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

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.

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 1 p -12 p , where 1 p -12 p 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. Nanomaterials 2022, 12, x FOR PEER REVIEW 2 of 20 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. Chart 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. Chart 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.
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, H b (r c ) is plotted versus H b (r c )-V b (r c )/2, where H b (r c ) and V b (r c ) 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 (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 Figure SA1 of the Appendix S1 of the Supporting Information for the definition of the QTAIM-DFA parameters of (R, θ) and (θ p , κ p ), along with Equations (SA3)-(SA6) and the footnotes of Table 1). The concept of the dynamic nature of the interactions was proposed based on (θ p , κ p ). The θ p and κ p 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.  (3)(4)(5)(6)(7)(8)(9)(10)(11)(12), Employing the Perturbed Structures Generated with CIV 1-3 .
Predicted Nature The perturbed structures necessary for QTAIM-DFA were generated by CIV [38], with the coordinates C i corresponding to the compliance constants C ii 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 C ij 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 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.

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 (C ii ) and the coordinates corresponding to C ii (C i ). 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 (S iw ) is generated by adding C i to the standard orientation of a fully optimized structure (S o ) in the matrix representation. The coefficient g iw in Equation (1) controls the structural difference between S iw and S o , g iw is determined to satisfy Equation (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). Five-digit C i values were used to predict S iw .
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 H b (r c ) values are plotted versus the H b (r c ) − V b (r c )/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) = (H b (r c ) − V b (r c )/2, H b (r c )) (R c 2 > 0.99999 as usual) [31].
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, ∆E(n) = E(n) − E(n − 1) for helicenes (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and ∆E(n p ) = E(n p ) − E(n p − 1) for [n]phenacenes (1 p -12 p ). The ∆E(n) values correspond to the energy differences in the formation of n from n − 1, and the ∆E(n p ) values similarly correspond to n p from (n p − 1). The E(n), E(n p ), ∆E(n), and ∆E(n p ) values were calculated on the energy surface, which are described by E ES (n), E ES (n p ), ∆E ES (n), and ∆E ES (n p ), respectively. The values were also calculated with the zero-point energies, which are described by E ZP (n), E ZP (n p ), ∆E ZP (n), and ∆E ZP (n p ). The values calculated with M06-2X/6-311+G(3d,p) are collected in Table S2 of the Supporting Information. The plot of ∆E ZP (n) versus ∆E ES (n) revealed an excellent correlation (y = 1.0042x + 0.6859; R c 2 = 0.980, see Figure S1 of the Supporting Information). As a result, ∆E ES (n) can be used to analyze the energy terms. Figure 1 shows the plots of ∆E ES (n) and ∆E ES (n p ) versus n. Both the ∆E ES (n) and ∆E ES (n p ) values (∆E ES (n; n p )) 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 ∆E ES (3; 3 p ) values are less than the ∆E ES (2; 2 p ) values; however, the ∆E ES (n; n p ) values increase from 3; 3 p to 4; 4 p . In the case of ∆E ES (n p ), the ∆E ES (4 p ) value is somewhat larger than ∆E ES (3 p ) but slightly smaller than ∆E ES (2 p ). The ∆E ES (n p ) value decreases again slightly from 4 p to 5 p . Then, the values are nearly constant for n p ≥ 5 p . 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 1 p -12 p . Namely, the n p system stabilizes almost constantly as the size of the species increase, especially for n p ≥ 5 p , although a change in ∆E ES (n p ) is detected for 2 p ≤ n p < 5 p .  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 The data points for ∆E ES (n) appear to be greater than those for ∆E ES (n p ) when n ≥ 4. The observations must be due to the severe steric repulsion in ∆E ES (n ≥ 4), where the plot for ∆E ES (n p ) corresponds to that without such severe steric repulsion. The ∆E ES (4) value is Nanomaterials 2022, 12, 321 6 of 19 much larger than those of ∆E ES (2) and ∆E ES (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 ∆E ES (n) values decrease in the following order: (8). The contribution of steric repulsion to ∆E ES (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. 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.

Survey of X--Y (X, Y = C and H) in 3-12 with the Molecular Graphs
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 b(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- The ∆E ES (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 ∆E ES (n) (8 ≤ n ≤ 12). The behaviour of ∆E ES (2)-∆E ES (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 ∆E ES (4) value is the largest among ∆E ES (2)-∆E ES (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 ∆E ES (n). The trend in ∆E ES (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 ∆E ES (8) and ∆E ES (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; R c 2 = 0.981), whereas the correlations of the plots for acenes versus helicenes were very poor (y = −0.685x + 0.995; R c 2 = 0.492 if calculated under the closed-shell singlet 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 (r BP ) were calculated for all X- * -Y of 3-12, along with the corresponding straight-line distances (R SL ). The values are collected in Table S4 of the Supporting Information, along with the differences between them (∆r BP = r BP -R SL ). The averaged values of ∆r BP were 0.2040, 0.4006, 0.0588, and 0.1451 Å for H bay - * -H bay , C bay - * -H bay , C bay - * -C bay , and C cape - * -C cape , respectively. As a result, ∆r BP for H bay - * -H bay and C bay - * -H bay were larger than 0.20 Å, while those for C bay - * -C bay and C cape - * -C cape were less than 0.15 Å. Therefore, the BPs corresponding to C bay - * -C bay and C cape - * -C cape can be roughly approximated as straight lines since the ∆r BP values are less than 0.20 Å (see also Figure S4 of the Supporting Information).

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 closedshell (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 inter- 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 (θ p , κ p ) for each X- * -Y of 3-12, along with the C ii 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. 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.
The C atoms in helicenes 3-12 were subdivided into C bay and C cape based on the positions of the atoms in the species, as were the H atoms into H bay and H cape . The bay and cape areas (positions) in the species are illustrated in Scheme 1. While both the C bay and C cape atoms of 3-12 participate in the interactions as BPs, only H bay atoms participate as BPs. The θ and θ p values for H- * -H, C- * -H, and C- * -C of 3-12, collected in Table 2, are all less than 90 • , except for θ p of 2 C bay - * -7 C bay in 10, where (θ, θ p ) = (70.5 • , 94.2 • ). The 2 C bay - * -7 C bay interaction in 10 is denoted by 10 ( 2 C bay - * -7 C bay ) (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 ( 2 C bay - * -7 C bay ), which is predicted to have a p-CS/t-HB nc nature. 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 ( 1 H bay - * -4 H bay ) has been much debated [69][70][71].  Table 1 and Figure S3 of the Supporting Information. Therefore, the BP (H bay - * -C bay ) in 6 and 9 should be analyzed carefully. One, one, four, four, seven, and eight different types of C- * -C interactions are detected The θ values for C cape - * -C cape are slightly smaller than those of C bay - * -C bay (by 0.5-2.2 • ), but the θ p values for C cape - * -C cape are much smaller than those of C bay - * -C bay (by 13.2-24.5 • ). In this case, θ p < θ for C cape - * -C cape , whereas θ p > θ for C bay - * -C bay . Interactions with θ p > θ are usually observed, but interactions with θ p < θ are rare.
Interactions with θ > θ p occur under some specific conditions. To examine the behaviour of θ and θ p in 7-12, the ∆θ p (=θ pθ) values are plotted versus θ p 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: R c 2 = 0.995). (A substantial correlation was not found in the plot of ∆θ p versus θ due to the very small range of θ). The two areas for C- * -C interactions with ∆θ p > 0 and ∆θ p < 0 are clearly illustrated by the green dotted lines in Figure 4 Figure 4 clearly shows that C bay - * -C bay and C cape - * -C cape in 9-12 belong to the areas where ∆θ p > 0 and ∆θ p < 0, respectively.  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  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 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 C cape - * -C cape in 9-12 appears to be similar to that on C bay - * -C bay in fully optimized structures. Namely, the C cape - * -C cape and C bay - * -C bay 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 θ p based on the behaviour of the interactions in the perturbed structures. The C bay - * -C bay interactions in the perturbed structures will be affected by steric compression, similar to the usual cases of interactions, whereas the C cape - * -C cape interactions will be inversely affected compared with the usual cases when measured by the θ p 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 (C i ) and 7:7 (C i ) 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 (C i ), 8:8 (C i ), and/or 10:10 (C i ) 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). Nanomaterials 2022, 12, x FOR PEER REVIEW 10 of 20 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.

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 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).
In the case of intramolecular interactions, 1 Hbay--5 Cbay and 3 Cbay--7 Hbay 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 ( 3 Cbay--7 Hbay) may be due to a structural change at 6:6 relative to 6. Similarly, 1 Hbay-- 6 Cbay and 3 Cbay-- 8 Hbay were detected at 7:7. The former was observed in 7, while 3 Cbay--8 Hbay in 7:7 appeared in place of 2 Cbay--7 Cbay in 7. The change in the In the case of intramolecular interactions, 1 H bay - * -5 C bay and 3 C bay - * -7 H bay 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 ( 3 C bay - * -7 H bay ) may be due to a structural change at 6:6 relative to 6. Similarly, 1 H bay - * -6 C bay and 3 C bay - * -8 H bay were detected at 7:7. The former was observed in 7, while 3 C bay - * -8 H bay in 7:7 appeared in place of 2 C bay - * -7 C bay 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 V xc term is the largest among several possibilities, where V xc 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 (noncovalent 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, θ p ) 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.
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 ν 1 appears to be very similar when calculated at both the M06-2X and LC-ωPBE levels, indicating that ν 1 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 ν 1 . The magnitudes of the displacements in the cape area seem (much) larger than those in the bay area in ν 1 . This will be instructive if the relationship is clarified for that between the magnitudes of the displacements and the ∆θ p values. This issue will be investigated in a future work. The very low energy of ν 1 in 7 suggests the basis set and level dependence can easily change the optimized structure.

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. H b (r c ) was plotted versus H b (r c ) − V b (r c )/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 (θ p , κ p ), where θ p corresponds to the tangent line and κ p is the curvature of the plot. The concept of the dynamic nature of the interactions was proposed based on (θ p , κ p ).
The interactions were analyzed by dividing the C atoms of 3-12 into C bay and C cape and the H atoms into H bay and H cape . While both C bay and C cape atoms of 3-12 take part in the interactions, only H bay atoms participate as BPs. The θ and θ p values for H- * -H, C- * -H, and C- * -C of 3-12 are all less than 90 • , except for 10 ( 2 C bay - * -7 C bay ), where (θ, θ p ) = (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 ( 2 C bay - * -7 C bay ), which is predicted to have a p-CS/t-HB nc nature. While the (θ, θ p ) values of C bay - * -C bay in 7-12 are (71.5-72.9 • , 79.5-87.8 • ), the values are (70.0-70.7 • , 66.3-69.7 • ) for C cape - * -C cape . The θ values for C cape - * -C cape are slightly smaller than those of C bay - * -C bay (by 0.5-2.2 • ), but the θ p values for C cape - * -C cape are much smaller than those of C bay - * -C bay (by 13.2-24.5 • ). In this case, θ < θ p for C bay - * -C bay , whereas θ > θ p for C cape - * -C cape . Interactions with θ < θ p are usually observed, whereas interactions with θ > θ p 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 ( 1 H bay - * -17' H cape ) has a nature close to p-CS/t-HB nc , since (θ, θ p ) = (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 ν 1 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 C ii −1 values. Detecting the interactions and predicting the nature of helicenes will provide a solid basis for investigating and applying the interactions in helicenes.