Behaviour of the XH-*-π and YX-*-π interactions (X, Y = F, Cl, Br and I) in the coronene π-system, as elucidated by QTAIM dual functional analysis with QC calculations

The dynamic and static nature of XH-*-π and YX-*-π in the coronene π-system (π(C24H12)) is elucidated by QTAIM dual functional analysis, where * emphasizes the presence of bond critical points (BCPs) in the interactions. The nature of the interactions is elucidated by analysing the plots of the total electron energy densities Hb(rc) versus Hb(rc) − Vb(rc)/2 [=(ħ2/8m)∇2ρb(rc)] for the interactions at BCPs, where Vb(rc) are the potential energy densities at the BCPs. The data for the perturbed structures around the fully optimized structures are employed for the plots in addition to those of the fully optimized structures. The plots are analysed using the polar coordinate of (R, θ) for the data of the fully optimized structures, while those containing the perturbed structures are analysed using (θp, κp), where θp corresponds to the tangent line of each plot and κp is the curvature. Whereas (R, θ) show the static nature, (θp, κp) represent the dynamic nature of the interactions. All interactions in X–H-*-π(C24H12) (X = F, Cl, Br and I) and Y–X-*-π(C24H12) (Y–X = F–F, Cl–Cl, Br–Br, I–I, F–Cl, F–Br and F–I) are classified by pure CS (closed shell) interactions and are characterized as having the vdW nature, except for X–H = F–H and Y–X = F–Cl, F–Br and F–I, which show the typical-HB nature without covalency. The structural features of the complexes are also discussed.


Introduction
Hydrogen bonds (HBs) and halogen bonds (XBs) are of current and continuous interest. HBs and XBs are fundamentally important for their ability to give rise to molecular association caused by the energy stabilization of the system. [1][2][3][4][5][6][7][8][9][10][11] The direction-control through the formation of HBs plays a crucial role in all elds of chemical and biological sciences. The opening and closing of the duplex DNA structure in active proliferation at around room temperature is a typical example of the effect of HBs. 12 HBs also play an important role in the very specic conformation of hormones with the HBs of the dimers controlling the characteristic biological properties. 13 Conventional HBs of the shared proton interaction type 4 are formed with atoms of the main group elements, which are usually not very strong in the neutral form (# approximately 40 kJ mol À1 ), 1,5 albeit usually stronger than the van der Waals (vdW) interactions. Contributions from the charge transfer (CT) interaction become more important as the strength of HBs increases in addition to the vdW interactions, where attractive electrostatic interactions and the dispersion force mainly contribute to form the vdW adducts. Conversely, the attractive interactions, between the electrophilic s*-orbitals of halogen or interhalogen molecules with the non-bonding orbitals (n-orbitals), must be the driving force for the formation of typical XBs. The nature of XBs has been discussed based on the theoretical background of the molecular orbital description for the bonding and the s-hole developed on the halogen atoms together with the stability based on the structural aspects. 14 XBs are applicable to a wide variety of elds in chemical and biological sciences, such as crystal engineering, supramolecular so matter and nanoparticles.
p-orbitals also give rise to similar HBs and XBs with hydrogen halides and halogen or inter-halogen molecules, respectively. Similar to the case of n-orbitals, p-orbitals act as electron donors to form such adducts. The p-electron systems usually construct planar molecules. Benzene and coronene 15 are the typical examples of the planar p-systems, together with graphene. Graphene shows unique physical properties. Graphene-based carbon allotropes, such as graphene, graphite, fullerenes 16 and carbon nanotubes, have attracted considerable attention owing their many potential applications in nanotechnology, including nanoelectronics, energy storage and (See eqn (S2) of the ESI † for the relation, (ħ 2 /8m)V 2 r b (r c ) ¼ H b (r c ) À V b (r c )/2.) Therefore, it seems difficult to characterize the CS interactions, such as van der Waals (vdW) interactions, 34,35 typical hydrogen bonds (t-HBs), 2,3,36,37 interactions in molecular complexes formed through charge transfer (CT-MCs), 38 trihalide ions (X 3 À ) 38 and interactions in trigonal bipyramidal adducts formed through CT (CT-TBPs). 38 Then, we proposed employing the signs of the rst derivatives of H b (r c ) À V b (r c )/2 and H b (r c ) (d(H b (r c ) À V b (r c )/2)/dr and dH b (r c )/dr, respectively) to characterize these interactions. The borderline between CT-MC and CT-TBP (containing X 3 À ) is dened by d(H b (r c ) À V b (r c )/2)/dr ¼ 0, while that between vdW and t-HB is by dH b (r c )/ dr ¼ 0, as shown by the experimental results, with the presumption that the CS interactions are reasonably characterized as expected. The proposed denitions for the classication of interactions are summarized in Table S1 of the ESI, † together with those tentatively proposed, 39 for convenience of discussion.
Recently, we proposed QTAIM dual functional analysis (QTAIM-DFA), 40-43 according to QTAIM. 26-29,44,45 QTAIM-DFA provides an excellent approach for evaluating, classifying and understanding weak to strong interactions in a unied form. [40][41][42][43] In QTAIM-DFA, H b (r c ) are plotted versus H b (r c ) À V b (r c )/2 [¼ (ħ 2 /8m) V 2 r b (r c )]. In our treatment, data for perturbed structures around fully optimized structures are employed for the plots, in addition to those from the fully optimized structures. 40-43 QTAIM-DFA can incorporate the classication of interactions by the signs of dr and dH b (r c )/dr with the denitions, tentatively proposed. 46 We have proposed the concept of "the dynamic nature of interactions" which originates from the data containing the perturbed structures. 40a,41-43 Data from the fully optimized structures correspond to the static nature of interactions. QTAIM-DFA is applied to typical chemical bonds and interactions and rough criteria are established. The rough criteria can distinguish the chemical bonds and interactions in question from other types of interactions. QTAIM-DFA and these criteria are explained in the ESI using Schemes S1 and S2, Fig. S1 and eqn (S1)-(S6). † The basic concept of the QTAIM approach is also surveyed.
We consider QTAIM-DFA to be well-suited to elucidate the dynamic and static nature of the p-HBs and p-XBs interactions in p(C 24 H 12 ), even though static behaviour of p-HBs in p(C 24 H 12 ) has been discussed. 47,48 In this study, we present the results of the investigations on the nature of the interactions. The interactions are classied and characterized based on the above criteria.

Methodological details in calculations
The structures were optimized using the Gaussian 09 programme package. 49 The basis set system (BSS) from the Sapporo Basis Set Factory 50 (BSS-S) was employed for the calculations. In the calculations with BSS-SA, the (7433211/743111/7411/2 + 1s1p) type was employed for I, the (743211/74111/721/2 + 1s1p) type for Br, the (63211/6111/31/2 + 1s1p) type for Cl and the Table 1 Structural parameters for X-H/p(C 24 H 12 ) and Y-X/p(C 24 H 12 ), optimized with M06-2X/BSS-SA a,b Y-X-*-p(C 24 H 12 ), (symmetry: type)  12 ) and X-H or Y-X. g Br-H being placed above the midpoint between 2 C and 3 C, which is dened by type IB 0 Cor . In this case, the r 1 value is measured from 2 C. h Close to the C s symmetry, where Cl in F-Cl pointing to 12 M, the midpoint between 1 C and 2 C.
(6211/311/21/2 + 1s1p) type for F with the (6211/311/21/2 + 1s1p) type for C and the (411/21/2 + 1s1p) type for H. BSS-SA was applied for the calculations at the M06-2X (M06-2X/BSS-SA) level of density functional theory (DFT). Optimized structures were conrmed by the frequency analysis. QTAIM functions were similarly calculated using the Gaussian 09 programme package 49 with the same method of the optimizations and the data were analysed with the AIM2000 51 and AIMAll 52 programmes. The results obtained at the M06-2X/BSS-SA level of theory will be mainly discussed in the text.
For BSS-SB, the (743321/74321/742 + 1s1p) type was employed for I, the (74321/7421/72 + 1s1p) type for Br, the (6321/ 621/3 + 1s1p) type for Cl and the (621/31/2 + 1s1p) type for F with the (621/31/2 + 1s1p) type for C and the (31/3 + 1s1p) type for H. The calculations were also performed at the M06-2X/BSS-SB level of theory to search for the potential energy surface minima as the pre-optimizations, when necessary. M06-2X/BSS-SB is also employed to conrm the minima and BPs with BCPs around the interactions in question, if they are not obtained satisfactorily with M06-2X/BSS-SA.
The results obtained using M06-2X/BSS-SB are discussed in Tables S1 and S2 of the ESI † and/or the text, if necessary. M06-2X/BSS-SA was also applied to the benzene p-system for convenience of comparison. The calculations were similarly performed using MP2/6-311G(d,p) 53,54 for the convenience of comparison. The results are collected in the ESI. † Normal coordinates of internal vibrations (NIV) obtained by the frequency analysis were employed to generate the perturbed structures. 41,42 This method is explained by eqn (1). A k-th perturbed structure (S kw ) was generated by the addition of the normal coordinates of the k-th internal vibration (N k ) to the standard orientation of the fully optimized structure (S o ) in the BCPs are denoted by red dots, RCPs by yellow dots, CCPs by green dots and BPs by pink lines. Carbon atoms are in black and hydrogen atoms are in grey, with fluorine, chlorine, bromine and iodine atoms in dark yellow, green, dark purple and purple, respectively. The contour plot of r(r) is also drawn for each on the plane containing the H-*-3 C(C 24 H 12 ) moiety for type IB Cor with the H-*-2 C(C 24 H 12 ) moiety for type IB 0 Cor or on the plane of the H-*-12 M(C 24 H 12 ) moiety for type IC Cor , where the contour plot is drawn on each plane. matrix representation. 55 The coefficient f kw in eqn (1) controls the difference in the structures between S kw and S o : f kw are determined to satisfy eqn (1) for the interaction in question, where r and r o show the distances under investigation in the perturbed and fully optimized structures, respectively, and a o is the Bohr radius (0.52918Å). 56 Namely, the perturbed structures with NIV correspond to those with r being elongated or shortened by 0.05a o or 0.1a o , relative to r o . N k of ve digits are used to predict S kw . We refer to this method to generate the perturbed structures as NIV.
In the QTAIM-DFA treatment, H b (r c ) are plotted versus H b (r c ) À V b (r c )/2 for ve data points of w ¼ 0, AE0.05 and AE0.1 in eqn (2). Each plot is analysed using a regression curve of the cubic function as shown in eqn (3)

Results and discussion
Optimizations of X-H/p(C 24 H 12 ) and Y-X/p(C 24 H 12 ) The structures were optimized for X-H/p(C 24 H 12 ) and Y-X/ p(C 24 H 12 ). The optimizations were initially performed with M06-2X/BSS-SB, assuming the C 1 symmetry. The X-H and Y-X Carbon atoms are in black and hydrogen atoms are in grey, with fluorine, chlorine, bromine and iodine atoms in dark yellow, green, dark purple and purple, respectively. The contour plot of r(r) is also drawn for each on the plane containing the X-*-3 C(C 24 H 12 ) moiety for type IB Cor or on the plane of X-*-12 M(C 24 H 12 ) moiety for type IC Cor , where the contour plot is drawn on each plane.
This journal is © The Royal Society of Chemistry 2018 Table 2 QTAIM functions and QTAIM-DFA parameters for X-H-*-p(C 24 H 12 ) and Y-X-*-p(C Y-X-*-p(C 24 H 12 ), (symmetry: type) Data are given at BCP, which is shown by X-*-p. c cV 2 components were placed in close proximity to types IA Cor , IB Cor and IC Cor together with type ID Cor (see Schemes 1 and 2) in the optimization processes, but the systematic search was not performed. Each adduct nally converged to a structure with the C 1 symmetry. The structures were optimized again with M06-2X/ BSS-SA. The optimized structures are conrmed by all positive frequencies aer the frequency analysis. Then, the C 1 structures with all positive frequencies were further optimized, assuming the C s symmetry in the cases where the C 1 structures appeared to be very close to the C s symmetry. The frequency analysis was also performed on the C s structures. The IB Cor and IC Cor types were predicted for X-H/p(C 24 H 12 ), while the IA Cor and IC Cor types were used for Y-X/p(C 24 H 12 ), when optimized with M06-2X/BSS-SA.
All positive frequencies were conrmed for all adducts, except for F-H/p( 3 C) (C 1 : IB Cor ), Cl-H/p( 12 M) (C 1 : IC Cor ), Cl-Cl/p( a C) (C 1 : IA Cor ), Cl-Cl/p( 12 M) (C 1 : IC Cor ) and F-Cl/ p( 12 M) (C 1 : IC Cor ). The motion of each imaginary frequency mainly corresponds to the angular displacements between p(C 24 H 12 ) and X-H or Y-X. In the case of Cl-H/p( 2 C) (C 1 : IB Cor ), the calculation converged to Cl-H/p( 12 M) (C 1 : IC Cor ), which did not give positive frequencies only aer the frequency analysis. Table 1 summarizes the structural parameters (r 1 , r 2 , q 1 , q 2 , f 1 and f 2 ) of X-H/p(C 24 H 12 ) and Y-X/p(C 24 H 12 ), dened in Scheme 1. The optimized structures are not shown in gures, but a number of them can be observed in Fig. 1 and 2. Cor ). This is a very gentle potential energy surface around the inter-conversion between Cl-H/p(C 24 H 12 ) (IC Cor ) and the related structure. Similarly, for the cases discussed above, all positive frequencies only were not predicted for F-H/p( 3 C) (C 1 : IB Cor ) and Cl-H/p( 12 M) (C 1 : IC Cor ) in X-H/p(C 24 H 12 ). This is also owing to the very gentle potential energy surface around the motions of the imaginary frequencies for Cl-Cl/p( a C) (C 1 : IA Cor ), Cl-Cl/p( 12 M) (C 1 : IC Cor ) and F-Cl/p( 12 M) (C 1 : IC Cor ) in Y-X/p(C 24 H 12 ). Nevertheless, with the exception of Cl-H/ p( 12 M) (C 1 : IC Cor ), positive frequencies only are predicted for these cases when the calculations are performed with M06-2X/ BSS-SB. The results are collected in Table S1 of the ESI. † The energy differences between X-H/p(C 24 H 12 ) and Y-X/ p(C 24 H 12 ) and the components, DE (¼ E(X-H/p(C 24 H 12 )/Y-X/ p(C 24 H 12 )) À (E(X-H/Y-X) + E(C 24 H 12 ))) (DE ES and DE Ent ), are also given in Table 1. IC Cor ) appear to deviate somewhat from the correlation (Fig. S2 of the ESI †). A much better correlation was obtained if the data for the ve species are omitted from the correlation (y ¼ 1.000x + 3.90: R c 2 ¼ 0.986 (n ¼ 16)). Therefore, DE ES can be used for the discussion of DE. Aer the elucidation of the structural feature of X-H/ p(C 24 H 12 ) and Y-X/p(C 24 H 12 ), molecular graphs, contour plots, negative Laplacians and trajectory plots are examined next.
Molecular graphs, contour plots, negative Laplacians and trajectory plots for X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) Cor ) or on the plane of H-*-12 M moiety for X-H-*-p(C 24 H 12 ) (IC Cor ), albeit partially. Fig. 2 shows the molecular graphs for the IA Cor and IC Cor types of F-X-*-p(C 24 H 12 ) (X ¼ F, Cl, Br and I), calculated with M06-2X/BSS-SA. The contour plot of r(r) is drawn for each adduct partially, similar to Fig. 1.
In Fig. 1, all expected BCPs are clearly observed, including those for the XH-*-p and YX-*-p interactions in question, together with ring critical points (RCPs) and cage critical points (CCPs), if such exist. The structural feature is visualized well by the molecular graphs. The BPs for H-*-p and X-*-p in question seem linear for most of X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ), although some seem somewhat bending. BCPs are well located at the (three-dimensional) saddle points of r(r). Negative Laplacians and trajectory plots are drawn for X-H-*-p(C 24 H 12 ), similar to Fig. 1 and are shown in Fig. S3 and S4 of the ESI, † respectively. Negative Laplacians and trajectory plots are also drawn for Y-X-*-p(C 24 H 12 ), similar to Fig. 2, and are shown in Fig. S5 and S6 of the ESI, † respectively. The behaviour of the BCPs is well-visualized through V 2 r(r) as shown in Fig. S3 and S5 of the ESI. † All BCPs in X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) are placed in the blue areas of the negative Laplacians; therefore, the interactions corresponding to the BCPs should be classied by the CS interactions. The space around the species around the interactions in question is well divided into atoms, as demonstrated in Fig. S4 and S6 of the ESI. † Survey of X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions, evaluated with M06-2X/BSS-SA How can the X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions be described? The interactions can be dened by the corresponding BPs, although we must be careful to use the correct terminology with this concept. 31 As shown in Fig. 1 and 2, BPs for the adducts appear to be straight, with the exception of X-H-*-p(C 24 H 12 ) (C 1 : IB Cor ) (X ¼ Br and I) and Y-X-*-p( 12 M) (C 1 : IC Cor ) (Y-X ¼ F-Cl and Br-Br). The lengths of BPs (r BP ) and the straight-line distances (R SL ) evaluated with M06-2X/BSS-SA, are collected in Table S3 of the ESI † together with the Dr BP (¼ r BP À R SL ) values. The Dr BP value are 0.68Å for F-Cl-*-p( 12 M) (C 1 : IC Cor ), 0.41Å for Br-Br-*-p( 12 M) (C 1 : IC Cor ), 0.35Å for Br-H-*-p( 2 C) (C 1 : IB Cor ) and 0.17Å for I-H-*-p( 3 C) (C 1 : IB Cor ). However, the Dr BP values are smaller than 0.064Å for X-H-*-p(C 24 H 12 ) and smaller than 0.015Å for Y-X-*-p(C 24 H 12 ) (C s : IA Cor ) (X, Y ¼ F, Cl, Br and I), as shown in Table S3. † Therefore, the H-*-p and X-*-p interactions in the coronene p-system can be approximated as straight lines, except for the four species, although Dr BP ¼ 0.064Å for F-H-*-p( 3 C) (C 1 : IB Cor ). The plot of r BP versus R SL for the adducts gave an excellent correlation (y ¼ 0.966x + 0.1079; R c 2 ¼ 0.999 (n ¼ 16)), if the data of the four species are neglected from the correlation (not shown in the gure). QTAIM functions are evaluated for the H-*-p and X-*-p interactions at BCPs in X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) (X, Y ¼ F, Cl, Br and I) using the M06-2X functional. The obtained values are presented in Table 2. Fig. 3 shows the plot of H b (r c ) versus H b (r c ) À V b (r c )/2 for the data in Table 2 and those from the perturbed structures around the fully optimized structures. All data in Fig. 3 appear in the region of H b (r c ) À V b (r c )/2 > 0 and H b (r c ) > 0, and therefore, all interactions in question are clas-sied by the pure CS interactions.
Nature of X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions, evaluated with M06-2X/BSS-SA Fig. 3 are analysed according to eqn (S3)-(S6) of the ESI, † which provide the QTAIM-DFA parameters of (R, q) and (q p , k p ). Table 2 collects the frequencies, correlated to NIV employed to generate the perturbed structures and the force constants, k f . The nature of the interactions in question is classied and characterized based on the QTAIM-DFA parameters, employing the standard values (criteria) as the reference. Table 3 summarizes the predicted nature of H-*-p in X-H-*-p(C 24 H 12 ) and X-*-p in Y-X-*-p(C 24 H 12 ), employing the q and q p values evaluated with M06-2X/BSS-SA.
As summarized in Table 3, the q and q p values in X-H-*-p(C 24 H 12 ) decrease in the order of X-H-¼ F-H-> Br-H-> I-H-, even though q p for X-H-¼ I-H-appears to be somewhat larger than that for the case of X-H-¼ Br-H-. The results show that q and q p in X-H-*-p(C 24 H 12 ) are controlled by the electronegativity of X. Namely, the values will be larger if the polarity of the X dÀ -H d+ type becomes larger. Conversely, q and q p in Y-X-*-p(C 24 H 12 ) become larger in the order of Y-X-¼ F-F-< Cl-Cl-< Br-Br-< I-I-< F-Cl-< F-Br-< F-I-. These results would be the reection of two factors. The rst is the soness of X. The q and q p values become larger with increasing soness of X. The Table 3 Nature of the H-*-p and X-*-p interactions in X-H-*-p(C 24 H 12 ) and Y-X-*-p (C 24 H   second factor is the polarity of Y dÀ -X d+ . The q and q p values increase with increasing polarity, resulting in the larger extension of s*(X-Y) at the X side. This is very interesting because the q and q p values are larger for Y-X-¼ F-Cl-, relative to the case of Y-X-¼ I-I-. The predicted nature is discussed next. It would be instructive to review the criteria before the detailed discussion of the nature for H-*-p and X-*-p. The criteria specify that q < 180 (H b (r c ) À V b (r c )/2 > 0) for the CS interactions and q > 180 (H b (r c ) À V b (r c )/2 < 0) for the SS interactions. The CS interactions for q < 180 are sub-divided into the pure CS interactions for 45 < q < 90 (H b (r c ) > 0) and the regular CS interactions for 90 < q < 180 (H b (r c ) < 0). The q p value plays an important role in characterizing the interactions. In the pure CS region of 45 < q < 90 , the character of interactions will be the vdW type for 45 < q p < 90 and the typical-HB type (t-HB) with no covalency (t-HB nc ) for 90 < q p < 125 , where q p ¼ 125 is tentatively given, corresponding to q ¼ 90 . The regular CS (90 < q < 180 ) and SS (180 < q) interactions are not discussed here, since the interactions in this region are not detected in this work.
The q values are less than 90 for all X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions examined in this work. Therefore, the H-*-p and X-*-p interactions are all classied by the pure CS interactions. On the other hand, the q p values are less than 90 for all interactions with the exception of F-H-*-p(C 24 H 12 ) of the IB Cor and IC Cor types and F-X-*-p(C 24 H 12 ) (X ¼ Cl, Br and I) of the IA Cor and IC Cor types. The interactions in X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) are all characterized as the vdW nature for those with q p < 90 . The interactions with q p > 90 are characterized to have the nature of typical hydrogen bonds with no covalency (t-HB nc ). However, the nature of the H-*-p interactions in F-H-*-p( 12 M) (C s : IC Cor ) should be examined carefully. The q p value is 129.1 , which is larger than 125 . The results suggest that the H-*-p interaction should be characterized as t-HB with covalency (t-HB wc ). However, the q value of 89.1 is less than 90 , therefore, the interaction must have no covalency. In this case, the q value should have the priority to the q p value in the prediction of the nature of the interaction, since q p is only given tentatively corresponding to q ¼ 90 . Therefore, the H-*-p interaction in F-H-*-p( 12 M) (C s : IC Cor ) would be better characterized as t-HB nc . However, the interaction appears to be close to the borderline area between t-HB nc and t-HB wc , since q ¼ 89.1 is close to 90 , while q p ¼ 129.1 > 125 .
The X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions (X, Y ¼ F, Cl, Br and I) were also analysed for the ID Cor type with M06-2X/BSS-SA (see Scheme 2). The results of this analysis are discussed next.
Nature of X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions of the ID Cor type, evaluated with M06-2X/BSS-SA Indeed, the ID Cor type is not optimized for X-H-*-p(C 24 H 12 ) with M06-2X/BSS-SA, even though they are optimized when calculated at the MP2 level. The nature of the Y-X-*-p(C 24 H 12 ) interactions around the main axis of p(C 24 H 12 ) is also very interesting. Therefore, X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) are optimized assuming the C 2v symmetry. The structural parameters are presented in Table S4 of the ESI, † and are dened in Scheme 2. Table S5 of the ESI † presents the QTAIM-DFA parameters of (R, q) and (q p , k p ) evaluated with M06-2X/BSS-SA, together with the frequencies correlated to NIV employed to generate the perturbed structures and the force constants k f .
The nature of the interactions in question is classied and characterized based on the QTAIM-DFA parameters, employing the standard values as the reference. Table 4 summarizes the predicted nature of the H-*-p and X-*-p interactions in X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) of the C 2v symmetry, respectively, employing the q and q p values evaluated with M06-2X/ BSS-SA. As summarized in Table 4, the q and q p values in X-H-*-p(C 24 H 12 ) (C 2v : ID Cor ) decrease in the order of X-H-¼ F-H-> Cl-H-> I-H-> Br-H-. On the other hand, the q p values in Y-X-*-p(C 24 H 12 ) increase in the order of F-F-< Cl-Cl-< I-I-and F-Cl-< Br-Br-< F-Br-< F-I-. The q and q p are smaller than 90 for all interactions in X-H-*-p(C 24 H 12 ) (C 2v : ID Cor ) and Y-X-*-p(C 24 H 12 ) (C 2v : ID Cor ) (see Table 4). Therefore, the H-*-p and X-*-p interactions are all classied by the pure CS interactions and are characterized to be of the vdW nature (p-CS/vdW).
Nature of X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) versus that of X-H-*-p(C 6 H 6 ) and Y-X-*-p(C 6 H 6 ) The Y-X-*-p(C 6 H 6 ) and X-H-*-p(C 6 H 6 ) interactions (X, Y ¼ F, Cl, Br and I) are similarly evaluated with M06-2X/BSS-SA. The results are presented in Table S6 and S7 of the ESI. † Fig. 4 shows the plots of q and q p for Y-X-*-p(C 24 H 12 ) versus those of Y-X- Table 4 Nature of the H-*-p and X-*-p interactions in X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) of the C 2v symmetry, respectively, evaluated with M06-2X/BSS-SA a Y-X-*-p(C 24 H 12  *-p(C 6 H 6 ) for convenience of comparison. As shown in Fig. 4, the q and q p values for Y-X-*-p( a C: C 24 H 12 ) (IA Cor ) appear to be somewhat smaller than those for Y-X-*-p(C 6 H 6 ) (C s : IB Bzn ), respectively, if those of the same Y-X are compared, whereas the values for Y-X-*-p( 12 M: C 24 H 12 ) (IC Cor ) are predicted to be larger than those for Y-X-*-p(C 6 H 6 ) (C s : IB Bzn ), respectively. Conversely, the q and q p values for Y-X-*-p(M o : C 24 H 12 ) (C 2v : ID Cor ) are very close to those for Y-X-*-p(C 6 H 6 ) (C 2v : ID Bzn ), respectively, if those of the same Y-X are compared. What is the reason for the predicted results shown in Fig. 4? The charge developed on the C and H atoms of benzene and coronene is examined as the possible origin of these results. Scheme 3 shows the charge evaluated based on the natural population analysis (Q n ) with MP2/6-311G(d,p). 57 The outside C C-H bonds in coronene are predicted to be substantially positively charged relative to the case of benzene, and the inside A C 6 atoms are almost neutral, resulting in the negative charge accumulated on the B C atoms (see, Scheme 3). The results show that the q and q p values for Y-X-*-p(C 24 H 12 ) would be larger than those for Y-X-*-p(C 6 H 6 ), respectively, if Q n for the former or around the interaction is smaller than for the latter. For the small range of the interactions in the adducts, the electron-electron repulsion may play a more important role in the strength of the X-*-p interaction rather than the attractive interaction such as the CT interaction.
The q and q p values for X-H-*-p(C 24 H 12 ) are similarly plotted versus those for X-H-*-p(C 6 H 6 ), as shown in Fig. S9 of the ESI. † In this case, the q and q p values increase in the order of p(M o : C 24 H 12 ) (ID Cor ) < p(C 6 H 6 ) # p( 3 C: C 24 H 12 ) (IB Cor ) < p( 12 M: C 24 H 12 ) (IC Cor ), if those of the same X-H are compared. These results appear to be in close agreement to those for Y-X-*-p(C 24 H 12 ) with Y-X-*-p(C 6 H 6 ) (see, Fig. S9 of the ESI †), even though small differences between the two cases are observed.
The H-*-p and X-*-p interactions in the bent p-systems are also of highly interest. An investigation of such interactions is currently in progress.

Conclusions
QTAIM-DFA was applied to the X-H-*-p(C 24 H 12 ) (X ¼ F, Cl, Br and I) and Y-X-*-p(C 24 H 12 ) (Y-X ¼ F-F, Cl-Cl, Br-Br, I-I, F-Cl, F-Br and F-I) interactions, which must be of fundamental importance. The structures were optimized mainly at the M06-2X/BSS-SA level of theory. Four types of structures were optimized for X-H/p(C 24 H 12 ) and Y-X/p (C 24 H 12 ) (types IA Cor , IB Cor , IC Cor and ID Cor ) (see, Schemes 1 and 2). The IB Cor and IC Cor types were predicted for X-H/p(C 24 H 12 ), while the IA Cor and IC Cor types were for Y-X/p(C 24 H 12 ), if optimized with M06-2X/BSS-SA. All BCPs expected are clearly observed in the molecular graphs drawn on the optimized structures.
QTAIM-DFA parameters of (R, q) and (q p , k p ) are calculated for H-*-p in X-H-*-p(C 24 H 12 ) and X-*-p in Y-X-*-p(C 24 H 12 ) by analysing the plots of H b (r c ) versus H b (r c ) À V b (r c )/2 at BCPs. The q values are smaller than 90 for all X-H-*-p(C 24 H 12 ) and Y-X-*-p(C 24 H 12 ) interactions, and are therefore classied as the pure CS interactions. The q p values are larger than 90 for F-H-*-p(C 24 H 12 ) of the IB Cor and IC Cor types and F-X-*-p(C 24 H 12 ) (X ¼ Cl, Br and I) of the IA Cor and IC Cor types; therefore, they have the t-HB nc nature. The H-*-p interaction in F-H-*-p(C 24 H 12 ) (C s : type IC Cor ) appear to be present close to the borderline area between t-HB nc and t-HB wc , since q ¼ 89.1 , which is close to 90 , while q p ¼ 129.1 > 125 . The H-*-p and X-*-p interactions other than above have the vdW nature due to q p < 90 . The q and q p values are smaller than 90 for all interactions in question in X-H-*-p(C 24 H 12 ) (C 2v : ID Cor ) and Y-X-*-p(C 24 H 12 ) (C 2v : ID Cor ). Therefore, the H-*-p and X-*-p interactions around the main axis of p(C 24 H 12 ) in the adducts are all predicted to have the nature of p-CS/vdW. The q and q p values for Y-X-*-p(M o : C 24 H 12 ) (C 2v : ID Cor ) are very close to the corresponding values for Y-X-*-p(C 6 H 6 ) (C 2v : ID Bzn ), respectively. Conversely, the q and q p values for Y-X-*-p( a C: C 24 H 12 ) (IA Cor ) appear to be somewhat smaller than the corresponding values for Y-X-*-p(C 6 H 6 ) (C s : IB Bzn ), respectively, whereas the values for Y-X-*-p( 12 M: C 24 H 12 ) (IC Cor ) are predicted to be larger than those for Y-X-*-p(C 6 H 6 ) (C s : IB Bzn ), respectively.

Conflicts of interest
The authors declare no conict of interest.