Radial Behavior of Electrostatic Potentials of Atoms and Ions Revisited: Isotropy and Anisotropy.

In this paper we revisit earlier work relating to monoatomic atoms and ions published by pioneers in the area of electrostatic potentials. We include plots of the radial distributions of the electrostatic potentials for spherically symmetric atoms and cations, and for singly, doubly and triply negative anions. For atoms with anisotropy in their densities and electrostatic potentials, such as the halonium cations, it is shown how the molecular surface approach for plotting electrostatic potentials complements that achieved by directional radial distributions.

Probing intramolecular interactions using molecular electrostatic potentials: changing electron density contours to unveil both attractive and repulsive interactions.

We focus on intramolecular interactions, using the electrostatic potential plotted on iso-density surfaces to lead the way. We show that plotting the electrostatic potential on varying iso-density envelopes much closer to the nuclei than the commonly used 0.001 a.u. contour can reveal the driving forces for such interactions, whether they be stabilizing or destabilizing. Our approach involves optimizing the structures of molecules exhibiting intramolecular interactions and then finding the contour of the electronic density which allows the interacting atoms to be separated; we call this the nearly-touching contour. The electrostatic potential allows then to identify the intramolecular interactions as either attractive or repulsive. The discussed 1,5- and 1,6-intramolecular interactions in o-bromophenol and o-nitrophenol are attractive, while the interactions between terminal methyl hydrogens in diethyl disulfides (as shown recently) and those between the closest hydrogens in planar biphenyl and phenanthrene are clearly repulsive in nature. For the attractive 1,4-interactions in trinitromethane and chlorotrinitromethane, and the 1,3-S⋯N and the 1,4-Si⋯N interactions in the ClH2Si(CH2)nNH2 series, the lack of (3,-1) bond critical points has often been cited as reason to not identify such interactions as attractive in nature. Here, by looking at the nearly-touching contours we see that bond critical points are neither necessary nor sufficient for attractive interactions, as others have pointed out, and in some instances also pointing to repulsive interactions, as the examples of planar biphenyl and phenanthrene discussed in this work show.

The Formation of σ-Hole Bonds: A Physical Interpretation.

This paper discusses two quite different computational experiments relating to the formation of σ-hole bonds A···B. The first involves looking at the complex at equilibrium and finding the contour X of the electronic density which allows the iso-density envelopes of A and B to be nearly touching. This contour increases, becoming closer to the nuclei, as the strength of the interaction increases. The second experiment involves allowing A and B to approach each other, with the aim of finding the distance at which their 0.001 a.u. iso-density envelopes are nearly merging into one envelope. What is found in the second experiment may be somewhat surprising, in that the ratio of the distance between interacting atoms at this nearly merging point-divided by the sum of the van der Waals radii of these atoms-covers a narrow range, typically between 1.2 and 1.3. It is intriguing to note that for the dataset presented, approaching molecules attracted to each other appear to do so unknowing of the strength of their ultimate interaction. This second experiment also supports the notion that one should expect favorable interactions, in some instances, to have close contacts significantly greater than the sums of the van der Waals radii.

Intramolecular Repulsion Visible Through the Electrostatic Potential in Disulfides: Analysis via Varying Iso-density Envelopes.

For a series of diethyl disulfide conformations, the nearly touching contours of the electrostatic potential plotted on iso-density molecular surfaces allow the assessment of intramolecular repulsion. The electrostatic potential is plotted on varying iso-density envelopes to find the nearly touching contours for which (a) the surface electrostatic potential does not show overlap between atoms or functional groups and (b) the typical features are visible (σ-hole, lone pair, hydrogen VS,max). When these nearly touching contours X are closer to the nuclei, the more electron density is excluded from the iso-density envelopes and the smaller are the volumes corresponding to these envelopes. Both the contours X and the corresponding volumes are found to correlate with relative conformational energy, reflecting the degree of intramolecular repulsion present in the various diethyl disulfides. Quantitative estimates of intramolecular repulsion can be made based on relationships between the nearly touching contour X vs relative energy and volume (of the nearly touching contour X) vs relative energy, obtained for series of diethyl disulfide conformers. These relations were used to analyze intramolecular repulsion in a set of disulfides broader than diethyl disulfide conformers. We have shown that the approach of varying electronic density contours can be used in the study of repulsive intramolecular interactions, hereby extending earlier work involving attractive intermolecular interactions.

Interpreting the variations in the kinetic and potential energies in the formation of a covalent bond.

We address the long-standing controversy as to the physical origin of covalent bonding, whether it involves a lowering of the potential energy or a lowering of the kinetic energy. We conclude that both of these do occur and contribute to the formation of the bond. The analysis is in terms of the virial theorem and the variations in the potential energy and the kinetic energy as the atoms approach each other. At large separations, the change in kinetic energy relative to the separated atoms is negative and stabilizing, while the corresponding potential energy change is positive and destabilizing. However, as the atoms approach their equilibrium separation, these rapidly reverse; the kinetic energy increases and the potential energy decreases, so that at equilibrium the net kinetic energy is positive and the net potential energy negative. At equilibrium, the bonding is due solely to the potential energy and is electrostatic.

Can Counter-Intuitive Halogen Bonding Be Coulombic?

We use the term "counter-intuitive" to describe an intermolecular interaction in which the electrostatic potentials of the interacting regions of the ground-state molecules have the same sign, both positive or both negative. In the present work, we consider counter-intuitive halogen bonding with nitrogen bases, in which both the halogen σ-hole and the nitrogen lone pair have negative potentials on their molecular surfaces. We show that these interactions can be treated as Coulombic despite the apparent repulsion between the ground-state molecules, provided that both electrostatics and polarization are explicitly taken into account. We demonstrate first that the energies of 20 counter-intuitive interactions with four nitrogen bases can be expressed very well in terms of just two molecular properties: the electrostatic potential of the halogen σ-hole and the average polarizability of the nitrogen base. Then we show that the same two properties can also represent the energies of an expanded data base that includes the 20 counter-intuitive plus an additional 20 weak and moderately-strong intuitive halogen bonding interactions (in which the σ-hole potentials are now positive).

The π-hole revisited.

It follows from the Schrödinger equation that the forces operating within molecules and molecular complexes are Coulombic, which necessarily entails both electrostatics and polarization. A common and important class of molecular complexes is due to π-holes. These are molecular regions of low electronic density that are perpendicular to planar portions of the molecular frameworks. π-Holes often have positive electrostatic potentials associated with them, which result in mutually polarizing attractive forces with negative sites such as lone pairs, π electrons or anions. In many molecules, π-holes correspond to a flattening of the electronic density surface but in benzene derivatives and in polyazines the π-holes are craters above and below the rings. The interaction energies of π-hole complexes can be expressed quite well in terms of regression relationships that account for both the electrostatics and the polarization. There is a marked gradation in the interaction energies, from quite weak (about -2 kcal mol-1) to relatively strong (about -40 kcal mol-1). Gradations are also evident in the ratios of the intermolecular separations to the sums of the respective van der Waals radii and in the gradual transition of the π-hole atoms from trigonal to quasi-tetrahedral configurations. These trends are consistent with the concept that chemical interactions form a continuum, from very weak to very strong.

The Neglected Nuclei.

Since the nuclei in a molecule are treated as stationary, it is perhaps natural that interpretations of molecular properties and reactivity have focused primarily upon the electronic density distribution. The role of the nuclei has generally received little explicit consideration. Our objective has been to at least partially redress this imbalance in emphasis. We discuss a number of examples in which the nuclei play the determining role with respect to molecular properties and reactive behavior. It follows that conventional interpretations based solely upon electronic densities and donating or withdrawing tendencies should be made with caution.

Electrostatics and Polarization in σ- and π-Hole Noncovalent Interactions: An Overview.

The energetics of σ- and π-hole interactions can be described very well in terms of electrostatics and polarization, consistent with their Coulombic natures. When both of these components are taken into account, very good correlations with quantum-chemically computed interaction energies are obtained. If polarization is only minor, as when the interactions are quite weak, then electrostatics can suffice, as represented by the most positive electrostatic potential associated with the σ- or π-hole. For stronger interactions, the combination of electrostatics plus polarization is very effective even for interaction energies considerably greater in magnitude than what is normally considered noncovalent bonding. Several procedures for treating polarization are summarized, including the use of point charges and the direct inclusion of electric fields.

Explicit Inclusion of Polarizing Electric Fields in σ- and π-Hole Interactions.

The interactions between a wide variety of molecules having σ-holes or π-holes and several nitrogen bases have been analyzed computationally. The σ- and π-hole atoms span groups III-VII of the periodic table. The interaction energies range from quite weak, typical of non-covalent bonding, to unusually strong: from -4.6 to -22.0 kcal/mol for σ-hole bonding and from -4.0 to -42.4 kcal/mol for π-hole bonding. The markedly greater strengths of some bonds does not imply that any new factors or types of bonding are involved; they simply reflect higher degrees of the polarization that is part of any Coulombic interaction. To explain the stronger bonding, this polarization must be explicitly taken into account. We show that the interaction energies can be related quite well to (a) the maximum positive electrostatic potentials associated with the σ- or π-holes on their molecular surfaces, (b) the polarizabilities of the nitrogen bases, and especially (c) the polarizing electric fields of the σ- or π-hole molecules at the positions of the nitrogens.

σ-holes and π-holes: Similarities and differences.

σ-Holes and π-holes are regions of molecules with electronic densities lower than their surroundings. There are often positive electrostatic potentials associated with them. Through these potentials, the molecule can interact attractively with negative sites, such as lone pairs, π electrons, and anions. Such noncovalent interactions, "σ-hole bonding" and "π-hole bonding," are increasingly recognized as being important in a number of different areas. In this article, we discuss and compare the natures and characteristics of σ-holes and π-holes, and factors that influence the strengths and locations of the resulting electrostatic potentials. © 2017 Wiley Periodicals, Inc.

The σ-Hole Coulombic Interpretation of Trihalide Anion Formation.

It is shown that the interactions of dihalogen molecules XY with halide anions Z- to form trihalide anions (XYZ)- can be satisfactorily described as Coulombic, involving the σ-holes on the atoms Y, but only if polarization is taken into account. We have approximated the polarizing effect of the halide anion Z- by means of a unit negative point charge. The CCSD/aug-cc-pVTZ computed interaction energies ΔE correlate well with the most positive electrostatic potentials associated with the induced σ-holes over a ΔE range of -12 to -63 kcal mol-1 . The (XYZ)- anions are more stable when the central atom is the largest, as has been observed, because the central atom is then the most polarizable, making the electrostatic potential associated with its σ-hole more positive.

A perspective on quantum mechanics and chemical concepts in describing noncovalent interactions.

Since quantum mechanical calculations do not typically lend themselves to chemical interpretation, analyses of bonding interactions depend largely upon models (the octet rule, resonance theory, charge transfer, etc.). This sometimes leads to a blurring of the distinction between mathematical modelling and physical reality. The issue of polarization vs. charge transfer is an example; energy decomposition analysis is another. The Hellmann-Feynman theorem at least partially bridges the gap between quantum mechanics and conceptual chemistry. It proceeds rigorously from the Schrödinger equation to demonstrating that the forces exerted upon the nuclei in molecules, complexes, etc., are entirely classically coulombic attractions with the electrons and repulsions with the other nuclei. In this paper, we discuss these issues in the context of noncovalent interactions. These can be fully explained in coulombic terms, electrostatics and polarization (which include electronic correlation and dispersion).

The influence of the metal cations and microhydration on the reaction trajectory of the N3 ↔ O2 thymine proton transfer: Quantum mechanical study.

This study involves the intramolecular proton transfer (PT) process on a thymine nucleobase between N3 and O2 atoms. We explore a mechanism for the PT assisted by hexacoordinated divalent metals cations, namely Mg2+ , Zn2+ , and Hg2+ . Our results point out that this reaction corresponds to a two-stage process. The first involves the PT from one of the aqua ligands toward O2. The implications of this stage are the formation of a hydroxo anion bound to the metal center and a positively charged thymine. To proceed to the second stage, a structural change is needed to allow the negatively charged hydroxo ligand to abstract the N3 proton, which represents the final product of the PT reaction. In the presence of the selected hexaaqua cations, the activation barrier is at most 8 kcal/mol. © 2017 Wiley Periodicals, Inc.

Close contacts and noncovalent interactions in crystals.

Close contacts, defined as interatomic separations less than the sum of the respective van der Waals radii, are commonly invoked to identify attractive nonbonded interactions in crystal lattices. While this is often effective, it can also be misleading because (a) there are significant uncertainties associated with van der Waals radii, and (b) it may not be valid to attribute the interactions solely to specific pairs of atoms. The interactions within crystal lattices are Coulombic, and the strongest positive and/or negative regions do not always correspond to the positions of atoms; they are sometimes located between atoms. Examples of both types are given and discussed, focusing in particular upon σ-hole interactions.

The σ-hole revisited.

A covalently-bonded atom typically has a region of lower electronic density, a "σ-hole," on the side of the atom opposite to the bond, along its extension. There is frequently a positive electrostatic potential associated with this region, through which the atom can interact attractively but noncovalently with negative sites. This positive potential reflects not only the lower electronic density of the σ-hole but also contributions from other portions of the molecule. These can significantly influence both the value and also the angular position of the positive potential, causing it to deviate from the extension of the covalent bond. We have surveyed these effects, and their consequences for the directionalities of subsequent noncovalent intermolecular interactions, for atoms of Groups IV-VII. The overall trends are that larger deviations of the positive potential result in less linear intermolecular interactions, while smaller deviations lead to more linear interactions. We find that the deviations of the positive potentials and the nonlinearities of the noncovalent interactions tend to be greatest for atoms of Groups V and VI. We also present arguments supporting the use of the 0.001 a.u. contour of the electronic density as the molecular surface on which to compute the electrostatic potential.

Hydrogen Bonding between Metal-Ion Complexes and Noncoordinated Water: Electrostatic Potentials and Interaction Energies.

The hydrogen bonding of noncoordinated water molecules to each other and to water molecules that are coordinated to metal-ion complexes has been investigated by means of a search of the Cambridge Structural Database (CSD) and through quantum chemical calculations. Tetrahedral and octahedral complexes that were both charged and neutral were studied. A general conclusion is that hydrogen bonds between noncoordinated water and coordinated water are much stronger than those between noncoordinated waters, whereas hydrogen bonds of water molecule in tetrahedral complexes are stronger than in octahedral complexes. We examined the possibility of correlating the computed interaction energies with the most positive electrostatic potentials on the interacting hydrogen atoms prior to interaction and obtained very good correlation. This study illustrates the fact that electrostatic potentials computed for ground-state molecules, prior to interaction, can provide considerable insight into the interactions.

σ-Hole bonding: a physical interpretation.

The anisotropic electronic densities of covalently-bonded Group IV-VII atoms frequently give rise to regions of positive electrostatic potential on the extensions of covalent bonds to these atoms. Through such positive "σ-holes," the atoms can interact attractively and highly directionally with negative sites such as the lone pairs of Lewis bases, anions, π electrons, etc. In the case of Group VII this is called "halogen bonding." Hydrogen bonding can be viewed as a less directional subset of σ-hole interactions. Since positive σ-holes often exist in conjunction with regions of negative potential, the atoms can also interact favorably with positive sites. In accordance with the Hellmann-Feynman theorem, all of these interactions are purely Coulombic in nature (which encompasses polarization and dispersion). The strength of σ-hole bonding increases with the magnitudes of the potentials of the positive σ-hole and the negative site; their polarizabilities must sometimes also be taken explicitly into account.

Complementarity of reaction force and electron localization function analyses of asynchronicity in bond formation in Diels-Alder reactions.

We have computationally compared three Diels-Alder cycloadditions involving cyclopentadiene and substituted ethylenes; one of the reactions is synchronous, while the others are slightly or highly asynchronous. Synchronicity and weak asynchronicity are characterized by the reaction force constant κ(ξ) having just a single minimum in the transition region along the intrinsic reaction coordinate ξ, while for high asynchronicity κ(ξ) has a negative maximum with minima on both sides. The electron localization function (ELF) shows that the features of κ(ξ) can be directly related to the formation of the new C-C bonds between the diene and the dienophile. There is thus a striking complementarity between κ(ξ) and ELF; κ(ξ) identifies the key points along ξ and ELF describes what is happening at those points.

Anisotropies in electronic densities and electrostatic potentials of Halonium Ions: focus on Chlorine, Bromine and Iodine.

CONTEXT: Why are the halonium cations so effective in forming strongly-bound complexes? We directed our research to address this question and we present electrostatic potential data for the valence-state halogen atoms X and halonium cations X+, where X = Cl, Br, I. The electron densities and electrostatic potentials of the halonium cations show considerably greater anisotropy than do the valence state halogens. The distances from the electrostatic potential surface maxima to the halogen nuclei are about 0.5 Å smaller than the distances from the electrostatic potential surface minima to the nuclei, giving the halonium cations each a more disk-like shape than the corresponding neutral valence state halogens. Their surface electrostatic potentials are totally consistent with the directionalities of halonium cations in complexes and the strengths of their interactions. To add perspective to this brief report, we have included calculations of the isotropic cation K+ and noble gas Kr. METHODS: The calculations of the electrostatic potentials of the valence states of the halogen atoms Cl, Br and I and the halonium cations Cl+, Br+ and I+, as well as K+ and Kr, on 0.001 au contours of their electronic densities were carried out with Gaussian O9 and the Wave Function Analysis - Surface Analysis Suite (WFA-SAS) at the M06-2X/6-31 + G(d,p) and M06-2X/3-21G* levels.