Type I–IV Halogen⋯Halogen Interactions: A Comparative Theoretical Study in Halobenzene⋯Halobenzene Homodimers

Abstract

In the current study, unexplored type IV halogen⋯halogen interaction was thoroughly elucidated, for the first time, and compared to the well-established types I–III interactions by means of the second-order Møller–Plesset (MP2) method. For this aim, the halobenzene⋯halobenzene homodimers (where halogen = Cl, Br, and I) were designed into four different types, parodying the considered interactions. From the energetic perspective, the preference of scouted homodimers was ascribed to type II interactions (i.e., highest binding energy), whereas the lowest binding energies were discerned in type III interactions. Generally, binding energies of the studied interactions were observed to decline with the decrease in the σ-hole size in the order, C₆H₅I⋯IC₆H₅ > C₆H₅Br⋯BrC₆H₅ > C₆H₅Cl⋯ClC₆H₅ homodimers and the reverse was noticed in the case of type IV interactions. Such peculiar observations were relevant to the ample contributions of negative-belt⋯negative-belt interactions within the C₆H₅Cl⋯ClC₆H₅ homodimer. Further, type IV torsional trans → cis interconversion of C₆H₅X⋯XC₆H₅ homodimers was investigated to quantify the π⋯π contributions into the total binding energies. Evidently, the energetic features illustrated the amelioration of the considered homodimers (i.e., more negative binding energy) along the prolonged scope of torsional trans → cis interconversion. In turn, these findings outlined the efficiency of the cis configuration over the trans analog. Generally, symmetry-adapted perturbation theory-based energy decomposition analysis (SAPT-EDA) demonstrated the predominance of all the scouted homodimers by the dispersion forces. The obtained results would be beneficial for the omnipresent studies relevant to the applications of halogen bonds in the fields of materials science and crystal engineering.

1. Introduction

The σ-hole interactions are of growing interest to researchers from various disciplines, owing to their wide applications in catalysis [1], anion recognition [2], biological systems [3,4], crystal materials [5,6], and drug discovery [7]. The occurrence of such interactions was mainly ascribed to the anisotropic distribution of the electron density on the surfaces of group IV-VII elements [8,9,10,11,12,13], resulting in the emergence of an electron depletion region coined as “σ-hole” [14,15]. Among σ-hole interactions, the halogen bond encouraged growing attention by dint of its decisive character in protein–ligand binding affinity [16,17,18], crystal engineering [19,20,21], and organo-catalysis [22,23]. Subsequently, versatile computational [24,25,26,27,28] and experimental [29,30,31,32] studies were devoted to characterizing halogen bonding interactions.

As an essential issue, IUPAC announced the definition of the halogen bond as follows: “A halogen bond occurs when there is evidence of a net attractive interaction between an electrophilic region associated with a halogen atom in a molecular entity and a nucleophilic region in another, or the same, molecular entity” [33]. From literature, the amphoteric nature of the halogen-containing molecules was thoroughly spotted, and their potentiality to simultaneously interact as a Lewis base and a Lewis acid was widely assessed, particularly in the case of like–like complexes [34,35,36,37,38,39,40,41].

Crucially, the ability of halogen-containing molecules to engage in halogen⋯halogen interactions was well-recognized within three different types [35,36,42,43,44]: type I (i.e., parallel displaced geometry): in which the A-halogen–halogen angle (θ₁) is nearly equal to halogen⋯halogen-A angle (θ₂); type II (i.e., L-geometry): in which the θ₁ angle is about 180° and θ₂ is about 90°; and type III (i.e., linear geometry), in which the θ₁ and θ₂ angles are typically equal to 180°, where A refers to the attached atom to halogen atom. Further, the literature recorded that the van der Waals, electrostatic, and dispersion forces were the dominant forces beyond type I, II, and III halogen⋯halogen interactions, respectively.

In addition to the three well-established types of halogen⋯halogen interactions, the unconventional Z-shaped structure of type IV interactions has recently been well-identified in the previously addressed chalcogen⋯chalcogen homodimers [45]. In this type, the θ₁ = C = Y⋯Y and θ₂ = Y⋯Y = C angles were set to be equal with a value of 90° in the Y = C = Y⋯Y = C = Y (i.e., Y = O, S, and Se) homodimers. Dispersion forces were documented with the most prevalent contributions to the total energy of this type of interactions [45]. Upon thorough literature review, the type IV halogen⋯halogen interactions have not been systematically elucidated yet.

Therefore, the current study was devoted to comparatively assessing the potentiality of halogen-containing molecules (C₆H₅X; where X = Cl, Br, and I) to participate in type I–IV halogen⋯halogen interactions within C₆H₅X⋯XC₆H₅ homodimers (Figure 1). Geometrical optimization, electrostatic potential (EP) analysis, and point-of-charge (PoC) approach calculations were carried out on the studied monomers. For the C₆H₅X⋯XC₆H₅ homodimers, a potential energy surface (PES) scan was performed in an exact orientation to obtain the four modeled types of interaction. Furthermore, quantum theory of atoms in molecules (QTAIM), noncovalent interaction (NCI) index analyses, and symmetry adapted perturbation theory-based energy decomposition analysis (SAPT-EDA) were adapted to elucidate the nature of the considered interactions. Such observations would be an opulent tool for identifying the complete nature of halogen⋯halogen interactions and their roles in supramolecular chemistry and materials science fields.

2. Results

2.1. Electrostatic Potential (EP) Analysis

Electrostatic potential (EP) analysis was recently deemed as an informative appliance to precisely scrutinize the electrophilicity and nucleophilicity over the molecular surface of the optimized monomers [46,47]. Thus, molecular electrostatic potential (MEP) maps were generated for the C₆H₅X molecules (where X = Cl, Br, and I) to graphically identify the electron-depletion and electron-rich regions. Additionally, maximum positive electrostatic potential (V_s,max) calculations were executed to quantify the maximum positive EP regions for all the studied molecules. MEP maps and V_s,max values of the considered monomers are displayed in Figure 2.

From the MEP maps depicted in Figure 2, the prominent blue regions were disclosed with variable sizes along the molecular surface of the halogen atom in the studied systems, outlining the occurrence of σ-holes. Conspicuously, the σ-hole size was perceived to increase in the order Cl < Br < I, according to increasing the atomic size of the halogen atom. The posterior results were ascribed to the ascending of the polarizability along with the lesser electronegativity of the heavier halogen atoms. From a quantitative perspective, the obtained σ-hole magnitude using V_s,max analysis was observed to be consistent with the σ-hole size pictured in MEP maps. Numerically, a positive EP region was detected with values of 10.1, 18.7, and 25.5 kcal/mol for C₆H₅Cl, C₆H₅Br, and C₆H₅I, respectively. These results were found to be in agreement with the previous findings [43].

2.2. Point-of-Charge (PoC) Calculations

The Point-of-charge (PoC) approach was documented as an informative tool for elucidating the electrostatic inclination of the group III-VIII element-containing molecules to participate in noncovalent interactions [48,49,50,51,52,53,54,55,56]. With the help of the PoC approach, negatively and positively charged points were utilized to parody the nucleophilic and electrophilic effects on the inspected molecular systems, respectively. In that context, molecular stabilization energy for C₆H₅X⋯PoC systems (where X = Cl, Br, and I) were estimated using values of ±0.50 au PoCs at a distance range of 2.5–5.0 Å with a step size of 0.1 Å along the x-axis and z-axis of the halogen atoms (i.e., the equatorial and axial surfaces) with ∠C-X⋯PoC angle of 180° and 90°, respectively.

As delineated in Figure 3, molecular stabilization energy curves generally outlined the predilection of most of the study’s halobenzene (C₆H₅X) monomers to engage in electrostatic interactions with negative and positive PoCs (i.e., the Lewis base and acid) along both x- and z-axes. Notably, all the studied C₆H₅X⋯PoC systems bolstered the nucleophilic effects (i.e., negative PoC) along the equatorial portions of halogen atoms (i.e., x-axis) over the outermost axial surface (i.e., z-axis), and the reverse was true for the electrophilic effects.

Regarding the negative PoC-based results, all the investigated C₆H₅X⋯PoC systems showed molecular stabilization energy (E_{stabilization}) in the presence of −0.50 au PoC along the x-axis. In contrast, molecular destabilization energy (E_{destabilization}) was recorded for all systems along the z-axis with an exception for the C₆H₅I molecule, which exhibited favorable E_{stabilization} with a value of −1.68 kcal/mol (

Table 1. Molecular stabilization/destabilization energies (E_{stabilization}/E_{destabilization}) of the C₆H₅X⋯PoC systems (where X = Cl, Br, and I) in the presence of ±0.50 au PoC at X⋯PoC distance of 2.5 Å along the x-axis and z-axis with ∠C-X⋯PoC angle of 180° and 90°, respectively.

System	Molecular Stabilization Energy (Estabilization/Edestabilization, kcal/mol)
	PoC= −0.50 au		PoC= +0.50 au
	x-Axis	z-Axis	x-Axis	z-Axis
C6H5Cl⋯PoC	−2.88	1.06	−4.39	–9.70
C6H5Br⋯PoC	−6.08	0.28	−3.25	–11.31
C6H5I⋯PoC	−10.99	−1.68	−2.14	–13.54

). Obviously, the molecular stabilization energies of C₆H₅X⋯PoC systems were perceived to increase coinciding with the atomic size of the halogen atom as follows: C₆H₅Cl⋯ < C₆H₅Br⋯ < C₆H₅I⋯PoC systems. As an illustration, E_{stabilization}/E_{destabilization} values were –2.88, –6.08, and –10.99 kcal/mol along the x-axis and 1.06, 0.28, and –1.68 kcal/mol along the z-axis for C₆H₅Cl⋯, C₆H₅Br⋯, C₆H₅I⋯PoC systems, respectively.

Turning to the positive PoC-based results, molecular stabilization energies were disclosed to decrease along the outermost equatorial surface according to the following order, C₆H₅Cl⋯ > C₆H₅Br⋯ > C₆H₅I⋯PoC. Evidently, an inverse correlation was obtained between E_{stabilization} and the halogen atomic size that might be ascribed to the elevated σ-hole size of the halogen atom. Contrarily, along the axial surface, E_{stabilization} was perceived to increase with decreasing the electronegativity of the halogen atoms. For instance, molecular stabilization energies were detected with values of –4.39, –3.25, and –2.14 kcal/mol along the x-axis accompanied by E_{stabilization} of –9.70, –11.31, and –13.54 kcal/mol along the z-axis of halogens of the C₆H₅Cl⋯, C₆H₅Br⋯, and C₆H₅I⋯PoC systems, respectively. It is worth mentioning that the PoC results obtained for C₆H₅X⋯PoC systems along the x-axis are consistent with a previous study in exhibiting direct and inverse correlations with the negative molecular stabilization energy in the presence of negative and positive PoCs, respectively [43].

2.3. Potential Energy Surface (PES) Scan

A potential energy surface (PES) scan was executed to thoroughly elucidate the ability of the C₆H₅X monomers (where X = Cl, Br, and I) to engage in type I–IV halogen⋯halogen interactions. PES scan was implemented for the C₆H₅X⋯XC₆H₅ homodimers at an X⋯X distance range of 2.5–5.0 Å with a step size of 0.1 Å. Colored binding energy (E_binding) maps were conceived to establish an in-depth insight into type I halogen⋯halogen interactions with a ∠C-X⋯X of 90° < θ₁ ≈ θ₂ < 180° (Figure 4). For type II, III, and IV halogen⋯halogen interactions, the binding energy curves were generated and are displayed in Figure 5. The MP2/aug–cc–pVDZ(PP) binding energies were assessed at the most favorable X⋯X distance and are listed along with the complexation parameters of type I–IV halogen⋯halogen interactions in

Table 2. Binding energies calculated (in kcal/mol) at MP2/aug–cc–pVDZ(PP) level of theory for C₆H₅X⋯XC₆H₅ homodimers (where X = Cl, Br, and I) via type I–IV halogen⋯halogen interactions at the most favorable parameters.

Homodimer		Complexation Parameters a			Binding Energy
		Distance (Å)	θ1b (Degree)	θ2b (Degree)	EMP2/aug−cc−pVDZ (kcal/mol)
Type I	C6H5Cl⋯ClC6H5	3.71	92.5°	92.5°	–1.12
	C6H5Br⋯BrC6H5	3.71	147.5°	147.5°	–1.15
	C6H5I⋯IC6H5	4.02	147.5°	147.5°	–1.49
Type II	C6H5Cl⋯ClC6H5	3.49	90°	180°	–1.32
	C6H5Br⋯BrC6H5	3.64			–1.80
	C6H5I⋯IC6H5	3.96			–2.20
Type III	C6H5Cl⋯ClC6H5	3.58	180°	180°	–0.60
	C6H5Br⋯BrC6H5	3.77			–0.74
	C6H5I⋯IC6H5	4.15			–0.79
Type IV	C6H5Cl⋯ClC6H5	3.68	90°	90°	–1.27
	C6H5Br⋯BrC6H5	3.97			–1.25
	C6H5I⋯IC6H5	4.43			–1.16

^a the most favorable parameters based on the binding energy maps and curves in Figure 4 and Figure 5, respectively; ^b θ₁ and θ₂ represented the ∠C-X⋯X and ∠X⋯X-C bond angles, respectively.

. The coordinates of the most preferential complexes are collected in Table S1.

From the colored maps illustrated in Figure 4, binding energies of type I interactions increased with enlarging the σ-hole size of halogen atoms in the order C₆H₅Cl⋯ClC₆H₅ < C₆H₅Br⋯BrC₆H₅ < C₆H₅I⋯IC₆H₅ homodimers. For example, the binding energies of the C₆H₅X⋯XC₆H₅ homodimers were –1.12, –1.15, and –1.49 kcal/mol in the case of X = Cl, Br, and I, respectively. The abovementioned pattern of the energetic features was in outstanding consistency with previously reported results, in which enhancement of the strength of type I halogen⋯halogen interactions was observed in the same order [57].

For C₆H₅Cl⋯ClC₆H₅ homodimer, the binding energy was found to generally increase by decreasing θ₁ and θ₂ until a certain angle of θ₁ = θ₂ = 147.5°. After that point, the E_binding fluctuated by decreasing the θ₁ and θ₂ values from 115° to 92.5°. For instance, binding energies were –0.58, –0.73, –0.36, and –1.12 in the case of the diminishing of θ₁ and θ₂ values from 177.5° to 147.5°, 115°, and 92.5°, respectively (Table 2). Meanwhile towering E_binding was disclosed for the C₆H₅Br⋯BrC₆H₅ and C₆H₅I⋯IC₆H₅ homodimers by decreasing θ₁ and θ₂ till it recorded its maxima at θ₁ = θ₂ = 147.5° and then decreased with diminishing θ₁ and θ₂ values. As an illustration, the E_binding of the C₆H₅I⋯IC₆H₅ homodimer was –0.76, –1.49, and –0.27 kcal/mol at θ₁ = θ₂ = 177.5°, 147.5°, and 92.5°, respectively. Conspicuously, these results illuminated the prevalence of negative-belt⋯negative-belt interactions within C₆H₅Cl⋯ClC₆H₅ homodimer (i.e., almost more favorable E_binding) as θ₁ and θ₂ became closer to 92.5°, relatively in comparison to the C₆H₅Br⋯BrC₆H₅ and C₆H₅I⋯IC₆H₅ counterparts.

Apparently, binding energies of type II interactions were spotted to drastically increase with the growth in the atomic size of the interacting halogen atoms, which was previously reported and confirmed by the obtained values [58]. Numerically, E_binding was estimated with values of –1.32, –1.80, and –2.20 kcal/mol for C₆H₅Cl⋯ClC₆H₅, C₆H₅Br⋯BrC₆H₅, and C₆H₅I⋯IC₆H₅ homodimers, respectively.

With respect to type III interactions, as an upshot to the presence of σ-hole along the molecular surface of the scouted molecules, specific directional bonding interactions between the two identically charged halogens were divulged, dubbed as σ-hole⋯σ-hole interactions [43]. The σ-hole⋯σ-hole interactions were discerned to debilitate with decreasing the σ-hole size (i.e., increasing the electronegativity) of the halogen atom that was in line with our previous work [43]. For instance, the E_binding of σ-hole⋯σ-hole interactions was assessed with values of –0.79, –0.74, and –0.60 kcal/mol for C₆H₅I⋯IC₆H₅, C₆H₅Br⋯BrC₆H₅, and C₆H₅Cl⋯ClC₆H₅ homodimers, respectively. Such observations might be interpreted as a consequence of the contributions of the polarization of one of the interacted halogen atoms by the other one.

Regarding unexplored type IV halogen⋯halogen interactions, a direct correlation was noticed between the binding energy and the atomic size of the halogen atoms, exhibiting the following order C₆H₅Cl⋯ClC₆H₅ > C₆H₅Br⋯BrC₆H₅ > C₆H₅I⋯IC₆H₅ homodimers. Evidently, binding energies of C₆H₅X⋯XC₆H₅ homodimers were detected with values of –1.27, –1.25, and –1.16 kcal/mol when X = Cl, Br, and I, respectively. These unconventional findings might be ascribed to the resurgent effect of negative-belt⋯negative-belt interactions on the E_binding of the studied homodimers. Hence, type IV-based results could be considered as an integrative window to the foregoing analog of type I interactions at θ₁ = θ₂ = 92.5°.

Overall, all the inspected halogen-containing molecules showed a propensity to participate in type I–IV halogen⋯halogen interactions. Clearly, the energetic features outlined that the highest binding energy was discerned to type II interactions, which is inconsistent with the chalcogen⋯chalcogen analogs that announced the preference of type IV interactions [53]. On the other hand, the lowest E_binding was relevant to type III interactions synchronically to chalcogen⋯chalcogen counterparts.

2.4. Quantum Theory of Atom in Molecules (QTAIM) Analysis

Quantum theory of atom in molecules (QTAIM), proclaimed by Bader et al., has been deemed as an authoritative tool to give a thorough foresight into the nature of intermolecular interactions [59]. For the inspected homodimers, QTAIM analysis was accomplished to uncover the occurrence of the halogen⋯halogen interactions by generating bond critical points (BCPs) and bond paths (BPs). Within the context of QTAIM, diverse BCP features, including electron density (ρ_b), Laplacian (∇²ρ_b), and total energy density (H_b), were minutely estimated for the C₆H₅X⋯XC₆H₅ homodimers at the most favorable parameters. The BPs and BCPs were generated and are depicted in Figure 6. The obtained electron density (ρ_b), Laplacian (∇²ρ_b), and total energy density (H_b) are listed in

Table 3. Topological parameters, including the electron density (ρ_b, au), Laplacian (∇²ρ_b, au), and total energy density (H_b, au), at bond critical points (BCPs) for C₆H₅X⋯XC₆H₅ homodimers (where X = Cl, Br, and I) in the fashion of type I–IV halogen⋯halogen interactions at the most favorable parameters.

	Homodimer	ρb (au)	∇2ρb (au)	Hb (au)
Type I	C6H5Cl⋯ClC6H5	0.00573	0.01575	0.00050
	C6H5Br⋯BrC6H5	0.00584	0.01839	0.00092
	C6H5I⋯IC6H5	0.00604	0.01499	0.00064
Type II	C6H5Cl⋯ClC6H5	0.00660	0.02206	0.00091
	C6H5Br⋯BrC6H5	0.00708	0.02255	0.00107
	C6H5I⋯IC6H5	0.00713	0.01773	0.00072
Type III	C6H5Cl⋯ClC6H5	0.00417	0.01567	0.00077
	C6H5Br⋯BrC6H5	0.00406	0.01470	0.00077
	C6H5I⋯IC6H5	0.00374	0.01070	0.00050
Type IV	C6H5Cl⋯ClC6H5	0.00603	0.01676	0.00052
	C6H5Br⋯BrC6H5	0.00516	0.01368	0.00065
	C6H5I⋯IC6H5	0.00439	0.00905	0.00036

.

As displayed in Figure 6, the potentiality of the scouted monomers to engage in halogen⋯halogen interactions was clearly affirmed through the presence of one bond path (BP) and bond critical point (BCP) between the interacting species. From the data listed in Table 3, positive values of electron density (ρ_b), Laplacian (∇²ρ_b), and total energy density (H_b) were observed, reflecting the closed-shell nature of the considered interactions. Obviously, the ρ_b, ∇²ρ_b, and H_b values were generally observed to be in coincidence with the energetic features. For instance, the ρ_b was estimated for type IV interactions with values of 0.00603, 0.00516, and 0.00439 au, along with binding energies of –1.27, –1.25, and –1.16 kcal/mol for C₆H₅Cl⋯ClC₆H₅, C₆H₅Br⋯BrC₆H₅, and C₆H₅I⋯IC₆H₅ homodimers, respectively.

2.5. Noncovalent Interactions (NCI) Analysis

The Noncovalent Interactions (NCI) index was endorsed to be a trustworthy method for providing a qualitative view of weak intermolecular interaction based on the electron density and its derivatives [60]. The 2D reduced density gradient (RDG) versus the electron density (ρ) multiplied by the sign of the second Hessian eigenvalue (λ₂) were generated for the studied homodimers. Besides, the 3D NCI plots, with a color scale of sign(λ₂)ρ from −0.035 (blue) to 0.020 (red), where λ₂ is the second eigenvalue of the Hessian matrix and ρ is the electron density, were then extracted. The obtained 2D and 3D NCI plots are shown in Figure 7.

According to Figure 7, green isosurfaces were observed within the C₆H₅X⋯XC₆H₅ homodimers, affirming the penchant of the considered halogen-containing molecules to participate in type I–IV halogen⋯halogen interactions. Moreover, the corresponding spikes of sign (λ₂)ρ at low densities emphasized the existence of attractive interactions (sign (λ₂)ρ < 0). From the synchronic perspective to the MP2 energetic features, the size of the green-coded regions and the shifting magnitude of the spikes towards the negative sign (λ₂)ρ were disclosed to be in line with the binding energy results registered in Table 2. Overall, the results of the QTAIM and NCI index analyses support previous studies that illustrated the interactions within the type II and III halogen⋯halogen complexes [43,58].

2.6. Symmetry-Adapted Perturbation Theory-Based Energy Decomposition Analysis (SAPT-EDA)

Symmetry-adapted perturbation theory-based energy decomposition analysis (SAPT-EDA) was fulfilled toward a further illustration of the physical nature of type I–IV halogen⋯halogen interactions. Using SAPT-EDA, the total binding energy was decomposed into its fundamental physical components, namely electrostatic (E_elst), induction (E_ind), dispersion (E_disp), and exchange (E_exch). SAPT-EDA calculations were performed for all of the four types of halogen⋯halogen interactions at the SAPT2+(3) level of truncation [61] using PSI4 code [62]. Total SAPT2+(3) energy accompanied with its physical components is collected in

Table 4. Electrostatic (E_elst), induction (E_ind), dispersion (E_disp), exchange (E_exch), total SAPT2+(3) binding energy (E_SAPT2+(3)), and the energy difference (∆∆E) between MP2 and SAPT2+(3) energies for C₆H₅X⋯XC₆H₅ homodimers (where X = Cl, Br, and I) in the fashion of type I–IV halogen⋯halogen interactions at the most favorable parameters ^a.

Homodimer		Eelst	Eind	Edisp	Eexch	ESAPT2+(3)b	∆∆E c
Type I	C6H5Cl⋯ClC6H5	–0.21	–0.17	–2.43	1.64	–1.17	0.41
	C6H5Br⋯BrC6H5	–0.74	–0.30	–2.12	1.91	–1.24	0.09
	C6H5I⋯IC6H5	–1.13	–0.53	–2.65	2.62	–1.68	0.19
Type II	C6H5Cl⋯ClC6H5	–0.08	–0.14	–2.19	1.26	–1.15	–0.17
	C6H5Br⋯BrC6H5	–0.67	–0.27	–2.03	1.74	–1.24	–0.56
	C6H5I⋯IC6H5	–1.61	–0.77	–3.59	3.60	–2.36	0.16
Type III	C6H5Cl⋯ClC6H5	0.09	–0.10	–1.44	0.77	–0.68	0.08
	C6H5Br⋯BrC6H5	0.17	–0.15	–1.75	0.91	–0.82	0.08
	C6H5I⋯IC6H5	0.32	–0.24	–1.99	0.98	–0.93	0.14
Type IV	C6H5Cl⋯ClC6H5	–0.28	–0.18	–2.67	1.83	–1.30	0.03
	C6H5Br⋯BrC6H5	–0.27	–0.18	–2.65	1.80	–1.30	0.05
	C6H5I⋯IC6H5	–0.33	–0.19	–2.54	1.77	–1.29	0.13

^a all energy terms are in kcal/mol; ^b $E_{\mathrm{SAPT}2+\left(3\right)}=E_{\mathrm{elst}}+E_{\mathrm{ind}}+E_{\mathrm{disp}}+E_{\mathrm{exch}}$; ^c $\Delta \Delta E=E_{\mathrm{MP}2/\mathrm{aug}-\mathrm{cc}-\mathrm{pVDZ}\left(\mathrm{PP}\right)}-E_{\mathrm{SAPT}2+\left(3\right)}$.

and shown on a graph in Figure 8.

As shown in Figure 8, the dispersion energy (E_disp) was the predominant force within all of the scouted types of halogen ⋯halogen interactions. Moreover, respectable contributions for the electrostatic (E_elst) and induction (E_ind) interactions were obtained in the four considered types of halogen⋯halogen interactions. On the other hand, unfavorable contributions for the exchange energy (E_exch) were estimated with positive values for all homodimers. For instance, in the case of type IV interaction for C₆H₅Cl⋯ClC₆H₅ homodimer, E_elst, E_ind, E_disp, and E_exch were assessed with values of −0.28, −0.18, −2.67, and 1.83 kcal/mol, respectively (Table 4).

From the data listed in Table 4, the electrostatic (E_elst), induction (E_ind), and dispersion (E_disp) forces were generally observed to be in coincidence with the MP2 energetic features. For example, the E_disp of type IV interactions were evaluated with values of –2.67, –2.65, and –2.54 kcal/mol accompanied with binding energies of –1.27, –1.25, and –1.16 kcal/mol for C₆H₅Cl⋯ClC₆H₅, C₆H₅Br⋯BrC₆H₅, and C₆H₅I⋯IC₆H₅ homodimers, respectively. Particularly, for σ-hole⋯σ-hole interactions (i.e., type III interactions), unfavorable E_elst was detected, that was observed to escalate with increasing the σ-hole size along with the molecular surface of the halogen atoms, as previously documented [43]. Evidently, E_elst were 0.09, 0.17, and 0.32 kcal/mol for C₆H₅Cl⋯ClC₆H₅, C₆H₅Br⋯BrC₆H₅, and C₆H₅I⋯IC₆H₅ homodimers, respectively. The precision of the employed level for the SAPT-EDA calculations was affirmed by quantifying the energy difference between the MP2 binding energy and the computed SAPT2+(3) energy (∆∆E) with nominal values.

2.7. π⋯π. Contribution into Type IV Halogen⋯Halogen Interaction

To gain a deep insight into π–π contributions within type IV halogen⋯halogen interactions, torsional trans → cis interconversion of C₆H₅X⋯XC₆H₅ homodimers (where X = Cl, Br, and I) were investigated (Figure 9). In this context, the torsional trans → cis interconversion was acclimated by scanning the dihedral angle Φ (C-X⋯X-C) in the range from 180º (i.e., trans configuration) to 0° (i.e., cis configuration) with an interval of 2.5°. Binding energy curves of torsional trans → cis interconversion of the inspected homodimers are illustrated in Figure 9. The negative energetic values for the trans and cis configurations and the maximum binding energy (E_max) within the prolonged range of the torsional trans → cis interconversion were estimated and are listed in

Table 5. Binding energies (in kcal/mol) for the trans (E_trans) and cis (E_cis) geometries in addition to the maximum negative binding energy (E_max) for the trans → cis interconversion of C₆H₅X⋯XC₆H₅ homodimers (where X = Cl, Br, and I) in the pattern of type IV halogen⋯halogen interaction.

Homodimer	trans Configuration	cis Configuration	Trans → cis Interconversion
	Etrans (kcal/mol)	Ecis (kcal/mol)	Emax (kcal/mol)	Φa (Degree)
C6H5Cl⋯ClC6H5	–1.27	–4.20	–4.91	25°
C6H5Br⋯BrC6H5	–1.25	–3.96	–4.25	20°
C6H5I⋯IC6H5	–1.16	–2.93	–2.93	10°

^a Torsional angle of C-X⋯X-C (see Figure 9 for details).

.

Looking at Figure 9, torsional trans → cis interconversion provided a privileged deal for all the inspected homodimers, in which the towering binding energies (i.e., more negative) were generally observed ongoing from Φ = 180° (i.e., trans configuration) to Φ = 0° (i.e., cis configuration). Evidently, binding energy was found to augment from –1.27, –1.25, –1.16 kcal/mol at Φ =180° till recorded at its maxima at Φ = 25°, 20°, and 10° with values of –4.91, –4.25, and –2.93 kcal/mol for C₆H₅Cl⋯ClC₆H₅, C₆H₅Br⋯BrC₆H₅, and C₆H₅I⋯IC₆H₅, respectively. Turning to the cis configuration, the binding energies were found to be increased according to the following order C₆H₅Cl⋯ClC₆H₅ > C₆H₅Br⋯BrC₆H₅ > C₆H₅I⋯IC₆H₅ with values of –4.20, –3.96, and –2.93 kcal/mol, respectively, coinciding with the trans configuration considerations. Astonishingly, the proceeding results accentuated that the cis configuration was outshined by the trans configuration (i.e., type IV interactions), highlighting the paramount contributions of the π–π interactions.

3. Computational Methods

In the presented study, the inclination of halobenzene molecules (C₆H₅X; where X = Cl, Br, and I) to participate in type I–IV halogen⋯halogen interactions were comparatively scrutinized. Geometrical structures of the investigated monomers were fully optimized at the MP2/aug–cc–pVDZ level of theory [63,64] for all the studied atoms except Br and I atoms that were treated with aug–cc–pVDZ(PP) basis set to consider the relativistic effects [65]. Upon the optimized monomers, the electrostatic potential (EP) analysis was carried out using an electron density contour of 0.002 au as a precious envelop that described the electrostatic potential of the chemical systems [7,36]. In that spirit, the MEP maps were generated, and the maximum positive electrostatic potential (V_s,max) values were computed with the help of Multiwfn 3.7 software [66]. Using the point-of-charge (PoC) approach, the propensity of the studied monomers to engage in type I-V halogen⋯halogen interactions from the electrostatic perspective was preciously appreciated [58,67,68,69]. In the PoC approach, negatively- and positively charged points with a value of ±0.50 au were employed to simulate the interactions of halobenzene with a Lewis base and acid, respectively. Accordingly, molecular stabilization energy (E_{stabilization}) was assessed at X–PoC distance ranging from 2.5 to 5.0 Å with a step size of 0.1 Å and ∠C-X–PoC angle of 180° and 90° along the x-axis and z-axis of the halogen atom (i.e., along the equatorial and axial surfaces of the halogen atom), respectively. Molecular stabilization energy (E_{stabilization}) of the C₆H₅⋯PoC systems was calculated according to the following equation [34,52,67,70,71]:

$$E_{\mathrm{stabilization}}=E_{\mathrm{halogen}-\mathrm{containing}\mathrm{molecule}\cdots \mathrm{PoC}}-E_{\mathrm{halogen}-\mathrm{containing}\mathrm{molecule}}$$

(1)

A potential energy surface (PES) scan was also implemented to elucidate the potentiality of the C₆H₅X⋯XC₆H₅ homodimers to participate in type I–IV halogen⋯halogen interactions. The optimized monomers were directed through a set of exact orientations parodying the studied interaction types without allowing the optimized monomers to be distorted. PES scan was performed for the considered homodimers at X–X distance range of 2.5–5.0 Å with a step size of 0.1 Å. Binding energy was then calculated as the difference between the energy of the homodimer and the sum of its monomers. The basis set superposition error (BSSE) was eliminated from the computed binding energies using the counterpoise correction (CP) method [72].

Quantum theory of atoms in molecules (QTAIM) was invoked to qualitatively elaborate the nature of the studied homodimers [73]. In the context of QTAIM, the bond paths (BPs) and bond critical points (BCPs) were generated. Various topological features, including electron density (ρ_b), Laplacian (∇²ρ_b) and total energy density (H_b), were also evaluated. Noncovalent interaction (NCI) index was then fulfilled to get a deeper pictorial insight into the nature of the considered interactions in the fashion of 2D-reduced density gradient (RDG) and 3D-colored NCI plots [60]. The QTAIM and NCI analyses were performed using Multiwfn 3.7 package [66] and portrayed using Visual Molecular Dynamics (VMD) software [74]. All calculations were executed at the MP2/aug–cc–pVDZ(PP) level of theory using Gaussian 09 software [75].

Symmetry-adapted perturbation theory-based energy decomposition analysis (SAPT-EDA) was incorporated as a robust tool to decompose the binding energy into its fundamental physical components. SAPT-EDA calculations were performed for the studied homodimers at SAPT2+(3) level of truncation [61] with the help of the PSI4 code [62]. The total SAPT2+(3) energy components, involving electrostatic (E_elst), induction (E_ind), dispersion (E_disp), and exchange energies (E_exch) were computed according to the following equations [76]:

$$E_{\mathrm{SAPT}2+\left(3\right)}^{}=E_{\mathrm{elst}}+E_{\mathrm{ind}}+E_{\mathrm{disp}}+E_{\mathrm{exch}}$$

(2)

where:

$$E_{\mathrm{elst}}=E_{\mathrm{elst}}^{\left(10\right)}+E_{\mathrm{elst},\mathrm{r}}^{\left(12\right)}+E_{\mathrm{elst},\mathrm{r}}^{\left(13\right)}$$

(3)

$$E_{\mathrm{ind}}=E_{\mathrm{ind},\mathrm{r}}^{\left(20\right)}+E_{\mathrm{exch}-\mathrm{ind},\mathrm{r}}^{\left(20\right)}+E_{\mathrm{ind}}^{\left(22\right)}+E_{\mathrm{exch}-\mathrm{ind}}^{\left(22\right)}+\mathsf{\delta }{E}_{\mathrm{HF},\mathrm{r}}^{\left(2\right)}$$

(4)

$$E_{\mathrm{disp}}=E_{\mathrm{disp}}^{\left(20\right)}+E_{\mathrm{exch}-\mathrm{disp}}^{\left(20\right)}+E_{\mathrm{disp}}^{\left(21\right)}+E_{\mathrm{disp}}^{\left(22\right)}\left(\mathrm{SDQ}\right)+E_{\mathrm{Est}.\mathrm{Disp}}^{\left(22\right)}\left(\mathrm{T}\right)+E_{\mathrm{disp}}^{\left(30\right)}$$

(5)

$$E_{\mathrm{exch}}=E_{\mathrm{exch}}^{\left(10\right)}+E_{\mathrm{exch}}^{\left(11\right)}+E_{\mathrm{exch}}^{\left(12\right)}$$

(6)

4. Conclusions

In this study, four different types of halogen⋯halogen interactions were comparatively scrutinized within the C₆H₅X⋯XC₆H₅ homodimers (where X = Cl, Br, and I), using a versatile range of quantum mechanical calculations. The occurrence of the σ-hole along the surface of the studied halogen atoms was found to be the common feature within the inspected molecules, as affirmed by MEP maps, which enabled them to behave as potent halogen bond donors. Astonishingly, PoC calculations generally assured the penchant of the considered molecules to electrostatically engage in halogen bonding interactions along the x- and z-axes. For the inspected homodimers, towering binding energies were evidently recorded with disparate magnitudes, alluding to the significant propensity of the scouted molecules to engage in type I–IV halogen⋯halogen interactions. Interestingly, for the explored interactions, type II interactions were reported as the superb geometry with favorable negative energies. In line with chalcogen⋯chalcogen interactions, type III halogen⋯halogen interactions were denoted as the most emaciated one. Generally, binding energy was noticed to increase coinciding with the atomic size of the halogen atoms. In particular, an alien behavior was uncovered within type IV interactions, in which an inversely proportional correlation was observed between the E_binding and the σ-hole size as an upshot to the varied negative-belt⋯negative-belt contributions. Moreover, the prolonged scope of the torsional trans → cis interconversion interactions was reported with favorable energetic values that was found to be in consistence with type IV interactions based-results for all the considered homodimers. SAPT-EDA results addressed the dispersion forces as the most preferential component within all the four studied interaction types. The full insight into the type I–IV halogen⋯halogen interactions that emerged from the current work would be fundamental guidance for the forthcoming studies pertinent to the crystal engineering and materials science fields.