# 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

**:**

_{4}–Br

_{12}, 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 Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}), but the core potentials stabilize them, relative to the case of 2Br

_{2}; this is possibly due to the reduced nuclear–electron distances, on average, for the dimers.

## 1. Introduction

**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(C

_{6}H

_{6}) 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].

_{2}) 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 Br

_{2}–Br

_{5}in the neutral and/or charged forms, together with Br

_{1}, so far [19,20]. Figure 1 draws the observed structure of Br

_{2}, 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 ((Br

_{2})

_{2}; Br

_{4}) and the linear alignment of four Br atoms in a double L-shaped trimer ((Br

_{2})

_{3}; Br

_{6}) in a planar Br

_{2}layer in addition to Br

_{2}itself. The linear four Br atoms are located in the two L-shaped dimers of Br

_{6}, overlapped at the central Br

_{2}. 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 Br

_{4}and Br

_{6}, respectively, where n

_{p}(Br) stands for the p-type nonbonding orbital of Br in the plane, perpendicular to the molecular Br

_{2}axis, and σ*(Br–Br) is the σ*-orbital of Br

_{2}. The crystal structures of Cl

_{2}and I

_{2}are very similar to that of Br

_{2}.

_{b}(

**r**

_{c}) are plotted versus H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 (=(ћ

^{2}/8m)∇

^{2}ρ

_{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].

^{2}ρ

_{b}(

**r**

_{c}) and H

_{b}(

**r**

_{c}), based on the QTAIM approach. The interactions are called shard shell (SS) interactions when ∇

^{2}ρ

_{b}(

**r**

_{c}) < 0 and closed-shell (CS) interactions when ∇

^{2}ρ

_{b}(

**r**

_{c}) > 0 [26]. In particular, CS interactions are called pure CS (p-CS) interactions when H

_{b}(

**r**

_{c}) > 0 and ∇

^{2}ρ

_{b}(

**r**

_{c}) > 0. We call interactions where H

_{b}(

**r**

_{c}) < 0 and ∇

^{2}ρ

_{b}(

**r**

_{c}) > 0 regular CS (r-CS) interactions, which clearly distinguishes these interactions from the p-CS interactions. The signs of ∇

^{2}ρ

_{b}(

**r**

_{c}) can be replaced by those of H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 because (ћ

^{2}/8m)∇

^{2}ρ

_{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.

_{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.

## 2. Methodological Details in Calculations

_{2}–Br

_{12}. 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 (=(ћ

^{2}/8m)∇

^{2}ρ

_{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.

**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}

_{o}+ wa

_{o}(w = (0), ±0.05 and ±0.1; a

_{o}= 0.52918 Å)

_{o}+ c

_{1}x + c

_{2}x

^{2}+ c

_{3}x

^{3}(R

_{c}

^{2}: square of correlation coefficient)

_{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

_{c}

^{2}(square of correlation coefficient) > 0.99999 in the norm) [25].

## 3. Results and Discussion

#### 3.1. Structural Optimizations of Polybromine Clusters, Br_{6}–Br_{12}

_{2}–Br

_{12}clusters were optimized with MP2/6-311+G(3df). The structural parameters for the optimized structures of minima for Br

_{2}–Br

_{6}and Br

_{8}–Br

_{12}are collected in Tables S1 and S2, respectively. Some transition states (TSs) for Br

_{4}and Br

_{6}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 Br

_{4}–Br

_{12}(C

_{s}-L

_{m}(m = 1–5)) and for the cyclic bromine clusters Br

_{4}–Br

_{12}, drawn on the optimized structures with MP2/6-311+G(3df) [51]. The energies for the formation of Br

_{4}–Br

_{6}and Br

_{8}–Br

_{12}are given in Tables S1 and S2, respectively, from the components (∆E = E(Br

_{2k}) − kE(Br

_{2})) 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

_{c}

^{2}(square of correlation coefficient) = 0.9999) [52]. Therefore, the ∆E

_{ES}values are employed for the discussion.

_{4}) is discussed first. Three structures were optimized for Br

_{4}as minima with some TSs. The minima are the L-shaped structure of C

_{s}symmetry (Br

_{4}(C

_{s}-L

_{1})) [19], the cyclic structure of C

_{2h}symmetry (Br

_{4}(C

_{2h})), and the tetrahedral type of D

_{2d}symmetry (Br

_{4}(D

_{2d})). A TS of the C

_{s}symmetry was detected between Br

_{4}(C

_{s}-L

_{1}) and Br

_{4}(C

_{2h}), and two TSs of the C

_{1}symmetry were between Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) and between Br

_{4}(D

_{2d}) and Br

_{4}(C

_{s}-L

_{1}). They are called TS (C

_{s}: C

_{s}, C

_{2h}), TS (C

_{1}: C

_{2h}, D

_{2d}), and TS (C

_{1}: D

_{2d}, C

_{s}), respectively. The three minima will be converted to each other through the three TSs. A TS between Br

_{4}(C

_{s}-L

_{1}) 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 Br

_{4}(C

_{2h}) and its topological isomer and between Br

_{4}(C

_{2d}) and its topological isomer.

_{4}(C

_{s}-L

_{1}), Br

_{4}(C

_{2h}), and Br

_{4}(D

_{2d}), together with the TSs TS (C

_{s}: C

_{s}, C

_{2h}), TS (C

_{s}: C

_{2h}, D

_{2d}), TS (C

_{1}: 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 Br

_{4}(C

_{s}-L

_{1}) seems very close to the border area between the vdW and typical hydrogen bond (t-HB) adducts. The driving force for the formation of Br

_{4}(C

_{s}-L

_{1}) must be Br

_{3}σ(3c–4e) of the n

_{p}(Br)→σ*(Br–Br) type. The interactions in Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) seem very different from those in Br

_{4}(C

_{s}-L

_{1}). The ΔE

_{ES}values of Br

_{4}(C

_{2h}) (−8.0 kJ mol

^{−1}) and Br

_{4}(D

_{2d}) (−9.1 kJ mol

^{−1}) are close to that for Br

_{4}(C

_{s}-L

_{1}) (−10.7 kJ mol

^{−1}). Moreover, the values for TS (C

_{s}: C

_{s}, C

_{2h}) (−7.4 kJ mol

^{−1}), TS (C

_{1}: C

_{2h}, D

_{2d}) (−7.6 kJ mol

^{−1}), TS (C

_{1}: 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.

_{6}, three structures of the linear C

_{s}symmetry (Br

_{6}(C

_{s}-L

_{2})), the linear C

_{2}symmetry (Br

_{6}(C

_{2})), and the cyclic C

_{3h}symmetry (Br

_{6}(C

_{3h}-c)) were optimized typically as minima. The linear Br

_{6}clusters of C

_{2h}symmetry (Br

_{6}(C

_{2h})) and C

_{2v}symmetry (Br

_{6}(C

_{2v})), similar to Br

_{6}(C

_{2}), were also optimized, of which the torsional angles, ϕ(

^{1}Br

^{2}Br

^{5}Br

^{6}Br) (=ϕ

_{3}), were 0° and 180°, respectively. One imaginary frequency was detected for each; therefore, they are assigned to TSs between Br

_{6}(C

_{2}) and the topological isomer on the different reaction coordinates. Further effort was not made to search for TSs.

_{ES}value for Br

_{6}(C

_{s}-L

_{2}) was predicted to be −22.6 kJ mol

^{−1}. The magnitude is slightly larger than the double value for Br

_{4}(C

_{s}-L

_{1}) (∆E

_{ES}= −10.7 kJ mol

^{−1}). Two types of σ (3c–4e) operate to stabilize Br

_{6}(C

_{s}-L

_{2}). One, σ(3c–4e), seems similar to that in Br

_{4}(C

_{s}-L

_{1}), 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 Br

_{6}(C

_{s}-L

_{2}). On the other hand, the linear interaction in Br

_{6}(C

_{2}) 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 Br

_{6}(C

_{2}) seems slightly smaller than that of Br

_{6}(C

_{s}-L

_{2}) but is very close to the double value for Br

_{4}(C

_{s}-L

_{1}). The magnitude of ∆E

_{ES}for Br

_{6}(C

_{3h}-c) is close to the triple value of Br

_{4}(C

_{s}-L

_{1}). One finds triply degenerated σ(3c–4e) interactions in Br

_{6}(C

_{3h}-c). The similarity in the interactions for Br

_{4}(C

_{s}-L

_{1}), Br

_{6}(C

_{2}), and Br

_{6}(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 r

_{2}and r

_{ω}, respectively).

#### 3.2. Molecular Graphs with Contour Plots of Polybromine Clusters

**r**) for the linear type of Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), drawn on the structures optimized with MP2/6-311+G(3df). Figure 4 draws the molecular graphs with contour maps of ρ(

**r**) for Br

_{4}–Br

_{12}, other than those for Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), 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

_{4}–Br

_{12}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 r

_{2}in Br

_{4}(C

_{2v}) (Δr

_{BP}= 0.014 Å), r

_{3}in Br

_{8}(S

_{4}-Wm) (0.014 Å), and r

_{2}in Br

_{10}(C

_{2}-c) (0.012 Å). Consequently, all BPs in Br

_{4}–Br

_{12}can be approximated as straight lines.

_{b}(

**r**

_{c}), H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 (=(ћ

^{2}/8m)∇

^{2}ρ

_{b}(

**r**

_{c})), and H

_{b}(

**r**

_{c}) values are calculated for the Br-∗-Br interactions at BCPs in the structures of Br

_{2}–Br

_{12}, optimized with MP2/6-311+G(3df) [53,54,55]. Table 1 collects the values for the noncovalent Br-∗-Br interactions in Br

_{4}–Br

_{12}of the C

_{s}-L

_{m}type. Table 2 summarizes the values for the noncovalent Br-∗-Br interactions in Br

_{4}–Br

_{12}, 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 Br

_{10}(C

_{s}-L

_{4}).

_{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 Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), Br

_{6}(C

_{2}), and Br

_{10}(C

_{2}) together with the C

_{ii}values. Table 2 collects the (R, θ) and (θ

_{p}, κ

_{p}) values for Br

_{4}–Br

_{12}, 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 Br

_{4}–Br

_{12}are collected in Table S4.

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

_{2}–Br

_{12}is discussed on the basis of the (R, θ, θ

_{p}) values, employing standard values as a reference (see Scheme SA3).

_{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).

_{p}) values are (0.0576 au, 184.3°, 190.9°) for the original Br

_{2}if evaluated with MP2/6-311+G(3df). Therefore, the nature of the Br-∗-Br bond in Br

_{2}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 Br

_{4}–Br

_{12}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 Br

_{4}–Br

_{12}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 r

_{2}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 Br

_{4}(C

_{s}-L

_{1}).

_{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 r

_{2}, while the interactions other than r

_{2}are predicted to have the t-HB

_{nc}nature (p-CS/t-HB

_{nc}) since θ

_{p}> 90°. The θ

_{p}values of r

_{2}for the C

_{s}-L

_{m}clusters will be less than 90°, irrespective of the angles between r

_{1}and r

_{2}, which are close to 180°. The θ

_{p}values will be larger than 90° for all noncovalent interactions other than r

_{2}. Table 1 contains the data for Br

_{10}(C

_{2}), of which θ

_{p}= 90.4° (> 90°) for r

_{2}and θ

_{p}= 87.1° (<90°) for r

_{4}, although Br

_{10}(C

_{2}) is not the C

_{s}-L

_{m}type. The results for r

_{2}seem reasonable based on the structure (cf. Figure 3), while those for r

_{4}would be complex. Table 1 summarizes the predicted nature.

_{4}–Br

_{12}, other than the C

_{s}-L

_{m}type clusters, θ

_{p}> 90° for r

_{2}in Br

_{8}(S

_{4}) (θ

_{p}= 93.4°) and Br

_{8}(S

_{4}-Wm) (θ

_{p}= 94.8°) and for r

_{2}, r

_{4}, and r

_{6}in Br

_{12}(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 r

_{2}and r

_{3}in Br

_{4}(C

_{2h}), r

_{2}in Br

_{4}(C

_{2v}), r

_{3}in Br

_{8}(S

_{4}-Wm), and r

_{7}and r

_{8}in Br

_{10}(C

_{2}-c). The results are summarized in Table 2.

_{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 summarizes the correlations among the QTAIM-DFA parameters.

#### 3.5. NBO Analysis for Br-∗-Br of Br_{4} (C_{s}-L_{1})–Br_{12} (C_{s}-L_{5})

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}) 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.

_{i}× F(i,j)

^{2}/(ε

_{j}− ε

_{i})

_{p}for the noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}). The values become larger in the order of P (r

_{2}: Br

_{4}(C

_{s}-L

_{1})) < P (r

_{2}: Br

_{6}(C

_{s}-L

_{2})–Br

_{12}(C

_{s}-L

_{5})) < P (r

_{ω}: Br

_{6}(C

_{s}-L

_{2})–Br

_{12}(C

_{s}-L

_{5})) < P (r

_{in}: Br

_{6}(C

_{s}-L

_{2})–Br

_{12}(C

_{s}-L

_{5})), 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 r

_{2}in Br

_{4}(C

_{s}-L

_{1}) < 17.7 ≤ E(2) ≤ 18.2 kJ mol

^{−1}for r

_{2}in Br

_{6}(C

_{s}-L

_{2})–Br

_{12}(C

_{s}-L

_{5}) < 19.5 ≤ E(2) ≤ 20.0 kJ mol

^{−1}for r

_{ω}in Br

_{6}(C

_{s}-L

_{2})– Br

_{12}(C

_{s}-L

_{5}) < 21.2 ≤ E(2) ≤ 22.0 kJ mol

^{−1}for r

_{in}in Br

_{8}(C

_{s}-L

_{3})–Br

_{12}(C

_{s}-L

_{5}).

_{ii}were also examined for noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}). 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 Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}) 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 Br

_{3}σ(3c–4e), the mutual interactions between Br

_{3}σ(3c–4e), and/or the steric effect from other bonds and interactions, placed proximity in space. The E(2) values for Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}) 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 Br_{4}

_{3}σ(3c–4e) of the n

_{p}(Br)→σ*(Br–Br) type plays an important role in the formation of Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}). However, there must exist some interactions, other than Br

_{3}σ(3c–4e), to stabilize the clusters. The ΔE

_{ES}values for Br

_{4}(C

_{2h}) (−8.0 kJ mol

^{−1}) and Br

_{4}(D

_{2d}) (−9.1 kJ mol

^{−1}) are not so different from that for Br

_{4}(C

_{s}-L

_{1}) (−10.7 kJ mol

^{−1}). However, Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) must consist of interactions other than σ(3c–4e). Indeed, Br

_{3}σ(3c–4e) of the n(Br)→σ*(Br–Br) type contributes to stabilizing Br

_{4}(C

_{s}-L

_{1}), but Br

_{4}(C

_{2h}) and Br

_{4}(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.

_{c}(i)) over all electrons, Σ

_{i}

^{n}H

_{c}(i), and the electron–electron repulsive terms, (Σ

_{i}

_{≠j}

^{n}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}

_{=1}

^{n}ε

_{i}, will be larger than E by (Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 since the electron–electron repulsions are doubly counted in Equation (6). Therefore, Σ

_{i}

^{n}H

_{c}(i) and (Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 are given separately by Equations (7) and (8), respectively. The ε

_{i}values for Br

_{4}(C

_{2h}), Br

_{4}(D

_{2d}), and 2Br

_{2}, together with Br

_{4}(C

_{s}-L

_{1}), 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

_{i}

^{n}H

_{c}(i) and Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 values for the formation of Br

_{4}(C

_{2h}), Br

_{4}(D

_{2d}), and Br

_{4}(C

_{s}-L

_{1}) are collected in Table S11.

_{i}

^{n}H

_{c}(i) + (Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2

_{i=1}

^{n}ε

_{i}= Σ

_{i}

^{n}H

_{c}(i) + (Σ

_{i≠j}

^{n}J

_{ij}− Σ

_{i≠j,‖}

^{n}K

_{ij})

_{i}

^{n}H

_{c}(i) = 2E − Σ

_{i=1}

^{n}ε

_{i}

_{i≠j}

^{n}J

_{ij}− Σ

_{i≠j,‖}

^{n}K

_{ij})/2 = Σ

_{i=1}

^{n}ε

_{i}− E

_{2k}) = P(Br

_{2k}) − kP(Br

_{2})

_{4}(C

_{s}-L

_{1}) is examined first. The σ(3c–4e) character in Br

_{4}(C

_{s}-L

_{1}) is confirmed by the natural charge evaluated with NPA (Qn), developed in the formation of Br

_{4}(C

_{s}-L

_{1}). 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 Br

_{4}(C

_{s}-L

_{1}) is almost localized on Br(4)–Br(3) or Br(2)–Br(1), except for a few cases. MOs in Br

_{4}(C

_{s}-L

_{1}) must be affected by the local charge. Each MO energy in Br

_{4}(C

_{s}-L

_{1}) seems higher than the corresponding value of 2Br

_{2}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}

_{=1}

^{n}ε

_{i}is evaluated to be −357.2 kJ mol

^{−1}for Br

_{4}(C

_{s}-L

_{1}). The results show that Br

_{4}(C

_{s}-L

_{1}) 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.

_{i}

^{n}H

_{c}(i) and Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 for Br

_{4}(C

_{s}-L

_{1}), Br

_{4}(C

_{2h}), and Br

_{4}(D

_{2d}), together with ΔE

_{ES}and ΔΣ

_{i}

_{=1}

^{n}ε

_{i}. In the case of Br

_{4}(C

_{s}-L

_{1}), ΔΣ

_{i}

^{n}H

_{c}(i) and Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 are evaluated to be 335.7 and −346.4 kJ mol

^{−1}, respectively, which stabilizes Br

_{4}(C

_{s}-L

_{1}) in total. Two Br

_{2}molecules in Br

_{4}(C

_{s}-L

_{1}) will supply a wider area for electrons without severe disadvantageous steric compression by the L-shaped structure in a plane. The structural feature of Br

_{4}(C

_{s}-L

_{1}) may reduce (or may not severely increase) the electron–electron repulsive terms, Δ((Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2), relative to the case of 2Br

_{2}, although ΔΣ

_{i}

^{n}H

_{c}(i) seems to destabilize it. The ΔΣ

_{i}

^{n}H

_{c}(i) + Δ(Σ

_{i}

_{≠j}

^{n}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 Br

_{4}(C

_{s}-L

_{1}), relative to 2Br

_{2}.

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) seem very different from that of Br

_{4}(C

_{s}-L

_{1}). The ΔΣ

_{i}

_{=1}

^{n}ε

_{i}terms for Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) are evaluated to be 587.5 and 908.1 kJ mol

^{−1}, respectively. Namely, Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) would be less stable than 2Br

_{2}if ΔΣ

_{i}

_{=1}

^{n}ε

_{i}are compared. Consequently, it is difficult to explain the stability of Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}), based on the MO energies. On the other hand, ΔΣ

_{i}

^{n}H

_{c}(i) of Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) are evaluated to be −603.5 and −926.3 kJ mol

^{−1}, respectively, whereas Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 are 595.5 and 917.2 kJ mol

^{−1}, respectively. As a result, the (ΔΣ

_{i}

^{n}H

_{c}(i) + Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2) values are −8.0 and −9.1 kJ mol

^{−1}for Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}), respectively, which correspond to their ΔE

_{ES}values (relative to 2E(Br

_{2})). The results show that the stabilizing effect of ΔΣ

_{i}

^{n}H

_{c}(i) overcomes the shorter electron–nuclear distances in the species on average. The shorter electron–electron distances must destabilize Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) through the factor of Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2, which is the inverse effect from the electron–nuclear interaction on ΔΣ

_{i}

^{n}H

_{c}(i). However, the effect of the shorter distances on ΔΣ

_{i}

^{n}H

_{c}(i) seems to contribute more effectively than the case of Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 in Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}), although they are not so large.

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) be rationalized through orbital interactions? The Δε

_{i}values of Br

_{4}(C

_{2h}) are positive for all occupied MOs, relative to the corresponding values of 2Br

_{2}, 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 Br

_{4}(C

_{2h}), but HOMO-4 (+40.8 kJ mol

^{−1}) is also formed in the π(Br

_{2})–π(Br

_{2}) mode. Similarly, HOMO (+13.7 kJ mol

^{−1}) is formed, together with HOMO-3 in the π*(Br

_{2}) + π*(Br

_{2}) mode. Therefore, all MOs seem not to contribute to stabilizing Br

_{4}(C

_{2h}) inherently. Nevertheless, HOMO, HOMO-4, and HOMO-7 seem to rationalize the appearance of BPs in Br

_{4}(C

_{2h}), along the diagonal line and shorter sides of the parallelogram, although all electrons contribute to the appearance of BPs in molecules.

_{i}of Br

_{4}(D

_{2d}) are positive for all occupied MOs, relative to the corresponding values of 2Br

_{2}, 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 Br

_{4}(D

_{2d}). HOMO-4 (+50.2 kJ mol

^{−1}) is formed through the π(Br

_{2})–π(Br

_{2}) mode in addition to HOMO-7. Similarly, HOMO (+13.9 kJ mol

^{−1}) is formed, accompanied by HOMO-3, in the π*(Br

_{2}) + π*(Br

_{2}) mode. Therefore, no MOs essentially stabilize Br

_{4}(D

_{2d}). However, the appearance of BPs along the longer and shorter diagonal lines of the tetrahedron of Br

_{4}(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.

## 4. Conclusions

_{4}–Br

_{10}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 Br

_{4}–Br

_{10}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 Br

_{4}–Br

_{12}is classified by the pure CS interactions (θ ≤ 76°). The noncovalent Br-∗-Br interactions in the linear type clusters of Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}) are predicted to have the p-CS/t-HB

_{nc}nature (90.6° ≤ θ

_{p}), except for r

_{2}, 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 r

_{2}in Br

_{8}(S

_{4}) (θ

_{p}= 93.5°) and Br

_{8}(S

_{4}-Wm) (θ

_{p}= 95.3°), which have the p-CS/t-HB

_{nc}nature.

_{3}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 Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}). 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 Br

_{4}(C

_{s}-L

_{1}), which can be explained based on the MO energies, ε

_{i}. However, it seems difficult to explain the stability of Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}) based on the energies. The Br

_{2}molecules must be stacked more effectively in Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}), resulting in shorter electronuclear distances on average. The energy lowering effect by ΔΣ

_{i}

^{n}H

_{c}(i), due to the effective stacking of 2Br

_{2}in Br

_{4}(C

_{2h}) and Br

_{4}(D

_{2d}), contributes to form the clusters, although the inverse contribution from Δ((Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2) must also be considered.

## Supplementary Materials

_{2}–Br

_{6}, Table S2: Structural parameters for Br

_{8}–Br

_{12}, Table S3: The bond path distances and the straight-line distances in the polybromide clusters, together with the differences between the two, Table S4: The ρ

_{b}(

**r**

_{c}), H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 (=(ћ

^{2}/8m)∇

^{2}ρ

_{b}(

**r**

_{c})), and H

_{b}(

**r**

_{c}) values and QTAIM-DFA parameters for Br-∗-Br in polybromine clusters of Br

_{2}–Br

_{12}, Table S5: Contributions from the donor–acceptor (NBO(i)→NBO(j)) interactions of the n(Br)→σ*(Br–Br) type in the optimized structures of Br

_{4}–Br

_{12}, calculated using NBO analysis, Table S6: MO energies of Br

_{4}(C

_{2h}), Table S7: MO energies of Br

_{4}(D

_{2d}), Table S8: MO energies of Br

_{2}(D

_{∞h}), Table S9: MO energies of Br

_{4}(C

_{s}-L

_{1}), Table S10: The Δε

_{i}values for Br

_{4}(C

_{s}-L

_{1}), relative to 2Br

_{2}(D

_{∞h}), Table S11: Energies for the Br

_{4}clusters and 2Br

_{2}, together with the differences between the two, Figure S1: Plot of ΔE

_{ZP}versus ΔE

_{ES}for Br

_{4}–Br

_{12}, relative to those of Br

_{2}, respectively, Figure S2: Plots of ΔE

_{ES}for Br

_{2}–Br

_{12}(C

_{s}-L

_{n}), Figure S3: Optimized structures for the cyclic bromine clusters of Br

_{8}–Br

_{12}, together with the linear type bromine cluster of Br

_{10}, Figure S4: Plot of θ and θ

_{p}versus R for the noncovalent Br-∗-Br interactions at the BCPs in the fully optimized structures of Br

_{4}–Br

_{12}, Figure S5: Plot of θ

_{p}versus θ for the noncovalent Br-∗-Br interactions at the BCPs in the fully optimized structures of Br

_{4}–Br

_{12}, Figure S6: Plot of E(2) versus R for noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), Figure S7: Plot of E(2) versus θ for noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), Figure S8: Plot of E(2) versus θ

_{p}for noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), Figure S9: MO

_{i}(i = 70, 67, 64, 35, and 30) and the energies relative to those corresponding to 2Br

_{2}, and Cartesian coordinates and energies of all the species involved in the present work. Appendix: Survey of QTAIM, closely related to QTAIM dual-functional analysis; Criteria for classification of interactions: behavior of typical interactions elucidated by QTAIM-DFA; Characterization of interactions.

## Author Contributions

_{b}(

**r**

_{c}), H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 (=(ћ

^{2}/8m)∇

^{2}ρ

_{b}(

**r**

_{c})), and H

_{b}(

**r**

_{c}) values and evaluated the QTAIM-DFA parameters and analyzed the data. S.H. and W.N. wrote the paper, while T.N. and E.T. organized the data to assist the writing. All authors have read and agreed to the published version of the manuscript.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 2.**Energy profile with molecular graphs for the structures of Br

_{4}clusters, optimized with MP2/6-311+G(3df).

**Figure 3.**Molecular graphs with contour plots of ρ(

**r**) for the linear-type bromine clusters of Br

_{4}–Br

_{12}, calculated with MP2/6-311+G(3df). (

**a**–

**e**) for the linear C

_{s}-L

_{m}type, (

**f**,

**g**) for the C

_{2}type, and (

**h**) for the notations of atoms, bonds, and angles, exemplified by B

_{12}(C

_{s}-L

_{5}). BCPs are denoted by red dots, and BPs (bond paths) are by pink lines. Bromine atoms are in reddish-brown.

**Figure 4.**Molecular graphs with contour plots of ρ(

**r**) for the cyclic bromine clusters of Br

_{4}–Br

_{12}, (

**a**–

**g**), calculated with MP2/6-311+G(3df). BCPs are denoted by red dots, RCPs (ring-critical points) by yellow dots, CCPs (cage-critical points) by blue dots, and BPs (bond paths) by pink lines. See ref. [55] for (

**a**).

**Figure 5.**QTAIM-DFA plots (H

_{b}(

**r**

_{c}) versus H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2) for the interactions in Br

_{10}(C

_{s}-L

_{4}), evaluated with MP2/6-311+G(3df); (

**a**) whole region, (

**b**) pure CS region, and (

**c**) SS region. Marks and colors are shown in the figure.

**Figure 6.**Plots of θ

_{p}and E(2) for the noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}). Colors are shown in the figure.

**Figure 7.**Plot of E(2) versus 1/C

_{ii}for the noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}).

**Figure 8.**Contributions from ΔΣ

_{i}

^{n}H

_{c}(i) (=ΔP =

**B**) and Δ(Σ

_{i}

_{≠j}

^{n}J

_{ij}− Σ

_{i}

_{≠j,‖}

^{n}K

_{ij})/2 (=ΔP =

**C**) to ΔE

_{ES}(=ΔP =

**D**, magnified by 10 times in the plot) for Br

_{4}(C

_{s}-L

_{1}), Br

_{4}(C

_{2h}), and Br

_{4}(D

_{2d}), relative to 2Br

_{2}, together with ΔΣ

_{i}

_{=1}

^{n}ε

_{i}(=ΔP =

**A**).

**Figure 9.**Energy profile for the formation of Br

_{4}(C

_{2h}), exemplified by HOMO, HOMO-3, HOMO-4, and HOMO-7.

**Figure 10.**Energy profile for the formation of Br

_{4}(D

_{2d}), exemplified by HOMO, HOMO-3, HOMO-4, and HOMO-7.

**Table 1.**The ρ

_{b}(

**r**

_{c}), H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 (=(ћ

^{2}/8m)∇

^{2}ρ

_{b}(

**r**

_{c})), and H

_{b}(

**r**

_{c}) values and QTAIM-DFA parameters for Br-∗-Br at BCPs in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}), together with Br

_{10}(C

_{2}) and Br

_{2}, evaluated with MP2/6-311+G(3df)

^{1}.

Species | BCP on | ρ_{b}(r_{c}) | c∇^{2}ρ_{b}(r_{c}) ^{2} | H_{b}(r_{c}) | R^{3} | θ^{4} | ||

(Symmetry) | (au) | (au) | (au) | (au) | (°) | |||

Br_{4} (C_{s}-L_{1}) | r_{2} | 0.0109 | 0.0045 | 0.0014 | 0.0048 | 72.5 | ||

Br_{6} (C_{s}-L_{2}) | r_{2} | 0.0113 | 0.0047 | 0.0014 | 0.0049 | 73.0 | ||

Br_{6} (C_{s}-L_{2}) | r_{4} | 0.0119 | 0.0049 | 0.0014 | 0.0051 | 73.7 | ||

Br_{8} (C_{s}-L_{3}) | r_{2} | 0.0114 | 0.0047 | 0.0014 | 0.0049 | 73.2 | ||

Br_{8} (C_{s}-L_{3}) | r_{4} | 0.0124 | 0.0050 | 0.0014 | 0.0052 | 74.4 | ||

Br_{8} (C_{s}-L_{3}) | r_{6} | 0.0120 | 0.0049 | 0.0014 | 0.0051 | 73.9 | ||

Br_{10} (C_{s}-L_{4}) | r_{2} | 0.0114 | 0.0047 | 0.0014 | 0.0049 | 73.2 | ||

Br_{10} (C_{s}-L_{4}) | r_{4} | 0.0125 | 0.0051 | 0.0014 | 0.0053 | 74.6 | ||

Br_{10} (C_{s}-L_{4}) | r_{6} | 0.0125 | 0.0051 | 0.0014 | 0.0053 | 74.6 | ||

Br_{10} (C_{s}-L_{4}) | r_{8} | 0.0120 | 0.0049 | 0.0014 | 0.0051 | 73.9 | ||

Br_{12} (C_{s}-L_{5}) | r_{2} | 0.0114 | 0.0047 | 0.0014 | 0.0049 | 73.2 | ||

Br_{12} (C_{s}-L_{5}) | r_{4} | 0.0126 | 0.0051 | 0.0014 | 0.0053 | 74.7 | ||

Br_{12} (C_{s}-L_{5}) | r_{6} | 0.0127 | 0.0051 | 0.0014 | 0.0053 | 74.7 | ||

Br_{12} (C_{s}-L_{5}) | r_{8} | 0.0126 | 0.0051 | 0.0014 | 0.0053 | 74.7 | ||

Br_{12} (C_{s}-L_{5}) | r_{10} | 0.0120 | 0.0049 | 0.0014 | 0.0051 | 73.9 | ||

Br_{6} (C_{2}) | r_{2} | 0.0104 | 0.0044 | 0.0014 | 0.0046 | 72.1 | ||

Br_{10} (C_{2}) | r_{2} | 0.0118 | 0.0048 | 0.0014 | 0.0050 | 73.6 | ||

Br_{10} (C_{2}) | r_{4} | 0.0106 | 0.0044 | 0.0014 | 0.0046 | 72.3 | ||

Species | C_{ii}^{5} | θ_{p:CIV}^{6} | κ_{p:CIV}^{7} | Predicted | ||||

(Symmetry) | (Å mdyn^{−1}) | (°) | (au^{−1}) | nature | ||||

Br_{4} (C_{s}-L_{1}) | 15.311 | 87.8 | 121.2 | p-CS/vdW ^{8} | ||||

Br_{6} (C_{s}-L_{2}) | 14.984 | 89.0 | 124.9 | p-CS/vdW ^{8} | ||||

Br_{6} (C_{s}-L_{2}) | 14.114 | 90.6 | 127.3 | p-CS/t-HB ^{9} | ||||

Br_{8} (C_{s}-L_{3}) | 14.826 | 89.2 | 125.0 | p-CS/vdW ^{8} | ||||

Br_{8} (C_{s}-L_{3}) | 13.590 | 92.2 | 132.0 | p-CS/t-HB ^{9} | ||||

Br_{8} (C_{s}-L_{3}) | 14.048 | 90.9 | 127.1 | p-CS/t-HB ^{9} | ||||

Br_{10} (C_{s}-L_{4}) | 14.751 | 89.4 | 126.2 | p-CS/vdW ^{8} | ||||

Br_{10} (C_{s}-L_{4}) | 13.445 | 92.6 | 133.2 | p-CS/t-HB ^{9} | ||||

Br_{10} (C_{s}-L_{4}) | 13.478 | 92.6 | 132.5 | p-CS/t-HB ^{9} | ||||

Br_{10} (C_{s}-L_{4}) | 13.983 | 91.1 | 128.4 | p-CS/t-HB ^{9} | ||||

Br_{12} (C_{s}-L_{5}) | 14.719 | 89.5 | 126.9 | p-CS/vdW ^{8} | ||||

Br_{12} (C_{s}-L_{5}) | 13.376 | 92.7 | 133.3 | p-CS/t-HB ^{9} | ||||

Br_{12} (C_{s}-L_{5}) | 13.334 | 93.0 | 134.3 | p-CS/t-HB ^{9} | ||||

Br_{12} (C_{s}-L_{5}) | 13.393 | 92.8 | 132.6 | p-CS/t-HB ^{9} | ||||

Br_{12} (C_{s}-L_{5}) | 13.962 | 91.1 | 128.8 | p-CS/t-HB ^{9} | ||||

Br_{6} (C_{2}) | 16.025 | 86.7 | 119.2 | p-CS/vdW ^{8} | ||||

Br_{10} (C_{2}) | 14.218 | 90.2 | 126.7 | p-CS/t-HB ^{9} | ||||

Br_{10} (C_{2}) | 16.378 | 87.2 | 120.0 | p-CS/vdW ^{8} |

^{1}The interactions in minima are shown.

^{2}c∇

^{2}ρ

_{b}(

**r**

_{c}) = H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2, where c = ħ

^{2}/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}= |d

^{2}y/dx

^{2}|/[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.

**Table 2.**The ρ

_{b}(

**r**

_{c}), H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2 (=(ћ

^{2}/8m)∇

^{2}ρ

_{b}(

**r**

_{c})), and H

_{b}(

**r**

_{c}) values and QTAIM-DFA parameters for Br-∗-Br at BCPs in Br

_{4}–Br

_{12}, other than the C

_{s}-L

_{m}structures, evaluated with MP2/6-311+G(3df)

^{1}.

Species | BCP on | ρ_{b}(r_{c}) | c∇^{2}ρ_{b}(r_{c}) ^{2} | H_{b}(r_{c}) | R^{3} | θ^{4} | ||

(Symmetry) | (au) | (au) | (au) | (au) | (°) | |||

Br_{4} (C_{2h}) | r_{2} | 0.0055 | 0.0022 | 0.0009 | 0.0024 | 67.2 | ||

Br_{4} (D_{2d}) | r_{2} | 0.0042 | 0.0017 | 0.0007 | 0.0018 | 66.0 | ||

Br_{6} (C_{3h}-c) | r_{2} | 0.0092 | 0.0038 | 0.0013 | 0.0040 | 70.7 | ||

Br_{8} (S_{4}) | r_{2} | 0.0128 | 0.0051 | 0.0014 | 0.0053 | 74.8 | ||

Br_{8} (S_{4}-Wm) ^{5} | r_{2} | 0.0136 | 0.0054 | 0.0013 | 0.0056 | 76.0 | ||

Br_{8} (S_{4}-Wm) ^{5} | r_{3} | 0.0038 | 0.0015 | 0.0007 | 0.0016 | 66.0 | ||

Br_{10} (C_{2}-c) | r_{2} | 0.0087 | 0.0035 | 0.0012 | 0.0037 | 70.5 | ||

Br_{10} (C_{2}-c) | r_{4} | 0.0097 | 0.0040 | 0.0014 | 0.0042 | 71.3 | ||

Br_{10} (C_{2}-c) | r_{6} | 0.0110 | 0.0044 | 0.0014 | 0.0046 | 73.0 | ||

Br_{10} (C_{2}-c) | r_{7} | 0.0049 | 0.0019 | 0.0008 | 0.0021 | 66.2 | ||

Br_{10} (C_{2}-c) | r_{8} | 0.0049 | 0.0018 | 0.0008 | 0.0020 | 66.6 | ||

Br_{12} (C_{i}) | r_{2} | 0.0129 | 0.0052 | 0.0014 | 0.0054 | 75.0 | ||

Br_{12} (C_{i}) | r_{4} | 0.0129 | 0.0052 | 0.0014 | 0.0054 | 75.0 | ||

Species | C_{ii}^{6} | θ_{p:CIV}^{7} | κ_{p:CIV}^{8} | Predicted | ||||

(Symmetry) | (Å mdyn^{−1}) | (°) | (au^{−1}) | nature | ||||

Br_{4} (C_{2h}) | 24.709 | 73.6 | 122.9 | p-CS/vdW ^{9} | ||||

Br_{4} (D_{2d}) | 40.402 | 69.6 | 136.3 | p-CS/vdW ^{9} | ||||

Br_{6} (C_{3h}-c) | 25.617 | 83.3 | 121.7 | p-CS/vdW ^{9} | ||||

Br_{8} (S_{4}) | 13.201 | 93.5 | 139.2 | p-CS/t-HB ^{10} | ||||

Br_{8} (S_{4}-Wm) ^{5} | 11.294 | 95.3 | 139.0 | p-CS/t-HB ^{10} | ||||

Br_{8} (S_{4}-Wm) ^{5} | 52.918 | 67.5 | 204.0 | p-CS/vdW ^{9} | ||||

Br_{10} (C_{2}-c) | 34.402 | 81.3 | 112.7 | p-CS/vdW ^{9} | ||||

Br_{10} (C_{2}-c) | 23.971 | 84.7 | 122.1 | p-CS/vdW ^{9} | ||||

Br_{10} (C_{2}-c) | 20.831 | 87.6 | 122.6 | p-CS/vdW ^{9} | ||||

Br_{10} (C_{2}-c) | 29.570 | 71.5 | 118.9 | p-CS/vdW ^{9} | ||||

Br_{10} (C_{2}-c) | 37.855 | 71.8 | 120.4 | p-CS/vdW ^{9} | ||||

Br_{12} (C_{i}) | 13.483 | 93.7 | 137.9 | p-CS/t-HB ^{10} | ||||

Br_{12} (C_{i}) | 13.482 | 93.7 | 137.3 | p-CS/t-HB ^{10} |

^{1}The interactions in minima are shown.

^{2}c∇

^{2}ρ

_{b}(

**r**

_{c}) = H

_{b}(

**r**

_{c}) − V

_{b}(

**r**

_{c})/2, where c = ħ

^{2}/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}= |d

^{2}y/dx

^{2}|/[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.

Entry | Correlation | a | b | R_{c}^{2} | n |
---|---|---|---|---|---|

1 | ΔE_{ZP} vs. ΔE_{ES} | 0.940 | 0.129 | 0.9999 | 20 ^{2} |

2 | θ vs. R | 2595.6 | 60.70 | 0.979 | 33 |

3 | θ_{p} vs. R | 6449.1 | 58.19 | 0.989 | 33 |

4 | θ_{p} vs. θ | 2.67 | −106.26 | 0.992 | 31 ^{3} |

5 | E(2) vs. C_{ii}^{−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. θ_{p} | 1.067 | 77.17 | 0.999 | 15 ^{4} |

^{1}The constants (a, b, R

_{c}

^{2}) 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 r

_{2}and r

_{3}in Br

_{4}(C

_{2h}).

^{4}For the noncovalent Br-∗-Br interactions in Br

_{4}(C

_{s}-L

_{1})–Br

_{12}(C

_{s}-L

_{5}).

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## Share and Cite

**MDPI and ACS Style**

Hayashi, S.; Nishide, T.; Tanaka, E.; Nakanishi, W.
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. *Molecules* **2021**, *26*, 2936.
https://doi.org/10.3390/molecules26102936

**AMA Style**

Hayashi S, Nishide T, Tanaka E, Nakanishi W.
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. *Molecules*. 2021; 26(10):2936.
https://doi.org/10.3390/molecules26102936

**Chicago/Turabian Style**

Hayashi, Satoko, Taro Nishide, Eiichiro Tanaka, and Waro Nakanishi.
2021. "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" *Molecules* 26, no. 10: 2936.
https://doi.org/10.3390/molecules26102936