σ -Holes vs. Buildups of Electronic Density on the Extensions of Bonds to Halogen Atoms

: Our discussion focuses upon three possible features that a bonded halogen atom may exhibit on its outer side, on the extension of the bond. These are (1) a region of lower electronic density (a σ -hole) accompanied by a positive electrostatic potential with a local maximum, (2) a region of lower electronic density (a σ -hole) accompanied by a negative electrostatic potential that also has a local maximum, and (3) a buildup of electronic density accompanied by a negative electrostatic potential that has a local minimum. In the last case, there is no σ -hole . We show that for diatomic halides and halogen-substituted hydrides, the signs and magnitudes of these maxima and minima can be expressed quite well in terms of the di ﬀ erences in the electronegativities of the halogen atoms and their bonding partners, and the polarizabilities of both. We suggest that the buildup of electronic density and absence of a σ -hole on the extension of the bond to the halogen may be an operational indication of ionicity.

The term "σ-hole", introduced by Clark et al. [1], refers to a local region of lower electronic density on the outer side of a bonded atom, on the extension of the bond to that atom. If the electronic density of a σ-hole is sufficiently low, there will be a positive electrostatic potential associated with it, through which the atom can interact attractively with negative sites such as lone pairs, π-electrons, and anions.
Such positive regions were first observed on the halogen atoms of some organic halides, by Brinck et al. in 1992 [2,3]. They pointed out that these unexpected positive regions provide an explanation of halogen bonding, the long-known but puzzling noncovalent interactions between bonded halogen atoms (which are presumed to be negative in character) and negative sites.
It must be emphasized that the interactions are with the positive potentials, not with the σ-holes, and σ-holes do not always give rise to positive electrostatic potentials. For instance, fluorines often have σ-holes with negative potentials, although less negative than their surroundings on the molecular surfaces. Such fluorines do not form halogen bonds (unless the negative site has a strong enough electric field to induce a positive potential [4,5]).
During the years 2007-2009, it was shown by Murray et al. [6][7][8] that localized positive electrostatic potentials are also often present on the extensions of bonds to atoms in Groups IV-VI, again allowing interactions with negative sites. This has led to understanding numerous instances of noncovalent bonding involving these atoms [9][10][11]. There have been several reviews of halogen bonding and other noncovalent interactions arising from σ-holes [9,[12][13][14][15].
What is the origin of a σ-hole on a bonded atom? When two atoms interact to form a bond, the charge distribution of each undergoes some degree of polarization due to the electric field of the other. Atoms in Groups V-VII commonly develop a lower electronic density and shorter radius on Since much of our discussion will deal with electrostatic potentials, we will briefly review this property. Any system of nuclei and electrons, such as a molecule or molecular complex, has an electrostatic potential V(r) at every point r in the surrounding space, given by Equation (1): Z A is the charge on nucleus A, located at R A and ρ(r) is the molecule's electronic density. Those regions in which V(r) is positive, reflecting dominant nuclear contributions, will interact attractively with nucleophiles (negative sites); regions of negative V(r), in which the effect of the electrons dominates, are attractive to electrophiles (positive sites). The electrostatic potential is a real physical property that is observable. It can be determined experimentally, by diffraction techniques [23][24][25], as well as computationally. It is important to recognize that the value of the potential at any point r reflects contributions from all of the nuclei and electrons of the molecule [26][27][28][29][30]. Among the consequences of this are that the electrostatic potential does not in general follow the electron density, i.e., electron-rich regions do not necessarily have negative potentials, and that the positive potentials due to σ-holes may not be directly at the σ-hole [19,[30][31][32].
In analyzing the noncovalent interactions of a molecule, V(r) is typically computed on its surface, as defined by the 0.001 au contour of its electronic density [33], and is labeled V S (r). The locally most positive and most negative values, of which there may be several on a given molecular surface, are designated as V S,max and V S,min , respectively.
In the present study, all calculations are at the density functional M06-2X/6-311G* level, with the Gaussian 09 [34] and the WFA-SAS [35] codes. The M06-2X functional was especially designed for noncovalent interactions and has been proven to be very effective for this purpose [36][37][38]. Extensive comparisons have demonstrated that the 6-311G* basis set gives reliable electrostatic potentials [39].

Electrostatic Potentials Associated with σ-Holes
What determines the sign and magnitude of the electrostatic potential due to a σ-hole? Two common generalizations invoke (a) the polarizability of the atom that has the σ-hole, and (b) the electron-attracting power of that atom relative to that of its bonding partner [9,12]. The more polarizable is the atom, and the more electron-attracting is its partner, the more positive will be the potential, and specifically the V S,mas , corresponding to the atom's σ-hole. First-row atoms, having low polarizabilities and high electronegativities, often have negative V S,mas associated with their σ-holes [9,12,40]. This is sometimes true of second-row atoms as well [40]. However, when such atoms are linked to strongly electron-attracting residues, their σ-holes do give rise to positive V S,max [6,[40][41][42]. For instance, the fluorines in both H 3 C-F and NC-F have σ-holes. However, the V S,max are −21 and 16 kcal/mol, respectively [40].
The above generalizations are overall quite useful for atoms in Groups IV-VII. However, significant exceptions and complications do arise. In the molecule FPBr(CN), for instance, the three σ-holes on the phosphorus have V S,max of 39 kcal/mol (Br-P bond), 35 kcal/mol (F-P bond), and 34 kcal/mol (NC-P) [30]. This is exactly the opposite of what would be anticipated from the fact that the CN group is the most electron-attracting and Br is the least. The arsenic in (H 3 C) 2 AsCN is expected to have three V S,max , corresponding to its three bonds, but the three positive regions overlap and result in just a single V S,max [32]. In heterocyclic molecules, the V S,mas anticipated on the extensions of the ring bonds frequently overlap and produce single V S,max near the centers of the bonds [19,32]. The whole situation is further analyzed in two extensive studies [43,44].
A basic problem, as already emphasized, is that the electrostatic potential at any point r is composed of contributions from all of the nuclei and electrons of the molecule [26][27][28][29][30], more so as they are closer to r. Thus, the location and value of the V S,max on an atom depend not only upon the atom itself and its bonding partner, but also upon other nearby atoms. This is less of a problem for halogen atoms, because they tend to protrude from a molecular framework and accordingly do not usually have other atoms in their immediate proximities. Our present analysis will accordingly focus upon halides.

Diatomic Molecules
We begin by looking at the computed electrostatic potentials of the diatomic molecules AX (Table 1), where A ranges from H to F, plus Cl and Br, and X is F, Cl, and Br. These molecules include singlets, doublets, and triplets. Our focus is upon the electrostatic potentials on the atoms X, on the extensions of the A-X bonds. By considering only diatomics, we eliminate the effects of atoms other than the bonding partners of the X atoms. Table 1 lists the extrema of V S (r) on the 0.001 au surfaces of the atoms X, on the extensions of the A-X bonds. These extrema include both maxima (V S,max ) and minima (V S,min ), and both the maxima and the minima can be either positive or negative. As far as we are aware, the possibility of electrostatic potential minima on the extensions of bonds to halogen atoms has received little or no attention in the past.
We will first discuss the maxima, both positive and negative. Most of the extrema in Table 1 are positive maxima. However, there are five maxima that are negative. These are the V S,max of the fluorines in HF, NF, ClF, and BrF, and that of the chlorine in BeCl. (The V S,max of the bromine in BeBr should be viewed as zero.) These negative V S,max are consistent with the generalizations mentioned in Section 3. Fluorine is the least polarizable and most electronegative of the elements, and chlorine also has a low polarizability and is much more electronegative than beryllium. All of these factors promote maxima being negative.
The relative electronegativities of A and X, χ A and χ B , are not, in themselves, sufficient to predict whether a V S,max on X will be positive or negative. The quantities (χ A − χ X ) are listed in Table 1, using the electronegativity values in Table 2. (The electronegativity scale is a modified version [45,46] of the widely-used one proposed by Allen [47,48].) The data in Table 1 show that X being more electronegative than A does not mean that a V S,max on X will be negative. There are nine molecules in Table 1 for which the quantity (χ A − χ X ) is negative but the V S,max of X is positive. Table 1. Properties of AX diatomic molecules (X = F, Cl and Br). V S (X) is computed maximum or minimum of V S (r) on X, on the extension of the A-X bond. The minima are explicitly indicated. χ A − χ X is difference in electronegativities of A and X. R 1 and R 2 are distances from X nucleus to 0.001 au contour, along the extension of the A-X bond (R 1 ) and to the lateral side of X (R 2 ) a .  All of the V S,max in Table 1, whether positive or negative, do correspond to σ-holes on the X atoms. This is shown by the dimensions of these atoms, also given in Table 1. For the molecules having a V S,max on X, the distance from the nucleus of X to the 0.001 au surface along the extension of the bond is in every case less than the distances to the lateral sides of X. This indicates a lower electronic density along the bond extension.
A particularly striking feature of Table 1 is that five molecules have minima (V S,min ), not maxima, along the extensions of the bonds to X. The five molecules are LiF, BeF, BF, LiCl, and LiBr. For all of these except BF the V S,min are negative; for BF it is essentially neutral, 1.1 kcal/mol. In each of these five molecules, the distance from the nucleus of X to the 0.001 au contour along the bond extension is greater than the distance to the lateral sides of X. This means that the electronic density is increased along the bond extension (the opposite of a σ-hole). Thus, the five molecules with V S,min on the X atom do not have σ-holes.
The contrast between molecules that do and do not have σ-holes is illustrated by Figure 1, which displays the computed electrostatic potentials on the molecular surfaces of (a) FCl and (b) LiCl. In FCl, there is a maximum on the chlorine on the extension of the F-Cl bond, with V S,max = 45.0 kcal/mol. The distance from the chlorine nucleus to the 0.001 au surface along the bond extension is 1.81 Å, compared to 2.12 Å perpendicular to the bond. This confirms the presence of a σ-hole on the chlorine.
do not have σ-holes.
The contrast between molecules that do and do not have σ-holes is illustrated by Figure 1, which displays the computed electrostatic potentials on the molecular surfaces of (a) FCl and (b) LiCl. In FCl, there is a maximum on the chlorine on the extension of the F-Cl bond, with VS,max = 45.0 kcal/mol. The distance from the chlorine nucleus to the 0.001 au surface along the bond extension is 1.81 Å, compared to 2.12 Å perpendicular to the bond. This confirms the presence of a σ-hole on the chlorine.  In LiCl, on the other hand, the chlorine has a minimum on the extension of the LiCl bond, with V S,min = −43.1 kcal/mol. The distance from the chlorine nucleus to the 0.001 au surface along the bond extension is 2.15 Å, and it is 2.05 Å perpendicular to the bond. This indicates an increase in electronic density on the outer side of the chlorine; there is no σ-hole on the chlorine. It can be seen in Figure 1 that the outer side of the chlorine is rounder in LiCl, Figure 1b than in FCl, Figure 1a; FCl displays the "polar flattening" that is associated with the presence of a σ-hole.
The five molecules in Table 1 that have a minimum (V S,min ) on the extension of the bond to X (LiF, LiCl, LiBr, BeF, and BF) are the ones that have the most electronegative X relative to A, i.e., (χ A − χ X ) has the most negative values. The electronic densities on their halogen atoms X are greater along the extensions of the bonds than on their lateral sides, and as we have already mentioned, the halogen atoms have no σ-holes.
Can the various V S,max and V S,min in Table 1, on the extensions of the bonds to the halogen atoms X, be represented analytically? We have already noted in Section 3 that the V S,max associated with σ-holes are linked to the relative electron attracting powers of X and its bonding partner A, and to the polarizability of X. Since the rearrangements of electronic density that lead to either a V S,max or a V S,min on X must involve A as well as X, it seems reasonable that the polarizability of A should also be taken explicitly into account [44].
Accordingly, we tested the three-parameter relationship Equation (2), using the electronegativity differences in Table 1 and the polarizabilities α in Table 2.
The result obtained with the NCSS statistical analysis package [50], shown in Figure 2, is very satisfactory. The correlation between predicted and computed V S,max or V S,min is R 2 = 0.943. The coefficients in Equation (2)  VS,max or VS,min = c1(χA-χX) + c2αA + c3αX + c4 (2) The result obtained with the NCSS statistical analysis package [50], shown in Figure 2, is very satisfactory. The correlation between predicted and computed VS,max or VS,min is R 2 = 0.943. The coefficients in Equation (2)   Note that the coefficient of the polarizability of X in Equation (2) is positive, while the coefficient of the polarizability of A is negative. These signs are consistent with the polarization of X tending to make X more positive and the polarization of A tending to make X more negative.

Substituted Hydrides, HnA-X
Can Equation (2) be applied to polyatomic molecules? As an initial test, we considered the substituted hydrides HnA-X, where A ranges from H to Br (excluding the noble gases and transition elements) and X is again F, Cl, and Br. The extrema of V(r) on the 0.001 au surfaces of the atoms X, on the extensions of the A-X bonds, are in Table 3. Note that the coefficient of the polarizability of X in Equation (2) is positive, while the coefficient of the polarizability of A is negative. These signs are consistent with the polarization of X tending to make X more positive and the polarization of A tending to make X more negative.

Substituted Hydrides, H n A-X
Can Equation (2) be applied to polyatomic molecules? As an initial test, we considered the substituted hydrides H n A-X, where A ranges from H to Br (excluding the noble gases and transition elements) and X is again F, Cl, and Br. The extrema of V(r) on the 0.001 au surfaces of the atoms X, on the extensions of the A-X bonds, are in Table 3.
A noteworthy feature of Table 3 is that there are nearly as many negative V S,max on the X atoms as there are positive ones. Furthermore there are many X with V S,min . In all of the molecules having a V S,min , the difference in electronegativities, χ A − χ X , is more negative than −1.50.
Nearly all of the fluorides have either a negative V S,max or a negative V S,min on the fluorine. The only exceptions are F 2 , which has a positive V S,max , and the molecules H 2 N-F, H 2 P-F, and HS-F, in which the negative potentials of the fluorines overlap with those of the lone pairs at the A atoms and no V S,max or V S,min can be attributed specifically to the fluorine.
When Equation (2) was applied to the H n A-X molecules in Table 3, using the data in Table 2, the result was very encouraging, as is shown in Figure 3. The R 2 is 0.953 and the coefficients are: c 1 = 18.89, c 2 = −0.9769, c 3 = 3.173, and c 4 = 10.80. The signs of the coefficients of the polarizabilities are consistent with the polarization effects noted above for the AX molecules. Table 3. Properties of H n A-X molecules (X = F, Cl, and Br). V S (X) is computed maximum or minimum of V S (r) on X, on the extension of the A-X bond. The minima are explicitly indicated. χ A − χ X is difference in electronegativities of A and X a . the polarizability of A, and the polarizability of X suffice to represent well all of the VS,max and VS,min of the diatomics AX ( Figure 2) and the substituted hydrides HnA-X ( Figure 3). It is actually somewhat surprising that Equation (2) is as effective for the hydrides as for the diatomics, given that the electronegativities and polarizabilities of free A atoms were used for the A in the HnA-X molecules. The fact that a covalently-bonded halogen can have a buildup of electronic charge on the extension of the bond has not, to our knowledge, been emphasized in the past. In such instances, the halogen does not have a σ-hole. Instead of a VS,max, it has a VS,min on its outer side, on the extension of the bond. Such VS,min are found only when the halogen atom is considerably more electronegative than its bond partner; a small difference is not sufficient to produce a VS,min (Tables 1 and 3).
A buildup of electronic density on the outer side of the halogen X indicates polarization of the electronic density of A toward X. This can be interpreted as resulting in some degree of ionic

Discussion and Summary
We have discussed three possible features that a bonded halogen atom can have on its outer side, on the extension of the bond: (1) A lower electronic density (a σ-hole) accompanied by a positive electrostatic potential with a V S,max . (2) A lower electronic density (a σ-hole) accompanied by a negative electrostatic potential with a V S,max . (3) A buildup of electronic density (no σ-hole), usually accompanied by a negative electrostatic potential with a V S,min .
It is especially notable that for each set of molecules, the diatomic halides AX and the substituted hydrides H n A-X, the various V S,max and V S,min can all be expressed by a single three-parameter relationship of the form of Equation (2). The difference between the electronegativities of A and X, the polarizability of A, and the polarizability of X suffice to represent well all of the V S,max and V S,min of the diatomics AX ( Figure 2) and the substituted hydrides H n A-X ( Figure 3). It is actually somewhat surprising that Equation (2) is as effective for the hydrides as for the diatomics, given that the electronegativities and polarizabilities of free A atoms were used for the A in the H n A-X molecules.
The fact that a covalently-bonded halogen can have a buildup of electronic charge on the extension of the bond has not, to our knowledge, been emphasized in the past. In such instances, the halogen does not have a σ-hole. Instead of a V S,max , it has a V S,min on its outer side, on the extension of the bond. Such V S,min are found only when the halogen atom is considerably more electronegative than its bond partner; a small difference is not sufficient to produce a V S,min (Tables 1 and 3).
A buildup of electronic density on the outer side of the halogen X indicates polarization of the electronic density of A toward X. This can be interpreted as resulting in some degree of ionic character, A + X − . While there is no unique or rigorous basis for distinguishing between covalency and ionicity, it may be that the formation of a V S,min can provide a useful operational distinction between them. On this basis, the ionic diatomics in Table 1 are LiF, BeF, BF, LiCl, and LiBr. In the H n A-X molecules, more of the A-X bonds have ionic character.
We recognize that the electronegativity differences χ A − χ X are only approximate measures of the relative effects of A and X upon each other. Perhaps a better means of doing this can be found. It is also clearly of interest to investigate the extension of the present analysis to atoms X other than halogens.
Author Contributions: The conceptualization and the formal analysis of this paper have been done by P.P. and J.S.M. The original draft preparation was done by P.P., while both P.P. and J.S.M. carried out review and editing. J.S.M. carried out the calculations and prepared the graphics for this paper.
Funding: This research received no external funding.