σ-Hole and LP-Hole Interactions of Pnicogen···Pnicogen Homodimers under the External Electric Field Effect: A Quantum Mechanical Study.

σ-Hole and lone-pair (lp)-hole interactions within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations were comparatively investigated on the pnicogen···pnicogen homodimers (PCl3)2, for the first time, under field-free conditions and the influence of the external electric field (EEF). The electrostatic potential calculations emphasized the impressive versatility of the examined PCl3 monomers to participate in σ-hole and lp-hole pnicogen interactions. Crucially, the sizes of σ-hole and lp-hole were enlarged under the influence of the positively directed EEF and decreased in the case of reverse direction. Interestingly, the energetic quantities unveiled more favorability of the σ-hole···lp-hole configuration of the pnicogen···pnicogen homodimers, with significant negative interaction energies, than σ-hole···σ-hole and lp-hole···lp-hole configurations. Quantum theory of atoms in molecules and noncovalent interaction index analyses were adopted to elucidate the nature and origin of the considered interactions, ensuring their closed shell nature and the occurrence of attractive forces within the studied homodimers. Symmetry-adapted perturbation theory-based energy decomposition analysis alluded to the dispersion force as the main physical component beyond the occurrence of the examined interactions. The obtained findings would be considered as a fundamental underpinning for forthcoming studies pertinent to chemistry, materials science, and crystal engineering.

Introduction

Noncovalent interactions have recently been the subject of many studies due to the recognition of their key role in drug discovery,[1−3] crystal engineering,[4,5] and materials science.[6−8] Among noncovalent interactions, hole bonds have recently evoked an exceptional interest. In a series of studies, the covalently bonded atoms of group IV–VII were reported with recognizable ability to form regions of depletion in electron density,[9−12] dubbed as σ-,[13] π-,[14] lone-pair (lp)-,[15] and radical (R•)-holes.[16] The holes of group IV–VII element-containing molecules can interact with Lewis bases, forming tetrel,[17−19] pnicogen,[20−25] chalcogen,[26−31] and halogen[13,31−36] bonds, respectively. Among hole-bonding complexes, the interactions of pnicogen-containing molecules with Lewis bases have found sustained attention of a variety of theoretical[37−42] and experimental studies[43−45] by dint of their significant importance in chemical reactions[46−48] and biological systems.[49]

More recently, like···like noncovalent interactions have triggered a rich trove of interest because of their genuine roles in material and crystal design.[50−53] In that spirit, many studies have recently been established to elucidate the features of like···like noncovalent interactions, in which the intermolecular interaction occurred between two similar atoms within the like···like configuration. The interactions involving the covalently bonded halogen atoms represent the most well-known type of like···like interactions.[54−60] Parallel to halogen···halogen interactions, the interactions of chalcogens,[61,62] tetrels,[63−65] and triels[66] within like···like configurations were thoroughly addressed. However, a great deal of interest has been directed toward the investigation of pnicogen bonds; there is a paucity in the literature pertinent to the pnicogen···pnicogen interactions.

One of the most crucial factors that affect noncovalent interactions is the external electric field (EEF). As a point of departure, Bandrauk et al. elucidated the effect of the intense local electric fields (ranging from ∼108 to ∼1010 V/m) arising from the surrounding medium of the biological systems.[67,68] Subsequently, a flurry of studies has been devoted to illustrating the EEF effect on variant noncovalent interactions.[69−72] Upon careful literature review, the EEF has demonstrated exceptional influence on the strength of the noncovalently bonded complexes. Nevertheless, there is a paucity in the literature relevant to its effect on versatile σ-hole and lp-hole interactions, in particular, within pnicogen-bearing complexes.

Accordingly, the current study was devoted to thoroughly elucidate the features of the σ-hole and lp-hole interactions of pnicogen···pnicogen homodimers (PCl3)2 within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations, for the first time, under field-free conditions and the influence of an EEF (Figure ). The employed EEF strengths were set to be 0.002, 0.004, and 0.008 au, which suited in the range of the electric field within the biological systems (i.e., 0.0002–0.0194 au). Versatile quantum mechanical calculations, including geometrical optimization, molecular electrostatic potential (MEP) maps, and surface electrostatic potential extrema (Vs,max), were carried out for the investigated PCl3 molecules under field-free and directed EEF conditions. Besides, the point-of-charge (PoC) approach was executed as an indicative tool for the electrostatic potentiality of the studied systems to attractively interact with Lewis bases and acids within a small scale (using PoC with values of −0.50 and +0.50 au, respectively). For pnicogen···pnicogen homodimers, the energetic quantities were thoroughly assessed using MP2 and CCSD/CBS levels of theory. Toward an in-depth insight, the quantum theory of atoms in molecules (QTAIM) and the noncovalent interaction (NCI) index were adopted to clarify the nature of the selected interactions from a topological perspective. The given results are not only substantial for the central understanding of pnicogen···pnicogen homodimers as essential molecular linkers but also informative for near-future technological applications pertinent to EEF.

Figure 1

Representation of (i) EEF effect on pnicogen σ-hole and lone-pair (lp)-hole and (ii) pnicogen···pnicogen homodimers within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations. The positive and negative signs represent the directionality of the employed EEF.

Computational Methods

The versatility of the pnicogen-bearing monomers PCl3 to engage in σ-hole and lp-hole interactions of pnicogen···pnicogen homodimers within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations was comparatively scrutinized (Figure ). The studied monomers and homodimers were first optimized under field-free conditions and the influence of EEF by the second-order Møller–Plesset perturbation theory (MP2) method[73] with the aug-cc-pVTZ basis set.[74−76] In geometry optimization of homodimers, no symmetry restrictions were considered. The utilized EEF was directed along the z-axis in both positive and negative directions, with values ranging from 0.002, 0.004, to 0.008 au (Figure ). No vibrational frequency calculations were carried out for the optimized complexes, which gave rise to the possibility that the structures were not energetic minima. Upon the optimized monomers, the MEP maps were generated to visualize the electrophilic and nucleophilic sites on the surfaces of the chemical systems. Also, the Vs,max at the σ-hole and lp-hole over the surface of the optimized PCl3 monomers were assessed with the help of the Multiwfn 3.7 package[77] using a 0.002 au electron density contour based on earlier recommendations.[54,78]

With the help of the PoC approach, an electrostatic model for pnicogen-based interactions was investigated, for the first time, under field-free conditions and the influence of the negatively and positively directed EEF. In the context of the PoC approach, the negatively and positively charged points were utilized with a value of ±0.50 au to mimic the roles of Lewis bases and acids,[16,33,79,80] respectively. Besides, the P···PoC distance effect was thoroughly studied in the range of 2.5–6.0 Å with a step size of 0.1 Å. Molecular stabilization energy (Estabilization) was then computed according to the following equation[15,81]

For the optimized homodimers, interaction energies were calculated under field-free conditions and the influence of the negatively and positively directed EEF, as the difference in energy between the complex and the sum of the monomers at the MP2/aug-cc-pVTZ level of theory. The interaction energies were then benchmarked at the CCSD(T)/CBS level and calculated as follows[82]where

The basis set superposition error (BSSE) was eliminated from the computed MP2 and CCSD(T) energetic quantities by incorporating the counterpoise correction (CC) procedure.[83] To elucidate the topological features of the studied interactions, QTAIM[84] was invoked. Using QTAIM, bond critical points (BCPs) and bond paths (BPs) were generated. Furthermore, the electron density (ρb) and Laplacian (∇2ρb) were evaluated. The noncovalent interaction (NCI) index was also adopted to investigate the origin of the σ-hole and lp-hole interactions within the investigated pnicogen···pnicogen homodimers based on the electron density and its derivatives.[85] QTAIM and NCI index analyses were performed using Multiwfn 3.7 software,[77] and their plots were graphed with Visual Molecular Dynamics (VMD) software.[86] All the remaining quantum mechanical calculations were carried out using Gaussian09 software.[87]

Furthermore, symmetry-adapted perturbation theory-based energy decomposition analysis (SAPT-EDA) was executed to elucidate the physical nature of the studied interactions. In that spirit, the physical energetic components, involving electrostatic (Eelst), induction (Eind), dispersion (Edisp), and exchange (Eexch), were assessed for the investigated homodimers with the help of the PSI4 code[88] at the SAPT2 level of truncation[89] using the aug-cc-PVTZ basis set. The total SAPT2 energy (ESAPT2) could be given according to the following equations[90]where

Results and Discussion

MEP Calculations

MEP maps have been considered as a real descriptor for the charge distribution along the molecular surfaces of the noncovalent bond donors and acceptors.[91−93] Whereby MEP maps, low and high electron densities are identified by the colored maps, where blue and red sites are prone to nucleophilic and electrophilic attacks, respectively. In the current study, MEP maps were generated under field-free conditions and the effect of EEF using 0.002 au electron density contour. Further quantitative evidence was introduced by evaluating Vs,max at the σ-hole and lp-hole over the surface of the optimized PCl3 monomer (Figure ). The MEP maps along with Vs,max values under the influence of the EEF are illustrated in Figure . The correlation of the EEF strength and direction with the Vs,max values at the σ-hole and lp-hole of the optimized PCl3 monomer is displayed in Figure .

Figure 2

MEP maps for PCl3 molecules as σ-hole and lp-hole under field-free conditions and the influence of the negatively and positively directed EEF with values ranging from 0.002 to 0.008 au. The electrostatic potential varies from −0.01 (red) to +0.01 (blue) au. The surface electrostatic potential extrema (Vs,max) are computed (in kcal/mol) at the σ-hole and lp-hole of the phosphorous atom.

Figure 3

Correlation between the EEF strength and the surface electrostatic potential extrema (Vs,max). The EEF’s positive and negative charges were adopted to illustrate the positive and negative directions, respectively.

Looking at Figure , the occurrence of pnicogen σ-hole and lp-hole on the surface of the investigated PCl3 molecules was obviously noticed. The prominent size of the blue region was detected in the case of σ-hole, outlining the further favorability of phosphorous, as a pnicogen bond donor, to interact via σ-hole rather than lp-hole. By applying EEF, as illustrated in Figure , the sizes of σ-holes and lp-holes were increased and decreased by orienting the employed EEF in the positive and negative directions, respectively. In the same context, the data shown in Figure consistently revealed the direct and reverse correlation between the positive value of the Vs,max and the strength of the negatively and positively directed EEF, respectively.

PoC Calculations

In the PoC approach, negatively and positively charged points are utilized to mimic the role of Lewis bases and acids in noncovalent interactions. The nucleophilic and electrophilic natures of the chemical systems are accordingly addressed from an electrostatic perspective in terms of molecular stabilization energy.[33] The PoC approach has recently been notarized as a trustworthy method for studying the σ-hole,[94−96] lp-hole,[15] π-hole,[97,98] and R•-hole[99] interactions from an electrostatic point of view.[15,99,100] With the help of the PoC approach, the ability of the PCl3 molecule to interact with Lewis bases and acids was investigated by employing negative and positive PoCs, respectively. ±σ-Hole and ±lp-hole tests were executed for the optimized PCl3 molecule under the influence of 0.000, ±0.002, ±0.004, ± 0.006, and ±0.008 au EEF at σ-hole··· and lp-hole···PoC distance in the range of 2.5–6.0 Å with a step size of 0.1 Å using a PoC value of ±0.50 au. Molecular stabilization energy curves were generated and are displayed in Figure . Table gathers molecular stabilization energies of the σ-hole··· and lp-hole···PoC systems at a distance of 2.5 Å.

Figure 4

Molecular stabilization energy curves for the PCl3···PoC systems calculated at σ-hole··· and lp-hole···PoC distance in the range of 2.5–6.0 Å under the field-free condition and the influence of the negatively and positively directed EEF with values ranging from 0.002 to 0.008 au in the presence of ±0.50 au PoC.

Table 1

Molecular Stabilization Energy (Estabilization) Values for the PCl3···PoC Systems Calculated at a σ-Hole··· and lp-Hole···PoC Distance of 2.5 Å under the Field-Free Condition and the Influence of the Negatively and Positively Directed EEF with Values Ranging from 0.002 to 0.008 au in the Presence of ±0.50 au PoC

		molecular stabilization energy (Estabilization, kcal/mol)
PCl3···PoC	EEF (au)	–0.50	+0.50
σ-hole···PoC	–0.008	–5.49	–2.92
	–0.004	–8.46	–2.33
	–0.002	–9.71	–1.82
	0.000	–10.84	–1.19
	+0.002	–11.85	–0.46
	+0.004	–12.74	0.38
	+0.008	–14.15	2.38
lp-hole···PoC	–0.008	–8.69	–14.61
	–0.004	–9.41	–11.93
	–0.002	–9.73	–10.54
	0.000	–10.03	–9.12
	+0.002	–10.29	–7.67
	+0.004	–10.53	–6.19
	+0.008	–10.91	–3.11

For –σ-hole and –lp-hole tests, it was noticed from the data in Figure that the optimized PCl3···PoC systems exhibited the most significant negative molecular stabilization energies in the presence of the positively directed EEF, followed by the absence of EEF, and finally the negatively directed EEF. From Table , as an illustration, the molecular stabilization energies of the σ-hole···PoC electrostatic model were found with values of −10.84, −11.85, and −9.71 kcal/mol under the influence of 0.000, +0.002, and −0.002 au EEF, respectively.

With regard to the effect of the EEF strength, the molecular stabilization energy increased with increasing the magnitude of the positively directed EEF value and decreased by applying the EEF along the reverse direction. For instance, in the case of σ-hole interactions, the molecular stabilization energies of the PCl3···PoC systems exhibited values of −11.85, −12.74, and −14.15 kcal/mol under the influence of +0.002, +0.004, and +0.008 au EEF, respectively. Conspicuously, the highly appreciated electrostatic interactions of the PCl3···PoC systems were observed by the occurrence of the substantial negative molecular stabilization energies in the case of σ-hole···PoC more than that of lp-hole···PoC.

Turning to the results of +σ-hole, it can be seen from the data in Figure that the molecular stabilization energies were progressively faded, and then the molecular destabilization energies boosted by applying the EEF along the positive direction. In contrast, the negatively directed EEF enhanced the strength of the PCl3···PoC systems.

In the presence of the +0.50 au PoC, the lp-hole electrostatic interactions exhibited the most considerable molecular stabilization energies, particularly under the influence of the −0.008 au EEF with a value of −14.61 kcal/mol. Such significant energies outlined the prominent contributions of the three coplanar atoms in the strength of lp-hole-based interactions. In all instances, the strength of the positively directed EEF exhibited direct and reversed correlations with the molecular stabilization energies of the PCl3···PoC systems in the presence of negative and positive PoC, respectively. The reversed pattern was detected for the strength of the negatively directed EEF.

Energetic Study

σ-Hole and lp-hole interactions of the (PCl3)2 homodimers were comparatively studied within the σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations (see Figure ). Geometrical optimization was first performed at the MP2/aug-cc-pVTZ level of theory for the investigated homodimers under the field-free condition and the influence of the negatively and positively directed EEF. Upon the optimized homodimers, interaction energies were computed at the same level of theory and are correlated with the EEF strength and direction in Figure .

Figure 5

Interaction energy of the (PCl3)2 homodimers within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations computed (in kcal/mol) at the MP2/aug-cc-pVTZ level of theory under the field-free condition and the influence of the negatively and positively directed EEF with values ranging from 0.002 to 0.008 au.

As shown in Figure , all the considered homodimers demonstrated potent potentiality to participate in pnicogen σ-hole and lp-hole interactions within the σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations. The superior negative interaction energies were ascribed to the (PCl3)2 homodimers within the σ-hole···lp-hole configuration, followed by σ-hole···σ-hole and lp-hole···lp-hole configurations. From Table , it can be seen that the interaction energies were −3.72, −3.55, and −2.98 kcal/mol for the (PCl3)2 homodimers within σ-hole···lp-hole, σ-hole···σ-hole, and lp-hole···lp-hole configurations, respectively, under the influence of +0.002 au EEF.

Table 2

Interaction Energies Calculated (in kcal/mol) at MP2/aug-cc-pVTZ, MP2/CBS, and CCSD(T)/CBS Levels of Theory of the (PCl3)2 Optimized Homodimers under the Field-Free Condition and the Influence of the Negatively and Positively Directed EEF with Values Ranging from 0.002 to 0.008 au

			EMP2/aug-cc-pVTZ (kcal/mol)
configuration	EEF (au)	distancea (Å)	BSSE-uncorrected	BSSE-corrected	estimated BSSE	EMP2/CBS (kcal/mol)	ECCSD(T)/CBS (kcal/mol)
σ-hole···σ-hole	–0.008	3.08	–6.25	–4.47	0.0028	–5.47	–3.50
	–0.004	3.15	–5.41	–3.72	0.0027	–4.58	–2.84
	–0.002	3.17	–5.21	–3.55	0.0026	–4.38	–2.71
	0.000	3.16	–5.16	–3.49	0.0026	–4.34	–2.66
	+0.002	3.16	–5.22	–3.55	0.0027	–4.40	–2.71
	+0.004	3.15	–5.39	–3.71	0.0027	–4.57	–2.84
	+0.008	3.08	–6.25	–4.50	0.0028	–5.50	–3.52
σ-hole···lp-hole	–0.008	4.59	–3.97	–2.97	0.0016	–3.54	–2.51
	–0.004	4.41	–4.18	–3.08	0.0017	–3.69	–2.60
	–0.002	4.13	–4.58	–3.32	0.0020	–3.98	–2.78
	0.000	4.10	–4.79	–3.49	0.0021	–4.16	–2.92
	+0.002	4.08	–5.06	–3.72	0.0021	–4.43	–3.15
	+0.004	4.03	–5.40	–4.00	0.0022	–4.78	–3.41
	+0.008	4.00	–6.36	–4.89	0.0023	–5.74	–4.26
lp-hole···lp-hole	–0.008	5.23	–4.13	–3.22	0.0015	–3.86	–2.78
	–0.004	5.23	–3.93	–3.04	0.0014	–3.66	–2.58
	–0.002	5.23	–3.88	–2.99	0.0014	–3.61	–2.54
	0.000	5.23	–3.87	–2.98	0.0014	–3.59	–2.52
	+0.002	5.23	–3.88	–2.98	0.0014	–3.60	–2.52
	+0.004	5.23	–3.93	–3.04	0.0014	–3.65	–2.57
	+0.008	5.23	–4.13	–3.22	0.0014	–3.86	–2.78

Distances between the two interacted phosphorous atoms of the pnicogen homodimers within the modeled σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations.

For σ-hole···σ-hole and lp-hole···lp-hole configurations, the directionality effect of the employed EEF nearly vanished, which might be interpreted as a consequence of their symmetrical nature. The enhancement of the interaction energy of the (PCl3)2 homodimers was detected by increasing the strength of the employed EEF. For example, the interaction energies showed values of −3.55, −3.71, and −4.50 kcal/mol for the optimized (PCl3)2 homodimers within the σ-hole···σ-hole configuration under the influence of ±0.002, ±0.004, and ±0.008 au EEF, respectively.

On the other hand, for an antisymmetric σ-hole···lp-hole configuration, it was observed that the interaction energies of the inspected homodimers increased and decreased by applying EEF along the positive and negative directions, respectively. The interaction energies exhibited a direct and an inverse correlation with the strength of the adopted EEF along positive and negative directions, respectively. Also, the intermolecular distances within the studied homodimers were noticed to be inversely and directly correlated with the strength of the negatively and positively directed EEF. Numerically, the intermolecular distances of the optimized (PCl3)2 homodimers within the σ-hole···lp-hole configuration under the influence of +0.008, +0.004, and +0.002 au EEF were 4.00, 4.03, and 4.08 Å, respectively.

Moreover, the benchmarking of the interaction energies was carried out for all the examined homodimers at the MP2/CBS and CCSD/CBS levels of theory. The computed MP2/CBS and CCSD(T)/CBS interaction energies are compiled in Table , revealing a near similarity between the interaction energy values computed at both levels of theory. Besides, the effect of the directed EEF on the BSSE-CC was evaluated (Table ). According to the data listed in Table , the EEF demonstrated a negligible effect on the computed BSSE values. Toward further accuracy, the effect of consideration of BSSE-CC in geometry optimization was examined for the (PCl3)2 homodimer within the three studied configurations. The examined homodimers were optimized with and without BSSE-CC, and the corresponding interaction energies were computed. According to the results, the difference between the interaction energies of the BSSE-corrected and BSSE-uncorrected optimized geometries were −0.08, −0.10, and −0.05 kcal/mol for σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations, respectively. Consequently, the obtained results affirmed that the consideration of BSSE-CC in the geometry optimization had a negligible effect on the computed interaction energies.

SAPT-EDA Calculations

The symmetry-adapted perturbation theory-based energy decomposition analysis (SAPT-EDA) has been confirmed as an informative tool for determining the physical nature of noncovalent interactions.[101,102] In the context of SAPT-EDA, the total interaction energy is directly decomposed into its four physical meaningful components, including electrostatic (Eelst), exchange (Eexch), induction (Eind), and dispersion (Edisp) forces (Figure ). For optimized homodimers, SAPT-EDA was carried out at the SAPT2 level of truncation using the PSI4 code,[88] and the released components are collected in Table .

Figure 6

Bar chart for physical components of total SAPT2 energy including electrostatic (Eelst), induction (Eind), dispersion (Edisp), and exchange (Eexch) components for pnicogen···pnicogen (PCl3)2 homodimers under the field-free condition and the effect of negatively and positively directed EEF.

Table 3

Electrostatic (Eelst), Induction (Eind), Dispersion (Edisp), and Exchange (Eexch) Energies of the Optimized (PCl3)2 Homodimers Calculated in kcal/mol at the SAPT2 Level of Truncation under the Field-Free Condition and the Influence of the Negatively and Positively Directed EEF with Values Ranging from 0.002 to 0.008 au

configuration	EEF (au)	Eelst (kcal/mol)	Eind (kcal/mol)	Edisp (kcal/mol)	Eexch (kcal/mol)	ETotal SAPT2 (kcal/mol)
σ-hole···σ-hole	–0.008	–8.04	–4.53	–10.63	19.47	–3.73
	–0.004	–6.64	–3.67	–9.68	16.24	–3.75
	–0.002	–6.24	–3.44	–9.40	15.32	–3.76
	0.000	–6.28	–3.46	–9.42	15.41	–3.75
	+0.002	–6.32	–3.48	–9.45	15.50	–3.76
	+0.004	–6.54	–3.61	–9.62	16.03	–3.75
	+0.008	–8.11	–4.56	–10.61	19.55	–3.73
σ-hole···lp-hole	–0.008	–2.44	–0.44	–6.89	6.36	–3.41
	–0.004	–2.67	–0.55	–7.31	6.86	–3.67
	–0.002	–3.95	–1.00	–8.00	9.01	–3.93
	0.000	–4.27	–1.10	–8.25	9.67	–3.94
	+0.002	–4.63	–1.23	–8.51	10.43	–3.94
	+0.004	–5.25	–1.44	–9.01	11.77	–3.92
	+0.008	–5.98	–1.73	–9.58	13.37	–3.92
lp-hole···lp-hole	–0.008	–2.54	–0.36	–7.42	6.92	–3.40
	–0.004	–2.49	–0.35	–7.35	6.79	–3.40
	–0.002	–2.49	–0.35	–7.35	6.78	–3.40
	0.000	–2.48	–0.35	–7.33	6.75	–3.40
	+0.002	–2.53	–0.35	–7.39	6.88	–3.40
	+0.004	–2.49	–0.35	–7.34	6.78	–3.40
	+0.008	–2.54	–0.36	–7.41	6.91	–3.40

From the data registered in Table , it can be seen that the studied interactions within all the inspected (PCl3)2 homodimers were dominated by the dispersion energy (Edisp), which were earlier reported in the case of halogen, chalcogen, and tetrel homodimers.[62,65,100] Besides, obvious contributions for the electrostatic (Eelst) and induction (Eind) interactions were also recognized in the designed pnicogen σ-hole and lp-hole interactions. However, exchange energy (Eexch) demonstrated positive values for all homodimers, outlining unfavorable exchange contributions to the strength of the explored interactions (Figure ). As an illustration, in the case of σ-hole···σ-hole homodimer under the field-free condition, the Eelst, Eind, Edisp, and Eexch values were −6.28, −3.46, −9.42, and 15.41 kcal/mol, respectively (Table ).

Consistent with the energetic findings listed in Table , the utilization of the EEF generally increased the contributions of the Eelst, Eind, and Edisp components for all the studied homodimers. For instance, the Edisp values of the homodimers within the σ-hole···lp-hole interactions were −8.25, −8.51, −9.01, and −9.58 kcal/mol under the influence of 0.000, +0.002, +0.004, and +0.008 au EEF strength, respectively.

In line with the MP2 results, the most prominent negative total SAPT2 energy was noticed in the case of σ-hole···lp-hole configuration, followed by σ-hole···σ-hole and lp-hole···lp-hole configurations. For instance, the total SAPT2 energies were −3.94, −3.76, and −3.40 kcal/mol for the (PCl3)2 homodimers within σ-hole···lp-hole, σ-hole···σ-hole, and lp-hole···lp-hole configurations, respectively, under the influence of +0.002 au EEF.

QTAIM Analysis

QTAIM has been documented as an intuitive technique that probes the nature of noncovalent interactions.[103−106] Using QTAIM, the BPs and the (3,–1) BCPs were generated to provide qualitative clues for the occurrence of σ-hole and lp-hole interactions (Figure ). Furthermore, the topological features, including electron density (ρb) and Laplacian (∇2ρb), were calculated and are collected in Table .

Figure 7

QTAIM diagrams for the optimized (PCl3)2 homodimers under the field-free condition and the influence of the negatively and positively directed EEF with values ranging from 0.002 to 0.008 au. Red dots represent the location of BCPs and BPs.

Table 4

Electron Density (ρb, au) and Laplacian (∇2ρb, au) at BCPs of the Optimized (PCl3)2 Homodimers under the Field-Free Condition and the Influence of the Negatively and Positively Directed EEF with Values Ranging from 0.002 to 0.008 au

configuration	EEF	ρb (au)	∇2ρb (au)
σ-hole···σ-hole	–0.008	0.0222	0.0354
	–0.004	0.0194	0.0343
	–0.002	0.0186	0.0338
	0.000	0.0187	0.0339
	+0.002	0.0188	0.0339
	+0.004	0.0192	0.0342
	+0.008	0.0223	0.0354
σ-hole···lp-hole	–0.008	0.0053	0.0177
	–0.004	0.0048	0.0163
	–0.002	0.0080	0.0247
	0.000	0.0087	0.0262
	+0.002	0.0095	0.0280
	+0.004	0.0071	0.0210
	+0.008	0.0078	0.0232
lp-hole···lp-hole	–0.008	0.0046	0.0155
	–0.004	0.0046	0.0156
	–0.002	0.0045	0.0150
	0.000	0.0045	0.0150
	+0.002	0.0047	0.0160
	+0.004	0.0047	0.0160
	+0.008	0.0049	0.0164

As displayed in Figure , all the studied homodimers were observed with variant numbers of BCPs and BPs that differ based on the configuration of the interacting species. Conspicuously, one, three, four, and six BCPs and BPs were noticed for the (PCl3)2 homodimers within the σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations, confirming the prominent contributions of the three coplanar atoms to the strength of the lp-hole based interactions. From Table , the closed shell nature of the σ-hole and lp-hole interactions within the considered homodimers was revealed by the relatively low values of ρb and the positive values of ∇2ρb.

Similar to energetic results, the ρb and ∇2ρb values were decreased and increased by employing the negatively and positively directed EEF, respectively. For example, the ρb values of the optimized (PCl3)2 homodimers within the σ-hole···lp-hole configuration under −0.002, 0.000, and +0.002 au EEF were 0.0080, 0.0087, and 0.0095 au, respectively.

NCI Analysis

Noncovalent interaction (NCI) index[85,107] was herein invoked to precisely illustrate the nature and origin of the examined intermolecular interactions based on the electron density and its derivatives. Within the context of the NCI index, 3D NCI plots were generated with a reduced density gradient (RDG) value of 0.50 au and are displayed for all the studied homodimers in Figure .

Figure 8

Noncovalent interaction (NCI) plots of the optimized (PCl3)2 homodimers under the field-free condition and the influence of the negatively and positively directed EEF with values ranging from 0.002 to 0.008 au. The isosurfaces are graphed with a RDG value of 0.50 au and colored according to sign(λ2)ρ with a range from −0.035 (blue) to 0.020 (red) au.

As shown in Figure , green regions were observed for all the considered complexes, ensuring the occurrence of attractive forces between the two interacting species. Obviously, the size of the green regions exhibited the same pattern as the energetic quantities (Table ). For example, the green-coded spheres of the interactions within the σ-hole···σ-hole and lp-hole···lp-hole configurations enlarged by increasing the magnitude of EEF in both positive and negative directions in the order 0.002 < 0.004 < 0.008 au. Also, the largest green-coded spheres were noticed in the case of (PCl3)2 homodimers within the σ-hole···lp-hole configuration, followed by σ-hole···σ-hole and lp-hole···lp-hole configurations that were in great consistency with the energetic affirmations and QTAIM observations. Overall, the findings of the QTAIM and NCI analyses remarkably confirmed the availability of (PCl3)2 homodimers to bond by σ-hole and lp-hole interactions within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations.

Conclusions

The potentiality of the σ-hole and lone-pair (lp)-hole pnicogen-containing molecules to form pnicogen···pnicogen homodimers (PCl3)2 within σ-hole···σ-hole, σ-hole···lp-hole, and lp-hole···lp-hole configurations was herein explored, for the first time, under field-free condition and the influence of the EEF. The following conclusions can be detected from MEP, Vs,max, PoC, interaction energy, SAPT-EDA, QTAIM, and NCI-based results: (i) the studied PCl3 molecule has the ability to form σ-hole and lp-hole with higher tendency for the anterior one, (ii) the positively directed EEF enlarged the sizes of the pnicogen σ-hole and lp-hole, while the negatively directed EEF exhibited reverse amplitude, (iii) the investigated molecule more preferentially forms the pnicogen···pnicogen homodimer (PCl3)2 within the σ-hole···lp-hole configuration compared to the other modeled configurations, (iv) the strength of the studied homodimers within the σ-hole···lp-hole configuration boosted by directed EEF along the positive direction and plunged along the negative one, (v) the symmetrical nature of σ-hole···σ-hole and lp-hole···lp-hole configurations conclusively decline the directionality effect of the applied EEF, and finally (vi) the dispersion energy was announced as the most prevalent forces dominated the σ-hole and lp-hole interactions within the studied pnicogen···pnicogen homodimers. These results would be informative for future research in the brain area of chemistry and materials science..