The nature of Br4σ(4c–6e) of the BBr-∗-ABr-∗-ABr-∗-BBr form is elucidated for SeC12H8(Br)SeBr---Br-Br---BrSe(Br)C12H8Se, the selenanthrene system, and the models with QTAIM dual functional analysis (QTAIM-DFA). Asterisks (∗) are employed to emphasize the existence of bond critical points on the interactions in question. Data from the fully optimized structure correspond to the static nature of interactions. In our treatment, data from the perturbed structures, around the fully optimized structure, are employed for the analysis, in addition to those from the fully optimized one, which represent the dynamic nature of interactions. The ABr-∗-ABr and ABr-∗-BBr interactions are predicted to have the CT-TBP (trigonal bipyramidal adduct formation through charge transfer) nature and the typical hydrogen bond nature, respectively. The nature of Se2Br5σ(7c–10e) is also clarified typically, employing an anionic model of [Br-Se(C4H4Se)-Br---Br---Br-Se(C4H4Se)-Br], the 1,4-diselenin system, rather than (BrSeC12H8)Br---Se---Br-Br---Br-Se(C12H8Se)-Br, the selenanthrene system.

1. Introduction

We have been much interested in the behavior of the linear interactions of the σ-type, higher than σ(3c–4e: three center-four electron interactions) [16], constructed by the atoms of heavier main group elements. We proposed to call such linear interactions the extended hypervalent interactions, σ(mc–ne: 4 ≤ m; m < n < 2m), after the hypervalent σ(3c–4e). The linear alignments of four chalcogen atoms were first demonstrated in the naphthalene system, bis[8-(phenylchalcogenyl)naphthyl]-1,1′-dichalcogenides [I: 1-(8-PhBEC10H6)AE-AE(C10H6BEPh-8′)-1′ (AE, BE = S and Se)] [712]. It was achieved through the preparation and the structural determination by the X-ray crystallographic analysis. The linear BE---AE-AE---BE interactions in I are proposed to be analysed as the σ(4c–6e) model not by the double AEBE2σ(3c–4e) model. σ(4c–6e) in I is characterized by the CT interaction of the np(BE) ⟶ σ∗(AE–AE)←np(BE) form [8, 1012], where np(BE) stands for the p-type nonbonding orbitals of BE and σ∗(AE-AE) are the σ∗ orbitals of AE-AE. The novel reactivity of σ(4c–6e) in I was also clarified [8].

σ(4c–6e) is the first member of σ(mc–ne: 4 ≤ m; m < n < 2m) [713]. The σ(4c–6e) interactions are strongly suggested to play an important role in the development of high functionalities in materials and in the key processes of biological and pharmaceutical activities, recently. The bonding is applied to a wide variety of fields, such as crystal engineering, supramolecular soft matters, and nanosciences [4, 1423]. The nature of BE---AE and AE-AE in BE---AE-AE---BE of σ(4c–6e) has been elucidated [2427] using the quantum theory of atoms in molecules (QTAIM) approach, introduced by Bader [2837]. The linear interactions of the σ(4c–6e) type will form if BE in is replaced by X, giving E2X2σ(4c–6e). The nature of E2X2σ(4c–6e) in the naphthalene system of 1-(8-XC10H6)E-E(C10H6X-8′)-1′ [II (E, X) = (S, Cl), (S, Br), (Se, Cl), and (Se, Br)] was similarly clarified very recently [38].

The σ(4c–6e) interaction will also be produced even if both BE and AE in are replaced by X. X4σ(4c–6e) should also be stabilized through CT of the np(X) ⟶ σ∗(X-X) ← np(X) form. The energy lowering of the system through the CT interaction must be the driving force for the formation of X4σ(4c–6e). X4σ(4c–6e) is the typical kind of halogen bonds, together with E2X2σ(4c–6e), which are of current and continuous interest [39]. Br4σ(4c–6e) has been clearly established in the selenanthrene system, SeC12H8(Br)SeBr---Br-Br---BrSe(Br)C12H8Se (1), through the preparation and the structural determination by the X-ray crystallographic analysis [39]. The atoms taking part in the linear interaction in question are shown in bold. The structure of (BrSeC12H8)Br---Se---Br-Br---Br-Se(C12H8Se)-Br (2) was also reported, in addition to 1, which is suggested to contain Se2Br5σ(7c–10e) since the seven atoms of Se2Br5 align almost linearly in crystals. Figure 1 shows the structures of 1 and 2 determined by the X-ray analysis and the approximate MO model for σ(4c–6e) and σ(7c–10e).

It is challenging to elucidate the nature of Br4σ(4c–6e) of the np(Br) ⟶ σ∗(Br-Br)←np(Br) form in 1 and Se2Br5σ(7c–10e) in 2, together with the related species. Figure 2 illustrates the process assumed for the formation of 1 and 2 from selenanthrene (S: SeC12H8Se). In this process, (SeC12H8)Br-Se-Br (3) should be formed first in the reaction of S with Br2, and then 3 reacts with Br2 to yield Br[Se(Br) (C12H8)]Se---Br-Br (4). The almost linear alignment of Br---Se---Br-Br in 4 could be analysed by the SeBr3σ(4c–6e) model, where the Br and Se atoms in 4 are placed in close proximity in space. While 1 containing Br4σ(4c–6e) forms in the reaction of (3 + Br2 + 3), the reaction of 3 + 4 yields 2, consisting Se2Br5σ(7c–10e). Both 1 and 2 are recognized as the Br2-included species. While XC4H4(Br)SeBr---Br-Br---BrSe(Br)C4H4X (5 (X = Se) and 6 (X = S)), models of 1, also consisted of Br4σ(4c–6e), Se2Br5σ(7c–10e) will appear typically in the anionic species, [Br-Se(Me2)-Br---Br---Br-Se(Me2)-Br] (7) and [Br-Se(SeC4H4)-Br---Br---Br-Se(C4H4Se)-Br] (8), models of 2. Species, 5, 6, 7, and 8, are shown in Figure 2, where 5, 6, and 8 belong to the 1,4-diselenin system.

What are the differences and similarities between X4σ(4c–6e), E4σ(4c–6e), and E2X2σ(4c–6e)? The nature of X4σ(4c–6e) in 1 (X = Br) is to be elucidated together with the models. Models, other than 5 and 6, are also devised to examine the stabilization sequence of Br4σ(4c–6e). H2Br4 (C2h) and Me2Br4 (C2h) have the form of R-Br---Br-Br---Br-R (RBr4R: R = H and Me), which are called the model group A (G(A)). The electronic efficiency to stabilize Br4σ(4c–6e) seems small for R in G(A). Br6 (C2h) is detected as the partial structure in the crystals of Br2 [40]. Br6 (C2h) in the crystals is denoted by Br6 (C2h)obsd. The optimized structure of Br6 (C2h) has one imaginary frequency, which belongs to G(A), together with Br6 (C2h)obsd. The optimized structure of Br6 retains the C2 symmetry, (Br6 (C2)), which also belongs to G(A). The CT interaction of the np(BBr) ⟶ σ∗(ABr-ABr) ⟵ np(BBr) form in Br4σ(4c–6e) will be much stabilized if the large negative charge is developed at the BBr atoms in Br-(R2)Se-BBr---ABr-ABr---BBr-Se(R2)-Br, where the ∠SeBBrABr is around 90°. The highly negatively charged BBr in Br-Se(R2)-BBr (R = H and Me) of σ(3c–4e) is employed to stabilize Br4σ(4c–6e), in this case. The models form G(B). The nature of Br4σ(4c–6e) in 5 and 6 is similarly analysed, which belongs to G(B). (D∞h) also belongs to G(B) although one imaginary frequency was predicted for , if optimized at the MP2 level. Figure 3 illustrates the story for the stabilization of Br4σ(4c–6e) in the sequence of the species, starting from G(A) to 1, via G(B). Figure 3 also shows the ABr-ABr and ABr---BBr distances (r(ABr-ABr) and r(ABr-BBr), respectively), together with the charge developed at BBr in the original species of R-BBr (Qn (BBr)), which construct R-BBr---ABr-ABr---BBr-R.

A chemical bond or interaction between atoms A and B is denoted by A-B, which corresponds to a bond path (BP) in the quantum theory of atoms in molecules (QTAIM) approach, introduced by Bader [2837]. We will use A-∗-B for BP, where the asterisk emphasizes the existence of a bond critical point (BCP, ∗) in A-B [28, 29]. (Dots are usually employed to show BCPs in molecular graphs. Therefore, A-•-B would be more suitable to describe the BP with a BCP. Nevertheless, A-∗-B is employed to emphasize the existence of a BCP on the BP in question in our case. BCP is a point along BP at the interatomic surface, where ρ(r) (charge density) reaches a minimum along the interatomic (bond) path, while it is a maximum on the interatomic surface separating the atomic basins). The chemical bonds and interactions are usually classified by the signs of Laplacian rho (∇2ρb(rc)) and Hb(rc) at BCPs, where ρb(rc) and Hb(rc) are the charge densities and total electron energy densities at BCPs, respectively (see Scheme S1 in Supplementary File). The relations between Hb(rc), ∇2ρb(rc), Gb(rc) (the kinetic energy densities), and Vb(rc) (the potential energy densities) are represented in equations (1) and (2):

How can the nature of Br4σ(4c–6e) and Se2Br5σ(7c–10e) be clarified? For the characterization of interactions in more detail, we recently proposed QTAIM dual functional analysis (QTAIM-DFA) [4247] for experimental chemists to analyze their own chemical bonds and interaction results based on their own expectations, according to the QTAIM approach [2837]. Hb(rc) is plotted versus Hb(rc) − Vb(rc)/2 (= (ћ2/8m)∇2ρb(rc)) at BCPs in QTAIM-DFA. The classification of interactions by the signs of ∇2ρb(rc) and Hb(rc) is incorporated in QTAIM-DFA. Data from the fully optimized structures correspond to the static natures of the interactions, which are analysed using the polar coordinate (R, θ), representation [42, 4446]. Each interaction plot, containing data from both the perturbed structures and the fully optimized one include a specific curve that provides important information about the interaction. This plot is expressed by (θp, κp), where θp corresponds to the tangent line of the plot and κp is the curvature. The concept of the dynamic nature of interactions has been proposed based on (θp, κp) [42, 44]. θ and θp are measured from the y-axis and the y-direction, respectively. We call (R, θ) and (θp, κp) QTAIM-DFA parameters, which are drawn in Figure 4, exemplified by (D∞h). While (R, θ) classifies the interactions, (θp, κp) characterizes them.

We proposed a highly reliable method to generate the perturbed structures for QTAIM-DFA very recently [48]. The method is called CIV, which employs the coordinates derived from the compliance force constants Cij for the internal vibrations. Compliance force constants Cij are defined as the partial second derivatives of the potential energy due to an external force, as shown in equation (3), where i and j refer to the internal coordinates and the force constants fi and fj correspond to i and j, respectively. The Cij values and the coordinates corresponding to the values can be calculated using the compliance 3.0.2 program, released by Brandhorst and Grunenberg [4952]. 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 the coordinate system:

QTAIM-DFA has excellent potential for evaluating, classifying, characterizing, and understanding weak to strong interactions according to a unified form. The superiority of QTAIM-DFA to elucidate the nature of interactions, employing the perturbed structures generated with CIV, is explained in the previous papers [48, 53] (see also Figure S2 and Table S2 in Supplementary File). QTAIM-DFA is applied to standard interactions and rough criteria that distinguish the interaction in question from others which are obtained. QTAIM-DFA and the criteria are explained in Supplementary File using Schemes S1–S3, Figures S1 and S2, Table S1, and equations (S1)–(S7). The basic concept of the QTAIM approach is also explained.

We consider QTAIM-DFA, employing the perturbed structures generated with CIV, to be well suited to elucidate the nature of Br4σ(4c–6e) in 1, Se2Br5σ(7c–10e) in 2, and the models derived from 1 and 2, together with the related linear interactions. The interactions in Br4σ(4c–6e) are denoted by BBr-∗-ABr-∗-ABr-∗-BBr, where the asterisk emphasizes the existence of a BCP in the interactions, so are those in Se2Br5σ(7c–10e). Herein, we present the results of the investigations on the extended hypervalent interactions in the species, together with the structural feature. Each interaction is classified and characterized, employing the criteria as a reference.

2. Methodological Details in Calculations

Calculations were performed employing the Gaussian 09 programs package [54]. The basis sets employed for the calculations were obtained, as implemented from Sapporo Basis Set Factory [55]. The basis sets of the (621/31/2), (6321/621/3), (74321/7421/72), and (743211/74111/721/2+1s1p) forms were employed for C, S, Se, and Br, respectively, with the (31/3) form for H. The basis set system is called BSS-A. All species were calculated employing BSS-A, and the Møller–Plesset second-order energy correlation (MP2) level [5658] was applied for the optimizations. Optimized structures were confirmed by the frequency analysis. The results of the frequency analysis were used to calculate the Cij values and the coordinates (Ci) corresponding to the values. The DFT level of CAM-B3LYP [59] was also applied when necessary. The QTAIM functions were analysed with the AIM2000 [60] and AIMAll [61] programs.

The method to generate perturbed structures with CIV is the same as that explained in the previous papers [48, 53]. As shown in equation (4), the i-th perturbed structure in question (Siw) is generated by the addition of the i-th coordinates (Ci), derived from Cij, to the standard orientation of a fully optimized structure (So) in the matrix representation. The coefficient fiw in equation (4) controls the structural difference between Siw and So: fiw is determined to satisfy equation (5) for r, where r and ro stand for the interaction distances in question in the perturbed and fully optimized structures, respectively, with ao = 0.52918 Å (Bohr radius). The Ci values of five digits are used to predict Siw:

In QTAIM-DFA, Hb(rc) is plotted versus Hb(rc) − Vb(rc)/2 for data of  = 0, ±0.05, and ±0.10 in equation (5). Each plot is analysed using a regression curve of the cubic function, as shown in equation (6), where (x, y) = (Hb(rc) − Vb(rc)/2 and Hb(rc)) ( (square of correlation coefficient) > 0.99999 in usual) [46].

3. Results and Discussion

3.1. Structural Optimizations

The structures of 1 (Ci) and 2 (C1) determined by the X-ray analysis are denoted by 1 (Ci)obsd and 2 (C1)obsd, respectively [39]. The structural parameters are shown in Tables S2 and S3 in Supplementary File, respectively. Figure 3 contains the selected structural parameters for 1 (Ci)obsd. The structures are optimized for G(A) of H2Br4 (C2h), Me2Br4 (C2h), Br6 (C2h), and Br6 (C2) and G(B) of H4Se2Br6 (Ci), Me4Se2Br6 (Ci), 5 (Ci), and 6 (Ci), together with 3 (Cs), 4 (Cs), 7 (C2h), 8 (C2h), and Br2 (D∞h). The optimized structural parameters are also collected in Tables S2 and S3 in Supplementary File. The frequency analysis was successful for the optimized structures, except for 1 (Ci)obsd and Br6 (C2h). All positive frequencies were obtained for 1 (Ci), if calculated with CAM-B3LYP/BSS-A, which confirms the structure. The Br---Br distances of Br4σ(4c–6e) in 1 (Ci) are somewhat longer if optimized at the CAM-B3LYP level, relative to 1 (Ci)obsd. While one imaginary frequency is detected in Br6 (C2h), Br6 (C2) has all positive frequencies. The optimized structures are not shown in figures, instead, some of them can be found in Figures 3 and 5, where the molecular graphs are drawn on the optimized structures. Figure 3 contains the optimized r(ABr-ABr) and r(ABr-BBr) distances for the models and the charge developed at BBr in the original R-BBr and Br-(R2)Se-BBr (Qn (BBr)), which give the models of G(A) and G(B), respectively. The r(ABr-BBr) values become shorter in the order shown in equation (7), if evaluated with MP2/BSS-A:

One imaginary frequency was also predicted for (D∞h) if optimized with MP2/BSS-A. (D∞h) seems to collapse to and Br, according to the imaginary frequency. The double negative charges in (D∞h) would be responsible for the results. The electrostatic repulsion between the double negative charges will operate to collapse it.

3.2. Energies for Formation of Br4σ(4c–6e) and NBO Analysis

Energies for the formation of R′Br4R′ from the components (2R′Br + Br2) (ΔE) are defined by equation (8). The ΔE values evaluated on the energy surface are denoted by ΔEES, while those corrected with the zero-point energies are by ΔEZP. The ΔEES and ΔEZP values for the optimized structures are given in Table S2 in Supplementary File. ΔEZP are excellently correlated to ΔEESEZP = 0.99ΔEES + 1.93: Rc2 = 0.9998, see Figure S3 in Supplementary File):

NBO analysis [62] was applied to ABr---BBr of the species to evaluate the contributions from CT to stabilize R′-BBr---ABr-ABr---BBr-R′. 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, according to equation (9), where qi is the donor orbital occupancy, εi and εj are diagonal elements (orbital energies), and F(i, j) is the off-diagonal NBO Fock matrix element. The results are collected in Table S4 in Supplementary File. The ΔEES values are very well correlated to E(2) for the optimized structures, except for (D∞h). (ΔEES = –0.71(2E(2)) + 7.17:  = 0.959, see Figure S4 in Supplementary File). (D∞h) is predicted to be less stable than the components.

Before application of QTAIM-DFA to Br4σ(4c–6e) and Se2Br5σ(7c–10e), molecular graphs were examined, as shown in the next section.

3.3. Molecular Graphs with Contour Plots for the Species Containing Br4σ(4c–6e), Se2Br5σ(7c–10e), and Related Linear Interactions

Figure 5 illustrates the molecular graphs of 5 (Ci), 6 (Ci), 7 (C2h), and 8 (C2h), drawn on the optimized structures, together with 1 (Ci)obsd and 2 (C1)obsd. Figure 5 also shows the contour plots of ρ(r) drawn on the suitable plane in the molecular graphs. BCPs are well demonstrated to locate on the (three-dimensional) saddle points of ρ(r). Molecular graphs of Me2Br4 (C2h), Br6 (C2), (D∞h), and Br(Me2)SeBr4Se(Me2)Br (Ci) are shown in Figure 3, which are drawn on the optimized structures.

3.4. Survey of Br4σ(4c–6e) and Se2Br5σ(7c–10e)

BPs in Br4σ(4c–6e) and Se2Br6σ(7c–10e) seem straight, as shown in Figures 3 and 5. To show the linearity more clearly, the lengths of BPs (rBP) for Br4σ(4c–6e) are calculated. The values are collected in Table S5 in Supplementary File, together with the corresponding straight-line distances (RSL). The table contains the values for Se2Br6σ(7c–10e) in 7 (C2h) and 8 (C2h). The differences between them (ΔrBP = rBPRSL) are less than 0.003 Å. The rBP values are plotted versus RSL, which are shown in Figure S5 in Supplementary File. The correlations are excellent, as shown in the figure. Therefore, Br4σ(4c–6e) and Se2Br6σ(7c–10e) in the species can be approximated by the straight lines.

QTAIM functions are calculated for Br4σ(4c–6e) at BCPs. Table 1 collects the values for the interactions. Hb(rc) is plotted versus Hb(rc) − Vb(rc)/2 for the data shown in Table 1, together with those from the perturbed structures generated with CIV. Figure 4 shows the plots for the ABr-∗-ABr and ABr-∗-BBr interactions in Br4σ(4c–6e) of the bromine species. The plots for ABr-∗-ABr appear in the region of Hb(rc) − Vb(rc)/2 > 0 and Hb(rc) < 0, for all species, except for the original Br2 (D∞h), of which the plot appears in the region of Hb(rc) − Vb(rc)/2 < 0 and Hb(rc) < 0. Therefore, the interactions are all classified by the regular-CS (closed shell) interactions, except for Br2 (D∞h), which is classified by the SS (shard shell) interaction. On the contrary, data of ABr-∗-BBr appear in the region of Hb(rc) − Vb(rc)/2 > 0 and Hb(rc) > 0 for all species, except for those in H4Se2Br6 (Ci), Me4Se2Br6 (Ci), 5 (Ci), and 6 (Ci), which appear in the region of Hb(rc) − Vb(rc)/2 > 0 and Hb(rc) < 0. As a result, ABr-∗-BBr is classified by the pure-CS interactions (p-CS) for all, except for the four species, of which ABr-∗-BBr is classified by the regular-CS interactions (r-CS). The ABr-∗-BBr interaction in (D∞h) is very close to the borderline between p-CS and r-CS since Hb(rc) = 0.0001 au for (D∞h), which is very close to zero. QTAIM-DFA parameters of (R, θ) and (θp, κp) are obtained by analysing the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 in Figure 4, according to equations (S3)–(S6). Table 1 collects the QTAIM-DFA parameters for Br4σ(4c–6e). The classification of interactions will also be discussed based on the (R, θ) values.

QTAIM functions are similarly calculated for Se2Br6σ(7c–10e) at BCPs, together with the related interactions. Hb(rc) is similarly plotted versus Hb(rc) − Vb(rc)/2 although not shown in the figures. Then, QTAIM-DFA parameters of (R, θ) and (θp, κp) are obtained by analysing the plots, according to equations (S3)–(S6). Table 2 collects the QTAIM-DFA parameters of (R, θ) and (θp, κp) for Br4σ(4c–6e).

3.5. Nature of Br4σ(4c–6e)

Interactions are characterized by (R, θ), which correspond to the data from the fully optimized structures. On the contrary, they are characterized employing (θp, κp) derived from the data of the perturbed structures around the fully optimized structures and the fully optimized ones. In this case, the nature of interactions is substantially determined based of the (R, θ, θp) values, while the κp values are used only additionally. It is instructive to survey the criteria before detail discussion. The criteria tell us that 180° < θ (Hb(rc) − Vb(rc)/2 < 0) for the SS interactions, 90° < θ < 180° (Hb(rc) < 0) for the r-CS interactions, and 45° < θ < 90° (Hb(rc) > 0) for p-CS interactions. The θp value characterizes the interactions. In the p-CS region of 45° < θ < 90°, the character of interactions will be the vdW type for 45° < θp < 90°, whereas it will be the typical HB type without covalency (t-HBnc) for 90° < θp < 125°, where θp = 125° is tentatively given for θ = 90°. The CT interaction will appear in the r-CS region of 90° < θ < 180°. The t-HB type with covalency (t-HBwc) appears in the region of 125° < θp < 150° (90° < θ < 115°), where (θ, θp) = (115°, 150°) is tentatively given as the borderline between t-HBwc and the CT-MC nature. The borderline for the interactions between CT-MC and CT-TBP types is defined by θp = 180°. θ = 150° is tentatively given for θp = 180°. Classical chemical bonds of SS (180° < θ) will be strong (Cov-s) when R > 0.15 au, whereas they will be weak (Cov-w) for R < 0.15 au. The classification and characterization of interactions are summarized in Table S1 and Scheme S3 in Supplementary File.

The ABr-∗-ABr and ABr-∗-BBr interactions of Br4σ(4c–6e) will be classified and characterized based on the (R, θ, θp) values, employing the standard values as a reference (see Scheme S2 in Supplementary File). R < 0.15 au for all interactions in Table 1; therefore, no Cov-s were detected in this work. The (θ, θp) values are (180.1°, 191.8°) for the original Br2 (D∞h) if evaluated with MP2/BSS-A. Therefore, the nature of Br-∗-Br in Br2 (D∞h) is classified by the SS interactions and characterized as the Cov-w nature, which is denoted by SS/Cov-w. The (θ, θp) values are (170.6–179.0°, 190.6–191.7°) for ABr-∗-ABr of Br4σ(4c–6e) in the optimized structures in Table 1, of which nature is r-CS/CT-TBP. The (θ, θp) values are (78.0–84.1°, 94.7–105.1°) for ABr-∗-BBr in the optimized structures of Br6 (C2), Br6 (C2h), and R2Br4 (C2h) (R = H and Me); therefore, the nature is predicted to be r-CS/t-HBwc. The nature of ABr-∗-BBr in R4Se2Br6 (Ci) (R = H and Me), 5 (Ci) and 6 (Ci), is r-CS/t-HBwc, judging from the (θ, θp) values of (90.9–92.8°, 116.4–122.5°). The calculated (θ, θp) values of ABr-∗-ABr and ABr-∗-BBr for the optimized structure of (D∞h) are (170.6°, 190.6°) and (89.5°, 118.2°), respectively. In this case, ABr-∗-ABr and ABr-∗-BBr are predicted to have the nature of r-CS/CT-TBP and p-CS/t-HBnc, respectively. However, ABr-∗-BBr is just the borderline region to the r-CS interactions with θ = 89.5°. The characteristic nature of the BE---AE-AE---BE interactions in (D∞h) would be controlled by the double negative charges in the species.

The results in Table 1 show that the ABr-∗-ABr interaction in Br4σ(4c–6e) becomes weaker, as the strength of the corresponding ABr-∗-BBr increases. The strength of ABr-∗-ABr becomes weaker in the order shown in equation (10), if evaluated by θ, while that of ABr-∗-BBr increases in the order shown in equation (11), if measured by θ. Very similar results were obtained by θp:

The orders shown in equations (10) and (11) seem to reasonably explain the characteristic behavior of Br4σ(4c–6e). The results must be the reflection of the np(BBr) ⟶ σ∗(ABr-ABr) ← np(BBr) form of Br4σ(4c–6e), where ABr-∗-ABr and ABr-∗-BBr become weaker and stronger, respectively, as the CT interaction increases. Br4σ(4c–6e) will be stabilized more effectively, if the negative charge is developed more at BBr. However, the two Br ligands in (D∞h) seem not so effective than that expected. This would come from the electrostatic repulsive factor between the double negative charges in (D∞h), as mentioned above.

The θ values for (ABr-∗-ABr and ABr-∗-BBr) in Br6 (C2h)obsd and 1 (Ci)obsd are (165.2°, 82.5°) and (175.3°, 87.7°), respectively. Therefore, ABr-∗-ABr and ABr-∗-BBr are classified by r-CS and p-CS, respectively. Both ABr-∗-ABr and ABr-∗-BBr in Br6 (C2h)obsd are predicted to be weaker than those in 1 (Ci)obsd, respectively. The results would be curious at the first glance, since ABr-∗-ABr will be weaker, if ABr-∗-BBr in BBr-∗-ABr-∗-ABr-∗-BBr becomes stronger, as mentioned above. They would be affected from the surrounding, such as the crystal packing effect. A Br2 molecule interacts with four bromine atoms adjacent to the Br2 molecule on the bc-plane in crystals, equivalently with 3.251 Å [40].

Similar investigations were carried out for I4σ(4c–6e), which will be discussed elsewhere (it is demonstrated that Br4σ(4c–6e) is predicted to be somewhat stronger than I4σ(4c–6e)).

3.6. Nature of Se2Br5σ(7c–10e)

The nature of Se2Br5σ(7c–10e) in 7 (C2h) and 8 (C2h) is elucidated, together with SeBr2σ(3c–4e) in 3 and SeBr4σ(4c–6e) in 4. The results are collected in Table 2. Figure 6 shows symmetric ψ184 (HOMO) and antisymmetric ψ185 (LUMO) of 8 (C2h), which correspond to ψ5 and ψ6 in σ(7c–10e), illustrated in Figure 1 although the Se atoms are contained in the linear Se2Br5σ(7c–10e) in 8 (C2h). The linear seven atomic orbitals on Se2Br5 are shown to construct ψ184 (HOMO) and ψ185 (LUMO) of 8 (C2h), which can be analysed as the Se2Br5σ(7c–10e) [39], so can the linear interaction in 7 (C2h), although not shown. The pseudolinear interaction of the seven atoms of 1 (C1)obsd could also be explained by the Se2Br5σ(7c–10e) model.

The results demonstrate that Se2Br5σ(7c–10e) stabilize well 7 (C2h) and 8 (C2h) although 1 (C1)obsd seems not so effective. The negative charge developed at the Br atom in 3 would not be sufficient to stabilize Se2Br5σ(7c–10e) in 1 (C1)obsd, relative to the case of the Br anion in 7 (C2h) and 8 (C2h), irrespective of the highly negatively charged Br atoms in SeBr2σ(3c–4e) of 3.

4. Conclusion

The intrinsic dynamic and static nature of Br4σ(4c–6e) is elucidated for 1 (Ci)obsd and the related species with QTAIM-DFA, employing the perturbed structures generated with CIV. The ABr-ABr interactions in BBr-∗-ABr-∗-ABr-∗-BBr of Br4σ(4c–6e) are weaker than Br-∗-Br in the optimized structure of Br2 (D∞h), which is predicted to have the SS/Cov-w nature. The ABr-ABr interactions in Br4σ(4c–6e) of the models are predicted to have the r-CS/CT-TBP nature, if optimized with MP2/BSS-A. The ABr-ABr interaction in 1 (Ci)obsd also appears in the r-CS region. On the contrary, the ABr-BBr interactions in Br6 (C2), Br6 (C2h), H2Br4 (C2h), and Me2Br4 (C2h) are predicted to have the p-CS/t-HBnc nature, whereas those in H4Se2Br4 (Ci), Me4Se2Br4 (Ci), 5 (Ci), and 6 (Ci) have the r-CS/t-HBwc nature, if evaluated with MP2/BSS-A. The ABr-∗-BBr interactions become stronger in the order of H2Br4 (C2h) < Br6 (C2h) ≤ Br6 (C2) < Me2Br4 (C2h) << Me4Se2Br6 (Ci) ≤ H4Se2Br6 (Ci) ≤ 5 (Ci) < 6 (Ci), which is the inverse order for ABr-∗-ABr, as a whole. The results are in accordance with the CT interaction of the np(BBr) ⟶ σ∗(ABr-ABr) ← np(BBr) form derived from Br4σ(4c–6e). The decreased binding force of ABr-∗-ABr must be transferred to ABr-∗-BBr in Br4σ(4c–6e). Namely, it is demonstrated that Br4σ(4c–6e) is stabilized as the strength of ABr-∗-BBr in Br4σ(4c–6e) increases, while ABr-∗-ABr becomes weakened relative to that in the original Br2 (D∞h). In this process, Br4σ(4c–6e) is totally stabilized. The ABr-∗-ABr and ABr-∗-BBr interactions in Br6 (C2h)obsd and 1 (Ci)obsd are classified by the r-CS and p-CS interactions, respectively, where the interactions in Br6 (C2h)obsd seem somewhat weaker than those in 1 (Ci)obsd. The Se2Br5σ(7c–10e) interactions are similarly elucidated for 2 (C1)obsd and the anionic models of 7 (C2h) and 8 (C2h). The Se2Br5σ(7c–10e) nature is clearly established for the optimized structures of 7 (C2h) and 8 (C2h), rather than 2 (C1)obsd. Extended hypervalent interactions of the σ(mc–ne: 4 ≤ m; m < n < 2m) type are shown to be well analysed and evaluated with QTAIM-DFA, employing the perturbed structures generated with CIV, exemplified by Br4σ(4c–6e) and Se2Br5σ(7c–10e).

Data Availability

The data used to support the findings of this study are available in the supplementary information files.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.


This work was partially supported by a Grant-in-Aid for Scientific Research (no. 17K05785) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

Supplementary Materials

Scheme S1: classification of interactions by the signs of ∇2ρb(rc) and Hb(rc), together with Gb(rc) and Vb(rc). Scheme S2: QTAIM-DFA: a plot of Hb(rc) versus Hb(rc) − Vb(rc)/2 for weak to strong interactions. Scheme S3: rough classification and characterization of interactions by θ and θp, together with kb(rc) (= Vb(rc)/Gb(rc)). QTAIM-DFA approach, computational data (Tables S2–S5 and Figures S3–S5), computation information and geometries of compounds, and graphical abstract. Figure S1: polar (R, θ) coordinate representation of Hb(rc) versus Hb(rc) − Vb(rc)/2, with (θp, κp) parameters. Figure S2: plot of Hb(rc) versus in r(1Cl-2Cl) = ro(1Cl-2Cl) +  for 1Cl-2Cl-3Cl (a) with the magnified picture of (a) (b) and that of Hb(rc) − Vb(rc)/2 versus (c). Typical hydrogen bonds without covalency and typical hydrogen bonds with covalency are abbreviated as t-HB without cov. and t-HB with cov., respectively, whereas Cov-w and Cov-s stand for weak covalent bonds and strong covalent bonds, respectively. Table S1: proposed definitions for the classification and characterization of interactions. (Supplementary Materials)