇NM₂ and B₆C₂M₂ cycles, as shown in following sections.

3.3 Metal–metal bonding

For an understanding on the behavior of the two metallic atoms, the Wiberg bond indexes (WBI)^51,52 are performed for both free metal dimers, and the M–M bonds in bimetallic boron cycles, and the calculated results are given in Table 5. The bond lengths of the free diatomic Sc₂ and Ti₂ amount to 2.8 and 1.94 Å,⁵³ respectively, and WBI values are found as 0.25 and 0.78. As seen from Table 5, the bond lengths calculated for the M₂ connection in all bimetallic boron cycles including B₈M₂²⁻, B₆C₂M and B₇NM₂ is ∼0.5 Å longer than those of free metal dimers, while the WBI values of M–M connections change according to the nature of the dopant. The B₈M₂²⁻ clusters have WBI values of ∼0.6 for Ti₂ and Sc₂, but these are of 0.9 and 0.5 for B₇NTi₂ and B₇NSc₂ clusters, respectively. For B₆C₂M₂, WBI values amount to 1.2 and 0.5 for Ti and Sc dopants, respectively. On the geometrical aspect, formation of bimetallic boron cycles tends to increase distance between two metal atoms. Although the WBI value of M₂ in B₈M₂²⁻ is reduced, the B₆C₂M₂ and B₇NM₂ result in larger values of WBI for the M₂ unit. According to the meaning of WBI, the presence of C or N dopants tends to enhance the strength of M–M bonds, in spite of their longer distances.

Table 5 Bond length of M–M, B–M, C–M, N–M, C–B and N–B (Å) and WBI values of M–M connections of the bimetallic boron cycles

Species	Parameter	Ti	Sc
a Value obtained for B8M2^2− cyclic structure.
M2	d(Exptl)	1.94	2.51
	d(M2)	1.90	2.6
	WBI	0.78	0.25
B8M2/B8M2^2−	d(M–M)	2.92, 2.49	2.94, 2.83
	d(B–M)^a	2.4	2.4
	WBI	0.64	0.65
B7NM2	d(M–M)	2.7	2.9
	d(B–M)	2.3	2.4
	d(N–M)	3.0	2.3
	d(N–B)	1.4	1.4
	WBIM–M	0.9	0.49
B6C2M2	d(M–M)	2.5	2.9
	d(B–M)	2.3	2.4
	d(C–M)	2.4	2.3
	d(C–B)	1.4	1.4
	WBIM–M	1.2	0.48

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3.4 Orbital interactions

To probe further the chemical bonding feature and thereby the stability of bimetallic boron cycles, it is imperative to consider the bonding model which raises from d–d interactions.⁵⁴ As for a representative example, Fig. 3 illustrates the orbital interactions of the Ti₂ dimer with the B₈²⁻ string. The results obtained for Ti₂ can be generalized for others bimetal-doped boron cycles.

Fig. 3 Orbital interaction diagram of the Ni₂ dimer with the B₈²⁻ string.

The Ti atom has an electron configuration of [Ar](3d)²(4s)², and as a consequence, the Ti₂ dimer has enough electrons to fully occupy an electronic configuration of [σ²_4sσ²_3dπ⁴]. The two σ MOs include the σ_4s orbital which is resulted from overlap of the 4s AOs, and the σ_3d which is a combination of 3d AOs. The anti-bonding δ*, π* and σ* MOs are filled, and thereby, the strength of metal–metal bond is reduced.

The delocalized MO pattern of the ring B₈²⁻ appears to satisfy a disk aromatic framework, which arises from a model of a particle in a circular box.⁵⁵ Within this model, the boundary condition gives two quantum numbers, namely the radial quantum number n and the rotational quantum number m. The radial quantum number has values of n = 1, 2, 3, … whereas m = 0, ±1, ±2, … which is denoted as m = σ, π, δ, …, respectively. The state with non-zero value of m shall thus be doubly degenerate. The lowest-lying eigenstates in ascending energy are 1σ, 1π, 1δ… etc. The full occupation of degenerate eigenstates which correspond to 2, 6, 12, 16, 20, … electrons leads to a disk aromatic character.

The orbital interaction of the B₈²⁻ disk configuration and the Ti₂ bond inherently enhances the stability of boron cycles, as displayed in Fig. 3. From there, the π*-MOs of the Ti₂ dimer whose occupation significantly reduces the strength of the Ti–Ti bond, are stabilized upon interaction with the doubly degenerate 1π levels of either the ring B₇⁻ or B₈, and thereby produces the π levels of the entire system. Interestingly, the δ*-MOs of Ti₂ enjoy a stabilizing overlap with the vacant 1δ level, and releases the doubly degenerate δ* levels, as given in Fig. 3. Interaction of the bonding σ_4s and the 2σ level ends up in a creation of the σ_4s MOs for the B₈Ti₂²⁻ bimetallic dianion. The π bonding MOs of Ti₂ have an enhanced overlap with the 2π level of the B₈²⁻ disk, whereas the 2δ-MOs of the B₈²⁻ counterpart undergo a stabilizing interaction with the δ-MOs of Ti₂ yielding the δ levels of the complex. As a result, the orbital configuration of the B₈Ti₂²⁻ cyclic structure as […(σ_4s)²(π)⁴(π*)⁴(δ)²(δ*)⁴] is now fully occupied by 16 electrons.

The electron configurations of the different bimetallic cyclic boron clusters are listed in Table 6. The π* and δ* levels of the isoelectronic cycles to B₈Ti₂²⁻ involving B₆C₂Ti₂ and B₇NTi₂ are fully occupied (Table 6). As a result, they tend to enhance the stability of cyclic structures, although they reduce the strength of Ti–Ti connection. The σ_4s and π levels are actually fully occupied, whereas the δ molecular orbitals are filled by two (2) electrons. These MOs increase the strength of Ti–Ti bond, and stabilize consequently the bimetallic cyclic boron clusters. The dianion B₈Sc₂²⁻ has the orbital configuration of …(σ_4s)²(π*)⁴(π)⁴(δ*)⁴ in which the π* and δ* MOs are fully occupied, as given in Table 2. The π* and δ* MOs of the isoelectronic boron cycles with B₈Sc₂²⁻ involving B₇NSc₂ and B₆C₂Sc₂ are also fulfilled by 4 electrons for each cluster. It is obvious that they are significant contributor to the stability of boron cycles containing the Sc-dopant. The full occupation of σ_4s and π levels contributes significantly to the thermodynamic stability of cyclic structure, due to the enhanced Sc–Sc connection. Therefore, a successive occupation of σ_4s, π, π* and δ* levels leads to the formation of bimetallic boron cycles. It is clear from Table 6 that the B₆C₂Ti₂ and B₇NTi₂ have different MO patterns, in which the appearance of C and N atoms, that are more electronegative than B atom, leads to a substantial change in the MO energy levels with expected splitting of the doubly degenerate MOs.

Table 6 Orbital configurations of the bimetallic boron cycles considered

Cluster	Orbital configuration
B8Sc2^2−	(σ4s)^2(π*)^4(π)^4(δ*)^4
B8Ti2^2−	(σ4s)^2(π*)^4(π)^4(δ*)^4(δ)^2
B7NSc2	(σ4s)^2(π*)^2(π)^2(π*)^2(π)^2(δ*)^4
B7NTi2	(π*)^2(σ4s)^2(π*)^2(π)^4(δ*)^4(σ3d)^2
B6C2Sc2	(σ4s)^2(π)^2(π*)^2(π)^2(π*)^2(δ*)^4(σ3d)^0
B6C2Ti2	(σ4s)^2(π)^2(π*)^2(π)^2(π*)^2(δ*)^4(σ3d)^2

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Fig. 4 displays a correlation diagram of three isoelectronic bimetallic boron cycles B₈Ti₂²⁻, B₆C₂Ti₂ and B₇NTi₂. As expected, the doubly degenerate levels of the B₈Ti₂²⁻ cycle including the π, π* and δ* are now split into two separate levels in both B₆C₂Ti₂ and B₇NTi₂ cycles. Due to the influence of the more electronegative C and N atoms, the energy of the σ_4s, π*, π, δ* and δ levels of B₆C₂Ti₂ and B₇NTi₂ become higher than those of the B₈Ti₂²⁻. Overall, the orbital interactions demonstrate that the high thermodynamic stability of bimetallic boron cycles is a consequence of two factors, namely, (i) the empty levels of B₈²⁻ involving 2π, 1δ and 2δ are occupied, and (ii) the bonding and anti-bonding MOs of M₂ are involved in stabilizing interactions with the levels of the disk B₈²⁻, which invariably enhance the stability of the resulting clusters. As a consequence, the B₈M₂^0/2−, B₇NM₂ and B₆C₂M₂ clusters are stabilized within the bimetallic cyclic motif.

Fig. 4 Orbital correlation diagram for the B₈Ti₂²⁻, B₆C₂Ti₂ and B₇NTi₂ clusters.

3.5 An analysis of the electron distribution

In order to explore further the chemical bonding phenomena, an analysis of the electron density using the electron localization indicator (ELI-D) technique⁵⁶ is performed, and the resulting partition of the densities is shown in Fig. 5. A similar localization domain pattern is thus observed for all bimetallic clusters considered. The di-synaptic basin V(B,B) is found for two B atoms of the B₈ ring in both B₈Sc₂²⁻ and B₈Ti₂²⁻, therefore the B atoms connect with each other via the classical 2e–2c bonds, and the latter go around the B₈ planar ring. For B₇NM₂, it has two localization domains located between N dopant and two B atoms yielding two 2e–2c bonds of the N with B atoms. The same behavior is observed in the mixed carbon–boron B₆C₂M₂ in which the C dopants connect with other B atoms by formation of 2e–2c bonds, as reflected by the presence of the V(C,B) basins. No localization domain, located in the regions between the metal M and the planar rings involving B₈, B₇N and B₆C₂, is observed, indicating a certain ionic character of the M–X bond with X = B, C and N. The localization centered on the M–M bonding is observed only at lower bifurcation values, due to their large radii. However, this does not imply that there is no direct M–M bonding.

Fig. 5 The ELI_D isosurface maps obtained at the bifurcation value of 1.4 for the B₈M₂²⁻, B₆C₂M₂ and B₇NM₂ bimetallic cyclic boron clusters, based on densities obtained at the TPSSh/cc-pVTZ level.

3.6 The aromaticity

The aromatic property of the bimetallic boron cycles considered is now analyzed using the ring current concept, which arises from a magnetic response of the electron density of the whole system. The magnetic response can be partitioned in terms of MO contributions. The contribution of each MO to the total ring current can be calculated, and thereby the participation of each MO to the aromatic character of the whole cluster can be established. The current density maps of both π and σ systems of electrons are produced on the B₈, B₆C₂ and B₇N molecular planes. Fig. 6 displays the ring current flows of the π, σ and total (involving π and σ electrons) for the bimetallic boron cycles, in their molecular plane. The ring current density maps of the π and σ orbitals are illustrated in Fig. 7.

Fig. 6 The total, π and σ ring current maps of isoelectronic systems B₈M₂²⁻, B₇NM₂ and B₆C₂M₂ clusters with M = Sc, Ti (B3LYP/6-311G(d)).

Fig. 7 The ring current maps of π-MOs of B₈Ti₂²⁻, B₇NTi₂ and B₆C₂Ti₂ clusters (B3LYP/6-311G(d)).

All boron cycles containing the Sc-dopant, including B₈Sc₂²⁻, B₆C₂Sc₂ and B₇NSc₂, exhibit a diatropic current density for both π and σ systems of electrons (Fig. 6a). A similar character is identified for both B₈Ti₂²⁻ and B₇NTi₂ structures whose diatropic ring current maps are observed for both π and σ electrons (Fig. 6b). Therefore they can be classified as having a double π and σ aromaticity, according to the magnetic criteria. For the neutral B₆C₂Ti₂ cluster, the ring current of π electrons is of diatropic nature in magnetic response, whereas its σ electrons produce a complicated ring current map. Therefore, the aromatic character of B₆C₂Ti₂ will be considered in a following section on the basis of MO contributions.

It is important to consider the contribution of each MO to ring current maps, in such a way that its participation to the aromaticity can be revealed. Each of the bimetallic boron cycles B₈M₂²⁻, B₇NM₂ and B₆C₂M₂ has 10 π electrons that populate the π* and δ* levels, and the MO-34. As a result, they can be classified as aromatic species, consistent with the classical (4N + 2) Hückel electron count. For a more complete understanding, the ring current maps of their π-MOs are calculated and displayed in Fig. 7 for boron cycles of Ti metal, and in Fig. S1 of the ESI† file for cycles containing Sc dopants. It is clear that the π ring current flows are mainly contributed, on the one hand, by two doubly degenerate MOs, namely the δ* and π* orbitals. On the other hand, the overlap of anti-bonding δ* and π*-MOs of the dimeric metal with δ and π levels of the disk configuration achieves the π diatropic current for bimetallic boron cycles. For σ electrons, the doubly degenerate π-MO brings in the main participation to diatropic ring current maps (Fig. 8), except for the B₈Ti₂²⁻. In Sc-containing boron cycles, the six σ electrons of boron cycles populating the σ_4s and π levels are again consistent with indication of ring current criterion. Subsequently, the (4N + 2) electron count also works for σ electrons of Sc clusters. Similar results are obtained for the B₆C₂Ti₂ and B₇NTi planar cycles in which the π levels give the main contribution to the ring current. The σ_3d MO of B₆C₂Ti₂ is active in the magnetic response and produces a complicated and unclear map for σ electrons. However, this MO can be ignored in the consideration of σ aromaticity due to the fact that it is mainly contributed by σ_3d of Ti₂ dimer, rather than a delocalized MO.

Fig. 8 The ring current maps of σ-MOs of B₈Ti₂²⁻, B₇NTi₂ and B₆C₂Ti₂ clusters (B3LYP/6-311G(d)).

In of the B₈Ti₂²⁻ cycle, there are 8 σ electrons arising from contribution of the doubly occupied δ-MO, beside the σ_4s and π levels, in such a way that this species can be regarded as antiaromatic according to the (4N + 2) electron count. The ring current map established for σ electrons of B₈Ti₂²⁻ clearly illustrates the diatropic nature, and as a consequence, the B₈Ti₂²⁻ cycle should be an aromatic species. On the other hand, the (4N + 2) rule does not work in the case of B₈Ti₂²⁻, simply because the δ-MO does not contribute to the ring current. Overall, the combination of the π-MOs (M₂) with π levels of the eight-membered ring does not only enhance the stability of the resulting MOs, but also induce an aromatic character of the bimetallic cyclic boron clusters.

4. Concluding remarks

In the present theoretical study, we investigated on the geometry, stability and aromaticity of a series of bimetal-doped boron clusters. The Sc and Ti dopants establish the bimetallic cyclic form with the eight-membered boron ring B₈²⁻ and their isoelectronic derivatives B₇N and B₆C₂, in which a perfect planar B₈²⁻, B₇N and B₆C₂ hetero-rings are coordinated with two transition metal atoms.

The thermodynamic stability of these bimetallic boron cycles, that are the global energy minimum of the corresponding systems, can be understood as the result of a stabilizing overlap between bonding and anti-bonding MOs of M₂ with different levels of eight-membered ring. The C and N elements, which are more electronegative than the B atom, also enjoy formation of planar eight-membered ring, via the classical 2c–2e bonding, rather than occupy a high coordination position. The double aromaticity feature, which comprises both σ and π aromaticity, is clearly supported by the magnetic responses of the electron densities within the planar cycles. Nevertheless, such an aromatic character is not always in line with the classical electron count for both delocalized electron systems, and this suggests that the latter cannot reliably be applied to predict the aromatic character of this class of compounds.

Acknowledgements

We thank Ton Duc Thang University (TDTU-DEMASTED) for support. MTN is indebted to the KU Leuven Research Council (GOA program) and FWO-Vlaanderen for continuing support. The authors are grateful to Dr Remco Havenith at University of Groningen, for assistance with the computations of ring current maps.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra04948f

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