Dynamic and Static Nature of XH-∗-π and YX-∗-π (X = F, Cl, Br, and I; Y = X and F) in the Distorted π-System of Corannulene Elucidated with QTAIM Dual Functional Analysis

Abstract

The dynamic and static nature of the XH-∗-π and YX-∗-π (X = F, Cl, Br, and I; Y = X and F) interactions in the distorted π-system of corannulene (π(C₂₀H₁₀)) is elucidated with a QTAIM dual functional analysis (QTAIM-DFA), where asterisks emphasize the presence of bond critical points (BCPs) on the interactions. The static and dynamic nature originates from the data of the fully optimized and perturbed structures, respectively, in QTAIM-DFA. On the convex side, H in F–H-∗-π(C₂₀H₁₀) and each X in Y–X-∗-π(C₂₀H₁₀) join to C of the central five-membered ring in π(C₂₀H₁₀) through a bond path (BP), while each H in X–H-∗-π(C₂₀H₁₀) does so to the midpoint of C=C in the central five-membered ring for X = Cl, Br, or I. On the concave side, each X in F–X-∗-π(C₂₀H₁₀) also joins to C of the central five-membered ring with a BP for X = H, Cl, Br, and I; however, the interactions in other adducts are more complex than those on the convex side. Both H and X in X–H-∗-π(C₂₀H₁₀) (X = Cl and Br) and both Fs in F–F-∗-π(C₂₀H₁₀) connect to the three C atoms in each central five-membered ring (with three BPs). Two, three, and five BPs were detected for the Cl–Cl, I–H, Br–Br, and I–I adducts, where some BPs do not stay on the central five-membered ring in π(C₂₀H₁₀). The interactions are predicted to have a vdW to CT-MC nature. The interactions on the concave side seem weaker than those on the convex side for X–H-∗-π(C₂₀H₁₀), whereas the inverse trend is observed for Y–X-∗-π(C₂₀H₁₀) as a whole. The nature of the interactions in the π(C₂₀H₁₀) adducts of the convex and concave sides is examined in more detail, employing the adducts with X–H and F–X placed on their molecular axis together with the π(C₂₄H₁₂) and π(C₆H₆) adducts.

1. Introduction

Hydrogen bonds (HBs) [1,2,3,4,5,6,7,8,9,10,11,12] and halogen bonds (XBs) [13,14,15,16] are fundamentally important because of their molecular association ability due to the stabilization of the energy system. HBs and XBs are applied in a wide variety of fields in the chemical and biological sciences [17,18,19] such as crystal engineering, supramolecular soft matter, and nanoparticles. The nature of HBs and XBs has also been discussed based on the theoretical background with the structural aspects [15,20] containing the σ-hole developed on the halogen atoms in XBs. The XH-∗-π and YX-∗-π adducts also form when π-orbitals interact with hydrogen halides and halogen or interhalogen molecules. They are referred to as π-HBs and π-XBs, respectively. In this case, the electrophilic σ*-orbitals of the molecules interact attractively with the π-orbitals, similarly to the case of the n-orbitals on the heteroatoms.

We recently reported the dynamic and static nature of π-HBs and π-XBs in the coronene π-system (π(C₂₄H₁₂)) [21,22], together with those in benzene (π(C₆H₆)) [23,24,25], naphthalene (π(C₁₀H₈)) [26], and anthracene (π(C₁₄H₁₀)) [27]. We have also been very interested in the behavior of π-HBs and π-XBs in distorted π-systems. π-Electron systems, such as π(C₂₄H₁₂), seem moderately rigid and moderately flexible. As a result, distorted π-systems such as corannulene (C₂₀H₁₀) [28,29,30,31] and sumanene (C₂₁H₁₂) [32,33] will form, where π(C₂₀H₁₀) and π(C₂₁H₁₂) are found in fullerenes as partial structures. It must be of particular interest to clarify the differences in the reactivity between the planar and distorted π-systems. The π-systems of coronene (π(C₂₄H₁₂)) and corannulene (π(C₂₀H₁₀)) must be the attractive candidates for the purpose. In the case of the bowl-shaped π(C₂₀H₁₀), the π-orbitals are extended to the convex (cv) side, while they will shrink to the concave (cc) side. This electronic structure will play an important role in the formation of the adducts between corannulene and XH and XY. These clarifications will allow us to anticipate the interactions of bowl-shaped aromatic ring compounds.

The main characteristics of the π-system in corannulene, together with the differences from that of coronene, are explained as follows. Both corannulene and coronene are the neutral aromatic hydrocarbons of the condensed benzene structures, where the central rings of corannulene and coronene contain five and six membered rings of the π-systems, respectively. The π-system of the five membered ring will be more negative, relative to the case of that of the six membered ring, since both π-systems tend to be stabilized by the formation of the 6π electron system. This factor can be examined based on the charge distributions in corannulene and coronene calculated based on the natural population analysis (NPA) [34]. Scheme 1 showed the charge (Qn) evaluated by NPA. The ^aC atom of the central five-membered ring in corannulene (Qn(^aC) = –0.016) is more negatively charged than that of coronene (Qn(^aC) = –0.009) (cf.: Qn(^fC) = –0.048 in corannulene and Qn(^gC) = –0.050 in coronene). On the other hand, Qn(C–H) (= 0.032) of the outside ring in corannulene is charged positively slightly more than that in coronene (Qn(C–H) = 0.031), where Qn(C–H) = 0.000 for benzene.

In the bowl-shaped π-system of C₂₀H₁₀, the π-orbitals are extended to the cv side, while they will shrink to the cc side. Namely, the electron density ρ(r) must be more widely extended on the cv side relative to the case of the cc side, as expected. The expectation can be visualized by the electron potential surface (EPS) on the cv and cc sides of corannulene. Figure 1 shows EPS on the cv and cc sides of corannulene, together with the lateral view. The electron–electron repulsive factor between π(C₂₀H₁₀) and B–A in B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X) will also play an important role in the interaction distances. The interaction distances are named r₁ (see Scheme 2). Such a repulsion could be larger on the cc side than that on the cv side in the bowl-shaped π(C₂₀H₁₀), although r₁ decreases as the CT interaction of the π(C₂₀H₁₀)→σ*(X–Y: X = X or F) type increases. The A and/or B dependence in r₁ is of interest.

The nature of the XH-∗-π in XH-∗-π(C₂₀H₁₀) and YX-∗-π in YX-∗-π(C₂₀H₁₀) will be elucidated, keeping in mind the above viewpoints, comparing that of the nature in the π(C₂₄H₁₂) (and π(C₆H₆)), after the optimizations of the adducts.

Scheme 2 illustrates the structures of corannulene adducts B–A⋯π(C₂₀H₁₀) (B–A = Y–X: X = F, Cl, Br, and I; Y = H, X, and F) to be clarified. Scheme 2 defines the types of structures together with the structural parameters. The optimized structures of B–A⋯π(C₂₀H₁₀) are referred to as type I_Cora if B–A interacts with the corannulene π-system through only one site of B–A. In this case, B–A is aligned close to the molecular axis of corannulene. Type I_Cora are referred to as type IA_Cora and type IB_Cora, respectively, if B–A interacts with a carbon atom or a midpoint of a C=C bond. The bowl-shaped π-system of π(C₂₀H₁₀) extends more widely over the outside area of cv but more narrowly to the inside area of cc. The structures of the adducts are controlled by the different electronic structures of the two sides. Type I_Cora are referred to as type I_Cora:cv and I_Cora:cc, respectively, if B–A interacts with π(C₂₀H₁₀) on the cv and cc sides, respectively. The structure is type IIB_Cora when B–A seems to interact with π(C₂₀H₁₀) through both sides of B–A, which is observed on the cc side.

Scheme 3 shows the structures of B–A⋯π (B–A = Y–X: X = F, Cl, Br, and I; Y = H, X, and F) for π of π(C₂₀H₁₀) (both the cv and cc sides), π(C₂₄H₁₂), and π(C₆H₆). The structures are limited to those for B–A in B–A⋯π being placed on the molecular axis. The structures are referred to as the ID type. The differences and similarities in the nature of the interactions between the planar and distorted π-systems together with the cv and cc sides are examined employing the ID type, shown in Scheme 3. The ID-type structure enables comparison of the nature under the same conditions, where the optimized structures are very different for the adducts with the three π-systems.

The nature of the interactions is analyzed employing the quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA) [35,36,37], which we proposed, after the QTAIM approach introduced by Bader [38,39]. A bond critical point (BCP, ∗) is an important concept in QTAIM, which appears on each bond path (PB). QTAIM functions for the interactions in question are calculated at the BCPs. ρ(r) at BCP is denoted by ρ_b(r_c), as are other QTAIM functions, such as the total electron energy densities H_b(r_c), potential energy densities V_b(r_c), and kinetic energy densities G_b(r_c). A chemical bond or interaction between A and B is denoted by A–B, which corresponds to a BP in QTAIM. We use A-∗-B for BP, where the asterisk emphasizes the existence of a BCP in A–B [35,36].

Equations (1) and (2) show the relationships among the functions (cf.: Virial theorem for Equation (2)) [38,39].

H_b(r_c) = G_b(r_c) + V_b(r_c)

(1)

(ћ²/8m)∇²ρ_b(r_c) = H_b(r_c) − V_b(r_c)/2 = G_b(r_c) + V_b(r_c)/2

(2)

H_b(r_c) is plotted versus H_b(r_c) − V_b(r_c)/2 in QTAIM-DFA. Data from the perturbed structures around fully optimized structures are employed for the plots, in addition to the fully optimized ones [35,36,37]. The perturbed structures in this work are generated by using the coordinates derived from the compliance constants of the internal vibrations C_ii [40,41,42,43,44]. The method is named CIV. CIV is recognized to be the most reliable method to generate the perturbed structures; however, it cannot be applied when BP starts at least one BCP. The perturbed structures are also generated by using the normal coordinates of the (best-fitted) internal vibrations [45,46]. The method is named NIV, which is also reliable. However, we must be careful when the (best-fitted) internal vibrations are not located in the interactions in question. Results with CIV and/or NIV are discussed in the text, selecting the more approximating one if there are some differences.

Data from the fully optimized structures are analyzed using the polar coordinate (R, θ) representation, which corresponds to the static nature of the interactions [35,36,37,40]. Each interaction plot, for the data from both the perturbed and the fully optimized structures, is expressed by (θ_p, κ_p). While θ_p corresponds to the tangent line of the plot, κ_p is the curvature. θ and θ_p are measured from the y-axis and the y-direction, respectively. We proposed the concept of the “dynamic nature of interactions” based on (θ_p, κ_p) [35,36,37,40]. We named (R, θ) and (θ_p, κ_p) the QTAIM-DFA parameters. (See also footnotes of

Table 1. QTAIM functions and QTAIM-DFA parameters for X–H-∗-π(C₂₀H₁₀) and Y–X-∗-π(C₂₀H₁₀) (X, Y = F, Cl, Br, and I), evaluated with MP2/BSS-A ^1,2, employing the perturbed structures generated with CIV and/or NIV.

Y–X-∗-π(C20H10) (Symmetry: Type)	ρb(rc)	c∇2ρb(rc) 3	Hb(rc)	R4	θ 5	Cii	θp 6	κp 7	Predicted Nature
	(eao–3)	(au)	(au)	(au)	(°)	(Å mdyn–1)	(°)	(au–1)
Convex side (with CIV)
F–H-∗-π(aC) (C1: IACora:cv)	0.0165	0.0064	0.0018	0.0067	74.7	17.674	123.6	362.5	p-CS/t-HBnc
F–F-∗-π(aC) (Cs: IACora:cv)	0.0167	0.0089	0.0024	0.0092	74.7	9.004	81.9	36.3	p-CS/vdw
Cl–Cl-∗-π(aC) (Cs: IACora:cv)	0.0227	0.0089	0.0006	0.0089	86.4	4.795	122.2	198.1	p-CS/t-HBnc
Br–Br-∗-π(aC) (Cs: IACora:cv)	0.0260	0.0085	−0.0007	0.0085	95.0	4.197	141.4	127.3	r-CS/t-HBwc
I–I-∗-π(aC) (Cs: IACora:cv)	0.0251	0.0071	−0.0013	0.0072	100.3	3.828	147.9	141.2	r-CS/t-HBwc
F–Cl-∗-π(aC) (Cs: IACora:cv)	0.0302	0.0104	−0.0013	0.0105	96.9	4.321	143.2	122.1	r-CS/t-HBwc
F–Br-∗-π(aC) (Cs: IACora:cv)	0.0341	0.0097	−0.0034	0.0102	109.3	3.036	159.2	75.5	r-CS/CT-MC
F–I-∗-π(aC) (C1: IACora:cv)	0.0332	0.0079	−0.0044	0.0091	119.2	2.487	165.0	52.6	r-CS/CT-MC
Convex side (with NIV)
F–H-∗-π(aC) (C1: IACora:cv)	0.0165	0.0064	0.0018	0.0067	74.7	102.1 8	111.4	307.7	p-CS/t-HBnc
Cl–H-∗-π(abM) (Cs: IBCora:cv) 9	0.0160	0.0057	0.0016	0.0060	74.9	82.0 8	88.5	117.3	p-CS/vdW
Br–H-∗-π(abM) (C1: IBCora:cv) 9	0.0169	0.0058	0.0014	0.0060	76.7	60.7 8	96.2	260.9	p-CS/t-HBnc
I–H-∗-π(abM) (Cs: IBCora:cv)	0.0175	0.0059	0.0013	0.0060	77.5	52.2 8	95.9	240.3	p-CS/t-HBnc
F–F-∗-π(aC) (Cs: IACora:cv)	0.0167	0.0089	0.0024	0.0092	74.7	75.6 8	81.8	35.4	p-CS/vdw
Cl–Cl-∗-π(aC) (Cs: IACora:cv)	0.0227	0.0089	0.0006	0.0089	86.4	79.2 8	120.6	136.5	p-CS/t-HBnc
Br–Br-∗-π(aC) (Cs: IACora:cv)	0.0260	0.0085	−0.0007	0.0085	95.0	65.9 8	139.3	125.3	r-CS/t-HBwc
I–I-∗-π(aC) (Cs: IACora:cv)	0.0251	0.0071	−0.0013	0.0072	100.3	60.0 8	145.7	129.1	r-CS/t-HBwc
F–Cl-∗-π(aC) (Cs: IACora:cv)	0.0302	0.0104	−0.0013	0.0105	96.9	93.3 8	141.8	101.6	r-CS/t-HBwc
F–Br-∗-π(aC) (Cs: IACora:cv)	0.0341	0.0097	−0.0034	0.0102	109.3	86.9 8	157.7	59.2	r-CS/CT-MC
F–I-∗-π(aC) (C1: IACora:cv)	0.0332	0.0079	−0.0044	0.0091	119.2	81.6 8	164.1	49.4	r-CS/CT-MC
Concave side (with CIV)
F–H-∗-π(aC) (C1: IACora:cc)	0.0144	0.0065	0.0021	0.0068	72.0	8.175	95.9	273.2	p-CS/t-HBnc
Cl–H-∗-π(aC) (Cs: IIACora:cc)	0.0162	0.0064	0.0015	0.0065	76.9	31.667	108.6	409.5	p-CS/t-HBnc
Br–H-∗-π(aC) (C1: IIACora:cc)	0.0174	0.0065	0.0014	0.0066	78.2	34.149	117.2	736.5	p-CS/t-HBnc
I–H-∗-π(fC) (C1: IIACora:cc)	0.0172	0.0062	0.0012	0.0063	78.9	57.414	93.5	200.2	p-CS/t-HBnc
F–F-∗-π(aC) (Cs: IIACora:cc)	0.0093	0.0048	0.0012	0.0050	76.3	12.665	84.6	6.4	p-CS/vdw
Br–Br-∗-π(aC) (Cs: IIBCora:cc)	0.0124	0.0054	0.0014	0.0056	75.4	9.442	86.3	85.5	p-CS/vdw
I–I-∗-π(aC) (Cs: IIBCora:cc)	0.0130	0.0049	0.0010	0.0050	78.7	4.975	91.2	117.9	p-CS/t-HBnc
F–Cl-∗-π(aC) (C1: IACora:cc)	0.0137	0.0065	0.0017	0.0067	75.6	6.162	95.0	140.9	p-CS/t-HBnc
F–Br-∗-π(aC) (C1: IACora:cc)	0.0139	0.0061	0.0014	0.0062	76.9	5.875	99.9	204.6	p-CS/t-HBnc
F–I-∗-π(aC) (Cs: IACora:cc)	0.0141	0.0054	0.0008	0.0054	81.1	5.412	109.2	324.3	p-CS/t-HBnc
Concave side (with NIV)
F–H-∗-π(aC) (C1: IACora:cc)	0.0144	0.0065	0.0021	0.0068	72.0	100.0 8	100.2	282.2	p-CS/t-HBnc
Cl–H-∗-π(aC) (Cs: IIACora:cc)	0.0162	0.0064	0.0015	0.0065	76.9	77.7 8	125.4	955.4	p-CS/t-HBnc
Br–H-∗-π(aC) (C1: IIACora:cc)	0.0174	0.0065	0.0014	0.0066	78.2	59.0 8	137.7	1830	p-CS/t-HBnc
I–H-∗-π(fC) (C1: IIACora:cc)	0.0172	0.0062	0.0012	0.0063	78.9	53.5 8	92.2	344.1	p-CS/t-HBnc
F–F-∗-π(aC) (Cs: IIACora:cc)	0.0093	0.0048	0.0012	0.0050	76.3	76.2 8	89.3	305.2	p-CS/vdw
Cl–Cl-∗-π(fC) (C1: IIACora:cc)	0.0118	0.0054	0.0017	0.0056	72.8	97.0 8	84.5	94.5	p-CS/vdw
Br–Br-∗-π(aC) (Cs: IIBCora:cc)	0.0124	0.0054	0.0014	0.0056	75.4	76.0 8	88.5	98.1	p-CS/vdw
I–I-∗-π(aC) (Cs: IIBCora:cc)	0.0130	0.0049	0.0010	0.0050	78.7	70.9 8	94.1	136.2	p-CS/t-HBnc
F–Cl-∗-π(aC) (C1: IACora:cc)	0.0137	0.0065	0.0017	0.0067	75.6	95.4 8	93.8	131.4	p-CS/t-HBnc
F–Br-∗-π(aC) (C1: IACora:cc)	0.0139	0.0061	0.0014	0.0062	76.9	79.2 8	97.1	168.1	p-CS/t-HBnc
F–I-∗-π(aC) (Cs: IACora:cc)	0.0141	0.0054	0.0008	0.0054	81.1	76.0 8	107.7	228.9	p-CS/t-HBnc

¹ See text for BSS-A. ² Data are given at BCP, which is shown by A-∗-π, where a one side interaction is shown if two are identical due to symmetry. ³ c∇²ρ_b(r_c) = H_b(r_c) − V_b(r_c)/2, where c = ћ²/8m. ⁴ R = (x² + y²)^1/2, where (x, y) = (H_b(r_c) − V_b(r_c)/2, H_b(r_c)). ⁵ θ = 90° − tan^–1 (y/x). ⁶ θ_p = 90° − tan^–1 (dy/dx). ⁷ κ_p = |d²y/dx²|/[1 + (dy/dx)²]^3/2. ⁸ The frequency corresponding to the interval vibration employed to generate the perturbed structures with NIV in cm^–1. ⁹ Perturbed structures are generated employing w = –0.05, –0.025, (0), 0.025, and 0.05 in Equation (4).

for the definition). QTAIM-DFA is applied to typical chemical bonds, and interactions and rough criteria are established. QTAIM-DFA and the criteria are explained in the Supplementary Materials using Schemes S1–S3, Figures S1 and S2, Table S1, and Equations (S1)–(S7). The basic concept of the QTAIM approach is also explained.

The analyzed results of QTAIM-DFA will help to elucidate the nature of the interactions and a better understanding of the adducts formed by the interactions. QTAIM-DFA must be one of the best methodologies to elucidate the nature of the interactions. QTAIM-DFA will not only classify the interactions but also elucidate the nature. With the results, we will be able to access to the nature of the interactions in the adduct between corannulene and XH, XX, and FX (X = F, Cl, Br, and I) by elucidating the nature of the interactions with the method.

The dynamic and static nature of π-HBs and π-XBs in the bowl-shaped corannulene π-system (π(C₂₀H₁₀)) is elucidated with QTAIM-DFA after the clarification of the structural feature. Herein, we present the results of the investigations on the nature of X–H-∗-π(C₂₀H₁₀), X–X-∗-π(C₂₀H₁₀), and F–X-∗-π(C₂₀H₁₀) (X = F, Cl, Br, and I), where the nature is classified and characterized by employing the criteria as a reference. The differences and similarities in the nature of the interactions in B–A-∗-π for π of π(C₂₀H₁₀) (both cv and cc sides), π(C₂₄H₁₂) [21], and π(C₆H₆) [23,24,25] are also discussed.

2. Methodological Details of the Calculations

Calculations were performed by employing the Gaussian 09 program package [47]. The 6-311G(2d,p) basis sets for C and H were employed for the calculations, with the basis sets of the 6-311 + G(3df) for F, Cl, and Br, and the (7433211/743111/7411/2 + 1s1p) type for I, implemented from the Sapporo Basis Set Factory [48]. The basis set system is named basis set system-A (BSS-A). The Møller–Plesset second-order energy correlation (MP2) level [49,50,51] was applied to BSS-A (MP2/BSS-A). Optimized structures were confirmed by a frequency analysis. QTAIM functions were calculated using the AIM2000 [52,53] and AIMAll [54] programs with the same method as the optimizations. The optimized structures were not corrected with the BSSE method. A natural bond orbital analysis (NBO) and natural population analysis (NPA) were calculated with M06-2X/BSS-A//MP2/BSS-A [34,55].

Equation (3) explains the process to generate the perturbed structures with CIV [40,41,42,43,44]. The coordinates derived from the C_ii values (C_i) are used to generate the i-th perturbed structures in question (S_iw). C_i is added to the standard orientation of the fully optimized structure (S_o) in the matrix representation. The coefficient g_iw in Equation (3) controls the difference in the structures between S_iw and S_o: g_iw are determined to satisfy Equation (4) for an interaction in question, where r and r_o show the interaction distances in question in the perturbed and fully optimized structures, respectively, with a_o of the Bohr radius (0.52918 Å). In the case of NIV, the process can be similarly explained by replacing C_i in Equation (3) to the (best-fitted) normal coordinates of the i-th internal vibration (N_i), which is formulated by S_iw = S_o + g_iw· N_i. The C_i and N_i values of five digits are used for the generation.

S_iw = S_o + g_iw· C_i

(3)

r = r_o + wa_o (w = (0), ±0.05, and ±0.1; a_o = 0.52918 Å)

(4)

y = c_o + c₁x + c₂x² + c₃x³ (R_c²: square of the correlation coefficient)

(5)

In the QTAIM-DFA treatment, H_b(r_c) is plotted versus H_b(r_c) − V_b(r_c)/2 for data of five points of w = 0, ±0.05, and ±0.1 in Equation (4) unless otherwise noted. Each plot is analyzed using a regression curve of the cubic function as shown in Equation (5), where (x, y) = (H_b(r_c) − V_b(r_c)/2, H_b(r_c)) (R_c² (square of correlation coefficient) > 0.99999 in usual) [37].

3. Results and Discussion

3.1. Optimizations of B–A⋯π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X)

The optimizations of B–A⋯π(C₂₀H₁₀), where B–A = X–H, X–X, and F–X (X = F, Cl, Br, and I), were started with MP2/BSS-A, putting B–A on various places close to the symmetry axis (^sM), some carbon atoms (^aC, ^fC, and/or ^gC) and/or the midpoint of the outside C=C bond (^ghM) on the cv and cc sides (see Scheme 2 and Scheme 3 for the definitions). On the cv side, all optimizations of B–A⋯π(C₂₀H₁₀) converged to type IA_Cora:cv, except for type IB_Cora:cv of X–H⋯π(C₂₀H₁₀) (X = Cl, Br, and I). For the cc side, they converged to type IA_Cora:cc for F–H⋯π(C₂₀H₁₀) and F–X⋯π(C₂₀H₁₀) (X = Cl, Br, and I), type IIA_Cora:cc for X–H⋯π(C₂₀H₁₀) (X = Cl, Br, and I) and X–X⋯π(C₂₀H₁₀) (X = F and Cl), and type IIB_Cora:cc for X–X⋯π(C₂₀H₁₀) (X = Br and I). The optimized C₁ structures were further optimized assuming the C_s structures when the C₁ structures were very close to the C_s symmetry. The structural parameters are summarized in Table S2 of the Supplementary Materials. The optimized structures are not shown in the figures, but they can be found as molecular graphs (see Figure 2 and Figure 3). They are drawn on the optimized structures.

Next, molecular graphs are examined after clarification of the structural feature of B–A⋯π(C₂₀H₁₀).

3.2. Molecular Graphs for B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X)

The molecular graphs are drawn on the optimized structures with MP2/BSS-A for X–H-∗-π(C₂₀H₁₀), X–X-∗-π(C₂₀H₁₀), and F–X-∗-π(C₂₀H₁₀), where X = F, Cl, Br, and I. Figure 2 and Figure 3 illustrate the molecular graphs on the cv and cc sides of π(C₂₀H₁₀), respectively. All BCPs expected are clearly detected, containing those for the XH-∗-π, XX-∗-π, and YX-∗-π interactions in question, together with the additional ones. The BPs in question appear clearly, with BCPs, ring critical points (RCPs), and cage critical points (CCPs), if any. The structural features of the species are well visualized by the molecular graphs.

In the reaction on the convex side of corannulene, only the structure of the monodentate coordination to the central five-membered ring was optimized with HX, XX, and FX. On the other hand, on the concave side, similarly, bidentate to pentadentate structures were optimized for coordination to the central five-membered ring, except for HF and FX. In the adducts of X–H-∗-π(C₂₀H₁₀), X–X-∗-π(C₂₀H₁₀), and F–X-∗-π(C₂₀H₁₀), electrons will flow from the corannulene to the components; thus, the electron density will increase on the end of the components.

As shown in Figure 2 and Figure 3, BPs for B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X) seem almost straight at first glance. BPs, as defined in QTAIM, is the connection of the minima of the electron density, where a BCP exists for each BP between two interacting atoms, although BP sometimes starts from a BCP of a bond. The BPs are not necessarily the shortest path. In other words, the lengths of BPs (r_BP) and the straight-line distances (R_SL) are approximately equal when the interaction characteristics are relatively simple, but they differ greatly when they are not. Such things often happen in the case of weak interactions. To further examine the behavior of the BPs, r_BP and R_SL were calculated. The values are collected in Table S4 of the Supplementary Materials, together with the differences between them Δr_BP (=r_BP − R_SL). The Δr_BP values are less than 0.10 Å for all BPs in question, except for Br–H-∗-π(C₂₀H₁₀) (C₁: type IIA_Cora:cc; 0.180 Å), Br–H-∗-π(C₂₀H₁₀) (C₁: type IB_Cora:cv; 0.681 Å), and Cl–Cl-∗-π(C₂₀H₁₀) (C₁: type IIA_Cora:cc; Δr_BP = 0.701 Å). The r_BP is plotted versus R_SL, which is shown in Figure S4 of the Supplementary Materials. The plot gave a very good correlation (y = 0.995x + 0.035: R_c² = 0.998), if omitted Br–H-∗-π(C₂₀H₁₀) (C₁: type IIA_Cora:cc), Br–H-∗-π(C₂₀H₁₀) (C₁: type IB_Cora:cv), and Cl–Cl-∗-π(C₂₀H₁₀) (C₁: type IIA_Cora:cc). Consequently, all BPs in question can be approximated as straight lines except for the three cases. There must exist some reasons for the large Δr_BP values. Typical cases where large Δr_BP values are observed are shown below. BPs often curve in the area (very) close to atoms. A BP appears when a maximum line of ρ(r) connects two atoms. In this case, the maximum line does not often direct toward the second atom just after it starts the first atom. Such a case is also observed in which a BP directs to a BCP of another bond, where it reaches not the BCP but an atom corresponding to the BCP.

3.3. Survey of B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X)

The energies for the formation of the adducts from the components, ΔE [= E(B–A∙∙∙π(C₂₀H₁₀))–(E(B–A) + E(C₂₀H₁₀)): B–A = X–H, X–X, and F–X], were calculated with MP2/BSS-A. The ΔE_ES and ΔE_ZP values are collected in Table S3 of the Supplementary Materials, where ΔE_ES and ΔE_ZP stand for ΔE on the energy surface and those with the collections by the zero-point energy, respectively, together with the second-perturbation energies corresponding to the donor-acceptor interaction from π(C=C) to σ*(X–H) or σ*(Y–X), calculated with M06-2X/BSS-A//MP2/BSS-A.

To confirm the validity of the argument using energy surfaces, we checked the correlation between ΔE_ZP and ΔE_ES. ΔE_ZP is plotted versus ΔE_ES, which is shown in Figure S3 of the Supplementary Materials. The plots gave (very) good correlations: y = 1.015x + 2.67: R_c² = 0.996 (n (number of data points) = 11) for cv, y = 1.025x + 3.08: R_c² = 0.999 (n = 11) for cc, and y = 1.021x + 2.90: R_c² = 0.998 (n = 22) for all. Therefore, ΔE_ES can be used for the discussion of ΔE.

Figure 4 shows the plots of ΔE_ES of B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X) on the cv and cc sides versus halogens (X) calculated with M06-2X/BSS-A//MP2/BSS-A. The energies for the formation of the adducts from the components were more stabilized on the concave side except for I–H-∗-π(^fC) (IIA_Cora:cc) and F–I-∗-π(^aC) (IA_Cora:cc) and become more stable as the atomic number of the halogen increases. I–H-∗-π(^fC) (IIA_Cora:cc) is out of trend because the H interacts with the outer carbon ^fC (Figure 2d). The stabilization energy is almost the same for F–I-∗-π(^aC) (IA_Cora:cc) and F–I-∗-π(^aC) (IA_Cora:cv). X–X-∗-π was also more stabilized than X–H-∗-π, and X–Y-∗-π was more stabilized than X–X-∗-π except for I–H-∗-π(^fC) (C₁: IIA_Cora:cc).

QTAIM functions were calculated for B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X). Table 1 lists the values. Figure 5 shows a plot of H_b(r_c) versus H_b(r_c)–V_b(r_c)/2 for the interactions in question. Data shown in Table 1 are employed for the plots together with those from the perturbed structures generated with CIV and NIV, although the interactions are limited to the main interactions. The nature of the interactions is clarified by analyzing the plots in Figure 5, according to Equations (S1)–(S4) of the Supplementary Materials.

3.4. Nature of B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X)

Table 1 lists the ρ(r), H_b(r_c) − V_b(r_c)/2, and H_b(r_c) values of the QTAIM functions. Table 1 collects the QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p), the analyzed results, compliance constants C_ii for CIV, employed to generate the perturbed structures, and/or the frequencies corresponding to the interval vibrations employed to generate the perturbed structures with NIV. The values for the main interactions are given in Table 1, while those for the additional interactions on the cv and cc side are in Table S8 of the Supplementary Materials, although the definition is tentative. The θ_p, values for the main interactions with NIV, are plotted versus those with CIV. The plot is shown in Figure S7 of the Supplementary Materials. The correlation is very good (y = 0.955x + 5.0; R_c² = 0.995), if all data is analyzed except for those from F–H-∗-π(C₂₀H₁₀) (cv), Cl–H-∗-π(C₂₀H₁₀) (cc), and Br–H-∗-π(C₂₀H₁₀) (cc). Only one BP connects the components of the adducts on the cv side, which results in the very good correlation. In the case of the cc side, the correlation is also very good, where the data of Cl–H-∗-π(C₂₀H₁₀) (cc) and Br–H-∗-π(C₂₀H₁₀) (cc) are neglected. The results come from the very complex interaction style on the cc side, where only one BP connects the components of the adducts for F–H-∗-π(C₂₀H₁₀) and F–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I), whereas the components of the adducts are connected by the multi-BPs for others.

The trends in θ and θ_p are summarized in Equations (6)–(13). The θ value of X–H-∗-π(C₂₀H₁₀) on the cv side seems smaller than the corresponding values on the cc side except for F–H-∗-π(C₂₀H₁₀). The θ value on the cv side becomes larger in the order shown in Equation (6). The order on the cc side shown in Equation (7) seems very similar to that on the cv side. In the case of Y–X-∗-π(C₂₀H₁₀), the θ value on the cv side is larger than the corresponding value on the cc side, except for F–F-∗-π(C₂₀H₁₀), as shown in Equations (8) and (9). The θ value of Y–X-∗-π(C₂₀H₁₀) on both the cv and cc sides becomes larger in a similar order, as shown in Equations (8) and (9), respectively. The θ values on the cc side seem much smaller than the corresponding values on the cv side except for F–F-∗-π(C₂₀H₁₀). The bend structure of Y–X-∗-π(C₂₀H₁₀) (C_s: IIA_Cora:cc) must be responsible for the much smaller θ values on the cc side. The θ_p values for X–H-∗-π(C₂₀H₁₀) and Y–X-∗-π(C₂₀H₁₀) show the same trends as those in the θ values.

Order in θ of X–H-∗-π(C₂₀H₁₀) on the cv side:

F–H- (θ = 74.7°: IA) ≈ Cl–H- (74.9°: IB) < Br–H- (76.7°: IB) < I–H- (77.5°: IB)

(6)

Order in θ of X–H-∗-π(C₂₀H₁₀) on the cc side:

F–H- (θ = 72.0°: IA) < Cl–H- (76.9°: IIA) < Br–H- (78.2°: IIA) < I–H- (78.9°: IIA)

(7)

Order in θ of Y–X-∗-π(C₂₀H₁₀) on the cv side:

F–F- (θ = 74.7°: IA) < Cl–Cl- (86.4°: IA) < Br–Br- (95.0°: IA) < F–Cl- (96.9°: IA) < I–I- (100.3°: IA) < F–Br- (109.3°: IA) < F–I- (119.2°: IA)

(8)

Order in θ of Y–X-∗-π(C₂₀H₁₀) on the cc side:

Cl–Cl- (θ = 72.8°: IIA) < Br–Br- (75.4°: IIB) ≈ F–F- (76.3°: IIA) < F–Cl- (75.6°: IA) < F–Br- (76.9°: IA) < I–I- (78.7°: IIB) < F–I- (81.1°: IA)

(9)

Order in θ_p of X–H-∗-π(C₂₀H₁₀) with NIV on the cv side:

Cl–H- (θ_p = 88.5°: IB) < I–H- (95.9°: IB) ≈ Br–H- (96.2°: IB) < F–H- (111.4°: IA)

(10)

Order in θ_p of X–H-∗-π(C₂₀H₁₀) with NIV on the cc side:

I–H- (θ_p = 92.2°: IIA) < F–H- (100.2°: IA) < Cl–H- (125.4°: IIA) < Br–H- (137.7°: IIA)

(11)

Order in θ_p of Y–X-∗-π(C₂₀H₁₀) with NIV on the cv side:

F–F- (θ_p = 81.8°: IA) < Cl–Cl- (120.6°: IA) < Br–Br- (139.3°: IA) < F–Cl- (141.8°: IA) < I–I- (145.7°: IA) < F–Br- (157.7°: IA) < F–I- (164.1°: IA)

(12)

Order in θ_p of Y–X-∗-π(C₂₀H₁₀) with NIV on the cc side:

Cl–Cl- (84.5°: IIA) < Br–Br- (88.5°: IIB) < F–F- (θ_p = 89.3°: IIA) < F–Cl- (93.8°: IA) < I–I- (94.1°: IIB) < F–Br- (97.1°: IA) < F–I- (107.7°: IA)

(13)

Order in θ_p of X–H-∗-π(C₂₀H₁₀) with CIV on the cc side:

I–H- (θ_p = 93.5°: IIA) < F–H- (95.9°: IA) < Cl–H- (108.6°: IIA) < Br–H- (117.2°: IIA)

(14)

Order in θ_p of Y–X-∗-π(C₂₀H₁₀) with CIV on the cv side:

F–F- (θ_p = 81.9°: IA) < Cl–Cl- (122.2°: IA) < Br–Br- (141.4°: IA) < F–Cl- (143.2°: IA) < I–I- (147.9°: IA) < F–Br- (159.2°: IA) < F–I- (165.0°: IA)

(15)

Order in θ_p of Y–X-∗-π(C₂₀H₁₀) with CIV on the cc side:

F–F- (θ_p = 84.6°: IIA) < Br–Br- (86.3°: IIB) < I–I- (91.2°: IIB) < F–Cl- (95.0°: IA) < F–Br- (99.9°: IA) < F–I- (109.2°: IIB)

(16)

The θ_p values with NIV are discussed first, then those with CIV, since data with CIV are lacking in some cases. The θ_p values of X–H-∗-π(C₂₀H₁₀) on both the cv and cc sides seem to increase normally except for F–H-∗-π(C₂₀H₁₀) on the cv side and I–H-∗-π(C₂₀H₁₀) on the cc side, as shown in Equations (10), (11), and (14), respectively. The values on the cc side seem larger than those on the cv side, as a whole. The bend IIA structures on the cc side versus the IB structures on the cv side for X–H-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) must be responsible for the trend in θ_p. The differences in the electronic structures around the cv and cc sides also contribute to the results.

The trends in θ_p with CIV are shown in Equations (12), (13), (15), and (16); the order in θ_p of Y–X-∗-π(C₂₀H₁₀) on the cc side seems similar to that on the cv side. However, the magnitudes of θ_p on the cv side are much larger than those on the cc side except for F–F-∗-π(C₂₀H₁₀). The bend structures on the cc sides are mainly responsible for the results.

The nature of the H-∗-π and X-∗-π interactions is discussed next. In QTAIM-DFA, the θ values classify the interactions in question, while the θ_p values characterize them. The θ values of 45° < θ < 180° (H_b(r_c) − V_b(r_c)/2 > 0) correspond to the closed shell (CS) interactions, which are subdivided into the pure CS (p-CS) interactions of 45° < θ < 90° (H_b(r_c) > 0) and the regular CS (r-CS) interactions of 90° < θ < 180° (H_b(r_c) < 0). All interactions in Table 1 are classified by the p-CS and r-CS interactions since 68.8° < θ < 119.2°. In the p-CS region, the character of interactions is the vdW type for 45° < θ_p < 90° and the typical hydrogen bond type with no covalency (t-HB_nc) for 90° < θ_p < 125°, where θ_p = 125° is tentatively given, corresponding to θ = 90°. The characteristics of the r-CS interactions are similarly defined. As a result, the (θ, θ_p) values of (75°, 90°), (90°, 125°), (115°, 150°), and (150°, 180°) can be considered to be the borderlines between the nature of interactions for vdW/t-HB_nc, t-HB_nc/t-HB_wc, t-HB_wc/CT-MC, and CT-MC/CT-TBP, respectively, where t-HB_wc, CT-MC, and CT-TBP represent t-HB with covalency, molecular complex formation through CT, and trigonal bipyramidal (TBP) adduct formation through CT, respectively. The parameters given in bold are superior to those in plain in the classification and characterization of interactions. The (θ, θ_p) values in Table 1 are larger than (72°, 89°) and less than (119°, 165°) for all H-∗-π and X-∗-π interactions. Therefore, the interactions will have the nature of p-CS/vdW, p-CS/t-HB_nc, r-CS/t-HB_wc, or r-CS/CT-MC.

The nature of the main interactions is discussed first. The θ_p values calculated with NIV are employed here. As shown in Table 1, the (θ, θ_p) values for F–H-∗-π(C₂₀H₁₀) (C₁: IA_Cora:cv) are (74.7°, 111.4°); therefore, the interaction is predicted to have the t-HB_nc nature that appeared in the p-CS region, which is denoted by p-CS/t-HB_nc. Both the θ and θ_p values are less than 90° for Cl–H-∗-π(C₂₀H₁₀) (C₁: IB_Cora:cv), F–F-∗-π(C₂₀H₁₀) (C_s: IA_Cora:cv), and X–X-∗-π(C₂₀H₁₀) (C_s: IIA_Cora:cc: X = F and Cl; C_s: IIB_Cora:cc: X = Br). Therefore, the interactions are predicted to have a p-CS/vdW nature. The (θ, θ_p:NIV) values are (86.4°, 120.6°) for Cl–Cl-∗-π(C₂₀H₁₀) (C_s: IA_Cora:cv), (74.7–77.5°, 95.9–111.4°) for X–H-∗-π(C₂₀H₁₀) (C₁: IA_Cora:cv: X = F; IB_Cora:cv: X = Br and I), and (72.0–81.1°, 92.2–137.7°) for X–H-∗-π(C₂₀H₁₀) (C₁: IA_Cora:cc: X = F; IIA_Cora:cc: X = Cl, Br, and I), I–I-∗-π(C₂₀H₁₀) (C_s: IIB_Cora:cc), and F–X-∗-π(C₂₀H₁₀) (C_s: IA_Cora:cc: X = Cl, Br, and I). Consequently, the interactions are predicted to have a p-CS/t-HB_nc nature. The Cl-∗-π interaction in Cl–Cl-∗-π(C₂₀H₁₀) (type IA_Cora:cv) seems close to the borderline area between p-CS and r-CS since θ = 86.4°, which is close to 90°. On the other hand, the (θ, θ_p) values are (95.0–100.3°, 139.3–145.7°) for X–X-∗-π(C₂₀H₁₀) (C_s: IA_Cora:cv: X = Br and I) and F–Cl-∗-π(C₂₀H₁₀) (C_s: IA_Cora:cv). As a result, the interactions are predicted to have a r-CS/t-HB_wc nature. The (θ, θ_p) values are (109.3–119.2°, 157.7–164.1°) for F–X-∗-π(C₂₀H₁₀) (C_s: IA_Cora:cv: X = Br and I). Therefore, the interactions are predicted to have a r-CS/CT-MC nature. Table 1 summarizes the predicted nature.

As shown in Table 1, the predicted nature of the interactions in question with CIV is the same as the corresponding one with NIV, where the differences in θ_p between those with NIV and CIV are (very) small (0.1° ≤ Δθ_p (= θ_p:NIV – θ_p:CIV) ≤ 4.7°) except for F–H-∗-π(C₂₀H₁₀) (C₁: IA_Cora:cv), Cl–H-∗-π(C₂₀H₁₀) (C_s: IIA_Cora:cc), and Br–H-∗-π(C₂₀H₁₀) (C₁: IIA_Cora:cc). The Δθ_p values for the three adducts are 12.2°, –16.8°, and –20,5°, respectively. The normal coordinates of the (best-fitted) internal vibrations would not be located on the interactions in question, which would be responsible for the large differences. The large differences in θ_p are fortunately burred in the larger θ_p ranges of the interactions in F–H-∗-π(C₂₀H₁₀) (C₁: IA_Cora:cv), Cl–H-∗-π(C₂₀H₁₀) (C_s: IIA_Cora:cc), and Br–H-∗-π(C₂₀H₁₀) (C₁: IIA_Cora:cc). However, the magnitudes of Δθ_p for the three adducts seem much larger than those expected. CIV cannot be applied for the interactions in which BPs start from BCP on C=C of π(C₂₀H₁₀). There must be other factors for the large magnitudes. Further investigations would be necessary for the detailed discussion on the large magnitudes of Δθ_p. Therefore, we will not discuss the differences further here.

3.5. Factors to Control Structures of B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X)

The molecular graphs for B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X), shown in Figure 2 and Figure 3, seem very different from those for B–A-∗-π(C₂₄H₁₂) (B–A = X–H, X–X, and F–X) [21], especially the role of the outside ring in the formation of the adducts. What factor controls the observed differences between the coronene and corannulene adducts? The factors were examined first based on the charge distributions in corannulene and coronene calculated with NPA, of which Qn values are shown in Scheme 1. In fact, HX and XY are located near the central 5-membered ring when complexed with corannulene (Figure 2 and Figure 3). Additionally, the formation of bidentate to pentadentate coordination complexes of HX and XY with corannulene on the concave side indicates that these interactions are attractive interactions. The calculated values seem consistent with the observed results, although the relation to the reactivity is complex.

How are the r₁ values of B–A-∗-π(C₂₀H₁₀) (B–A = X–H, X–X, and F–X)? The r₁ values are plotted versus X = F, Cl, Br, and I, separately by X–H, X–X, and F–X and the cv and cc sides (totally six cases). Figure 6 illustrates the plot, which shows the clear trend in r₁ of B–A-∗-π(C₂₀H₁₀). The order in r₁ is summarized in Equation (17), where the r₁ values of the same X are compared. Next, the trends in r₁ are individually discussed.

Order in r₁ of B–A-∗-π(C₂₀H₁₀):

B–A- (cv or cc) = X–H- (cv) < X–H- (cc) < F–X- (cv) < X–X- (cv) < F–X- (cc) < X–X- (cc)

(17)

The r₁ values on the cc side (r₁(cc)) are (very) close to r₁(cv) (r₁(cv) ≈ r₁(cc)) for X–H-∗-π(C₂₀H₁₀) if r₁ of the same X are compared. On the other hand, r₁(cc) is (much) larger than r₁(cv) (r₁(cv) ≪ r₁(cc)) for X–X-∗-π(C₂₀H₁₀) and F–X-∗-π(C₂₀H₁₀) (X = F, Cl, Br, and/or I). The structures of X–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) are the IIB_Cora:cc type on the cc side, whereas they are the IB_Cora:cc type on the cv side. The difference in the structures must be the reason for r₁ (cv) ≪ r₁ (cc) in X–X-∗-π(C₂₀H₁₀). In the case of F–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I), the structures are the IA_Cora type on both the cc and cv sides. Therefore, the difference in the steric repulsion between the cc and cv sides is the reason for the calculated results. The steric repulsion on the concave side of the corannulene may not be as large. However, when HX or XY approaches corannulene from the concave side, it is easy to predict that the steric interaction will be large.

The r₁ values of F–X-∗-π(C₂₀H₁₀) are smaller than those of X–X-∗-π(C₂₀H₁₀) on the cv side; if r₁ of the same X is compared, so is the cc side. The accepting ability of F–X should be larger than that of X–X, which results in r₁(F–X-∗-π(C₂₀H₁₀)) < r₁(X–X-∗-π(C₂₀H₁₀)). The X dependence in r₁ can be discussed by the data in Figure 6. The r₁ values of X–H-∗-π(C₂₀H₁₀) on the cv side seem almost constant of X–H- = F–H- < Cl–H- ≈ Br–H- ≈ I–H-, whereas the values on the cc side become larger (gradually) in the order of X–H- = F–H- < Cl–H- ≈ Br–H- < I–H-. The positive charge developed on H in X–H seems to mainly control the r₁ values on the cv side, where Qn(H) = 0.568, 0.331, 0.218, and 0.098 for F–H, Cl–H, Br–H, and I–H, respectively, if evaluated with the NPA under the M06-2X/BSS-A//MP2/BSS-A. The r₁ values become larger in the order of X = F < Cl < Br < I for both sides of cv and cc in X–X-∗-π(C₂₀H₁₀) and F–X-∗-π(C₂₀H₁₀). The atomic sizes of X must be an important factor for the X dependence on r₁, where the size should be proportional to the number of electrons on X in X–X and F–X. The atomic sizes can be approximated by the vdW radii of atoms, which are 2.40, 2.94, 3.50, 3.70, and 3.96 Å for H, F, Cl, Br, and I, respectively.

In the corannulene adducts with HX of the convex side, the H-∗-π(C₂₀H₁₀) distances of XH-∗-π(C₂₀H₁₀) are HF < HCl < HBr < HI, where HX acts as the monodentate. However, this order is inversely related to the acidity of XH. The size of the halogen may be one of the reasons. The overall trend for the concave and convex sides is the same. However, it is likely on the concave side that a portion of each interaction force will be consumed by the atom on the opposite side by taking bidentate to pentadentate configuration, relative to the convex side. This is because the interaction force of the atoms on the opposite side would be weakened by the bidentate to pentadentate coordination. This means that the distance between atoms on the concave side is longer than that on the convex side.

The electron–electron repulsive factor between π(C₂₀H₁₀) and Y–X (Y = X and F) in Y–X-∗-π(C₂₀H₁₀) is also expected to play an important role in r₁. The repulsion could be larger on the cc side than on the cv side in the bowl-shaped π(C₂₀H₁₀), resulting in the relative values of r₁(cv) ≪ r₁(cc). The electron density ρ(r) must be more widely extended on the cv side relative to the case of the cc side, as expected. The higher negative area on the cv side of π(C₂₀H₁₀) corresponds to the larger distribution of ρ(r) relative to the case of the cc side. The results shown in Figure 6 can be well understood based on the EPS shown in Figure 1, where r₁ decreases as the CT interaction of the π(C₂₀H₁₀)→σ*(X–Y: X = X or F) type increases.

3.6. Meaning of the QTAIM-DFA Parameters and the Related Values

What is the meaning of the QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p) for the adducts collected in Table 1? However, it seems often difficult to compare the values with those derived from other methods, since the QTAIM approach (and QTAIM-DFA) are analyzed using the values at BCP on BP, whereas analyzed values from other methods seem (very) different from the case of QTAIM-DFA. The charge of an atom (A) in a molecule calculated based on the QTAIM approach (QTAIM(A)) is intrinsically very different from that with NPA (Qn(A)), for instance [38]. Nevertheless, the meaning of the QTAIM-DFA parameters is clarified by comparing other physical parameters.

First of all, ΔE are plotted versus θ, separately by the cv and cc sides (see Table 1). Figure 7 shows the plots. The plots give (very) good correlations, which are shown in the figure. The plots for the cv side are shown by Figure 7a, which are analyzed as the two correlations. The first correlation consists of X–H-∗-π(C₂₀H₁₀) (X = F, Cl, Br, and I) and X–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) (y = –0.916x + 43.5; R_c² = 0.992) and the second one of F–X-∗-π(C₂₀H₁₀) (X = F, Cl, Br, and I) (y = –1.056x + 61.6; R_c² = 0.999). The correlations are close with each other; therefore, they could be recognized as a correlation, where the components in all adducts are connected by only one BP for each. In the case of the cc side (Figure 7b), the plots were analyzed as the four groups. The first group consists of X–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) (y = –2.099x + 94.5; R_c² = 0.997), while X–H-∗-π(C₂₀H₁₀) (X = F, Cl, and Br) forms the second group (y = –3.451x + 220.7; R_c² = 1.000). The third group contains F–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) (y = –2.490x + 138.3; R_c² = 0.945), while data for I–H-∗-π(C₂₀H₁₀) and F–F-∗-π(C₂₀H₁₀) deviate from the above correlations. The X–X bonds in the first group seem close to parallel to the averaged molecular plane of C₂₀H₁₀. In the case of F–H-∗-π(C₂₀H₁₀) in the second group, F–H and π(C₂₀H₁₀) are connected by only one BP in the adduct. Indeed, the components of X–H-∗-π(C₂₀H₁₀) (X = Cl and Br) are connected by three BPs, but the structures seem close with each other. The components of F–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) are connected through only one BP; therefore, the structures are very close with each other.

The θ values are shown to well correlate to the ΔE values, if the data are appropriately analyzed separately by the structures. The results may show that θ appears proportional to ΔE in the weak CS interaction region of vdW, t-HB, and CT-MC. However, the θ values of the main interactions are employed for the plots.

It is necessary to search for such a parameter that covers all interactions for those multi-BPs that connect the components. We examined the C_ii^–1 values for the purpose, since the total values of C_ii^–1 could be calculated according to Equation (18).

C_ii^–1_total = Σ_k C_ii^–1_k

(18)

The ΔE values of the adducts are plotted versus C_ii^–1 or C_ii^–1_total for the adducts. Figure 8 shows the plot, which is analyzed separately by the cv and cc adducts. Figure 8a,b show the plots for the cv and cc adducts, respectively. The plots for the cv side were analyzed as two correlations, similarly to the case of Figure 7a. Data from X–X-∗-π(C₂₀H₁₀) (X = F, Cl, Br, and I) form the first correlation (y = –208.2x + 6.3; R_c² = 0.999), while the second correlation y = –116.6x−15.9; R_c² = 0.993) consists of those from F–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) and F–H-∗-π(C₂₀H₁₀). In the case of the cc side, the plots were analyzed as three groups. The first group consists of the data from X–H-∗-π(C₂₀H₁₀) (X = Cl and Br) and X–X-∗-π(C₂₀H₁₀) (X = F and Br) (y = –358.1x + 39.8; R_c² = 0.956) and those from F–X-∗-π(C₂₀H₁₀) (X = Cl, Br, and I) and F–H-∗-π(C₂₀H₁₀) form the second one (y = –565.4x + 41.8; R_c² = 0.996), whereas those from I–H-∗-π(C₂₀H₁₀) and I–I-∗-π(C₂₀H₁₀) seem to deviate from the correlations. The I atom may interact uniquely with the cc side of π(C₂₀H₁₀). The strength of the interaction would extend over the range of the correlation line, perhaps due to its softness and the (very) large size.

The C_ii^–1 or C_ii^–1_total values are demonstrated to correlate well with the ΔE values, if the data are appropriately analyzed separately by the structures, again. The results show that the C_ii^–1 or C_ii^–1_total values are proportional to the ΔE values, as expected [19]. The plots in Figure 7a seem very close to those of Figure 8a, as a whole. The results may show that the θ values can be recognized as a parameter for ΔE, if only one BP connects the components of the cv adducts in the weak CS interaction region of vdW, t-HB, and CT-MC. In the case of the cc side, the plots in Figure 7b seem very different from those in Figure 8b, in the numbers of the plots and the members for the groups. The results must be the reflection of the different physical meanings between θ and C_ii^–1 (C_ii^–1_total). Of course, the contributions from the weaker BPs are considered in C_ii^–1_total but only the strongest one is in θ, which must also be a very important factor to explain the differences in the plots between Figure 7b and Figure 8a.

3.7. Differences in the Nature of B–A-∗-π (B–A = X–H and F–X) among π of π(C₂₀H₁₀), π(C₂₄H₁₂) and π(C₆H₆)

The ID structures defined in Scheme 2 are optimized assuming the C_5v symmetry for the π(C₂₀H₁₀) adducts, the C_6v symmetry for the π(C₂₄H₁₂), and the C_2v symmetry for the π(C₆H₆) adducts. (The optimizations did not converge well under the C_6v symmetry for the π(C₆H₆) adducts.) The structural parameters are collected in Table S6 of the Supplementary Materials. The QTAIM functions and the similarly calculated QTAIM-DFA parameters are collected in Tables S7–S10 of the Supplementary Materials. The values of ΔE_ES for X–H-∗-π and F–X-∗-π adducts with π(C₂₀H₁₀), π(C₂₄H₁₂), and π(C₆H₆) are also collected in Table S11 of the Supplementary Materials.

Table 2. θ and θ_p values for H-∗-π and X-∗-π interactions with π(C₂₀H₁₀), π(C₂₄H₁₂), and π(C₆H₆), evaluated with MP2/BSS-A, together with values from corresponding adducts with π(C₆H₆), respectively ¹.

Y–X-∗-π(C20H10) (Symmetry: Type)	θ 2	θp3	Δθ 4	Δθp5	Y–X-∗-π(C24H12/C6H6) (Symmetry: Type)	θ 2	θp 3	Δθ 4	Δθp 5
	(°)	(°)	(°)	(°)		(°)	(°)	(°)	(°)
Convex side of π(C20H10)					Y–X-∗-π(C24H12)
F–H-∗-π(aC) (C5v: IDCora:cv) 6,7	66.1	68.9	−1.3	−0.6	F–H-∗-π(aC) (C6v: IDCor) 6,7	67.0	68.8	−0.4	−0.7
Cl–H-∗-π(aC) (C5v: IDCora:cv) 6,7	71.5	74.4	−0.7	2.3	Cl–H-∗-π(aC) (C6v: IDCor) 6,7	72.0	73.7	−0.2	1.6
Br–H-∗-π(aC) (C5v: IDCora:cv) 6,7	72.5	75.8	−0.3	3.1	Br–H-∗-π(aC) (C6v: IDCor) 6,7	72.7	75.2	−0.1	2.5
I–H-∗-π(aC) (C5v: IDCora:cv) 6,7	73.6	77.7	0.1	3.6	I–H-∗-π(aC) (C6v: IDCor) 6,7	73.7	77.6	0.2	3.5
F–F-∗-π(aC) (C5v: IDCora:cv) 6,7	70.5	73.5	−1.4	−1.9	F–F-∗-π(aC) (C6v: IDCor) 6,7	71.3	74.9	−0.6	−0.5
F–Cl-∗-π(aC) (C5v: IDCora:cv) 6,7	69.6	77.6	1.2	5.4	F–Cl-∗-π(aC) (C6v: IDCor) 6,7	69.1	75.8	0.7	3.6
F–Br-∗-π(aC) (C5v: IDCora:cv) 6,7	71.7	81.3	2.1	7.2	F–Br-∗-π(aC) (C6v: IDCo) 6,7	70.8	78.8	1.2	4.7
F–I-∗-π(aC) (C5v: IDCora:cv) 7,8	76.5	90.8	3.5	11.6	F–I-∗-π(aC) (C6v: IDCor) 6,7	75.1	86.8	2.1	7.6
Concave side of π(C20H10)					Y–X-∗-π(C6H6)
F–H-∗-π(aC) (C5v: IDCora:cc) 6,7	68.2	76.0	0.8	6.5	F–H-∗-π(aC) (C6v: IDBzn) 6,7	67.4	69.5	–	–
Cl–H-∗-π(aC) (C5v: IDCora:cc) 6,7	74.8	84.8	2.6	12.7	Cl–H-∗-π(aC) (C6v: IDBzn) 6,7	72.2	72.1	–	–
Br–H-∗-π(aC) (C5v: IDCora:cc) 6,7	75.9	86.8	3.1	14.1	Br–H-∗-π(aC) (C6v: IDBzn) 6,7	72.8	72.7	–	–
I–H-∗-π(aC) (C5v: IDCora:cc) 6,7	77.8	88.1	4.3	14.0	I–H-∗-π(aC) (C6v: IDBzn) 6,7	73.5	74.1	–	–
F–F-∗-π(aC) (C5v: IDCora:cc) 6,7	72.7	75.7	0.8	0.3	F–F-∗-π(aC) (C6v: IDBzn) 6,7	71.9	75.4	–	–
F–Cl-∗-π(aC) (C5v: IDCora:cc) 6,7	73.1	83.8	4.7	11.6	F–Cl-∗-π(aC) (C6v: IDBzn) 6,7	68.4	72.2	–	–
F–Br-∗-π(aC) (C5v: IDCora:cc) 6,7	74.5	86.4	4.9	12.3	F–Br-∗-π(aC) (C6v: IDBzn) 6,7	69.6	74.1	–	–

¹ See text for BSS-A. ² θ = 90°–tan^–1 (y/x), where (x, y) = (H_b(r_c)–V_b(r_c)/2, H_b(r_c)). ³ θ_p = 90°–tan^–1 (dy/dx). ⁴ Δθ = (θ for the π(C₂₀H₁₀) or π(C₂₄H₁₂) adducts) − (θ for the π(C₆H₆) adducts). ⁵ Δθ_p = (θ_p for the π(C₂₀H₁₀) or π(C₂₄H₁₂) adducts) − (θ_p for the π(C₆H₆) adducts). ⁶ Predicted to be vdW interactions appearing in the p-CS region. ⁷ Two imaginary frequencies being predicted for each. ⁸ Predicted to be t-HB_nc interactions appeared in the p-CS region.

shows the θ and θ_p values for the X–H-∗-π and F–X-∗-π interactions together with the Δθ and Δθ_p values. The Δθ and Δθ_p values are given from those of the π(C₆H₆) adducts. They are defined as Δθ = (θ for the π(C₂₀H₁₀) or π(C₂₄H₁₂) adducts) − (θ for the π(C₆H₆) adducts) and Δθ_p = (θ_p for the π(C₂₀H₁₀) or π(C₂₄H₁₂) adducts) − (θ_p for the π(C₆H₆) adducts).

The trends in θ and θ_p are discussed, employing the Δθ and Δθ_p values. Equations (19)–(22) summarize the trends in Δθ and Δθ_p, where the values for the F–H and F–F adducts are omitted, since the trends of the values seem very different from others in some cases.

Order for Δθ in X–H-∗-π:

π(C₂₀H₁₀: cv) (–0.7° ≤ Δθ ≤ 0.1°) ≤ π(C₂₄H₁₂) (–0.2° ≤ Δθ ≤ 0.2°) < π(C₆H₆) (Δθ = 0.0°) < π(C₂₀H₁₀: cc) (2.6°≤ Δθ ≤ 4.3°)

(19)

Order for Δθ in F–X-∗-π:

π(C₆H₆) (Δθ = 0.0°) < π(C₂₄H₁₂) (0.7° ≤ Δθ ≤ 2.1°) ≤ π(C₂₀H₁₀: cv) (1.2° ≤ Δθ ≤ 3.5°) < π(C₂₀H₁₀: cc) (4.7° ≤ Δθ ≤ 5.5°)

(20)

Order for Δθ_p in X–H-∗-π:

π(C₆H₆) (Δθ_p = 0.0°) < π(C₂₄H₁₂) (1.6° ≤ Δθ_p ≤ 3.5°) ≤ π(C₂₀H₁₀: cv) (2.3° ≤ Δθ_p ≤ 3.6°) < π(C₂₀H₁₀: cc) (12.7° ≤ Δθ_p ≤ 14.0°)

(21)

Order for Δθ_p in F–X-∗-π:

π(C₆H₆) (Δθ_p = 0.0°) < π(C₂₄H₁₂) (3.6° ≤ Δθ_p ≤ 7.6°) < π(C₂₀H₁₀: cv) (5.4° ≤ Δθ_p ≤ 11.6°) < π(C₂₀H₁₀: cc) (11.6° ≤ Δθ_p ≤ 15.4°)

(22)

Equation (19) shows the order for Δθ in X–H-∗-π. The order seems reasonable. However, the position of π(C₆H₆) should be considered, of which the order would be expected to appear between π(C₂₀H₁₀: cc) and π(C₂₄H₁₂). The order for Δθ in F–X-∗-π is given in Equation (20). The order seems curious at first glance. However, the order can be understood if the order is explained separately by the partial order of π(C₆H₆) < π(C₂₄H₁₂) < π(C₂₀H₁₀: cv) and < π(C₂₀H₁₀: cc).

Equation (21) shows the order for Δθ_p in X–H-∗-π. The order seems close to that for Δθ in F–X-∗-π shown in Equation (20); therefore, the order can be explained as discussed for the trend in Equation (20). The order for Δθ_p in F–X-∗-π is given in Equation (22). The order is very close to those in Equations (20) and (21), again. Therefore, the order is well understood by the partial order of π(C₆H₆) < π(C₂₄H₁₂) < π(C₂₀H₁₀: cc) < π(C₂₀H₁₀: cv), again.

The magnitudes of Δθ_p in Equation (22) seem larger than those in Equation (21), as a whole, and the magnitudes of Δθ_p in Equations (21) and (22) seem larger than the magnitudes of Δθ in Equation (20). The dynamic nature of X–H-∗-π and F–X-∗-π in the adducts with π(C₂₀H₁₀) on both the cv and cc sides would be (very) flexible by the structural changes.

The trends in the behavior of Δθ and Δθ_p show that the process for the formation of the adducts is much more complex than expected based on the electronic structures of π(C₆H₆), π(C₂₄H₁₂), π(C₂₀H₁₀: cv), and π(C₂₀H₁₀: cc), which were simply imaged. The magnitudes of Δθ and Δθ_p for the F–H and F–F adducts with π(C₂₀H₁₀) and π(C₂₄H₁₂), from those with π(C₆H₆), seem much smaller than those for the other adducts. The results may show that the nature of the interactions in question would not be affected so much for the F–H and F–F adducts relative to the case of other adducts.

4. Conclusions

What is the nature of the XH-∗-π and YX-∗-π interactions in a distorted π-system? The nature of such interactions is elucidated, exemplified by the corannulene π-system (π(C₂₀H₁₀)) with QTAIM-DFA, where the π(C₂₀H₁₀) orbitals extend wider over the cv side but narrower on the cc side. In the optimized structures of XH-∗-π(C₂₀H₁₀) and YX-∗-π(C₂₀H₁₀) (X = F, Cl, Br, and I; Y = X and F) with MP2/BSS-A, only one side of the atom, H of X–H or X of Y–X, joins π(C₂₀H₁₀) on the cv side. However, both sides of the atoms connect to π(C₂₀H₁₀) on the cc side in some cases. In the case of the cv adducts, all XH-∗-π(C₂₀H₁₀) (I_Cora:cv) interactions are predicted to have the p-CS/t-HB_nc nature except for FH-∗-π(C₂₀H₁₀), of which the nature is p-CS/vdW. For YX-∗-π(C₂₀H₁₀) (I_Cora:cv) interactions, the p-CS/vdW and p-CS/t-HB_nc nature is predicted for YX = FF and ClCl, respectively, the p-CS/t-HB_nc nature is predicted for YX = BrBr, II, and FCl, and the p-CS/CT-MC nature is predicted for YX = FBr and FI. For the cc adducts, the predicted nature seems more complex. Therefore, the descriptions are limited to the main interactions, which are given plainly in Table 2, to avoid complexity. All XH-∗-π(C₂₀H₁₀) interactions are predicted to have a p-CS/t-HB_nc nature, irrespective of the (IA_Cora:cc) structure for X = F and the (IIA_Cora:cc) structure for X = Cl, Br, and I. While the FX-∗-π(C₂₀H₁₀) interactions are predicted to have the p-CS/t-HB_nc nature for X = Cl, Br, and I, the p-CS/vdW nature is for X = F. The optimized structures are (IA_Cora:cc) for the former and (IIA_Cora:cc) for the latter. In the case of the XX-∗-π(C₂₀H₁₀) (II_Cora:cc) interactions, the p-CS/vdW nature is predicted for X = Cl and Br, whereas the p-CS/t-HB_nc nature is predicted for X = I. Indeed, the predicted nature of the interactions is controlled mainly by the cv and cc sides of π(C₂₀H₁₀) but are determined depending on the structures of the adducts. Therefore, they are more complex on the cc side. The predicted nature seems to increase in the order of X = F < Cl < Br < I for YX-∗-π(C₂₀H₁₀) (Y = F and X).

The differences and similarities are also clarified for the XH-∗-π and YX-∗-π interactions with the bowl-shaped π(C₂₀H₁₀) system and the planar π(C₆H₆)) and π(C₂₄H₁₂) systems. Indeed, the structures and the nature of the interactions seems well understood based in QTAIM-DFA, but the results show a much more complex process for the formation of the adducts than that expected based on the simply imaged electronic structures. The results will provide a useful guideline to analyze the nature of the interactions in the distorted π-systems and the adducts.