Noble gas supported B₃⁺ cluster: formation of strong covalent noble gas–boron bonds

Abstract

The stability of noble gas (Ng) bound B₃⁺ clusters is assessed via an in silico study, highlighting their structure and the nature of the Ng–B bonds. Ar to Rn atoms are found to form exceptionally strong bonds with B₃⁺ having each Ng–B bond dissociation energy in the range of 15.1–34.8 kcal mol⁻¹ in B₃Ng₃⁺ complexes with a gradual increase in moving from Ar to Rn. The computed thermochemical parameters like enthalpy and free energy changes for the Ng dissociation processes from B₃Ng₃⁺ also support the stability of Ar to Rn analogues for which the corresponding dissociation processes are endergonic in nature even at room temperature. The covalent nature of the Ng–B bonds is indicated by the localized natural Ng–B bond orbitals and high Wiberg bond indices (0.57–0.78) for Ng–B bonds. Electron density analysis also supports the covalency of these Ng–B bonds where the electron density is accumulated in between Ng and B centres. The orbital interaction energy is the main contributor (ca. 63.0–64.4%) of the total attraction energy in Ng–B bonds. Furthermore, the Ng–B bonding can be explained in terms of a donor–acceptor model where the Ng (HOMO) → B₃Ng₂⁺ (LUMO) σ-donation has the major contribution.

---

Introduction

The first member of group 13, boron (B), has received much attention from inorganic chemists¹ as well as from theoreticians.² Boron, being an electron deficient element, shows interesting unusual chemical bonding patterns in boron clusters. In recent years, one-, two-centre as well as multi-centre boron species were experimentally synthesized which were found to be stabilized by σ-donor and π-acceptor ligands³ exhibiting bizarre chemical bonding patterns. Tricoordinated borylene complexes were synthesized and generally designated using the formula: L → B(R) ← L,^1a,4 where L = N-heterocyclic carbene (NHC), cyclic alkyl amino carbene (CAAC), carbon monoxide (CO); R = H or bulky aryl group. Here, the single boron atom acts as a Lewis base. Diatomic B₂, having a B≡B triple bond, was also stabilized and isolated in L → B≡B ← L⁵ (L = NHC, CAAC, CO, BO⁻). All of the mentioned ligands are of σ-donor and π-acceptor type in nature. The B≡B triple bond length is correlated with the π-acceptor strength of L. Next discussion comes with triangular boron clusters. Theoretical calculations show that B₃⁺ is the smallest π-aromatic^2a,6 system where the highest occupied molecular orbital (HOMO) is delocalized over the three B atoms. Recently, theoretically⁷ NHC stabilized triangular B₃⁺ cluster is predicted by Tai et al. but its experimental verification is still awaited. However, [Ga₃R₃][M₂]⁸ and [Al₃R₃][M₂]⁹ (M = Na, K; R = bulky aryl group) have been synthesized and isolated. Very recently, Frenking and co-workers¹⁰ showed the formation of boron-carbonyl (B₃(CO)₃⁺) and boron-dinitrogen (B₃(NN)₃⁺) cations by using a pulsed laser and detected it in mass spectra. In those works, the bonding analyses showed that these B₃L₃⁺ complexes are stabilized predominantly by L → B₃L₂⁺ σ-donation.

The long controversy about the capability of noble gas (Ng) to form chemical bonds with other elements was ended in 1962 when Neil Bartlett reported the synthesis of xenon hexafluoroplatinate, Xe⁺(PtF₆)⁻.¹¹ In a very short interval, scientists reported the syntheses of XeF₂,¹² XeF₄,¹³ RnF₂,¹⁴ and KrF₂.¹⁵ These pioneering works opened a new window for the search of new Ng compounds. In the last two decades, while a huge contribution to this field was made by the experimentalists¹⁶ through the syntheses of different Ng compounds, the new prediction of Ng compounds along with the nature of bonding analyses in those synthesized and predicted compounds were also made by theoreticians.¹⁷

In general, there are two types of Ng compounds, first one being Ng insertion compounds within X–Y bond (XNgY) where X and Y cover a large range including H,^18a–c F,^18b,d,e Cl,^18c Br,^18c I,^18c CN,^18c BN⁻,^18d BN–R (R = H, CH₃, CCH, CHCH₂, F, and OH),^18e BO,^18f BS,^18g CO⁺,^18h CS⁺,¹⁸ⁱ OSi⁺,^18j CCH^18k etc. and the second one being donor–acceptor type of compounds (NgXY) such as, NgLiX (X = H, F),^17j NgBeX (X = O, S, CO₃, SO₄, NBO, NCN),^17d,h,v and NgMX (M = Cu, Ag, Au; X = F, Cl, Br, CN, O),^{16n–s,17p,w} where the Ng atom acts as an electron donor. Although a few number of Ng insertion compounds with Ng–B chemical bonds were reported in the literature, the number of examples of the latter type is rather scanty. First Ng inserted boron compound, FXeBF₂ was reported by Goetschel and Loos in 1979.^18l Hu's group predicted a series of Ng inserted boron compounds such as FNgBO (Ng = Ar, Kr, Xe)^18f and FNgBNR (Ng = Ar, Kr, Xe; R = H, CH₃, CCH, CHCH₂, F, OH).^18e The stability of anionic FNgBN⁻ (Ng = He–Rn) compounds,^18d which are isoelectronic with FNgBO,^18f was analyzed by Grandinetti and coworkers. Very recently, the stability of FNgBS^18g and HNgCS⁺¹⁸ⁱ was also investigated by Ghanty and coworkers. In these Ng inserted boron compounds, the bonds between B and Ng atoms are of covalent type as confirmed by different quantum chemical calculations. On the other hand, there are only two types of reported NgB species which have the donor–acceptor type of bonding. While the potential energy surfaces of BNg⁺ (Ng = He, Ne and Ar)^18m species are explored in 1971, Cooks and coworkers detected BNg⁺ (Ng = Ar, Kr and Xe)¹⁸ⁿ in mass spectrometer by halogen/Ng exchange reaction in 1999. Ng supported boron–boron triple bond in Ng–B≡B–Ng (Ng = Ar, Kr) was also studied computationally and it shows remarkably strong Ng–B bonds.^18o

Recently inert N₂ gas is stabilized with elongated N–N bond length in B₃(NN)₃⁺.¹⁰ This result shows B₃⁺ is capable to bond with inert system. So it will be worth checking the binding ability of this B₃⁺ cluster with inert gases and analyze the bonding nature in these complexes. In this work, we have reported the high stability of B₃Ng₃⁺ clusters (Ng = Ar–Rn), thereby exploring the possibility of B₃⁺ detection in Ng matrices. The dissociation energy, changes in enthalpy (ΔH) and Gibbs' free energy (ΔG) for the dissociation processes are computed to check the efficacy of the Ng binding ability of B₃⁺ cluster. The nature of Ng–B bonding is analyzed through the natural bond orbital (NBO), electron density and energy decomposition (EDA) analyses. NBO analysis shows the presence of localized bond orbital in Ng–B bond, whereas the electron density analysis implies the Ng–B bonds to be of covalent type. Finally, EDA in conjunction with natural orbital for chemical valence (NOCV) indicates that the B₃Ng₃⁺ complexes are mainly stabilized by Ng (HOMO) → B₃Ng₂⁺ (LUMO) σ-donation.

Computational details

All the reported geometries of the studied B₃Ng₃⁺ complexes are optimized using MP2 (ref. 19) and CCSD(T)²⁰ methods along with Ahlrichs' basis sets, def2-TZVP and def2-QZVP.²¹ The effective core potential is used for the core electrons of Xe (ecp-28)²² and Rn (ecp-46).²² The harmonic vibrational frequency calculations are also done at the MP2/def2-TZVP level to find the nature of the stationary point and to compute the zero point energy (ZPE) correction and the thermochemical parameters. Due to expensive numerical frequency calculation at the CCSD(T)/def2-TZVP level, we have computed the frequency of the B₃Ar₃⁺ complex only to check that the minimum energy structure is not an artifact of the MP2 level of computation. The absence of any imaginary frequency reveals that the studied complexes are minima on the corresponding potential energy surfaces. Here the ZPE corrected dissociation energy (D₀) values are provided. The dissociation of B₃Ng₃⁺ is considered in two ways. In one way, B₃Ng₃⁺ dissociates into B₃⁺ and 3Ng atoms, thus D₀ is computed as,

D₀ = E₀(B₃⁺) + 3E₀(Ng) − E₀(B₃Ng₃⁺) (1)

where E₀ are the ZPE corrected energies of B₃⁺, Ngs and B₃Ng₃⁺. Here, we have also computed the D₀ values at the complete basis set (CBS) limit. For this purpose, the energies obtained using the def2-TZVP and def2-QZVP basis sets are used and extrapolated them using the equation:where E(X) and E(∞) are the energies obtained at X zeta basis sets (X = 3 for def2-TZVP and 4 for def2-QZVP) and at CBS limit, respectively. For def2 basis set with cardinal values 3 and 4, the α is found to be 7.88.²³

In the second process, the successive dissociation is also considered via formation of B₃Ng₂⁺, B₃Ng⁺ and lastly B₃⁺ where in each step one Ng atom gets detached. The optimization followed by frequency calculations are also done for B₃Ng₂⁺ and B₃Ng⁺ complexes at the MP2/def2-TZVP level. Natural population analysis (NPA)²⁴ and Wiberg bond index (WBI)²⁵ calculations are performed at the MP2/def2-TZVP level to compute the charge (q) at each atomic centre and the bond order, respectively, in NBO²⁶ scheme. All of the above calculations are carried out using the Gaussian 09 program package.²⁷

The electron density descriptors at the bond critical point (r_c) of Ng–B bond are calculated by the topological analysis of the electron density²⁸ at the MP2/def2-TZVP/WTBS level using Multiwfn program.²⁹ The all electron WTBS basis set is used for Xe and Rn atoms.³⁰

The EDA³¹ is carried out in conjunction with NOCV at the revPBE-D3/TZ2P//MP2/def2-TZVP level using the ADF (2013.01) program package.³² Grimme and coworkers³³ in their benchmark study have argued that the revPBE-D3 functional is one of the currently used most robust GGA functionals for related systems. Zeroth order regular approximation (ZORA)³⁴ is adopted to implement scalar relativistic effects for the heavier atoms like Xe and Rn. The EDA scheme decomposes the interaction energy (ΔE^int) between the fragments into four contributing energy terms as,

ΔE^int = ΔE^Pauli + ΔE^elstat + ΔE^orb + ΔE^disp (3)

The repulsive Pauli interaction energy (ΔE^Pauli) originated from the repulsion between the occupied orbitals at the interacting fragments. The term ΔE^elstat corresponds to the classical electrostatic energy between that fragments and it is generally attractive in nature. The next one is the attractive orbital interaction energy, ΔE^orb, which arises from the charge transfer and mixing of the occupied and unoccupied orbitals on the fragments and polarization effect. The ΔE^disp takes care of the dispersion energy correction towards the interacting systems which is an important contribution for heavier elements. The sum of the three attractive energy factors predominates over the ΔE^Pauli and stabilizes the overall system. The EDA–NOCV^31,35 calculation further splits the total differential density Δρ(r) (corresponding to ΔE^orb) into the deformation density (Δρⁱ(r)) and also shows the flow of the electron density. Each deformation density corresponds to the pair wise donation and back donation of electron density between those two fragments and ΔE^orb_i is the associated energy.

Result and discussion

Geometries and dissociation energies

All the minimum energy structures of B₃Ng₃⁺ are found to be at D_3h point group symmetry and in ¹A₁′ electronic state as depicted in Fig. 1. The results associated with the dissociation of B₃Ng₃⁺ into B₃⁺ and 3Ng atoms at the MP2/def2-TZVP, MP2/def2-QZVP, and MP2/CBS levels are provided in Table 1, whereas the same at the CCSD(T)/def2-TZVP level are presented in Table 1-SI.† The ZPE corrected dissociation energy values (D₀) for the dissociation of B₃Ng₃⁺ into B₃⁺ and 3Ng atoms are in the range of 45.4–104.3 kcal mol⁻¹ for Ar–Rn analogues (or in the range of 15.1–34.8 kcal mol⁻¹ for each Ng–B bond) at the MP2/def2-TZVP level, whereas D₀ values for the He and Ne analogues are very small as 0.4 and 3.4 kcal mol⁻¹, respectively (see Table 2-SI†). As D₀ values are very small for He and Ne complexes, we have moved all the results associated with He and Ne to the ESI (see Tables 2-SI–6-SI and Fig. 1-SI and 2-SI†). The results at the MP2/def2-QZVP level also show very similar D₀ values (difference ranging within 0.3–1.8 kcal mol⁻¹) with those at the MP2/def2-TZVP level. Further, the same values at the CBS limit are almost same as those obtained by the def2-QZVP basis set. Nevertheless, as the difference in D₀ values for each Ng–B bond is only 0.1–0.7 kcal mol⁻¹ between def2-TZVP and CBS, we have continued with the def2-TZVP basis set.

Fig. 1 Pictorial depiction of B₃⁺ and B₃Ng₃⁺ (Ng = He–Rn) complexes. All of the above structures are in D_3h point group of symmetry and in ¹A₁′ electronic state. The M–Ng bond distances are in Å unit and the values within first, second and square brackets are computed at the MP2/def2-TZVP, MP2/def2-QZVP and CCSD(T)/def2-TZVP level, respectively.

Table 1 ZPE corrected dissociation energy (D₀, kcal mol⁻¹) of Ng–B bonds, enthalpy change (ΔH, kcal mol⁻¹), and free energy change (ΔG, kcal mol⁻¹) at 298 K for the dissociation process: B₃Ng₃⁺ → B₃Ng₂⁺ + 3Ng at the MP2/def2-TZVP, MP2/def2-QZVP and MP2/CBS levels

Complex		D0	ΔH	ΔG
B3Ar3^+	MP2/def2-TZVP	45.4	46.8	24.4
	MP2/def2-QZVP	44.9	46.3	23.9
	MP2/CBS	44.9	46.3	23.8
B3Kr3^+	MP2/def2-TZVP	64.5	65.8	43.2
	MP2/def2-QZVP	64.8	66.1	43.6
	MP2/CBS	64.8	66.1	43.6
B3Xe3^+	MP2/def2-TZVP	88.4	89.6	67.6
	MP2/def2-QZVP	89.4	90.7	68.7
	MP2/CBS	89.6	90.8	68.8
B3Rn3^+	MP2/def2-TZVP	104.3	105.4	83.6
	MP2/def2-QZVP	102.5	103.7	81.8
	MP2/CBS	102.3	103.4	81.5

---

On the other hand, D₀ values for each Ng–B bond are found to be overestimated at the MP2 level by an amount of 0.2–4.2 kcal mol⁻¹ compared to the CCSD(T) level, except for the case of He (see Table 1-SI†). The D₀ values at the CCSD(T) level are computed by using the ZPE correction obtained from the MP2 level. Only for B₃Ar₃⁺, the ZPE correction obtained from the CCSD(T) level is included and the corresponding D₀ value is found to be 2.1 kcal mol⁻¹ lower than that at the MP2 level. The corresponding Ng–B bond distance is also found to be longer by ∼0.03 Å at the CCSD(T) level compared to the MP2 level. The increasing order of D₀ values from Ar to Rn analogues is obvious from its increase in the polarizability of the Ng atom along the group. The thermochemical parameters like ΔH and ΔG are also computed at 298 K and at one atmospheric pressure. The calculated ΔH values are found to be highly positive for all Ar–Rn analogues implying highly endothermic nature of the dissociation processes. The corresponding ΔG values reveal that the B₃Ng₃⁺ complexes are stable enough at room temperature. Note that the dissociation of B₃Ng₃⁺ into B₃⁺ and 3Ng atoms is entropically favorable process, but the high positive ΔH predominates over the TΔS term, making the dissociation process non-spontaneous one.

The energy difference (ΔE_H–L) between the HOMO and LUMO is a stability indicator of a system. The high value of ΔE_H–L means that the given system is reluctant either to add an electron in LUMO or to remove an electron from HOMO. The ΔE_H–L value for B₃⁺ is found to be 8.61 eV (see Table 2), whereas the increased ΔE_H–L values are 11.08, 10.92, 10.54 and 10.10 eV for Ar, Kr, Xe and Rn bound analogues, respectively. Thus, upon binding with Ng atoms the system becomes more stable than the bare one.

Table 2 HOMO–LUMO energy gap (ΔE_H–L, eV), NPA charges on B and Ng centres (q, au), Wiberg bond indices of Ng–B and B–B bonds (WBI), Ng–B and B–B bond distances (r, Å) in B₃Ng₃⁺ at the MP2/def2-TZVP level

Complex	ΔEH–L	qB	qNg	WBINg–B	WBIB–B	rNg–B	rB–B	r^covNg–B^a
a r^covNg–B is calculated by taking summation of the covalent radii given in ref. 36.
B3^+	8.61	0.33			1.56		1.603
B3Ar3^+	11.08	−0.03	0.37	0.57	1.48	1.979	1.540	1.900
B3Kr3^+	10.92	−0.12	0.45	0.67	1.46	2.092	1.551	2.000
B3Xe3^+	10.54	−0.23	0.56	0.78	1.44	2.240	1.555	2.240
B3Rn3^+	10.10	−0.24	0.57	0.78	1.44	2.319	1.557	2.340

---

Successive dissociation

To get further insight into the stability of intermediate complexes, we have also computed D₀, ΔH and ΔG for the successive detachment of Ng atoms from B₃Ng₃⁺ which leads to the formation of B₃Ng₂⁺, B₃Ng⁺ and at last the bare B₃⁺ cluster (see Table 3 and Fig. 2). The related geometrical parameters are provided in Table 6-SI.† The D₀, ΔH and ΔG values associated with the first, second and third dissociations of the Ng atoms are abbreviated as 1, 2 and 3 in their superscript, respectively. The D₀ value for the first dissociation is the lowest followed by the second dissociation then the third one, i.e., D¹₀ < D²₀ < D³₀. The thermochemical parameters also follow the same order. These results imply that the successive Ng binding ability drops significantly with the gradual increase in the number of Ng atoms. In fact, the strength of Ng–B bonds becomes almost half in additional Ng binding with B₃Ng⁺. It may also be noted that the first Ar detachment process from B₃Ar₃⁺ is exergonic by −0.6 kcal mol⁻¹ at room temperature, which indicates that slightly lower temperature is needed to make B₃Ar₃⁺ viable. The decrease in Ng–B bond strength gradually becomes larger in going from Ar to Rn. This is related to the degree of Ng → B electron donation populating the electron density within B₃⁺ moiety. For first Ng, such electron donation is significantly high as it is shifted to a positively charged B₃ moiety. However, as a result of this electron donation the electropositive nature of B₃ diminishes and consequently for the second and third Ng atoms such donation gets increasingly more reduced, resulting in a lower Ng–B bond strength with an increase in the number of Ng atoms (see Table 3). Note that such Ng → B electron donation is a major stabilizing factor for such complexes as obtained from the EDA (vide infra). Further, since the extent of Ng → B electron donation is increasing larger along Ar–Rn, such drop in D₀ value is more prominent along the same. With an increase in number of Ng atoms, the repulsion between two neighbouring Ng atoms would also contribute to some extent in reducing the Ng–B bond energy and it would be greater for the heavier Ng atoms.

Table 3 ZPE corrected successive dissociation energy (D₀, kcal mol⁻¹) of Ng–B bonds, enthalpy change (ΔH, kcal mol⁻¹) at 298 K, free energy change (ΔG, kcal mol⁻¹) for the respective dissociation process at 298 K, NPA charges on B and Ng centres (q, au), Wiberg bond indices of Ng–B (WBI) of their respective mother moiety at the MP2/def2-TZVP level

Ng	B3Ng3^+^a → B3Ng2^+ + Ng			B3Ng2^+ → B3Ng^+ + Ng						B3Ng^+ → B3^+ + Ng
	D^10	ΔH^1	ΔG^1	D^20	ΔH^2	ΔG^2	qB^b	qNg	WBI	D^30	ΔH^3	ΔG^3	qB^b	qNg	WBI
a For the NPA charges on B and Ng centres (q, au), Wiberg bond indices of Ng–B (WBI) of B3Ng3^+ see Table 2.b qB is the charge of that boron atom which is bonded to Ng-atom; the superscript 1, 2 and 3 denote the first, second and third dissociation process.
Ar	7.0	7.2	−0.6	14.0	14.5	6.9	−0.09	0.42	0.64	24.4	25.1	18.0	−0.18	0.47	0.71
Kr	10.7	10.9	3.0	20.0	20.4	12.9	−0.19	0.52	0.75	33.8	34.5	27.4	−0.30	0.58	0.82
Xe	15.3	15.5	8.0	27.2	27.6	20.1	−0.30	0.64	0.86	45.9	46.5	39.5	−0.43	0.72	0.93
Rn	19.1	19.3	11.8	32.0	32.4	24.9	−0.31	0.66	0.87	53.2	53.8	46.9	−0.44	0.75	0.95

---

Fig. 2 The successive dissociation of Ng atom from B₃Ng₃⁺ via B₃Ng₂⁺, B₃Ng⁺ and to the bare B₃⁺ cluster.

Mixed complexes

The possibility of mixed Ng bound B₃⁺ cluster is studied by considering two types of Ng bound B₃⁺ systems, B₃XYZ⁺ an B₃X₂Y⁺ (X, Y and Z are Ng atoms). In the first type, all the Ng atoms are different and in the second case one Ng atom differs from the others two. The corresponding results are provided in Table 4 and the related geometries are presented in Fig. 3 and 4. The results show that in the presence of heavier Ng atom in B₃XYZ⁺ and B₃X₂Y⁺, the D₀ value for the lighter Ng–B bonds decreases, whereas the heaviest Ng–B bond gets stronger. As for example, the D₀ value for Ar–B bond decreases to 3.2 kcal mol⁻¹ in B₃ArKrXe⁺, 2.6 kcal mol⁻¹ in B₃ArKrRn⁺, 2.0 kcal mol⁻¹ in B₃ArXe₂⁺ and 4.4 kcal mol⁻¹ in B₃Ar₂Xe⁺ from 7.0 kcal mol⁻¹ in B₃Ar₃⁺. A similar result is also noted for Kr. But the Xe–B bond in these mixed complexes (when Rn is not present) becomes stronger when compared with Xe–B bond in B₃Xe₃⁺ complex. The D₀ value of Xe–B bond in B₃Xe₃⁺ complex is 15.3 kcal mol⁻¹, which is enhanced by 5.3 kcal mol⁻¹ in B₃ArKrXe⁺, 3.6 kcal mol⁻¹ in B₃ArXe₂⁺ and 6.9 kcal mol⁻¹ in B₃Ar₂Xe⁺ complexes. Rn also follows the same trend as Xe, but the relative increase in Rn–B bond energy is higher than those in Xe analogues. Therefore, the present results show that if a mixture of Ngs remains present in the systems, only pure complexes of type B₃X₃⁺ would be formed exclusively where the complexes with heaviest Ng atoms would be present in the largest proportion. Even if the complexes like B₃XYZ⁺ and B₃X₂Y⁺ are formed, they would be in very small proportion as the lighter Ng atoms would be spontaneously replaced by the heavier one.

Table 4 ZPE corrected dissociation energy (D₀, kcal mol⁻¹) of Ng–B bonds, enthalpy change (ΔH, kcal mol⁻¹), and free energy change (ΔG, kcal mol⁻¹) at 298 K for the dissociation process: B₃XYZ⁺ → B₃XY⁺ + Z (X, Y and Z = Ng, which includes Ar, Kr, Xe and Rn) at the MP2/def2-TZVP level

Dissociation process	D0	ΔH	ΔG
B3ArKrXe^+ → B3KrXe^+ + Ar	3.2	3.4	−3.6
B3ArKrXe^+ → B3ArXe^+ + Kr	10.6	10.8	3.7
B3ArKrXe^+ → B3ArKr^+ + Xe	20.6	20.7	13.7
B3ArKrRn^+ → B3KrRn^+ + Ar	2.6	2.8	−4.2
B3ArKrRn^+ → B3ArRn^+ + Kr	9.9	10.1	3.1
B3ArKrRn^+ → B3ArKr^+ + Rn	26.2	26.2	19.4
B3ArXe2^+ → B3Xe2^+ + Ar	2.0	2.2	−4.7
B3ArXe2^+ → B3ArXe^+ + Xe	18.9	19.0	11.8
B3Ar2Xe^+ → B3ArXe^+ + Ar	4.4	4.6	−2.9
B3Ar2Xe^+ → B3Ar2^+ + Xe	22.2	22.3	15.4
B3Kr2Xe^+ → B3KrXe^+ + Kr	9.1	9.3	1.6
B3Kr2Xe^+ → B3Kr2^+ + Xe	18.9	19.0	11.9

---

Fig. 3 The dissociation of different B₃XYZ⁺ producing B₃XY⁺ and Z (X, Y and Z are Ng atoms, which includes Ar, Kr, Xe and Rn). Values in first, second and third braces indicates D₀, ΔH and ΔG values respectively at the MP2/def2-TZVP level. ​All the energy values are in kcal mol⁻¹.

Fig. 4 The dissociation of different B₃X₂Y⁺ producing B₃X₂⁺ and Y or B₃XY⁺ and X (X and Y are Ng atoms, which includes Ar, Kr, Xe). Values in first, second and third braces indicates D₀, ΔH and ΔG values respectively at the MP2/def2-TZVP level. ​All the energy values are in kcal mol⁻¹.

NBO analysis

The total charge on the bare B₃⁺ system is equally divided among the three B atoms as 0.33|e| on each B centre. Upon binding with Ng atoms, huge changes in the electronic charge distribution occur (see Table 2). The Ng atoms acquire positive charges on it, whereas the B centres become negatively charged for the Ar to Rn analogous. From this result one can undoubtedly draw the picture of Ng → B charge transfer. The gradual increase in the charge transfer from Ar to Rn is described by the increase in the polarizability of the Ng centres. The WBI values range from 0.57 to 0.78 for Ar to Rn. The high WBI values indicate that there must be a covalent type of interaction in between the B and Ng atoms. The optimized Ar–B and Kr–B bond lengths are found to be closer to the sum of the corresponding covalent radii of the atoms while the Xe–B bond length is equal to the sum of their covalent radii and Rn–B bond length is shorter than the sum of their covalent radii.³⁶ Thus, there must be a covalent interaction between the Ng and B centres. Detailed NBO analysis shows the formation of three Ng–B bond orbitals which are equivalent due to symmetry for a given B₃Ng₃⁺ complex as shown in Fig. 5. The occupancies of these NBOs are found to be close to 2.0, which again proves that the Ng–B is a covalent bond. The Ng–B NBOs are polarized with 72.5–82.4% towards the Ng atom (see Table 5) and p-orbitals take major part in the bonding with 72.0–82.9% from Ng side and 77.5–78.6% from B side.

Fig. 5 The plot of natural bond orbitals (NBO) of Ng–B bonds in B₃Ng₃⁺ (Ng = Ar–Rn) clusters.

Table 5 The NBO occupancy (|e|), contribution from the atoms constructing that NBO and atomic contributions towards that NBO for B₃Ng₃⁺ (Ng = Ar–Rn) complexes are presented at the MP2/def2-TZVP level

Complex	NBO	Occ.	Atomic contribution	Atomic orbitals contribution to NBO
				Ng	B
B3Ar3^+	B–Ar	1.999	B (17.6%)	s (27.8%)	s (20.5%)
			Ar (82.4%)	p (72.0%)	p (78.6%)
B3Kr3^+	B–Kr	1.999	B (21.7%)	s (24.1%)	s (21.2%)
			Kr (78.3%)	p (75.8%)	p (77.9%)
B3Xe3^+	B–Xe	1.999	B (26.9%)	s (21.0%)	s (21.7%)
			Xe (73.1%)	p (78.8%)	p (77.3%)
B3Rn3^+	B–Rn	1.998	B (27.5%)	s (17.0%)	s (21.7%)
			Rn (72.5%)	p (82.9%)	p (77.5%)

---

Electron density descriptors

The Ng–B bond is further discussed in the light of electron density analysis proposed by Bader.²⁸ A covalent bond can be characterized in terms of the accumulation of the electron density (ρ(r_c)) in between the two atomic centres. The Laplacian of the electron density (∇²ρ(r_c)) at r_c is negative for a covalent bond i.e., ∇²ρ(r_c) < 0, whereas ∇²ρ(r_c) > 0 indicates charge depletion at r_c, hence a noncovalent bond. However, there are many examples like, systems with heavier atom³⁷ and some covalent compounds like F₂ (ref. 37d) and CO,²⁸ where the criterion based on ∇²ρ(r_c) fails to predict the covalent bond correctly. Cremer and Kraka^37d suggested the use of local energy density, H(r_c) < 0 as another indicator to describe the covalency of a bond, where H(r_c) = G(r_c) + V(r_c), and G(r_c) and V(r_c) are local kinetic energy density and local potential energy density, respectively. The ∇²ρ(r_c) < 0 is found for the Xe–B and Rn–B bonds whereas the Ar–B and Kr–B bonds have ∇²ρ(r_c) > 0, but H(r_c) < 0, indicating the covalency of these bonds (see Table 6). The contour plots of ρ(r_c) and ∇²ρ(r) for the B₃Ng₃⁺ complexes are depicted in Fig. 6 in which black solid lines shows the ρ(r_c); blue solid lines show the region of ∇²ρ(r) > 0 and magenta dashed lines highlight the area of ∇²ρ(r) < 0 (see Fig. 1-SI† for He and Ne). It is obvious from the contour plots of ρ(r_c) that the electron density of Ngs becomes polarized towards B centres and the degree of deformation in electron density gradually increases in moving towards the heavier Ng atoms. The plots of ∇²ρ(r) reveal that in all the cases, electron density is accumulated in between B and Ng centres and the region of such accumulation becomes gradually larger along Ar to Rn. In cases of Ar and Kr, the bond critical point lies slightly outside the region of electron density accumulation.

Table 6 Electron density descriptors (au) at the bond critical points (r_c) in between Ng and B atoms in B₃Ng₃⁺ (Ng = Ar–Rn) obtained from the wave functions generated at the MP2/def2-TZVP/WTBS//MP2/def2-TZVP level (all electron WTBS basis set is used only for Xe and Rn)

Complex	ρ(rc)	∇^2ρ(rc)	G(rc)	Vρ(rc)	H(rc)
B3Ar3^+	0.066	0.157	0.086	−0.132	−0.046
B3Kr3^+	0.071	0.056	0.071	−0.127	−0.057
B3Xe3^+	0.080	−0.062	0.055	−0.125	−0.070
B3Rn3^+	0.082	−0.124	0.032	−0.095	−0.063

---

Fig. 6 Contour plots of (a) electron density and (b) Laplacian of the electron density (∇²ρ(r)) of B₃Ng₃⁺ (Ng = Ar–Rn) complexes at the MP2/def2-TZVP/WTBS level (black solid lines in (a) shows ρ(r) contours; blue solid lines and magenta dashed lines in (b) show the region with ∇²ρ(r) > 0 and region with ∇²ρ(r) < 0 respectively).

Energy decomposition analysis

The EDA calculations of the B₃Ng₃⁺ systems are carried out taking B₃Ng₂⁺ as one fragment and one Ng atom as other (see Table 6). The positive and negative sign in the energy terms stand for the repulsive and attractive nature of the energies respectively. The maximum contribution towards the Ng–B bond stabilization comes from the ΔE^orb, which ranges from 63.0–64.4% followed by ΔE^elstat (34.4–35.4%). The ΔE^disp (1.2–1.8%) contributes the lowest towards the total stabilization. The plots of deformation density are given in Fig. 7 and 2-SI.† Here, the red colour stands for Δρ(r) < 0 and blue one is for Δρ(r) > 0, i.e., the electron density flows from red to blue region. The Ng → B₃Ng₂⁺ σ-donation is found to be the major contribution towards the ΔE^orb for all Ng atoms. In the Ng → B₃Ng₂⁺ σ-donation the shift in electron density, Δρ(σ₁) occurs from Ng atom which accumulates in between Ng and B centre that further gets shifted towards B₃⁺ ring. Detailed molecular orbital consideration of the Ng and B₃Ng₂⁺ fragments, highlights that the electron density shifts from the HOMO of Ng (choosing p_z orbital) to the LUMO of the B₃Ng₂⁺ (see Fig. 8). The energy corresponding to this Ng → B₃Ng₂⁺ σ-donation is abbreviated as ΔE^σ₁, which ranges from 72.9–78.1% of ΔE^orb for Ar–Rn analogues. ΔE^σ₁ gradually increases from Ar to Rn as similar to the D₀. The shortening of the B–B bond upon complexation with Ng can also be explained from the shape of Δρ(σ₁). Here the σ-donation clearly shows the accommodation of electron density in between the B–B bond of the B₃⁺ ring. This accumulated electron density acts as an extra glue for the B–B bond which shortens the B–B bond. The next pair-wise orbital interaction comes from B₃Ng₂⁺ → Ng back donation in which two orbital terms are originated from the B₃Ng₂⁺ → Ng π-back donation(Δρ(π₁), ΔE^π₁; Δρ(π₂), ΔE^π₂) and one is B₃Ng₂⁺ → Ng σ-back donation (Δρ(σ₂), ΔE^σ₂) (see Table 7 and Fig. 7). The Δρ(π₁) and Δρ(π₂) correspond to shifting of electron density from the HOMO of B₃Ng₂⁺ to a vacant d orbital of Ng and HOMO−1 to another vacant d orbital of Ng, respectively. On the other hand, the electron density shift from HOMO−2 on B₃Ng₂⁺ towards a vacant p orbital on Ng which is responsible for B₃Ng₂⁺ → Ng σ-back donation.

Fig. 7 Plots of deformation densities, Δρ(r), of the pair-wise orbital interactions in of B₃Ng₃⁺ (Ng = Ar–Rn) complexes at the revPBE-D3/TZ2P//MP2/def2-TZVP level. The associated orbital interaction energies are given in kcal mol⁻¹. The colour code of the charge flow is red → blue.

Fig. 8 Plots of the deformation densities, Δρ(r), and shape of the most important interacting orbitals of the pair-wise orbital interactions between B₃Ar₂⁺ and Ar in B₃Ar₃⁺. The associated orbital interaction energies are given in kcal mol⁻¹. The colour code of the charge flow is red → blue.

Table 7 EDA results of the B₃Ng₃⁺ (Ng = Ar–Rn) complexes considering Ng as one fragment and B₃Ng₂⁺ as another fragment at the revPBE-D3/TZ2P//MP2/def2-TZVP level. All energy terms are in kcal mol^−1a

Complex	ΔE^Pauli	ΔE^elstat	ΔE^orb	ΔE^disp	ΔE^int	ΔE^σ1	ΔE^π1	ΔE^σ2	ΔE^π2	ΔE^rest
a The percentage values within the parentheses show the contribution towards the total attractive interaction ΔE^elstat + ΔE^orb + ΔE^disp.
B3Ar3^+	74.9	−29.0 (34.4)	−54.2 (64.4)	−1.1 (1.2)	−9.4	−39.5 (72.9)	−4.7 (8.7)	−4.5 (8.9)	−4.2 (7.7)	−0.9
B3Kr3^+	84.9	−34.4 (35.2)	−62.0 (63.5)	−1.3 (1.4)	−12.8	−45.7 (73.7)	−5.5 (8.8)	−5.0 (8.1)	−4.6 (7.4)	−1.2
B3Xe3^+	93.2	−38.3 (34.6)	−70.7 (63.9)	−1.7 (1.5)	−17.6	−53.3 (75.3)	−6.1 (8.7)	−5.3 (7.5)	−4.9 (6.9)	−1.1
B3Rn3^+	93.2	−39.5 (35.2)	−70.7 (63.0)	−2.0 (1.8)	−19.1	−55.2 (78.1)	−5.8 (8.1)	−4.6 (6.5)	−4.6 (6.5)	−0.6

---

Conclusion

B₃⁺ cluster shows a high tendency towards complexation with Ar–Rn atoms having each Ng–B bond dissociation energy in the range of 13.2–29.8 kcal mol⁻¹ with gradual increase along Ar to Rn. The corresponding changes in enthalpy and free energy values for the dissociation processes highlight the thermochemical stability of B₃Ng₃⁺ complexes. Localized Ng–B natural bond orbitals are noted in between Ng and B centres and consequently high Wiberg bond indices (0.57–0.78) are also found for Ng–B bonds, indicating the covalent nature of these bonds. Further, the electron density analysis also supports the covalent nature of the bonds where the local electron density is negative for all Ng–B bonds (Ng = Ar–Rn) and the Laplacian of electron density (∇²ρ(r_c)) is negative for Xe/Rn–B bonds. Nevertheless, there is always an accumulation of electron density in between Ng and B centres as noted from the contour plots of ∇²ρ(r_c). The energy decomposition analysis shows that the Ng–B bonds are mainly stabilized by of the orbital interaction energy (ca. 63.0–64.4%) and the electrostatic interaction energy (ca. 34.4–35.4%). Furthermore, the decomposition of orbital term into σ- and π-contributions reveals that the Ng–B bonding can be represented by a donor–acceptor model where the Ng (HOMO) → B₃Ng₂⁺ (LUMO) σ-donation is a major contributor (ca. 72.9–78.1%) of the orbital term for Ar–Rn analogues. The σ- and π-back donations also occur from the occupied MOs of B₃Ng₂⁺ to unoccupied MOs of Ng.

Acknowledgements

PKC would like to thank DST, New Delhi for the J. C. Bose National Fellowship. RS and SM thank UGC, New Delhi for their fellowships. The work in México was supported by Conacyt (Grant CB-2015-252356). MO thanks Conacyt for his master fellowship.

References

1. (a) H. C. Brown, J. Org. Chem., 1981, 46, 4599; (b) R. Kinjo, B. Donnadieu, M. A. Celik, G. Frenking and G. Bertrand, Science, 2011, 333, 610; (c) Y. Segawa, M. Yamashita and K. Nozaki, Science, 2006, 314, 113; (d) N. Miyaura and A. Suzuki, Chem. Rev., 1995, 95, 2457; (e) T. Wideman and L. G. Sneddon, Inorg. Chem., 1995, 34, 1002; (f) A. N. Alexandrova, A. I. Boldyrev, H.-J. Zhai and L.-S. Wang, Coord. Chem. Rev., 2006, 250, 2811.
2. (a) D. Y. Zubarev, Analysis of Chemical Bonding in Clusters by Means of the Adaptive Natural Density Partitioning, All Graduate Theses and Dissertations, Paper 13, http://www.digitalcommons.usu.edu/etd/13, 2008; (b) M. Kimura, S. Suzuki, N. Shimakura, J. P. Gu, G. Hirsch, R. J. Buenker and I. Shimamura, Phys. Rev. A, 1996, 54, 3029; (c) Y. S. Tergiman, F. Fraija and M. C. Bacchus-Montabonel, , Int. J. Quantum Chem., 2002, 89, 322; (d) H.-J. Zhai, L.-S. Wang, A. N. Alexandrova, A. I. Boldyrev and V. G. Zakrzewski, J. Phys. Chem. A, 2003, 107, 9319; (e) T. R. Galeev, C. Romanescu, W.-L. Li, L.-S. Wang and A. I. Boldyrev, J. Chem. Phys., 2011, 135, 104301; (f) A. P. Sergeeval, B. B. Averkiev, H.-J. Zhai, A. I. Boldyrev and L.-S. Wang, J. Chem. Phys., 2011, 134, 224304; (g) W. Huang, A. P. Sergeeva, H.-J. Zhai, B. B. Averkiev, L.-S. Wang and A. I. Boldyrev, Nat. Chem., 2010, 2, 202; (h) I. A. Popov, V. F. Popov, K. V. Bozhenko, I. Černušák and A. I. Boldyrev, J. Chem. Phys., 2013, 139, 114307; (i) T. Heine and G. Merino, Angew. Chem., Int. Ed., 2012, 51, 4275; (j) G. Merino and T. Heine, Angew. Chem., Int. Ed., 2012, 51, 10226; (k) J. M. Mercero, A. I. Boldyrev, G. Merino and J. M. Ugalde, Chem. Soc. Rev., 2015, 44, 6519; (l) A. C. Castro, E. Osorio, J. L. Cabellos, E. Cerpa, E. Matito, M. Sola, M. Swart and G. Merino, Chem.–Eur. J., 2014, 16, 4583; (m) L. Liu, D. Moreno, E. Osorio, A. C. Castro, S. Pan, P. K. Chattaraj, T. Heine and G. Merino, RSC Adv., 2016, 6, 27177; (n) G. Martínez-Guajardo, J. L. Cabellos, A. Díaz-Celaya, S. Pan, R. Islas, P. K. Chattaraj, T. Heine and G. Merino, Sci. Rep., 2015, 5, 11287; (o) D. Moreno, S. Pan, G. Martínez-Guajardo, L. Lei-Zeonjuk, R. Islas, E. Osorio, P. K. Chattaraj, T. Heine and G. Merino, Chem. Commun., 2014, 50, 8140.
3. (a) D. Martin, Y. Canac, V. Lavallo and G. Bertrand, J. Am. Chem. Soc., 2014, 136, 5023; (b) H. Braunschweig, C.-W. Chiu, T. Kupfer and K. Radacki, Inorg. Chem., 2011, 50, 4247.
4. (a) G. Frenking, Nature, 2015, 552, 297; (b) H. Braunschweig, R. D. Dewhurst, F. Hupp, M. Nutz, K. Radacki, C. W. Tate, A. Vargas and Q. Ye, Nature, 2015, 522, 327.
5. (a) H. Braunschweig, R. D. Dewhurst, K. Hammond, J. Mies, K. Radacki and A. Vargas, Science, 2012, 336, 1420; (b) J. Böhnke, H. Braunschweig, P. Constantinidis, T. Dellermann, W. C. Ewing, I. Fischer, K. Hammond, F. Hupp, J. Mies, H.-C. Schmitt and A. Vargas, J. Am. Chem. Soc., 2015, 137, 1766; (c) H. Braunschweig, T. Dellermann, R. D. Dewhurst, W. C. Ewing, K. Hammond, J. O. C. Jimenez-Halla, T. Kramer, I. Krummenacher, J. Mies, A. K. Phukan and A. Vargas, Nat. Chem., 2013, 5, 1025; (d) M. Zhou, N. Tsumori, Z. Li, K. Fan, L. Andrews and Q. Xu, J. Am. Chem. Soc., 2002, 126, 12936; (e) S.-D. Li, H.-J. Zhai and L.-S. Wang, J. Am. Chem. Soc., 2008, 130, 2573.
6. (a) T. B. Tai, N. M. Tam and M. T. Nguyen, Theor. Chem. Acc., 2012, 131, 1241; (b) D. Y. Zubarev and A. I. Boldyrev, J. Comput. Chem., 2007, 28, 251.
7. T. B. Tai and M. T. Nguyen, Angew. Chem., Int. Ed., 2013, 52, 4554.
8. (a) X.-W. Li, W. T. Pennington and G. H. Robinson, J. Am. Chem. Soc., 1995, 117, 7578; (b) X.-W. Li, Y. Xie, P. Schreiner, K. D. Gripper, R. C. Crittendon, C. Campana, H. F. Schaefer and G. H. Robinson, Organometallics, 1996, 15, 3798.
9. R. J. Wright, M. Brynda and P. P. Power, Angew. Chem., Int. Ed., 2006, 45, 5953.
10. J. Jin, G. Wang, M. Zhou, D. M. Andrada, M. Hermann and G. Frenking, Angew. Chem., Int. Ed., 2016, 55, 2078.
11. (a) N. Bartlett, Proc. Chem. Soc., 1962, 218; (b) L. Graham, O. Graudejus, N. K. Jha and N. Bartlett, Coord. Chem. Rev., 2000, 197, 321.
12. R. Hoppe, W. Daehne, H. Mattauch and K. Roedder, Angew. Chem., Int. Ed. Engl., 1962, 1, 599.
13. H. H. Claassen, H. Selig and J. G. Malm, J. Am. Chem. Soc., 1962, 84, 3593.
14. P. R. Fields, L. Stein and M. H. Zirin, J. Am. Chem. Soc., 1962, 84, 4164.
15. J. J. Turner and G. C. Pinetel, Science, 1963, 140, 974.
16. (a) C. A. Thompson and L. Andrews, J. Am. Chem. Soc., 1994, 116, 423; (b) J. Li, B. E. Bursten, B. Liang and L. Andrews, Science, 2002, 295, 2242; (c) B. Liang, L. Andrews, J. Li and B. E. Bursten, J. Am. Chem. Soc., 2002, 124, 9016; (d) X. Wang, L. Andrews, J. Li and B. E. Bursten, Angew. Chem., Int. Ed., 2004, 43, 2554; (e) V. I. Feldman, F. F. Sukhov and A. Y. Orlov, Chem. Phys. Lett., 1997, 280, 507; (f) L. Khriachtchev, H. Tanskanen, M. Pettersson, M. Räsänen, J. Ahokas, H. Kunttu and V. Feldman, J. Chem. Phys., 2002, 116, 5649; (g) V. I. Feldman, F. F. Sukhov, A. Yu. Orlov and I. V. Tyulpina, J. Am. Chem. Soc., 2003, 125, 4698; (h) M. Pettersson, J. Lundell, L. Khriachtchev, L. Isamieni and M. Räsänen, J. Am. Chem. Soc., 1998, 120, 7979; (i) M. Pettersson, J. Lundell, L. Khriachtchev and M. Räsänen, J. Chem. Phys., 1998, 109, 618; (j) M. Pettersson, L. Khriachtchev, J. Lundell and M. Räsänen, J. Am. Chem. Soc., 1999, 121, 11904; (k) L. Khriachtchev, M. Pettersson, N. Runeberg, J. Lundell and M. Räsänen, Nature, 2000, 406, 874; (l) L. Khriachtchev, M. Pettersson, A. Lignell and M. Räsänen, J. Am. Chem. Soc., 2001, 123, 8610; (m) L. Khriachtchev, M. Pettersson, J. Lundell, H. Tanskanen, T. Kiviniemi, N. Runeberg and M. Räsänen, J. Am. Chem. Soc., 2003, 125, 1454; (n) S. A. Cooke and M. C. L. Gerry, J. Am. Chem. Soc., 2004, 126, 17000; (o) J. M. Michaud, S. A. Cooke and M. C. L. Gerry, Inorg. Chem., 2004, 43, 3871; (p) J. M. Thomas, N. R. Walker, S. A. Cooke and M. C. L. Gerry, J. Am. Chem. Soc., 2004, 126, 1235; (q) S. A. Cooke and M. C. L. Gerry, Phys. Chem. Chem. Phys., 2004, 6, 3248; (r) S. A. Cooke and M. C. L. Gerry, J. Am. Chem. Soc., 2004, 126, 17000; (s) J. M. Michaud and M. C. L. Gerry, J. Am. Chem. Soc., 2006, 128, 7613; (t) D. S. Brock, H. P. A. Mercier and G. J. Schrobilgen, J. Am. Chem. Soc., 2013, 135, 5089; (u) G. L. Smith, H. P. A. Mercier and G. J. Schrobilgen, Inorg. Chem., 2011, 49, 12359; (v) J. R. Debackere, H. P. A. Mercier and G. J. Schrobilgen, J. Am. Chem. Soc., 2014, 136, 3888.
17. (a) W. Koch, B. Liu and G. Frenking, J. Chem. Phys., 1990, 92, 2464; (b) G. Frenking, W. Koch, F. Reichel and D. Cremer, J. Am. Chem. Soc., 1990, 112, 4240; (c) S. Pan, A. Gupta, S. Mandal, D. Moreno, G. Merino and P. K. Chattaraj, Phys. Chem. Chem. Phys., 2015, 17, 972; (d) R. Saha, S. Pan, G. Merino and P. K. Chattaraj, J. Phys. Chem. A, 2015, 119, 6746; (e) J. Lundell, A. Cohen and R. B. Gerber, J. Phys. Chem. A, 2002, 106, 11950; (f) W. A. Grochala, Phys. Chem. Chem. Phys., 2012, 14, 14860; (g) G. Frenking, W. Koch, D. Cremer, J. Gauss and J. F. Liebman, J. Phys. Chem., 1988, 93, 3410; (h) S. Pan, D. Moreno, J. L. Cabellos, J. Romero, A. Reyes, G. Merino and P. K. Chattaraj, J. Phys. Chem. A, 2014, 118, 487; (i) L. Khriachtchev, M. Räsänen and R. B. Gerber, Acc. Chem. Res., 2009, 42, 183; (j) G. Frenking, W. J. Gauss and D. Cremer, J. Am. Chem. Soc., 1988, 110, 8007; (k) G. Leroy and M. Sana, Theor. Chim. Acta, 1990, 77, 383; (l) S. Pan, M. Contreras, J. Romero, A. Reyes, G. Merino and P. K. Chattaraj, Chem.–Eur. J., 2013, 19, 2322; (m) S. Pan, R. Saha, S. Mandal and P. K. Chattaraj, Phys. Chem. Chem. Phys., 2016, 18, 11661; (n) L. Operti, R. Rabezzana, F. Turco, S. Borocci, M. Giordani and F. Grandinetti, Chem.–Eur. J., 2011, 17, 10682; (o) S. Borocci, M. Giordani and F. Grandinetti, J. Phys. Chem. A, 2015, 119, 2383; (p) S. Pan, A. Gupta, R. Saha, G. Merino and P. K. Chattaraj, J. Comput. Chem., 2015, 36, 2168; (q) S. Pan, D. Moreno, S. Ghosh, G. Merino and P. K. Chattaraj, J. Comput. Chem., 2016, 37, 226; (r) S. Pan, S. Jalife, M. Kumar, V. Subramanian, G. Merino and P. K. Chattaraj, ChemPhysChem, 2013, 14, 2511; (s) S. Pan, D. Moreno, G. Merino and P. K. Chattaraj, ChemPhysChem, 2014, 15, 3554; (t) S. Pan, R. Saha and P. K. Chattaraj, New J. Chem., 2015, 39, 6778; (u) P. Szarek and W. Grochala, J. Phys. Chem. A, 2015, 119, 2483; (v) Q. Zhang, M. Chen, M. Zhou, D. M. Andrada and G. Frenking, J. Phys. Chem. A, 2015, 119, 2543; (w) S. Pan, R. Saha, A. Kumar, A. Gupta, G. Merino and P. K. Chattaraj, Int. J. Quantum Chem., 2016, 116, 1016.
18. (a) C. Zhu, K. Niimi, T. Taketsugu, M. Tsuge, A. Nakayama and L. Khriachtchev, J. Chem. Phys., 2015, 142, 054305; (b) C. Cheng and L. Sheng, Comput. Theor. Chem., 2012, 989, 39; (c) J. Ahokas, K. Vaskonen, J. Eloranta and H. Kunttu, J. Phys. Chem. A, 2000, 104, 9506; (d) P. Antoniotti, S. Borocci, N. Bronzolino, P. Cecchi and F. Grandinetti, J. Phys. Chem. A, 2007, 111, 10144; (e) J.-L. Chen, C.-Y. Yang, H.-J. Lin and W.-P. Hu, Phys. Chem. Chem. Phys., 2013, 15, 9701; (f) T.-Y. Lin, J.-B. Hsu and W.-P. Hu, Chem. Phys. Lett., 2005, 402, 514; (g) A. Ghosh, S. Dey, D. Manna and T. K. Ghanty, J. Phys. Chem. A, 2015, 119, 5732; (h) T. Jayasekharan and T. K. Ghanty, J. Chem. Phys., 2008, 129, 184302; (i) A. Ghosh, D. Manna and T. K. Ghanty, J. Phys. Chem. A, 2015, 119, 2233; (j) P. Sekhar, A. Ghosh and T. K. Ghanty, J. Phys. Chem. A, 2015, 119, 11601; (k) L. Khriachtchev, H. Tanskanen, J. Lundell, M. Pettersson, H. Kiljunen and M. Räsänen, J. Am. Chem. Soc., 2003, 125, 4696; (l) C. T. Goetschel and K. R. Loos, J. Am. Chem. Soc., 1972, 94, 3018; (m) J. F. Liebman and L. C. Allen, Inorg. Chem., 1972, 11, 114; (n) J. T. Koskinen and R. Graham Cooks, J. Phys. Chem. A, 1999, 103, 9567; (o) A. Papakondylis, E. Miliordos and A. Mavridis, J. Phys. Chem. A, 2004, 108, 4335.
19. (a) C. Møller and M. S. Plesset, Phys. Rev., 1934, 46, 618; (b) M. Head-Gordon and J. A. Pople, Chem. Phys. Lett., 1988, 153, 503.
20. (a) J. Čížek, J. Chem. Phys., 1966, 45, 4256; (b) J. Čížek, Adv. Chem. Phys., 1969, 14, 35; (c) R. J. Bartlett and M. Musial, Rev. Mod. Phys., 2007, 79, 291.
21. F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys., 2005, 7, 32975.
22. K. A. Peterson, D. Figgen, E. Goll, H. Stoll and M. Dolg, J. Chem. Phys., 2003, 119, 11113.
23. (a) A. Anoop, W. Thiel and F. Neese, J. Chem. Theory Comput., 2010, 6, 3137; (b) Orca manual, Version 3.0, p. 56, https://orcaforum.cec.mpg.de/OrcaManual.pdf.
24. A. E. Reed, R. B. Weinstock and F. Weinhold, J. Chem. Phys., 1985, 83, 735.
25. K. B. Wiberg, Tetrahedron, 1968, 24, 1083.
26. A. E. Reed, L. A. Curtiss and F. Weinhold, Chem. Rev., 1988, 88, 899.
27. M. J. Frisch, et al., Gaussian 09, Revision C.01, Gaussian, Inc., Wallingford CT, 2009.
28. R. F. W. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, Oxford, UK, 1990.
29. T. Lu and F. W. Chen, J. Comput. Chem., 2012, 33, 580.
30. S. Huzinaga and M. Klobukowski, Chem. Phys. Lett., 1993, 212, 260–264.
31. M. P. Mitoraj, A. Michalak and T. A. Ziegler, J. Chem. Theory Comput., 2009, 5, 962.
32. (a) E. J. Baerends, et al., ADF2013.01, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, 2013; (b) G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. F. Guerra, S. J. A. Van Gisbergen, J. G. Snijders and T. Ziegler, J. Comput. Chem., 2001, 22, 931.
33. L. Goerigk and S. Grimme, Phys. Chem. Chem. Phys., 2011, 13, 6670.
34. E. van Lenthe, A. Ehlers and E.-J. Baerends, J. Chem. Phys., 1999, 110, 8943.
35. G. F. Caramori, R. M. Piccoli, M. Segal, A. Muñoz-Castro, R. Guajardo-Maturana, D. M. Andradae and G. Frenking, Dalton Trans., 2015, 44, 377.
36. B. Cordero, V. Gómez, A. E. Platero-Prats, M. Revés, J. Echeverría, E. Cremades, F. Barragán and S. Alvarez, Dalton Trans., 2008, 2832–2838.
37. (a) P. Macchi, D. M. Proserpio and A. Sironi, J. Am. Chem. Soc., 1998, 120, 13429; (b) P. Macchi, L. S. Garlaschelli, S. Martinengo and A. Sironi, J. Am. Chem. Soc., 1999, 121, 10428; (c) I. V. Novozhilova, A. V. Volkov and P. Coppens, J. Am. Chem. Soc., 2003, 125, 1079; (d) D. Cremer and E. Kraka, Angew. Chem., Int. Ed., 1984, 23, 627.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra16188j

---