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Mar 3, 2017 - Musher1,2 and developed by others,3−20 although “hypervalent systems” are not well-described in the literature.21,22 The linear al...
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Linear Four-Chalcogen Interactions in Radical Cationic and Dicationic Dimers of 1,5-(Dichalcogena)canes: Nature of the Interactions Elucidated by QTAIM Dual Functional Analysis with QC Calculations Satoko Hayashi,* Kengo Nagata, Shota Otsuki, and Waro Nakanishi* Department of Material Science and Chemistry, Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan S Supporting Information *

ABSTRACT: The dynamic and static nature of extended hypervalent interactions of the BE···AE···AE···BE type are elucidated for four center−seven electron interactions (4c−7e) in the radical cationic dimers (1·+) and 4c−6e in the dicationic dimers (12+) of 1,5-(dichalcogena)canes (2: AE(CH2CH2CH2)2BE: AE, BE = S, Se, Te, and O). The quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA) is applied for the analysis. Total electron energy densities Hb(rc) are plotted versus Hb(rc) − Vb(rc)/2 [= (ℏ2/8m)∇2ρb(rc)] at bond critical points (BCPs) of the interactions, where Vb(rc) values show potential energy densities at BCPs. Data from the fully optimized structures correspond to the static nature of the interactions. Those from the perturbed structures around the fully optimized ones are also plotted, in addition to those of the fully optimized ones, which represent the dynamic nature of interactions. The BE···AE−AE···BE interactions in 12+ are stronger than the corresponding ones in 1·+, respectively. On the one hand, for 12+ with AE, BE = S, Se, and Te, AE···AE are all classified by the shared shell interactions and predicted to have the weak covalent nature, except for those in 1a2+ (AE = BE = S) and 1d2+ (AE = BE = Se), which have the nature of regular closed shell (r-CS)/trigonal bipyramidal adduct formation through charge transfer (CT-TBP). On the other hand, AE···BE are predicted to have the nature of r-CS/molecular complex formation through charge transfer for 1a2+, 1b2+ (AE = Se; BE = S), and 1d2+ or r-CS/CT-TBP for 1c2+ (AE = Te; BE = S), 1e2+ (AE = Te; BE = Se), and 1f2+ (AE = BE = Te). The B E···AE−AE···BE interactions in 1·+ and 12+ are well-analyzed by applying QTAIM-DFA.

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is characterized by the np(BE)→σ*(AE−AE)←np(BE) interaction, of which driving force is the charge transfer (CT) from np(BE) of high donating ability to σ*(AE−AE) of high accepting ability, where np(BE) stands for the p-type lone-pair orbitals of B E. AE2BE2 σ(4c−6e) can also be recognized as a chalcogen bonding,12,32−36 which are of current and continuous interest. The dispersion and the σ-hole interaction may play an important role for the formation and stabilization of AE2BE2 σ(4c−6e).37−56 Figure 1 shows the observed structure of I with (AE, BE) = (Se, S) and the approximate molecular orbital (MO) model for E4 σ(4c−6e). Species containing σ(mc−ne) (4 ≤ m; m < n < 2m) are glowing recently. σ(4c−6e) is the first member of σ(mc−ne) (4 ≤ m; m < n < 2m), which have been recognized to play an important role not only in the development of high functionalities of materials but also in the biological and pharmaceutical reactions.11−20 It is challenging to elucidate the

INTRODUCTION Much attention has been paid to the linear σ-type bonds/ interactions, constructed by the atoms of heavier main-group elements. The linear σ-type bonds/interactions by the three atoms have been called three center−four electron bonds/ interactions (σ(3c−4e)) and/or hypervalent bonds/interactions, which have been originally proposed by Pimentel and Musher1,2 and developed by others,3−20 although “hypervalent systems” are not well-described in the literature.21,22 The linear alignment of four chalcogen atoms has been demonstrated at the naphthalene 1,8-positions in bis[8-(phenylchalcogena)naphthyl]-1,1′-dichalcogenides [I: 1-(8-PhBEC10H6)AE−AE(C10H6BEPh-8′)-1′ ((AE, BE) = (S, S), (S, Se), (Se, S), and (Se, Se))], through the structural determination by the X-ray crystallographic analysis.23−27 We proposed that the linear four A B E2 E2 interactions in I should be analyzed by the σ(4c−6e) model (AE2BE2 σ(4c−6e)), instead of the doubly degenerated σ(3c−4e). The linear interactions longer than σ(3c−4e) are called “extended hypervalent bonds/interactions”, σ(mc−ne), where 4 ≤ m and m < n < 2m (m center−n electron bonds/ interactions), after σ(3c−4e).11,12,17,23−31 AE2BE2 σ(4c−6e) in I © XXXX American Chemical Society

Received: January 20, 2017 Revised: March 3, 2017 Published: March 3, 2017 A

DOI: 10.1021/acs.jpca.7b00667 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Chart 1. Radical cationic dimers (1·+ (22·+)) and dicationic dimers (12+ (222+)) of 1,5-(dichalcogena)canes (2), together with the radical cations (2·+) and dications (22+)

Figure 1. Observed structure of 1-(8-PhBEC10H6)AE−AE(C10H6BEPh8′)-1′ (I (AE, BE) = (Se, S)) (a)23 and approximate MO model for E4 σ(4c−6e) (b).

Figure 2 draws the structures of 1b2+ (2b22+: GUYRUO)77 and 1d2+ (2d22+: CAXKET),86 determined by the X-ray

nature of σ(4c−6e) constructed by the chalcogen atoms (AE, B E = S, Se, Te, and/or O).23−27 The nature of AE2BE2 σ(4c−6e) has been investigated for the neutral species in I (AE, BE = S, Se) and the related models.23 How is the nature of AE2BE2 σ(4c−6e) in the charged species? What are the similarities and differences between the behavior of AE2BE2 σ(4c−6e) in the charged species and those in the neutral ones? The purpose of this paper is to elucidate the nature of the BE···AE−AE···BE interactions of AE2BE2 σ(4c−6e) in the charged species, which enables us to understand the similarities and differences in the interactions between the neutral and charged spesies. The interactions of AE2BE2 σ(4c−7e) in the charged species will also be clarified. The research groups of Furukawa57−69 and Glass70−78 energetically studied the chemistry of radical cationic dimers (1·+ (22·+)) and dicationic dimers (12+ (222+)) of 1,5(dichalcogena)canes (2: AE(CH2CH2CH2)2BE: AE, BE = S, Se, Te, and/or O), together with radical cations and dications of 2 (2·+ and 22+, respectively). The transannular interactions between chalcogen atoms in 2·+ and 22+ play a crucial role in the chemistry of 2.79−83 The charged species 12+ (222+) are the excellent candidates for the formation of AE2BE2 σ(4c−6e) in 12+, together with 1·+ (22·+) for AE2BE2 σ(4c−7e) in 1·+. Chart 1 illustrates 1·+ (1a·+−1j·+; 2a2·+−2j2·+) and 12+ (1a2+−1j2+; 2a22+−2j22+), where (AE, BE) = a (S, S), b (Se, S), c (Te, S), d (Se, Se), e (Te, Se), f (Te, Te), g (O, O), h (S, O), i (Se, O), and j (Te, O), together with 2a−2j, 2a·+−2j·+, and 2a2+−2j2+. A E and BE are chosen to satisfy χ(AE) > χ(BE), where χ(AE) and χ(BE) stand for the electronegativity of AE and BE, respectively, proposed by Allured-Rockov.84,85 While 1·+ are expected to form in the reaction of 2·+ with neutral 2, 12+ will form in the reaction of 2·+ with 2·+ and/or the reaction of 22+ with neutral 2. The behavior of σ(4c−7e) in 1a·+−1j·+ and σ(4c−6e) in 1a2+−1j2+ are clarified, subsequent to AE2BE2 σ(4c−6e) in I (AE, B E = S, Se) and the related models.23 It is inevitable to examine the structural feature of 1a·+−1j·+ and 1a2+−1j2+, containing the MO representations, to clarify the whole picture of AE2BE2 σ(4c−7e) and A E 2 B E 2 σ(4c−6e) in the species. The B E··· A E− A E··· B E interactions will also be denoted by B ′E···A′E−AE···BE, if necessary.

Figure 2. Observed structures of 1b2+ (2b22+: GUYRUO)77 (a) and 1d2+ (2d22+: CAXKET)86 (b).

crystallographic analysis. Four atoms of Se2S2 in 1b2+ and Se4 in 1d2+ are demonstrated to align linearly. The linear alignments of four AE2BE2 atoms in 1a2+−1j2+ can be analyzed by the σ(4c−6e) model (see Figure 1), so can those in 1a·+− 1j·+ by the σ(4c−7e) model. While ψ4 in Figure 1b is vacant in σ(4c−6e), it is occupied by a single electron in σ(4c−7e). It is necessary to apply the unrestricted method to the odd electron system, such as 1a·+−1j·+, in the calculations. We recently reported the nature of the transannular AE···BE interactions in 2a−2j, 2a·+−2j·+, and 2a2+−2j2+,87 after examination of the behavior of AE−BE in the neutral and charged forms of HAE− BEH, MeAE− BEMe, and cyclo1,2-AE−BE(CH2)3.88 The transannular interactions between np(AE) and np(BE) in 2 can be described by σ(2c−4e). The orbital interactions between np(AE) and np(BE) of the σ(2c− 4e) type in 2 will occur effectively, if the two directions of np(AE) and np(BE) are in the same line (on a coplane). The requirements are almost satisfied in 2 and (much) better satisfied in 2·+ and 22+. The interactions in 2·+ and 22+ are characterized by σ(2c−3e) and σ(2c−2e), respectively. B

DOI: 10.1021/acs.jpca.7b00667 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A Figure 3 illustrates σ(2c−4e), σ(2c−3e), and σ(2c−2e) for 2, 2·+, and 22+, respectively. The transannular interaction between

interactions in the change of the distances. On the basis of the behavior of the perturbed structures, we proposed the concept of dynamic nature of interactions, although the nature is defined at the fully optimized structures. QTAIM-DFA is applied to typical chemical bonds and interactions, and rough criteria are established. The criteria will distinguish the chemical bonds and interactions under consideration from others. QTAIM-DFA will provide an excellent possibility to evaluate, classify, and understand weak to strong interactions in a unified form.122−129 QTAIM-DFA and the criteria are explained in the Supporting Information, employing Schemes S1 and S2, Figure S1, and eqs S1−S7. The basic concept of the QTAIM approach is also surveyed.116−121 Theoretical investigations on the phenomena arising from the extended hypervalent interactions are successively increasing. However, it is still of high importance to clarify the causality in the phenomena of the interactions, with physical necessity. Indeed, some structures of 12+ are determined so far, but the nature of AE2BE2 σ(4c−7e) in 1·+ and AE2BE2 σ(4c−6e) in 12+ are to be elucidated further, for the better understanding of the phenomena caused by the interactions. QTAIM-DFA is applied to the interactions of AE2BE2 σ(4c−7e) in 1a·+−1j·+ and A B E2 E2 σ(4c−6e) in 1a2+−1j2+, where BE···AE−AE···BE will be denoted by BE-*-AE-*-AE-*-BE, where asterisks emphasize the existence of BCPs on BPs. Herein, we present the results of the investigations on the nature of AE2BE2 σ(4c−7e) in 1a·+−1j·+ and AE2BE2 σ(4c−6e) in 1a2+−1j2+, together with the structural feature, containing the MO representations. The results are closely related to those for AEBE σ(2c−3e) in 2a·+−2j·+ and A B E E σ(2c−2e) 2a2+−2j2+, which we reported recently. The dynamic and static nature of the interactions will be discussed by employing the criteria, as a reference. Methodological Details in Calculations. Calculations are performed using the Gaussian 09 program package.130 Various basis-set sytems (BSS-A to BSS-F) were examined for the analysis of BE···AE−AE···BE in 1·+ and 12+. The results are summarized in Table S1 of the Supporting Information. Various levels (MP2, 131−133 M06-2X, 134 M06, 134 LCwPBE,135 CAM-B3LYP,136,137 and B3LYP138−140) were also examined. In BSS-A, the 6-311+G(3df) basis sets141−144 are employed for S, Se, and O, the basis set of the (7433111/ 743111/7411/2 + 1s1p1d1f) type is for Te, obtained from the Sapporo basis set factory,145,146 and the 6-311+G(d,p) basis sets are for C and H. Higher basis sets are employed for S, Se, Te, and O, since it is necessary to evaluate the nature of chalcogen−chalcogen interactions more accurately. The basis set for Te belongs to the triple-ζ system, as well as those for O, S, and Se. BSS-A is mainly employed at the M06-2X level in this work (M06-2X/BSS-A). Unrestricted method is applied to the odd electron systems. Structures were confirmed by the frequency analysis performed on the optimized structures with the same method. The CAM-B3LYP level is employed to confirm all positive frequencies for the optimized structures, when all positive frequencies are not obtained for the optimized structures at the M06-2X level. QTAIM functions were calculated using the Gaussian 09 program package130 with the same basis sets at the same level of the optimizations. The results were analyzed with the AIM2000 program.147 Normal coordinates of internal vibrations (NIV) obtained by the frequency analysis were employed to generate the perturbed structures.116−121 The method is explained in eq 1. A kth perturbed structure in question (Skw) was generated by the addition of the normal coordinates of the kth internal

Figure 3. σ(2c−4e), σ(2c−3e), and σ(2c−2e) interactions in 2, 2·+, and 22+, respectively. Electrons in HOMO of 2 are shown in blue, which will be removed in the one- or two-electron reduction of 2, while those in HOMO−1 are in red, which will not be removed in the reduction.

np(AE) and np(BE) in 2 is expected to form the σ(AE···BE) and σ*(AE···BE) orbitals, which are both filled with two electrons, as illustrated for 2 in Figure 3. The σ*(AE···BE) and σ(AE···BE) orbitals correspond to highest occupied molecular orbital (HOMO) and next one (HOMO−1), respectively. Therefore, one electron will be removed from σ*(AE···BE) of 2 in the oneelectron reduction, and two electrons will also be from σ*(AE···BE) of 2 in the two-electron reduction. 2·+ and 22+ are produced in the reduction processes, respectively (see Figure 3). The σ*(AE···BE) orbitals of 2 will yield lowest unoccupied molecular orbital (LUMO) and HOMO in 2·+, in the one-electron reduction of 2, since 2·+ is singly occupied and treated as the doublet. Similarly, σ*(AE···BE) and σ(AE···BE) of 2 will give LUMO and HOMO, respectively, in 22+, in the twoelectron reduction of 2, since MOs in 22+ are filled with two electrons and treated as the singlet. σ*(AE···BE) in 2·+ and 22+ will easily accept electrons in np(E) of 2, which is the driving force for the formation of AE2BE2 σ(4c−7e) in 1·+ (22·+) and A B E2 E2 σ(4c−6e) in 12+ (222+), respectively. The quantum theory of atoms-in-molecules (QTAIM) approach, introduced by Bader,89−98 enables us to analyze the nature of chemical bonds and interactions.28,99−114 The bond critical point (BCP)115 is an important concept in QTAIM, which is a point along the bond path (BP) at the interatomic surface where the charge density ρ(r) reaches a minimum. ρ(r) at BCP is denoted by ρb(rc) and so are other QTAIM functions. A lot of QTAIM investigations have been reported so far;28,99−114 however, those from a viewpoint of experimental chemists seem not so many. We searched for such method that enables experimental chemists to be able to analyze their own results, concerning chemical bonds and interactions, by their own images and, recently, proposed QTAIM dual functional analysis (QTAIM-DFA).116−121 In our treatment, data from the perturbed structures are employed, in addition to those of the fully optimized one, where the interaction distances in question in the perturbed structures are elongated or shortened suitably, relative to those in the fully optimized structures.116−121 Data from the perturbed structures provide information about the C

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Table 1. Structural Parameters around B′E···A′E−AE···BE for Radical Cationic (1a·+−1j·+) and Dicationic (1a2+−1j2+) Dimers, Together with the Symmetry, Optimized with BSS-A at the M06-2X Levela species (AE, BE) for B ′E···A′E−AE···BE

r0(AE, A′E) (Å)

Δr0(AE, A′E)b (Å)

1a·+ (S, S) 1b·+ (Se, S) 1c·+ (Te, S)c 1c·+ (Te, S)d 1d·+ (Se, Se) 1e·+ (Te, Se)

2.8908 3.0759 3.3552 3.4688 3.1213 3.3810

−0.71 −0.72 −0.76 −0.65 −0.68 −0.74

1f·+ (Te, Te) 1g·+ (O, O) 1h·+ (S, O) 1i·+ (Se, O) 1j·+ (Te, O)

3.4204 2.3745 2.6577e 2.7713 2.9868 3.3145

−0.70 −0.67 −0.38e −0.83 −0.81 −0.81

1a2+ (S, S) 1b2+ (Se, S) 1c2+ (Te, S)f 1c2+ (Te, S)d 1d2+ (Se, Se) 1e2+ (Te, Se) 1f2+ (Te, Te) 1g2+ (O, O)

2.4013 2.6260 2.9710 2.9876 2.6769 2.9989 3.0513 2.4583

−1.20 −1.17 −1.15 −1.13 −1.12 −1.12 −1.07 −0.58

1h2+ (S, O) 1i2+ (Se, O) 1j2+ (Te, O)

2.2135 2.4786 2.8858

−1.39 −1.32 −1.23

r0(AE, BE) (Å)

radical cationic dimers 3.1315 3.2262 3.3571 3.3615 3.2867 3.4036 3.4652e 3.5767 2.6303 153.66e 2.8753 2.9444 3.0035 dicationic dimers 2.7481 2.8178 2.9101 2.9310 2.9195 3.0321 3.2167 2.4892 2.4909e 2.4206 2.4706 2.5087

Δr0(AE, BE)b (Å)

∠A′EAEBE (deg)

ϕ(B′EA′EAEBE) (deg)

symmetry

−0.47 −0.47 −0.50 −0.50 −0.51 −0.56 −0.49e −0.54 −0.41

165.98 156.84 143.05 147.90 159.08 142.42 147.89e 146.92 159.52

180.00 180.00 180.00 180.00 180.00 −171.74

Ci Ci Ci C1 Ci C1

180.00 −84.10

Ci C1

−0.44 −0.48 −0.58

159.39 150.21 138.96

180.00 180.00 180.00

Ci Ci Ci

−0.85 −0.88 −0.95 −0.93 −0.88 −0.93 −0.90 −0.55 −0.55e −0.90 −0.95 −1.07

176.63 170.89 162.26 163.33 171.45 162.85 162.48 151.25 158.64e 175.17 168.46 160.50

180.00 180.00 180.00 180.00 180.00 180.00 180.00 −94.27

Ci Ci C1 C1 Ci Ci Ci C1

180.00 180.00 180.00

Ci Ci Ci

a

BSS-A; the 6-311+G(3df) basis set for O, S, and Se and that of the (7433111/743111/7411/2 + 1s1p1d1f) type for Te with the 6-311+G(d,p) basis set for C and H. See Table S1 of the Supporting Information for the BSSs. bΔr0(AE, A′E/BE: 1x*) = r0(AE, A′E/BE: 1x*) − ΣrvdW(AE, A′E/BE); x = a−j and * = ·+ or 2+. cTwo negative frequencies are predicted, of which motions correspond to the distortion between the components (1c·+ and 1c), but the effect on r0(AE, A′E/BE) seems very small. dOptimized at the CAM-B3LYP level. eValues corresponding to (A′E, B′E). fOne negative frequency is predicted, of which motion corresponds to the distortion between the components (1c·+ and 1c·+), but the effect on r0(AE, A′E/BE) seems very small.

in eq 3, where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)) (Rc2 > 0.999 99 is usual).116−121

vibration (Nk) to the standard orientation of a fully optimized structure (So) in the matrix representation.148 The coefficient f kw in eq 1 is determined to satisfy eq 2, for an interaction in question. The perturbed structures with NIV correspond to the structures in the zero-point internal vibrations, of which interaction distances in question are elongated or shortened, relative to the values given in eq 2, where r and r0 stand for the distances in the perturbed and fully optimized structures, respectively, with a0 of Bohr radius (0.529 18 Å).116−121 Nk of five digits are employed to predict Skw. Skw = So + fkw ·Nk

r = ro + wao

RESULTS AND DISCUSSION Basis Sets and Levels Employed for the Evaluations. QTAIM functions would be affected (much) from the interaction distances in question. Therefore, it is desirable to employ such methods that reproduce well the observed distances, if the predicted results are discussed in relation to the observed ones. The magnitudes of Δrco (= ro:calcd − robsd) less than 0.013 Å are desirable for QTAIM-DFA, which corresponds to half of the intervals of adjacent (data) points in the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 (0.05a0/2 = 0.013 Å). The magnitudes of Δrco less than 0.05a0 (0.026 Å) would be acceptable, although the counterions around the cationic species and/or some crystal packing effect may have an effect on the observed distances.82,83,90 The AE−AE and AE···BE distances in BE···AE−AE···BE predicted with various BSSs and levels, which are denoted by r0(AE, AE) and r0(AE, BE), respectively, were compared with the observed values, exemplified by 1b2+ ((AE, BE) = (Se, S)) and 1d2+ (Se, Se). The results are summarized in Tables S2−S4 of the Supporting Information. The r0(ASe, ASe) and r0(AS, BSe) values predicted for 1b2+ were 0.024 and 0.092 Å longer and

(1)

(w = 0, ±0.05, and ± 0.1; ao

= 0.529 18Å)

y = co + c1x + c 2x 2 + c3x 3 coefficient)

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(2)

(R c 2: square of correlation (3)

In the QTAIM-DFA treatment, Hb(rc) values are plotted versus Hb(rc) − Vb(rc)/2 for data of five (data) points of w = 0, ±0.05, and ±0.1 in eq 2. Each plot for an interaction is analyzed using a regression curve assuming the cubic function as shown D

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Figure 4. Plots of Δr0(AE, A′E) and Δr0(AE, BE) for 1a·+−1j·+ and 1a2+−1j2+, evaluated with M06-2X/BSS-A. Averaged values being employed for Δr0(AE, BE) of the C1 structures in the plot.

E−A′E and AE···BE (A′E···B′E) directions. We believe that 1c·+ (Te, S) and 1c2+ (Te, S) of the Ci symmetry are both energy minima, but their potential energy surfaces around AE−A′E and A E···BE would be too gradual to give all positive frequencies, if optimized with M06-2X/BSS-A. 1c·+ (Te, S) and 1c2+ (Te, S) gave all positive frequencies with CAM-B3LYP/BSS-A, retaining the Ci symmetry. The results support that the very gradual potential energy surfaces around B′E···A′E−AE···BE for the species, if evaluated with M06-2X/BSS-A. Table 1 collects the evaluated r0(AE, A′E) and r0(AE, BE) (r0(A′E, B′E)) values for 1a·+−1j·+ and 1a2+−1j2+, together with the Δr0(AE, A′E) and Δr0(AE, BE) (Δr0(A′E, B′E)) values, which are measured from the sum of vdW radii149 (Δr0(AE, A′E) = r0(AE, A′E) − ∑rvdW(AE, A′E), for an example). Table 1 contains the angles of ∠A′EAEBE and/or ∠AEA′EB′E and the torsional angles of ϕ( B ′ E A ′ E A E B E), together with the symmetries. The magnitudes of Δr0(AE, A′E) are larger than that of Δr0(AE, BE) for 1a·+−1j·+ and 1a2+−1j2+, and the magnitudes of Δr0(AE, A′E) and Δr0(AE, BE) in 1a2+−1j2+ are larger than the corresponding values in 1a·+−1j·+, respectively, except for Δr0(AE, A′E) of 1g2+ (O, O) versus that of 1g·+ (O, O). The reason is considered based on the MO theory (cf. Figure 1b). No attractive interactions between any adjacent atoms are expected for the hypothetical σ(4c−8e), where ψ1−ψ4 in σ(4c− 8e) are all filled with two electrons. In this case, four chalcogen atoms exist, as if they were four independent atoms, although two chalcogen atoms in 1a−1j are connected by the CH2CH2CH2 chains. While the inverse character of ψ4 will be brought to σ(4c−7e) in 1a·+−1j·+ by only one electron, such character will be appeared in σ(4c−6e) of 1a2+−1j2+ by two electrons. The antibonding character in ψ4 is actually larger for the inside AE−A′E than that for the outside AE···BE (A′E···B′E) in B′E···A′E−AE···BE, irrespective of the approximate MO model (see Figure 1b). Therefore, the larger magnitudes of Δr0(AE, A ′E), relative to Δr0(AE, BE), are reasonably explained by the A

shorter than the observed values, respectively, if evaluated at the MP2 level with BSS-A. Indeed, Δrco(ASe, ASe) of 0.024 Å seems acceptable, but Δrco(AS, BSe) of −0.092 Å would be too short. In the case of 1d2+, the predicted r0(ASe, ASe) and r0(ASe, B Se) values were 0.101 and 0.058 Å longer and shorted than the observed values, respectively, if evaluated with MP2/BSS-A. Consequently, on the one hand, the MP2 level would not be suitable for the purpose. On the other hand, the r0(ASe, ASe) and r0(ASe, BS) values predicted for 1b2+ were 0.021 and 0.012 Å shorter than the observed values, respectively, and the r0(ASe, A Se) and r0(ASe, BSe) values predicted for 1d2+ were 0.011 and 0.029 Å longer than the observed values, respectively, if calculated with M06-2X/BSS-A. The Δrco values predicted with M06-2X/BSS-A seem most widely acceptable in the optimizations of 1b2+ and 1d2+. The M06 and CAM-B3LYP levels seem difficult to be accepted, due to Δrco(ASe, ASe) of 0.046 Å and Δrco(ASe, BSe) of 0.043 Å for 1d2+ at the M06 level and Δrco(ASe, BSe) of 0.047 Å for 1d2+ at the CAM-B3LYP level (see Table S3 of the Supporting Information). Therefore, the structures of 1a·+−1j·+ and 1a2+−1j2+ are optimized with M06-2X/BSS-A, and the optimized structures are confirmed by the frequency analysis. The basis set dependence seems very small, if optimized at the M06-2X level, as shown in Table S4 of the Supporting Information. The results are in sharp contrast to the case of the MP2 level. The results strongly suggest that lower basis set systems can be applied to the (much) longer linear interactions of chalcogen atoms. Structural Feature in 1·+ and 12+. Species of 1a·+−1j·+ and 1a2+−1j2+ are optimized with M06-2X/BSS-A, retaining the Ci symmetry, except for 1e·+ (Te, Se), 1g·+ (O, O), and 1g2+ (O, O), which have the C1 symmetry. An imaginary frequency is predicted for each of 1c·+ (Te, S) and 1c2+ (Te, S), if optimized retaining the Ci symmetry. The motion of the imaginary frequency corresponds to the rotation around the Te−Te bond with the negligible stretching motion of the E

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Figure 5. Plot of relative energies (ΔE) in the formation of 1a·+−1j·+ and 1a2+−1j2+ from 2a−2j, 2a·+−2j·+, and/or 2a2+−2j2+, evaluated with M062X/BSS-A.

inverse character of ψ4. It is also well-explained for the larger magnitudes of Δr0(AE, A′E) and Δr0(AE, BE) in 1a2+−1j2+, relative to the corresponding values in 1a·+−1j·+, respectively. Figure 4 shows the plot of Δr0(AE, A′E) and Δr0(AE, BE) for 1a·+−1j·+ and 1a2+−1j2+, given in Table 1. While the ratios of Δr0(AE, A′E)/Δr0(AE, BE) in 1a·+−1j·+ decrease in the order of A E = BE = O (1.6) > S (1.5) > Se (1.33) ≥ Te (1.30), those in 1a2+−1j2+ do similarly in the order of AE = BE = S (1.4) > Se (1.3) > Te (1.2) > O (1.1). The behavior of the ratios for AE = B E = O is contrastive between 1g·+ and 1g2+. The ratios of Δr0(AE, A′E: 1a2+−1j2+)/Δr0(AE, A′E: 1a·+−1j·+) are 1.7−2.0, if the same (AE, A′E) are compared, except for Δr0(AE, A′E: 1g2+ (O, O))/Δr0(AE, A′E: 1g·+ (O, O)), for which the ratio is 0.9. In the case of Δr0(AE, BE: 1a2+−1j2+)/Δr0(AE, BE: 1a·+−1j·+), the ratios are 1.5−1.7, if the same (AE, BE) are compared, except for Δr0(AE, BE: 1g2+ (O, O))/Δr0(AE, BE: 1g·+ (O, O)), of which the ratios are 1.3−1.4. The trends can be wellexplained based on the character of ψ4 in the σ(4c−6e) and σ(4c−7e) models. Namely, the linear alignments of B ′E···A′E−AE···BE in 1a·+−1j·+ and 1a2+−1j2+ are well-analyzed by the σ(4c−6e) and σ(4c−7e) models, respectively. However, the character of O4 in 1g2+ seems very different from others. The ∠BEAEA′E values should also be considered to discuss the behavior of AE2BE2 σ(4c−7e) in 1a·+−1j·+ and AE2BE2 σ(4c−6e) in 1a2+−1j2+. The ∠BEAEA′E value larger than 150° is desirable for the linear alignment of AE2BE2 to be recognized as σ(4c−7e) or σ(4c−6e),6,14−16 although tentative. The values are less than 150° for 1c·+ (Te, S) (143°), 1e·+ (Te, Se) (145° in the average), 1f·+ (Te, Te) (147°), and 1j·+ (Te, O) (139°). They should be analyzed by the borderline mechanism between σ(4c−7e) and some models other than σ(4c−7e). Nevertheless, they are also tentatively analyzed as the extension of the σ(4c−7e) model in this paper, except for 1g·+ (O, O). They may correspond to the initial stage to form σ(4c−7e). The ∠BEAEA′E values are predicted to be larger than 150° for 1a2+− 1j2+. Therefore, the linear interactions in 1a2+−1j2+ are analyzed as σ(4c−6e), except for 1g2+ (O, O).

How are the relative energies in the formation of 1a·+−1j·+ and 1a2+−1j2+ from 2a−2j, 2a·+−2j·+, and/or 2a2+−2j2+? Figure 5 shows the plots of the relative energies, which correspond to the energy differences between the species specified in the figure (see Table S5 of the Supporting Information). While the formation of 2a·+−2j·+ from 2a−2j by the removal of an electron destabilizes the species by 6.4−7.6 eV with 8.6 eV for 2g·+ (O, O), that of 2a2+−2j2+ by the removal of two electrons destabilizes the species by 17.1−20.0 eV with 23.2 eV for 2g2+ (O, O). The magnitudes of the destabilizations are larger for the case from 2g (O, O) than those from others. While 1a·+− 1j·+ are more stable than (2a·+ + 2a)−(2j·+ + 2j), respectively, by 0.6−1.1 eV, with 0.4 eV for 1g·+, 1a2+−1j2+ are more stable than (2a2+ + 2a)−(2j2+ + 2j), respectively, by 2.7−4.0 eV, with 2.3 eV for 1g2+, as shown in Figure 5. The smaller stabilization energies in the formation of 1g·+ and 1g2+ may come from the less stable nature of the linear O4 interactions in the species. The magnitudes of the stabilization seem larger for the species with AE ≠ BE than the case of AE = BE, although slightly. As shown in Figure 5, 1a2+−1j2+ are predicted to be less stable than the corresponding two radicals, (2a·+ + 2a·+)−(2j·+ + 2j·+), respectively, by 1.1−1.9 eV with 3.6 eV for 1g2+. One may imagine that 1a2+−1j2+ would exist as the corresponding equivalent two radicals, at first glance. However, we must consider the differences between experimental and calculation conditions. Dications are surrounded by counterions and/or solvent molecules, under experimental conditions. The electrostatic interactions must also be very important to stabilize the ionic species. However, the species are calculated under the isolated conditions in vacuum. The ionic species were calculated using another model with M06-2X/BSS-A. Figure 6 shows the results of the calculations (see Table S6 of the Supporting Information). It is predicted that [1d·+•BF4−] is more stable than [2d·+•BF4−] + 2d by 0.48 eV and that [BF 4 − •1d 2+ •BF 4 − ] is more stable than [BF4−•2d2+•BF4−] + 2d by 1.37 eV, where [BF4−•1d2+•BF4−] is denoted by [1d 2+ •2BF 4 − ] in the figure. Similarly, F

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(5.75 eV, as in Figure 6) is also much smaller than that predicted for [2d2+ + 2BF4−] (18.00 eV, as shown in Figure 5, where BF4− is not shown). The reaction processes are similarly followed with MP2/BSSA, just to be sure in the energy basis. Figure 6 also draws the results with MP2/BSS-A, which supports well those with M062X/BSS-A. The smaller ΔE values are predicted at the MP2 level relative to the case of the M06-2X level, especially for [BF4−•1d2+•BF4−]. Survey of the BE···AE−AE···BE Interactions in 1·+ and 2+ 1 . How is the structural feature of 1·+ and 12+, containing the energy diagram and the character of MOs? What MOs of 1a·+− 1j·+ and 1a2+−1j2+ correspond to ψ1−ψ4 of E4 σ(4c−7e) and E4 σ(4c−6e), respectively? (See Figure 1b for E4 σ(4c−6e).) The B E···AE−AE···BE interactions in 1·+ and 12+ are discussed in relation to the character of the selected MOs and the energy diagram. Figure 7 draws the energy profile for the formation of σ(4c−7e) and σ(4c−6e), exemplified by 1d·+ (Se, Se) from 2d·+ (Se, Se) with 2d (Se, Se) and 1d2+ (Se, Se) from 2d2+ (Se, Se) with 2d (Se, Se), respectively, evaluated with M06-2X/BSSA. Figure 8 illustrates the corresponding MOs for σ(4c−7e) and σ(4c−6e), exemplified by 1d·+ (Se, Se) and 1d2+ (Se, Se), respectively. While ψ58 (HOMO) in 2d (Se, Se) splits into ψ58‑β (LUMO) and ψ58‑α (HOMO) in 2d·+ (Se, Se), ψ57 (HOMO−1) in 2d (Se, Se) does into ψ57‑β and ψ55‑α in 2d·+ (Se, Se), after one electron removal, as shown in Figure 7. MOs of σ(4c−7e) in 1d·+ (Se, Se) are formed through the interactions of MOs in 2d·+ (Se, Se) with those in 2d (Se, Se). Similarly, ψ58 (HOMO) in 2d (Se, Se) changes to ψ58 (LUMO) in 2d2+ (Se, Se), after the two-electron removal from 2d (Se, Se). However, the (np(SeA) + np(SeB)) character in ψ57 (HOMO−1) moves into ψ52 and ψ54 of 2d2+ (Se, Se), through the interactions of the original ψ57 (HOMO−1) of 2d2+ (Se, Se) with some MOs of the framework in 2d2+ (Se, Se), due to the energy lowering by

Figure 6. Effect of counterions in the formation of a radical cation and a dication, exemplified by 2d* (Se, Se) with BF4− (* = ·+ and 2+). The ΔE values are plotted, for the process evaluated with M06-2X/BSS-A and MP2/BSS-A, where the energies of the removed electrons are neglected.

[2d·+•BF4−] + [2d·+•BF4−] is evaluated to be less stable than [BF4−•1d2+•BF4−] by 1.50 eV. Consequently, the disproportionation of [BF4−•1d2+•BF4−] to 2[2d·+•BF4−] seems unlikely. The very large effect of counterions is demonstrated in this evaluation. While the ΔE value for the formation of [2d·+•BF4−] from [2d + BF4−] (2.94 eV, as shown in Figure 6) is much smaller than that predicted for [2d·+ + BF4−] (6.78 eV, as shown in Figure 5, where BF4− is not shown), although the contributions from the removed electrons are neglected. Similarly, the ΔE value in the formation of [BF4−•2d2+•BF4−]

Figure 7. Energy profile for some MOs of 1d•+ (Se, Se) formed from 2d•+ (Se, Se) and 2d (Se, Se) and those of 1d2+ (Se, Se) from 2d2+ (Se, Se) and 2d (Se, Se), evaluated with M06-2X/BSS-A. G

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Figure 8. MOs illustrated for σ(4c−7e) and σ(4c−6e), exemplified by 1d·+ (Se, Se) and 1d2+ (Se, Se), evaluated with M06-2X/BSS-A. The 0.020 au−3/2 isosurface is rendered for MOs.

Figure 9. Molecular graphs for 1·+ and 12+, exemplified by 1d·+ (Se, Se) (a) and 1d2+ (Se, Se) (b) evaluated with M06-2X/BSS-A. BCPs are denoted by red dots, ring critical points (RCPs) by yellow dots, cage critical points (CCPs) by green dots, and BPs by pink lines. Carbon atoms are in black, hydrogen atoms are in gray, and selenium atoms are in pink.

After clarification of the structural feature in 1·+ and 12+, containing the MO representations, in relation to the components, next extension is to elucidate the nature of B E···AE−AE···BE in 1·+ and 12+. The interactions will be analyzed by applying QTAIM-DFA. Molecular Graphs, Contour Plots, Negative Laplacians, and Trajectory Plots around BE-*-AE-*-AE-*-BE in 1·+ and 12+. Figure 9 shows the molecular graphs of 1·+ and 12+, exemplified by 1d·+ (Se, Se) and 1d2+ (Se, Se). All BCPs expected are detected, containing those between the Se···Se atoms. Figure 10 shows the contour plots of ρ(r) drawn on the

the double positive charges developed around the chalcogen atoms in 2d2+ (Se, Se). As shown in Figure 8, MOs of ψ 116 (LUMO), ψ115 (HOMO), ψ114, and ψ107 in 1d2+ (Se, Se) correspond rather clealy to ψ4−ψ1 of E4 σ(4c−6e), respectively, although ψ107 contains some character from the backbone of 2d2+ (Se, Se), maybe due to the substantial low energy of ψ107. In the case of σ(4c−7e) in 1d·+ (Se, Se), ψ116‑α (HOMO), ψ115‑α, ψ114‑α, and ψ111‑α can be chosen for the α-spin series of ψ4−ψ1 in σ(4c− 7e), while ψ116‑β (LUMO), ψ115‑β, ψ114‑β, and ψ113‑β seem to correspond to ψ4−ψ1 of σ(4c−7e) in the β-spin series (Figure 8). H

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Figure 10. Contour plots of ρ(r) for 1·+ and 12+ drawn on the planes, containing BE-*-AE-*-AE-*-BE, exemplified by 1d·+ (Se, Se) (a) and 1d2+ (Se, Se) (b), evaluated with M06-2X/BSS-A. BCPs on the plane are shown by red dots, those outside of the plane show in dark pink dots, RCPs on and outside the plane show by blue squares and light blue ones, respectively. CCPs are shown by green squares, and BPs on the plane by black lines and those outside of the plane are indicated by gray lines. Atoms on and outside the plane are by black dots and gray ones, respectively. The contours (eao−3) are at 2l (l = ± 8, ± 7, ..., 0) with 0.0047 (heavy line).

planes, containing ASe-*-ASe for 1d·+ (Se, Se) and 1d2+ (Se, Se). Fortunately, the planes contain whole interactions of BSe*-ASe-*-ASe-*-BSe. BCPs are placed at the (three-dimensional) saddle points of ρ(r) (see Figure 10). Negative Laplacians and trajectory plots of 1·+ and 12+ are illustrated in Figures S2 and S3 of the Supporting Information, respectively, exemplified by 1d·+ (Se, Se) and 1d2+ (Se, Se), similarly to the case of Figure 10. The behavior of BCPs is well-described in Figure S2, and the space around the species is reasonably divided into atoms of the species in Figure S3, for an example. The Se−Se stretching modes are depicted in Figure S4 of the Supporting Information, exemplified by 1d·+ (Se, Se) and 1d2+ (Se, Se). They are used when the perturbed structures around the fully optimized structures are generated. Interactions can be unambiguously described by BPs. BPs seem almost straight as shown in Figures 9 and 10. Namely, the B E-*-AE-*-AE-*-BE interactions in 1d·+ (Se, Se) and 1d2+ (Se, Se) can be approximated by the straight lines. To confirm the linearity of the interactions, the lengths of BPs (rBP) and the corresponding straight-line distances (RSL) for AE-*-BE and AE*-AE in 1a·+−1j·+ and 1a2+−1j2+ are collected in Table S4 of the Supporting Information, together with the differences between them (ΔrBP = rBP − RSL). Magnitudes of ΔrBP are less than 0.01 Å for AE-*-BE and less than 0.005 Å for AE-*-AE. Namely, A B E2 E2 σ(4c−7e) in 1a·+−1j·+ and AE2BE2 σ(4c−6e) in 1a2+− 1j2+ are well-described by the straight lines. The plot of rBP versus RSL gave an excellent correlation (rBP = 1.003RSL − 0.0062, Rc2 = 0.999 97), although it is not shown. Application of QTAIM-DFA to AE-*-BE and AE-*-AE in 1a·+−1j·+ and 1a2+−1j2+. Table 2 collects the values for the QTAIM functions of ρb(rc), Hb(rc) − Vb(rc)/2, Hb(rc), and kb(rc) (= Vb(rc)/Gb(rc)) for AE-*-BE and AE-*-AE at BCPs in 1a·+−1j·+ and 1a2+−1j2+. Figure 11 shows the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for the fully optimized data in Table 2, together with those of the perturbed structures around the fully optimized ones. The plots are analyzed according to eqs S3−S6 of the Supporting Information, which gives the QTAIM-DFA parameters of (R, θ) and (θp, κp). Table 2 also collects the values of (R, θ) and (θp, κp) for AE-*-BE and AE*-AE in 1a·+−1j·+ and 1a2+−1j2+, together with the frequencies (ν) and force constants (kf) corresponding to AE-*-BE and AE*-AE in question. Nature of AE-*-BE and AE-*-AE in 1a·+−1j·+ and 1a2+− 1j2+, Elucidated by (R, θ) and (θp, κp). The nature of AE-*-BE and AE-*-AE in 1a·+−1j·+ and 1a2+−1j2+ is examined on the basis of QTAIM-DFA parameters of (R, θ) and (θp, κp), employing the standard values as a reference (see the criteria

shown in Scheme S2 of the Supporting Information). Before detail discussion, it is instructive to survey the criteria briefly, closely related to those in this work. Interactions are called closed shell (CS) and shared shell (SS) interactions for 45° < θ < 180° (0 < Hb(rc) − Vb(rc)/2 = (ℏ2/8m)∇2ρb(rc)) and 180° < θ < 206.6° (0 > Hb(rc) − Vb(rc)/2 = (ℏ2/8m)∇2ρb(rc)), respectively. The CS interactions are subdivided into pure CS and regular CS for 45° < θ < 90° (0 < Hb(rc)) and 90° < θ < 180° (Hb(rc) < 0), respectively. The θp values play an important role to characterize the interactions in question. In the pure CS region of 45° < θ < 90°, the character of interactions will be the van der Waals (vdW) type for 45° < θp < 90°, whereas it has the nature of typical hydrogen bond (HB) with no covalency (t-HBnc) for 90° < θp < 125°, where θp of 125° is tentatively given corresponding to θ = 90°.51 The CT interactions will appear in the regular CS region of 90° < θ < 180°. The typical HB interactions with covalency (t-HBwc) appear in range of 125° < θp < 150° (90° < θ < 115°), where the point of (θ, θp) = (115°, 150°) is tentatively set up for the borderline between t-HBwc and molecular complex formation through charge transfer (CT-MC). The borderline between CT-MC and triginal bipyramidal adduct formation through charge transfer (CT-TBP) is defined by θp = 180°. Therefore, CT-MC and CT-TBP will appear in the ranges of 150° < θp < 180° (115° < θ < 150°) and 180° < θp < 190° (150° < θ < 180°), respectively. While θ = 150° and θp = 190° are given tentatively corresponding to θp = 180° and θ = 180°, respectively, where (θ, θp) = (180°, 190°) is the borderline between CT-TBP and covalent bonds (Cov) of the SS interactions. R contributes to subclassify Cov of the SS interactions. Classical chemical bonds of SS will be strong when R > 0.15 au (Cov-s); therefore, they are weak when R < 0.15 au (Cov-w). Table 3 summarizes the nature of AE-*-BE and AE-*-AE in 1a·+−1j·+ and 1a2+−1j2+, classified and characterized by the (R, θ, θp) values. The Cov-s interactions are not detected for the chalcogen−chalcogen interactions in this work, since R < 0.053 au (