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Article

Intrinsic Dynamic and Static Nature of π···π Interactions in Fused Benzene-Type Helicenes and Dimers, Elucidated with QTAIM Dual Functional Analysis

by
Taro Nishide
and
Satoko Hayashi
*
Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan
*
Author to whom correspondence should be addressed.
Nanomaterials 2022, 12(3), 321; https://doi.org/10.3390/nano12030321
Submission received: 15 December 2021 / Revised: 11 January 2022 / Accepted: 14 January 2022 / Published: 19 January 2022
(This article belongs to the Special Issue Novel Self-Assembled Organic Nanoarchitectures Stabilized by Selective and Directional Interactions Engineering and Properties)
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Versions Notes

Abstract

:
The intrinsic dynamic and static nature of the π···π interactions between the phenyl groups in proximity of helicenes 3–12 are elucidated with the quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA). The π···π interactions appear in C-∗-C, H-∗-H, and C-∗-H, with the asterisks indicating the existence of bond critical points (BCPs) on the interactions. The interactions of 3–12 are all predicted to have a p-CS/vdW nature (vdW nature of the pure closed-shell interaction), except for 2Cbay-∗-7Cbay of 10, which has a p-CS/t-HBnc nature (typical-HBs with no covalency). (See the text for definition of the numbers of C and the bay and cape areas). The natures of the interactions are similarly elucidated between the components of helicene dimers 6:6 and 7:7 with QTAIM-DFA, which have a p-CS/vdW nature. The characteristic electronic structures of helicenes are clarified through the natures predicted with QTAIM-DFA. Some bond paths (BPs) in helicenes appeared or disappeared, depending on the calculation methods. The static nature of Ccape-∗-Ccape is very similar to that of Cbay-∗-Cbay in 9–12, whereas the dynamic nature of Ccape-∗-Ccape appears to be very different from that of Cbay-∗-Cbay. The results will be a guide to design the helicene-containing materials of high functionality.

Graphical Abstract

1. Introduction

Helicenes, which are ortho-fused polycyclic aromatic or heteroaromatic compounds with all rings angularly arranged to form helically shaped molecules, are of current and continuing interest. Helicenes are chiral; as a result, they are expected to have specific functionalities. Recently, helicenes have been widely applied in various fields [1,2,3,4,5], such as organic semiconductors [6,7,8,9,10], asymmetric catalysis [11,12,13,14,15,16], and molecular recognition [17,18,19,20,21,22], due to their diverse functionalities in materials. Many studies have also been reported on self-assembly phenomena at metal surfaces [23,24,25,26] caused by interactions with the π-orbitals of helicenes. The π-orbitals will cause intramolecular π···π interactions between adjacent aromatic rings of helicenes, which play an important role in effective interactions. It is crucial to clarify the nature of π···π interactions for future high-functioning material developments based on helicenes. The discussion in this paper will be limited to π···π interactions between the aromatic rings, the nature of which needs to be clarified, since helicenes of the fused benzene type were chosen as the target.
The noncovalent distances between the aromatic planes in close proximity to the helicenes were determined as the total effect of the attractive and repulsive forces between the atoms on the planes. The restoring forces from the deviated planarity in the helicenes should be a main factor for the attractive and repulsive forces due to π-orbital overlapping in the helicenes. The noncovalent distances between the planes in close proximity to the helicenes are defined as the balanced distances of the two factors. The noncovalent intramolecular distances between atoms in close proximity to the helicenes must be (much) shorter than the noncovalent intermolecular distances between the unrestricted nonhelical aromatic species. The shorter distances in helicenes result from the π···π interactions between the planes in close proximity in space, which operate under very severe conditions. Clarifying the nature of the π···π interactions in helicenes under such severe conditions will enable us to understand the factors that control the structures and the nature of the interactions. The results will also provide a starting point for understanding the nature of π···π interactions and will hint at designs for materials with high functionality based on the interactions.
We have been particularly interested in the π···π interactions that operate under severe conditions, as these should be the factors that control the fine details of the structures. Interactions are also expected to result in materials with high functionalities. The nature of π···π interactions under such severe conditions was investigated in a series of fused benzene-type helicenes 1–12 and concave-type dimers 6:6–8:8 and 10:10, where 1–3 are analyzed as helicenes in this paper, although they are usually not. Scheme 1 shows the structures of helicenes 1–12, dimers 6:6–8:8, 10:10, and [n]phenacenes 1p–12p, where 1p–12p are the comparative compounds and p stands for phenacenes. The bay and cape areas used in this paper are also illustrated. We have previously reported the nature of the benzene π···π interactions in cyclophanes [27] (see also [28,29]). The π···π interactions in the helicenes must correspond to the extended π···π interactions of the species.
The π···π interactions in the helicenes were analyzed with QTAIM dual functional analysis (QTAIM-DFA [30,31,32,33,34,35]), which we proposed based on the QTAIM approach introduced by Bader [36,37]. The π···π interactions will be reproduced on the bond paths (BPs) between atoms, where a bond critical point (BCP, ∗) appears on each BP. The π···π interactions in helicenes are typically described by BPs with BCPs of the H-∗-H, C-∗-H, and C-∗-C forms. The asterisk indicates the existence of a BCP in each BP [36,37]. In QTAIM-DFA, Hb(rc) is plotted versus Hb(rc)–Vb(rc)/2, where Hb(rc) and Vb(rc) are the total electron energy densities and potential energy densities, respectively, at the BCPs of the interactions in question. In our treatment, data from the fully optimized structures and the perturbed structures surrounding the fully optimized structures are used for the plots.
Data from the fully optimized structures in the plots were analyzed using polar coordinate (R, θ) representation, which corresponds to the static nature of the interactions [30,31,32,33,34,35]. Data from both the perturbed and fully optimized structures are expressed by (θp, κp), where θp corresponds to the tangent line and κp is the curvature of the plot. θ and θp are measured from the y-axis and the y-direction, respectively. (See Figure SA1 of the Appendix S1 of the Supporting Information for the definition of the QTAIM-DFA parameters of (R, θ) and (θp, κp), along with Equations (SA3)–(SA6) and the footnotes of Table 1). The concept of the dynamic nature of the interactions was proposed based on (θp, κp). The θp and κp for the major bonds seem to be controlled by the characters of the local bonds in question: The influence from the behaviors of the minor bonds would not be so severe for usual cases.
The perturbed structures necessary for QTAIM-DFA were generated by CIV [38], with the coordinates Ci corresponding to the compliance constants Cii for the internal vibrations [39,40,41,42,43,44]. The basic concept for the compliance constants was introduced by Taylor and Pitzer [45], followed by Konkoli and Cremer [46]. The Cij are defined as the partial second derivatives of the potential energy due to an external force [47,48,49], where i and j refer to internal coordinates. The dynamic nature of the interactions based on perturbed structures with CIV is described as the “intrinsic dynamic nature of interactions” because the coordinates are invariant to the choice of coordinate system. QTAIM-DFA and the criteria obtained by applying QTAIM-DFA with CIV to standard interactions are explained in the Appendix S1 of the Supporting Information using Schemes SA1–SA3, Figure SA1 and SA2, Table SA1, and Equations (SA1)–(SA7).
In this work, we present the results of the investigations into the natures of the π···π interactions in 1–12, 6:6–8:8, and 10:10, although some are discussed in the Supporting Information or calculated only for comparison. The interactions are classified and characterized by using the criteria as a reference. The structural features and the energy profile are also discussed to provide a solid basis for the discussion.

2. Methodological Details of the Calculations

Calculations were performed with the Gaussian 09 program package [50]. The 6-311+G(3d,p) basis set was used for the calculations at the DFT level of M06-2X [51] (M06-2X/6-311+G(3d,p)). The optimized structures were confirmed by frequency analysis. The results of the frequency analysis were used to calculate the compliance constants (Cii) and the coordinates corresponding to Cii (Ci). Calculations were also performed with M06-2X/6-311+G(2d,p) and LC-ωPBE/6-311+G(2d,p) [52] to examine the basis set and level dependence, containing the optimized π···π distances, on the results. The results with M06-2X/6-311+G(3d,p) are discussed in the text, while the results with M06-2X/6-311+G(2d,p) and LC-ωPBE/6-311+G(2d,p) are discussed mainly in the Supporting Information. We should be careful with the basis set and level dependence on the QTAIM-DFA parameters, which has been examined carefully [53]. Similar methodology was also employed for the theoretical studies of the π-stacking [54,55].
Equation (1) explains the method for generating perturbed structures with CIV [38]. The i-th perturbed structure in question (Siw) is generated by adding Ci to the standard orientation of a fully optimized structure (So) in the matrix representation. The coefficient giw in Equation (1) controls the structural difference between Siw and So, giw is determined to satisfy Equation (2) for r, where r and ro stand for the interaction distances in question in the perturbed and fully optimized structures, respectively, with ao = 0.52918 Å (Bohr radius). Five-digit Ci values were used to predict Siw.
Siw = So + giw·Ci
r = ro + wao (w = (0), ±0.025 and ±0.05; ao = 0.52918 Å)
y = co + c1x + c2x2 + c3x3 (Rc2: square of the correlation coefficient)
The QTAIM functions were calculated using the same basis set system as in the optimizations, unless otherwise noted, and were analyzed with the AIM2000 [56,57] and AIMAll [58] programs. The Hb(rc) values are plotted versus the Hb(rc) − Vb(rc)/2 values for five data points in Equation (2) in QTAIM-DFA: w = 0, ±0.025, and ±0.05. Each plot was analyzed using a cubic function regression curve, as shown in Equation (3), where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)) (Rc2 > 0.99999 as usual) [31].

3. Results and Discussion

3.1. Structural Features of 1–12 and Their Energy Profile

The structures of 1–12 were optimized with M06-2X/6-311+G(3d,p), M06-2X/6-311+G(2d,p), and LC-ωPBE/6-311+G(2d,p), retaining C2 symmetry. The selected noncovalent X···Y distances (X, Y = C, H) in the optimized structures with M06-2X/6-311+G(3d,p), M06-2X/6-311+G(2d,p), and LC-ωPBE/6-311+G(2d,p) are shown in Table S1 of the Supporting Information, along with the observed values [59,60,61,62,63,64,65,66].
How can the behaviour of the energies of the helicenes be explained? The energies of the helicenes were compared with the energies of [n]phenacene, a nonhelical species, evaluated with M06-2X/6-311+G(3d,p). The energy profiles will be discussed based on the energy differences, ΔE(n) = E(n) − E(n − 1) for helicenes (1–12) and ΔE(np) = E(np) − E(np − 1) for [n]phenacenes (1p–12p). The ΔE(n) values correspond to the energy differences in the formation of n from n − 1, and the ΔE(np) values similarly correspond to np from (np− 1). The E(n), E(np), ΔE(n), and ΔE(np) values were calculated on the energy surface, which are described by EES(n), EES(np), ΔEES(n), and ΔEES(np), respectively. The values were also calculated with the zero-point energies, which are described by EZP(n), EZP(np), ΔEZP(n), and ΔEZP(np). The values calculated with M06-2X/6-311+G(3d,p) are collected in Table S2 of the Supporting Information. The plot of ΔEZP(n) versus ΔEES(n) revealed an excellent correlation (y = 1.0042x + 0.6859; Rc2 = 0.980, see Figure S1 of the Supporting Information). As a result, ΔEES(n) can be used to analyze the energy terms.
Figure 1 shows the plots of ΔEES(n) and ΔEES(np) versus n. Both the ΔEES(n) and ΔEES(np) values (ΔEES(n; np)) decrease when n increases from 2 to 3. The extension of the π system appears to contribute more to the formation of phenanthrene from naphthalene than the repulsive noncovalent H···H interaction. The ΔEES(3; 3p) values are less than the ΔEES(2; 2p) values; however, the ΔEES(n; np) values increase from 3; 3p to 4; 4p. In the case of ΔEES(np), the ΔEES(4p) value is somewhat larger than ΔEES(3p) but slightly smaller than ΔEES(2p). The ΔEES(np) value decreases again slightly from 4p to 5p. Then, the values are nearly constant for np ≥ 5p. The results show that the repulsive energy from the noncovalent H···H interaction does not appear to be as severe as the stabilization factor from the extended π systems in 1p–12p. Namely, the np system stabilizes almost constantly as the size of the species increase, especially for np ≥ 5p, although a change in ΔEES(np) is detected for 2p ≤ np < 5p.
The data points for ΔEES(n) appear to be greater than those for ΔEES(np) when n ≥ 4. The observations must be due to the severe steric repulsion in ΔEES(n ≥ 4), where the plot for ΔEES(np) corresponds to that without such severe steric repulsion. The ΔEES(4) value is much larger than those of ΔEES(2) and ΔEES(3). The results can be explained by considering the much larger contribution from the repulsive noncovalent H···H interaction in 4 than in 3. This consideration is supported by the optimized structure of 4, drawn in Figure 2, as the molecular graph type. The ΔEES(n) values decrease in the following order: ΔEES(4) > ΔEES(5) > ΔEES(6) > ΔEES(7) > ΔEES(8). The contribution of steric repulsion to ΔEES(n) due to noncovalent interactions is expected to increase as n increases in this process. However, the observed results are the opposite of what was expected. Therefore, the observed trend should be attributed to the increased energy-lowering effect by the extended π systems in 4–8 relative to the repulsive interactions.
The ΔEES(n) value becomes somewhat larger again from n = 8 to 9 and 9 to 10, and then decreases again from 10 to 11 and 11 to 12. The subtle conditions in the steric repulsion contribute to the complex behaviour of ΔEES(n) (8 ≤ n ≤ 12). The behaviour of ΔEES(2)–ΔEES(12) shown in Figure 1 should be affected both by the repulsive factor of the noncovalent H-∗-H, C-∗-H, and C-∗-C interactions and by the energy-lowering factor of the extended π system. The ΔEES(4) value is the largest among ΔEES(2)–ΔEES(12). The results are of great interest since the repulsive noncovalent H···H interaction in 4 from 3 appears to be very large among 2–12 when evaluated by ΔEES(n). The trend in ΔEES(n) seems to be in good agreement with those reported by Rulíšek et al., calculated with PBE-D/TZVP//PBE-D/6-31G(d), except for ΔEES(8) and ΔEES(9) [67].
It is also instructive to analyze the aromaticities of acenes, phenacenes, and helicenes after investigating the energy profiles. The structures of acenes, phenacenes, and helicenes are illustrated in Chart S1 of the Supporting Information, together with the definition of the ring positions. The aromaticities were analyzed by the HOMA (harmonic oscillator model of aromaticity) method [68]. The HOMA values are collected in Table S3 of the Supporting Information. The HOMA values of the acenes and phenacenes are plotted versus those of the helicenes, which are shown in Figure S2 of the Supporting Information. The plot of the data for phenacenes versus those for helicenes gave a very good correlation (y = 0.964x + 0.042; Rc2 = 0.981), whereas the correlations of the plots for acenes versus helicenes were very poor (y = −0.685x + 0.995; Rc2 = 0.492 if calculated under the closed–shell singlet conditions and y = −0.399x + 0.880; Rc2 = 0.317 under the open–shell singlet conditions). The very good correlation of the former demonstrates that the aromaticities of the helicenes appear to be very similar to those of the phenacenes, irrespective of the very severe steric deformations in the structures of helicenes. However, the very poor correlations with the negative correlation constants show that the aromaticities of the helicenes are very different from those of acenes.

3.2. Survey of X-∗-Y (X, Y = C and H) in 3–12 with the Molecular Graphs

Figure 2 shows the molecular graphs, exemplified by 4, 7, 9, 11, and 12. Many BPs with BCPs are detected in the π···π interactions between the phenyl rings in close proximity to the helicenes. The molecular graphs for helicenes 3–12, except for 4, 7, 9, 11, and 12, are shown in Figure S3 of the Supporting Information.
The BPs corresponding to X-∗-Y (X, Y = C and H) appear almost straight, as shown in Figure 2 and Figure S4 of the Supporting Information, although some appear somewhat bent. To examine the linearity of the BPs further, the lengths of the BPs (rBP) were calculated for all X-∗-Y of 3–12, along with the corresponding straight-line distances (RSL). The values are collected in Table S4 of the Supporting Information, along with the differences between them (ΔrBP = rBP – RSL). The averaged values of ΔrBP were 0.2040, 0.4006, 0.0588, and 0.1451 Å for Hbay-∗-Hbay, Cbay-∗-Hbay, Cbay-∗-Cbay, and Ccape-∗-Ccape, respectively. As a result, ΔrBP for Hbay-∗-Hbay and Cbay-∗-Hbay were larger than 0.20 Å, while those for Cbay-∗-Cbay and Ccape-∗-Ccape were less than 0.15 Å. Therefore, the BPs corresponding to Cbay-∗-Cbay and Ccape-∗-Ccape can be roughly approximated as straight lines since the ΔrBP values are less than 0.20 Å (see also Figure S4 of the Supporting Information).
The QTAIM functions were calculated at BCPs on X-∗-Y of 3–12 with M06-2X/6-311+G(3d,p). Table 1 collects the ρb(rc), Hb(rc) − Vb(rc)/2, and Hb(rc) values for one of the X-∗-Y if it is doubly degenerated due to the C2 symmetry of the optimized structures. Figure 3 shows the plots of Hb(rc) versus Hb(rc) – Vb(rc)/2 for each X-∗-Y, exemplified by 3–6, 8, 10, and 12, where H-∗-H was detected in 3 and 4 and C-∗-H and C-∗-C were detected in 8, 10, and 12. (See Figure S5 of the Supporting Information for 7, 9, and 11).
The plots were analyzed according to Equations (SA3)–(SA6) of the Supporting Information. Table 1 also collects the QTAIM-DFA parameters of (R, θ) and (θp, κp) for each X-∗-Y of 3–12, along with the Cii values corresponding to the interactions in question. The (θp, κp) values, evaluated with CIV, should be denoted by (θp:CIV, κp:CIV), respectively. However, (θp, κp) will be used in place of (θp:CIV, κp:CIV) to simplify the notation. The QTAIM functions and QTAIM-DFA parameters calculated with M06-2X/6-311+G(2d,p) and LC-ωPBE/6-311+G(2d,p) are collected in Tables S5 and S6 of the Supporting Information respectively.

3.3. Nature of Each X-∗-Y in 3–12

The criteria shown in Scheme SA3 and Table SA1 of the Supporting Information indicate that the interactions in the range of 45° < θ < 90° should be classified as pure closed-shell (p-CS) interactions. In the p-CS region of 45° < θ < 90°, the character of the interactions will be the vdW type for 45° < θp < 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° < θp < 125° (75° < θ < 90°), where θ = 75° and θp = 125° are tentatively given for θp = 90° and θ = 90°, respectively.
The C atoms in helicenes 3–12 were subdivided into Cbay and Ccape based on the positions of the atoms in the species, as were the H atoms into Hbay and Hcape. The bay and cape areas (positions) in the species are illustrated in Scheme 1. While both the Cbay and Ccape atoms of 3–12 participate in the interactions as BPs, only Hbay atoms participate as BPs. The θ and θp values for H-∗-H, C-∗-H, and C-∗-C of 3–12, collected in Table 2, are all less than 90°, except for θp of 2Cbay-∗-7Cbay in 10, where (θ, θp) = (70.5°, 94.2°). The 2Cbay-∗-7Cbay interaction in 10 is denoted by 10 (2Cbay-∗-7Cbay) (see also Table 1). Therefore, the H-∗-H, C-∗-H, and C-∗-C interactions of 3–12 are all classified as p-CS interactions and characterized to have a vdW nature, which is denoted by p-CS/vdW, except for 10 (2Cbay-∗-7Cbay), which is predicted to have a p-CS/t-HBnc nature.
Next, the interactions were individually examined. The (θ, θp) values are (71.3°, 72.8°) and (71.8°, 74.5°) for 3 (1Hbay-∗-4Hbay) and 4 (1Hbay-∗-5Hbay), respectively. The θ values for 3 (1Hbay-∗-4Hbay) and 4 (1Hbay-∗-5Hbay) are larger than those of A-∗-HF (A = He, Ne, and Ar: (θ, θp) = (59.9–70.9°, 64.0–88.0°)), whereas the θp values are larger than those of A-∗-HF (A = He and Ar). The interaction in 4 is estimated to be slightly stronger than that in 3, although the real image of 3 (1Hbay-∗-4Hbay) has been much debated [69,70,71]. The detection of BPs with BCPs for 3 (1Hbay-∗-4Hbay) would not show enough strength for the interaction. It could be the mathematical results of the treatment. Nevertheless, 3 (1Hbay-∗-4Hbay) and 4 (1Hbay-∗-5Hbay) are discussed as very weak interactions in this work because the (θ, θp) values are larger than those of A-∗-HF (A = He, Ne, and Ar). Double Hbay-∗-Cbay interactions are detected for each of 5–12, with (θ, θp) values of (70.7–71.6°, 76.7–80.8°). The (θ, θp) values are very close to those of A-∗-HF (A = He, Ne, and Ar). The BP (Hbay-∗-Cbay) in 6 and 9 connect the Hbay and Cbay atoms. However, they are not located at the nearest positions, as shown in Table 1 and Figure S3 of the Supporting Information. Therefore, the BP (Hbay-∗-Cbay) in 6 and 9 should be analyzed carefully.
One, one, four, four, seven, and eight different types of C-∗-C interactions are detected for 7–12, respectively. The (θ, θp) values for C-∗-C in 7–12 are (70.0–72.9°, 66.3–94.2°). It appears better to separately examine the values for two groups of Cbay-∗-Cbay and Ccape-∗-Ccape. While the (θ, θp) values of Cbay-∗-Cbay in 7–12 are (71.5–72.9°, 79.5–87.8°), the values are (70.0–70.7°, 66.3–69.7°) for Ccape-∗-Ccape. The θ values for Ccape-∗-Ccape are slightly smaller than those of Cbay-∗-Cbay (by 0.5–2.2°), but the θp values for Ccape-∗-Ccape are much smaller than those of Cbay-∗-Cbay (by 13.2–24.5°). In this case, θp < θ for Ccape-∗-Ccape, whereas θp > θ for Cbay-∗-Cbay. Interactions with θp > θ are usually observed, but interactions with θp < θ are rare.
Interactions with θ > θp occur under some specific conditions. To examine the behaviour of θ and θp in 7–12, the Δθp (=θp – θ) values are plotted versus θp for C-∗-C, H-∗-H and C-∗-H in 3–12. Figure 4 shows this plot. The plot showed a very good correlation for all data (y = 0.918x – 64.88: Rc2 = 0.995). (A substantial correlation was not found in the plot of Δθp versus θ due to the very small range of θ). The two areas for C-∗-C interactions with Δθp > 0 and Δθp < 0 are clearly illustrated by the green dotted lines in Figure 4. The Δθp values for the interactions are positive if the θp values are larger than 70.7°, whereas Δθp < 0 if θp < 70.7°. Figure 4 clearly shows that Cbay-∗-Cbay and Ccape-∗-Ccape in 9–12 belong to the areas where Δθp > 0 and Δθp < 0, respectively.
It seems difficult to clearly explain the results shown in Figure 4; however, our explanation is as follows: The static nature of the interactions described by θ should be a measure of the strength of the interactions. If so, the steric compression on Ccape-∗-Ccape in 9–12 appears to be similar to that on Cbay-∗-Cbay in fully optimized structures. Namely, the Ccape-∗-Ccape and Cbay-∗-Cbay interactions in the fully optimized structures of 9–12 would be affected similarly to steric compression, according to the θ values. On the other hand, the dynamic nature of the interactions is defined by θp based on the behaviour of the interactions in the perturbed structures. The Cbay-∗-Cbay interactions in the perturbed structures will be affected by steric compression, similar to the usual cases of interactions, whereas the Ccape-∗-Ccape interactions will be inversely affected compared with the usual cases when measured by the θp values at the BCPs of the interactions.

3.4. Nature of Each X-∗-Y in 6:6 and 7:7

What is the behaviour of the interactions when the helicenes form concave-type dimers? The behaviour was elucidated, exemplified by 6:6 (Ci) and 7:7 (Ci) with M06-2X/6-311+G(3d,p). Figure 5 shows molecular graphs of 6:6 and 7:7. Five and four independent BPs with BCPs were detected in 6:6 and 7:7, respectively, between the components of H-∗-H and C-∗-H, as well as two independent BPs with BCPs for the intramolecular C-∗-H interactions in each component of 6:6 and 7:7. The behaviour of the interactions was also investigated for 7:7 (Ci), 8:8 (Ci), and/or 10:10 (Ci) with M06-2X/6-311+G(2d,p) and LC-ωPBE/6-311+G(2d,p). The results are collected in Tables S7 and S8 of the Supporting Information. The QTAIM functions were similarly calculated for the intermolecular interactions at the BCPs on the BPs of 6:6 and 7:7 with M06-2X/6-311+G(3d,p).
Table 2 collects the ρb(rc), Hb(rc) − Vb(rc)/2, and Hb(rc) values for one of the doubly degenerate interactions due to the Ci symmetry of the optimized structures. Figure 6 shows the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for each interaction between the components at 6:6 and 7:7. (The plots for 8:8 and 10:10 are shown in Figure S7 of the Supporting Information, and the data are collected in Table S8 of the Supporting Information).
The plots were analyzed similarly to the case of 3–12. Table 2 also collects the QTAIM-DFA parameters of (R, θ) and (θp, κp) for the intermolecular interactions in question at 6:6 and 7:7, together with the Cii values corresponding to the interactions in question. The (θ, θp) values for the three H-∗-H and two C-∗-C intermolecular independent interactions of 6:6 are (70.9–74.3°, 74.9–88.1°) and (68.3–69.2°, 68.0–70.8°), respectively. The (θ, θp) values for the couple of H-∗-H and two C-∗-H intermolecular independent interactions at 7:7 are (68.9–70.0°, 72.7–75.5°) and (71.4–73.8°, 73.2–73.3°), respectively. The θ and θp values for the intermolecular H-∗-H, C-∗-H, and C-∗-C interactions at 6:6 and 7:7 are all less than 90°; therefore, the interactions are all predicted to have a p-CS/vdW nature (see Table 2). The interactions appear to be very weak, based on the (θ, θp) values. However, (θ, θp) = (72.6°, 88.1°) for 6:6 (1Hbay-∗-17’Hcape), of which nature seems close to p-CS/t-HBnc.
In the case of intramolecular interactions, 1Hbay-∗-5Cbay and 3Cbay-∗-7Hbay were detected at 6:6. The former was also observed in 6, whereas the latter was newly detected in 6:6. The new appearance of 6 (3Cbay-∗-7Hbay) may be due to a structural change at 6:6 relative to 6. Similarly, 1Hbay-∗-6Cbay and 3Cbay-∗-8Hbay were detected at 7:7. The former was observed in 7, while 3Cbay-∗-8Hbay in 7:7 appeared in place of 2Cbay-∗-7Cbay in 7. The change in the optimized structures between 7 and 7:7 would again be responsible for the results. However, clarifying the reason for the appearance/disappearance of BPs is very complex and difficult in helicenes, and it is beyond the scope of this work.
Highly theoretical treatment must be necessary to clarify the reason for the appearance and disappearance of BPs/BCPs. Pendás and coworkers discussed BPs as privileged exchange channels, using the interacting quantum atom (IQA) framework [72]. They have investigated how BPs between an atom A and atoms B in its environment appear to be determined by competition among the A–B exchange correlation energies that always contribute to stabilize the A–B interactions. And they have predicted that a BP is found between two atoms by examining a number of archetypal simple systems: (1) there is no other competing atom in its vicinity, so there must be a direct exchange route between them or (2) its Vxc term is the largest among several possibilities, where Vxc stands for a quantum-mechanical correction coming from the exchange correlation second-order density [72]. It has also indicated that interaction energies between both atoms cannot be universally used to predict the existence of a BP between them [73]. Moreover, they are not correlated to distances or to the density values at BCPs. On the contrary, the exchange contribution is shown to be an appropriate descriptor [73]. Similarly, theoretical treatments are applied to various interactions, employing QTAIM-defined an atomic interaction line (AIL: Presence or absence), IQA-defined interaction energy and its components, NCI (non-covalent interactions)-defined isosurfaces, and deformation density [74]. The reason for the appearance and disappearance of BPs/BCPs in the helicenes would be rationalized by applying above theory [27].
The (θ, θp) values for the intramolecular interactions at 6:6 and 7:7 are (70.8–71.9°, 78.3–80.0°). As a result, the interactions are all predicted to have a p-CS/vdW nature (see Table 2). The predicted natures of the interactions in 6:6 and 7:7 appear to be similar to those in 6 and 7, perhaps due to the very weak nature of both dimers and monomers.

4. Basis Set and Level Dependence of the Predicted Natures

The basis set and level dependence of the predicted natures was investigated, exemplified by 7 and 7:7, to attempt to determine the reason why the optimized structures can easily change. Table 3 shows the QTAIM-DFA parameters of (R, θ) and (θp, κp) and the Cii values, calculated with M06-2X/6-311+G(3d,p), M06-2X/6-311+G(2d,p), and LC-ωPBE/6-311+G(2d,p). Table 3 includes the distances in question as well as some internal vibration(s) νn corresponding to the interactions in question, which are closely related to (θp, κp). Figure 7 shows the motions of the internal vibrations for ν1 of 7 and 7:7 calculated with M06-2X/6-311+G(3d,p), M06-2X/6-311+G(2d,p), and LC-ωPBE/6-311+G(3d,p). (Other motions are shown in Figure S9 of the Supporting Information).
The calculated r(1Hbay···6Cbay) and r(2Cbay···7Cbay) distances were 2.590 Å and 2.975 Å, respectively, for 7, when calculated with M06-2X/6-311+(3d,p), while the values were 2.590 Å and 3.003 Å, respectively, when calculated with M06-2X/6-311+(2d,p). The differences are less than 0.001 Å for r(1Hbay···6Cbay) and 0.028 Å for r(2Cbay···7Cbay). The results show that the structure of 7 optimized with M06-2X/6-311+(2d,p) appears to be nearly identical to that optimized with M06-2X/6-311+(3d,p). On the other hand, the r(1Hbay···6Cbay) and r(2Cbay···7Cbay) of 7 were 2.561 Å and 3.227 Å, respectively, if calculated with LC-ωPBE/6-311+(2d,p). The difference was −0.029 Å for the former but 0.224 Å for the latter relative to the corresponding values calculated with M06-2X/6-311+(2d,p). The structure of 7 optimized with LC-ωPBE/6-311+(2d,p) appears to be (very) different from that optimized with M06-2X/6-311+(2d,p), especially around r(2Cbay···7Cbay). In the case of 1Hbay···5Cbay, the distance was optimized to be 2.468 Å with LC-ωPBE/6-311+(2d,p), which is shorter than the r(1Hbay···6Cbay) distance optimized with M06-2X/6-311+(2d,p) (2.590 Å) by 0.122 Å. BPs (with BCPs) were detected for 1Hbay-∗-6Cbay and 2Cbay-∗-7Cbay in 7 if calculated with M06-2X/6-311+(3d,p) and M06-2X/6-311+(2d,p), while a BP was detected for 1Hbay-∗-5Cbay if calculated with LC-ωPBE/6-311+(2d,p). The 2Cbay-∗-7Cbay and 1Hbay-∗-5Cbay distances in 7, optimized with LC-ωPBE/6-311+(2d,p), were (much) longer and shorter than those optimized with M06-2X/6-311+(2d,p), respectively.
The differences in the optimized distances appear to be the main factor for the appearance/disappearance of the BPs, although predicting the appearance/disappearance of the BPs is very difficult and complex. Despite such different results, the motion of ν1 appears to be very similar when calculated at both the M06-2X and LC-ωPBE levels, indicating that ν1 is a good measure for imaging the dynamic nature of the π···π interactions in 7 among the internal vibrations. Small differences in the dynamic nature of the interactions predicted at both the M06-2X and LC-ωPBE levels result from the (very) similar motion of ν1. The magnitudes of the displacements in the cape area seem (much) larger than those in the bay area in ν1. This will be instructive if the relationship is clarified for that between the magnitudes of the displacements and the Δθp values. This issue will be investigated in a future work. The very low energy of ν1 in 7 suggests the basis set and level dependence can easily change the optimized structure.
In the case of 7:7, the r(20Hcape···18’Hcape), r(20Hcape···20’Hbay), r(20Hcape···2’Ccape), and r(8Hcape···3’Ccape) distances were 2.543 Å, 2.716 Å, 2.677 Å, and 2.964 Å, respectively, when calculated with M06-2X/6-311+(3d,p), while the values were 2.564 Å, 2.729 Å, 2.688 Å, and 2.986 Å, respectively, when calculated with M06-2X/6-311+(2d,p). The differences are 0.011–0.022 Å, which are less than approximately 0.02 Å. The results show that the optimized structures of 7:7 are very similar with both M06-2X/6-311+(3d,p) and M06-2X/6-311+(2d,p). On the other hand, the distances are optimized to be 2.947 Å, 3.015 Å, 3.066 Å, and 3.659 Å with LC-ωPBE/6-311+(2d,p). The differences with the corresponding values of M06-2X/6-311+(2d,p) are 0.286–0.674 Å. Namely, the structure of 7:7 optimized with LC-ωPBE/6-311+(2d,p) appears to be very different from that optimized with M06-2X/6-311+(2d,p), similar to the case of 7.
The 20Ccape-∗-18’Hcape distance was optimized to be 3.236 Å, which is longer than r(20Hcape···18’Hcape) (2.947 Å) by 0.288 Å with LC-ωPBE/6-311+(2d,p). However, a BP was detected for 20Ccape-∗-18’Hcape but not for 20Hcape-∗-18’Hcape. The difference in the atomic size between C and H, such as the van der Waals radii, may be responsible for the predicted results, in this case. The 20Ccape-∗-18’Hcape, 20Hcape-∗-20’Hbay, 20Hcape-∗-2’Ccape, and 8Hcape-∗-3’Ccape distances at 7:7, optimized with LC-ωPBE/6-311+(2d,p), were much longer than the corresponding distances, optimized with M06-2X/6-311+(2d,p). BPs were detected for 20Hcape-∗-18’Hcape, 20Hcape-∗-20’Hbay, 20Hcape-∗-2’Ccape, and 18Hcape-∗-3’Ccape when calculated with M06-2X/6-311+(2d,p), while they were detected for 20Ccape-∗-18’Hcape, 20Hcape-∗-20’Hbay, 20Hcape-∗-2’Ccape, and 8Hcape-∗-3’Ccape when calculated with LC-ωPBE/6-311+(2d,p). The differences in the optimized distances appear to be the main factor for the appearance/disappearance of the BPs, similar to the case of 7.
Table 3 contains the ν1 values for 7 and 7:7, calculated with M06-2X/6-311+(3d,p), M06-2X/6-311+(2d,p), and LC-ωPBE/6-311+(2d,p). Table 3 also contains some vibrations closely related to the interactions in question (corresponding to the perturbed structures) at 7:7, where most candidates were found to be less than νn of ν20. The ν1 values for 7 were 48.4 cm−1, 46.6 cm−1, and 40.2 cm−1 when calculated with the three methods, respectively. The ν1 motion of 7 appears to be very similar when calculated with the three methods. In the case of 7:7, the frequencies of ν1 calculated with M06-2X/6-311+G(3d,p) and M06-2X/6-311+G(2d,p) were 14.0 cm−1 and 13.1 cm−1, respectively, while the value calculated with LC-ωPBE was 2.1 cm−1. Very large differences are predicted for ν1 at 7:7 when calculated at the M06-2X and LC-ωPBE levels. The ν1 value with the motion should correspond to the strength of the interactions in the direction of the perturbed structures.
The (θ, θp) values of 7:7 are (67.9–73.0°, 69.4–75.2°) and (62.9–67.2°, 67.4–73.9°) with M06-2X/6-311+(2d,p) and LC-ωPBE/6-311+(2d,p), respectively. The differences seem large relative to the case of 7, with (θ, θp) values of (70.4°, 81.6°) and (70.5°, 82.2°) when calculated with M06-2X/6-311+(2d,p) and LC-ωPBE/6-311+(2d,p), respectively, although 7 (1Hbay-∗-6Cbay) was detected with M06-2X/6-311+(2d,p) and 7 (1Hbay-∗-5Cbay) was detected with LC-ωPBE/6-311+(2d,p). The (optimized) structures of 7:7 would be affected more easily by surroundings containing the calculation methods than the case of 7. The basis set and level dependence of the interactions in 7 and 7:7 can help us to better understand the interactions in helicenes.
The unit of Cii [Å mdyn−1] is the inverse of that of the force constant, which corresponds to the frequency. Therefore, the strengths of the interactions should be roughly inversely proportional to the Cii values. As shown in Table 3, the Cii values for the π···π interactions of 7 are 3.1–5.5 Å mdyn−1 for Hbay-∗-Cbay and Cbay-∗-Cbay with the three methods. The values for the π···π interactions of 7:7 are 10.0–27.9, 11.4–31.4, and 52.7–96.4 Å mdyn−1 for Hcape-∗-Hcape and Hcape-∗-Ccape when calculated with M06-2X/6-311+(3d,p), M06-2X/6-311+(2d,p), and LC-ωPBE/6-311+(2d,p), respectively. This consideration explains the above results.

5. Conclusions

It is challenging to clarify the natures of π···π interactions in helicenes since the interactions are factors that control the fine details of structures and are expected to give rise to specific functionalities for the species. The repulsive interactions between the benzene rings in helicenes must be very strong; therefore, the π···π interactions would be considered strong. The π···π interactions in the helicenes are described by the H-∗-H, C-∗-H, and C-∗-C forms with BPs and BCPs. The π···π interactions in helicenes 1–12, as well as in dimers 6:6 and 7:7, were analyzed with QTAIM-DFA after clarifying the structural features and the energy profile. Hb(rc) was plotted versus Hb(rc) − Vb(rc)/2, and the data from the fully optimized structures and the perturbed structures around the fully optimized structures were used in QTAIM-DFA. Data from the fully optimized structures in the plots correspond to the static nature of the interactions, which are analyzed using polar coordinate (R, θ) representation. Data from both the perturbed and fully optimized structures are expressed by (θp, κp), where θp corresponds to the tangent line and κp is the curvature of the plot. The concept of the dynamic nature of the interactions was proposed based on (θp, κp).
The interactions were analyzed by dividing the C atoms of 3–12 into Cbay and Ccape and the H atoms into Hbay and Hcape. While both Cbay and Ccape atoms of 3–12 take part in the interactions, only Hbay atoms participate as BPs. The θ and θp values for H-∗-H, C-∗-H, and C-∗-C of 3–12 are all less than 90°, except for 10 (2Cbay-∗-7Cbay), where (θ, θp) = (70.5°, 94.2°). Therefore, the H-∗-H, C-∗-H, and C-∗-C interactions of 3–12 are all predicted to have a p-CS/vdW nature, except for 10 (2Cbay-∗-7Cbay), which is predicted to have a p-CS/t-HBnc nature. While the (θ, θp) values of Cbay-∗-Cbay in 7–12 are (71.5–72.9°, 79.5–87.8°), the values are (70.0–70.7°, 66.3–69.7°) for Ccape-∗-Ccape. The θ values for Ccape-∗-Ccape are slightly smaller than those of Cbay-∗-Cbay (by 0.5–2.2°), but the θp values for Ccape-∗-Ccape are much smaller than those of Cbay-∗-Cbay (by 13.2–24.5°). In this case, θ < θp for Cbay-∗-Cbay, whereas θ > θp for Ccape-∗-Ccape. Interactions with θ < θp are usually observed, whereas interactions with θ > θp are rare.
The H-∗-H, C-∗-H, and C-∗-C interactions of dimers 6:6 and 7:7 were similarly analyzed. The interactions were predicted to have a p-CS/vdW nature, although 6:6 (1Hbay-∗-17’Hcape) has a nature close to p-CS/t-HBnc, since (θ, θp) = (72.6°, 88.1°). The interactions at 3–12 and 6:6 and 7:7 were predicted to be much weaker than expected. The very low energy of ν1 of 7:7 supports the very weak nature predicted for interactions and the easy dependence of the levels on the nature of the interactions. The strength of the interactions can also be estimated by the Cii−1 values. Detecting the interactions and predicting the nature of helicenes will provide a solid basis for investigating and applying the interactions in helicenes.

Supplementary Materials

The following are available online at https://www.mdpi.com/article/10.3390/nano12030321/s1. Table S1: The observed and calculated C···C length (robsd and rcalcd, respectively), which are located in the bay area for 3–5 and the bay and cape area between adjacent aromatic rings for 6–12, together with the differences, Δrcalcd (=rcalcd:XY − robsd:XY) in each C···C for 3–12, elucidated with various methods. Table S2: The total energy EES and zero-point energy EZP values for n; np, where n = 1 to 12, along with the ΔEES and ΔEZP(n; np) values, evaluated with M06-2X/6-311+G(3d,p). Table S3: The HOMA indices for [n]acenes, [n]phenacenes, and [n]helicenes, evaluated with M06-2X/6-311+G(3d,p). Table S4: The observed and calculated X-∗-Y lengths (Robsd:XY and Rcalcd:XY, respectively; X, Y = C and H) and the length of the bond paths (rBP:XY) and the corresponding straight-line distances (RSL:XY), together with the differences, ΔRcalcd:XY (=Rcalcd:XY – Robsd:XY) in each X-∗-Y for 3–12, 6:6 and 7:7, evaluated with M06-2X/6-311+G(3d,p), where RSL:XY = Rcalcd:XY, together with 8:8 and 10:10, calculated with M06-2X/6-311+G(2d,p). Table S5: QTAIM functions and QTAIM-DFA parameters for the fused benzene-type helicenes of monomers (3–12 (C2)), together with the nature of each noncovalent interaction, elucidated with M06-2X/6-311+G(2d,p). Table S6: QTAIM functions and QTAIM-DFA parameters for the fused benzene-type helicenes of monomers (3–12 (C2)), along with the nature of each noncovalent interaction, elucidated with LC-ωPBE/6-311+G(2d,p). Table S7: The EES (au), ΔEES (kJ mol−1), and ΔEZP (kJ mol−1) values for 6:6–8:8 and 10:10, evaluated with various methods. Table S8: QTAIM functions and QTAIM-DFA parameters for the concave-type dimer of helicenes (8:8–10:10 (Ci)), together with the nature of each noncovalent interaction, elucidated with M06-2X/6-311+G(2d,p). Figure S1: Plots of ΔEZP(n) versus ΔEES(n), calculated with M06-2X/6-311+G(3d,p). Figure S2: Plots of HOMA indices for [n]acenes (n = 4–12) at closed-shell singlet state, [n]acenes (n = 7–12) at open-shell singlet state, and [n]phenacenes (n = 4–12) versus those for [n]helicenes (n = 4–12), calculated with MP2/6-311+G(3d,p). Figure S3: Molecular graphs for 3–6, 8, and 10 calculated with M06-2X/6-311+G(3d,p). Figure S4: Plots of rBP versus RSL for Hbay-∗-Hbay, Cbay-∗-Hbay, Cbay-∗-Cbay, and Ccape-∗-Ccape in 3–12, calculated with M06-2X/6-311+G(3d,p). Figure S5: Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for H-∗-H, C-∗-H, and C-∗-C for 7, 9, and 11. Figure S6: Molecular graphs for 8:8 (Ci) and 10:10 (Ci) calculated with M06-2X/6-311+G(2d,p). Figure S7: Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for H-∗-H, C-∗-H, and C-∗-C for 8:8 and 10:10. Figure S8: The internal vibration motions of νn for 7:7 (Ci) from the top view. Figure S9: The internal vibration motions of νn for 7:7 (Ci).

Author Contributions

S.H. formulated the project. T.N. and S.H. optimized all compounds. T.N. calculated the ρb(rc), Hb(rc) − Vb(rc)/2 (=(ℏ2/8m)∇2ρb(rc)), and Hb(rc) values, evaluated the QTAIM-DFA parameters, and analyzed the data. S.H. and T.N. organized the data for writing, and T.N. wrote the paper. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data are contained within the article or the Supplementary Materials.

Acknowledgments

The authors are very grateful to Waro Nakanishi of Wakayama University for discussing the dynamic and static nature of π···π interactions in helicenes and dimers, elucidated with QTAIM-DFA.

Conflicts of Interest

The authors declare no conflict of interest.

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Nanomaterials 12 00321 sch001 550
Scheme 1. Helicenes, 1–12, dimers, 6:6–8:8 and 10:10, and [n]phenacenes, 1p–12p. The bay and cape areas in 1–12 are illustrated. The number of C is shown, where the number of H is the same for C–H. Benzene, naphthalene, and phenanthrene are defined corresponding to n = 1, 2, and 3, respectively.
Scheme 1. Helicenes, 1–12, dimers, 6:6–8:8 and 10:10, and [n]phenacenes, 1p–12p. The bay and cape areas in 1–12 are illustrated. The number of C is shown, where the number of H is the same for C–H. Benzene, naphthalene, and phenanthrene are defined corresponding to n = 1, 2, and 3, respectively.
Nanomaterials 12 00321 sch001
Nanomaterials 12 00321 g001 550
Figure 1. Plot of ΔEES(n) and ΔEES(np) versus n, evaluated with M06-2X/6-311+G(3d,p), where ΔEES(n) = EES(n) − EES(n − 1) and ΔEES(np) = EES(np) − EES(np − 1).
Figure 1. Plot of ΔEES(n) and ΔEES(np) versus n, evaluated with M06-2X/6-311+G(3d,p), where ΔEES(n) = EES(n) − EES(n − 1) and ΔEES(np) = EES(np) − EES(np − 1).
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Nanomaterials 12 00321 g002 550
Figure 2. Molecular graphs for 4 (a), 7 (b), 9 (c), 11 (d), and 12 (e), calculated with M06-2X/6-311+G(3d,p), where BPs with BCPs corresponding to intramolecular noncovalent interactions are detected. The BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, CCPs (cage critical points) by green dots, and BPs by pink lines. The carbon atoms are in black and the hydrogen atoms are in grey.
Figure 2. Molecular graphs for 4 (a), 7 (b), 9 (c), 11 (d), and 12 (e), calculated with M06-2X/6-311+G(3d,p), where BPs with BCPs corresponding to intramolecular noncovalent interactions are detected. The BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, CCPs (cage critical points) by green dots, and BPs by pink lines. The carbon atoms are in black and the hydrogen atoms are in grey.
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Nanomaterials 12 00321 g003 550
Figure 3. Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for H-∗-H, C-∗-H, and C-∗-C, exemplified by those in 3–6, 8, 10, and 12. (a) Whole picture; (b) Magnified picture of the C-∗-H area; (c) Magnified picture of the C-∗-C bay area; (d) Magnified picture of the C-∗-C cape area. The definitions of (R, θ) and (θp, κp) are illustrated, exemplified by H-∗-H in 4.
Figure 3. Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for H-∗-H, C-∗-H, and C-∗-C, exemplified by those in 3–6, 8, 10, and 12. (a) Whole picture; (b) Magnified picture of the C-∗-H area; (c) Magnified picture of the C-∗-C bay area; (d) Magnified picture of the C-∗-C cape area. The definitions of (R, θ) and (θp, κp) are illustrated, exemplified by H-∗-H in 4.
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Nanomaterials 12 00321 g004 550
Figure 4. Plot of Δθp versus θp for H-∗-H, C-∗-H, and C-∗-C in 3–12, evaluated with M06-2X/6-311+G(3d,p), where Δθp = (θp – θ).
Figure 4. Plot of Δθp versus θp for H-∗-H, C-∗-H, and C-∗-C in 3–12, evaluated with M06-2X/6-311+G(3d,p), where Δθp = (θp – θ).
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Nanomaterials 12 00321 g005 550
Figure 5. Molecular graphs for helicene dimers, 6:6 (a) and 7:7 (b), calculated with M06-2X/6-311+G(3d,p), where BPs with BCPs corresponding to intra- and intermolecular noncovalent interactions are detected. The BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, CCPs (cage critical points) by green dots, and BPs by pink lines. The carbon atoms are in black and the hydrogen atoms are in grey.
Figure 5. Molecular graphs for helicene dimers, 6:6 (a) and 7:7 (b), calculated with M06-2X/6-311+G(3d,p), where BPs with BCPs corresponding to intra- and intermolecular noncovalent interactions are detected. The BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, CCPs (cage critical points) by green dots, and BPs by pink lines. The carbon atoms are in black and the hydrogen atoms are in grey.
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Nanomaterials 12 00321 g006 550
Figure 6. Plots of Hb(rc) versus Hb(rc) – Vb(rc)/2 for H-∗-H, C-∗-H, and C-∗-C at 6:6 (Ci) and 7:7 (Ci), calculated with M06-2X/6-311+G(3d,p).
Figure 6. Plots of Hb(rc) versus Hb(rc) – Vb(rc)/2 for H-∗-H, C-∗-H, and C-∗-C at 6:6 (Ci) and 7:7 (Ci), calculated with M06-2X/6-311+G(3d,p).
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Nanomaterials 12 00321 g007 550
Figure 7. The internal vibrational motions of ν1 for 7 (C2) and 7:7 (Ci). (a) For 7 (C2) calculated with M06-2X/6-311+G(3d,p); (b) for 7 (C2) calculated with M06-2X/6-311+G(2d,p); (c) for 7 (C2) calculated with LC-ωPBE/6-311+G(2d,p); (d) for 7:7 (Ci) calculated with M06-2X/6-311+G(3d,p); (e) for 7:7 (Ci) calculated with M06-2X/6-311+G(2d,p); (f) for 7:7 (Ci) calculated with LC-ωPBE/6-311+G(2d,p).
Figure 7. The internal vibrational motions of ν1 for 7 (C2) and 7:7 (Ci). (a) For 7 (C2) calculated with M06-2X/6-311+G(3d,p); (b) for 7 (C2) calculated with M06-2X/6-311+G(2d,p); (c) for 7 (C2) calculated with LC-ωPBE/6-311+G(2d,p); (d) for 7:7 (Ci) calculated with M06-2X/6-311+G(3d,p); (e) for 7:7 (Ci) calculated with M06-2X/6-311+G(2d,p); (f) for 7:7 (Ci) calculated with LC-ωPBE/6-311+G(2d,p).
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Table
Table 1. QTAIM Functions and QTAIM-DFA Parameters Evaluated for the Fused Benzene-Type Helicenes of Monomers (3–12), Employing the Perturbed Structures Generated with CIV 1–3.
Table 1. QTAIM Functions and QTAIM-DFA Parameters Evaluated for the Fused Benzene-Type Helicenes of Monomers (3–12), Employing the Perturbed Structures Generated with CIV 1–3.
Speciesρb(rc)c∇2ρb(rc) 4Hb(rc)R 5θ 6Cii 7θp 8κp 9Predicted
Nature
X-∗-Y(eao–3)(au)(au)(au)(°)(Å mdyn−1)(°)(au−1)
3 (1Hbay-∗-4Hbay)0.01300.00600.00200.006371.33.2972.812.4p-CS/vdW
4 (1Hbay-∗-5Hbay)0.01650.00780.00260.008271.86.6774.511.4p-CS/vdW
5 (1Hbay-∗-6Cbay)0.01310.00630.00210.006671.66.1280.8136p-CS/vdW
6 (1Hbay-∗-5Cbay)0.01310.00600.00200.006371.33.8279.931.8p-CS/vdW
7 (1Hbay-∗-6Cbay)0.01350.00630.00210.006771.55.4777.9188p-CS/vdW
7 (2Cbay-∗-7Cbay)0.01140.00510.00170.005471.93.3380.6128p-CS/vdW
8 (1Hbay-∗-6Cbay)0.01300.00610.00210.006571.15.5176.7189p-CS/vdW
8 (2Cbay-∗-7Cbay)0.01170.00520.00170.005571.82.0186.428.5p-CS/vdW
9 (1Hbay-∗-5Cbay)0.01340.00620.00220.006670.73.4079.5627p-CS/vdW
9 (2Cbay-∗-7Cbay)0.01130.00500.00160.005372.11.8381.6119p-CS/vdW
9 (3Cbay-∗-8Cbay)0.01220.00530.00160.005672.91.8785.0196p-CS/vdW
9 (4Ccape-∗-22Ccape) 100.00550.00200.00070.002270.75.2566.9135p-CS/vdW
9 (6Ccape-∗-23Ccape)0.00610.00210.00070.002270.48.5169.437.7p-CS/vdW
10 (1Hbay-∗-6Cbay)0.01370.00640.00210.006771.65.7479.2123p-CS/vdW
10 (2Cbay-∗-7Cbay) 110.01130.00500.00180.005370.51.8694.22890p-CS/t-HBnc
10 (3Cbay-∗-8Cbay)0.01140.00500.00160.005372.11.7882.0182p-CS/vdW
10 (6Ccape-∗-23Ccape)0.00590.00200.00070.002170.49.0269.77.0p-CS/vdW
10 (7Ccape-∗-25Ccape)0.00610.00220.00080.002470.33.4268.010.3p-CS/vdW
11 (1Hbay-∗-6Cbay)0.01360.00630.00210.006771.65.4179.8113p-CS/vdW
11 (2Cbay-∗-7Cbay)0.01160.00510.00170.005471.51.8487.1142p-CS/vdW
11 (3Cbay-∗-8Cbay)0.01150.00500.00160.005372.01.9182.6166p-CS/vdW
11 (4Cbay-∗-9Cbay)0.01110.00490.00160.005271.71.6979.5155p-CS/vdW
11 (4Ccape-∗-22Ccape)0.00530.00190.00070.002070.16.8168.313.5p-CS/vdW
11 (6Ccape-∗-23Ccape)0.00590.00200.00070.002170.210.2169.77.5p-CS/vdW
11 (7Ccape-∗-25Ccape)0.00590.00220.00080.002370.23.5867.87.2p-CS/vdW
11 (9Ccape-∗-26Ccape)0.00620.00210.00070.002270.45.8069.564.7p-CS/vdW
12 (1Hbay-∗-6Cbay)0.01360.00630.00210.006771.54.8480.5103p-CS/vdW
12 (2Cbay-∗-7Cbay)0.01150.00510.00170.005371.51.7787.8349p-CS/vdW
12 (3Cbay-∗-8Cbay)0.01170.00510.00160.005372.31.7483.6241p-CS/vdW
12 (4Cbay-∗-9Cbay)0.01100.00480.00160.005171.51.7280.787.8p-CS/vdW
12 (4Ccape-∗-22Ccape)0.00550.00200.00070.002270.24.8066.3697p-CS/vdW
12 (6Ccape-∗-23Ccape)0.00600.00210.00080.002270.06.7868.08.6p-CS/vdW
12 (7Ccape-∗-25Ccape)0.00590.00220.00080.002370.03.4168.23.3p-CS/vdW
12 (9Ccape-∗-26Ccape)0.00590.00200.00070.002170.36.9569.424.7p-CS/vdW
12 (10Ccape-∗-28Ccape)0.00560.00200.00070.002270.54.0968.665.4p-CS/vdW
1 Calculated with M06-2X/6-311+G(3d,p). 2 Data are given at the BCPs. 3 All interactions are predicted to have the p-CS/vdW nature, except for 10 (2Cbay-∗-7Cbay), which has the p-CS/t-HBnc nature. 4 c∇2ρb(rc) = Hb(rc) − Vb(rc)/2, where c = ℏ2/8m. 5 R = (x2 + y2)1/2, where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)). 6 θ = 90° − tan−1 (y/x). 7 Cij = ∂2E/∂fi∂fj, where i and j refer to internal coordinates, and the external force components acting on the system fi and fj correspond to i and j, respectively. 8 θp = 90° − tan−1 (dy/dx). 9 κp = ?d2y/dx2?/[1 + (dy/dx)2]3/2. 10 Data from w = ±0.0125, ±0.025, and 0 were used for the plot since BCPs were not detected at w = ±0.05. 11 Data from w = −0.05, −0.0375, −0.025, −0.0125, and 0 were used for the plot since BCPs were not detected when w > 0.
Table
Table 2. QTAIM Functions and QTAIM-DFA Parameters Evaluated for the Fused Benzene-Type Helicenes of Concave-Type Dimers (6:6 and 7:7), Employing the Perturbed Structures Generated with CIV 1–3.
Table 2. QTAIM Functions and QTAIM-DFA Parameters Evaluated for the Fused Benzene-Type Helicenes of Concave-Type Dimers (6:6 and 7:7), Employing the Perturbed Structures Generated with CIV 1–3.
Speciesρb(rc)c∇2ρb(rc)Hb(rc)RθCiiθpκpPredicted
Nature
X-∗-Y(eao–3)(au)(au)(au)(°)(Å mdyn−1)(°)(au−1)
6:6 (1Hbay-∗-17’Hcape) 40.00450.00180.00060.001972.625.0288.15163p-CS/vdW
6:6 (1Hcape-∗-17’Hcape)0.00610.00220.00060.002374.335.6376.2184.2p-CS/vdW
6:6 (1Hcape-∗-16’Hcape)0.00510.00190.00070.002070.942.0174.969.9p-CS/vdW
6:6 (15Ccape-∗-17’Ccape) 50.00650.00250.00090.002769.212.6470.81066p-CS/vdW
6:6 (16Ccape-∗-17’Ccape)0.00660.00260.00100.002868.38.6368.053.5p-CS/vdW
6:6 (1Hbay-∗-5Cbay)0.01280.00570.00190.006071.93.81778.332.3p-CS/vdW
6:6 (3Cbay-∗-7Hbay)0.01340.00610.00200.006471.73.41479.329.5p-CS/vdW
7:7 (20Hcape-∗-18’Hcape)0.00730.00310.00120.003368.914.6175.524.6p-CS/vdW
7:7 (20Hcape-∗-20’Hcape)0.00540.00220.00080.002470.027.9072.736.5p-CS/vdW
7:7 (20Hcape-∗-2’Ccape)0.00790.00300.00100.003271.410.0173.3125.7p-CS/vdW
7:7 (18Hcape-∗-3’Ccape)0.00500.00160.00050.001673.817.3673.2181.8p-CS/vdW
7:7 (1Hbay-∗-6Cbay)0.01320.00620.00210.006570.85.55780.093.5p-CS/vdW
7:7 (3Cbay-∗-8Hbay)0.01380.00650.00210.006871.95.02478.8154.1p-CS/vdW
1 Calculated with M06-2X/6-311+G(3d,p). 2 Data are given at the BCPs. 3 See footnotes of Table 1 for the QTAIM-DFA parameters and Cii. 4 Data from w = −0.0375, −0.025, −0.0125, 0, and 0.0125 were used for the plot, since BCPs for 6:6 (1Hbay-∗-17’Hcape) were not detected when w > 0.0125. 5 Data from w = −0.05, −0.0375, −0.025, −0.0125, and 0 were used for the plot, since BCPs for 6:6 (15Ccape-∗-17’Ccape) were not detected when w > 0.
Table
Table 3. QTAIM Functions and QTAIM-DFA Parameters Evaluated for the Fused Benzene-Type Helicene (7) and the Concave-Type Dimer (7:7), Employing the Perturbed Structures Generated with CIV, together with the X-∗-Y Distances and the Corresponding Internal Vibrations, with the Frequencies Closely Related to the Interactions in Question 1,2.
Table 3. QTAIM Functions and QTAIM-DFA Parameters Evaluated for the Fused Benzene-Type Helicene (7) and the Concave-Type Dimer (7:7), Employing the Perturbed Structures Generated with CIV, together with the X-∗-Y Distances and the Corresponding Internal Vibrations, with the Frequencies Closely Related to the Interactions in Question 1,2.
Speciesr(X···Y)RθCiiθpκpPredicted
Nature
X-∗-Y(Å)(au)(°)(Å mdyn−1)(°)(au−1)
7 (M06-2X/6-311+G(3d,p): ν1 = 43.4 cm−1)
1Hbay-∗-6Cbay2.59020.006771.55.4777.9187.5p-CS/vdW
2Cbay-∗-7Cbay2.97510.005471.93.3380.6128.2p-CS/vdW
7 (M06-2X/6-311+G(2d,p): ν1 = 46.6 cm−1)
1Hbay-∗-6Cbay2.58960.006770.45.4981.639.9p-CS/vdW
2Cbay-∗-7Cbay3.00250.005469.93.0678.7120.2p-CS/vdW
7 (LC-ωPBE/6-311+G(2d,p): ν1 = 40.1 cm−1) 3,4
1Hbay-∗-5Cbay2.46810.006670.53.7282.246.4p-CS/vdW
7:7 (M06-2X/6-311+G(3d,p): ν1 = 14.0 cm−1; ν4 = 24.2 cm−1; ν5 = 29.3 cm−1; ν11 = 81.2 cm−1)
20Hcape-∗-18’Hcape2.54230.003368.914.6175.524.6p-CS/vdW
20Hcape-∗-20’Hcape2.71550.002470.027.9072.736.5p-CS/vdW
20Hcape-∗-2’Ccape2.67690.003271.410.0173.3125.7p-CS/vdW
18Hcape-∗-3’Ccape2.96400.001673.817.3673.2181.8p-CS/vdW
7:7 (M06-2X/6-311+G(2d,p): ν1 = 13.0 cm−1; ν4 = 22.1 cm−1; ν5 = 27.3 cm−1; ν11 = 78.6 cm−1)
20Hcape-∗-18’Hcape2.56430.003267.916.6375.260.1p-CS/vdW
20Hcape-∗-20’Hcape2.72870.002468.629.2071.942.2p-CS/vdW
20Hcape-∗-2’Ccape2.68810.003170.211.4469.412.4p-CS/vdW
18Hcape-∗-3’Ccape2.98570.001673.031.4373.2181.9p-CS/vdW
7:7 (LC-ωPBE/6-311+G(2d,p): ν1 = 2.1 cm−1; ν5 = 15.8 cm−1; ν6 = 28.8 cm−1; ν8 = 48.2 cm−1) 5,6
20Ccape-∗-18’Hcape3.23570.001563.865.6367.440.4p-CS/vdW
20Hcape-∗-20’Hcape3.01480.001463.696.4266.990.7p-CS/vdW
20Hcape-∗-2’Ccape3.06560.001467.252.6670.010.9p-CS/vdW
18Hcape-∗-3’Ccape3.65940.000562.973.9973.9394.0p-CS/vdW
1 See footnotes of Table 1 for the QTAIM-DFA parameters and Cii. 2 The motions of the internal vibrations are shown in Figure 7 and Figure S9 of the Supporting Information. 3 BPs and BCPs were not detected for 1Hbay-∗-6Cbay and 2Cbay-∗-7Cbay. 4 r(1Hbay-∗-6Cbay) = 2.5609 Å and r(2Cbay-∗-7Cbay) = 3.2265 Å. 5 BPs and BCPs were not detected for 20Hcape-∗-18’Hcape. 6 r(20Hcape-∗-18’Hcape) = 2.9473 Å.
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Nishide, T.; Hayashi, S. Intrinsic Dynamic and Static Nature of π···π Interactions in Fused Benzene-Type Helicenes and Dimers, Elucidated with QTAIM Dual Functional Analysis. Nanomaterials 2022, 12, 321. https://doi.org/10.3390/nano12030321

AMA Style

Nishide T, Hayashi S. Intrinsic Dynamic and Static Nature of π···π Interactions in Fused Benzene-Type Helicenes and Dimers, Elucidated with QTAIM Dual Functional Analysis. Nanomaterials. 2022; 12(3):321. https://doi.org/10.3390/nano12030321

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Nishide, Taro, and Satoko Hayashi. 2022. "Intrinsic Dynamic and Static Nature of π···π Interactions in Fused Benzene-Type Helicenes and Dimers, Elucidated with QTAIM Dual Functional Analysis" Nanomaterials 12, no. 3: 321. https://doi.org/10.3390/nano12030321

APA Style

Nishide, T., & Hayashi, S. (2022). Intrinsic Dynamic and Static Nature of π···π Interactions in Fused Benzene-Type Helicenes and Dimers, Elucidated with QTAIM Dual Functional Analysis. Nanomaterials, 12(3), 321. https://doi.org/10.3390/nano12030321

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