Revisiting the covalent nature of halogen bonding: a polarized three-center four-electron bond

Abstract

As an important intermolecular interaction, halogen bonding has been studied extensively, but its nature still suffers from controversy without one uniform essence. Electrostatics, charge transfer, polarization and dispersion are emphasized, but the covalent nature is usually overlooked except for the strong halogen bonding species I₃⁻, which is widely accepted as a result of a three-center four-electron (3c-4e) interaction. In our study, the potential energy surface of I₃⁻ has been evaluated to explore the dissociation from I₃⁻ to I₂⋯I⁻. We found that different from an equivalent 3c-4e bond in I₃⁻, I₂⋯I⁻ can be rationalized by a polarized one. In addition, when the orbitals are polarized, it is exactly what traditional charge transfer or the popular σ-hole picture describes. I₃⁻ can be described by the Lewis theory model with the middle I⁺ cation serving as the Lewis acid and two terminal I⁻ anions acting as Lewis base. Therefore, we further extended this model to a series of I-containing species with chemical composition of L–I⁺–L, F⁻–I⁺–L and H₃P–I⁺–L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) to explore the nature of halogen bonding. When the forces of two bases around I⁺ are the same, it corresponds to an equivalent 3c-4e bond, such as I₃⁻. Otherwise, it is a polarized multicenter bond, such as I₂⋯I⁻. This work gives a new insight into the nature of halogen bonding compounds: besides the well-known I₃⁻, the nature of the other species is also a multicenter bond, existing as equivalent and polarized 3c-4e bonds, respectively.

---

1. Introduction

Halogen atoms are usually considered as species of high electron density owing to their high electronegativity and interaction with electron-deficient groups. In addition, halogen-containing compounds can also interact with electron-rich species, which is known as halogen bonding.^1–6 The ability of halogen atoms to attractively interact with electron donors was recognized as early as 1814, when the adduct reaction between iodine and ammonia was observed.⁷ Decades later in 1863, the complex was first purified and identified as a NH₃⋯I₂ adduct by Guthrie.⁸ Similar adducts involving bromine and chlorine were obtainedshortly after.⁹ Until now, halogen bonding has been under active investigation.^10–21 It plays an essential role in controlling the construction of supramolecular complexes in solid,^2,3,22,23 liquid,^24–27 and gas phases.²⁸ Besides, halogen bonding has led to a range of applications in the biological field,^3,29 such as drug design³⁰ and protein–ligand complexation.³¹

Halogen bonding has a general form of Y–X⋯D^5,32 (Scheme 1), where X is a halogen atom, and it functions as the electrophilic species (halogen bond donor), and D is a donor of electron density (halogen bond acceptor). The distance of X⋯D is shorter than the sum of their van der Waals radii, and the YX bond length is slightly elongated than the YX monomer, and the Y–X⋯D angle is approximately linear.^2–4

Scheme 1 General scheme for the formation of a halogen bond.

Though studied extensively, the exact nature of halogen bonding is still disputed. Classically, the chemical nature of the interaction was explained as charge transfer by Mulliken in 1950.³³ More precisely, in a charge transfer picture, halogen bonding is described as a donation from the donor lone pair into the σ* antibonding orbital.^34–40 Some computational studies have found a good correlation between the degree of charge transfer and the halogen bonding strength.^40,41 Besides, a more widespread model of this kind of interaction is based on the electrostatic effect. The electron density in a halogen atom is anisotropically distributed when the atom is covalently bonded to other atoms, resulting in regions with positive potential at the terminus of the YX bond (δ⁺ in Scheme 1), which is introduced as a “σ-hole” in 2007 based on computational studies.^42,43 The σ-hole perspective states that halogen bonding is an electrostatic attraction between the σ-hole and the negative potential of a Lewis base.^2–4 This concept is frequently used, but pure electrostatic interactions have been questioned.^44,45 Huber et al. have found that in case of CX₃I (X = F, Cl, Br and I), the σ-hole trend is exactly opposite to the trend in binding energy.⁴⁶ Furthermore, polarization and dispersion contributions cannot be overlooked.^2,3,47,48 Energy decomposition analysis is widely used by theoretical methods to analyze the components of different interactions.^32,41,49 Overall, the nature depends upon the individual interacting species,³ and no model can explain all observed features.

Halogen bonding is comparable in interaction energy (from 5 to 200 kJ mol⁻¹) to the well-known hydrogen bonding,^50–54 and I₃⁻ is the strongest.^4,5,55 It is widely accepted that I₃⁻ can be considered as the result of a three-center four-electron (3c-4e) interaction.^56–61 The 3c-4e bonding pattern for I₃⁻ can be rationalized by an MO picture in Fig. 1a. Three MOs are bonding (σ), non-bonding (n) and antibonding (σ*) orbitals formed primarily by σ_pz and orbitals of I₂ and p_z orbital of I⁻. I⁻ lone pair contributes two electrons, and I–I bonding pair contributes another two, for a total of 4e. These two pairs of electrons occupy σ and n orbitals, leaving the σ* orbital vacant. A similar case is the short strong hydrogen bond, represented by [F–H–F]⁻, which can also be considered as an equivalent multicenter interaction, and was recently verified experimentally by Tokmakoff et al.⁶²

Fig. 1 Molecular orbitals scheme for the formation of I₃⁻ by (a) (I₂ + I⁻) model, where I₃⁻ results from the overlap of orbitals of I₂ and I⁻, and (b) (I⁺ + 2I⁻) model, where I₃⁻ results from the overlap of orbitals of two I⁻ and one I⁺.

In our opinion, as shown in Fig. 1b, I₃⁻ can be described by the Lewis theory model with the middle I⁺ cation serving as the Lewis acid and two terminal I⁻ anions acting as the Lewis base, each ion providing one p_z orbital to form bonding (σ), non-bonding (n) and antibonding (σ*) orbitals. Similarly, 4e occupy σ and n orbitals. We speculate that its nature of an equivalent 3c-4e bond is ascribed to the same forces coming from two terminal I⁻. Inspired by this, in this study, we performed theoretical studies on species composed of I⁺ and two bases (L–I⁺–L, F⁻–I⁺–L and H₃P–I⁺–L, where L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) to explore the nature of the halogen bonding. We found that besides I₃⁻, other halogen bond compounds can also be well explained by multicenter bonds by analyzing their structural and electronic properties. Thus, we propose a new perspective on halogen bonding: its nature should be a multicenter bond, corresponding to equivalent and polarized 3c-4e bonds, respectively.

2. Computational methods

To confirm the computational method for our system, we performed calculations under several density functional theory (DFT) and the CCSD(T) theoretical levels to compare the values of the bond length, frequency and interaction energy with the corresponding experimental results for I₂. As depicted in Table 1, the I–I bond length at the M06-2X⁶³/def2tzvpp level (2.652 Å) was close to that at the CCSD(T)/dhf-QZVP level (2.659 Å), and they were slightly shorter than the experimental value (2.666 Å).⁶⁴ Compared with other methods, M06-2X/def2-TZVPP was considered to be appropriate, and our calculations on halogen bonding in this study were all given at the M06-2X/def2tzvpp level.

Table 1 Benchmark on the bond length, frequency of the I–I bond and binding energy of molecular I₂

Theoretical level	RI–I (Å)	FreqI–I (cm^−1)	Eb^a (kcal mol^−1)
a Binding energy of I2 is defined based on the calculation of Eb = E(I2) − 2E(I), where E(I2) or E(I) suggests the corresponding single point energies.b Refer to the ref. 64.
Experiment^b	2.666	215.5	−35.9
CCSD(T)/dhf-QZVP	2.659	224.4	−44.9
B3LYP-D3(BJ)/dhf-QZVP	2.694	213.5	−46.3
B3LYP-D3(BJ)/def2-TZVPP	2.688	214.1	−46.5
M06-2X/dhf-QZVP	2.651	233.3	−33.7
M06-2X/def2-TZVPP	2.652	232.5	−33.8

---

All the geometric structures were fully relaxed at the M06-2X/def2tzvpp level of theory. Frequencies of all the halogen bonding compounds were all checked to ensure that they were true local minimums. All bonding energies were calculated including basis set superposition error (BSSE) corrections⁶⁵ and were referenced to the optimized geometries of the fragments that compose the molecule.

Noncovalent interactions (NCI)⁶⁶ indexes, which is an approach to detect the strength of the weak interactions based on the electron density and its derivatives was selected. The results are shown in the scatter plots of the reduced density gradient (RDG) versus the electron density (ρ) multiplied by the sign of λ₂ (sign(λ₂)ρ). The low-density regime represents non-covalent/weak interactions. Negative values (sign(λ₂)ρ < 0) indicate an attractive interaction, while positive values (sign(λ₂)ρ > 0) are indicative of a steric repulsion. Spike mentioned in this study corresponds to the value of sign(λ₂)ρ of interaction. In general, more negative values of NCI spikes indicate a stronger interaction between the fragments.

Chemical bonding analysis was carried out using the adaptive natural density partitioning (AdNDP)⁶⁷ method. It is a tool that provides the partitioning of electron density. It allows an electron pair to be delocalized over n atoms with n ranging from one to the total number of atoms in the whole molecule. This method accepts only those bonding elements whose occupation numbers (ON) exceed the specified threshold values, which are usually chosen to be close to 2.00|e|.

All DFT calculations and natural bond orbital (NBO)^68,69 analysis were performed on the Gaussian 16 package.⁷⁰ Wiberg bond index (WBI)⁷¹ and NCI analysis were performed by the Multiwfn 3.4 program.⁷² Molecular visualization was performed using the MOLEKEL 5.4 software.⁷³

3. Results and discussion

3.1 Structural and electronic properties of I₃⁻

It is well known that I₂ can react with I⁻ easily in a solution, forming the strong halogen bonding complex I₃⁻. As shown in Fig. 2a, we scanned the energy of I₃⁻ by changing the distance of I(1)⋯I(2) and I(2)⋯I(3) from 2.4 Å to 6.0 Å. It is clear that the most stable structure is a linear molecule with two equal I⋯I length of 2.92 Å (structure I). The black dash line can be seen as the dissociation pathway of I₃⁻ to I₂⋯I⁻, and we chose structures II and III with the distance of I(1)⋯I(2) of 4.00 and 6.00 Å, respectively, to investigate the changes in the chemical bonding. Fig. 2b gives the 3c-4e chemical bonding patterns of structures I, II and III by the AdNDP method.

Fig. 2 (a) Potential energy surface of I₃⁻, bonding and non-bonding orbitals of structures I, II and III (as labeled) by the AdNDP analysis based on (b) the 3c-4e model and (c) traditional model (I₂ + I⁻).

For structure I, there are two symmetric 3c-2e bonds with idealized occupation numbers 2.00|e|, corresponding to bonding and non-bonding bonds composed of three 5p orbitals of I atoms. When the length of I(1)⋯I(2) is fixed at 4.00 Å, the I(2)⋯I(3) distance is shortened to 2.73 Å (structure II). Different from the symmetric 3c-2e bonds for structure I, more electrons concentrate on I(1) and I(2) in the bonding orbital, while more electrons focus on I(2) and I(3) in the non-bonding orbital. Therefore, I₂⋯I⁻ can also be explained by the 3c-4e bond model, just like I₃⁻, and the difference is that the 3c-4e bond is polarized. Furthermore, when the I(1)⋯I(2) distance is further elongated to 6.00 Å (structure III), the 3c-4e bonds is totally polarized, corresponding to one lone pair on I(1) and one 2c-2e σ orbital between I(2) and I(3). Taken together, we can obtain that during dissociation, electrons of the 3c-2e bonding orbital in I₃⁻ undergoes a shift to I(1)⋯I(2) and finally focus on I(1), while that of 3c-2e non-bonding orbitals finally concentrate on the σ bond between I(2) and I(3). Therefore, we can conclude that besides I₃⁻, interactions in other products during dissociation should also be ascribed to multicenter bonds.

Traditionally, halogen bonding complexes should be explained as a charge transfer from the donor lone pair of I⁻ to the σ* orbital of I₂ or electrostatic attraction between I⁻ and σ-hole of I₂. For comparison, Fig. 2c gives the AdNDP bonding orbitals of structures I, II and III based on the I₂ + I⁻ model (2c-2e/1c-2e). In structure III, I⁻ is far enough with I₂ molecule, so it is a typical compound with a weak halogen bond. In this situation, the 3c-4e model is very similar to the 2c-2e/1c-2e model in the bonding picture. In structure II, the halogen bond [I(1)⋯I(2)] is strengthened, and the non-Lewis component of the bonding orbitals according to the I₂ + I⁻ model increases, where the ON value of 1c-2e orbital decreases to 1.84|e|. The lower ON value suggests that electron transfer from I⁻ to I₂ should be considered. This phenomenon becomes more obvious for the equilibrium structure (I), the ON value of non-bonding orbital being quite small (1.49|e|). Therefore, the polarized 3c-4e bond model is consistent with the traditional 2c-2e/1c-2e model for the cases with a weak halogen bond (structure III), whereas it is more reasonable to describe the cases with a strong halogen bond (structures I and II).

To further confirm the essence of the multicenter bond discussed above, other properties corresponding to products obtained from the dissociation of I₃⁻ were studied, as shown in Fig. 3. The bond order (Fig. 3a) of I(1)⋯I(2) decreases with the increase in the I(1)⋯I(2) distance, and gradually approaches to zero. Moreover, those of I(2)⋯I(3) increase gradually and finally reach 1.5. Besides, the spike values of the NCI analysis (Fig. 3b) indicate that the interaction between I(2) and I(3) becomes stronger during dissociations with more negative values, and NBO charges (Fig. 3c) show that most charges finally concentrate on I(1). All the results are in good accordance with the AdNDP results discussed above, and suggest the same conclusion: interaction between I(1) and I(2) being weakened, whereas that between I(2) and I(3) being strengthened during dissociation, corresponding to the symmetric 3c-4e bond changing to a polarized 3c-4e one.

Fig. 3 Values of WBIs (a), NCI spikes (b) and NBO charges (c) during the dissociation of I₃⁻.

3.2 Structural and electronic properties of L–I⁺–L

I₃⁻ can be described by the Lewis theory model with the middle I⁺ cation serving as the Lewis acid and two terminal I⁻ anions acting as the Lewis base. Inspired by this, a series of electron donors L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) were chosen to interact with I⁺ to explore whether multicenter bonding can also be applied to other equivalent halogen bonding compounds (L–I⁺–L). Binding energies of I⁺ with different bases were calculated (Table 2, E_b = E(I⁺–L) − E(I⁺) − E(L)), spanning over a wide range, from −16.25 to −252.40 kcal mol⁻¹. Based on the binding energy, it is clear that the strength of the Lewis base decreases in the order: OH⁻ > F⁻ > Cl⁻ > Br⁻ > I⁻ > PH₃ > NH₃ > H₂S > HI > H₂O > HBr > HCl.

Table 2 Binding energy of I⁺ with various bases

Bases	OH^−	F^−	Cl^−	Br^−	I^−	PH3
Eb (kcal mol^−1)	−252.40	−227.47	−206.16	−201.61	−201.29	−89.88
Bases	NH3	H2S	HI	H2O	HBr	HCl
Eb (kcal mol^−1)	−64.66	−52.12	−40.10	−29.15	−25.74	−16.25

---

Optimized structures of I⁺–L and L–I⁺–L are shown in Fig. 4. It can be observed that all the L–I⁺–L angles are nearly linear with two equal I⁺⋯L distances, which are obviously longer than those in L–I⁺ monomers, similar to that in I₃⁻. Besides, WBIs in Fig. 5 suggest that the values of bond order between I⁺ and L in L–I⁺–L complexes are less than those in the corresponding I⁺–L monomers, which is in accordance with the changes in the bond lengths.

Fig. 4 Optimized structures of L–I⁺ and L–I⁺–L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) complexes. Bond length (Å) is labelled in red.

Fig. 5 Values of WBIs of L–I⁺–L and I⁺–L complexes with different bases (OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) as the electron donors.

Furthermore, it can be easily observed from the AdNDP analysis (Fig. 6) that all complexes can be well described by two symmetric bonding and non-bonding orbitals with reliable occupation numbers (1.93–2.00|e|), just like I₃⁻. Therefore, as long as two forces around the central I⁺ are the same, it corresponds to an equivalent 3c-4e bond interaction.

Fig. 6 Bonding (right) and non-bonding (left) 3c-2e orbitals of L–I⁺–L complexes by the AdNDP analysis.

3.3 Structural and electronic properties of F⁻–I⁺–L

Frequently, halogen bonding compounds do not own a perfect structural symmetry. Thus, we further extended our study to more general species. Electron donors on one side of I⁺ was fixed to F⁻, and the other side was different bases, as shown in Table 2, forming halogen boding complexes of F⁻–I⁺–L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl). The optimized structures (Fig. 7) show that the angles of F⋯I⋯L approach to 180°, and F⋯I distances are all longer than that in the FI monomer of 1.90 Å (Fig. 7) at the same theoretical level.

Fig. 7 Optimized structures of F⁻–I⁺ and F⁻–I⁺–L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) complexes. Bond length (Å) is labelled in red.

As expected, the chemical bonding analysis (Fig. 8) indicated that there are two 3c-2e orbitals for each species with idealized occupation numbers (1.95–2.00|e|). In case of OH⁻, F⁻, Cl⁻ and Br⁻ as electron donors, all bonding and non-bonding orbitals are evident and in good symmetry. When the electron donors change to I⁻, PH₃ and NH₃, it is clear that 3c-4e orbitals are polarized, just like those in I₂⋯I⁻, bonding orbitals being mainly focused on the I⁺–L segment, while non-bonding orbitals concentrating on the other terminals. In terms of the rest of the complexes, owing to their weak binding ability with I⁺, the polarization becomes clearer. Two 3c-2e orbitals of each are totally polarized to one σ bond between F and I and one lone pair electrons at the other end. In addition, NCI spikes (Fig. 9a) are in good agreement with the AdNDP results. As the bases change from OH⁻ to HCl, interaction between F and I become stronger with more negative values, while those between I and L become weaker. WBIs in Fig. 9b further support the conclusion; when the binding ability of bases is strong enough (OH⁻, F⁻, Cl⁻, Br⁻, I⁻), values of the bond order of F⁻–I⁺ are close to those of I⁺–L. However, there are big gaps for weak bases, and electrons mainly concentrate on F⁻–I⁺. Therefore, we can infer that the multicenter bond can be extended to rationalize more general halogen bonding compounds.

Fig. 8 Bonding (right) and non-bonding (left) 3c-2e orbitals of F⁻–I⁺–L complexes by the AdNDP analysis.

Fig. 9 Values of (a) NCI spikes and (b) WBIs of F⁻–I⁺ and I⁺–L in F⁻–I⁺–L complexes with different bases (OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) as the electron donors.

3.4 Structural and electronic properties of H₃P–I⁺–L

Finally, we fixed the medium base PH₃ to investigate the halogen bonding property with various bases: H₃P–I⁺–L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl). Similarly, the optimized structures (Fig. 10) show that two bases interact with I⁺ at 180° angles. Compared with the P–I bond length of 2.33 Å in the H₃P–I⁺ monomer (Fig. 10), P⋯I distances are elongated when bonding to other electron donors and vary within a wide range from 2.35 to 3.20 Å.

Fig. 10 Optimized structures of H₃P–I⁺ and H₃P–I⁺–L (L = OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) complexes. Bond length (Å) is labelled in red.

Chemical bonding analysis with two 3c-2e orbitals for each are depicted in Fig. 11. Differently from the F⁻–I⁺–L complexes in Fig. 8, because OH⁻, F⁻, Cl⁻ and I⁻ are stronger bases than PH₃, orbitals of these species are directly polarized, and more σ character focus on the I⁺–L segments in the bonding orbitals, whereas the non-bonding orbitals serve as the lone pair electrons. Symmetric bonding and non-bonding orbitals emerge when I⁺ interacts with two equal PH₃ groups. When the bases are further weakened (NH₃, SH₂, HI, H₂O, HBr and HCl), the orbitals are polarized again. Differently from the first four strong groups (OH⁻, F⁻, Cl⁻ and I⁻) as bases, more electrons concentrating on the H₃P–I⁺ segments and finally act as 2c-2e σ bonds in the 3c-2e non-bonding orbitals. Moreover, in the 3c-2e bonding orbitals, more electrons focus on the I⁺–L parts and finally serve as the lone pair electrons. NCI and bond order analysis in Fig. 12 show that stronger interactions (more negative values in spikes and more positive values in WBIs) located on the I⁺–L segments in case of OH–, F–, Cl– and I–, but when it comes to NH₃, SH₂, HI, H₂O, HBr and HCl, stronger interactions transfer to the P–I⁺ segments, following the same trend with the AdNDP methods. Therefore, the multicenter bond is further illustrated to be capable to be applied to more halogen bonding compounds.

Fig. 11 Bonding (right) and non-bonding (left) 3c-2e orbitals of PH₃–I⁺–L complexes by the AdNDP analysis.

Fig. 12 Values of (a) NCI spikes and (b) WBIs of I⁺–L and P–I⁺ in H₃P–I⁺–L complexes with different bases (OH⁻, F⁻, Cl⁻, Br⁻, I⁻, PH₃, NH₃, H₂S, HI, H₂O, HBr and HCl) as the electron donors.

4. Conclusions

A series of complexes formed by I⁺ and representative electron donors as well as I₃⁻ have been examined theoretically to gain a deeper insight into the nature of halogen bonding. It can be easily observed that during the dissociation of I₃⁻, equivalent 3c-4e bonds gradually changed to polarized orbitals, and finally served as one 2c-2e σ bond and one lone pair of electrons. We concluded that both I₃⁻ and I₂⋯I⁻ can be described by a multicenter bond, and the traditional charge transfer or σ-hole model should be incorporated into a polarized 3c-4e bond. In addition, the other I-containing complexes can also be explained by the 3c-4e bond, all of them having two 3c-2e orbitals with reliable ON values. When the forces of two bases around I⁺ are the same, it possesses clear bonding and non-bonding orbitals, corresponding to the equivalent 3c-4e bond. When the forces are different, it is a polarized 3c-4e bond. Thus, our study revisits the covalent nature of halogen bonding and comes to the conclusion that not only the well-known I₃⁻, but also all halogen bonding compounds are dominant by a multicenter bond, existing equivalent and polarized 3c-4e bonds, respectively. We hope that it can be applied to explain more halogen bonding complexes.

Conflicts of interest

The authors declare no competing financial interests.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 21873001) and by the Foundation of Distinguished Young Scientists of Anhui Province.

References

1. C. B. Aakeroy, M. Fasulo, N. Schultheiss, J. Desper and C. Moore, J. Am. Chem. Soc., 2007, 129, 13772.
2. G. Cavallo, P. Metrangolo, R. Milani, T. Pilati, A. Priimagi, G. Resnati and G. Terraneo, Chem. Rev., 2016, 116, 2478–2601.
3. L. C. Gilday, S. W. Robinson, T. A. Barendt, M. J. Langton, B. R. Mullaney and P. D. Beer, Chem. Rev., 2015, 115, 7118–7195.
4. P. Metrangolo, F. Meyer, T. Pilati, G. Resnati and G. Terraneo, Angew. Chem., Int. Ed., 2008, 47, 6114–6127.
5. P. Metrangolo, H. Neukirch, T. Pilati and G. Resnati, Acc. Chem. Res., 2005, 38, 386–395.
6. P. Von R. Schleyer and R. West, J. Am. Chem. Soc., 1959, 81, 3164–3165.
7. M. Colin, Ann. Chim., 1814, 91, 252–272.
8. F. Guthrie, J. Chem. Soc., 1863, 16, 239–244.
9. I. Remsen and J. Norris, Am. Chem. J., 1896, 18, 90–95.
10. J. M. Guevara Vela, D. Ochoa Resendiz, M. A. Costales Castro, R. Hernández Lamoneda and Á. Martín Pendás, ChemPhysChem, 2018, 19, 2512–2517.
11. D. M. Ivanov, A. S. Novikov, I. V. Ananyev, Y. V. Kirina and V. Y. Kukushkin, Chem. Commun., 2016, 52, 5565–5568.
12. A. C. Legon, Phys. Chem. Chem. Phys., 2010, 12, 7736–7747.
13. C. Z. Liu, S. Koppireddi, H. Wang, D. W. Zhang and Z. T. Li, Angew. Chem., Int. Ed., 2019, 58, 226–230.
14. M. Niyas, R. Ramakrishnan, V. Vijay, E. Sebastian and M. Hariharan, J. Am. Chem. Soc., 2019, 141, 4536–4540.
15. P. Politzer and J. S. Murray, ChemPhysChem, 2013, 14, 278–294.
16. P. Politzer, J. S. Murray and T. Clark, Phys. Chem. Chem. Phys., 2013, 15, 11178–11189.
17. S. Scheiner, CrystEngComm, 2013, 15, 3119–3124.
18. S. Scheiner, CrystEngComm, 2019, 21, 2875–2883.
19. A. Vanderkooy, A. K. Gupta, T. Földes, S. Lindblad, A. Orthaber, I. Pápai and M. Erdélyi, Angew. Chem., Int. Ed. Engl., 2019, 131, 9110–9114.
20. A. R. Voth, P. Khuu, K. Oishi and P. S. Ho, Nat. Chem., 2009, 1, 74.
21. L. Meazza, J. A. Foster, K. Fucke, P. Metrangolo, G. Resnati and J. W. Steed, Nat. Chem., 2013, 5, 42–47.
22. T. Caronna, R. Liantonio, T. A. Logothetis, P. Metrangolo, T. Pilati and G. Resnati, J. Am. Chem. Soc., 2004, 126, 4500–4501.
23. A. Farina, S. V. Meille, M. T. Messina, P. Metrangolo, G. Resnati and G. Vecchio, Angew. Chem., Int. Ed., 1999, 38, 2433–2436.
24. G. Bergamaschi, L. Lascialfari, A. Pizzi, M. I. M. Espinoza, N. Demitri, A. Milani, A. Gori and P. Metrangolo, Chem. Commun., 2018, 54, 10718–10721.
25. B. Hawthorne, H. Fan-Hagenstein, E. Wood, J. Smith and T. Hanks, Int. J. Spectrosc., 2013, 2013, 216518–216527.
26. J. Xu, X. Liu, T. Lin, J. Huang and C. He, Macromolecules, 2005, 38, 3554–3557.
27. T. M. Beale, M. G. Chudzinski, M. G. Sarwar and M. S. Taylor, Chem. Soc. Rev., 2013, 42, 1667–1680.
28. N. Cheetham and A. Pullin, Chem. Commun., 1967, 233–234.
29. R. Wilcken, M. O. Zimmermann, A. Lange, A. C. Joerger and F. M. Boeckler, J. Med. Chem., 2013, 56, 1363–1388.
30. Y. Lu, Y. Wang and W. Zhu, Phys. Chem. Chem. Phys., 2010, 12, 4543–4551.
31. H. Matter, M. Nazaré, S. Güssregen, D. W. Will, H. Schreuder, A. Bauer, M. Urmann, K. Ritter, M. Wagner and V. Wehner, Angew. Chem., Int. Ed., 2009, 48, 2911–2916.
32. T. Xia, D. Li and L. Cheng, Chem. Phys., 2020, 539, 110978.
33. R. S. Mulliken, J. Am. Chem. Soc., 1950, 72, 600–608.
34. H. A. Bent, Chem. Rev., 1968, 68, 587–648.
35. D. Bulfield and S. M. Huber, Chem.–Eur. J., 2016, 22, 14434–14450.
36. O. Hassel, Science, 1970, 170, 497–502.
37. S. M. Huber, J. D. Scanlon, E. Jimenez-Izal, J. M. Ugalde and I. Infante, Phys. Chem. Chem. Phys., 2013, 15, 10350–10357.
38. A. E. Reed, F. Weinhold, R. Weiss and J. Macheleid, J. Phys. Chem., 1985, 89, 2688–2694.
39. S. W. Robinson, C. L. Mustoe, N. G. White, A. Brown, A. L. Thompson, P. Kennepohl and P. D. Beer, J. Am. Chem. Soc., 2015, 137, 499–507.
40. C. Wang, D. Danovich, Y. Mo and S. Shaik, J. Chem. Theory Comput., 2014, 10, 3726–3737.
41. L. P. Wolters and F. M. Bickelhaupt, ChemistryOpen, 2012, 1, 96–105.
42. P. Politzer, P. Lane, M. C. Concha, Y. Ma and J. S. Murray, J. Mol. Model., 2007, 13, 305–311.
43. P. Politzer, J. S. Murray and M. C. Concha, J. Mol. Model., 2007, 13, 643–650.
44. Y.-X. Lu, J.-W. Zou, Y.-H. Wang, Y.-J. Jiang and Q.-S. Yu, J. Phys. Chem. A, 2007, 111, 10781–10788.
45. K. E. Riley, J. S. Murray, P. Politzer, M. C. Concha and P. Hobza, J. Chem. Theory Comput., 2009, 5, 155–163.
46. S. M. Huber, E. Jimenez-Izal, J. M. Ugalde and I. Infante, Chem. Commun., 2012, 48, 7708–7710.
47. K. E. Riley and P. Hobza, Phys. Chem. Chem. Phys., 2013, 15, 17742–17751.
48. J. Thirman, E. Engelage, S. M. Huber and M. Head-Gordon, Phys. Chem. Chem. Phys., 2018, 20, 905–915.
49. B. Pinter, N. Nagels, W. A. Herrebout and F. De Proft, Chem.–Eur. J., 2013, 19, 519–530.
50. L. J. Prins, D. N. Reinhoudt and P. Timmerman, Angew. Chem., Int. Ed., 2001, 40, 2382–2426.
51. T. Steiner, Angew. Chem., Int. Ed., 2002, 41, 48–76.
52. B. Chen, I. Ivanov, M. L. Klein and M. Parrinello, Phys. Rev. Lett., 2003, 91, 215503.
53. C. Laurence and M. Berthelot, Perspect. Drug Discov. Des., 2000, 18, 39–60.
54. C. F. Matta, J. Hernández-Trujillo, T. H. Tang and R. F. Bader, Chem.–Eur. J., 2003, 9, 1940–1951.
55. E. Corradi, S. V. Meille, M. T. Messina, P. Metrangolo and G. Resnati, Angew. Chem., Int. Ed., 2000, 39, 1782–1786.
56. G. A. Landrum, N. Goldberg and R. Hoffmann, J. Chem. Soc., Dalton Trans., 1997, 3605–3613.
57. M. L. Green and G. Parkin, Dalton Trans., 2016, 45, 18784–18795.
58. G. C. Pimentel, J. Chem. Phys., 1951, 19, 446–448.
59. K. Sonnenberg, L. Mann, F. A. Redeker, B. Schmidt and S. Riedel, Angew. Chem., Int. Ed., 2020, 59, 5464–5493.
60. A. Y. Rogachev and R. Hoffmann, Inorg. Chem., 2013, 52, 7161–7171.
61. S. Shaik, New J. Chem., 2007, 31, 2015–2028.
62. B. Dereka, Q. Yu, N. H. C. Lewis, W. B. Carpenter, J. M. Bowman and A. Tokmakoff, Science, 2021, 371, 160–164.
63. Y. Zhao and D. G. Truhlar, Acc. Chem. Res., 2008, 41, 157–167.
64. K. P. Huber and G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979.
65. A. Rosa, A. W. Ehlers, E. J. Baerends, J. G. Snijders and G. t. Velde, J. Phys. Chem., 1996, 100, 5690–5696.
66. E. R. Johnson, S. Keinan, P. Mori-Sánchez, J. Contreras-García, A. J. Cohen and W. Yang, J. Am. Chem. Soc., 2010, 132, 6498–6506.
67. D. Y. Zubarev and A. I. Boldyrev, Phys. Chem. Chem. Phys., 2008, 10, 5207–5217.
68. A. E. Reed and F. Weinhold, J. Chem. Phys., 1983, 78, 4066–4073.
69. F. Weinhold, J. Comput. Chem., 2012, 33, 2363–2379.
70. M. Frisch, G. Trucks, H. Schlegel, G. Scuseria, M. Robb, J. Cheeseman, G. Scalmani, V. Barone, G. Petersson and H. Nakatsuji, Gaussian, Inc., Wallingford CT: 2016.
71. L. K. Harper, A. L. Shoaf and C. A. Bayse, ChemPhysChem, 2015, 16, 3886–3892.
72. T. Lu and F. Chen, J. Comput. Chem., 2012, 33, 580–592.
73. U. Varetto, MOLEKEL 5.4, Swiss National Supercomputing Centre, Manno, Switzerland, 2009.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1ra05695f

---