alt="equation"/>
ESP is an application of Coulomb's law and is a physical property that can be determined experimentally by diffraction techniques or computationally [138]. Given an electron density function ρ(r′), V(r) is the ESP at any measured point r (Eq. (1.1)). ZA is the charge on the nucleus that is located at RA. |RA − r| is the distance of the positive charge from r, and likewise |r ′ − r| is the distance of the electronic charge from r, where r′ is the integration variable over all space. A positive V(r) indicates that effects by the nucleus are dominant or that the nucleus is not entirely shielded by the electron cloud. A negative V(r) indicates that the electron density, in the form of electron pairs, π‐bonds, etc., is dominant. ESP is frequently computed and viewed as a map covering the surface of a molecule. This surface is arbitrarily selected; however the most common surface to map is an outer contour of electron density, as it accurately encompasses lone pairs, strained bonds, and π‐electrons (Figures 1.2 and 1.14). Typically, the ρ(r′) = 0.001 au (electrons/bohr3) contour is used, but other similar contours at 0.0015 or 0.002 au will also achieve the same ends [138]. The ESP values along this surface are then set to a color gradient directly on the molecule in question, and the extremes are typically represented as blue and red (Figures 1.14 and 1.2). While ρ(r′) and V(r) are in Eq. (1.1), there is a distinct difference between the values. ρ(r′) is dependent on only electrons, while V(r) incorporates contributions from all nuclei and electrons. As such, Politzer and Murray caution: It cannot be assumed that high (low) electronic densities correspond to negative (positive) electrostatic potentials. The potential in a given region is the net result of negative contributions from the electrons and positive ones from the nuclei of the entire molecule, their effects of course being greater as they are closer to the region in question [140]. In other words, ESP maps do not necessarily correlate with overall electron density.
Disclaimer aside, ESP maps are still highly informative. They have helped justify the amphoteric behavior of halogens observed in the solid state, where electrophiles approach the halogen “side‐on” orthogonal to the CX bond and nucleophiles “head‐on” in line with the CX bond [4]. In particular, ESP studies by Politzer and Murray [5,141] led to the establishment of the σ‐hole concept, which has proven to be a widely valuable tool for conceptualizing the halogen bond and has contributed to the renaissance of other σ‐hole‐type interactions like chalcogen and pnictogen bonding [142]. Additionally, the ease of constructing ESP maps has led to their use in predicting relative halogen bond strength. For example, Politzer showed that the iodine VS,max values of iodobenzene derivatives largely positively correlate with their interaction energies with acetone [139] (Figure 1.14; Table 1.2). This relationship has been demonstrated a number of times theoretically [130,143] and has led to the use of VS,max values as predictors of solid‐state structures [72,75,144] and performance in solution [145]. Widespread application of VS,max and ESP maps has likely contributed to the halogen bond being mistakenly viewed as a purely electrostatic interaction; however other components are frequently important to fully describe the interaction [146]. For example, there are a number of cases where a more positive VS,max does not correlate with a stronger halogen bond [147].
Figure 1.14 Computed ESP maps on 0.001 au molecular surfaces of (a) iodobenzene, (b) meta‐difluoroiodobenzene, (c) ortho‐difluoroiodobenzene, and (d) pentafluoroiodobenzene. Color ranges, in kcal/mol, are red, greater than 20; yellow, between 20 and 10; green, between 10 and 0; and blue, negative. Black hemispheres denote the positions of the iodine VS,max.
Source: From Riley et al. [139]. © 2011 Springer Nature.
Table 1.2 Table of iodine VS,max values and interaction energies (ΔE) of iodobenzene derivatives with acetone.
Source: Adapted from Riley et al. [139]. Copyright 2011 John Wiley & Sons.
| Interaction angle | |||
| At (X⋯O<span class="dbondb"</span>C) = 180° | At optimum X⋯O<span class="dbondb"</span>C angle | ||
| System | VS,max (kcal/mol) | ΔE (kcal/mol) | ΔE (kcal/mol) |
| Iodobenzene | 17.3 | −2.44 | −3.22 |
| meta‐Difluoroiodobenzene | 26.1 | −3.38 | −4.13 |
| ortho‐Difluoroiodobenzene | 25.5 | −3.64 | −4.71 |
| para‐Fluoroiodobenzene | 35.9 | −4.88 | −5.97 |
1.4.3 Limitations on Electrostatic Potential
While ESP is an effective tool for predicting and conceptualizing interactions, there are limitations. Obviously, contacts that are not primarily electrostatic in nature cannot be accurately predicted, such as those reliant on polarization or charge transfer. Furthermore, ESP maps are only for isolated molecules and therefore do not account for other nuances when two molecules come together. For example, ESP maps do not calculate changes in electron distribution resulting from polarization due to incoming molecules. Therefore, to more accurately predict the strength of a halogen bond, more involved computational techniques that factor additional variables should be considered.
1.4.4 Atomic Orbital Theory and the σ‐Hole
Formation of the σ‐hole and the halogen bond interaction can also be described using atomic orbital theory. To paraphrase Clark, Murray, and Politzer, the electron‐deficient σ‐hole is caused by depleted occupancy in the outer lobe of a p‐orbital of a covalent bond [8]. The halogen “X” has an s2px2py2pz1 electronic configuration where the RX bond is on the z‐axis. In this electron configuration, two p‐orbitals are filled, and one is half filled, thus highlighting the depleted electron density in the pz orbital. This picture becomes
