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

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 Cs 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 (C2h) and Br4 (D2d), 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.


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(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].
Structures of halogen molecules (X 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 charged forms, together with Br1, so far [19,20]. Figure 1 draws t Br2, for example. The bromine molecules seem to exist as a zig-za chains in crystals. One would find the linear alignment of three dimer ((Br2)2; Br4) and the linear alignment of four Br atoms in a ((Br2)3; Br6) in a planar Br2 layer in addition to Br2 itself. The linear in the two L-shaped dimers of Br6, overlapped at the central Br2 mers seem to construct the zig-zag type infinite chains, the linea linear infinite chains. The attractive np(Br)→σ*(Br-Br) σ(3c-4e) (t interaction of the σ-type) and np(Br)→σ*(Br-Br)←np(Br) σ(4c-6 portant role to stabilize Br4 and Br6, respectively, where np(Br) st bonding orbital of Br in the plane, perpendicular to the molecula is the σ*-orbital of Br2. The crystal structures of Cl2 and I2 are ver We have been very interested in the behavior of halogen clusters, together with the structures. How can the interactions in be clarified? We propose QTAIM dual-functional analysis (QTA on the quantum theory of atoms in molecules (QTAIM) approa [26,27] to classify and characterize the various interactions effe DFA, Hb(rc) are plotted versus Hb(rc) − Vb(rc)/2 (= (ћ 2 /8m)∇ 2 ρb(rc) (s supplementary materials), where ρb(rc), Hb(rc), and Vb(rc) stand total electron energy densities, and potential energy densities, re cal points (BCPs, * ) on the bond paths (BPs) in this paper [26]. Th at BCPs will be similarly denoted by Gb(rc) [26]. A chemical bond o Br and Br is denoted by Br- * -Br in this work, where the asterisk of a BCP on a BP for Br-Br [26,27]. In our treatment, data from t tures are plotted together with those from the perturbed structur mized ones. The static nature of the interactions corresponds to optimized structures, which are analyzed using polar coordin [21][22][23][24][25]. On the other hand, the dynamic nature originates base the perturbed and fully optimized structures [21][22][23][24][25]. The plot where θp corresponds to the tangent line and κp is the curvature measured from the y-axis and the y-direction, respectively. We c Figure 1. Structure of Br 2 , determined by X-ray crystallographic analysis [17].
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 (=( 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].
Interactions are classified by the signs of ∇ 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 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 halogenhalogen 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.
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].
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 c 2 (square of correlation coefficient) > 0.99999 in the norm) [25].

Structural Optimizations of Polybromine Clusters, Br 6 -Br 12
Structures of the neutral Br 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.
The behavior of the neutral dibromine clusters (Br 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. Figure 2 draws the energy profiles for the optimized structures of minima, Br 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 ∆E 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 ∆E 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. The ΔEES value for Br6 (Cs-L2) was predicted to be -22.6 kJ mol -1 . The magnitude is slightly larger than the double value for Br4 (Cs-L1) (∆EES = −10.7 kJ mol -1 ). Two types of σ (3c-4e) operate to stabilize Br6 (Cs-L2). One, σ(3c-4e), seems similar to that in Br4 (Cs-L1), but the other would be somewhat different. Namely, the second interaction would contribute to ∆EES somewhat more than that of the first one in the formation of Br6 (Cs-L2). On the other hand, the linear interaction in Br6 (C2) can be explained by σ(4c-6e) of the np(Br)→σ*(Br-Br)←np(Br) type. The magnitude of ∆EES of Br6 (C2) seems slightly smaller than that of Br6 (Cs-L2) but is very close to the double value for Br4 (Cs-L1). The magnitude of ∆EES for Br6 (C3h-c) is close to the triple value of Br4 (Cs-L1). One finds triply degenerated σ(3c-4e) interactions in Br6 (C3h-c). The similarity in the interactions for Br4 (Cs-L1), Br6 (C2), and Br6 (C3h-c) will be discussed again later. The magnitudes of ∆EES become proportionally larger to the size of the clusters, as shown in Figures S1 and S2. The ΔEES values are plotted versus k in Br2k (2 ≤ k ≤ 6) for the Cs-Lm type. The results are shown in Figure S2. Contributions from inner σ(3c-4e) (named rin) to ΔEES seem slightly larger than those from σ(3c-4e) in the front end and end positions (named r2 and rω, 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. for Br4 (Cs-L1)-Br12 (Cs-L5), calculated with MP2/6-311+G(3df) [50] (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). In the case of Br 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.

Molecular Graphs with Contour Plots of Polybromine Clusters
The ∆E 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 ∆E 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 ∆E ES seem slightly larger than those from σ(3c-4e) in the front end and end positions (named r 2 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. Figure 3 illustrates the molecular graphs with contour maps of ρ(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).

Survey of the Br- * -Br Interactions in Polybromine Clusters
As shown in Figures 2-4, the BPs in Br4-Br12 seem almost straight. The linearity is confirmed by comparing the lengths of BPs (rBP) with the corresponding straight-line distances (RSL). The rBP and RSL values are collected in Table S3, together with the differences

Survey of the Br- * -Br Interactions in Polybromine Clusters
As shown in Figures 2-4, the BPs in Br4-Br12 seem almost straight. The linearity is confirmed by comparing the lengths of BPs (rBP) with the corresponding straight-line distances (RSL). The rBP and RSL values are collected in Table S3, together with the differences

Survey of the Br- * -Br Interactions in Polybromine Clusters
As shown in Figures 2-4, the BPs in Br 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, ∆r BP (=r BP − R SL ). The magnitudes of ∆r BP are less than 0.01 Å, except for r 2 in Br 4 (C 2v ) (∆r 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. The , 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 Tables 1 and 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 ).

The Nature of Br- * -Br Interactions in Polybromine Clusters
The nature of the covalent and noncovalent Br- * -Br interactions in Br 2 -Br 12 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 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 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.
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 summarizes the correlations among the QTAIM-DFA parameters. 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 ).
To further examine the behavior of noncovalent Br- * -Br interactions, NBO analysis is applied to the interactions.
Relations between E(2) and C 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). Relations between E(2) and Cii were also examined for noncovalent Br- * -Br interactions in Br4 (Cs-L1)-Br12 (Cs-L5). The E(2) values were plotted versus Cii -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 np(Br)→σ*(Br-Br) type in Br4 (Cs-L1)-Br12 (Cs-L5) are well evaluated, not only by E(2) but also by Cii -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 (Cs-L1)-Br12 (Cs-L5) were also plotted versus R, θ, and θp, shown in Figures S6-S8, respectively. The plots give very good correlations, which are given in Table 3 (Entries 6-8).
The total energy for a species (E) is given by the sum of the core terms (Hc(i)) over all
The total energy for a species (E) is given by the sum of the core terms (H 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 (∆P) in the formation of Br 2k from the components are evaluated according to Equation (9). The ∆Σ i n H c (i) and ∆(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.
The nature of noncovalent Br-Br interactions in Br 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  Figure S9. In total, ∆Σ 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. Figure 8 shows the plots of ∆Σ i n H c (i) and ∆(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 ∆E ES and ∆Σ i=1 n ε i . In the case of Br 4 (C s -L 1 ), ∆Σ i n H c (i) and ∆(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, ∆((Σ i =j n J ij − Σ i =j, n K ij )/2), relative to the case of 2Br 2 , although ∆Σ i n H c (i) seems to destabilize it. The ∆Σ i n H c (i) + ∆(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 .
The nature of interactions in the charged clusters is also o tions are in progress.

Conclusions
The intrinsic dynamic and static nature of noncovalent Brcidated for Br4-Br10 with MP2/6-311+G(3df). QTAIM-DFA was tion. Hb(rc) were plotted versus Hb(rc) − Vb(rc)/2 for the interac optimized structures, together with those from the perturbed s CIV. The nature of the covalent Br- * -Br bonds in Br4-Br10 is pred w nature if calculated with MP2/6-311+G(3df). On the other ha covalent Br- * -Br interactions in Br4-Br12 is classified by the pure The noncovalent Br- * -Br interactions in the linear type clusters are predicted to have the p-CS/t-HBnc nature (90.6° ≤ θp), except the first end, which have the p-CS/vdW nature, although it is ve between the two (θp ≤ 89.4°). In the case of the cyclic clusters interactions will have the p-CS/vdW nature (θp ≤ 88.4°), except f The nature of interactions in the charged clusters is also of interest. Such investigations are in progress.

Conclusions
The intrinsic dynamic and static nature of noncovalent Br- * -Br interactions was elucidated for Br 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.
The energies for Br 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 ∆Σ 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 ∆((Σ i =j n J ij − Σ i =j, n K ij )/2) must also be considered.

Supplementary Materials:
The following are available online, Table S1: Structural parameters for Br 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 S5: Contributions from the donoracceptor (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 ∆ε 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 ∆E ZP versus ∆E ES for Br 4 -Br 12 , relative to those of Br 2 , respectively, Figure S2: Plots of ∆E 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