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

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.

What are the differences and similarities between X 4 σ(4c-6e), E 4 σ(4c-6e), and E 2 X 2 σ(4c-6e)? e nature of X 4 σ(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 Br 4 σ(4c-6e). H 2 Br 4 (C 2h ) and Me 2 Br 4 (C 2h ) have the form of R-Br---Br-Br---Br-R (RBr 4 R: R � H and Me), which are called the model group A (G(A)). e electronic efficiency to stabilize Br 4 σ(4c-6e) seems small for R in G(A). Br 6 (C 2h ) is detected as the partial structure in the crystals of Br 2 [40]. Br 6 (C 2h ) in the crystals is denoted by Br 6 (C 2h ) obsd . e optimized structure of Br 6 (C 2h ) has one imaginary frequency, which belongs to G(A), together with Br 6 (C 2h ) obsd . e optimized structure of Br 6 retains the C 2 symmetry, (Br 6 (C 2 )), which also belongs to G(A). e CT interaction of the n p ( B Br) ⟶ σ * ( A Br-A Br) ⟵ n p ( B Br) form in Br 4 σ(4c-6e) will be much stabilized if the large negative charge is developed at the B Br atoms in Br-(R 2 )Se-B Br---A Br-A Br---B Br-Se(R 2 )-Br, where the ∠Se B Br A Br is around 90°. e highly negatively charged B Br in Br-Se(R 2 )-B Br (R � H and Me) of σ(3c-4e) is employed to stabilize Br 4 σ(4c-6e), in this case. e models form G(B). e nature of Br 4 σ(4c-6e) in 5 and 6 is similarly analysed, which belongs to G(B). Br 2− 4 (D ∞h ) also belongs to G(B) although one imaginary frequency was predicted for Br 2− 4 , if optimized at the MP2 level. Figure 3 illustrates the story for the stabilization of Br 4 σ(4c-6e) in the sequence of the species, starting from G(A) to 1, via G(B). Figure 3 also shows the A Br-A Br and A Br---B Br distances (r( A Br-A Br) and r( A Br-B Br), respectively), together with the charge developed at B Br in the original species of R-B Br (Qn ( B Br)), which construct R-B Br---A Br-A Br---B Br-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 [28][29][30][31][32][33][34][35][36][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. erefore, 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). e chemical bonds and interactions are usually classified by the signs of Laplacian rho (∇ 2 ρ b (r c )) and H b (r c ) at BCPs, where ρ b (r c ) and H b (r c ) are the charge densities and total electron energy densities at BCPs, respectively (see Scheme S1 in Supplementary File). e relations between H b (r c ), ∇ 2 ρ b (r c ), G b (r c ) (the kinetic energy densities), and V b (r c ) (the potential energy densities) are represented in equations (1) and (2): How can the nature of Br 4 σ(4c-6e) and Se 2 Br 5 σ(7c-10e) be clarified? For the characterization of interactions in more detail, we recently proposed QTAIM dual functional analysis (QTAIM-DFA) [42][43][44][45][46][47] for experimental chemists to analyze their own chemical bonds and interaction results based on their own expectations, according to the QTAIM approach [28][29][30][31][32][33][34][35][36][37].
e classification of interactions by the signs of ∇ 2 ρ b (r c ) and H b (r c ) 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][45][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.
is plot is expressed by (θ p , κ p ), where θ p corresponds to the tangent line of the plot and κ p is the curvature. e concept of the dynamic nature of interactions has been proposed based on (θ p , κ p ) [42,44]. θ and θ p are measured from the y-axis and the y-direction, respectively. We call (R, θ) and (θ p , κ p ) QTAIM-DFA parameters, which are drawn in Figure 4, exemplified by Br 2− 4 (D ∞h ). While (R, θ) classifies the interactions, (θ p , κ p ) characterizes them.
We proposed a highly reliable method to generate the perturbed structures for QTAIM-DFA very recently [48]. e method is called CIV, which employs the coordinates derived from the compliance force constants C ij for the internal vibrations. Compliance force constants C ij 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 f i and f j correspond to i and j, respectively. e C ij values and the coordinates corresponding to the values can be calculated using the compliance 3.0.2 program, released by Brandhorst and Grunenberg [49][50][51][52]. e 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. e 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). e 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 Br 4 σ(4c-6e) in 1, Se 2 Br 5 σ(7c-10e) in 2, and the models derived from 1 and 2, together with the related linear interactions. e interactions in Br 4 σ(4c-6e) are denoted by B Br- * -A Br- * -A Br- * -B Br, where the asterisk emphasizes the existence of a BCP in the interactions, so are those in Se 2 Br 5 σ(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.
All species were calculated employing BSS-A, and the Møller-Plesset second-order energy correlation (MP2) level [56][57][58] was applied for the optimizations. Optimized structures were confirmed by the frequency analysis. e results of the frequency analysis were used to calculate the C ij values and the coordinates (C i ) corresponding to the values. e DFT level of CAM-B3LYP [59] was also applied when necessary.
e QTAIM functions were analysed with the AIM2000 [60] and AIMAll [61] programs. e 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 (S iw ) is generated by the addition of the i-th coordinates (C i ), derived from C ij , to the standard orientation of a fully optimized structure (S o ) in the matrix representation.
e coefficient f iw in equation (4) controls the structural difference between S iw and S o : f iw is determined to satisfy equation (5) for r, where r and r o stand for the interaction distances in question in the perturbed and fully optimized structures, respectively, with a o � 0.52918Å (Bohr radius). e C i values of five digits are used to predict S iw : .05, and ±0.10 in equation (5). Each plot is analysed using a regression curve of the cubic function, as shown in equation (6)

Structural Optimizations.
e structures of 1 (C i ) and 2 (C 1 ) determined by the X-ray analysis are denoted by 1   (C i ) obsd and 2 (C 1 ) obsd , respectively [39].
e structural parameters are shown in Tables S2 and S3 in Supplementary File, respectively. Figure 3 contains the selected structural parameters for 1 (C i ) obsd . e structures are optimized for G(A) of H 2 Br 4 (C 2h ), Me 2 Br 4 (C 2h ), Br 6 (C 2h ), and Br 6 (C 2 ) and G(B) of H 4 Se 2 Br 6 (C i ), Me 4 Se 2 Br 6 (C i ), 5 (C i ), and 6 (C i ), together with 3 (C s ), 4 (C s ), 7 (C 2h ), 8 (C 2h ), and Br 2 (D ∞h ). e optimized structural parameters are also collected in Tables S2 and S3 in Supplementary File. e frequency analysis was successful for the optimized structures, except for 1 (C i ) obsd and Br 6 (C 2h ). All positive frequencies were obtained for 1 (C i ), if calculated with CAM-B3LYP/BSS-A, which confirms the structure. e Br---Br distances of Br 4 σ(4c-6e) in 1 (C i ) are somewhat longer if optimized at the CAM-B3LYP level, relative to 1 (C i ) obsd . While one imaginary frequency is detected in Br 6 (C 2h ), Br 6 (C 2 ) has all positive frequencies. e 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.  (7), if evaluated with MP2/BSS-A: One imaginary frequency was also predicted for Br 2− 4 (D ∞h ) if optimized with MP2/BSS-A. Br 2− 4 (D ∞h ) seems to collapse to Br − 3 and Br − , according to the imaginary frequency. e double negative charges in Br 2− 4 (D ∞h ) would be responsible for the results. e electrostatic repulsion between the double negative charges will operate to collapse it.

Energies for Formation of Br 4 σ(4c-6e) and NBO Analysis.
Energies for the formation of R′Br 4 R′ from the components (2R′Br + Br 2 ) (ΔE) are defined by equation (8) Figure S3 in Supplementary File): NBO analysis [62] was applied to A Br---B Br of the species to evaluate the contributions from CT to stabilize R′-B Br---A Br-A Br---B Br-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 q i is the donor orbital occupancy, ε i and ε j are diagonal elements (orbital energies), and F(i, j) is the off-diagonal NBO Fock matrix element.
e results are collected in Table S4 in Supplementary File. e ΔE ES values are very well correlated to E(2) for the optimized structures, except for Br 2− 4 (D ∞h ). (ΔE ES � -0.71(2E(2)) + 7.17: R 2 c � 0.959, see Figure S4 in Supplementary File). Br 2− 4 (D ∞h ) is predicted to be less stable than the components.
Before application of QTAIM-DFA to Br 4 σ(4c-6e) and Se 2 Br 5 σ(7c-10e), molecular graphs were examined, as shown in the next section. Containing Br 4 σ(4c-6e), Se 2 Br 5 σ(7c-10e), and Related Linear Interactions. Figure 5 illustrates the molecular graphs of 5 (C i ), 6 (C i ), 7 (C 2h ), and 8 (C 2h ), drawn on the optimized structures, together with 1 (C i ) obsd and 2 (C 1 ) 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 Me 2 Br 4 (C 2h ), Br 6 (C 2 ), Br 2− 4 (D ∞h ), and Br(Me 2 )SeBr 4 Se(Me 2 )Br (C i ) are shown in Figure 3, which are drawn on the optimized structures.  Table 1, together with those of the perturbed structures generated with CIV. Marks and colours for the species are shown in the figure.   e table contains the values for Se 2 Br 6 σ(7c-10e) in 7 (C 2h ) and 8 (C 2h ). e differences between them (Δr BP � r BP -R SL ) are less than 0.003Å. e r BP values are plotted versus R SL , which are shown in Figure S5 in Supplementary File. e correlations are excellent, as shown in the figure. erefore, Br 4 σ(4c-6e) and Se 2 Br 6 σ(7c-10e) in the species can be approximated by the straight lines.

Survey of Br
QTAIM functions are calculated for Br 4 σ(4c-6e) at BCPs. Table 1 collects the values for the interactions. H b (r c ) is plotted versus H b (r c ) − V b (r c )/2 for the data shown in Table 1, together with those from the perturbed structures generated with CIV. Figure 4 Table 1 collects the QTAIM-DFA parameters for Br 4 σ(4c-6e). e classification of interactions will also be discussed based on the (R, θ) values.
QTAIM functions are similarly calculated for Se 2 Br 6 σ(7c-10e) at BCPs, together with the related interactions. H b (r c ) is similarly plotted versus H b (r c ) − V b (r c )/2 although not shown in the figures. en, QTAIM-DFA parameters of (R, θ) and (θ p , κ p ) are obtained by analysing the plots, according to equations (S3)-(S6). Table 2 collects the QTAIM-DFA parameters of (R, θ) and (θ p , κ p ) for Br 4 σ(4c-6e). (R, θ), which correspond to the data from the fully optimized structures. On the contrary, they are characterized employing (θ p , κ p ) 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, θ p ) values, while the κ p values are used only additionally. It is instructive to survey the criteria before detail discussion. e criteria tell us that 180°< θ (H b (r c ) − V b (r c )/2 < 0) for the SS interactions, 90°< θ < 180°(H b (r c ) < 0) for the r-CS interactions, and 45°< θ < 90°(H b (r c ) > 0) for p-CS interactions. e θ p value characterizes the interactions. In the p-CS region of 45°< θ < 90°, the character of interactions will be the vdW type for 45°< θ p < 90°, whereas it will be the typical HB type without covalency (t-HB nc ) for 90°< θ p < 125°, where θ p � 125°is tentatively given for θ � 90°. e CT interaction will appear in the r-CS region of 90°< θ < 180°. e t-HB type with covalency (t-HB wc ) appears in the region of 125°< θ p < 150°(90°< θ < 115°), where (θ, θ p ) � (115°, 150°) is tentatively given as the borderline between t-HB wc and the CT-MC nature. e borderline for the interactions between CT-MC and CT-TBP types is defined by θ p � 180°. θ � 150°is tentatively given for θ p � 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. e classification and characterization of interactions are summarized in Table S1 and Scheme S3 in Supplementary File. e A Br- * -A Br and A Br- * -B Br interactions of Br 4 σ(4c-6e) will be classified and characterized based on the (R, θ, θ p ) 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. e results in Table 1 show that the A Br- * -A Br interaction in Br 4 σ(4c-6e) becomes weaker, as the strength of the corresponding A Br- * -B Br increases. e strength of A Br- * -A Br becomes weaker in the order shown in equation (10), if evaluated by θ, while that of A Br- * -B Br increases in the order shown in equation (11), if measured by θ. Very similar results were obtained by θ p : θ for A Br − * − A Br: Br 2 D ∞h > H 2 Br 4 C 2h ≥ Br 6 C 2 and C 2h > Me 2 Br 4 C 2h > H 4 Se 2 Br 6 C i ≥ Me 4 Se 2 Br 6 C i
. h) e Br-Br distance in Br 2 was optimized to be 2.2756Å with MP2/BSS-A, which was very close to the observed distance in the gas phase (2.287Å) [63].
However, the values are shorter than those determined by the X-ray crystallographic analysis (2.491Å) [40] by 0.210Å. e noncovalent Br---Br distance is 3.251Å in crystal, which is shorter than the sum of the van der Waals radii [64]    Br- * -B Br become weaker and stronger, respectively, as the CT interaction increases. Br 4 σ(4c-6e) will be stabilized more effectively, if the negative charge is developed more at B Br. However, the two Br − ligands in Br 2− 4 (D ∞h ) seem not so effective than that expected. is would come from the electrostatic repulsive factor between the double negative charges in Br 2− 4 (D ∞h ), as mentioned above. e θ values for ( A Br- * -A Br and A Br- * -B Br) in Br 6 (C 2h ) obsd and 1 (C i ) obsd are (165.2°, 82.5°) and (175.3°, 87.7°), respectively. erefore, A Br- * -A Br and A Br- * -B Br are classified by r-CS and p-CS, respectively. Both A Br- * -A Br and A Br- * -B Br in Br 6 (C 2h ) obsd are predicted to be weaker than those in 1 (C i ) obsd , respectively. e results would be curious at the first glance, since A Br- * -A Br will be weaker, if A Br- * -B Br in B Br- * -A Br- * -A Br- * -B Br becomes stronger, as mentioned above. ey would be affected from the surrounding, such as the crystal packing effect. A Br 2 molecule interacts with four bromine atoms adjacent to the Br 2 molecule on the bc-plane in crystals, equivalently with 3.251Å [40].

Conclusion
e intrinsic dynamic and static nature of Br 4 σ(4c-6e) is elucidated for 1 (C i ) obsd and the related species with QTAIM-DFA, employing the perturbed structures generated with CIV. e A Br-A Br interactions in B Br- * -A Br- * -A Br- * -B Br of Br 4 σ(4c-6e) are weaker than Br- * -Br in the optimized structure of Br 2 (D ∞h ), which is predicted to have the SS/Cov-w nature. e A Br-A Br interactions in Br 4 σ(4c-6e) of the models are predicted to have the r-CS/CT-TBP nature, if optimized with MP2/BSS-A. e A Br-A Br interaction in 1 (C i ) obsd also appears in the r-CS region. On the contrary, the A Br-B Br interactions in Br 6 (C 2 ), Br 6 (C 2h ), H 2 Br 4 (C 2h ), and Me 2 Br 4 (C 2h ) are predicted to have the p-CS/t-HB nc nature, whereas those in H 4 Se 2 Br 4 (C i ), Me 4 Se 2 Br 4 (C i ), 5 (C i ), and 6 (C i ) have the r-CS/t-HB wc nature, if evaluated with MP2/BSS-A. e A Br- * -B Br interactions become stronger in the order of H 2 Br 4 (C 2h ) < Br 6 (C 2h ) ≤ Br 6 (C 2 ) < Me 2 Br 4 (C 2h ) << Me 4 Se 2 Br 6 (C i ) ≤ H 4 Se 2 Br 6 (C i ) ≤ 5 (C i ) < 6 (C i ), which is the inverse order for A Br- * -A Br, as a whole. e results are in accordance with the CT interaction of the n p ( B Br) ⟶ σ * ( A Br-A Br) ← n p ( B Br) form derived from Br 4 σ(4c-6e).
e decreased binding force of A Br- * -A Br must be transferred to A Br- * -B Br in Br 4 σ(4c-6e). Namely, it is demonstrated that Br 4 σ(4c-6e) is stabilized as the strength of A Br- * -B Br in Br 4 σ(4c-6e) increases, while A Br- * -A Br becomes weakened relative to that in the original Br 2 (D ∞h ). In this process, Br 4 σ(4c-6e) is totally stabilized. e A Br- * -A Br and A Br- * -B Br interactions in Br 6 (C 2h ) obsd and 1 (C i ) obsd are classified by the r-CS and p-CS interactions, respectively, where the interactions in Br 6 (C 2h ) obsd seem somewhat weaker than those in 1 (C i ) obsd . e Se 2 Br 5 σ(7c-10e) interactions are similarly elucidated for 2 (C 1 ) obsd and the anionic models of 7 (C 2h ) and 8 (C 2h ). e Se 2 Br 5 σ(7c-10e) nature is clearly established for the optimized structures of 7 (C 2h ) and 8 (C 2h ), rather than 2 (C 1 ) 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 Br 4 σ(4c-6e) and Se 2 Br 5 σ(7c-10e).

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.