Linear Multiselenium Interactions in Dicationic Oligomers of 1,5-(Diselena)canes: Behavior of Semc σ(mc c-ne e) (6≤mc ≤16) Elucidated with QTAIM Dual Functional Analysis.

The intrinsic dynamic and static nature mc center-ne electron interactions of the σ-type σ(mc c-ne e) were elucidated for the Se-Se interactions in dicationic oligomers of Se(CH2 CH2 CH2 )2 Se (1 (Se, Se)) [n2+ (Se, Se): n=1-8], especially for mc ≥6, where n2+ (Se, Se: n=1-8) are abbreviated by n2+ (n=1-8), respectively. QTAIM dual functional analysis (QTAIM-DFA) was applied to the interactions. Perturbed structures generated using coordinates derived from the compliance constants (Cii ) were employed for QTAIM-DFA. Each Se-*-Se in 12+ and 22+ has the nature of CT-TBP (trigonal bipyramidal adduct formation through CT) and Cov-w (weak covalent), respectively, which supply the starting points of the investigations. The asterisk emphasizes the existence of a bond critical point on the interaction. All Se-*-Se in 32+ are classified by the regular closed shell (r-CS) interactions and characterized as CT-MC (molecular complex formation through CT), which are denoted as r-CS/CT-MC, except for the central interaction, of which nature is r-CS/CT-TBP. Most interactions in 42+ -82+ are r-CS/t-HBwc (typical-HB with covalency) but some are pure-CS/t-HBnc (t-HB with no covalency). The linear Se2n 2+ interactions in 22+ -82+ seem close to those without any limitations, since the nature of Se-*-Se inside and outside of (CH2 CH2 CH2 )2 are very similar with each other. The linear Se2n 2+ interactions in 32+ -82+ are shown to be analyzed as σ(mc c-ne e: 6≤mc ≤16), not by the accumulated σ(3c-4e).

Introduction

We have proposed the concept of “extended hypervalent bonds/interactions of the σ‐type, σ(mc‐ne) (m center‐n electron interactions: m≥4; m

It is challenging to clarify the nature of σ(mc‐ne: m≥6). Each σ(mc‐ne: m≥6) consists of σ‐type linear MOs of ψ1−ψ, where ψ (1≤i≤m) has the nodal planes (n np) of i−1. Figure 1 illustrates the approximate MO model, exemplified by E10 σ(10c–18e). MOs in E10 σ(10c–18e) consist of ψ1−ψ10, which contain n np of 0–9, respectively. Dicationic oligomers of 1,5‐(dichalcogena)canes [E(CH2CH2CH2)2E’: 1 (E, E’), where E, E’=S, Se, and Te] seem an excellent candidate to supply σ(mc‐ne: m≥6; m

Figure 1

Approximate MO model for E10 σ(10c–18e). Colors correspond to the relative signs of AOs.

Scheme 1

Dicationic oligomers 2+ (E, E’: n=1–8) of 1,5‐(dichalcogena)canes [1 (E, E’)], together with 1 (E, E’) and the radical cations 1 .+ (E, E’), where * (Se, Se: n=1–8; =null, ⋅+, and 2+) are abbreviated by n, respectively.

Scheme 2

Notation of E and E–E (E=Se) in 2+, exemplified by 7 2+ for n of odd (a) and 8 2+ for n of even (b). (Numbering starting from the central positions.)

We have reported the nature of the chalcogen‐chalcogen interactions in 2 .+ (E, E’) and 2 2+ (E, E’), together with that in 1 (E, E’), 1 .+ (E, E’), and 1 2+ (E, E’). The radical cationic dimers 2 .+ (E, E’) and dicationic dimers 2 2+ (E, E’) are shown to be stable, and the interactions are well clarified with the QTAIM approach. Are the higher dicationic oligomers of 1 [ 2+ (E, E’: 3≤n≤8)] stable? What is the nature of the chalcogen‐chalcogen interactions in 2+ (E, E’: 3≤n≤8)? The nature of the chalcogen‐chalcogen interactions in 2+ (E, E’: 3≤n≤8) is to be elucidated, with the structural feature and the stability.

QTAIM dual functional analysis (QTAIM‐DFA), which we proposed based on the QTAIM approach introduced by Bader, is employed to elucidate the nature. The charge density (ρ()) at a bond critical point (BCP, * ) on the bond path (BP) is denoted by ρ b( c) in this paper, as are other QTAIM functions such as the total electron energy densities H b( c), potential energy densities V b( c), and kinetic energy densities G b( c) at BCPs. H b( c) are plotted versus H b( c)−V b( c)/2 (=(ħ 2/8 m)∇2 ρ b( c); see eq (SA2) in the Supporting Information) in QTAIM‐DFA. In our treatment, data from the perturbed structures around the fully optimized structures are employed, in addition to those from the fully optimized ones. Data from the fully optimized structures in the plots are analyzed using the polar coordinate (R, θ) representation, which corresponds to the static nature of the interactions. Each interaction plot for the data from both the perturbed and fully optimized structures is expressed by (θ p, κ p), where θ p corresponds to the tangent line and κ p is the curvature of the plot. θ and θ p are measured from the y‐axis and the y‐direction, respectively. (See also Figure 7 for the definition of QTAIM‐DFA parameters of (R, θ) and (θ p, κ p), drawn exemplified by 1Se‐*‐2Se of 5 2+.) The concept of the dynamic nature of interactions has been proposed based on (θ p, κ p).

Figure 7

Plots of H b( c) versus H b( c)−V b( c)/2 for Se‐*‐Se in 5 2+.

The perturbed structures necessary for QTAIM‐DFA are generated with CIV (QTAIM‐DFA with CIV), which we proposed recently. The coordinates corresponding to the compliance constants C for the internal vibrations are employed in CIV.[ , , , ] CIV is shown to be a highly reliable method to generate the perturbed structures. The dynamic nature of interactions based on the perturbed structures with CIV is described as the “intrinsic dynamic nature of interactions” since the coordinates are invariant to the choice of coordinate system. QTAIM‐DFA with CIV is applied to standard interactions and rough criteria that distinguish the interaction in question from others that are obtained. QTAIM‐DFA and the criteria are explained in the Appendix of the Supporting Information using Schemes SA1–SA3, Figures SA1 and SA2, Table SA1, and eqs (SA1)–(SA7). The basic concept of the QTAIM approach is also explained.

Theoretical investigations on the phenomena arising from σ(mc‐ne: m≥4) seem successively increasing.[ , , , , ] However, it is still of high importance to clarify the causality in the phenomena from which the interactions arise, with physical necessity. Indeed, the knowledge of the behavior of σ(4c–6e) and σ(5c–6e) has increased, but the nature of σ(mc‐ne: m≥6) seems still to be in the dark, while the alignments of multichalcogen atoms are often observed in crystals. We elucidated the intrinsic dynamic and static nature of each Se‐*‐Se in σ(mc‐ne: 6≤m≤16) in 2+ (3≤n≤8), by QTAIM‐DFA with CIV. Each Se‐*‐Se in 2+ (3≤n≤8) is classified and characterized employing the criteria as a reference. The nature of each Se‐*‐Se in σ(mc‐ne: 6≤m≤16) of 2+ (3≤n≤8) is discussed in a unified form, together with those for 1 2+ and 2 2+, as the starting points for 2+ (3≤n≤8). The structural feature and the stability are also discussed.

Computational Methods

Calculations were performed employing the Gaussian 09 program package. The 6–311G(3d) basis set was employed for Se with the 6–311G(d) basis set for C and H at the DFT level of M06‐2X. The basis set system is called BSS‐A (M06‐2X/BSS‐A), in this paper. To examine the basis set and level dependence on the nature, MP2 /BSS‐B was also applied to 1 and 1 2+ and M06‐2X/BSS‐B to 2 2+, where the 6–311+G(3df) basis set was employed for Se with the 6‐311+G(d,p) basis set for C and H in BSS‐B. The optimized structures were confirmed by the frequency analysis. The results of the frequency analysis are used to obtain the compliance constants (C) and the coordinates corresponding to C (). NBO analysis was applied under M06‐2X/BSS‐A.

The method to generate the perturbed structures with CIV is explained in eq (1). The i th perturbed structure in question (S ) is generated by the addition of the i th coordinates derived from C (C ) to the standard orientation of a fully optimized structure (S o) in the matrix representation. The coefficient g in eq (1) controls the structural difference between and S o: g is determined to satisfy eq (2) 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=0.52918 Å (Bohr radius). The values of five digits are used to predict .

QTAIM functions were calculated using the same basis set system as in the optimizations, unless otherwise noted, and were analyzed with the AIM2000 and AIMAll programs. The H b( c) values are plotted versus H b( c)−V b( c)/2 for five data points of w=0, ±0.025, and ±0.05 in eq (2) in QTAIM‐DFA. Each plot is analyzed using a regression curve of the cubic function, shown in eq (3), where (x, y)=(H b( c)−V b( c)/2, H b( c)) (R 2>0.99999 in usual).

Results and Discussion

Structural Feature of n 2+ (n=1–8: E=E’=Se)

The structures of 2+ (n=3–8) were optimized with M06‐2X/BSS‐A, retaining the C 2 or higher symmetry, together with 1, 1 2+ and 2 2+. The optimized structures are not shown in figures but some of them can be found in molecular graphs drawn on the optimized structures (see Figure 6 and Figure S1 of the Supporting Information). The Se‐Se distances in the optimized structures of 2+ (n=1–8) are collected in Table S1 of the Supporting Information. Charges developed on the Se atoms (Qn(Se)) are calculated for the optimized structures of 2+ (n=1–8), employing the natural population analysis (NPA). The Qn(Se) values are summarized in Table S2 of the Supporting Information. Energies for the formation of 2+ (n=2–8) from the components (1 2++(n−1) ⋅ 1) [ΔE( 2+)=E( 2+)−(E(1)+(n−1)E(1))] are also calculated. The ΔE( 2+) values are given in Table S3 of the Supporting Information.

Figure 6

Molecular graphs with contour maps of ρ b( c) drawn on the, where all selenium atoms are located, plane for 5 2+ (a) and 8 2+ (b), calculated with M06‐2X/BSS‐A.

What is the behavior of r(Se, Se) in 3 2+–8 2+? Figure 2 shows the plot of each r(Se, Se) values for 3 2+–8 2+, together with 1, 1 2+ and 2 2+, collected in Table S1 of the Supporting Information. While all r(Se, Se) values in 3 2+–8 2+ and 2 2+ are longer than the value of 1 2+, they are shorter than the value of 1, except for r(1Se, 1’Se) in 6 2+, r(1Se, 2Se) in 7 2+, and r(1Se, 1’Se) and r(2Se, 3Se) in 8 2+. The central r(1Se, 1’Se) distances become longer in the order of (1 2+<2 2+<) 3 2+<4 2+<5 2+<7 2+ (<1)<6 2+<8 2+. The distance becomes longer if it goes more outside in 2 2+–4 2+, whereas the zig‐zag type behavior is predicted for r(Se, Se) in 5 2+–8 2+. While r(3Se, 4Se) is longest for 5 2+, r(1Se, 2Se) is longest for 7 2+ and the r(1Se, 1’Se) values are longest for 6 2+ and 8 2+. The behavior for 5 2+ and 7 2+ seems intermediate between the two groups. In 6 2+–8 2+, some r(Se, Se) distances at approximately the central positions are longer than that of 1, while the distances of five most outside positions in 6 2+–8 2+ are shorter than that of 1. These results may suggest that the linear Se‐Se interactions in 6 2+–8 2+ can be analyzed separated by the three parts, two outside parts and a remaining central one. The behavior of the central part seems very complex for 6 2+–8 2+.

Figure 2

Plots of each r(Se, Se) for 3 2+–8 2+, together with 1, 1 2+, and 2 2+.

What factors operate to stabilize the Se‐Se interactions in 3 2+–8 2+, together with 1, 1 2+ and 2 2+? Two electron removal from 1 forms the stable Se2 σ(2c–2e) of 1 2+, as discussed above. The positive charges developed at the Se atoms calculated with NPA (Qn(Se)) are analyzed, next. Figure 3 shows the plot of Qn(Se) for 3 2+–8 2+, together with 1, 1 2+ and 2 2+, which are collected in Table S2 of the Supporting Information (see Scheme 2 for the notation of the Se atoms). All Qn(Se) values for 2 2+–8 2+ are larger than the value of 1 but smaller than that of 1 2+. Qn(Se) shows a characteristic behavior depending on the lengths of the alignments, n in 2+. The values are largest at the central Se atoms and become smaller if the Se atom goes to more outside positions from the central position(s) for 2 2+–5 2+, although the trend is not so sharp around the central positions in 5 2+. However, Qn(Se) values are smallest at the central position, and the values become larger, then reach to maximum and then decrease again, as the Se atom goes to more outside positions from the central position(s) for 6 2+–8 2+. The results may also suggest that it is better to analyze the interactions separately as the three parts for 6 2+–8 2+. Nevertheless, the energy lowering effect in the formation of 6 2+–8 2+ seems not so different from the effect in the formation of 3 2+–5 2+, if the effect per component is compared, as a whole (see, Figure 5).

Figure 3

Plots of each Qn(Se) for 3 2+–8 2+, together with 1, 1 2+, and 2 2+.

Figure 5

Energies for the formation of 2+ from [1 2++(n−1) ⋅ 1] (n=3–8 and 2), evaluated with M06‐2X/BSS‐A.

How are the Se‐Se distances correlated to Qn(Se)? The Se‐Se interaction seems stronger, as the Se atoms at both sides of the interaction are more positively charged. Namely, the r(Se, Se) values for Se‐Se are expected to be linearly correlated to (Qn(Se)+Qn(Se)). The r(Se, Se) values in Table S1 of the Supporting Information are plotted versus (Qn(Se)+Qn(Se)) in Table S2 of the Supporting Information, evaluated with M06‐2X/BSS‐A. Figure 4 shows the plot, which gives very good correlations if it is analyzed as two correlations. The two groups (G(B) and G(R)) are shown by black circles and red triangles, respectively, in the plot. They give very good correlations (y=−1.182x+4.268 (R c 2=0.992) for G(B) and y=−1.776x+4.876 (R c 2=0.995) for G(R)), although the data points for (1Se, 1’Se) of 1 2+ and 2 2+ are omitted from the correlations, which are shown in blue. The colors in the plot of Figure 4 are the same as those for the r(Se, Se) values in Table S1 of the Supporting Information, respectively. Data from (Se, Se) belong to G(B) if (Se, Se) are contained in the cyclic Se(CH2CH2CH2)2 Se system, except for those in 1 2+.

Figure 4

Plots of r(Se, Se) versus (Qn(Se)+Qn(Se)) in 3 2+–8 2+, evaluated with M06‐2X/BSS‐A. Data from G(B) are shown by black circles for the inside Se‐Se interactions in Se(CH2CH2CH2)2 Se. Data from G(R) are shown by red triangles if (Se, Se) belong to the outside ones. Data from 1 2+ and 2 2+ are also shown by the hollow blue cycle and triangle, respectively.

However, data from (Se, Se) form G(R) if (Se, Se) belong to the adjacent two cyclic Se(CH2CH2CH2)2Se systems, except for those in 2 2+. The results show that r(Se, Se) becomes shortened proportionally to the increase in Qn(Se)+ Qn(Se). Namely, the Se‐Se interaction becomes stronger as Qn(Se)+Qn(Se) increases, irrespective of the increase of the electrostatic repulsion between Se and Se, which must weaken the interaction. Data for 1 2+ and 2 2+ slightly deviate from the correlations, maybe due to the highly strong nature of the 1Se‐1’Se interactions. The results should correlate well with the ΔE values discussed below.

What is the behavior of the relative energies ΔE in the formation of 2+ (n=3–8) from the components? Figure 5 shows the plot of ΔE ( 2+) versus n of 2+ (n=3–8), evaluated with M06‐2X/BSS‐A, together with ΔE (2 2+). The ΔE value of 3 2+ is −3.60 eV, the magnitude of which is larger than that of 2 2+ by 0.79 eV. While 2 2+ is stabilized by only one molecules of 1, 1 2+ in 3 2+ is stabilized by two molecules of 1. However, the magnitude of ΔE per the component in 3 2+ (−1.20 eV=−3.60 eV/3) is smaller than that of Δ2 2+ (−1.41 eV=−2.81 eV/2). The magnitudes of ΔΔE ( 2+) [=E( 2+)−ΔE(( 1)2+)] increase almost constantly by approximately 0.38 eV, when they go from 3 2+ to 8 2+. The results show that the dicationic trimmer 3 2+ is well stabilized and so are the higher oligomers, 4 2+–8 2+.

After clarification of the structural feature of 1 2+–8 2+, the next extension is to elucidate the nature of the Se‐*‐Se interactions, by applying QTAIM‐DFA.

Molecular Graphs with Contour Plots around Se‐*‐Se

Figure 6 shows the molecular graphs with contour plots of 5 2+ and 8 2+, for example. All BCPs expected are detected, including the CPs between the Se‐Se atoms and those for very weak interactions in the components and those between them. BCPs between Se atoms appear at the (three‐dimensional) saddle points of ρ(). The BCPs for 2 2+–7 2+, other than 5 2+, are drawn in Figure S1 of the Supporting Information. All BCPs expected for the Se‐Se interactions are detected.

Survey of Se‐*‐Se in n 2+ (n=1–8)

BPs, corresponding to Se‐*‐Se, seem straight, as shown in Figure 6 and Figure S1 of the Supporting Information. To examine the linearity of the BPs further, the lengths of the BPs (r BP) in question are calculated for all Se‐*‐Se of 2+ (n=1–8), together with the corresponding straight‐line distances (R SL). The values are collected in Table S1 of the Supporting Information, together with the differences between them (Δr BP=r BP−R SL).

The Δr BP values are less than 0.04 Å. As a result, BPs corresponding to all Se‐*‐Se of 2+ (n=1–8) can be approximated as the straight lines (see also Figure S2 of the Supporting Information).

QTAIM functions are calculated for 1 2+–8 2+ at BCPs on Se‐*‐Se with M06‐2X/BSS‐A. Table 1 collects the ρ b( c), H b( c)−V b( c)/2, and H b( c) values. Figure 7 shows the plots of H b( c) versus H b( c)−V b( c)/2 for each Se‐*‐Se, exemplified by 5 2+. (See Figure S3 of the Supporting Information for 6 2+ and 7 2+.) All data for the optimized structures appear in the regular CS region, showing the CT nature of the interactions. The plots are analyzed according to eqs (SA3)–(SA6) of the Supporting Information. Table 1 collects the QTAIM‐DFA parameters of (R, θ) and (θ p, κ p) for each Se‐*‐Se of 3 2+–8 2+, 1, 1 2+ and 2 2+, together with the C values corresponding to the interactions in question. The (θ p, κ p) values, evaluated with CIV, should be denoted by (θ p:CIV, κ p:CIV), respectively. However, (θ p, κ p) will be used in place of (θ p:CIV, κ p:CIV) to simplify the notation.

Table 1

QTAIM Functions and QTAIM‐DFA Parameters Evaluated for the Dicationic Oligomers of Cyclo‐1,5‐Se(CH2CH2CH2)2Se (1), 3 2+–8 2+, 1 2+, and 2 2+, together with 1, Employing the Perturbed Structures Generated with CIV.[a,b]

Species	ρ b(r c)	c∇2 ρ b(r c)[c]	H b(r c)	R [d]	θ [e]	Cii [f]	θ p [g]	κ p [h]	Predicted
uSe‐*‐vSe	[ea o −3]	[au]	[au]	[au]	[°]	[Å mdyn−1]	[°]	[au−1]	Nature
3 2+
1Se‐*‐1’Se	0.0489	0.0036	−0.0099	0.0105	159.7	2.08	183.4	28.3	r‐CS/CT‐TBP
1Se‐*‐2Se	0.0379	0.0043	−0.0053	0.0069	141.0	4.41	177.1	60.8	r‐CS/CT‐MC
2Se‐*‐3Se	0.0220	0.0047	−0.0010	0.0048	101.9	4.48	151.9	203	r‐CS/CT‐MC
4 2+
1Se‐*‐1’Se	0.0380	0.0042	−0.0054	0.0068	142.0	5.46	178.8	32.2	r‐CS/CT‐MC
1Se‐*‐2Se	0.0342	0.0046	−0.0044	0.0064	133.5	3.35	175.5	48.9	r‐CS/CT‐MC
2Se‐*‐3Se	0.0218	0.0040	−0.0010	0.0042	103.5	10.69	153.6	181	r‐CS/CT‐MC
3Se‐*‐4Se	0.0163	0.0042	0.0000	0.0042	90.0	4.40	128.4	396	p‐CS/t‐HBnc
5 2+
1Se‐*‐1’Se	0.0262	0.0047	−0.0021	0.0051	114.1	4.86	170.2	85.3	r‐CS/CT‐MC
1Se‐*‐2Se	0.0231	0.0041	−0.0012	0.0042	106.8	15.96	161.2	155	r‐CS/CT‐MC
2Se‐*‐3Se	0.0235	0.0046	−0.0014	0.0048	106.7	9.37	166.8	133	r‐CS/CT‐MC
3Se‐*‐4Se	0.0172	0.0037	−0.0002	0.0037	92.7	11.94	140.4	457	r‐CS/t‐HBwc
4Se‐*‐5Se	0.0168	0.0043	0.0000	0.0043	90.5	5.09	130.5	400	r‐CS/t‐HBwc
6 2+
1Se‐*‐1’Se	0.0120	0.0030	0.0004	0.0031	82.6	16.45	107.7	373	p‐CS/t‐HBnc
1Se‐*‐2Se	0.0164	0.0042	0.0000	0.0042	90.2	4.19	131.6	475	r‐CS/t‐HBwc
2Se‐*‐3Se	0.0156	0.0035	0.0001	0.0035	89.2	8.68	129.1	538	p‐CS/t‐HBnc
3Se‐*‐4Se	0.0217	0.0046	−0.0010	0.0047	102.1	5.99	155.8	196	r‐CS/CT‐MC
4Se‐*‐5Se	0.0187	0.0038	−0.0004	0.0038	95.8	11.56	144.1	321	r‐CS/t‐HBwc
5Se‐*‐6Se	0.0176	0.0043	−0.0002	0.0043	92.4	5.01	135.2	332	r‐CS/t‐HBwc
7 2+
1Se‐*‐1’Se	0.0149	0.0041	0.0002	0.0041	87.1	4.96	121.1	461	p‐CS/t‐HBnc
1Se‐*‐2Se	0.0107	0.0028	0.0005	0.0028	80.5	23.12	101.2	429	p‐CS/t‐HBnc
2Se‐*‐3Se	0.0164	0.0043	0.0000	0.0043	90.1	4.58	131.6	468	r‐CS/t‐HBwc
3Se‐*‐4Se	0.0150	0.0035	0.0001	0.0035	87.9	9.87	124.9	519	p‐CS/t‐HBnc
4Se‐*‐5Se	0.0201	0.0045	−0.0006	0.0046	98.2	5.12	149.7	256	r‐CS/t‐HBwc
5Se‐*‐6Se	0.0170	0.0037	−0.0001	0.0037	92.0	8.83	135.7	454	r‐CS/t‐HBwc
6Se‐*‐7Se	0.0174	0.0044	−0.0001	0.0044	91.6	4.45	132.9	354	r‐CS/t‐HBwc
8 2+
1Se‐*‐1’Se	0.0102	0.0027	0.0005	0.0028	79.6	24.16	98.2	395	p‐CS/t‐HBnc
1Se‐*‐2Se	0.0148	0.0041	0.0002	0.0041	87.0	3.98	120.4	443	p‐CS/t‐HBnc
2Se‐*‐3Se	0.0113	0.0029	0.0004	0.0030	81.4	19.50	104.1	355	p‐CS/t‐HBnc
3Se‐*‐4Se	0.0162	0.0042	0.0000	0.0042	89.7	4.06	129.6	459	p‐CS/t‐HBnc
4Se‐*‐5Se	0.0145	0.0034	0.0002	0.0034	87.0	11.53	121.7	455	p‐CS/t‐HBnc
5Se‐*‐6Se	0.0187	0.0044	−0.0004	0.0044	95.2	5.20	143.0	311	r‐CS/t‐HBwc
6Se‐*‐7Se	0.0156	0.0036	0.0001	0.0036	89.0	7.47	126.6	477	p‐CS/t‐HBnc
7Se‐*‐8Se	0.0163	0.0042	0.0000	0.0042	89.6	4.10	126.9	383	p‐CS/t‐HBnc
1 2+
1Se‐*‐1’Se	0.0964	‐0.0051	−0.0386	0.0390	187.5	0.56	191.8	0.9	SS/Cov‐w
2 2+
1Se‐*‐1’Se	0.0642	0.0015	−0.0170	0.0171	175.1	1.49	190.2	7.5	r‐CS/CT‐TBP[i]
1Se‐*‐2Se	0.0411	0.0048	−0.0067	0.0082	144.2	2.18	177.7	42.4	r‐CS/CT‐MC
1
1Se‐*‐1’Se	0.0114	0.0035	0.0006	0.0035	81.0	4.32	100.6	166	p‐CS/t‐HBnc

[a] Calculated with M06‐2X/BSS‐A. [b] Data are given at BCPs. [c] c∇2 ρ b( c)=H b( c)−V b( c)/2, where c=ħ 2/8 m. [d] R=(x 2+y 2)1/2, where (x, y)=(H b( c)−V b( c)/2, H b( c)). [e] θ=90°−tan−1 (y/x). [f] Defined in eq (R1) of the references. [g] θ p=90°−tan−1 (dy/dx). [h] κ p=|d2 y/dx 2|/[1+(dy/dx)2]3/2. [i] The borderline between the r‐CS/CT‐TBP nature and the SS/Cov‐w nature.

ρ b( c)

c∇2 ρ b( c)[c]

H b( c)

C [f]

Se‐*‐Se

[°]

[°]

Nature of each Se‐*‐Se in n 2+ (n=1–8)

The central 1Se‐*‐1’Se interaction is strongest in 3 2+–5 2+ and 2 2+, among the interactions in the same species. The strength of 1Se‐*‐1’Se becomes weaker in the order of (1 2+>2 2+>) 3 2+>4 2+>5 2+. The Se‐*‐Se interaction in the same species becomes weaker, if it goes from the central position to a more outside position, namely, 1Se‐*‐1’Se>1Se‐*‐2Se>2Se‐*‐3Se>3Se‐*‐4Se>4Se‐*‐5Se for 3 2+–5 2+ and/or 2 2+, although the order changes as in 2Se‐*‐3Se>1Se‐*‐2Se for 5 2+. However, 1Se‐*‐1’Se is weakest for 6 2 and 8 2+, while 1Se‐*‐2Se is weaker than 2Se‐*‐3Se for 7 2+. The strength of Se‐*‐Se shows ripple‐like changes from the center to the outside for 6 2+–8 2+. The behavior of Se‐*‐Se of 5 2+ seems intermediate between the two groups of 3 2+–4 2+ and 6 2+–8 2+. The 1Se‐*‐1’Se interaction is strongest in 5 2+ but the strength of Se‐*‐Se waves slightly from the central position to a more outside position. The behavior of Se‐*‐Se detected by the QTAIM parameters should be closely related to the behavior of r(Se, Se), although there seem to be some differences in the magnitudes (see Figure 2).

Before a detailed discussion of the nature, it is instructive to survey the criteria shown in Scheme SA3 and Table SA1 of the Supporting Information. While θ classifies the interactions, θ p characterizes them. The criteria tell us that 45°<θ<180° (00) for the pure CS interactions (p‐CS) and 90°<θ<180° (H b( c)<0) for the regular CS interactions (r‐CS). In the p‐CS region of 45°<θ<90°, the character of the interactions will be the vdW type for 45°<θ p<90° (45°<θ<75°), whereas the character of the interactions will be the typical hydrogen bond type (t‐HB) with no covalency (t‐HBnc) for 90°<θ p<125° (75°<θ<90°), where θ=75° and θ p=125° are tentatively given for θ p=90° and θ=90°, respectively. The CT interaction will appear in the r‐CS region of 90°<θ<180°. The t‐HB interactions with covalency (t‐HBwc) appear in the range of 125°<θ p<150° (90°<θ<115°), where (θ, θ p)=(115°, 150°) is tentatively given as the borderline between the t‐HBwc and CT‐MC (molecular complex formation through CT) nature. The borderline between CT‐MC and CT‐TBP (TBP adduct formation through CT) types of the interactions is defined by θ p=180° (θ=150°), where θ=150° is tentatively given, corresponding to θ p=180°. The borderline between CT‐TBP and Cov‐w (weak covalent bonds) is defined by θ=180° (θ p=190°), where θ p=190° is tentatively given, corresponding to θ=180°. As a result, the (θ, θ p) values of (75°, 90°), (90°, 125°), (115°, 150°), (150°, 180°), and (180°, 190°) correspond to the borderlines between the nature of interactions for vdW/t‐HBnc, t‐HBnc/t‐HBwc, t‐HBwc/CT‐MC, CT‐MC/CT‐TBP, and CT‐TBP/Cov‐w, respectively. The parameters, described in bold, are superior to those tentatively given parameters in the classification and/or characterization of interactions. The strong covalent bonds (Cov‐s) of SS (180°<θ) are not detected in this work, since R in Table 1 are smaller than 0.05 au (<0.15 au), where the borderline between Cov‐w and Cov‐s is defined by R=0.15 au. Consequently, the nature of each Se‐*‐Se in Table 1 can be classified and characterized based on the (θ, θ p) values.

Each Se‐*‐Se interaction of 3 2+–8 2+ in Table 1 is now classified and characterized based on the (θ, θ p) values, evaluated with M06‐2X/BSS‐A, employing the QTAIM‐DFA parameters of the standard interactions as a reference. Before discussion of Se‐*‐Se in 3 2+–8 2+, the nature of Se‐*‐Se in 1, 1 2+ and 2 2+, are surveyed, as the starting points. The (θ, θ p) values for 1Se‐*‐1’Se in 1 are (81.0°, 100.6°). Therefore, the interaction is classified by the p‐CS interaction and characterized to have the t‐HBnc nature, which is denoted by p‐CS/t‐HBnc. However, the (θ, θ p) values are (187.5°, 191.8°) for 1Se‐*‐1’Se in 1 2+; therefore, it is classified by the SS interaction and characterized as the Cov‐w nature (SS/Cov‐w). The results show that the weak σ(2c–4e) interaction of p‐CS/t‐HBnc for 1Se‐*‐1’Se in 1 becomes the strong σ(2c–2e) interaction of SS/Cov‐w in 1 2+, through the removal of two electrons from the σ*(1Se‐*‐1’Se) orbital. The 1Se‐*‐1’Se, 1Se‐*‐2Se, and 1’Se‐*‐2’Se interactions in 2 2+ construct σ(4c–6e). The (θ, θ p) values for 1Se‐*‐1’Se and 1Se‐*‐2Se are (175.1°, 190.2°) and (144.2°, 177.7°), respectively. Therefore, the 1Se‐*‐1’Se and 1Se‐*‐2Se interactions are predicted to have the r‐CS/CT‐TBP and r‐CS/CT‐MC natures, respectively.

The QTAIM‐DFA parameters for 1Se‐*‐1’Se in 1 and 1 2+ evaluated with MP2/BSS‐B and for 1Se‐*‐1’Se and 1Se‐*‐2Se in 2 2+ evaluated with M06‐2X/BSS‐B are collected in Table S4 of the Supporting Information, together with the QTAIM functions and the predicted nature. The predicted nature for each interaction with M06‐2X/BSS‐A is the same as that corresponding interaction in 1 and 1 2+ with MP2/BSS‐B and in 2 2+ with M06‐2X/BSS‐B, although there are some differences in the calculated parameters. Consequently, the small differences between the QTAIM‐DFA parameters calculated with M06‐2X/BSS‐A and those obtained with MP2/BSS‐B and M06‐2X/BSS‐B are confirmed not to damage our discussion so much on the nature of the interactions. The nature of each Se‐*‐Se in 3 2+–8 2+, together with 1 2+ and 2 2+, will be discussed based on the parameters evaluated with M06‐2X/BSS‐A, which enables us to discuss the nature of Se‐*‐Se in 1–8, in a unified form.

Each Se‐*‐Se interaction in 3 2+–8 2+ is expected to have a nature between that of 1 and 1 2+. The (θ, θ p) values for 1Se‐*‐1’Se of 3 2+ are (159.7°, 183.4°), therefore, it is predicted to have the r‐CS/CT‐TBP nature. The values for 1Se‐*‐2Se and 2Se‐*‐3Se of 3 2+ are (101.9–141.0°, 151.9–177.1°), therefore, they are predicted to have the r‐CS/CT‐MC nature. In the case of 4 2+, the (θ, θ p) values for 1Se‐*‐1’Se, 1Se‐*‐2Se, and 2Se‐*‐3Se are (103.5–142.0°, 153.6–178.8°), which are predicted to have the r‐CS/CT‐MC nature. The values for 3Se‐*‐4Se are (90.0°, 128.4°), therefore, it is just on the borderline area between p‐CS/t‐HBnc and r‐CS/t‐HBwc. The (θ, θ p) values for 1Se‐*‐1’Se, 1Se‐*‐2Se, and 2Se‐*‐3Se of 5 2+ are (106.7–114.1°, 161.2–170.2°), therefore, they are predicted to have the r‐CS/CT‐MC nature. The values for 3Se‐*‐4Se and 4Se‐*‐5Se are (90.5–92.7°, 130.5–140.4°), and they are predicted to have the r‐CS/t‐HBwc nature, although 3Se‐*‐4Se seems close to the borderline area between p‐CS/t‐HBnc and r‐CS/t‐HBwc.

In the case of 6 2+, while the (θ, θ p) values for 1Se‐*‐1’Se and 2Se‐*‐3Se are (82.6–89.2°, 107.7–129.1°), which are predicted to have the p‐CS/t‐HBnc nature, the values for 1Se‐*‐2Se, 4Se‐*‐5Se, and 5Se‐*‐6Se are (90.2–95.8°, 131.6–144.1°), which are predicted to have the r‐CS/t‐HBwc nature. The (θ, θ p) values for 3Se‐*‐4Se are (102.1°, 155.8°), which is predicted to have the r‐CS/CT‐MC nature. Similar to the case of 6 2+, the (θ, θ p) values for 1Se‐*‐1’Se, 1Se‐*‐2Se, and 3Se‐*‐4Se of 7 2+ are (80.5–87.9°, 101.2–124.9°), which are predicted to have the p‐CS/t‐HBnc nature, whereas the (θ, θ p) values for 2Se‐*‐3Se, 4Se‐*‐5Se, 5Se‐*‐6Se, and 6Se‐*‐7Se are (90.1–98.2°, 131.6–149.7°), which are predicted to have the r‐CS/t‐HBwc nature. However, 2Se‐*‐3Se (θ=90.1°) is very close to the borderline area to the p‐CS/t‐HBnc nature, and 4Se‐*‐5Se (θ p=149.7°) is close to the borderline area for the p‐CS/CT‐MC nature. Contrary to the cases of 6 2+ and 7 2+, the Se‐*‐Se interactions of 8 2+ are all predicted to have the p‐CS/t‐HBnc nature by the (θ, θ p) values of (79.6–89.7°, 98.2–129.6°), except for 5Se‐*‐6Se with (θ, θ p)=(95.2°, 143.0°), which is predicted to have the nature of p‐CS/t‐HBwc, although 3Se‐*‐4Se and 7Se‐*‐8Se seem close to the borderline area for the r‐CS/t‐HBwc nature with (θ, θ p)=(89.6–89.7°, 126.9–129.6°).

Weak interactions of the Se‐H and H–H types are also observed between the components in 6 2+–8 2+, which are analyzed similarly. The results are shown in Table S5 of the Supporting Information. The interactions are all predicted to have the p‐CS/vdW nature.

Relations Between QTAIM‐DFA Parameters and C

What are the relations among the QTAIM‐DFA parameters? Before the discussion, the parameters of R, θ, and θ p are plotted versus ρ(), which are shown in Figure S4 of the Supporting Information. The parameters increase monotonically as the increase of ρ(), although the plots are convex downward for R and convex upward for θ and θ p. The plot for R versus ρ() would be imaged as two streams. Then, θ and θ p are plotted versus R, separately by the Se‐*‐Se interactions inside or outside Se(CH2CH2CH2)2Se in 3 2+–8 2+, together with 1, 1 2+ and 2 2+. Figure 8 shows the plot, which reveals the two streams of the data (points) for R, θ, and θ p, (see also Figure 4). The results show that the (CH2CH2CH2)2 chains between the Se atoms in Se(CH2CH2CH2)2Se of 1 2+–8 2+ and 1 affect the parameters depending on the inside or outside positions of the components. However, the magnitudes in the differences of the parameters are very small. As a result, σ(mc‐ne: 4≤m≤16) formed in 3 2+–8 2+ and 2 2+ seem not to be affected so much from the chain. Namely, the σ‐type linear Se2 interactions in 2 2+–8 2+ would be close to those formed without any restrictions.

Figure 8

Plots of θ and θ p versus R for Se‐*‐Se in 1 2–8 2 and 1.

The R values are next plotted versus 1/C for all Se‐*‐Se in 3 2+–8 2+, together with 1 2+ and 2 2+. The plot, shown in Figure S5 of the Supporting Information, gave a good correlation (R=0.0207(1/C)+0.0009: R c 2=0.956). The results show that the strength of Se‐*‐Se can be estimated not only by R but also by 1/C, although roughly. The θ values show similar correlation to 1/C, although the data from Se‐*‐Se in 1 2+ deviate from the correlation. The correlation between θ p and 1/C seems poor. The ρ b( c) values also show good correlation to 1/C, although the data from Se‐*‐Se in 1 2+ deviate from the correlation, similar to the case of θ.

NBO Analysis for each Se‐*‐Se Interaction in 12+–82+

How does the CT term of each Se‐*‐Se contribute to stabilize 3 2+–8 2+ and 2 2+? The second‐order perturbation energies of E(2) are examined for the n(Se)→σ*(Se‐Se) 3c–4e type interactions in 2 2+–8 2+ by the NBO analysis.

The CT contributions in the intramolecular n(Se)→σ*(Se‐Se) interactions of 3 2+–8 2+ and 2 2+ seem very different from the typical cases. The linear Se2 interactions of 3 2+–8 2+ and 2 2+ construct the conjugate system of the σ‐type, and the SeSeSe angles of the interactions are often smaller than 150°, which we proposed tentatively as a lower limit for the linear interactions. The positive charge on the species would accelerate the mixing of the orbitals between those on Se and the frameworks constructed by C and H. There seem to be many candidates for the CT interactions in the linear Se2 interactions of 2 2+–8 2+. However, such CT terms were detected for the typical cases in 3 2+–5 2+ and 7 2+ (and 2 2+) but not in 6 2+ and 8 2+. It would be difficult to specify the n(Se)→σ*(Se‐Se) 3c–4e type interactions among many candidates in the linear σ‐type Se2 interactions of 2 2+–8 2+. The detection of the non‐adjacent n(Se)→σ*(Se‐Se) 3c–4e type interactions, such as np(3Se)→σ*(1Se‐1’Se) in 3 2+–5 2+ and np(3Se)→σ*(5Se‐6Se) and np(2Se)→σ*(5Se‐6Se) in 5 2+ must also be derived from the conjugated linear σ‐type Se2 interactions of 2 2+–8 2+. Detected E(2) values for 2 2+–5 2+ and 7 2+, are collected in Table S6 of the Supporting Information, where only one side of the interactions is considered for the symmetric species. Large E(2) values are predicted for np(Se)→σ*(Se‐Se) 3c–4e, which is consistent for the interactions being formed in the conjugated linear σ‐type Se2 interactions.

Only fairly good correlations were obtained between E(2) and 1/C ii, irrespective of other cases. The results show that the energies for the conjugated linear σ‐type Se2 interactions of 3 2+–8 2+ and 2 2+ cannot be fractionalized well to each evaluated by 1/C, namely the interactions in 3 2+–8 2+ should be analyzed as σ(mc‐ne: 6≤m≤16).

Conclusions

The nature of the extended hypervalent interactions of σ(mc‐ne: m≥6) has been investigated. Such interactions are elucidated for 3 2+–8 2+, together with 2 2+. The magnitudes of ΔE in the formation of 2+ from [1 2++(n−1) ⋅ 1] (n=2–8) are shown to increase almost constantly by approximately 0.38 eV. The r(Se, Se) values must be closely related to the stability of the dicationic oligomers (ΔE), which correlate well with (Qn(Se)+Qn(Se)) for 3 2+–8 2+, together with 1, 1 2+, and 2 2+. Very good correlations are obtained, if analyzed as two correlations, although the data of 1 2+ are omitted. The results strongly suggest that the Se‐Se interactions will be stronger than the electrostatic repulsion of the positive charges developed at both sides of Se‐Se in 3 2+–8 2+, together with 1, 1 2+, and 2 2+. The electrostatic terms can be estimated by Qn(Se)+Qn(Se), which must also correlate with the contributions of Se2 σ(2c–2e).

QTAIM‐DFA with CIV is applied to elucidate the intrinsic dynamic and static nature of the Se‐*‐Se interactions in 1 2+–8 2+ with 1, after clarification of the structural feature. The nature of the Se‐*‐Se interactions in 1 2+–8 2+ with 1 is summarized as follows. The Se‐*‐ Se interaction in the same species becomes weaker if it goes from the central position to a more outside position for 2 2+–5 2+, although the order changes as in 2Se‐*‐3Se>1Se‐*‐2Se for 5 2+. However, 1Se‐*‐1’Se is weakest in 6 2+ and 8 2+, while 1Se‐*‐2Se is weakest in 7 2+. The strength of Se‐*‐Se shows ripple‐like changes from the central to the outside for 6 2+–8 2+. The predicted nature of each Se‐*‐Se in 1 2+–8 2+ is summarized in Table 1. The nature of Se‐*‐Se predicted by the QTAIM parameters should be closely related to that in r(Se, Se), as a whole. The (CH2CH2CH2)2 chains in 1 2+–8 2+ and 1 affect the QTAIM‐DFA parameters, although slightly. Therefore, σ(mc‐ne: 4≤m≤16) formed in 2 2+–8 2+ should be close to those formed without any limitations. The strength of Se‐*‐Se described by R can be estimated by 1/C ii. The E(2) values based on NBO revealed the specific behavior of the conjugated σ‐type Se 2+ interactions in 2 2+–8 2+. The results show that the linear Se 2+ interactions of 6≤n≤16 should be analyzed as σ(mc‐ne: 6≤m≤16), not by the accumulated σ(3c–4e).

Indeed, 3 2–8 2+ are demonstrated to be energetically stable, similarly to the case of 2 2+, however, 3 2+–8 2+ are not isolated, yet. The entropy term must play an important role under the experimental conditions, in addition to the enthalpy term. The entropy term contributes much to isolate 1 2+ and 2 2+. The term will work more negatively as n in 2+ becomes larger. As a result, the next target to isolate 2+ should be n=3, if the entropy term is considered. But 2+ (n>3) could be isolated, if the specific stabilization conditions, such as those from the crystal packing effect and/or the counter ions, are satisfied.

Conflict of interest

The authors declare no conflict of interest.

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Supplementary

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