Intrinsic Dynamic and Static Nature of Halogen Bonding in Neutral Polybromine Clusters, with the Structural Feature Elucidated by QTAIM Dual-Functional Analysis and MO Calculations

Abstract

The intrinsic dynamic and static nature of noncovalent Br-∗-Br interactions in neutral polybromine clusters is elucidated for Br₄–Br₁₂, applying QTAIM dual-functional analysis (QTAIM-DFA). The asterisk (∗) emphasizes the existence of the bond critical point (BCP) on the interaction in question. Data from the fully optimized structures correspond to the static nature of the interactions. The intrinsic dynamic nature originates from those of the perturbed structures generated using the coordinates derived from the compliance constants for the interactions and the fully optimized structures. The noncovalent Br-∗-Br interactions in the L-shaped clusters of the C_s symmetry are predicted to have the typical hydrogen bond nature without covalency, although the first ones in the sequences have the vdW nature. The L-shaped clusters are stabilized by the n(Br)→σ*(Br–Br) interactions. The compliance constants for the corresponding noncovalent interactions are strongly correlated to the E(2) values based on NBO. Indeed, the MO energies seem not to contribute to stabilizing Br₄ (C_2h) and Br₄ (D_2d), but the core potentials stabilize them, relative to the case of 2Br₂; this is possibly due to the reduced nuclear–electron distances, on average, for the dimers.

1. Introduction

Halogen bonding is of current and continuous interest [1,2]. A lot of information relevant to halogen bonding has been accumulated so far [3]. Halogen bonding has been discussed on the basis of the shorter distances between halogen and other atoms in crystals [4,5,6]. The short halogen contacts are found in two types: symmetric (type I) and bent (type II) geometries. The bonding has also been investigated in the liquid [7,8] and gas [9] phases. The nature of halogen bonding has been discussed based on the theoretical background on the molecular orbital description for the bonding and the σ-hole developed on the halogen atoms, together with the stability of the structural aspects [10]. We also reported the dynamic and static nature of Y–X---π(C₆H₆) interactions recently [11]. Halogen bonding is applied to a wide variety of fields in chemical and biological sciences, such as crystal engineering, supramolecular soft matters, and nanoparticles. Efforts have been made to unify and categorize the accumulated results and establish the concept of halogen bonding [3,12,13,14,15].

Structures of halogen molecules (X₂) have been reported, as determined by X-ray crystallographic analysis for X = Cl, Br, and I [16,17,18]. The behavior of bromine–bromine interactions has been reported for the optimized structures of Br₂–Br₅ in the neutral and/or charged forms, together with Br₁, so far [19,20]. Figure 1 draws the observed structure of Br₂, for example. The bromine molecules seem to exist as a zig-zag structure in the infinite chains in crystals. One would find the linear alignment of three Br atoms in an L-shaped dimer ((Br₂)₂; Br₄) and the linear alignment of four Br atoms in a double L-shaped trimer ((Br₂)₃; Br₆) in a planar Br₂ layer in addition to Br₂ itself. The linear four Br atoms are located in the two L-shaped dimers of Br₆, overlapped at the central Br₂. While the L-shaped dimers seem to construct the zig-zag type infinite chains, the linear four Br atoms construct linear infinite chains. The attractive n_p(Br)→σ*(Br–Br) σ(3c–4e) (three center–four electron interaction of the σ-type) and n_p(Br)→σ*(Br–Br)←n_p(Br) σ(4c–6e) must play a very important role to stabilize Br₄ and Br₆, respectively, where n_p(Br) stands for the p-type nonbonding orbital of Br in the plane, perpendicular to the molecular Br₂ axis, and σ*(Br–Br) is the σ*-orbital of Br₂. The crystal structures of Cl₂ and I₂ are very similar to that of Br₂.

We have been very interested in the behavior of halogen bonding in polyhalogen clusters, together with the structures. How can the interactions in the polyhalogen clusters be clarified? We propose QTAIM dual-functional analysis (QTAIM-DFA) [21,22,23,24,25] based on the quantum theory of atoms in molecules (QTAIM) approach introduced by Bader [26,27] to classify and characterize the various interactions effectively [28]. In QTAIM-DFA, H_b(r_c) are plotted versus H_b(r_c) − V_b(r_c)/2 (=(ћ²/8m)∇²ρ_b(r_c) (see Equation (SA2) in the supplementary materials), where ρ_b(r_c), H_b(r_c), and V_b(r_c) stand for the charge densities, total electron energy densities, and potential energy densities, respectively, at bond critical points (BCPs, ∗) on the bond paths (BPs) in this paper [26]. The kinetic energy densities at BCPs will be similarly denoted by G_b(r_c) [26]. A chemical bond or an interaction between Br and Br is denoted by Br-∗-Br in this work, where the asterisk emphasizes the existence of a BCP on a BP for Br–Br [26,27]. In our treatment, data from the fully optimized structures are plotted together with those from the perturbed structures around the fully optimized ones. The static nature of the interactions corresponds to the data from the fully optimized structures, which are analyzed using polar coordinate (R, θ) representation [21,22,23,24,25]. On the other hand, the dynamic nature originates based on the data from both the perturbed and fully optimized structures [21,22,23,24,25]. The plot is expressed by (θ_p, κ_p), where θ_p corresponds to the tangent line and κ_p is the curvature of the plot. θ and θ_p are measured from the y-axis and the y-direction, respectively. We call (R, θ) and (θ_p, κ_p) the QTAIM-DFA parameters [29].

Interactions are classified by the signs of ∇²ρ_b(r_c) and H_b(r_c), based on the QTAIM approach. The interactions are called shard shell (SS) interactions when ∇²ρ_b(r_c) < 0 and closed-shell (CS) interactions when ∇²ρ_b(r_c) > 0 [26]. In particular, CS interactions are called pure CS (p-CS) interactions when H_b(r_c) > 0 and ∇²ρ_b(r_c) > 0. We call interactions where H_b(r_c) < 0 and ∇²ρ_b(r_c) > 0 regular CS (r-CS) interactions, which clearly distinguishes these interactions from the p-CS interactions. The signs of ∇²ρ_b(r_c) can be replaced by those of H_b(r_c) − V_b(r_c)/2 because (ћ²/8m)∇²ρ_b(r_c) = H_b(r_c) − V_b(r_c)/2 (see Equation (SA2) in the supporting information). Indeed, H_b(r_c) − V_b(r_c)/2 = 0 corresponds to the borderline between the classic covalent bonds of SS and the noncovalent interactions of CS, but H_b(r_c) = 0 appears to be buried in the noncovalent interactions of CS. As a result, it is difficult to characterize the various CS interactions based on the signs of H_b(r_c) − V_b(r_c)/2 and/or H_b(r_c). In QTAIM-DFA, the signs of the first derivatives of H_b(r_c) − V_b(r_c)/2 and H_b(r_c) (d(H_b(r_c) − V_b(r_c)/2)/dr and dH_b(r_c)/dr, respectively, where r is the interaction distance) are used to characterize CS interactions, in addition to those of H_b(r_c) − V_b(r_c)/2 and H_b(r_c), after analysis of the plot. While the former corresponds to (θ_p, κ_p), the latter does to (R, θ). The analysis of the plots enables us to characterize the various CS interactions more effectively. Again, the details are explained later.

The perturbed structures necessary for QTAIM-DFA can be generated. Among them, a method employing the coordinates corresponding to the compliance constants C_ii for internal vibrations is shown to be highly reliable to generate the perturbed structures [30,31,32,33,34,35,36,37,38,39]. The method, which we proposed recently, is called CIV. The dynamic nature of interactions based on the perturbed structures with CIV is described as the “intrinsic dynamic nature of interactions” since the coordinates are invariant to the choice of coordinate system. Rough criteria that distinguish the interaction in question from others are obtained by applying QTAIM-DFA with CIV to standard interactions. QTAIM-DFA and the criteria are explained in the appendix of the supplementary materials using Schemes SA1–SA3, Figures SA1 and SA2, Table SA1, and Equations (SA1)–(SA7). The basic concept of the QTAIM approach is also explained.

QTAIM-DFA, using the perturbed structures generated with CIV, is well-suited to elucidate the intrinsic dynamic and static nature of halogen–halogen interactions in the polyhalogen clusters. As the first step to clarify the nature of various types of halogen–halogen interactions in the polyhalogen clusters, the nature of each bromine–bromine interaction in the neutral polybromine clusters is elucidated by applying QTAIM-DFA. Various types of structures and interactions are found in the optimized structures of polybromine clusters, other than those observed in the crystals. Here, we present the results of investigations on the polybromine clusters, together with the structural feature, elucidated with QTAIM-DFA and QC calculations.

2. Methodological Details in Calculations

The structures were optimized by employing Gaussian 09 programs [40]. The 6-311+G(3df) basis [41,42,43,44] set was applied to optimize the structures of neutral polybromine clusters, Br₂–Br₁₂. The Møller–Plesset second-order energy correlation (MP2) level [45,46,47] was applied for the optimizations. Optimized structures were confirmed by frequency analysis. The results of the frequency analyses were employed to calculate the C_ij values and coordinates corresponding to C_ii [30,34,35,36]. The ρ_b(r_c), H_b(r_c) − V_b(r_c)/2 (=(ћ²/8m)∇²ρ_b(r_c)), and H_b(r_c) values were calculated using the Gaussian 09 program package [40], with the same method applied to the optimizations. Data were analyzed with the AIM2000 [48,49] and AIMAll [50] programs.

Coordinates corresponding to the compliance constants for an internal coordinate i of the internal vibrations (C_i) were employed to generate the perturbed structures necessary in QTAIM-DFA [21,22,23,24,25]. Equation (1) explains the method to generate the perturbed structures with CIV. An i-th perturbed structure in question (S_iw) was generated by the addition of the coordinates (C_i) corresponding to C_ii to the standard orientation of a fully optimized structure (S_o) in the matrix representation. The coefficient g_iw in Equation (1) controls the difference in structures between S_iw and S_o: g_iw are determined to satisfy Equation (2) for the interaction in question, where r and r_o show the distances in question in the perturbed and fully optimized structures, respectively, with a_o of Bohr radius (0.52918 Å) [21,22,23,24,25,30].

S_iw = S_o + g_iw × C_i

(1)

r = r_o + wa_o (w = (0), ±0.05 and ±0.1; a_o = 0.52918 Å)

(2)

y = c_o + c₁x + c₂x² + c₃x³   (R_c²: square of correlation coefficient)

(3)

In the QTAIM-DFA treatment, H_b(r_c) are plotted versus H_b(r_c) − V_b(r_c)/2 for the data of five points of w = 0, ±0.05, and ±0.1 in Equation (2). Each plot is analyzed using a regression curve of the cubic function, as shown in Equation (3), where (x, y) = (H_b(r_c) − V_b(r_c)/2, H_b(r_c)) (R_c² (square of correlation coefficient) > 0.99999 in the norm) [25].

3. Results and Discussion

3.1. Structural Optimizations of Polybromine Clusters, Br₆–Br₁₂

Structures of the neutral Br₂–Br₁₂ clusters were optimized with MP2/6-311+G(3df). The structural parameters for the optimized structures of minima for Br₂–Br₆ and Br₈–Br₁₂ are collected in Tables S1 and S2, respectively. Some transition states (TSs) for Br₄ and Br₆ were also calclaterd. The notation of C_s-L_m (m = 1–5) is used for the linear L-shaped clusters of the C_s symmetry, where m stands for the number of noncovalent interactions in Br_2m+2 (m = 1–5). Cyclic structures are also optimized, retaining the higher symmetries. The optimized structures are not shown in figures, but they can be found in the molecular graphs with the contour maps of ρ(r) for the linear-type bromine clusters Br₄–Br₁₂ (C_s-L_m (m = 1–5)) and for the cyclic bromine clusters Br₄–Br₁₂, drawn on the optimized structures with MP2/6-311+G(3df) [51]. The energies for the formation of Br₄–Br₆ and Br₈–Br₁₂ are given in Tables S1 and S2, respectively, from the components (∆E = E(Br_2k) − kE(Br₂)) on the energy surfaces (∆E_ES) and those with the collections of zero-point energies (∆E_ZP). The ∆E_ZP values were plotted versus ∆E_ES. The plot is shown in Figure S1, which gives an excellent correlation (y = 0.940x + 0.129; R_c² (square of correlation coefficient) = 0.9999) [52]. Therefore, the ∆E_ES values are employed for the discussion.

The behavior of the neutral dibromine clusters (Br₄) is discussed first. Three structures were optimized for Br₄ as minima with some TSs. The minima are the L-shaped structure of C_s symmetry (Br₄ (C_s-L₁)) [19], the cyclic structure of C_2h symmetry (Br₄ (C_2h)), and the tetrahedral type of D_2d symmetry (Br₄ (D_2d)). A TS of the C_s symmetry was detected between Br₄ (C_s-L₁) and Br₄ (C_2h), and two TSs of the C₁ symmetry were between Br₄ (C_2h) and Br₄ (D_2d) and between Br₄ (D_2d) and Br₄ (C_s-L₁). They are called TS (C_s: C_s, C_2h), TS (C₁: C_2h, D_2d), and TS (C₁: D_2d, C_s), respectively. The three minima will be converted to each other through the three TSs. A TS between Br₄ (C_s-L₁) and its topological isomer was also detected, which is called TS (C_2v: C_s, C_s); however, further effort was not made to search for similar TSs between Br₄ (C_2h) and its topological isomer and between Br₄ (C_2d) and its topological isomer.

Figure 2 draws the energy profiles for the optimized structures of minima, Br₄ (C_s-L₁), Br₄ (C_2h), and Br₄ (D_2d), together with the TSs TS (C_s: C_s, C_2h), TS (C_s: C_2h, D_2d), TS (C₁: C_2d, C_s), and TS (C_2v: C_s, C_s). The optimized structures are not shown in the figures, but they can be found in the molecular graphs shown in Figure 2, illustrated on the optimized structures. All BCPs expected are detected clearly, together with RCPs and a CCP [26]. The ΔE_ES value of −10.7 kJ mol⁻¹ for the formation of Br₄ (C_s-L₁) seems very close to the border area between the vdW and typical hydrogen bond (t-HB) adducts. The driving force for the formation of Br₄ (C_s-L₁) must be Br₃ σ(3c–4e) of the n_p(Br)→σ*(Br–Br) type. The interactions in Br₄ (C_2h) and Br₄ (D_2d) seem very different from those in Br₄ (C_s-L₁). The ΔE_ES values of Br₄ (C_2h) (−8.0 kJ mol⁻¹) and Br₄ (D_2d) (−9.1 kJ mol⁻¹) are close to that for Br₄ (C_s-L₁) (−10.7 kJ mol⁻¹). Moreover, the values for TS (C_s: C_s, C_2h) (−7.4 kJ mol⁻¹), TS (C₁: C_2h, D_2d) (−7.6 kJ mol⁻¹), TS (C₁: D_2d, C_s) (−7.0 kJ mol⁻¹), and TS (C_2v: C_s, C_s) (−8.7 kJ mol⁻¹) are not so different from those for the minima.

In the case of Br₆, three structures of the linear C_s symmetry (Br₆ (C_s-L₂)), the linear C₂ symmetry (Br₆ (C₂)), and the cyclic C_3h symmetry (Br₆ (C_3h-c)) were optimized typically as minima. The linear Br₆ clusters of C_2h symmetry (Br₆ (C_2h)) and C_2v symmetry (Br₆ (C_2v)), similar to Br₆ (C₂), were also optimized, of which the torsional angles, ϕ(¹Br²Br⁵Br⁶Br) (=ϕ₃), were 0° and 180°, respectively. One imaginary frequency was detected for each; therefore, they are assigned to TSs between Br₆ (C₂) and the topological isomer on the different reaction coordinates. Further effort was not made to search for TSs.

The ΔE_ES value for Br₆ (C_s-L₂) was predicted to be −22.6 kJ mol⁻¹. The magnitude is slightly larger than the double value for Br₄ (C_s-L₁) (∆E_ES = −10.7 kJ mol⁻¹). Two types of σ (3c–4e) operate to stabilize Br₆ (C_s-L₂). One, σ(3c–4e), seems similar to that in Br₄ (C_s-L₁), but the other would be somewhat different. Namely, the second interaction would contribute to ∆E_ES somewhat more than that of the first one in the formation of Br₆ (C_s-L₂). On the other hand, the linear interaction in Br₆ (C₂) can be explained by σ(4c–6e) of the n_p(Br)→σ*(Br–Br)←n_p(Br) type. The magnitude of ∆E_ES of Br₆ (C₂) seems slightly smaller than that of Br₆ (C_s-L₂) but is very close to the double value for Br₄ (C_s-L₁). The magnitude of ∆E_ES for Br₆ (C_3h-c) is close to the triple value of Br₄ (C_s-L₁). One finds triply degenerated σ(3c–4e) interactions in Br₆ (C_3h-c). The similarity in the interactions for Br₄ (C_s-L₁), Br₆ (C₂), and Br₆ (C_3h-c) will be discussed again later. The magnitudes of ∆E_ES become proportionally larger to the size of the clusters, as shown in Figures S1 and S2. The ΔE_ES values are plotted versus k in Br_2k (2 ≤ k ≤ 6) for the C_s-L_m type. The results are shown in Figure S2. Contributions from inner σ(3c–4e) (named r_in) to ΔE_ES seem slightly larger than those from σ(3c–4e) in the front end and end positions (named r₂ and r_ω, respectively).

After examination of the optimized structures, the next extension is to clarify the nature of Br-∗-Br interactions by applying QTAIM-DFA. The contour plots are discussed next.

3.2. Molecular Graphs with Contour Plots of Polybromine Clusters

Figure 3 illustrates the molecular graphs with contour maps of ρ(r) for the linear type of Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅), drawn on the structures optimized with MP2/6-311+G(3df). Figure 4 draws the molecular graphs with contour maps of ρ(r) for Br₄–Br₁₂, other than those for Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅), calculated with MP2/6-311+G(3df) [53,54] (see also Figure S3). All BCPs expected are detected clearly, together with RCPs and a CCP containing those for noncovalent Br-∗-Br interactions, which are located at the (three-dimensional) saddle points of ρ(r).

3.3. Survey of the Br-∗-Br Interactions in Polybromine Clusters

As shown in Figure 2, Figure 3 and Figure 4, the BPs in Br₄–Br₁₂ seem almost straight. The linearity is confirmed by comparing the lengths of BPs (r_BP) with the corresponding straight-line distances (R_SL). The r_BP and R_SL values are collected in Table S3, together with the differences between them, Δr_BP (=r_BP − R_SL). The magnitudes of Δr_BP are less than 0.01 Å, except for r₂ in Br₄ (C_2v) (Δr_BP = 0.014 Å), r₃ in Br₈ (S₄-Wm) (0.014 Å), and r₂ in Br₁₀ (C₂-c) (0.012 Å). Consequently, all BPs in Br₄–Br₁₂ can be approximated as straight lines.

The ρ_b(r_c), H_b(r_c) − V_b(r_c)/2 (=(ћ²/8m)∇²ρ_b(r_c)), and H_b(r_c) values are calculated for the Br-∗-Br interactions at BCPs in the structures of Br₂–Br₁₂, optimized with MP2/6-311+G(3df) [53,54,55].

Table 1. The ρ_b(r_c), H_b(r_c) − V_b(r_c)/2 (=(ћ²/8m)∇²ρ_b(r_c)), and H_b(r_c) values and QTAIM-DFA parameters for Br-∗-Br at BCPs in Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅), together with Br₁₀ (C₂) and Br₂, evaluated with MP2/6-311+G(3df) ¹.

Species	BCP on	ρb(rc)		c∇2ρb(rc) 2	Hb(rc)	R3		θ4
(Symmetry)		(au)		(au)	(au)	(au)		(°)
Br4 (Cs-L1)	r2	0.0109		0.0045	0.0014	0.0048		72.5
Br6 (Cs-L2)	r2	0.0113		0.0047	0.0014	0.0049		73.0
Br6 (Cs-L2)	r4	0.0119		0.0049	0.0014	0.0051		73.7
Br8 (Cs-L3)	r2	0.0114		0.0047	0.0014	0.0049		73.2
Br8 (Cs-L3)	r4	0.0124		0.0050	0.0014	0.0052		74.4
Br8 (Cs-L3)	r6	0.0120		0.0049	0.0014	0.0051		73.9
Br10 (Cs-L4)	r2	0.0114		0.0047	0.0014	0.0049		73.2
Br10 (Cs-L4)	r4	0.0125		0.0051	0.0014	0.0053		74.6
Br10 (Cs-L4)	r6	0.0125		0.0051	0.0014	0.0053		74.6
Br10 (Cs-L4)	r8	0.0120		0.0049	0.0014	0.0051		73.9
Br12 (Cs-L5)	r2	0.0114		0.0047	0.0014	0.0049		73.2
Br12 (Cs-L5)	r4	0.0126		0.0051	0.0014	0.0053		74.7
Br12 (Cs-L5)	r6	0.0127		0.0051	0.0014	0.0053		74.7
Br12 (Cs-L5)	r8	0.0126		0.0051	0.0014	0.0053		74.7
Br12 (Cs-L5)	r10	0.0120		0.0049	0.0014	0.0051		73.9
Br6 (C2)	r2	0.0104		0.0044	0.0014	0.0046		72.1
Br10 (C2)	r2	0.0118		0.0048	0.0014	0.0050		73.6
Br10 (C2)	r4	0.0106		0.0044	0.0014	0.0046		72.3
Species	Cii5		θp:CIV6		κp:CIV7		Predicted
(Symmetry)	(Å mdyn−1)		(°)		(au−1)		nature
Br4 (Cs-L1)	15.311		87.8		121.2		p-CS/vdW 8
Br6 (Cs-L2)	14.984		89.0		124.9		p-CS/vdW 8
Br6 (Cs-L2)	14.114		90.6		127.3		p-CS/t-HB 9
Br8 (Cs-L3)	14.826		89.2		125.0		p-CS/vdW 8
Br8 (Cs-L3)	13.590		92.2		132.0		p-CS/t-HB 9
Br8 (Cs-L3)	14.048		90.9		127.1		p-CS/t-HB 9
Br10 (Cs-L4)	14.751		89.4		126.2		p-CS/vdW 8
Br10 (Cs-L4)	13.445		92.6		133.2		p-CS/t-HB 9
Br10 (Cs-L4)	13.478		92.6		132.5		p-CS/t-HB 9
Br10 (Cs-L4)	13.983		91.1		128.4		p-CS/t-HB 9
Br12 (Cs-L5)	14.719		89.5		126.9		p-CS/vdW 8
Br12 (Cs-L5)	13.376		92.7		133.3		p-CS/t-HB 9
Br12 (Cs-L5)	13.334		93.0		134.3		p-CS/t-HB 9
Br12 (Cs-L5)	13.393		92.8		132.6		p-CS/t-HB 9
Br12 (Cs-L5)	13.962		91.1		128.8		p-CS/t-HB 9
Br6 (C2)	16.025		86.7		119.2		p-CS/vdW 8
Br10 (C2)	14.218		90.2		126.7		p-CS/t-HB 9
Br10 (C2)	16.378		87.2		120.0		p-CS/vdW 8

¹ The interactions in minima are shown. ² c∇²ρ_b(r_c) = H_b(r_c) − V_b(r_c)/2, where c = ħ²/8m. ³ R = [(H_b(r_c) − V_b(r_c)/2)² + H_b(r_c)²]^1/2. ⁴ θ = 90° − tan⁻¹[H_b(r_c)/(H_b(r_c) − V_b(r_c)/2)]. ⁵ Defined in Equation (R1) in the text. ⁶ θ_p = 90° − tan⁻¹(dy/dx), where (x, y) = (H_b(r_c) − V_b(r_c)/2, H_b(r_c)). ⁷ κ_p = |d²y/dx²|/[1 + (dy/dx)²]^3/2. ⁸ The pure CS interaction of the vdW nature. ⁹ The pure CS interaction of the HB nature without covalency.

collects the values for the noncovalent Br-∗-Br interactions in Br₄–Br₁₂ of the C_s-L_m type.

Table 2. The ρ_b(r_c), H_b(r_c) − V_b(r_c)/2 (=(ћ²/8m)∇²ρ_b(r_c)), and H_b(r_c) values and QTAIM-DFA parameters for Br-∗-Br at BCPs in Br₄–Br₁₂, other than the C_s-L_m structures, evaluated with MP2/6-311+G(3df) ¹.

Species	BCP on	ρb(rc)		c∇2ρb(rc) 2	Hb(rc)	R3		θ4
(Symmetry)		(au)		(au)	(au)	(au)		(°)
Br4 (C2h)	r2	0.0055		0.0022	0.0009	0.0024		67.2
Br4 (D2d)	r2	0.0042		0.0017	0.0007	0.0018		66.0
Br6 (C3h-c)	r2	0.0092		0.0038	0.0013	0.0040		70.7
Br8 (S4)	r2	0.0128		0.0051	0.0014	0.0053		74.8
Br8 (S4-Wm) 5	r2	0.0136		0.0054	0.0013	0.0056		76.0
Br8 (S4-Wm) 5	r3	0.0038		0.0015	0.0007	0.0016		66.0
Br10 (C2-c)	r2	0.0087		0.0035	0.0012	0.0037		70.5
Br10 (C2-c)	r4	0.0097		0.0040	0.0014	0.0042		71.3
Br10 (C2-c)	r6	0.0110		0.0044	0.0014	0.0046		73.0
Br10 (C2-c)	r7	0.0049		0.0019	0.0008	0.0021		66.2
Br10 (C2-c)	r8	0.0049		0.0018	0.0008	0.0020		66.6
Br12 (Ci)	r2	0.0129		0.0052	0.0014	0.0054		75.0
Br12 (Ci)	r4	0.0129		0.0052	0.0014	0.0054		75.0
Species	Cii6		θp:CIV7		κp:CIV8		Predicted
(Symmetry)	(Å mdyn−1)		(°)		(au−1)		nature
Br4 (C2h)	24.709		73.6		122.9		p-CS/vdW 9
Br4 (D2d)	40.402		69.6		136.3		p-CS/vdW 9
Br6 (C3h-c)	25.617		83.3		121.7		p-CS/vdW 9
Br8 (S4)	13.201		93.5		139.2		p-CS/t-HB 10
Br8 (S4-Wm) 5	11.294		95.3		139.0		p-CS/t-HB 10
Br8 (S4-Wm) 5	52.918		67.5		204.0		p-CS/vdW 9
Br10 (C2-c)	34.402		81.3		112.7		p-CS/vdW 9
Br10 (C2-c)	23.971		84.7		122.1		p-CS/vdW 9
Br10 (C2-c)	20.831		87.6		122.6		p-CS/vdW 9
Br10 (C2-c)	29.570		71.5		118.9		p-CS/vdW 9
Br10 (C2-c)	37.855		71.8		120.4		p-CS/vdW 9
Br12 (Ci)	13.483		93.7		137.9		p-CS/t-HB 10
Br12 (Ci)	13.482		93.7		137.3		p-CS/t-HB 10

¹ The interactions in minima are shown. ² c∇²ρ_b(r_c) = H_b(r_c) − V_b(r_c)/2, where c = ħ²/8m. ³ R = [H_b(r_c) − V_b(r_c)/2)² + H_b(r_c)²]1/2. ⁴ θ = 90° − tan⁻¹[H_b(r_c)/(H_b(r_c) − V_b(r_c)/2)]. ⁵ Image from windmill. ⁶ Defined in Equation (R1) in the text. ⁷ θ_p = 90° − tan⁻¹(dy/dx), where (x, y) = (H_b(r_c) − V_b(r_c)/2, H_b(r_c)). ⁸ κ_p = |d²y/dx²|/[1 + (dy/dx)²]^3/2. ⁹ The pure CS interaction of the vdW nature. ¹⁰ The pure CS interaction of the HB nature without covalency.

summarizes the values for the noncovalent Br-∗-Br interactions in Br₄–Br₁₂, other than those of the C_s-L_m type. H_b(r_c) are plotted versus H_b(r_c) − V_b(r_c)/2 for the data shown in Table 1 and Table 2, together with those from the perturbed structures generated with CIV. Figure 5 shows the plots for the noncovalent Br-∗-Br interactions and covalent Br-∗-Br bonds, exemplified by Br₁₀ (C_s-L₄).

QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p) are obtained by analyzing the plots of H_b(r_c) versus H_b(r_c) − V_b(r_c)/2, according to Equations (S3)–(S6). Table 1 collects the QTAIM-DFA parameters for the noncovalent Br-∗-Br interactions of Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅), Br₆ (C₂), and Br₁₀ (C₂) together with the C_ii values. Table 2 collects the (R, θ) and (θ_p, κ_p) values for Br₄–Br₁₂, other than those given in Table 1, together with the C_ii values. The (R, θ) and (θ_p, κ_p) values for the covalent Br-∗-Br bonds in Br₄–Br₁₂ are collected in Table S4.

3.4. The Nature of Br-∗-Br Interactions in Polybromine Clusters

The nature of the covalent and noncovalent Br-∗-Br interactions in Br₂–Br₁₂ is discussed on the basis of the (R, θ, θ_p) values, employing standard values as a reference (see Scheme SA3).

It is instructive to survey the criteria shown in Scheme SA3 before detailed discussion. The criteria tell us that 180° < θ (H_b(r_c) − V_b(r_c)/2 < 0) for the SS interactions and θ < 180° (H_b(r_c) − V_b(r_c)/2 > 0) for the CS interactions. The CS interactions are subdivided into pure CS interactions (p-CS) of 45° < θ < 90° (H_b(r_c) > 0) and regular CS interactions (r-CS) of 90° < θ < 180° (H_b(r_c) < 0). The θ_p value predicts the character of interactions. In the pure CS region of 45° < θ < 90°, the character of interactions will be the vdW type for 45° < θ_p < 90° and the typical-HB type with no covalency (t-HB_nc) for 90° < θ_p < 125°, where θ_p = 125° approximately corresponds to θ = 90°. The classical chemical covalent bonds of SS (180° < θ) will be strong when R > 0.15 au (Cov-s: strong covalent bonds), whereas they will be weak for R < 0.15 au (Cov-w: weak covalent bonds).

The (R, θ, θ_p) values are (0.0576 au, 184.3°, 190.9°) for the original Br₂ if evaluated with MP2/6-311+G(3df). Therefore, the nature of the Br-∗-Br bond in Br₂ is classified by the SS interactions (θ > 180°) and characterized to have a Cov-w nature (θ_p > 180° and R < 0.15 au). The nature is denoted by SS/Cov-w. The (R, θ, θ_p) values for the covalent Br-∗-Br bonds in Br₄–Br₁₂ are (0.0472–0.0578 au, 182.0–184.4°, 190.4–192.1°); therefore, their nature is predicted to be SS/Cov-w. The nature of the covalent Br-∗-Br bonds seems unchanged in the formation of the clusters [53,54]. The noncovalent Br-∗-Br interactions in Br₄–Br₁₂ are all classified by pure CS interactions since θ ≤ 76° (<< 90°) [53,54]. The θ_p values in the C_s-L_m clusters change systematically. The θ_p values for r₂ in Br_2k (C_s-L_m) (k = 2–6) are predicted to be in the range of 89.1° ≤ θ_p ≤ 89.6°, with θ_p = 87.9° for Br₄ (C_s-L₁).

However, the values for r_n-2 in Br_2k (C_s-L_m) (k = 2–6) are in the range of 90.6° ≤ θ_p ≤ 91.2° and the values for noncovalent interactions, other than edge positions, are in the range of 92.1° ≤ θ_p ≤ 93.0°. Namely, the noncovalent Br-∗-Br interactions are predicted to have the vdW nature (p-CS/vdW) for r₂, while the interactions other than r₂ are predicted to have the t-HB_nc nature (p-CS/t-HB_nc) since θ_p > 90°. The θ_p values of r₂ for the C_s-L_m clusters will be less than 90°, irrespective of the angles between r₁ and r₂, which are close to 180°. The θ_p values will be larger than 90° for all noncovalent interactions other than r₂. Table 1 contains the data for Br₁₀ (C₂), of which θ_p = 90.4° (> 90°) for r₂ and θ_p = 87.1° (<90°) for r₄, although Br₁₀ (C₂) is not the C_s-L_m type. The results for r₂ seem reasonable based on the structure (cf. Figure 3), while those for r₄ would be complex. Table 1 summarizes the predicted nature.

In the case of the noncovalent Br-∗-Br interactions in Br₄–Br₁₂, other than the C_s-L_m type clusters, θ_p > 90° for r₂ in Br₈ (S₄) (θ_p = 93.4°) and Br₈ (S₄-Wm) (θ_p = 94.8°) and for r₂, r₄, and r₆ in Br₁₂ (C_i) (93.4° ≤ θ_p ≤ 93.7°). The interactions would have the t-HB_nc nature (p-CS/t-HB_nc). Very weak noncovalent Br-∗-Br interactions are also detected. The ranges of 64.2° ≤ θ ≤ 66.6° and 66.2° ≤ θ_p ≤ 71.2° are predicted for r₂ and r₃ in Br₄ (C_2h), r₂ in Br₄ (C_2v), r₃ in Br₈ (S₄-Wm), and r₇ and r₈ in Br₁₀ (C₂-c). The results are summarized in Table 2.

What are the relationships between the QTAIM-DFA parameters for the noncovalent Br-∗-Br interactions? The θ and θ_p values are plotted versus R. The plots are shown in Figure S4; they give very good correlations. The θ_p values are plotted versus θ. The plot is shown in Figure S5; it also gives a very good correlation.

Table 3. Correlations in the plots ¹.

Entry	Correlation	a	b	Rc2	n
1	ΔEZP vs. ΔEES	0.940	0.129	0.9999	20 2
2	θ vs. R	2595.6	60.70	0.979	33
3	θp vs. R	6449.1	58.19	0.989	33
4	θp vs. θ	2.67	−106.26	0.992	31 3
5	E(2) vs. Cii−1	535.5	−18.22	0.997	15 4
6	E(2)vs. R	9760.9	−29.92	0.983	15 4
7	E(2) vs θ	2.446	−160.88	0.996	15 4
8	E(2) vs. θp	1.067	77.17	0.999	15 4

¹ The constants (a, b, R_c²) are the correlation constant, the y-intercept, and the square of the correlation coefficient, respectively, in y = ax + b. ² Containing TS species. ³ Neglecting the data of r₂ and r₃ in Br₄ (C_2h). ⁴ For the noncovalent Br-∗-Br interactions in Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅).

summarizes the correlations among the QTAIM-DFA parameters.

To further examine the behavior of noncovalent Br-∗-Br interactions, NBO analysis is applied to the interactions.

3.5. NBO Analysis for Br-∗-Br of Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅)

The noncovalent Br-∗-Br interactions in Br₄(C_s-L₁)–Br₁₂ (C_s-L₅) are characterized by σ(3c–4e) of the n(Br)→σ*(Br–Br) type. NBO analysis [56] was applied to the n(Br)→σ*(Br–Br) interactions with MP2/6-311+G(3df). For each donor NBO (i) and acceptor NBO (j), the stabilization energy E(2) is calculated based on the second-order perturbation theory in NBO. The E(2) values are calculated according to Equation (4), where q_i is the donor orbital occupancy, ε_i, ε_j are diagonal elements (orbital energies), and F(i,j) is the off-diagonal NBO Fock matrix element. The values are obtained separately by the contributions from n_s(Br)→σ*(Br–Br) and n_p(Br)→σ*(Br–Br), which are summarized in Table S5. The total values corresponding to n_s+p(Br)→σ*(Br–Br) (=n_s(Br)→σ*(Br–Br) + n_p(Br)→σ*(Br–Br)) were calculated, which are also summarized in Table S5. The total values are employed for the discussion.

E(2) = q_i × F(i,j)²/(ε_j − ε_i)

(4)

Figure 6 shows the plots of E(2) and θ_p for the noncovalent Br-∗-Br interactions in Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅). The values become larger in the order of P (r₂: Br₄ (C_s-L₁)) < P (r₂: Br₆ (C_s-L₂)–Br₁₂ (C_s-L₅)) < P (r_ω: Br₆ (C_s-L₂)–Br₁₂ (C_s-L₅)) < P (r_in: Br₆ (C_s-L₂)–Br₁₂ (C_s-L₅)), where P means E(2) or θ_p, while r_ω and r_in stand for the last end and the inside noncovalent interactions, respectively, in the sequence (see Figure 2 and Figure 3). The values for P = E(2) are as follows: E(2) = 16.6 kJ mol⁻¹ for r₂ in Br₄ (C_s-L₁) < 17.7 ≤ E(2) ≤ 18.2 kJ mol⁻¹ for r₂ in Br₆ (C_s-L₂)–Br₁₂ (C_s-L₅) < 19.5 ≤ E(2) ≤ 20.0 kJ mol⁻¹ for r_ω in Br₆ (C_s-L₂)– Br₁₂ (C_s-L₅) < 21.2 ≤ E(2) ≤ 22.0 kJ mol⁻¹ for r_in in Br₈ (C_s-L₃)–Br₁₂ (C_s-L₅).

Relations between E(2) and C_ii were also examined for noncovalent Br-∗-Br interactions in Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅). The E(2) values were plotted versus C_ii⁻¹ for the noncovalent interactions. Figure 7 shows the plot. The plot gives a very good correlation, which is shown in Table 3 (Entry 5). The results show that the energies for σ(3c–4e) of the n_p(Br)→σ*(Br–Br) type in Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅) are well evaluated, not only by E(2) but also by C_ii⁻¹. Similar relations would be essentially observed for the interactions in the nonlinear clusters; however, the analyses will be much complex due to the unsuitable structures for the NBO analysis, such as the deviations in the interaction angles expected for Br₃ σ(3c–4e), the mutual interactions between Br₃ σ(3c–4e), and/or the steric effect from other bonds and interactions, placed proximity in space. The E(2) values for Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅) were also plotted versus R, θ, and θ_p, shown in Figures S6–S8, respectively. The plots give very good correlations, which are given in Table 3 (Entries 6–8).

3.6. MO Descriptions for Noncovalent Br-∗-Br Interactions in Br₄

As discussed above, Br₃ σ(3c–4e) of the n_p(Br)→σ*(Br–Br) type plays an important role in the formation of Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅). However, there must exist some interactions, other than Br₃ σ(3c–4e), to stabilize the clusters. The ΔE_ES values for Br₄ (C_2h) (−8.0 kJ mol⁻¹) and Br₄ (D_2d) (−9.1 kJ mol⁻¹) are not so different from that for Br₄ (C_s-L₁) (−10.7 kJ mol⁻¹). However, Br₄ (C_2h) and Br₄ (D_2d) must consist of interactions other than σ(3c–4e). Indeed, Br₃ σ(3c–4e) of the n(Br)→σ*(Br–Br) type contributes to stabilizing Br₄ (C_s-L₁), but Br₄ (C_2h) and Br₄ (D_2d) are shown to be stabilized by the σ(Br–Br)→σ*Ry(Br) interaction by NBO, where Ry stands for the Rydberg term, although not shown.

The total energy for a species (E) is given by the sum of the core terms (H_c(i)) over all electrons, Σ_iⁿ H_c(i), and the electron–electron repulsive terms, (Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2, as shown by Equation (5), where H_c(i) consists of the kinetic energy and electron–nuclear attractive terms for electron i. E contains the nuclear–nuclear repulsive terms, although not clearly shown in Equation (5). As shown in Equation (6), the sum of MO energy for electron i, ε_i, over all electrons, Σ_i=1ⁿ ε_i, will be larger than E by (Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 since the electron–electron repulsions are doubly counted in Equation (6). Therefore, Σ_iⁿ H_c(i) and (Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 are given separately by Equations (7) and (8), respectively. The ε_i values for Br₄ (C_2h), Br₄ (D_2d), and 2Br₂, together with Br₄ (C_s-L₁), are collected in Tables S6–S9, respectively, for convenience of discussion. Parameters (ΔP) in the formation of Br_2k from the components are evaluated according to Equation (9). The ΔΣ_iⁿ H_c(i) and Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 values for the formation of Br₄ (C_2h), Br₄ (D_2d), and Br₄ (C_s-L₁) are collected in Table S11.

E = Σ_iⁿ H_c(i) + (Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2

(5)

Σ_i=1ⁿ ε_i = Σ_iⁿ H_c(i) + (Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)

(6)

Σ_iⁿ H_c(i) = 2E − Σ_i=1ⁿ ε_i

(7)

(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 = Σ_i=1ⁿ ε_i − E

(8)

ΔP(Br_2k) = P(Br_2k) − kP(Br₂)

(9)

The nature of noncovalent Br---Br interactions in Br₄ (C_s-L₁) is examined first. The σ(3c–4e) character in Br₄ (C_s-L₁) is confirmed by the natural charge evaluated with NPA (Qn), developed in the formation of Br₄ (C_s-L₁). The evaluated Qn values are Br(1: −0.0128|e⁻|)–Br(2: −0.0002|e⁻|)---Br(3: −0.0010|e⁻|)–Br(4: 0.0140|e⁻|); therefore, Qn(Br(4)–Br(3)) and Qn(Br(2)–Br(1)) are +0.013|e⁻| and −0.013|e⁻|, respectively. Each MO in Br₄ (C_s-L₁) is almost localized on Br(4)–Br(3) or Br(2)–Br(1), except for a few cases. MOs in Br₄ (C_s-L₁) must be affected by the local charge. Each MO energy in Br₄ (C_s-L₁) seems higher than the corresponding value of 2Br₂ by 10–20 kJ mol⁻¹ if the MO is localized on Br(2)–Br(1), lower by 15–25 kJ mol⁻¹ on Br(3)–Br(4), and slightly lower by 0–5 kJ mol⁻¹ if the MO is localized on the whole molecule. We should be careful since it depends on the phase in MO and the position of the Br atom(s). Typical cases are shown in Figure S9. In total, ΔΣ_i=1ⁿ ε_i is evaluated to be −357.2 kJ mol⁻¹ for Br₄ (C_s-L₁). The results show that Br₄ (C_s-L₁) is stabilized in the formation of the dimer from the components through the lowering of MO energies in total, which is consistent with those evaluated with NBO, as discussed above.

Figure 8 shows the plots of ΔΣ_iⁿ H_c(i) and Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 for Br₄ (C_s-L₁), Br₄ (C_2h), and Br₄ (D_2d), together with ΔE_ES and ΔΣ_i=1ⁿ ε_i. In the case of Br₄ (C_s-L₁), ΔΣ_iⁿ H_c(i) and Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 are evaluated to be 335.7 and −346.4 kJ mol⁻¹, respectively, which stabilizes Br₄ (C_s-L₁) in total. Two Br₂ molecules in Br₄ (C_s-L₁) will supply a wider area for electrons without severe disadvantageous steric compression by the L-shaped structure in a plane. The structural feature of Br₄ (C_s-L₁) may reduce (or may not severely increase) the electron–electron repulsive terms, Δ((Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2), relative to the case of 2Br₂, although ΔΣ_iⁿ H_c(i) seems to destabilize it. The ΔΣ_iⁿ H_c(i) + Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 value is equal to −10.7 kJ mol⁻¹, which corresponds to the stabilization energy of Br₄ (C_s-L₁), relative to 2Br₂.

The energy profiles of Br₄ (C_2h) and Br₄ (D_2d) seem very different from that of Br₄ (C_s-L₁). The ΔΣ_i=1ⁿ ε_i terms for Br₄ (C_2h) and Br₄ (D_2d) are evaluated to be 587.5 and 908.1 kJ mol⁻¹, respectively. Namely, Br₄ (C_2h) and Br₄ (D_2d) would be less stable than 2Br₂ if ΔΣ_i=1ⁿ ε_i are compared. Consequently, it is difficult to explain the stability of Br₄ (C_2h) and Br₄ (D_2d), based on the MO energies. On the other hand, ΔΣ_iⁿ H_c(i) of Br₄ (C_2h) and Br₄ (D_2d) are evaluated to be −603.5 and −926.3 kJ mol⁻¹, respectively, whereas Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 are 595.5 and 917.2 kJ mol⁻¹, respectively. As a result, the (ΔΣ_iⁿ H_c(i) + Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2) values are −8.0 and −9.1 kJ mol⁻¹ for Br₄ (C_2h) and Br₄ (D_2d), respectively, which correspond to their ΔE_ES values (relative to 2E(Br₂)). The results show that the stabilizing effect of ΔΣ_iⁿ H_c(i) overcomes the shorter electron–nuclear distances in the species on average. The shorter electron–electron distances must destabilize Br₄ (C_2h) and Br₄ (D_2d) through the factor of Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2, which is the inverse effect from the electron–nuclear interaction on ΔΣ_iⁿ H_c(i). However, the effect of the shorter distances on ΔΣ_iⁿ H_c(i) seems to contribute more effectively than the case of Δ(Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2 in Br₄ (C_2h) and Br₄ (D_2d), although they are not so large.

How can the BPs in Br₄ (C_2h) and Br₄ (D_2d) be rationalized through orbital interactions? The Δε_i values of Br₄ (C_2h) are positive for all occupied MOs, relative to the corresponding values of 2Br₂, except for HOMO-3 (−5.5 kJ mol⁻¹), HOMO-6 (−2.9 kJ mol⁻¹), HOMO-7 (−35.8 kJ mol⁻¹), and HOMO-13 (−1.1 kJ mol⁻¹). Figure 9 illustrates the interactions to produce HOMO, HOMO-3, HOMO-4, and HOMO-7. Indeed, HOMO-7 seems to contribute well to stabilizing Br₄ (C_2h), but HOMO-4 (+40.8 kJ mol⁻¹) is also formed in the π(Br₂)–π(Br₂) mode. Similarly, HOMO (+13.7 kJ mol⁻¹) is formed, together with HOMO-3 in the π*(Br₂) + π*(Br₂) mode. Therefore, all MOs seem not to contribute to stabilizing Br₄ (C_2h) inherently. Nevertheless, HOMO, HOMO-4, and HOMO-7 seem to rationalize the appearance of BPs in Br₄ (C_2h), along the diagonal line and shorter sides of the parallelogram, although all electrons contribute to the appearance of BPs in molecules.

Similarly, Δε_i of Br₄ (D_2d) are positive for all occupied MOs, relative to the corresponding values of 2Br₂, except for HOMO-3 (−1.9 kJ mol⁻¹), HOMO-7 (−39.2 kJ mol⁻¹), and HOMO-13 (−0.5 kJ mol⁻¹). Figure 10 illustrates the interactions to produce HOMO, HOMO-3, HOMO-4, and HOMO-7 in Br₄ (D_2d). HOMO-4 (+50.2 kJ mol⁻¹) is formed through the π(Br₂)–π(Br₂) mode in addition to HOMO-7. Similarly, HOMO (+13.9 kJ mol⁻¹) is formed, accompanied by HOMO-3, in the π*(Br₂) + π*(Br₂) mode. Therefore, no MOs essentially stabilize Br₄ (D_2d). However, the appearance of BPs along the longer and shorter diagonal lines of the tetrahedron of Br₄ (D_2d) seem to be rationalized by HOMO-7, together with HOMO-3 and HOMO-4, modifying the BPs, although BPs will appear as the whole properties of molecules.

The nature of interactions in the charged clusters is also of interest. Such investigations are in progress.

4. Conclusions

The intrinsic dynamic and static nature of noncovalent Br-∗-Br interactions was elucidated for Br₄–Br₁₀ with MP2/6-311+G(3df). QTAIM-DFA was applied to the investigation. H_b(r_c) were plotted versus H_b(r_c) − V_b(r_c)/2 for the interactions at BCPs of the fully optimized structures, together with those from the perturbed structures, generated with CIV. The nature of the covalent Br-∗-Br bonds in Br₄–Br₁₀ is predicted to have the SS/Cov-w nature if calculated with MP2/6-311+G(3df). On the other hand, the nature of the noncovalent Br-∗-Br interactions in Br₄–Br₁₂ is classified by the pure CS interactions (θ ≤ 76°). The noncovalent Br-∗-Br interactions in the linear type clusters of Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅) are predicted to have the p-CS/t-HB_nc nature (90.6° ≤ θ_p), except for r₂, outside the ones of the first end, which have the p-CS/vdW nature, although it is very close to the border area between the two (θ_p ≤ 89.4°). In the case of the cyclic clusters, the noncovalent Br-∗-Br interactions will have the p-CS/vdW nature (θ_p ≤ 88.4°), except for r₂ in Br₈ (S₄) (θ_p = 93.5°) and Br₈ (S₄-Wm) (θ_p = 95.3°), which have the p-CS/t-HB_nc nature.

The energies for Br₃ σ(3c–4e) of the n_p(Br)→σ*(Br–Br) type are well evaluated by not only E(2) but also C_ii⁻¹ for Br₄ (C_s-L₁)–Br₁₂ (C_s-L₅). E(2) correlates very well to C_ii⁻¹. The CT interactions of the n_p(Br)→σ*(Br–Br) type must contribute to form Br₄ (C_s-L₁), which can be explained based on the MO energies, ε_i. However, it seems difficult to explain the stability of Br₄ (C_2h) and Br₄ (D_2d) based on the energies. The Br₂ molecules must be stacked more effectively in Br₄ (C_2h) and Br₄ (D_2d), resulting in shorter electronuclear distances on average. The energy lowering effect by ΔΣ_iⁿ H_c(i), due to the effective stacking of 2Br₂ in Br₄ (C_2h) and Br₄ (D_2d), contributes to form the clusters, although the inverse contribution from Δ((Σ_i≠jⁿ J_ij − Σ_i≠j,‖ⁿ K_ij)/2) must also be considered.