External Electric Field Effect on the Strength of σ-Hole Interactions: A Theoretical Perspective in Like⋯Like Carbon-Containing Complexes

For the first time, σ-hole interactions within like⋯like carbon-containing complexes were investigated, in both the absence and presence of the external electric field (EEF). The effects of the directionality and strength of the utilized EEF were thoroughly unveiled in the (F-C-F3)2, (F-C-H3)2, and (H-C-F3)2 complexes. In the absence of the EEF, favorable interaction energies, with negative values, are denoted for the (F-C-F3)2 and (H-C-F3)2 complexes, whereas the (F-C-H3)2 complex exhibits unfavorable interactions. Remarkably, the strength of the applied EEF exhibits a prominent role in turning the repulsive forces within the latter complex into attractive ones. The symmetrical nature of the considered like⋯like carbon-containing complexes eradicated the effect of directionality of the EEF. The quantum theory of atoms in molecules (QTAIM), and the noncovalent interaction (NCI) index, ensured the occurrence of the attractive forces, and also outlined the substantial contributions of the three coplanar atoms to the total strength of the studied complexes. Symmetry-adapted perturbation theory (SAPT) results show the dispersion-driven nature of the interactions.

The efficacious role of the directionality and strength of the external electric field (EEF) on the intermolecular noncovalent interactions were precisely illustrated [42][43][44][45][46][47]. Initially, the EEF effect on Group VII σ-hole interactions was explained through tunning the Cl⋯N traditional halogen bond to a shared chlorine bond, or an ion-pair bond [48]. Subsequently, the elucidation of the EEF effect on the σ-hole interactions was expanded to involve group IV-VIII elements-Lewis base interactions [45,46,49]. Nevertheless, the impact of EEF on the σ-hole interactions within the like⋯like complexes is still ambiguous.
Accordingly, for the first time, σ-hole interactions within like⋯like carbon-containing complexes were delicately studied and comparatively explored, in both the absence and the presence of the positively-and negatively-directed EEF ( Figure 1). Three carboncontaining complexes, namely, (F-C-F3)2, (F-C-H3)2, and (H-C-F3)2, were precisely investigated. A plethora of quantum mechanical calculations, including molecular electrostatic potential (MEP) and surface electrostatic potential extrema (Vs,max), were performed for the considered monomers. Further, for the inspected like⋯like complexes, interaction energy, quantum theory of atoms in molecules (QTAIM), and noncovalent interaction (NCI) index analyses were executed. To pinpoint the physical nature of the investigated interactions, the symmetry-adapted perturbation theory (SAPT) analysis was adopted. The findings of this study provide versatile and noteworthy contributions to enhance the understanding of the effect of the EEF on the σ-hole interactions within like⋯like complexes.  Representation of (i) the external electric field (EEF) effect on carbon σ-hole and (ii) the like· · · like carbon-containing complexes. Positive and negative signs represent the directionality of the employed EEF.

MEP and V s,max Calculations
Recent versatile studies demonstrate the molecular electrostatic potential (MEP) as a reliable technique to provide a powerful clue for the charge distribution over the molecular surface [50][51][52]. MEP maps were generated for the optimized carbon-containing molecules in the absence and the presence of the positively-and negatively-directed EEF, with values ranging from 0.004 to 0.020 au ( Figures S1 and S2). Moreover, the surface electrostatic potential extrema (V s,max ) was assessed in order to present quantitative evidence for the molecular electrostatic potential. MEP maps, along with V s,max values, are depicted in Figure 2 for the optimized molecules in the absence of EEF (i.e., EEF = 0.000), and in the presence of +0.040 and −0.040 au EEF. The intercorrelation of the V s,max values with the direction and magnitude of the applied EEF is represented in Figure 3.

MEP and Vs,max Calculations
Recent versatile studies demonstrate the molecular electrostatic potential (MEP) as a reliable technique to provide a powerful clue for the charge distribution over the molecular surface [50][51][52]. MEP maps were generated for the optimized carboncontaining molecules in the absence and the presence of the positively-and negativelydirected EEF, with values ranging from 0.004 to 0.020 au ( Figures S1 and S2). Moreover, the surface electrostatic potential extrema (Vs,max) was assessed in order to present quantitative evidence for the molecular electrostatic potential. MEP maps, along with Vs,max values, are depicted in Figure 2 for the optimized molecules in the absence of EEF (i.e., EEF = 0.000), and in the presence of +0.040 and −0.040 au EEF. The intercorrelation of the Vs,max values with the direction and magnitude of the applied EEF is represented in Figure 3.  for F-C-F 3 , F-C-H 3 , and H-C-F 3 optimized molecules in the absence of EEF (i.e., EEF = 0.000 au), and the presence of the positively-and negatively-directed EEF (i.e., +0.004 and −0.004 au, respectively). The electrostatic potential varies from −0.01 (red) to +0.01 (blue) au. The surface electrostatic potential extrema (V s,max ) at the investigated σ-holes are given in kcal/mol. From the MEP maps depicted in Figures S1 and S2, positive, blue-coded electrostatic potential regions (i.e., σ-hole) are detected along the outer surface of the carbon atoms, in varying sizes. As seen in Figure 2, the most prominent σ-hole size is found in the case of F-C-F 3, followed by F-C-H 3 and H-C-F 3 , outlining the direct correlation between the σ-hole size and the electronegativity of the covalently bonded atoms to the central carbon atom. Numerically, in the absence of EEF, the V s,max values increase as follows: H-C-F 3 < F-C-H 3 < F-C-F 3, with V s,max values of 15.4, 26.0, and 31.6 kcal/mol, respectively.
Turning to the effect of EEF directionality, as illustrated in Figure 2, the sizes of the σ-holes increase and decrease by orienting the employed EEF in the positive and negative directions, respectively. In the same context, Figure 3 consistently reveals the direct and reverse correlation between the positive value of the surface electrostatic potentials extrema (V s,max ), and the strength of the positively-and negatively-directed EEF, respectively.
Using the directional EEF, the σ-hole size exhibits a superior behavior (i.e., becomes more positive) by applying the EEF along the positive direction, whereas an inversed pattern is observed under the effect of the negatively-directed EEF. Illustratively, for the H-C-F 3 molecule, the V s,max value in the absence of EEF increases from 15.4 kcal/mol to 18.6 kcal/mol, and decreases from 15.4 kcal/mol to 12.2 kcal/mol, with the implementation of +0.004 au and −0.004 au EEF, respectively. Turning to the effect of EEF strength, direct and inverse correlations are disclosed between the σ-hole size and the magnitude of the From the MEP maps depicted in Figures S1 and S2, positive, blue-coded electrostati potential regions (i.e., σ-hole) are detected along the outer surface of the carbon atoms, in varying sizes. As seen in Figure 2, the most prominent σ-hole size is found in the case o F-C-F3, followed by F-C-H3 and H-C-F3, outlining the direct correlation between the σ-hol size and the electronegativity of the covalently bonded atoms to the central carbon atom Numerically, in the absence of EEF, the Vs,max values increase as follows: H-C-F3 < F-C-H < F-C-F3, with Vs,max values of 15.4, 26.0, and 31.6 kcal/mol, respectively.
Turning to the effect of EEF directionality, as illustrated in Figure 2, the sizes of th σ-holes increase and decrease by orienting the employed EEF in the positive and negativ directions, respectively. In the same context, Figure 3 consistently reveals the direct and reverse correlation between the positive value of the surface electrostatic potential extrema (Vs,max), and the strength of the positively-and negatively-directed EEF respectively.
Using the directional EEF, the σ-hole size exhibits a superior behavior (i.e., become more positive) by applying the EEF along the positive direction, whereas an inversed pattern is observed under the effect of the negatively-directed EEF. Illustratively, for th H-C-F3 molecule, the Vs,max value in the absence of EEF increases from 15.4 kcal/mol to 18.6 kcal/mol, and decreases from 15.4 kcal/mol to 12.2 kcal/mol, with the implementation o +0.004 au and −0.004 au EEF, respectively. Turning to the effect of EEF strength, direct and inverse correlations are disclosed between the σ-hole size and the magnitude of th positively-and negatively-directed EEF, respectively. For example, the Vs,max value of th H-C-F3 molecule increases to 18.6, 22.0, 25.2, 28.3, and 31.7 kcal/mol when the positively directed EEF increases to 0.004, 0.008, 0.012, 0.016, and 0.020 au, respectively.

Interaction Energy
The σ-hole interactions of the like⋯like carbon-containing complexes wer thoroughly studied in the absence and the presence of the EEF (see Figure 1). Initially, th

Interaction Energy
The σ-hole interactions of the like· · · like carbon-containing complexes were thoroughly studied in the absence and the presence of the EEF (see Figure 1). Initially, the (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes were optimized at the MP2/aug-cc-pVTZ level of theory. For the optimized complexes, the interaction energies (E 1 ) were evaluated at the same level of theory, and then benchmarked at the CCSD(T)/CBS level of theory (E 2 ) ( Table 1). The correlation between the interaction energies with the EEF direction and strength is graphically represented in Figure 4. Table 1. Interaction energies calculated (in kcal/mol) at MP2/aug-cc-pVTZ (E 1 ) and CCSD(T)/CBS (E 2 ) levels of theory for the optimized (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes in the absence and the presence of the positively-and negatively-directed external electric field (EEF), with values ranging from 0.004 to 0.020 au.  Molecules 2022, 27, x FOR PEER REVIEW 5 of ( Table 1). The correlation between the interaction energies with the EEF direction an strength is graphically represented in Figure 4. EEF a (au)  With regard to the effect of the EEF-based results, it is worth noting that th implementation of the positively-and negatively-directed EEF exhibit the same patter on the considered like⋯like complexes (Figure 4). This similar amplitude could b ascribed to the domination of the symmetrical nature of the complexes und consideration, which led in turn to the elimination of the directionality effect of th applied EEF. With regard to the effect of the EEF-based results, it is worth noting that the implementation of the positively-and negatively-directed EEF exhibit the same pattern on the considered like· · · like complexes ( Figure 4). This similar amplitude could be ascribed to the domination of the symmetrical nature of the complexes under consideration, which led in turn to the elimination of the directionality effect of the applied EEF.
The foregoing observations also highlight the role of EEF in turning the repulsive interactions (i.e., positive interaction energy) into attractive ones (i.e., negative interaction energy) within the studied complexes. Numerically, the MP2 interaction energies of the (F-C-H 3 ) 2 complex become more favorable, and turn from 0.02 kcal/mol (in the absence of EEF) to 0.00, −0.04, −0.12, −0.23, and −0.37 kcal/mol in the presence of ±0.004, ±0.008, ±0.012, ±0.016, and ±0.020 au EEF, respectively. Moreover, an inverse correlation is discerned between the C-C intermolecular distance and the strength of the applied EEF, alluding to the interaction energy enhancement.
Regarding energetic quantities, the MP2/aug-cc-PVTZ interaction energy is benchmarked at the CCSD(T)/CBS level of theory (Table 1), and both quantities are comparable, outlining a nearly similar trend.

QTAIM Analysis
The quantum theory of atoms in molecules (QTAIM) was presented as an informative tool to precisely characterize both the origin and nature of the noncovalent interactions, based on the electron density features [53,54]. In the current study, QTAIM analysis was implemented to indicate the origin of the studied σ-hole interactions through generating the bond critical points (BCPs) and bond paths (BPs). Figure 5 displays the QTAIM diagrams for the considered complexes in the absence and the presence of the positively-and negativelydirected EEE. The topological parameters, including the electron density (ρ b ), Laplacian (∇ 2 ρ b ), and total energy density (H b ) were estimated and are collected in Table S1. As illustrated in Figure 5, all the considered like⋯like complexes demonstrate six BCPs and BPs between the three coplanar substituents in each interacting monomer. These observations affirm the prominent contributions of the attractive forces between coplanar substituents over carbon analogs, which is in line with our earlier affirmations [31,55,56].
From Table S1, positive values are denoted for the electron density (ρb), Laplacian (∇ 2 ρb), and total energy density (Hb), ensuring the closed-shell nature of the investigated σ-hole interactions. Interestingly, the computed topological parameters highlight the EEF effect on the studied interactions, with growing ρb, ∇ 2 ρb, and Hb values with an increasing EEF magnitude, which is in line with the energetic findings (Table S1)  As illustrated in Figure 5, all the considered like· · · like complexes demonstrate six BCPs and BPs between the three coplanar substituents in each interacting monomer. These observations affirm the prominent contributions of the attractive forces between coplanar substituents over carbon analogs, which is in line with our earlier affirmations [31,55,56].

NCI Analysis
The noncovalent interaction (NCI) index [45,57] was considered as a dependable index to characterize the noncovalent interactions, based on the electron density and its derivatives. The 3D NCI plots were generated for the (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes using (λ 2 )ρ ranging from −0.035 (blue) to 0.020 (red), where the second eigenvalue of the hessian matrix and the electron density are represented by λ 2 and ρ, respectively. Figure S3 shows the 3D NCI plots for the optimized complexes in the absence and the presence of the positively-and negatively-directed EEF.
Based on the data displayed in Figure S3, the potentiality of the inspected carboncontaining molecules to participate in σ-hole interactions is assured and detected by the existence of green-colored isosurfaces between the two interacting molecules. Furthermore, the occurrence of circular-shaped green isosurface between the three coplanar substituents demonstrates their contributions, which is in line with the QTAIM affirmations. The superior effect of the EEF on the strength of the investigated complexes is evidently unveiled via increasing the green isosurfaces, by increasing the strength of the applied EEF.

SAPT Analysis
Symmetry-adapted perturbation theory (SAPT) analysis was settled on as an authoritative tool to elucidate the physical forces beyond the occurrence of the noncovalent interactions [58]. SAPT was carried out for the (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes at the SAPT2+ level of truncation ( Figure 6). The total SAPT2+ energies, accompanied by the fundamental components of all the studied complexes, are compiled in Table S2.
For all the inspected complexes, the E disp is found to be the most dominant force within the inspected interactions ( Figure 6). In comparison, the contributions of the E elst and E ind, along with E exch, are generally limited. Illustratively, the E elst , E ind , E disp, and E exch are −0.15, −0.04, −1.89, and 1.17 kcal/mol, respectively, in the case of the (F-C-F 3 ) 2 optimized complex in the absence of EEF (Table S2).
As listed in Table S2, the E disp component exhibit the prevalent contributions of the studied complexes to the total energies in the absence and the presence of EEF. Notably, the contributions of E disp are found to be enhanced in line with the interaction energy pattern as follows: (F-C-H 3 ) 2 < (H-C-F 3 ) 2 < (F-C-F 3 ) 2. For example, for (F-C-F 3 ) 2 , (H-C-F 3 ) 2 , and (F-C-H 3 ) 2 , the E disp , calculated in the presence of ±0.020 au EEF, has values of −2.06, −1.84, and −1.39 kcal/mol, along with interaction energies of −1.09, −0.62, and −0.37 kcal/mol, respectively. Evidently, the favorable contributions of the E elst , E ind , and E disp increase upon utilizing the positively-and negatively-directed EEF, which is in coincidence with the interaction energy findings (Table 1). For instance, the E disp of the (F-C-F 3 ) 2 complex is −1.90, −1.94, −1.96, −2.02, and −2.06 kcal/mol upon applying an EEF of ±0.004, ±0.008, ±0.012, ±0.016, and ±0.020 au, respectively. The accuracy of the considered level for SAPT analysis is appreciated through assessing the energy difference between the MP2 energy and the computed SAPT2+ energy (∆∆E) ( Table S2). The resulting tiny energy difference (∆∆E) outlines the accuracy of the utilized SAPT level of truncation.
Symmetry-adapted perturbation theory (SAPT) analysis was settled on as authoritative tool to elucidate the physical forces beyond the occurrence of t noncovalent interactions [58]. SAPT was carried out for the (F-C-F3)2, (F-C-H3)2, and (H-F3)2 complexes at the SAPT2+ level of truncation ( Figure 6). The total SAPT2+ energi accompanied by the fundamental components of all the studied complexes, are compil in Table S2. For all the inspected complexes, the Edisp is found to be the most dominant for within the inspected interactions ( Figure 6). In comparison, the contributions of the E and Eind, along with Eexch, are generally limited. Illustratively, the Eelst, Eind, Edisp, and E are −0.15, −0.04, −1.89, and 1.17 kcal/mol, respectively, in the case of the (F-C-F3)2 optimiz complex in the absence of EEF (Table S2).
As listed in Table S2, the Edisp component exhibit the prevalent contributions of t studied complexes to the total energies in the absence and the presence of EEF. Notab

Computational Methods
The inclination of carbon-containing molecules to engage in σ-hole interactions within like· · · like complexes is elucidated in the absence and the presence of the positively-and negatively-directed EEF (Figure 1). In the current study, (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 were chosen as the carbon-containing complexes. The EEF strength was employed with values ranging from 0.004 to 0.020 au, with an interval of 0.004 au. Geometrical optimization was executed at the MP2/aug-cc-pVTZ level of theory [59][60][61]. Molecular electrostatic potential (MEP) analysis was accomplished through generating MEP maps, and then assessing the surface electrostatic potential extrema (V s,max ), using 0.002 au electron density contour. The value of electron density contour was selected to provide a precise characterization for the electrostatic potential on the molecular surfaces, as previously reported [62,63]. The extraction of the V s,max values was also carried out using Multiwfn 3.7 software [64].
Within the optimized like· · · like complexes, the interaction energies were evaluated in the absence and the presence of the positively-and negatively-directed EEF, with values of 0.004, 0.008, 0.012, 0.016, and 0.020 au. The interaction energies were estimated as the difference in energy between the complex and the sum of the monomers. The benchmarking of the interaction energies was executed at the CCSD(T)/CBS level of theory [65,66], according to the idea that correlation energy is roughly proportional to X −3 for basis sets of the aug-cc-pVXZ type [67], using the following equations: where: By using the counterpoise (CP) correction procedure [68], the basis set superposition error (BSSE) was eradicated from the calculated interaction energies. The origin of the considered interactions was illustrated by generating the bond critical points (BCPs) and bond paths (PBs) with the utilization of the quantum theory of atoms in molecules (QTAIM) [61]. In the context of QTAIM, a variety of topological parameters, including electron density (ρ b ), Laplacian (∇ 2 ρ b ), and total energy density (H b ), were computed. NCI index analysis was also invoked, and the corresponding NCI plots were portrayed. The NCI isosurfaces were distinguished by the coloring scale of electron density (ρ) that distinguished the attractive forces (i.e., −0.035 au (blue)) from the repulsive ones (i.e., 0.020 au (red)) [69]. The QTAIM and NCI calculations were carried out via Multiwfn 3.7 software [64], and visualized using the Visual Molecular Dynamics (VMD) package [70]. All quantum mechanical calculations were performed using Gaussian 09 software [71].

Conclusions
The predilection of carbon-containing molecules to engage in σ-hole interactions within the (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes was inspected in the absence and the presence of the external electric field (EEF). In the absence of EEF, the MP2 energetic quantities addressed the occurrence of negative interaction energies for the (F-C-F 3 ) 2 and (H-C-F 3 ) 2 complexes with a higher favorability for the former. Upon the implementation of EEF along the positive and negative directions, the interaction energy escalates with the same magnitude, indicating the neglected effect of the EEF directionality on the strength of the like· · · like symmetrical complexes. In addition, the investigated like· · · like complexes demonstrate a supreme penchant to engage in favorable interactions when the applied EEF strength increases. The QTAIM results assert the closed-shell nature of the studied interactions. The SAPT calculations reveal the domination of the dispersion forces within all the studied complexes. These findings proclaim the prominent role of the EEF in enhancing the strength of the noncovalent interactions within like· · · like complexes, providing a fundamental linchpin for future studies related to crystal engineering and materials science.

Supplementary Materials:
The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/molecules27092963/s1, Figure S1. Molecular electrostatic potential (MEP) maps plotted at an 0.002 au electron density contour for F-C-F 3 , F-C-H 3 , and H-C-F 3 optimized molecules in the presence of the positive EEF, ranged from +0.008 to +0.020; Figure S2. Molecular electrostatic potential (MEP) maps plotted at an 0.002 au electron density contour for F-C-F 3 , F-C-H 3 , and H-C-F 3 optimized molecules in the presence of the negative EEF, ranged from −0.008 to −0.020; Figure S3. NCI plots of the optimized (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes in the absence and the presence of the positively-and negatively-directed external electric field (EEF), with values ranging from 0.004 to 0.020 au; Table S1. Electron density (ρ b , au), Laplacian (∇ 2 ρ b , au), and total energy density (H b , au) at bond critical points (BCPs) of the optimized (F-C-F 3 ) 2 , (F-C-H 3 ) 2 , and (H-C-F 3 ) 2 complexes in the absence and the presence of the positively-and negatively-directed external electric field (EEF), with values ranging from 0.004 to 0.020 au; Table S2. Electrostatic (E elst ), induction (E ind ), dispersion (E disp ), exchange (E exch ), and the estimated total SAPT energy (E Total SAPT2+ ) of the optimized (F-C-