Interactions of Substituted Nitroaromatics with Model Graphene Systems: Applicability of Hammett Substituent Constants To Predict Binding Energies

Abstract

Applicability of Hammett parameters (σ_m and σ_p) was tested in extended π-systems in gas phase. Three different model graphene systems, viz. 5,5-graphene (GR), 3-B-5,5-graphene (3BGR), and 3-N-5,5-graphene (3NGR), were designed as extended π-systems, and interactions of various nitrobenzene derivatives (mainly m- and p-substituted together with some multiple substitutions) on such platforms were monitored using density functional theory (M06/cc-pVDZ, M06/cc-pVTZ, M06/sp-aug-cc-pVTZ) and Møller–Plesset second-order perturbation (MP2/cc-pV-DZ) theory. Offset face to face (OSFF) stackings were found to be the favored orientations, and reasonable correlations were found between binding energies (ΔE_B) and the ∑|σ_m| values of the substituted nitrobenzenes. It was proposed previously that |σ_m| contains information about the substituents’ polarizability and controls electrostatic and dispersion interactions. The combination of ∑|σ_m| and molar refractivity (as ∑M_r) or change in polarizability (Δα: with respect to benzene) of nitrobenzene derivatives generated statistically significant correlation with respect to ΔE_B, thereby supporting the hypothesis related to the validity of |σ_m| correlations. The |σ_p| parameters also maintain similar correlations for the various p-substituted nitrobenzene derivatives together with several multiply-substituted nitrobenzene derivatives. The correlation properties in such cases are similar to the |σ_m| cases, and the energy partition analysis for both the situations reveled importance of electrostatic and dispersion contributions in such interactions. The applicability of Hammett parameters was observed previously on the restricted parallel face to face orientation of benzene···substituted benzene systems, and the present results show that such an idea could be used to predict ΔE_B values in OSFF orientations, if the scaffolds are designed in such a way that substituted benzene systems cannot escape their π-clouds.

1. Introduction

Noncovalent interactions of aromatics are important in a wide range of chemical and biological processes.¹⁻⁷ These include enzyme–substrate recognition,^4,8 protein structure and function,⁹⁻¹¹ DNA/RNA base stacking¹² and intercalation,^13,14 organic reaction development,¹⁵ and organic materials.¹⁶⁻¹⁹ Arene–arene interactions play a key role in enzyme–substrate reactions. A significant number of experimental^4,20−28 and theoretical investigations²⁹⁻³⁶ are available in this connection explaining the role of such interactions. High-level theoretical investigation demonstrated that dispersion forces primarily dominate the observed binding,³⁰⁻³⁹ although the initial studies of Hunter and Sanders⁴⁰ in this direction indicated the major force of interactions to be electrostatic in origin.

Computational study of interactions between benzene and monosubstituted benzenes in parallel face to face (PFF) orientation showed that there exists a correlation of binding energies (ΔE_B) with Hammett σ_m values of monosubstituted aromatics.⁴¹ The results were interpreted as support to the proposed domination of the dispersion and electrostatic interactions in such cases. Still the Hammett constants σ_m or ∑σ_m for multisubstituted aromatics were not found to be sufficient to predict the arene–arene binding energies. More recently, Watt et al.⁴² have shown a good correlation between the ΔE_B of arene–arene complexes and the ∑|σ_m| values of the substituted arenes. These studies were carried out under constrained geometric conditions to keep the interacting molecules in strict PFF orientations. Obviously, such an orientation was chosen to maximize π–π (origin of most of the dispersive forces) and electrostatic interactions. The authors also found the contribution of both electrostatic and dispersion energies as important guiding factors to impose good correlation with the ∑|σ_m| parameters. The importance of ∑|σ_m| parameters was also demonstrated by Cormier and Lewis for the interactions of Li⁺ and Na⁺ with cyclopentadienyl anions.⁴³

No reasonable explanation exists as to why Hammett constant (σ_m) and related parameters like ∑σ_m, and ∑|σ_m| should have any correlation with ΔE_B. The origin of the idea of using Hammett constants to predict arene–arene attraction was explored in numerous experimental studies.^8−10,12 The main assumption here is that such an attraction is largely electrostatic in nature (Hunter and Sanders),⁴⁰ and a parameter that describes the electronic effects of substituents on benzoic acid acidity,⁴⁴ the Hammett constant, should work to model such electronic/electrostatic arene–arene interactions. Several investigations in recent times have, however, found that this assumption is not correct. Watt and co-workers⁴² carried out detailed analysis in this respect and proposed that the ∑|σ_m| values contain information that regular Hammett paramers σ_m and ∑σ_m are lacking. They concluded that it is probably the dispersion contribution to the overall binding. The hypothesis did originate from the results of multiple regression analysis correlating ΔE_B with ∑σ_m and ∑M_r (M_r: refractivity of molecules) parameters. The correlation was very promising, and because M_r implicitly involves polarizability of molecules, the hypothesis of the involvement of dispersion effect in ∑|σ_m| was qualitatively substantiated. A recent review by Lewis and co-workers⁴⁵ summarized the different ideas related to this problem, but their explanations are not yet conclusive and the authors still supported their earlier conclusion.⁴²

The Hammett constant σ_p had also been found to correlate with ΔE_B in arene–arene interactions for multiply-substituted benzene systems. Wheeler and Houk⁴¹ suggested that the correlation of σ_p with ΔE_B in such cases relates to the polarization of the π-system of the substituted ring. Subsequent investigations of Watt et al.⁴² also suggested that ∑σ_p and ∑M_r (replaced further by lipophilicity parameter ∑π) for substituted benzenes in PFF orientations of the arene–arene systems show similar correlation to ΔE_B as discussed for the σ_m parameter cases, although the correlations were not as good.

There are two other orientations in arene–arene interactions, viz. offset face to face (OSFF) and edge to face (ETF). The interacting moieties in OSFF orientations were more stable than the rest of the geometric orientations.³⁶ In the case of benzene–benzene dimers, OSFF and ETF orientations were almost equally stable.²⁹⁻³⁵ There could be, of course, an infinite number of OSFF and ETF arene–arene dimer orientations. The OSFF orientations can vary by how offset the two arenes are with respect to each other (in an infinite number of directions), and the ETF arene–arene dimer orientations can vary in the angle between the two aromatics and whether the vertical aromatic has an edge, a C atom, or some intermediate orientation, interacting with the horizontal aromatic. So, there is no unique orientation for the most stable benzene···substituted benzene interactions. The interacting systems (OSFF) were not considered in most of the previous studies of Hammett correlations with ΔE_B, as face to face π–π interactions would be lost and Hammett constant correlation might not be that important. There have been, of course, several arene–substituted arene interaction studies in recent times to show the importance of OSFF orientation of the interacting systems in explaining structural aspects of different π–π stackings.^46,47 Now, if one changes the scaffold to a larger conjugated π-system like graphene (instead of benzene), the substituted benzene moieties in parallel orientations (either in PFF or OSFF) cannot escape π–π stacking. Thus, the hypotheses of the correlations of Hammett constants with ΔE_B could be valid here, regardless of the nature of the parallel orientations of the substituted benzenes against the graphene scaffold. Moreover, it will also be interesting to monitor the effect of doping on the graphene sheet for such interactions. Controlled doping of the single-walled carbon nanotubes by boron atoms does not have marked change in the π-character⁴⁴ of the system and it is expected to be the same for the graphene sheet also. In the present situation, it will be interesting to see if such doping has any effect on the Hammett correlations as discussed above.

The work presented here addresses the idea of Hammett constants’ (σ_m, σ_p, and related parameters) correlation with the ΔE_B of various nitrobenzene derivatives on model graphene scaffolds. The nitrobenzene derivatives were chosen because of the availability of the experimental Gibbs free energy of binding of a few of such compounds with the graphene systems in OSFF orientation (discussed in detail in Section 2). These experimental values are useful to calibrate the computed binding energy data. The model graphene system was designed from 5,5-graphene. Singly B- and N-doped graphenes were also designed from this model for further use as scaffolds with graphene-like π-orbital character. The B- and N-doped graphene materials are quite well known,⁴⁸⁻⁵⁰ and previous theoretical studies also showed that they preserve the properties of the pristine graphene materials.⁵⁰ The singly doped model B- and N-graphenes in the present studies are thus expected to influence the nature of interactions with nitroaromatics (due to the influence of B and N atoms) without disturbing the π-character of the whole system with respect to pristine graphene. The preservation of π–π stacking character was observed in case of interactions between N-doped graphene systems.^51,52 The size of the graphene systems was chosen such that the nitrobenzene derivatives retain OSFF orientations during interactions avoiding edge effect. It would be shown that dispersion forces together with electrostatic interactions play important roles in such binding, as was suggested in arene–arene interactions. Although the Hammett σ-parameters of nitrobenzene derivatives show good correlations in terms of ∑|σ_x| (x: m or p) during their interactions with 5,5-graphene, two variable equations with a combination involving ∑σ_x (x: m or p) and ∑M_r are found to be more convincing. Moreover, because the parameter M_r involves polarizability, the computed shift of polarizabilities (Δα) of nitrobenzene derivatives (with respect to benzene) in combination with ∑σ_x (x: m or p) parameter also shows convincing correlation in this respect. Although doped 5,5-graphenes show similar interactions to nitrobenzene derivatives, the correlations with Hammett parameters are not as good as the 5,5-graphene cases. Probably, change of electrostatic interactions with respect to pristine graphene system plays a role here. Energy decomposition analysis has been carried out to account for the role of electrostatic and dispersion interactions in such complexes. The results will also show their influence in correlating the binding energies to Hammett σ-parameters.

2. Methods of Computation

The model 5,5-graphene (GR) system is shown in Figure 1. The system is designed in such a way that it has a mirror plane with respect to the x-axis. Thus, full optimization of the system does not disturb the overall symmetry (D_2h) of this model graphene unit. The edges of the graphene unit are terminated with H-atoms to avoid edge effect during interactions with the chosen nitrobenzene systems. The B- and N-doped model systems were designed by substituting C₃ of the central ring (C_2v-symmetry), as shown in Figure 1, and we designate them as 3-B-5,5-graphene (3BGR) and 3-N-5,5-graphene (3NGR), respectively. The substituted nitrobenzene dimers (X–C₆H₄–NO₂) were chosen to have X = H, CH₃, OCH₃, OC₂H₅, Cl, Br, I, OH, CN, and NH₂ substituents at m- and p-positions. Moreover, three compounds with substituents 3-NO₂-4-OH, 3,5-di-NO₂-4-OH, and 3,5-di-NO₂-4-CH₃ (with respect to the NO₂ group of nitrobenzene) were also considered for investigating the interaction properties. The interaction geometries between the GR and substituted nitrobenzene molecules were computed, allowing the systems to relax fully from initial PFF orientation. A similar approach was adopted for the interactions with 3BGR and 3NGR scaffolds. There could be more than one local minimum in such optimizations, but we have located the desired minima through optimizations from different starting parallel OSFF orientations. Geometry optimizations of nitroaromatics, graphene scaffolds, and their complexes were carried out using density functional theory (DFT)⁵³ with M06 functional⁵⁴ and cc-pVDZ basis set of atoms (cc-pVDZ-PP ECP (effective core potential) basis set was used for iodine). The M06-2X functional⁵⁴ could have been another choice here, but because both M06 and M06-2X generate similar results for such noncovalent interactions, we have used the M06 functional only. M06-2X was used for the initial calibrations. The minimum-energy structures in the respective cases were confirmed through frequency calculations. These results were used to compute the binding energy (ΔE_B), enthalpy (ΔH_B°), and Gibbs free energy (ΔG_B°) of binding (at 298K) of the complexes. The calculations included counterpoise (CP) and zero-point energy (ZPE) corrections. Calculations of thermodynamic properties were carried out by applying the ideal gas, rigid rotator, and harmonic oscillator approximations.⁵⁵

Figure 1.

(A–C) Highest occupied molecular orbitals of the optimized structure (DFT/M06/cc-pVDZ level) of GR, 3BGR, and 3NGR, respectively. (D–F) Molecular electrostatic potential (MEP) pictures of the optimized structures (DFT/M06/cc-pVDZ level) of GR···m-nitroaniline, 3BGR···p-nitrophenol, and 3NGR···m-nitrophenol, respectively. The various colored regions on the surfaces are deep blue (highly positive, >0.1 au), light blue (<0.1 and >0.05 au), green (0.0 au), and yellow (<−0.1 and >−0.05 au). The optimized geometries of GR (G), 3BGR (H), and 3NGR (I) (DFT/M06/cc-pVDZ level) are also added for more clarity of the scaffold structures.

The computed ΔE_B values (ΔE_Bs) were further refined using higher-level cc-pVTZ and sp-aug-cc-pVTZ basis sets at the DFT/M06 level. There were convergence problems in using aug-cc-pVTZ basis sets directly in the energy calculations of the GR/3BGR/3NGR···nitrobenzene (and nitrobenzene derivative) complexes (probably because of linear dependence). The higher-angular-momentum diffuse functions (d and f) of the main atoms and the hydrogen diffuse functions were causing such problems, and these diffuse functions were eliminated in the final sp-aug-cc-pVTZ basis sets to achieve energy convergence. In the case of iodine, the cc-pVTZ-PP and sp-aug-cc-pVDZ-PP ECP basis sets were used for energy calculations. Single-point calculations were carried out on the optimized geometries at the DFT/cc-pVDZ level to estimate the CP corrections. The ZPE and the energies for the individual components were obtained from the frequency analysis of the optimized structures at the DFT/cc-pVDZ level for the computations of ΔE_Bs. The ΔE_Bs from all of these computations, and ΔH_B° and ΔG_B° at the DFT/cc-pVDZ level as well, were further corrected through inclusion of dispersion energy of interactions (ΔE_disp). The magnitude of ΔE_disp was estimated at the DFT/cc-pVTZ level (using optimized geometries from the DFT/cc-pVDZ calculations) through inclusion of Grimme’s empirical dispersion (GD3)⁵⁶ and obtained as the difference of interaction energies (with CP corrections) with and without the dispersion effects. The computed ΔE_disp values were added to the estimated binding energies. This is a valid approximation as the geometries are the same at all of these basis set levels. We have further estimated the ΔE_B (CP- and ZPE-corrected) through single-point Møller–Plesset second-order perturbation (MP2)⁵⁷ calculations (cc-pVDZ basis sets) using the geometries and ZPE corrections from the DFT/cc-pVDZ results. The results generated a direct estimation of ΔE_disp and used in energy partitioning analysis.

Energy partitioning analyses were carried out to estimate the contribution of electrostatic interactions in such complex formations. These were computed using a hybrid variational–perturbational interaction energy decomposition scheme.⁵⁸ The SCF interaction energy is partitioned into first-order electrostatic (E_el⁽¹⁰⁾), Heitler–London exchange (E_ex), and higher-order delocalization (ΔE_del^HF) energy terms.

1

The MP2 calculations generate an estimation of ΔE_disp, which could be designated as E_MP⁽²⁾. It includes the dispersion and correlation contributions to the Hartree–Fock components, and is calculated using the supermolecular approach as the difference of MP2 energy corrections of the supermolecule and the monomers (eq 2).

2

The energy terms on the right-hand side of eq 2 represent the difference between the MP2 and Hartree–Fock energies of the supermolecule (AB) and the monomers (A and B). All of the interaction energy terms are calculated consistently in the dimer-centered basis set and are therefore free from the basis set superposition error (BSSE) due to the full counterpoise correction. The contribution of the multipolar electrostatic interactions in the cation−π complexes has been calculated using the distributed atomic multipolar expansion of the charge distributions of monomers.⁵⁹ The multipolar expansion technique is based on the numerically equivalent spherical harmonic formulations of Stone et al.^60,61 The optimized structures at the DFT/B3LYP/cc-pVDZ level were used for such calculations. The results of such energy decomposition calculations are not rigorous and would be valid for interpretation purposes only, as there is an element of arbitrariness in any of such decomposition schemes. All of the computations were carried out using Gaussian 09 code.⁶² The interaction energy decomposition scheme implemented in the GAMESS code⁶³ was used for energy partitioning analyses and computation of the multipolar components of the total electrostatic interaction energies.⁶⁴ Molecular graphics have been generated using the GaussView05 software.⁶⁵

3. Results and Discussion

3.1. Comparison of the Binding Energies of Nitroaromatics with GR, 3BGR, and 3NGR Scaffolds

The π-orbitals of GR, 3BGR, and 3NGR (Figure 1) show that the π-network of these systems are capable of holding the nitroaromatics through long-range π–π interactions. The optimized structures of such stackings are available in Figures S1–S4 (Supporting Information), and a general model of such interactions, depicting distance between the two π-systems, is shown in Scheme 1. As expected, all of the optimized structures show OSFF arrangements and the aromatic nitro compounds do not get out of the GR/3BGR/3NGR platforms due their extended π-network. The binding energies (ΔE_B, ΔH_B°, and ΔG_B°) were initially computed (as discussed in Section 2) at the DFT/M06/cc-pVDZ level. The higher basis set (cc-pVTZ and sp-aug-cc-pVTZ) and MP2/cc-pVDZ calculations were carried out to monitor the accuracy of the ΔE_B values through higher electron correlation effects. These strategies to compute ΔE_B error limits are essential, as the primary objective is to use such results to monitor the validity of Hammett correlation in such interactions. The binding energy values for various interactions at the DFT/cc-pVDZ level are shown in Tables 1 and 2 for the m- and p-substituted nitrobenzenes, respectively, together with di- and tri-substituted nitrobenzenes. The ΔE_B at higher basis sets and at the MP2/cc-pVDZ level, for the similar cases, are shown in Tables 3 and 4, respectively.

Scheme 1. Schematic Representation of the Interacting Distance (d) between Nitrobenzene Derivatives and the Graphene Systems (GR/3BGR/3NGR).

The distance is measured between the center of mass (CM) of the aromatic ring of the nitrobenzene derivatives and the graphene systems. The whole graphene systems were not considered to compute CM. The part over which the aromatic ring of the interacting nitrobenzene derivatives were lying (almost in parallel orientation) was used to compute the CM of the graphene systems.

Table 1. Computed Binding Energies (ΔE_B, kcal/mol) and Related Thermodynamic Parameters (ΔH_B°, and ΔG_B°, kcal/mol) of Various m-Substituted and Several Di- and Tri-Substituted Nitrobenzene Derivatives Interacting with Model (5,5)-Graphene (GR) and its B- and N-Doped Scaffolds (3BGR and 3NGR, Respectively) at the M06/cc-pVDZ Level^a,b.

	GR			3BGR			3NGR
substituents	ΔEB	ΔHB°	ΔGB°	ΔEB	ΔHB°	ΔGB°	ΔEB	ΔHB°	ΔGB°
–Hc	–13.4	–14.6	–2.7	–14.3	–14.1	–1.2	–15.0	–15.0	–1.1
m-NH2	–17.5	–17.5	–3.5	–17.7	–17.6	–3.7	–17.6	–17.4	–4.3
m-OH	–16.4	–15.9	–4.0	–15.8	–15.7	–2.1	–15.5	–15.6	–0.8
m-OCH3	–16.2	–17.2	–4.7	–18.8	–18.4	–5.6	–18.2	–18.1	–3.9
m-OC2H5	–17.6	–17.3	–3.7	–18.7	–17.9	–3.8	–18.1	–18.2	–2.9
m-CH3	–16.0	–16.6	–2.9	–17.1	–16.9	–3.8	–17.3	–17.0	–3.9
m-NO2	–19.1	–19.2	–3.7	–19.3	–18.9	–5.2	–19.0	–19.1	–3.5
m-CN	–17.7	–17.3	–5.0	–18.6	–18.3	–4.7	–18.6	–18.4	–4.5
m-Cl	–17.3	–16.8	–4.2	–16.9	–16.4	–4.0	–16.0	–16.8	–3.3
m-Br	–18.1	–17.9	–4.1	–18.3	–16.6	–4.9	–18.0	–17.9	–3.5
m-I	–19.1	–19.0	–6.3	–18.5	–18.0	–3.5	–18.7	–18.1	–5.7
3-NO2, 4-OH	–19.4	–19.1	–10.0	–20.3	–19.7	–13.2	–19.3	–18.7	–11.5
3,5-di-NO2, 4-OH	–23.4	–23.8	–13.5	–23.6	–24.1	–14.1	–24.9	–24.7	–17.4
3,5-di-NO2, 4-CH3	–23.9	–23.5	–16.2	–23.5	–23.8	–13.4	–23.9	–23.8	–15.4
benzened	–4.4	–3.7	5.0	–5.1	–4.5	6.0	–5.7	–4.8	4.2

^aSimilar values for the interactions with benzene are also included for comparison.

^bThe ΔE_B, ΔH_B°, and ΔG_B° values are BSSE- and ZPE-corrected. See the text for details.

^cNitrobenzene.

^dBenzene interacting with GR/3BGR/3NGR.

Table 2. Computed Binding Energies (ΔE_B, kcal/mol) and Related Thermodynamic Parameters (ΔH_B° and ΔG_B°, kcal/mol) of Various p-Substituted Nitrobenzene Derivatives Interacting with Model (5,5)-Graphene (GR) and Its B- and N-Doped Scaffolds (3BGR and 3NGR, Respectively) at the M06/cc-pVDZ Level^a.

	GR			3BGR			3NGR
substituents	ΔEB	ΔHB°	ΔGB°	ΔEB	ΔHB°	ΔGB°	ΔEB	ΔHB°	ΔGB°
p-NH2	–17.7	–17.4	–5.1	–17.4	–16.9	–5.2	–16.5	–16.5	–2.6
p-OH	–15.9	–16.0	–1.4	–17.9	–17.7	–4.1	–15.6	–15.7	–0.3
	(−15.8)b	(−15.9)b	(−1.2)b
p-OCH3	–18.4	–18.2	–4.9	–18.1	–18.0	–3.9	–18.7	–18.7	–4.2
p-OC2H5	–19.9	–19.8	–6.0	–19.4	–19.6	–4.1	–18.8	–19.2	–2.9
p-CH3	–17.6	–16.7	–5.8	–17.4	–16.5	–5.6	–19.3	–17.8	–5.4
p-NO2	–18.1	–18.1	–4.0	–18.5	–18.2	–5.5	–19.5	–19.1	–6.7
p-CN	–15.8	–17.7	–7.0	–18.9	–18.5	–5.6	–18.4	–18.0	–5.4
p-Cl	–16.7	–17.6	–5.7	–17.6	–17.1	–5.1	–16.5	–16.1	–3.2
p-Br	–17.2	–17.1	–6.6	–17.7	–17.3	–5.7	–18.5	–17.9	–6.3
p-I	–19.0	–18.8	–5.6	–18.2	–18.0	–4.4	–17.7	–17.4	–6.3

^aThe ΔE_B, ΔH_B°, and ΔG_B° values are BSSE- and ZPE-corrected. See the text for details.

^bComputed values using full-geometry optimization using empirical dispersion corrections.

Table 3. Computed Binding Energies (ΔE_B, kcal/mol) and Related Thermodynamic Parameters (ΔH_B° and ΔG_B°, kcal/mol) of Various m-Substituted and Several Di- and Tri-Substituted Nitrobenzene Derivatives Interacting with Model (5,5)-Graphene (GR) and Its B- and N-Doped Scaffolds (3BGR and 3NGR, Respectively) at the M06/cc-pVTZ (TZ), M06/sp-aug-cc-pVTZ (ATZ), and MP2/cc-pVDZ (MP2) Levels^a.

	ΔEB (GR)			ΔEB (3BGR)			ΔEB (3NGR)
substituents	TZ	ATZ	MP2	TZ	ATZ	MP2	TZ	ATZ	MP2
–Hb	–13.4	–13.4	–12.3	–13.7	–13.7	–9.6	–14.8	–14.8	–12.1
m-NH2	–17.6	–17.8	–14.8	–17.6	–17.0	–14.3	–17.5	–17.7	–17.5
m-OH	–16.0	–16.1	–14.2	–15.8	–15.8	–10.9	–15.5	–15.5	–12.2
m-OCH3	–16.0	–16.1	–14.1	–18.3	–18.3	–14.7	–17.9	–17.8	–14.3
m-OC2H5	–16.9	–16.9	–13.8	–18.5	–17.7	–12.7	–17.7	–18.3	–17.1
m-CH3	–15.7	–15.7	–13.7	–16.9	–16.9	–12.0	–16.8	–16.9	–13.5
m-NO2	–18.8	–18.8	–16.7	–18.9	–19.0	–16.9	–18.6	–18.6	–17.1
m-CN	–16.8	–16.9	–16.2	–18.3	–18.3	–16.5	–17.9	–17.9	–17.5
m-Cl	–16.6	–16.7	–15.3	–16.6	–16.6	–13.1	–16.0	–15.5	–12.3
m-Br	–17.4	–17.4	–15.5	–17.6	–17.5	–15.3	–17.1	–17.1	–15.7
m-I	–18.4	–18.3	–15.5	–18.4	–17.9	–15.6	–17.7	–17.6	–15.5
3-NO2, 4-OH	–18.9	–19.0	–17.1	–20.3	–20.2	–17.4	–19.3	–18.5	–14.6
3,5-di-NO2, 4-OH	–23.1	–24.0	–20.2	–23.6	–21.8	–19.6	–24.9	–26.1	–21.0
3,5-di-NO2, 4-CH3	–23.4	–23.5	–20.7	–23.6	–22.1	–20.1	–23.9	–24.2	–22.0

^aThe ΔE_B, ΔH_B°, and ΔG_B° values are BSSE- and ZPE-corrected. See the text for details.

^bNitrobenzene.

Table 4. Computed Binding Energies (ΔE_B, kcal/mol) and Related Thermodynamic Parameters (ΔH_B° and ΔG_B°, kcal/mol) of Various p-Substituted Nitrobenzene Derivatives Interacting with Model (5,5)-Graphene (GR) and Its B- and N-Doped Scaffolds (3BGR and 3NGR, Respectively) at the M06/cc-pVTZ (TZ), M06/sp-aug-cc-pVTZ (ATZ), and MP2/cc-pVDZ (MP2) Levels^a.

	ΔEB (GR)			ΔEB (3BGR)			ΔEB (3NGR)
substituents	TZ	ATZ	MP2	TZ	ATZ	MP2	TZ	ATZ	MP2
p-NH2	–17.2	–17.4	–14.8	–17.1	–17.1	–14.3	–16.2	–16.3	–15.5
p-OH	–16.2	–16.3	–14.2	–17.0	–17.4	–10.9	–15.7	–15.7	–12.2
p-OCH3	–18.0	–18.1	–14.1	–17.9	–17.9	–14.7	–18.4	–18.4	–14.3
p-OC2H5	–19.3	–19.4	–13.8	–19.5	–19.5	–12.7	–18.7	–17.8	–17.1
p-CH3	–16.8	–17.0	–13.7	–17.1	–17.1	–12.0	–18.3	–18.2	–13.5
p-NO2	–17.9	–17.9	–16.7	–18.2	–18.2	–16.9	–19.9	–20.4	–17.1
p-CN	–14.9	–15.0	–16.2	–18.6	–18.6	–16.5	–18.0	–18.1	–17.5
p-Cl	–15.9	–15.9	–15.3	–17.2	–17.2	–13.1	–15.8	–15.7	–12.3
p-Br	–16.4	–16.5	–15.5	–17.1	–16.8	–15.3	–17.3	–17.3	–15.7
p-I	–18.3	–18.3	–15.5	–17.8	–17.7	–15.6	–17.2	–17.8	–15.5

^aThe ΔE_B, ΔH_B°, and ΔG_B° values are BSSE- and ZPE-corrected. See the text for details.

The ΔE_B’s computed at different levels are more or less similar, although the computed values at higher basis sets and at the MP2 level are slightly lower than those at the DFT/cc-pVDZ computations. This could be due to the fixed geometry (from DFT/cc-pVDZ results) used in such calculations. All of these binding energies (DFT levels) are dispersion (ΔE_disp)-corrected, as discussed in Section 2. The approximation used in dispersion correction (Section 2) could be justified by computation of binding energies through full-geometry optimizations with the inclusion of the empirical GD3 function. The results for the GR···p-nitrophenol (DFT/cc-pVDZ level) case is included in Table 2, and the computed binding energies do not differ from those using the approximation as discussed. The Gaussian code actually performs such geometry optimization through the inclusion of GD3 empirical dispersion energy to the nuclear repulsion term in each optimization cycle. Thus, in the case of fully optimized geometry (without dispersion correction), addition of dispersion energy at the final stage is almost equivalent to the full-geometry optimization technique, including dispersion effect, and thus the empirical approach for dispersion correction for binding energies is fully logical.

There are not too many experiments to justify the computed binding energies for the present systems. In recent years, experiments have been carried out to estimate adsorption energies (ΔG_ad) of nitrobenzene, m-dinitrobenzene, and p-nitrotoluene on graphene surface.⁶⁶ The adsorption isotherm was computed using Freundlich and Langmuir techniques. The experimental equilibrium binding constants K_F and K_L from Freundlich and Langmuir isotherms, respectively, could be used to compute ΔG_ad values (ΔG_ads) for these systems. The magnitudes of ΔG_ads for nitrobenzene, m-nitrobenzene, and p-nitrotoluene are −5.3, −5.8, and −5.8 kcal/mol, respectively, using K_F (7247.34 ± 228.49 19195.1 ± 752.0, 18645.5 ± 737.6 [(mg/kg)/(mg/L)^N; N = 1 here] for the respective cases),⁶⁶ whereas they are, respectively, −4.2, −5.1, and −5.4 using K_L (0.0118 ± 0.00216, 0.0569 ± 0.0128, 0.0939 ± 0.0176 (L/mg) for the respective cases).⁶⁶ Our computed ΔG_B° values (Tables 1 and 2) show that they are not very different from the experiment (−2.6, −3.7, and −5.8 kcal/mol for the respective systems). The exact matching is not possible, as the computed values are in the gas phase. The overestimated CP correction through double-ζ basis set could also contribute to such deviation from experiment. The computed values, of course, maintain the same trend in ΔG_B° with respect to the experiment, and the results thus give us confidence that the computed binding energies could be used for Hammett correlation analysis.

3.2. Binding Energies and Hammett Correlation

There are several interesting features related to the binding energies of aromatic nitro compounds on GR, 3BGR, and 3NGR. The binding energies for o- and p-substituted nitroaromatics cases are more or less similar on all of these scaffolds. The ΔE_B values of the nitroaromatics are much higher than the interactions of benzene with GR, BGR, and NGR. These values are available in Table 1, and it should be noted that the positive ΔG_B° values predict the benzene···GR/BGR/NGR interactions to be relatively unstable. The general trend is that the ΔE_Bs (ΔH_B° and ΔG_B° values as well) increase with the increase of the number of substituents in the nitroaromatic ring. It should be noted that regardless of the ortho or meta orientation of the group X in the nitroaromatics, ΔE_B is always greater than that for the nitrobenzene···GR/3BGR/3NGR interactions. The computed ΔE_Bs for the various disubstituted nitroaromatics (Tables 1–4) further show that it increases with various m- and p-substituents depending on their electron redistribution effect on the aromatic ring, although the changes of distance (Table 5) between the interacting systems are not very significant. These distances were measured as the distance between the center of mass of the aromatic rings of the nitroaromatics and the GR/3BGR/3NGR scaffold, as shown in Scheme 1. With further increase of substituents in the nitroaromatics (Tables 1–4), there are further increment in the ΔE_B values. These trends were also observed previously by several authors in the case of benzene···substituted benzene dimers.^41,42 The molecular electrostatic potential (MEP) maps for three representative cases are shown in Figure 1. Figure S5 (Supporting Information) shows the MEP maps of several more of such interactions together with the corresponding nitroaromatic components. The pictures show that the individual MEPs of the nitroaromatics do not show any significant change with respect to the composite system. This particular feature indicates that these π–π interactions are mostly dominated by dispersion and the electrostatic contributions for stabilizing effect.

Table 5. Interacting Distances (d, Å)^a between GR/3BGR/3NGR and Various m- and p-Substituted and Several Di- and Tri-Substituted Nitrobenzene Derivatives for Their Optimized Geometries at the DFT/M06/cc-pVDZ Level of Computations.

	d (GR)			d (3BGR)			d (3NGR)
substituents	m-	p-	others	m-	p-	others	m-	p-	others
–Hb	3.34			3.38			3.24
–NH2	3.41	3.33		3.40	3.34		3.26	3.24
–OH	3.28	3.40		3.32	3.36		3.22	3.26
–OCH3	3.46	3.37		3.44	3.45		3.33	3.32
–OC2H5	3.44	3.50		3.46	3.46		3.34	3.33
–CH3	3.48	3.36		3.42	3.47		3.36	3.40
–NO2	3.30	3.30		3.47	3.47		3.26	3.30
–CN	3.41	3.37		3.41	3.35		3.33	3.32
–Cl	3.40	3.36		3.41	3.37		3.37	3.27
–Br	3.44	3.34		3.49	3.40		3.41	3.34
–I	3.42	3.44		3.48	3.52		3.40	3.45
3-NO2, 4-OH			3.33			3.41			3.34
3,5-di-NO2, 4-OH			3.27			3.28			3.26
3,5-di-NO2, 4-CH3			3.38			3.39			3.34

^aThe definition of the distance d is explained in Scheme 1.

^bNitrobenzene.

It is customary to plot ΔE_Bs against Hammett σ-parameters⁴⁴ to investigate the validity of Hammett correlation in cases of π–π stacking. Several authors did such analysis for the benzene-substituted benzene interactions, and a brief account of the validity of ∑|σ_m| parameters instead of ∑σ_m was given in Introduction in relation to the previous findings.⁴² In the present cases also, the plots of ΔE_Bs versus ∑σ_m and ∑σ_p parameters (Table S1) do not produce decent linear correlations, although the correlations between ΔE_B and ∑σ_m are somewhat better than those of the corresponding ∑σ_p cases (Figure S6, Supporting Information). These types of correlations could be partly successful if only electron-withdrawing monosubstituted cases are considered, as was done by Houk and Wheeler in their earlier studies on benzene···substituted benzene dimers.⁴¹ Because in the present investigations we are studying only the cases of aromatic nitro-derivatives, the consideration of such specific features is out of scope here.

The plots of ΔE_Bs versus ∑|σ_m| (Table S1) for the nitroaromatics produce reasonably good linear correlations for all of the interacting dimers of nitroaromatics with GR/3BGR/3NGR (r: correlation coefficient > 0.92 for all three scaffolds). ΔE_Bs at various levels, viz. DFT/M06 (cc-pVDZ, cc-pVTZ, and sp-aug-cc-pVTZ basis sets) and MP2/cc-pVDZ, show more or less similar correlations. Only the correlations through DFT/M06/cc-pVDZ calculations are shown here (Figure 2), and the others are documented in the Supporting Information (Figure S7). This policy is also adopted for the other analysis throughout the manuscript (considering the total amount of analyzed data). The plots of ΔE_Bs versus ∑|σ_p| (Table S1) for the nitroaromatics are not as impressive as the ∑|σ_m| cases, but they still show reasonable correlation in the present molecular domain (Figure 2, cc-pVTZ). The best r value in most of the cases is >0.9. The linear correlation is slightly weaker for these nitrobenzene interactions with 3NGR. The results of such correlation using other basis sets and MP2/cc-pVDZ are shown in Figure S8 (Supporting Information), and they also indicate that linear correlation exits in such cases.

Figure 2.

Correlations of ΔE_B with ∑|σ_m| (A) and ∑|σ_p| (B) for the interactions between GR, 3BGR, and 3NGR with various m- and p-substituted and several di- and tri-substituted nitrobenzene derivatives (Tables 1 and 2) at the DFT/M06/cc-pVDZ level. (A) Correlations of GR (blue line and black dots; ΔE_B = −4.22∑|σ_m| – 12.92, n = 14, r = 0.9), 3BGR (red line; ΔE_B = −3.85∑|σ_m| – 13.84, n = 14, r = 0.91), and 3NGR (green line; ΔE_B = −4.04∑|σ_m| – 13.69, n = 14, r = 0.87) (r: correlation coefficient) interacting with m-substituted and other higher substituted nitrobenzene derivatives. (B) Correlations of GR (blue line; ΔE_B = −4.26∑|σ_p| – 11.98, n = 14, r = 0.91), 3BGR (red line; ΔE_B = −3.86∑|σ_p| – 13.45, n = 14, r = 0.91), and 3NGR (green line; ΔE_B = −3.92∑|σ_p| – 12.90, n = 14, r = 0.80) interacting with p-substituted and other higher substituted nitrobenzene derivatives. Similar correlations through M06/cc-pVTZ, M06/sp-aug-cc-pVTZ, and MP2/cc-pVDZ calculations are shown in Figures S6 and S7 (Supporting Information).

The Hammett equation⁴⁴ usually predicts a linear correlation of ΔG_B° against the σ-parameters from the relation

3

where K_B should be taken as the binding constant for the interactions between the graphene (and doped graphene) and the substituted aromatic system (substituted nitrobenzenes for our specific cases) and K₀ is the same constant for any reference system. From eq 3, one can easily show by multiplying both sides with RT (R: universal gas constant, T: absolute temperature) that

4

This is a linear relation between ΔG_B° and σ. Because ΔG = ΔH – TΔS (H: enthalpy and S: entropy), one can generate similar relation between ΔH_B° and σ (eq 5).

5

We have neglected the difference (TΔS_B – TΔS₀), as it is too small. The plots of ∑|σ_m| parameters of nitrobenzene derivatives versus ΔG_B°/ΔH_B° generate linear correlations, which are shown in Figure S9 (Supporting Information) as predicted in eqs 4 and 5. Linear correlations were observed for all of the three scaffolds, viz. GR, 3BGR, and 3NGR. Similar correlations were also observed with respect to the ∑|σ_p| parameters (Figure S9, Supporting Information). The ΔG₀°/ΔH₀° terms in eqs 4 and 5 are absorbed in the correlation equations in the intercept parameters and the prefactor term of σ. Because ΔH_B and ΔE_B are more or less similar, the use of ΔE_B to monitor the validity of Hammett correlation is justified.

One can compare the strength of such Hammett correlations from the slopes of the linear correlations (called ρ values) as shown in the caption of Figure 2. If ρ > 1, it could be interpreted as a measure of strong Hammett correlation (following the original interpretation of ρ as reaction constant for the reactions of various m- and p-substituted benzene derivatives).⁶⁷ The slopes of Hammett correlations of ΔE_Bs versus both ∑|σ_m| (slopes for GR: 4.22, 3BGR: 3.85, 3NGR: 4.04) and ∑|σ_p| (slopes for GR: 4.26, 3BGR: 3.86, 3NGR: 3.92) are available in Figure 2. According to the concept of ρ, the Hammett correlation is quite strong in such cases and the strengths are more or less equivalent, although correlations in the case of GR are always slightly stronger. Such analysis could be extended for other correlations also (as presented in Figures S7–S9, Supporting Information). The comparison of such ρ values with arene···arene interactions is not straightforward, as a few results are available with σ_m correlation only. Wheeler and Houk⁴¹ presented the correlation equation between the interaction energies and σ_m values of benzene···substituted benzene complexes. The computed ρ value for this case is 2.72, which indicated a strong correlation comparable to our cases.

The logic behind the use of ∑|σ_m|/∑|σ_p| parameters in such correlations has been discussed earlier (Introduction section), and it was proposed that these parameters might take into account the contribution of electrostatic interactions because ∑σ_m/∑σ_p parameters together with ∑M_r (M_r: molecular refractivity) correlate jointly with ΔE_B. Because M_r implicitly contains molecular polarizability α through the relation M_r = 4/3(πNα) (N: Avogadro number), the contribution of electrostatic interactions was indirectly predicted through such correlations. In the present cases, we have found that ΔE_B could be satisfactorily correlated with the ∑|σ_m|/∑|σ_p| and ∑M_r values of various nitroaromatics⁶⁸ (Table S2) through multiple correlations, as shown below (eqs 6–11).

GR···nitroaromatics (M06/cc-pVDZ)

6

7

3BGR···nitroaromatics (M06/cc-pVDZ)

8

9

3NGR···nitroaromatics (M06/cc-pVDZ)

10

11

The F statistics in the above correlations show that correlations are significant and most satisfactory for the GR···nitroaromatics interactions. The deviations observed in the 3BGR and 3NGR could be attributed to the slight deviations of these scaffolds from ideal π-characters, but still the correlations are significant. The correlations using other basis sets (cc-pVTZ and sp-aug-cc-pVTZ) and MP2/cc-pVDZ are shown in Table S3 (Supporting Information). It should be noted that all of the M06 level correlations are more or less similar. The correlations at the MP2/cc-pVDZ level are somewhat weaker, but still show that the correlations at the M06 level are validated in such analysis. The somewhat weaker correlations at the MP2 level are not unexpected, as the geometry of the complexes was not optimized at this level.

The implicit contribution of α in M_r has further prompted us to investigate the direct effect of this parameter (as Δα with respect to benzene) in such correlations. The parameter α is directly available in quantum chemical calculations in isotropic form [α = 1/3(α_xx + α_yy + α_zz)], and is an experimentally observable quantity. Moreover, because Δα represents the effect of electronic redistribution due to substitution of various functional groups in benzene, correlation with Δα will directly indicate the role of electrostatic and dispersion contributions in the present interactions. The correlation presented below (eqs 12–17) shows that ΔE_B correlates satisfactorily with ∑|σ_m|/∑|σ_p| and Δα (Table S2).

GR···nitroaromatics (M06/cc-pVDZ)

12

13

3BGR···nitroaromatics (M06/cc-pVDZ)

14

15

3NGR···nitroaromatics (M06/cc-pVDZ)

16

17

Table S4 (Supporting Information) contains correlations using other basis sets (cc-pVTZ and sp-aug-cc-pVTZ) at M06 and MP2 (cc-pVDZ) levels. The correlations in most of the cases are impressive (r > 0.9), although in a few cases, the MP2 results are not as good as the M06 cases. The situation is similar to that of ∑|σ_m|/∑|σ_p|, ∑M_r multiple correlations with ΔE_B. Although the parameters, viz. r and F statistics, show impressive correlations, reproducibility of ΔE_B from the above correlations need to be further tested. Figures S10–S13 (Supporting Information) show the correlations of ΔE_B with the predicted ΔE_B from the multiple correlation equations, as discussed above. The linear correlations using various basis sets and techniques (M06 and MP2) are quite satisfactory in our present molecular domain. The MP2 calculations in some cases (mostly in p-substituted compounds) have shown slightly weaker correlations (r < 0.9, Figures S10–S13). These deviations were mostly observed during interactions of such compounds with 3BGR and 3NGR. Previously, lipophilicity parameter π (instead of M_r) was tried as ∑π in the correlation analysis of benzene···substituted benzene interactions.⁴² We did not take into account such correlations, as we are only studying the gas-phase situations.

3.3. Energy Decomposition Calculations

The success of ∑σ_m/∑σ_p, M_r/Δα parameters correlating ΔE_B’s in the multiple correlations eqs 6–17, shows that the contribution of the dispersion of electrostatic interactions is important in such correlations and also generates the logic behind the intrinsic contributions of such interaction terms in the correlations of ΔE_B’s with ∑|σ_m|/∑|σ_p|. Such correlations also generate further need to quantitatively analyze the contributions of various interactions terms in such complex formation. Tables S5 and S6 (Supporting Information) show the interaction energy components for m- and p-substituted nitrobenzenes, respectively (together with derivatives with multiple substituents) at the Hartree–Fock (HF) level, as discussed in eq 1, and the contribution of the dispersion interactions (ΔE_disp) ( E_MP2⁽²⁾ in eq 2). As it could be seen from Tables S5 and S6, the total interactions are repulsive at the HF level. The attractive electrostatic (E_el) and delocalization (ΔE_del^HF) are offset by the repulsive Heitler–London exchange term (E_ex). The main stabilization arises from the contribution of the ΔE_disp term, which is actually contribution from dispersion and higher-order correlations with the HF components.

Figure 3 represents a bar chart representation of the contributions of the attractive E_el⁽¹⁰⁾ and ΔE_disp terms in the interactions between GR/3BGR/3NGR and the various m-substituted nitrobenzenes as well as the derivatives with multiple substituents (as shown in Table S5). Although ΔE_disp shows substantial contribution in stabilization of the complexes in MP2 calculations, E_el also plays an important role here. This is evident from the magnitude of the repulsive E_ex^HL term in such calculations. If the ΔE_disp, E_ex, and ΔE_del^HF terms are added, the total stabilization coming out through such contributions becomes almost equal to the contribution from E_el. Figure 3 also contains the contribution of empirical ΔE_disp term in the DFT calculations. These contributions are much smaller than ΔE_disp in MP2 calculations (E_MP2⁽²⁾). This is quite obvious from the fact that the DFT calculations already include electron correlation effects to show some stabilization of the complexes. The empirical ΔE_disp here is only the unaccounted dispersion effect in such calculations. This effect (empirical ΔE_disp) actually makes the ΔE_B values in DFT calculations more or less similar with respect to the MP2 results (Tables 1–4), although the MP2 values are somewhat lower. Interaction energy components for the p-substituted nitrobenzenes are shown in Table S6 (Supporting Information). Similar to the cases for m-substituted compounds, the p-substituted nitrobenzene derivatives also show that the interactions with GR, 3BGR, and 3NGR are mostly due to E_el and ΔE_disp contributions.

Figure 3.

Bar chart graphs comparing the contributions of E_el⁽¹⁰⁾ and ΔE_disp in GR (A)/3BGR (B)/3NGR (C) interactions with various m-substituted and several di- and tri-substituted nitrobenzene derivatives (Table 1) through energy decomposition analysis. The red and blue bars, respectively, represent ΔE_disp (MP2) and E_el contributions. The empirical dispersion contributions at the M06/cc-pVDZ level (green bars) are also included for comparison. The magnitudes of these parameters with other decomposition energy components are available in Table S5 (Supporting Information). The nitrobenzene derivatives (a–n) are same in all of the three panels and are in the same order (from the top) as in Table 1. The results for p- and higher nitrobenzene derivatives are available in the Supporting Information (Table S6).

The symmetry-adapted perturbation theory (SAPT) analysis⁶⁹ of the interactions between benzene and substituted benzenes at the DFT level⁴² indicated that the sum of the energy components ΔE_disp + ΔE_ex + ΔE_ind (ΔE_ex: exchange component; ΔE_ind: induction component) is more or less constant for such interactions. The ΔE_ex in SAPT and the E_ex^HL component in our present study are more or less similar, but the ΔE_ind and ΔE_del components are not totally similar. Although ΔE_del^HF is associated with the relaxation of electron densities of monomers upon interaction restrained by the Pauli principle,^58,59 (charge delocalization together with charge-transfer interactions), ΔE_ind is associated with the interactions associated with interactions arising from the charges due to the deformation of the monomer units. Hence, ΔE_disp + E_ex + ΔE_del^HF is not constant and varies depending on the associated charge-transfer terms in ΔE_del. The E_el⁽¹⁰⁾ term increases the binding for both the strong electron-withdrawing and electron-donating substituents and makes almost equal contribution with respect to ΔE_disp + E_ex + ΔE_del^HF (Tables S5 and S6). Thus, ∑σ_m (or ∑σ_p) together with ∑M_r or Δα terms is needed to predict ΔE_B in the present cases. This observation also concurs with the results of previous SAPT analysis on the benzene···substituted benzene interactions.⁴²

It is to be remembered that these results hold for the gas-phase cases. Because our computed ΔG_B° have close resemblance with several experimental results for molecules adsorbed on graphene (as discussed earlier), the propositions for the correlations, as suggested, could also hold for condensed phase. No comments could be made regarding the solvent effect as the GR systems chosen do not have any solubility information. Thus, the proposition of Cockfroft and Hunter regarding⁷⁰ the absolute dominance of the electrostatic term (in benzene···substituted benzene interactions) cannot be tested here. But it is obvious that E_el⁽¹⁰⁾ is one of the dominating factors in stabilizing such interactions. The E_el term has some more features in such interactions, and it was also the primary observation of Hunter and Sanders⁴⁰ in relation to the benzene···substituted benzene interactions. Our analysis showed that the stabilizing effects in E_el⁽¹⁰⁾ are mostly coming from charge–charge interactions. But the dipole–dipole, dipole–quadruple, and quadruple–quadruple terms also have contributions. The contribution of the other contributing terms are not very significant. Because the nitrobenzene derivatives chosen for such interactions mostly have appreciable dipole moments (Table S2) and polarizabilities, these contributions of higher-order electrostatic terms in E_el are justified. The exceptions in the present cases are p-dinitrobenzene, p-cyanonitrobenzene, and a few other nitrobenzene derivatives with multiple substitutions, where the dipole moment is either zero or very low (Table S2). In such cases, the contribution of the E_el⁽¹⁰⁾ component is also low (Tables S5 and S6). The charge–charge and dipole–quadruple terms mostly contribute to the stabilizing E_el component.

4. Conclusions

We have investigated the applicability of Hammett parameters in extended π-systems, where geometry restrictions were not needed. Three different model graphene systems, viz. GR, 3BGR, and 3NGR, were designed as extended π-systems, and interactions of various nitroaromatics were monitored using M06/cc-pVDZ, M06/cc-pVTZ, M06/sp-aug-cc-pVTZ, and MP2/cc-pVDZ levels of calculation. The applicability of Hammett substituent constants (σ_m or σ_p) to predict binding energies for π–π interactions has been investigated previously on various benzene···m- and p-substituted benzene complexes.

There were several viewpoints regarding the validity of such an approach, but later it was shown that the sum of the absolute values of σ_m (∑|σ_m|) did reasonably good job in such prediction. The ∑σ_p parameters also did show similar linear correlations to ΔE_B, although such correlations were weaker with respect to ∑|σ_m| parameters. The ∑|σ_m| parameters were assumed to contain the effect of dispersion (or polarizability) as these parameters together with ∑M_r parameters showed good correlations with ΔE_B. The geometry of the complexes in such studies were restricted to PFF orientation to enforce π–π interactions for the applicability of the Hammett parameters. The optimized structures (M06/cc-pVDZ level), in our present investigation, showed these structures to be in OSFF orientation, but because of the extended π-systems of GR, 3BGR, and 3NGR, the nitroaromatics were always experiencing π–π interactions. The ΔE_B in all of the three different interactions showed good correlations with ∑|σ_m| and ∑|σ_p| parameters. Moreover, the convincing multiple regression analysis using ∑σ_m and ∑σ_p and M_r and Δα showed that, like benzene···substituted benzene cases, these interactions have contributions from dispersion and electrostatic components.

The energy decomposition analysis for the interactions showed that electrostatic interactions together with ΔE_disp are important factors in such interactions. The E_el⁽¹⁰⁾ components in such interactions have almost equal stabilizing contribution with respect to the ΔE_disp + E_ex + ΔE_del^HF interaction energy components, and this contribution varies depending on the ∑|σ_m| and ∑|σ_p| values. Furthermore, E_el always shows stabilizing effect regardless of the nature of substituents of the nitrobenzene derivatives. A further analysis of the E_el⁽¹⁰⁾ terms indicated the contributions of the dipole–dipole, dipole–quadruple, and quadruple–quadruple terms in shaping up the total electrostatic contributions in such interactions and thereby giving some physical background to the importance of ∑M_r and Δα multiple regression equations to predict ΔE_Bs. Most of these observations are more or less similar to the conclusions of the previous investigations on benzene···substituted benzene systems, which indicates the applicability of Hammett correlation constants to predict ΔE_B in OSFF orientations, if scaffolds are designed in such a way that substituted benzene systems cannot escape their π-cloud. The present observations are valid in a limited domain of molecules, and a much wider range of benzene derivatives are to be investigated to achieve a general validity of such hypothesis.

Supporting Information Available

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsomega.7b01912.

• Complete reference of Gaussian 09 code; Tables S1–S6; Figures S1–S14 (PDF)

The authors declare no competing financial interest.