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.

Are HOMO-LUMO gaps reliable indicators of explosive impact sensitivity?

A high priority in designing and evaluating proposed explosives is to minimize sensitivity, i.e., vulnerability to unintended detonation due to an accidental stimulus, such as impact. In order to establish a capability for predicting impact sensitivity, there have been numerous attempts to correlate it with some molecular or crystal property or properties. One common approach has been to relate impact sensitivity to the difference between the energies of the highest-occupied and lowest-unoccupied molecular orbitals of the explosive molecule, the "HOMO-LUMO gap." In the present study, we tested this approach for a series of twelve explosive nitroaromatics, using four different computational methods. We found that the HOMO-LUMO gap does not appear to be a reliable indicator of relative impact sensitivity. Since detonation initiation involves a series of steps, all of which influence sensitivity; it seems more realistic to try to identify fundamental factors and general trends related to sensitivity ‒ an approach that has already had some success ‒ rather than to seek correlations with one or two specific properties.

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.

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).

A general model for the solubilities of gases in liquids.

The solubility of a compound is one of its most important properties. Here, regression relationships are presented for solubilities of a series of gases in water and in four organic solvents, treating each solvent separately. The solubilities are related to the Coulombic intermolecular interactions arising from the intrinsic polarities of the solute molecules and the polarities induced in them by the solvent. As a measure of intrinsic polarity, a statistical quantity defined in terms of the solute's molecular electrostatic potential is used, and the measure of induced polarity is taken to be the solute's molecular polarizability. Regression analyses show that solubility in water is best expressed in terms of just the intrinsic polarities of the solutes, but for the organic solvents, it is necessary to take into account both the intrinsic and the induced polarities of the solutes. If the dielectric constant of the solvent is included in the regression analysis, then a single relationship can encompass all four organic solvents. Solute molecular volumes were not found to contribute significantly to the present relationships.

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 Si···N intramolecular interactions.

We report a computational study of two series of molecules, one having the Si-O-N linkage and the other with the Si-(CH2)n-N linkage, where n = 1-4. The silicons have various substituents-combinations of H, CH3, F, Cl and CF3. Many of these compounds have been prepared and characterized experimentally. The Si···N distances were found to be relatively short, which may denote a noncovalent interaction or may simply be dictated by the molecular structures. This and the nature of the interaction (if any) were the subjects of our study. We addressed these issues computationally, determining optimized geometries, energies and electrostatic potentials at the density functional M06-2X/6-31 + G(d,p) level. We conclude that there is an attractive Coulombic Si···N interaction in each case, although it is sometimes quite weak. It involves the lone pair of the nitrogen and the positive σ-hole potential produced on the silicon by the bonding from the substituent that is anti to the nitrogen. This accounts for the key features of these molecules, such as the dependence of the Si···N distances upon the electron-withdrawing power of the anti substituent and the effects of gauche chlorines in weakening the interactions. When the Si···N interactions are disrupted by rotation of the N(CH3)2 group or by conversion of the molecules to open-chain conformers, the Si···N distances lengthen and are essentially the same regardless of the substituent in the anti position. These observations confirm the presence of Si···N interactions in the original molecules.

An Occam's razor approach to chemical hardness: lex parsimoniae.

The term "chemical hardness" refers to the resistance to deformation of the electronic density of a system; the greater this resistance, the "harder" the system. Polarizability, a physical property, is an inverse measure of resistance to deformation and thus should be inversely related to hardness. This is indeed generally accepted. Hardness has been postulated to be the second derivative of a system's energy with respect to its number of electrons, despite the fact that this involves the differentiation of a noncontinuous function. This second derivative is typically approximated as the difference between the ionization energy I and the electron affinity A of the ground-state system, which results in ambiguity in that many molecules do not form stable negative ions. For atoms, the quantity I - A does vary approximately inversely with polarizability, but this is only because the electron affinity is usually relatively low and ionization energy is known to be inversely related to polarizability for atoms. However, molecular polarizability depends primarily upon volume, and so does not show an acceptable inverse correlation with I - A. Since both hardness and polarizability refer to the same property of a system-its resistance to deformation of the electronic density, we propose that the reciprocal of polarizability be taken to be a measure of hardness. We show that polarizabilities that are not known can be estimated quite accurately in terms of the average local ionization energies on the atomic or molecular surfaces and, for molecules, their volumes.

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 Hellmann-Feynman theorem: a perspective.

The Hellmann-Feynman theorem has, with a few exceptions, not been exploited to the degree that it merits. This is due, at least in part, to a widespread failure to recognize that its greatest value may be conceptual rather than numerical, i.e., in achieving insight into molecular properties and behavior. In this brief overview, we shall discuss three examples of significant concepts that have come out of the Hellmann-Feynman theorem: (1) The forces exerted upon the nuclei in molecules are entirely Coulombic in nature. (2) The total energies of atoms and molecules can be expressed rigorously in terms of just the electrostatic potentials at their nuclei that are produced by the electrons and other nuclei. (3) Dispersion forces are due to the attractions of nuclei to their own polarized electronic densities. To summarize, energy and force analyses should not be viewed as competitive but rather as complementary.

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.

Electronegativity-a perspective.

Electronegativity is a very useful concept but it is not a physical observable; it cannot be determined experimentally. Most practicing chemists view it as the electron-attracting power of an atom in a molecule. Various formulations of electronegativity have been proposed on this basis, and predictions made using different formulations generally agree reasonably well with each other and with chemical experience. A quite different approach, loosely linked to density functional theory, is based on a ground-state free atom or molecule, and equates electronegativity to the negative of an electronic chemical potential. A problem that is encountered with this approach is the differentiation of a noncontinuous function. We show that this approach leads to some results that are not chemically valid. A formulation of atomic electronegativity that does prove to be effective is to express it as the average local ionization energy on an outer contour of the atom's electronic density.

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 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.

Halogen bonding in hypervalent iodine and bromine derivatives: halonium salts.

Halogen bonds have been identified in a series of ionic compounds involving bromonium and iodonium cations and several different anions, some also containing hypervalent atoms. The hypervalent bromine and iodine atoms in the examined compounds are found to have positive σ-holes on the extensions of their covalent bonds, while the hypervalent atoms in the anions have negative σ-holes. The positive σ-holes on the halogens of the studied halonium salts determine the linearity of the short contacts between the halogen and neutral or anionic electron donors, as usual in halogen bonds.

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.