Bioinorganic Chemistry and Applications
Volume 2020 (2020), Article ID 2901439, 14 pages
https://doi.org/10.1155/2020/2901439

Research Article

Dynamic and Static Nature of Br₄σ(4c–6e) and Se₂Br₅σ(7c–10e) in the Selenanthrene System and Related Species Elucidated by QTAIM Dual Functional Analysis with QC Calculations

Satoko Hayashi, Taro Nishide, and Waro Nakanishi

Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan

Correspondence should be addressed to Satoko Hayashi; hayashi3@sys.wakayama-u.ac.jp and Waro Nakanishi; nakanisi@sys.wakayama-u.ac.jp

Received 22 January 2019; Revised 15 June 2019; Accepted 19 November 2019; Published 24 July 2020

Academic Editor: Claudio Pettinari

Copyright © 2020 Satoko Hayashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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) [1–6], 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)] [7–12]. 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, 10–12], 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) [7–13]. 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, 14–23]. The nature of BE---AE and AE-AE in BE---AE-AE---BE of σ(4c–6e) has been elucidated [24–27] using the quantum theory of atoms in molecules (QTAIM) approach, introduced by Bader [28–37]. 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).

(a)

(b)

(c)

(d)

Figure 1: Structure of 1 determined by the X-ray crystallographic analysis (a) and the approximate MO model for σ(4c–6e) (b); structure of 2 (c) and the approximate MO model for σ(7c–10e) (d).

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.

Figure 2: Process assumed for the formation of Br₄σ(4c–6e) in 1 from Se(C₁₂H₈)Se (S) via 3 and Se₂Br₅σ(7c–10e) in 2 via 3 and 4. 5 and 6 with Br₄σ(4c–6e), models of 1, and 7 and 8 with Se₂Br₅σ(7c–10e), models of 2, are also shown. Atoms taking part in the linear interactions are shown by red.

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.

Figure 3: Sequence in the stabilization of Br₄σ(4c–6e), starting from those in G(A) to 1 via those of G(B).

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 [28–37]. 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) [42–47] for experimental chemists to analyze their own chemical bonds and interaction results based on their own expectations, according to the QTAIM approach [28–37]. 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, 44–46]. 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.

(a)

(b)

Figure 4: QTAIM-DFA plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for ABr-∗-ABr (a) and ABr-∗-BBr (b) in Br₄σ(4c–6e) of the species in Table 1, together with those of the perturbed structures generated with CIV. Marks and colours for the species are shown in the figure.

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 [49–52]. 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 [56–58] 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:

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Molecular graphs of 5 (Ci) (a), 6 (Ci) (b), 7 (C2h) (c), and 8 (C2h) (d) drawn on the structures optimized at the MP2 level, together with 1 (Ci)obsd (e) and 2 (C₁)obsd (f). Contour plots of ρ(r) are also drawn on the planes containing the linear interactions. BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, and CCPs (cage critical points) by green dots. BPs (bond paths) are drawn as pink lines and the secondary ones as pink dots. They are associated with the BCPs. Carbon and hydrogen atoms are shown in black and gray, respectively. The contours () are at 2l (l = ±8, ±7, …, and 0).

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 ΔEES (ΔEZP = 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 = rBP–RSL) 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.

Table 1: QTAIM functions and QTAIM-DFA parameters for BBr-∗-ABr-∗-A′Br-∗-B′Br at BCPs in Br₄σ(4c–6e), together with ABr-∗-ABr in Br₂, evaluated with MP2/BSS-Aa).

Species (symmetry)
Interaction X-∗-Y
ρb(rc) ()
c∇²ρb(rc)b) (au)
Hb(rc) (au)
kb(rc)c)
Rd) (au)
θe) (°)
Cij (Å mdyn⁻¹)
θp:CIVf) (°)
κp:CIVg) (au⁻¹)
Predicted nature

Br₂ (D∞h)h)
Br-∗-Br
0.1130
−0.0001
−0.0497
−2.005
0.0497
180.1
0.4
191.8
1.8
SS/Cov-wi)

(D∞h)j)
ABr-∗-ABr
0.0922
0.0052
−0.0313
−1.751
0.0317
170.6
0.8
190.6
3.6
r-CS/CT-TBPk)

ABr-∗-BBr
0.0198
0.0058
0.0001
−0.995
0.0058
89.5
−19.6
118.2
146
p-CS/t-HBncl)

Br₆ (C₂)
ABr-∗-ABr
0.1099
0.0010
−0.0466
−1.960
0.0467
178.8
0.4
191.3
2.3
r-CS/CT-TBPk)

ABr-∗-BBr
0.0131
0.0049
0.0009
−0.899
0.0050
79.6
14.3
97.8
105
p-CS/t-HBncl)

Br₆ (C2h)m)
ABr-∗-ABr
0.1099
0.0010
−0.0466
−1.961
0.0467
178.8
0.4
191.7
1.7
r-CS/CT-TBPk)

ABr-∗-BBr
0.0131
0.0049
0.0009
−0.899
0.0050
79.6
14.3
97.6
97
p-CS/t-HBncl)

Br₆ (C2h)obsdn)
ABr-∗-ABr
0.0765
0.0053
−0.0200
−1.654
0.0207
165.2

r-CS

ABr-∗-BBr
0.0156
0.0055
0.0007
−0.929
0.0055
82.5

p-CS

H₂Br₄ (C2h)
ABr-∗-ABr
0.1101
0.0008
−0.0468
−1.966
0.0468
179.0
0.4
191.7
2.0
r-CS/CT-TBPk)

ABr-∗-BBr
0.0118
0.0045
0.0010
−0.881
0.0046
78.0
15.5
94.7
100
p-CS/t-HBncl)

Me₂Br₄
ABr-∗-ABr
0.1076
0.0016
−0.0444
−1.932
0.0445
177.9
0.4
191.4
1.8
r-CS/CT-TBPk)

ABr-∗-BBr
0.0164
0.0057
0.0006
−0.945
0.0057
84.1
10.5
105.1
100
p-CS/t-HBncl)

H₄Se₂Br₆ (Ci)
ABr-∗-ABr
0.1028
0.0031
-0.0400
−1.866
0.0401
175.6
0.5
191.5
2.5
r-CS/CT-TBPk)

ABr-∗-BBr
0.0220
0.0068
−0.0002
−1.016
0.0068
91.9
9.9
117.4
75
r-CS/t-HBwco)

Me₄Se₂Br₆ (Ci)
ABr-∗-ABr
0.1028
0.0032
−0.0400
−1.862
0.0402
175.4
0.5
191.4
3.7
r-CS/CT-TBPk)

ABr-∗-BBr
0.0212
0.0067
−0.0001
−1.008
0.0067
90.9
9.9
116.4
101
r-CS/t-HBwco)

5 (Ci)p)
ABr-∗-ABr
0.1016
0.0036
−0.0389
−1.844
0.0391
174.7
0.5
191.5
4.4
r-CS/CT-TBPk)

ABr-∗-BBr
0.0226
0.0070
−0.0003
−1.023
0.0070
92.7
6.8
118.0
557
r-CS/t-HBwco)

ABr-∗-BBrq)
0.0226
0.0070
−0.0003
−1.023
0.0070
92.7
6.8
118.0
551
r-CS/t-HBwco)

5 (Ci)r)
ABr-∗-ABr
0.1047
0.0020
−0.0383
−1.905
0.0384
177.0
0.5
191.3
2.5
r-CS/CT-TBPk)

ABr-∗-BBr
0.0145
0.0048
0.0008
−0.904
0.0048
80.1
15.9
97.4
102
p-CS/t-HBnco)

6 (Ci)
ABr-∗-ABr
0.1014
0.0037
−0.0388
−1.841
0.0389
174.6
0.5
191.5
36
r-CS/CT-TBPk)

ABr-∗-ABrq)
0.1014
0.0037
−0.0388
−1.841
0.0389
174.6
0.5
191.6
36
r-CS/CT-TBPk)

ABr-∗-BBrq)
0.0227
0.0070
−0.0004
−1.024
0.0071
92.8
42.1
122.5
2474
r-CS/t-HBwco)

6 (Ci)r)
ABr-∗-ABr
0.1044
0.0021
−0.0380
−1.901
0.0381
176.9
0.5
191.3
2.6
r-CS/CT-TBPk)

ABr-∗-BBr
0.0147
0.0048
0.0008
−0.907
0.0049
80.3
16.6
97.8
103
p-CS/t-HBnco)

1 (Ci)r)
ABr-∗-ABr
0.1063
0.0013
−0.0398
−1.939
0.0398
178.1
0.5
191.7
2.3
r-CS/CT-TBPk)

ABr-∗-BBr
0.0123
0.0043
0.0010
−0.868
0.0044
76.9
18.3
91.8
100
p-CS/t-HBncl)

1 (Ci)obsds)
ABr-∗-ABr
0.1019
0.0032
−0.0393
−1.860
0.0394
175.3

r-CS

ABr-∗-BBr
0.0200
0.0066
0.0003
−0.979
0.0066
87.7

p-CS

a)See the text for BSS. b)c∇²ρb(rc) = Hb(rc) − Vb(rc)/2, where c = ћ²/8m. c)kb(rc) = Vb(rc)/Gb(rc). d)R = (x² + y²)1/2, where (x, y) = (Hb(rc) − Vb()/2, Hb()). e)θ = 90° − tan⁻¹ (y/x).f)θp = 90°– tan⁻¹(dy/dx). g)κp = |d²y/dx²|/[1 + (dy/dx)²]3/2. h)The Br-Br distance in Br₂ was optimized to be 2.2756 Å with MP2/BSS-A, which was very close to the observed distance in the gas phase (2.287 Å) [63]. However, the values are shorter than those determined by the X-ray crystallographic analysis (2.491 Å) [40] by 0.210 Å. The noncovalent Br---Br distance is 3.251 Å in crystal, which is shorter than the sum of the van der Waals radii [64] by 0.45 Å. i)The SS interaction of the weak covalent nature. j)With one imaginary frequency for the vibration mode of the SGU symmetry. k)The regular-CS interaction of the CT-TBP nature. l)The pure-CS interaction of the HB nature with no covalency. m)With one imaginary frequency for the rotational mode around the linear Br₄ interaction. n)See ref. [40] o)The regular-CS interaction of the HB nature with covalency. p)With one imaginary frequency for the vibration mode of the AU symmetry. q) = (0), ±0.025, and ±0.05. r)At the CAM-B3LYP level. s)See ref. [39].

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).

Table 2: QTAIM functions and QTAIM-DFA parameters for ABr-∗-ASe-∗-BBr-∗-CBr-∗-DBr-∗-BSe-∗-EBr at BCPs in 7 (C2h), 8 (C2h), and 2 (C₁)obsd, together with ABr-∗-ASe-∗-BBr in 3 (Cs) and ABr-∗-ASe-∗-BBr-∗-CBr-∗-DBr in 4 (Cs), evaluated with MP2 BSS‐Aa).

Species (symmetry)
Interaction X-∗-Y
ρb(rc) ()
c∇²ρb(rc)b) (au)
Hb(rc) (au)
kb(rc)c)
Rd) (au)
θe) (°)
Cij (Å mdyn⁻¹)
θp:CIVf) (°)
κp:CIVg) (au⁻¹)
Predicted nature

7 (C2h)
ASe-∗-ABrh)
0.0423
0.0080
−0.0056
−1.258
0.0098
124.8
6.3
169.9
55
r-CS/CT-MCi)

ASe-∗-BBrj)
0.0825
0.0043
−0.0264
−1.753
0.0267
170.7
1.2
192.2
2.2
r-CS/CT-TBPk)

BBr-∗-CBrl)
0.0335
0.0086
−0.0022
−1.115
0.0088
104.6
9.4
145.5
102
r-CS/t-HBwcm)

8 (C2h)
ASe-∗-ABrh)
0.0492
0.0085
−0.0079
−1.318
0.0116
133.0
2.3
172.7
53
r-CS/CT-MCi)

ASe-∗-BBrj)
0.0662
0.0075
−0.0158
−1.513
0.0175
154.6
2.2
187.7
17
r-CS/CT-TBPk)

BBr-∗-CBrl)
0.0398
0.0092
−0.0038
−1.171
0.0100
112.5
4.2
151.5
54
r-CS/t-HBwcm)

2 (C₁)obsdn)
ABr-∗-ASe
0.0219
0.0065
−0.0005
−1.039
0.0065
94.6

r-CS

ASe-∗-BBr
0.0576
0.0102
−0.0113
−1.356
0.0152
137.8

r-CS

BBr-∗-CBr
0.0952
0.0068
−0.0337
−1.713
0.0343
168.6

r-CS

CBr-∗-DBr
0.0183
0.0062
0.0005
−0.961
0.0062
85.7

p-CS

DBr-∗-BSe
0.0818
0.0063
−0.0271
−1.682
0.0278
166.9

r-CS

BSe-∗-EBr
0.0753
0.0072
−0.0220
−1.604
0.0231
161.9

r-CS

3 (Cs)
ABr-∗-ASe
0.0737
0.0061
−0.0214
−1.636
0.0223
164.0
0.8
185.8
8.4
r-CS/CT-TBPk)

ASe-∗-BBr
0.0678
0.0069
−0.0177
−1.562
0.0189
158.7
1.0
183.0
18
r-CS/CT-TBPk)

4 (Cs)
ABr-∗-ASe
0.0131
0.0042
0.0004
−0.945
0.0042
84.0
6.4
105.1
84
p-CS/t-HBnco)

ASe-∗-BBr
0.0425
0.0088
−0.0052
−1.229
0.0103
120.6
4.6
163.2
63
r-CS/CT-MCi)

BBr-∗-CBr
0.0933
0.0059
−0.0321
−1.732
0.0326
169.6
0.9
192.1
5.6
r-CS/CT-TBPk)

a)See the text for BSS. b)c∇²ρb(rc) = Hb(rc) − Vb(rc)/2, where c = ћ²/8m. c)kb(rc) = Vb(rc)/Gb(rc). d)R = (x² + y²)1/2, where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)). e)θ = 90° − tan⁻¹ (y/x). f)θp = 90° − tan⁻¹ (dy/dx). g)κp = |d²y/dx²|/[1 + (dy/dx)²]3/2. h)Because it has Ci symmetry, it is the same as BSe-∗-EBr. i)The regular-CS interaction of the CT-MC nature. j)The same as BSe-∗-DBr. k)The regular-CS interaction of the CT-TBP nature. l)The same as CBr-∗-DBr. m) The pure-CS interaction of the HB nature with no covalency. n)See ref. [39]. o)The regular-CS interaction of the HB nature with no covalency.

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.

Figure 6: Molecular orbitals for σ(7c–10e). ψ₁₈₄ (HOMO) and ψ₁₈₅ (LUMO) of 8 (C2h).

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.

Acknowledgments

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)

References

G. C. Pimentel, “The bonding of trihalide and bifluoride ions by the molecular orbital method,” The Journal of Chemical Physics, vol. 19, no. 4, pp. 446–448, 1951.

J. I. Musher, “The chemistry of hypervalent molecules,” Angewandte Chemie International Edition in English, vol. 8, no. 1, pp. 54–68, 1969.

K.-Y. Akiba, Chemistry of Hypervalent Compounds, Wiley-VCH, New York, USA, 1999.

A. J. Mukherjee, S. S. Zade, H. B. Singh, and R. B. Sunoj, “Organoselenium chemistry: role of intramolecular interactions†,” Chemical Reviews, vol. 110, no. 7, pp. 4357–4416, 2010.

S. Patai and Z. Rappoport, The Chemistry of Organic Selenium and Tellurium Compounds, vol. 1, Wiley, New York, USA, 1986.

J. C. Martin and M. M. Chau, “Anchimerically accelerated bond homolysis. VII: simultaneous participation of three neighboring groups in the transition state for a radical-forming perester decomposition: new route to hypervalent compounds of sulfur and iodine,” Journal of the American Chemical Society, vol. 96, no. 10, pp. 3319–3321, 1974.

W. Nakanishi, S. Hayashi, and T. Arai, “Linear alignment of four sulfur atoms in bis[(8-phenylthio)naphthyl] disulfide: contribution of linear S4 hypervalent four-centre six-electron bond to the structure,” Chemical Communications, vol. 2002, no. 20, pp. 2416-2417, 2002.

W. Nakanishi, S. Hayashi, S. Morinaka, T. Sasamori, and N. Tokitoh, “Extended hypervalent E′---E-E---E′ 4c-6e (E, E′ = Se, S) interactions: structure, stability and reactivity of 1-(8-PhE′C10H6)EE(C10H6E′Ph-8′)-1′,” New Journal of Chemistry, vol. 32, no. 11, pp. 1881–1889, 2008.

W. Nakanishi, S. Hayashi, and S. Toyota, “Four-center six-electron interaction versus lone Pair−Lone pair interaction between selenium atoms in naphthalene peri positions,” The Journal of Organic Chemistry, vol. 63, no. 24, pp. 8790–8800, 1998.

W. Nakanishi, “Hypervalent chalcogen compounds (chapter 10.3),” in Handbook of Chalcogen Chemistry: New Perspectives in Sulfur, Selenium and Tellurium, F. A. Devillanova, Ed., pp. 644–668, Royal Society of Chemistry, London, UK, 2006.

W. Nakanishi and S. Hayashi, “Hypervalent chalcogen compounds (chapter 12.3),” in Handbook of Chalcogen Chemistry: New Perspectives in Sulfur, Selenium and Tellurium, F. A. Devillanova and W.-W. du Mont, Eds., vol. 2, pp. 335–372, Royal Society of Chemistry, Cambridge, UK, 2nd edition, 2013.

W. Nakanishi, S. Hayashi, M. Hashimoto, M. Arca, M. C. Aragoni, and V. Lippolis, in The Chemistry of Organic Selenium and Tellurium Compounds, Z. Rappoport, Ed., vol. 4, pp. 885–972, Wiley, New York, USA, 2014.

W. Nakanishi, S. Hayashi, and N. Itoh, “Extended hypervalent 5c−6e interactions: linear alignment of five C−Se---O---Se−C atoms in anthraquinone and 9-methoxyanthracene bearing arylselanyl groups at the 1,8-positions,” The Journal of Organic Chemistry, vol. 69, no. 5, pp. 1676–1684, 2004.

A. Panda, G. Mugesh, H. B. Singh, and R. Butcher, “Synthesis, structure, and reactivity of organochalcogen (Se, Te) compounds derived from 1-(N,N-Dimethylamino)naphthalene and N,N-dimethylbenzylamine,” Organometallics, vol. 18, no. 10, pp. 1986–1993, 1999.

M. Kulcsar, A. Beleaga, C. Silvestru et al., “Solid state structure and solution behaviour of organoselenium(ii) compounds containing 2-{E(CH₂CH₂)2NCH₂}C₆H₄ groups (E = O, NMe),” Dalton Transactions, vol. 2007, no. 21, pp. 2187–2196, 2007.

M. C. Durrant, “A quantitative definition of hypervalency,” Chemical Science, vol. 6, no. 11, pp. 6614–6623, 2015.

M. S. Schmøkel, S. Cenedese, J. Overgaard et al., “Testing the concept of hypervalency: charge density analysis of K₂SO₄,” Inorganic Chemistry, vol. 51, no. 15, pp. 8607–8616, 2012.

Z. Rappoport, The Chemistry of Organic Selenium and Tellurium Compounds, vol. 4, Wiley, New York, USA, 2014.

P. Kilian, F. R. Knight, and J. D. Woollins, “Naphthalene and related systemsperi-substituted by group 15 and 16 elements,” Chemistry—A European Journal, vol. 17, no. 8, pp. 2302–2328, 2011.

R.-F. Hu, Y.-H. Wen, J. Zhang, Z.-J. Li, and Y.-G. Yao, “4,4′-bipyridylium bis(hydrogen 2,2′-dithiodibenzoate) dihydrate,” Acta Crystallographica Section E Structure Reports Online, vol. 60, no. 11, pp. o2029–o2031, 2004.

S. Wang, H. Mao-Lin, and F. Chen, “catena-Poly[[[aquazinc(II)]-μ-2,2′-dithiodibenzoato] bis(N,N-dimethylformamide)],” Acta Crystallographica Section E Structure Reports Online, vol. 60, no. 4, pp. m413–m415, 2004.

X.-J. Wang, Z.-F. Chen, B.-S. Kang et al., “Chemistry of 2-mercaptophenol (H2mp) . Part iii. One dimensional arrays of binuclear anions of Mo (VI) linked by bis (2-hydroxyphenyl) disulfide via hydrogen bondings,” Polyhedron, vol. 18, no. 5, pp. 647–655, 1999.

T. Mak, W. Yip, W. Chan, G. Smith, and C. Kennard, “The crystal structure of bis(2-chlorophenyl) disulfide,” Australian Journal of Chemistry, vol. 42, no. 8, pp. 1403–1406, 1989.

Y. Tsubomoto, S. Hayashi, and W. Nakanishi, “Dynamic and static behavior of the E-E′ bonds (E, E′ = S and Se) in cystine and derivatives, elucidated by AIM dual functional analysis,” RSC Advances, vol. 5, no. 15, pp. 11534–11540, 2015.

W. Nakanishi, Y. Tsubomoto, and S. Hayashi, “Nature of S2Se2 σ(4c-6e) at naphthalene 1,8-positions and models, elucidated by QTAIM dual functional analysis,” RSC Advances, vol. 6, no. 95, pp. 93195–93204, 2016.

S. Hayashi, Y. Tsubomoto, and W. Nakanishi, “Behavior of the E-E' bonds (E, E' = S and Se) in glutathione disulfide and derivatives elucidated by quantum chemical calculations with the quantum theory of atoms-in-molecules approach,” Molecules, vol. 23, no. 2, pp. 443–451, 2018.

Y. Tsubomoto, S. Hayashi, W. Nakanishi, L. K. Mapp, and S. J. Coles, “High-resolution X-ray diffraction determination of the electron density of 1-(8-PhSC10H6)SS(C10H6SPh-8′)-1′ with the QTAIM approach: evidence for S4 σ(4c-6e) at the naphthalene peri-positions,” RSC Advances, vol. 8, no. 18, pp. 9651–9660, 2018.

R. F. W. Bader, Atoms in Molecules. A Quantum Theory, Oxford University Press, Oxford, UK, 1990.

C. F. Matta and R. J. Boyd, An Introduction to the Quantum Theory of Atoms in Molecules in the Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, Wiley-VCH, Weinheim, Germany, 2007.

F. Biegler-König and J. Schönbohm, “Update of the AIM2000-program for atoms in molecules,” Journal of Computational Chemistry, vol. 23, no. 15, pp. 1489–1494, 2002.

F. Biegler-König, J. Schönbohm, and D. Bayles, “Software news and updates, AIM2000—a program to analyze and visualize atoms in molecules,” Journal of Computational Chemistry, vol. 22, no. 5, pp. 545–559, 2001.

R. F. W. Bader, “A bond path: a universal indicator of bonded interactions,” The Journal of Physical Chemistry A, vol. 102, no. 37, pp. 7314–7323, 1998.

R. F. W. Bader, “A quantum theory of molecular structure and its applications,” Chemical Reviews, vol. 91, no. 5, pp. 893–928, 1991.

R. F. W. Bader, “Atoms in molecules,” Accounts of Chemical Research, vol. 18, no. 1, pp. 9–15, 1985.

T. H. Tang, R. F. W. Bader, and P. J. MacDougall, “Structure and bonding in sulfur-nitrogen compounds,” Inorganic Chemistry, vol. 24, no. 13, pp. 2047–2053, 1985.

R. F. W. Bader, T. S. Slee, D. Cremer, and E. Kraka, “Description of conjugation and hyperconjugation in terms of electron distributions,” Journal of the American Chemical Society, vol. 105, no. 15, pp. 5061–5068, 1983.

F. W. Biegler-könig, R. F. W. Bader, and T.-H. Tang, “Calculation of the average properties of atoms in molecules II,” Journal of Computational Chemistry, vol. 3, no. 3, pp. 317–328, 1982.

Y. Tsubomoto, S. Hayashi, W. Nakanishi, T. Sasamori, and N. Tokitoh, “Nature of E₂X₂σ(4c-6e) of the X---E-E---X type at naphthalene 1,8-positions and model, elucidated by X-ray crystallographic analysis and QC calculations with the QTAIM approach,” Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials, vol. 73, no. 2, pp. 265–275, 2017.

W. Nakanishi, S. Hayashi, S. Yamaguchi, and K. Tamao, “First Br₄ four centre-six electron and Se₂Br₅ seven centre-ten electron bonds in nonionic bromine adducts of selenanthrene,” Chemical Communications, vol. 2004, no. 2, pp. 140-141, 2004.

B. M. Powell, K. M. Heal, and B. H. Torrie, “The temperature dependence of the crystal structures of the solid halogens, bromine and chlorine,” Molecular Physics, vol. 53, no. 4, pp. 929–939, 1984.

H. Vogt, D. Wulff-Molder, and M. Meisel, “Synthese und kristallstruktur der (p-brombenzyl)triphenylphosphonium-bromide: [(p-Br-C₆H₄-CH₂)P(C₆H₅)₃]⁺B⁻;[(p-Br-C₆H₄-CH₂)P(C₆H₅)³]⁺Br²⁻ und [(p-Br-C₆H₄-CH₂)P(C₆H₅)₃]⁺Br³⁻,” Zeitschrift für Naturforschung B, vol. 51, no. 10, pp. 1443–1448, 1996.

W. Nakanishi, S. Hayashi, and K. Narahara, “Polar coordinate representation of hb(rc) versus (ℏ²/8m)∇²ρb(rc) at BCP in AIM analysis: classification and evaluation of weak to strong interactions,” The Journal of Physical Chemistry A, vol. 113, no. 37, pp. 10050–10057, 2009.

W. Nakanishi, S. Hayashi, and K. Narahara, “Atoms-in-molecules dual parameter analysis of weak to strong interactions: behaviors of electronic energy densities versus laplacian of electron densities at bond critical points,” The Journal of Physical Chemistry A, vol. 112, no. 51, pp. 13593–13599, 2008.

W. Nakanishi and S. Hayashi, “Atoms-in-Molecules dual functional analysis of weak to strong interactions,” Current Organic Chemistry, vol. 14, no. 2, pp. 181–197, 2010.

W. Nakanishi and S. Hayashi, “Dynamic behaviors of interactions: application of normal coordinates of internal vibrations to AIM dual functional analysis,” The Journal of Physical Chemistry A, vol. 114, no. 27, pp. 7423–7430, 2010.

W. Nakanishi, S. Hayashi, K. Matsuiwa, and M. Kitamoto, “Applications of normal coordinates of internal vibrations to generate perturbed structures: dynamic behavior of weak to strong interactions elucidated by atoms-in-molecules dual functional analysis,” Bulletin of the Chemical Society of Japan, vol. 85, no. 12, pp. 1293–1305, 2012.

W. Nakanishi and S. Hayashi, “Role of dG/dw and dV/dw in AIM analysis: an approach to the nature of weak to strong interactions,” The Journal of Physical Chemistry A, vol. 117, no. 8, pp. 1795–1803, 2013.

W. Nakanishi and S. Hayashi, “Perturbed structures generated using coordinates derived from compliance constants in internal vibrations for QTAIM dual functional analysis: intrinsic dynamic nature of interactions,” International Journal of Quantum Chemistry, vol. 118, no. 11, Article ID e25590, 2018.

K. Brandhorst and J. Grunenberg, “Efficient computation of compliance matrices in redundant internal coordinates from Cartesian Hessians for nonstationary points,” The Journal of Chemical Physics, vol. 132, no. 18, Article ID 184101, 2010.

K. Brandhorst and J. Grunenberg, “How strong is it? The interpretation of force and compliance constants as bond strength descriptors,” Chemical Society Reviews, vol. 37, no. 8, pp. 1558–1567, 2008.

J. Grunenberg, “III-defined concepts in chemistry: rigid force constants vs. compliance constants as bond strength descriptors for the triple bond in diboryne,” Chemical Science, vol. 6, no. 7, pp. 4086–4088, 2015.

The Unit of Cij is Å mdyn⁻¹, Å³ Mdyn⁻¹ in Reference 50 Should Read Å mdyn⁻¹.

S. Hayashi, T. Nishide, and W. Nakanishi, “Behavior of multi-HBs in acetic acid dimer and related species: QTAIM dual functional analysis employing perturbed structures generated using coordinates from compliance force constants,” Bulletin of the Chemical Society of Japan, vol. 92, no. 1, pp. 87–96, 2019.

M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., Gaussian 09 (Revision D.01), Gaussian, Inc., Wallingford, CT, USA, 2009.

T. Noro, M. Sekiya, and T. Koga, “Segmented contracted basis sets for atoms H through Xe: Sapporo-(DK)-nZP sets (n = D, T, Q),” Theoretical Chemistry Accounts, vol. 131, no. 2, pp. 1124–1131, 2012.

C. Møller and M. S. Plesset, “Note on an approximation treatment for many-electron systems,” Physical Review, vol. 46, no. 7, pp. 618–622, 1934.

J. Gauss, “Effects of electron correlation in the calculation of nuclear magnetic resonance chemical shifts,” The Journal of Chemical Physics, vol. 99, no. 5, pp. 3629–3643, 1993.

J. Gauss, “Accurate calculation of NMR chemical shifts,” Berichte der Bunsengesellschaft für physikalische Chemie, vol. 99, no. 8, pp. 1001–1008, 1995.

Y. Zhao and D. G. Truhlar, “The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals,” Theoretical Chemistry Accounts, vol. 120, no. 1–3, pp. 215–241, 2008.

F. Biegler-König, “Calculation of atomic integration data,” Journal of Computational Chemistry, vol. 21, no. 12, pp. 1040–1048, 2000.

T. A. Keith, AIMAll (Version 17.11.14), TK Gristmill Software, Overland Park, KS, USA, 2017, http://aim.tkgristmill.com/.

E. D. Glendening, J. K. Badenhoop et al., Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, USA, 2013.

Y. Iwasawa, Ed.“The chemical society of Japan edition (chapter 11),” in Kagakubinran Kisohen II, Maruzen, Tokyo, Japan, 5th edition, 2004.

A. Bondi, “van der Waals volumes and radii,” The Journal of Physical Chemistry, vol. 68, no. 3, pp. 441–451, 1964.