Toward a physical understanding of electron-sharing two-center bonds. II. Pseudo-potential based analysis of diatomic molecules

Insights into the variations of kinetic and potential energies in a multi-bond reaction: the reaction electronic flux perspective

CONTEXT

The debate over whether kinetic energy (KE) or potential energy (PE) are the fundamental energy components that contribute to forming covalent bonds has been enduring and stimulating over time. However, the supremacy of these energy components in reactions where multiple bonds are simultaneously formed or broken has yet to be explored. In this study, we use the reaction electronic flux (REF), an effective tool for investigating changes in driving electronic activity when bond formation or dissociation occurs in a chemical reaction, to examine the fluctuations in the KE and PE in a multi-bond reaction. To that end, the activation of CO2 by low-valent group 14 catalysts through a concerted σ -bond metathesis mechanism is analyzed. The findings of this preliminary study suggest that the REF can be utilized as a tool to rationalize alterations in the KE and PE in a multi-bond reaction. Specifically, analyses across the reaction coordinate reveal that changes in the KE and PE precede activation in the REF, stimulating the electronic activity where bond formation or dissociation processes dominate.

METHODS

The activation of CO2 by the low-valent LEH catalysts (L = N,N'-bis(2,6-diisopropyl phenyl)- β -diketiminate; E = Si, Ge, Sn, and Pb) was studied along the reaction coordinate at the M06-2X/6-31 G(d,p)-LANL2DZ(E) level of theory. The respective minimum energy path calculations were obtained using the intrinsic reaction coordinate (IRC) procedure. The reaction electronic flux (REF) was calculated through the computation of the electronic chemical potential using the frontier molecular orbital approximation. Mayer bond orders along the reaction coordinate have been determined using the NBO 3.1 program in Gaussian16. Most of the reaction coordinate quantities reported in this study (REF, KE, PE, among others) have been determined using the Kudi program and custom Python scripts.

Reassessing the Composition of Hybrid Orbitals in Contemporary VB Calculations

Large variations in the ratios between the p and s components of individual hybrid orbitals that have been observed in contemporary ab initio VB calculations are reassessed, and links are established to specific energy terms that drive bond formation. It is demonstrated that the ratios between the p and s components for individual hybrid orbitals are not indicative of the overall hybridization status of the relevant atom, which exhibits only relatively small variations with the level of theory, irrespective of whether or not non-dynamical and dynamical electron correlation effects are accounted for. An alternative orbital representation that turns out to be far more consistent with the overall hybridization of the relevant atom is examined. The chosen test cases, which can be compared with the classical sp³, sp², and sp hybridization models for a central carbon atom, are CH₄ (T_d), trigonal CH₃ (D_3h), and triplet CH₂ distorted from its ground state geometry so as to be linear (D_∞h).

Analysis of chemical bonding of the ground and low-lying states of Mo2 and of Mo2Clx complexes, x = 2 - 10

In the present study, we perform accurate calculations via multireference configuration interaction and coupled cluster methodologies on the dimolybdenum molecule in conjunction with complete series of correlation and weighted core correlation consistent basis sets up to quintuple size. The bonding, dissociation energies, and spectroscopic parameters of the seven states that correlate to the ground state products are calculated. The ground state has a sextuple chemical bond and each of the calculated excited state has one less bond than the previous one. The calculated values for the ground(X1Σg+ ) state of Mo2 have been extrapolated to the complete basis set limits. Our final values, re=1.9324 Å and De(D0)=4.502{plus minus}0.007(4.471{plus minus}0.009) eV, are in excellent agreement with the experimental values of re=1.929, 1.938(9) Å and D0=4.476(10) eV. The Mo2 in 13Σg+ state is a weakly bound dimer, forming 5s...5pz bonds, with De=0.120 eV at re=3.53 Å. All calculated excited states (except 13Σg+) have a highly multireference character (C0=0.25-0.55). The ordering of the molecular bonding orbitals changes as the spin is increased from quintet to septet state. The quite low bond dissociation energy of the ground state is due to the splitting of the molecular bonding orbitals in two groups differing in energy by ~3 eV. Finally, the bond breaking of Mo2, as the multiplicity of spin is increased, is analyzed in parallel with the Mo-Mo bond breaking in a series of Mo2Clx complexes when x is increased. Physical insight into the nature of the sextuple bond and its low dissociation energy is provided.

The Nature of the Polar Covalent Bond

Quantum chemical calculations using density functional theory are reported for the diatomic molecules LiF, BeO, and BN. The nature of the interatomic interactions is analyzed with the EDA-NOCV method, and the results are critically discussed and compared with data from QTAIM, NBO and Mayer approaches. Polar bonds, like nonpolar bonds, are caused by the interference of wave functions, which lead to an accumulation of electronic charge in the bonding region. Polar bonds generally have a larger percentage of electrostatic bonding to the total attraction, but nonpolar bonds may also possess large contributions from Coulombic interaction. The term "ionic contribution" refers to VB structures and is misleading because it refers to separate fragments with negligible overlap that occur only in the solid state and in solution, not in a molecule. The EDA-NOCV method gives detailed information about the individual orbital contributions, which can nicely be identified by visual inspection of the associated deformation densities. It is very important, particularly for polar bonds to distinguish between the interatomic interactions of the final dissociation products after bond rupture and the interactions between the fragments in the eventually formed bond.

The role of references and the elusive nature of the chemical bond

Chemical bonding theory is of utmost importance to chemistry, and a standard paradigm in which quantum mechanical interference drives the kinetic energy lowering of two approaching fragments has emerged. Here we report that both internal and external reference biases remain in this model, leaving plenty of unexplored territory. We show how the former biases affect the notion of wavefunction interference, which is purportedly recognized as the most basic bonding mechanism. The latter influence how bonding models are chosen. We demonstrate that the use of real space analyses are as reference-less as possible, advocating for their use. Delocalisation emerges as the reference-less equivalent to interference and the ultimate root of bonding. Atoms (or fragments) in molecules should be understood as a statistical mixture of components differing in electron number, spin, etc.

The theory of chemical bonding relies on arbitrary references. Here the authors report a fundamental study on the chemical bond showing that considering the binding fragments as objects in real space enables to eliminate inherent biases.

Electronic densities and valence bond wave functions

Valence bond (VB) wave functions are studied from the density point of view. The density is plotted as a difference with the quasi-state built on the same orbitals. The densities of the components of the VB wave function are also shown. The breathing orbital effect leads to small modifications of the density. It is shown that while the densities of ionic and covalent components are the same, their coupling ends-up in modifications of the electronic density.

Orbital contraction and covalent bonding

According to Ruedenberg's classic treatise on the theory of chemical bonding [K. Ruedenberg, Rev. Mod. Phys. 34, 326-376 (1962)], orbital contraction is an integral consequence of covalent bonding. While the concept is clear, its quantification by quantum chemical calculations is not straightforward, except for the simplest of molecules, such as H₂ ⁺ and H₂. This paper proposes a new, yet simple, approach to the problem, utilizing the modified atomic orbital (MAO) method of Ehrhardt and Ahlrichs [Theor. Chim. Acta 68, 231 (1985)]. Through the use of MAOs, which are an atom-centered minimal basis formed from the molecular and atomic density operators, the wave functions of the species of interest are re-expanded, allowing the computation of the kinetic energy (and any other expectation value) of free and bonded fragments. Thus, it is possible to quantify the intra- and interfragment changes in kinetic energy, i.e., the effects of contraction. Computations are reported for a number of diatomic molecules H₂, Li₂, B₂, C₂, N₂, O₂, F₂, CO, P₂, and Cl₂ and the polyatomics CH₃-CH₃, CH₃-SiH₃, CH₃-OH, and C₂H₅-C₂H₅ (where the single bonds between the heavy atoms are studied) as well as dimers of He, Ne, Ar, and the archetypal ionic molecule NaCl. In all cases, it is found that the formation of a covalent bond is accompanied by an increase in the intra-fragment kinetic energy, an indication of orbital contraction and/or deformation.

Understanding the Uniqueness of 2p Elements in Periodic Tables

The Periodic Table, and the unique chemical behavior of the first element in a column (group), were discovered simultaneously one and a half centuries ago. Half a century ago, this unique chemistry of the light homologs was correlated to the then available atomic orbital (AO) radii. The radially nodeless 1s, 2p, 3d, 4f valence AOs are particularly compact. The similarity of r(2s)≈r(2p) leads to pronounced sp-hybrid bonding of the light p-block elements, whereas the heavier p elements with n≥3 exhibit r(ns) ≪ r(np) of approximately -20 to -30 %. Herein, a comprehensive physical explanation is presented in terms of kinetic radial and angular, as well as potential nuclear-attraction and electron-screening effects. For hydrogen-like atoms and all inner shells of the heavy atoms, r(2s) ≫ r(2p) by +20 to +30 %, whereas r(3s)≳r(3p)≳r(3d), since in Coulomb potentials radial motion is more radial orbital expanding than angular motion. However, the screening of nuclear attraction by inner core shells is more efficient for s than for p valence shells. The uniqueness of the 2p AO is explained by this differential shielding. Thereby, the present work paves the way for future physical explanations of the 3d, 4f, and 5g cases.

The Valence Orbitals of the Alkaline-Earth Atoms

Quantum chemical calculations of the alkaline-earth oxides, imides and dihydrides of the alkaline-earth atoms (Ae=Be, Mg, Ca, Sr, Ba) and the calcium cluster Ca6 H9 [N(SiMe3 )2 ]3 (pmdta)3 (pmdta=N,N,N',N'',N''-pentamethyldiethylenetriamine) have been carried out by using density functional theory. Analysis of the electronic structures by charge and energy partitioning methods suggests that the valence orbitals of the lighter atoms Be and Mg are the (n)s and (n)p orbitals. In contrast, the valence orbitals of the heavier atoms Ca, Sr and Ba comprise the (n)s and (n-1)d orbitals. The alkaline-earth metals Be and Mg build covalent bonds like typical main-group elements, whereas Ca, Sr and Ba covalently bind like transition metals. The results not only shed new light on the covalent bonds of the heavier alkaline-earth metals, but are also very important for understanding and designing experimental studies.

Clarifying the quantum mechanical origin of the covalent chemical bond

Lowering of the electron kinetic energy (KE) upon initial encounter of radical fragments has long been cited as the primary origin of the covalent chemical bond based on Ruedenberg's pioneering analysis of H[Formula: see text] and H₂ and presumed generalization to other bonds. This work reports KE changes during the initial encounter corresponding to bond formation for a range of different bonds; the results demand a re-evaluation of the role of the KE. Bonds between heavier elements, such as H₃C-CH₃, F-F, H₃C-OH, H₃C-SiH₃, and F-SiF₃ behave in the opposite way to H[Formula: see text] and H₂, with KE often increasing on bringing radical fragments together (though the total energy change is substantially stabilizing). The origin of this difference is Pauli repulsion between the electrons forming the bond and core electrons. These results highlight the fundamental role of constructive quantum interference (or resonance) as the origin of chemical bonding. Differences between the interfering states distinguish one type of bond from another.

The Basics of Covalent Bonding in Terms of Energy and Dynamics

We address the paradoxical fact that the concept of a covalent bond, a cornerstone of chemistry which is well resolved computationally by the methods of quantum chemistry, is still the subject of debate, disagreement, and ignorance with respect to its physical origin. Our aim here is to unify two seemingly different explanations: one in terms of energy, the other dynamics. We summarize the mechanistic bonding models and the debate over the last 100 years, with specific applications to the simplest molecules: H2+ and H2. In particular, we focus on the bonding analysis of Hellmann (1933) that was brought into modern form by Ruedenberg (from 1962 on). We and many others have helped verify the validity of the Hellmann-Ruedenberg proposal that a decrease in kinetic energy associated with interatomic delocalization of electron motion is the key to covalent bonding but contrary views, confusion or lack of understanding still abound. In order to resolve this impasse we show that quantum mechanics affords us a complementary dynamical perspective on the bonding mechanism, which agrees with that of Hellmann and Ruedenberg, while providing a direct and unifying view of atomic reactivity, molecule formation and the basic role of the kinetic energy, as well as the important but secondary role of electrostatics, in covalent bonding.

Adhesion, forces and the stability of interfaces

Weak molecular interactions (WMI) are responsible for processes such as physisorption; they are essential for the structure and stability of interfaces, and for bulk properties of liquids and molecular crystals. The dispersion interaction is one of the four basic interactions types – electrostatics, induction, dispersion and exchange repulsion – of which all WMIs are composed. The fact that each class of basic interactions covers a wide range explains the large variety of WMIs. To some of them, special names are assigned, such as hydrogen bonding or hydrophobic interactions. In chemistry, these WMIs are frequently used as if they were basic interaction types. For a long time, dispersion was largely ignored in chemistry, attractive intermolecular interactions were nearly exclusively attributed to electrostatic interactions. We discuss the importance of dispersion interactions for the stabilization in systems that are traditionally explained in terms of the “special interactions” mentioned above. System stabilization can be explained by using interaction energies, or by attractive forces between the interacting subsystems; in the case of stabilizing WMIs, one frequently speaks of adhesion energies and adhesive forces. We show that the description of system stability using maximum adhesive forces and the description using adhesion energies are not equivalent. The systems discussed are polyaromatic molecules adsorbed to graphene and carbon nanotubes; dimers of alcohols and amines; cellulose crystals; and alcohols adsorbed onto cellulose surfaces.

The Virial Theorem and Covalent Bonding

A long-held view of the origin of covalent binding is based on the notion that electrostatic forces determine the stability of a system of charged particles and that, therefore, potential energy changes drive the stabilization of molecules. A key argument advanced for this conjecture is the rigorous validity of the virial theorem. Rigorous in-depth analyses have however shown that the energy lowering of covalent bonding is due to the wave mechanical drive of electrons to lower their kinetic energy through expansion. Since the virial theorem applies only to systems with Coulombic interaction potentials, its relevance as a foundation of the electrostatic view is tested here by calculations on analogues of the molecules H₂⁺ and H₂, where all 1/ r interaction potentials are replaced by Gaussian-type potentials that yield one-electron "atoms" with realistic stability ranges. The virial theorem does not hold in these systems, but covalent bonds are found to form nonetheless, and the wave mechanical bonding analysis yields analogous results as in the case of the Coulombic potentials. Notably, the key driving feature is again the electron delocalization that lowers the interatomic kinetic energy component. A detailed discussion of the role of the virial theorem in the context of covalent binding is given.

Covalent Bonding and Charge Shift Bonds: Comment on "The Carbon-Nitrogen Bonds in Ammonium Compounds Are Charge Shift Bonds"

The paper by Gershoni-Poranne and Chen (R. Gershoni-Poranne, P. Chen, Chem. Eur. J. 2017, 23, 4659) gives an incorrect definition of covalent bonding. Furthermore, the assignment of so-called charge shift bonds in ammonium compounds has no physical foundation and is conceptually redundant.

Covalent Bonding in the Hydrogen Molecule

This work addresses the continuing disagreement between two schools of thought concerning the mechanism of covalent bonding. According to Hellmann, Ruedenberg, and Kutzelnigg, covalent bonding is a quantum mechanical phenomenon whereby lowering of the kinetic energy associated with electron sharing, i.e., delocalization, is the key stabilization mechanism. The opposing view of Slater, Feynman, and Bader has maintained that the source of stabilization is electrostatic potential energy lowering due to electron density redistribution to binding regions between nuclei. Following our study of H₂⁺ we present an analogous detailed study of H₂ where bonding involves an electron pair with repulsion and correlation playing a significant role in its properties. We use a range of different computational approaches to study and reveal the relevant contributions to bonding as seen in the electron density and corresponding kinetic and potential energy distributions. The energetics associated with the more complex electronic structure of H₂, when examined in detail, clearly agrees with the analysis of Ruedenberg; i.e., covalent bonding is due to a decrease in the interatomic kinetic energy resulting from electronic delocalization. Our results support the view that covalent bonding is a quantum dynamical phenomenon requiring a properly quantized kinetic energy to be used in its description.

Chemical bonding: the orthogonal valence-bond view

Chemical bonding is the stabilization of a molecular system by charge- and spin-reorganization processes in chemical reactions. These processes are said to be local, because the number of atoms involved is very small. With multi-configurational self-consistent field (MCSCF) wave functions, these processes can be calculated, but the local information is hidden by the delocalized molecular orbitals (MO) used to construct the wave functions. The transformation of such wave functions into valence bond (VB) wave functions, which are based on localized orbitals, reveals the hidden information; this transformation is called a VB reading of MCSCF wave functions. The two-electron VB wave functions describing the Lewis electron pair that connects two atoms are frequently called covalent or neutral, suggesting that these wave functions describe an electronic situation where two electrons are never located at the same atom; such electronic situations and the wave functions describing them are called ionic. When the distance between two atoms decreases, however, every covalent VB wave function composed of non-orthogonal atomic orbitals changes its character from neutral to ionic. However, this change in the character of conventional VB wave functions is hidden by its mathematical form. Orthogonal VB wave functions composed of orthonormalized orbitals never change their character. When localized fragment orbitals are used instead of atomic orbitals, one can decide which local information is revealed and which remains hidden. In this paper, we analyze four chemical reactions by transforming the MCSCF wave functions into orthogonal VB wave functions; we show how the reactions are influenced by changing the atoms involved or by changing their local symmetry. Using orthogonal instead of non-orthogonal orbitals is not just a technical issue; it also changes the interpretation, revealing the properties of wave functions that remain otherwise undetected.

Covalent bonds are created by the drive of electron waves to lower their kinetic energy through expansion

An analysis based on the variation principle shows that in the molecules H2 (+), H2, B2, C2, N2, O2, F2, covalent bonding is driven by the attenuation of the kinetic energy that results from the delocalization of the electronic wave function. For molecular geometries around the equilibrium distance, two features of the wave function contribute to this delocalization: (i) Superposition of atomic orbitals extends the electronic wave function from one atom to two or more atoms; (ii) intra-atomic contraction of the atomic orbitals further increases the inter-atomic delocalization. The inter-atomic kinetic energy lowering that (perhaps counter-intuitively) is a consequence of the intra-atomic contractions drives these contractions (which per se would increase the energy). Since the contractions necessarily encompass both, the intra-atomic kinetic and potential energy changes (which add to a positive total), the fact that the intra-atomic potential energy change renders the total potential binding energy negative does not alter the fact that it is the kinetic delocalization energy that drives the bond formation.

The source of chemical bonding

Developments in the application of quantum mechanics to the understanding of the chemical bond are traced with a view to examining the evolving conception of the covalent bond. Beginning with the first quantum mechanical resolution of the apparent paradox in Lewis's conception of a shared electron pair bond by Heitler and London, the ensuing account takes up the challenge molecular orbital theory seemed to pose to the classical conception of the bond. We will see that the threat of delocalisation can be overstated, although it is questionable whether this should be seen as reinstating the issue of the existence of the chemical bond. More salient are some recent developments in a longstanding discussion of how to understand the causal aspects of the bonding interaction--the nature of the force involved in the covalent link--which are taken up in the latter part of the paper.

100th anniversary of Bohr's model of the atom

In the fall of 1913 Niels Bohr formulated his atomic models at the age of 27. This Essay traces Bohr's fundamental reasoning regarding atomic structure and spectra, the periodic table of the elements, and chemical bonding. His enduring insights and superseded suppositions are also discussed.

Interference energy in C-H and C-C bonds of saturated hydrocarbons: dependence on the type of chain and relationship to bond dissociation energy

Interference energy for C-H and C-C bonds of a set of saturated hydrocarbons is calculated by the generalized product function energy partitioning (GPF-EP) method in order to investigate its sensitivity to the type of chain and also its contribution to the bond dissociation energy. All GPF groups corresponding to chemical bonds are calculated by use of GVB-PP wave functions to ensure the correct description of bond dissociation. The results show that the interference energies are practically the same for all the C-H bonds, presenting only small variations (0.5 kcal.mol(-1)) due to the structural changes in going from linear to branched and cyclic chains. A similar trend is verified for the C-C bonds, the sole exception being the cyclopropane molecule, for which only the C-C bond exhibits a more significant variation. On the other hand, although the interference energy is quantitatively the most important contribution to the bond dissociation energy (DE), one cannot predict DE only from the bond interference energy. Differences in the dissociation energies of C-C and C-H bonds due to structural changes in the saturated hydrocarbons can be mainly attributed to quasi-classical effects.

Chemical potential of molecules contrasted to averaged atomic electronegativities: alarming differences and their theoretical rationalization

We present the first large-scale empirical examination of the relation of molecular chemical potentials, μ(0)(mol) = -½(I(0) + A(0))(mol), to the geometric mean (GM) of atomic electronegativities, <χ(0)(at)>(GM) = <½(I(0) + A(0))(at)>(GM), and demonstrate that μ(0)(mol) ≠ -<χ(0)(at)>(GM). Out of 210 molecular μ(0)(mol)values considered more than 150 are not even in the range min{μ(0)(at)} < μ(0)(mol) < max{μ(0)(at)} spanned by the μ(0)(at) = -χ(0)(at) of the constituent atoms. Thus the chemical potentials of the large majority of our molecules cannot be obtained by any electronegativity equalization scheme, including the "geometric mean equalization principle", ½(I(0) + A(0))(mol) = <½(I(0) + A(0))(at)>(GM). For this equation the root-mean-square of relative errors amounts to SE = 71%. Our results are at strong variance with Sanderson's electronegativity equalization principle and present a challenge to some popular practice in conceptual density functional theory (DFT). The influences of the "external" potential and charge dependent covalent and ionic binding contributions are discussed and provide the theoretical rationalization for the empirical facts. Support is given to the warnings by Hinze, Bader et al., Allen, and Politzer et al. that equating the chemical potential to the negative of electronegativity may lead to misconceptions.