Unicorns in the world of chemical bonding models

All-body concept and quantified limits of cooperativity and related effects in homodromic cyclic water clusters from a molecular-wide and electron density-based approach

We strongly advocate distinguishing cooperativity from cooperativity-induced effects. From the MOWeD-based approach, the origin of all-body cooperativity is synonymous with physics- and quantum-based processes of electron (e) delocalization throughout water clusters. To this effect, over 10 atom-pairs contribute to the total e-density at a BCP(H,O) between water molecules in a tetramer. Intermolecular all-body e-delocalization, that is, cooperativity, is an energy-minimizing process that fully explains non-additive increase in stability of a water molecule in clusters with an increase in their size. A non-linear change in cooperativity and cooperativity-induced effects, such as (i) structural (e.g., a change in d(O,O)) or topological intra- and intermolecular properties in water clusters (e.g., electron density or potential energy density at bond critical points) is theoretically reproduced by the proposed expression. It predicted the limiting value of delocalized electrons by a H2O molecule in homodromic cyclic clusters to be 1.58e. O-atoms provide the vast majority of electrons that "travel throughout a cluster predominantly on a privileged exchange quantum density highway" (⋅⋅⋅O-H⋅⋅⋅O-H⋅⋅⋅O-H⋅⋅⋅) using Bader's classical bond paths as density bridges linking water molecules. There are, however, additional electron exchange channels that are not seen on molecular graphs as bond paths. A 3D visual representation of the "privileged" and "additional" exchange channels as well as detailed intra- and inter-molecular patterns of e-sharing and (de)localizing is presented. The energy stabilizing contribution made by three O-atoms of neighboring water molecules was found to be large (-597 kcal/mol in cyclic hexamer) and 5 times more significant than that of a classical O-H⋅⋅⋅O intermolecular H-bond.

Can Aromaticity Be Evaluated Using Atomic Partitions Based on the Hilbert-Space?

Aromaticity is a fundamental concept in chemistry that explains the stability and reactivity of many compounds by identifying atoms within a molecule that form an aromatic ring. Reliable aromaticity indices focus on electron delocalization and depend on atomic partitions, which give rise to the concept of an atom-in-the-molecule (AIM). Real-space atomic partitions present two important drawbacks: a high computational cost and numerical errors, limiting some aromaticity measures to medium-sized molecules with rings up to 12 atoms. This restriction hinders the study of large conjugated systems like porphyrins and nanorings. On the other hand, traditional Hilbert-space schemes are free of the latter limitations but can be unreliable for the large basis sets required in modern computational chemistry. This paper explores AIMs based on three robust Hilbert-space partitions - meta-Löwdin, Natural Atomic Orbitals (NAO), and Intrinsic Atomic Orbitals (IAO) - which combine the advantages of real-space partitions without their disadvantages. These partitions can effectively replace real-space AIMs for evaluating the aromatic character. For the first time, we report multicenter index (MCI) and Iring values for large rings and introduce ESIpy, an open-source Python code for aromaticity analysis in large conjugated rings.

Electronic Current Density Induced by Uniform Magnetic Fields in Clarenes

Some planar and non-planar clarenes have been studied using maps of magnetically induced quantum-mechanical current density and tools from differential topology to assess their magnetic response in connection with recent results by Du and Wang. Bond current strengths have been computed to estimate quantitative measures. Isosurfaces of the divergence of induced Lorentz force density have been shown to provide useful additional criteria, especially in the case of non-planar clarenes.

σ-Arsolido complexes

The σ-stannyl complexes [M(SnnBu3)(CO)n(η5-C5H5)] (n = 3, M = Mo, W; n = 2, M = Fe) serve as mild reagents for the installation of σ-arsolyl ligands in transmetallation reactions with As-chloro-arsoles ClAsC4R4 (R = Me, Ph) to afford [M(σ-AsC4R4)(CO)n(η5-C5H5)]. The reaction of [Cr(SnnBu3)(CO)3(η5-C5H5)] with ClAsC4Ph4 most likely proceeds in a similar manner but is immediately followed by rapid formation of (AsC4Ph4)2 and [Cr2(CO)6(η5-C5H5)2]. The reaction of [Mo(SnnBu3)(CO)3(η5-C5H5)] with ClAsC4(SiMe3)-2,5-Me2-3,4 is accompanied by monodesilylation to afford [Mo{σ-AsC4(SiMe3)-2-Me2-3,4}(CO)3(η5-C5H5)]. The slow reaction of [Fe(SnnBu3)(CO)2(η5-C5H5)] with ClAsC4Me4 produced only traces of [Fe(σ-AsC4Me4)(CO)2(η5-C5H5)] due to competition with the Diels-Alder type dimerisation of the haloarsole. Although attempts to decarbonylate the σ-arsolyl complexes were unsuccessful, computational analysis suggests that the trigonal 'XL' arsolenium coordination mode is viable.

The Separability Problem in Molecular Quantum Systems: Information-Theoretic Framework for Atoms in Molecules

Even though molecules are fundamentally quantum entities, the concept of a molecule retains certain classical attributes concerning its constituents. This includes the empirical separability of a molecule into its three-dimensional, rigid structure in Euclidean space, a framework often obtained through experimental methods like X-Ray crystallography. In this work, we delve into the mathematical implications of partitioning a molecule into its constituent parts using the widely recognized Atoms-In-Molecules (AIM) schemes, aiming to establish their validity within the framework of Information Theory concepts. We have uncovered information-theoretical justifications for employing some of the most prevalent AIM schemes in the field of Chemistry, including Hirshfeld (stockholder partitioning), Bader's (topological dissection), and the quantum approach (Hilbert's space definition). In the first approach we have applied the generalized principle of minimum relative entropy derived from the Sharma-Mittal two-parameter functional, avoiding the need for an arbitrary selection of reference promolecular atoms. Within the ambit of topological-information partitioning, we have demonstrated that the Fisher information of Bader's atoms conform to a comprehensive theory based on the Principle of Extreme Physical Information avoiding the need of employing the Schwinger's principle, which has been proven to be problematic. For the quantum approach we have presented information-theoretic justifications for conducting Löwdin symmetric transformations on the density matrix to form atomic Hilbert spaces generating orthonormal atomic orbitals with maximum occupancy for a given wavefunction.

The aromatic nature of auracycles and diauracycles based on calculated ring-current strengths

We have calculated the magnetically induced current density susceptibility for gold-containing organometallic molecular rings using the gauge-including magnetically induced currents (GIMIC) method. The aromatic nature has been determined by calculating the strength of the magnetically induced ring current susceptibility, which is often called ring current. To our knowledge, we show here for the first time that gold-containing organometallic rings may be aromatic or antiaromatic sustaining ring currents in the presence of an external magnetic field. The calculated aromatic character of the rings agrees with the aromatic nature one expects when using Hückel's aromaticity rules. The studied auracycles and diauracycles with 4n electrons in the conjugated orbitals generally sustain a weak paratropic ring current, whereas those having 4n + 2 electrons in the conjugated orbitals sustain a diatropic ring current that is almost as strong as that of benzene. The number of electrons are obtained by assuming that each C, N and Au atom of the ring contribute one electron, and a H atom connected to a N atom in the ring increases the number of electrons by one. An electron-attracting ligand at Au removes one electron from the ring. Formation of a short Au-Au bonding diauracycles reduces the number of electrons in the ring by two.

The σ+π dual aromaticity of typical bi-tetrazole ring molecule TKX-50

Two complexes of dihydroxylammonium 5,5'-bistetrazole-1,1'-diolate (TKX-50) were employed to evaluate the aromaticity of their tetrazole rings via deep analysis such as the electronic structure, the ZZ component of the natural chemical shielding tensor (NICSZZ) and component orbitals, localized orbital locator purely contributed by σ-orbitals (LOL-σ) and localized orbital locator purely contributed by π-orbitals (LOL-π), the anisotropy of the induced current density (AICD) and the ZZ component of iso-chemical shielding surface (ICSSZZ) of these tetrazole rings thereof. The conclusion shows: that all tetrazole rings and bi-tetrazole rings in complexes have strong σ and a comparable strength π double aromaticity; all these magnetic shields almost symmetrically increase from the central axis to the tetrazole ring atoms; tetrazole rings in complex II show a little stronger dual aromaticity than that in complex I mainly due to the different orientation of the fragment 2 encompassing two hydroxylamine groups resulting in different effects on the contributions of σ orbitals and π orbitals to total aromaticity of tetrazole rings thereof; the difference in aromaticity is fundamentally caused by the atoms O with stronger electron-withdrawing than atom N in fragment 2 interact with bi-tetrazole ring through O in complex I but through N in complex II.

Aromaticity: Quo Vadis

Aromaticity is one of the most deeply rooted concepts in chemistry. But why, if two-thirds of existing compounds can be classified as aromatic, is there no consensus on what aromaticity is? σ-, π-, δ-, spherical, Möbius, or all-metal aromaticity… why are so many attributes needed to specify a property? Is aromaticity a dubious concept? This perspective aims to reflect where the aromaticity community is and where it is going.

Unicorns, Rhinoceroses and Chemical Bonds

The nascent field of computationally aided molecular design will be built around the ability to make computation useful to synthetic chemists who draw on their empirically based chemical intuition to synthesize new and useful molecules. This fact poses a dilemma, as much of existing chemical intuition is framed in the language of chemical bonds, which are pictured as possessing physical properties. Unfortunately, it has been posited that calculating these bond properties is impossible because chemical bonds do not exist. For much of the computationalchemistry community, bonds are seen as mythical-the unicorns of the chemical world. Here, we show that this is not the case. Using the same formalism and concepts that illuminated the atoms in molecules, we shine light on the bonds that connect them. The real space analogue of the chemical bond becomes the bond bundle in an extended quantum theory of atoms in molecules (QTAIM). We show that bond bundles possess all the properties typically associated with chemical bonds, including an energy and electron count. In addition, bond bundles are characterized by a number of nontraditional attributes, including, significantly, a boundary. We show, with examples drawn from solid state and molecular chemistry, that the calculated properties of bond bundles are consistent with those that nourish chemical intuition. We go further, however, and show that bond bundles provide new and quantifiable insights into the structure and properties of molecules and materials.

Gold(I)···Lanthanide(III) Bonds in Discrete Heterobimetallic Compounds: A Combined Computational and Topological Study

The chemical nature of the ligand-unsupported gold(I)-lanthanide(III) bond in the proposed [LnIII(η5-Cp)2][AuIPh2] (Ln-Au; LnIII = LaIII, EuIII, or LuIII; Cp = cyclopentadienide; Ph = phenyl) models is examined from a theoretical viewpoint. The covalent bond-like Au-Ln distances (Au-La, 2.95 Å; Au-Eu, 2.85 Å; Au-Lu, 2.78 Å) result from a strong interaction between the oppositely charged fragments (ΔEintMP2 > 600 kJ mol-1), including the aforementioned metal-metal bond and additional LnIII-Cipso and C-H···π interactions. The Au-Ln bond has been characterized as a chemical bond rather than a strong metallophilic interaction with the aid of energy decomposition analysis, interaction region indicator, and quantum theory of atoms in molecules topological tools. The chemical nature of the Au-Ln bond cannot be fully ascribed to a covalent or an ionic model; an intermediate situation or a charge shift bond is proposed. The [AuIPh2]- anion has also been identified as a suitable lanthanide(III) emission sensitizer for La-Au and Lu-Au.

Demarcating Noncovalent and Covalent Bond Territories: Imine-CO2 Complexes and Cooperative CO2 Capture

Chemical bond territory is rich with covalently bonded molecules wherein a strong bond is formed by equal or unequal sharing of a quantum of electrons. The noncovalent version of the bonding scenarios expands the chemical bonding territory to a weak domain wherein the interplay of electrostatic and π-effects, dipole-dipole, dipole-induced dipole, and induced dipole-induced dipole interactions, and hydrophobic effects occur. Here we study both the covalent and noncovalent interactive behavior of cyclic and acyclic imine-based functional molecules (XN) with CO₂. All parent XN systems preferred the formation of noncovalent (nc) complex XN···CO₂, while more saturated such systems (XN') produced both nc and covalent (c) complexes XN'⁺-(CO₂)^-. In all such cases, crossover from an nc to c complex is clearly demarcated with the identification of a transition state (ts). The complexes XN'···CO₂ and XN'⁺-(CO₂)^- are bond stretch isomers, and they define the weak and strong bonding territories, respectively, while the ts appears as the demarcation point of the two territories. Cluster formation of XN with CO₂ reinforces the interaction between them, and all become covalent clusters of general formula (XN⁺-(CO₂)^-)_n. The positive cooperativity associated with the NH···OC hydrogen bond formation between any two XN'⁺-(CO₂)^- units strengthened the N-C coordinate covalent bond and led to massive stabilization of the cluster. For instance, the stabilizing interaction between the XN unit with CO₂ is increased from 2-7 kcal/mol range in a monomer complex to 14-31 kcal/mol range for the octamer cluster (XN'⁺-(CO₂)^-)₈. The cooperativity effect compensates for the large reduction in the entropy of cluster formation. Several imine systems showed the exergonic formation of the cluster and are predicted as potential candidates for CO₂ capture and conversion.

Clarifying notes on the bonding analysis adopted by the energy decomposition analysis

We discuss the fundamental aspects of the EDA-NOCV method and address some critical comments that have been made recently. The EDA-NOCV method unlike most other methods focuses on the process of bond formation between the interacting species and not just only on the analysis of the finally formed bond. This is demonstrated using LiF as an example. There is a difference between the interactions between the initial species which form the bond and are also the final product of bond cleavage, and the interactions between the fragments in the eventually formed molecule. The flexibility of the method allows the choice of the interacting fragments which helps to identify the charge and electron configuration of the fragments which describe the bond. This is very helpful in cases where the bond may be described with several Lewis structures. We reject the idea that it would be a disadvantage to have "bond path functions" as the energy components in the EDA, which actually indicate the variability of the method. The bonding analysis in a different sequence of the bond formation gives important results for the various questions that can be asked. This is demonstrated by using CH₂, CO₂ and the formation of a guanine quartet as examples. The fact that a bond is always defined by the bound molecule, the fragments, and their states is universal and deeply physical, as we show here again for various examples. The results of the EDA-NOCV method are in full accordance with the physical mechanism of the chemical bond as revealed by Ruedenberg.

Anatomy of Classical Boron-Boron Bonding: Overlap and sp Dissonance

Classical bonding is predominantly understood using the insipid spⁿ hybridization for σ-bonds as well as π bonds and their delocalized variants. Because hybridization ignores intricate differences in the energy and size of valence atomic orbitals, its naïve application to classically bonded boron atoms leads to numerous surprises in bond strengths, frontier MOs/bands, and even geometry. Here we show that the sp dissonance caused by size mismatch between the valence s and p orbitals of boron plays a crucial role in its bonding, subtly distinct from that of carbon and silicon. Unlike the heavier p block elements, boron prefers to actively engage its compact 2s orbitals in bonding. This leads to the overreach of p-p σ-type overlap that reduces its magnitude in the entire B─B bonding range. Consequently, the π-type overlap remains substantial, making its electronic structure visibly distinct in saturated and unsaturated regimes. The deltahedral frameworks offer a compromise by breaking this symmetry-enforced dichotomy of classical σ- and π-type bonding and following alternate electron counts that suit the electron deficiency of the boron. The pathological anatomy of classical B─B σ-bonding also explains the origins of puzzling metallic character and disorder in their classical boride networks even with ideal electron count, unlike deltahedral borides. The implications of sp dissonance are illustrated in classical boron networks of various hybridizations, explaining the unusual preference for unique sp³ lattice with strained four-membered rings in CrB₄, origins of observed σ holes in MgB₂ that lead to its superconducting nature, and the absence of Peierls distortion in LiB.

Yet another perspective on hole interactions

Hole interactions are known by different names depending on the key atom of the bond (halogen bond, chalcogen bond, hydrogen bond, etc.), and the geometry of the interaction (σ if in line, π if perpendicular to the Lewis acid plane). However, its origin starts with the creation of a Lewis acid by an underlying covalent bond, which forms an electrostatic depletion and a virtual antibonding orbital, which can create non-covalent interactions with Lewis bases. In this (maybe subjective) perspective, we will claim that hole interactions must be defined via the molecular orbital origin of the molecule. Under this premise we can better explore the richness of such bonding patterns. For that, we will study old, recent and new systems, trying to pinpoint some misinterpretations that are often associated with them. We will use as exemplars the triel bonds, a couple of metal complexes, a discussion on convergent σ-holes, and many cases of anti-electrostatic hole interactions.

Pauling's Conceptions of Hybridization and Resonance in Modern Quantum Chemistry

We employ the tools of natural bond orbital (NBO) and natural resonance theory (NRT) analysis to demonstrate the robustness, consistency, and accuracy with which Linus Pauling's qualitative conceptions of directional hybridization and resonance delocalization are manifested in all known variants of modern computational quantum chemistry methodology.

Energetic Descriptors of Steric Hindrance in Real Space: An Improved IQA Picture*

Steric hindrance (SH) plays a central role in the modern chemical narrative, lying at the core of chemical intuition. As it however happens with many successful chemical concepts, SH lacks an underlying physically sound root, and multiple mutually inconsistent approximations have been devised to relate this fuzzy concept to computationally derivable descriptors. We here argue that being SH related to spatial as well as energetic features of interacting systems, SH can be properly handled if we chose a real space energetic stance like the Interacting Quantum Atoms (IQA) approach. Drawing on previous work by Popelier and coworkers (ChemistryOpen 8, 560, 2019) we build an energetic estimator of SH, referred to as E_ST . We show that the rise in the self-energy of a fragment that accompanies steric congestion is a faithful proxy for the chemist's SH concept if we remove the effect of charge transfer. This can be done rigorously, and the E_ST here defined provides correct sterics even for hydrogen atoms, where the plain use of deformation energies leads to non-chemical results. The applicability of E_ST is validated in several chemical scenarios, going from atomic compressions to archetypal S_N2 reactions. E_ST is shown to be a robust steric hindrance descriptor.

Assessing the Orbital Contribution in the "Spodium Bond" by Natural Orbital for Chemical Valence-Charge Displacement Analysis

The term "spodium bond" (SpB) has been recently proposed to describe the noncoordinative interaction that can be established between a polarized group 12 metal and a mild Lewis base (LB). Most of the systems showing short metal-donor distances compatible with SpB are characterized by the coexistence of multiple weak interactions, including hydrogen and halogen bonding, making the assessment of real importance of SpB difficult. Here, we show that the relative importance of each contribution can be probed by dissecting the orbital component of the interaction through the extended transition state-natural orbital for chemical valence-charge displacement analysis (ETS-NOCV-CD). The latter gives useful information about relative energies and electrons involved, for model systems ([(thiourea)₂MX₂]···LB; M = Zn, Cd, and Hg; X = Cl and I; and LB = CH₂S, CH₂O, CH₃CN, and CO) and a variety of structures extracted from experimentally characterized adducts, allowing us to demonstrate the lack of a direct correlation between a favorable metal-base distance and the presence of an orbital contribution for the SpB.

Experimental Quantum Chemistry: A Hammett-inspired Fingerprinting of Substituent Effects

The quantum mechanically calculable Q descriptor is shown to be a potent quantifier of chemical reactivity in complex molecules - it shows a strong correlation to experimentally derived field effects in non-aromatic substrates and Hammett σm and σp parameters. Models for predicting substituent effects from Q are presented and applied, including on the elusive pentazolyl substituent. The presented approach enables fast computational estimation of substituent effects, and, in extension, medium-throughput screening of molecules and compound design. An experimental dataset is suggested as a candidate benchmark for aiding the general development and comparison of electronic structure analyses. It is here used to evaluate the experimental quantum chemistry (EQC) framework for chemical bonding analysis in larger molecules.

Generalization of Intrinsic Orbitals to Kramers-Paired Quaternion Spinors, Molecular Fragments, and Valence Virtual Spinors

Localization of molecular orbitals finds its importance in the representation of chemical bonding (and antibonding) and in the local correlation treatments beyond mean-field approximation. In this paper, we generalize the intrinsic atomic and bonding orbitals [G. Knizia, J. Chem. Theory Comput. 2013, 9, 11, 4834-4843] to relativistic applications using complex and quaternion spinors, as well as to molecular fragments instead of atomic fragments only. By performing a singular value decomposition, we show how localized valence virtual orbitals can be expressed on this intrinsic minimal basis. We demonstrate our method on systems of increasing complexity, starting from simple cases such as benzene, acrylic acid, and ferrocene molecules, and then demonstrate the use of molecular fragments and inclusion of relativistic effects for complexes containing heavy elements such as tellurium, iridium, and astatine. The aforementioned scheme is implemented into a standalone program interfaced with several different quantum chemistry packages.

Beyond the Classical Electron-Sharing and Dative Bond Picture: Case of the Spin-Polarized Bond

Chemical bonds are traditionally assigned as electron-sharing or donor-acceptor/dative. External criteria such as the nature of the dissociation process, energy partitioning schemes, or quantum chemical topology are invoked to assess the bonding situation. However, for systems with marked multi-reference character, this binary categorization might not be precise enough to render the bonding properties. A third scenario can be foreseen: spin polarized bonds. To illustrate this, the case of a NaBH3 - cluster is presented. According to the analysis NaBH3 - exhibits a strong diradical character and cannot be classified as either electron-sharing or a dative bond. Elaborated upon are the common problems of popular bonding descriptions. Additionally, a simple model, based on the bond order and local spin indicators, which discriminates between all three bonding situations, is provided.

Metallaaromatic Chemistry: History and Development

Since the prediction of the existence of metallabenzenes in 1979, metallaaromatic chemistry has developed rapidly, due to its importance in both experimental and theoretical fields. Now six major types of metallaromatic compounds, metallabenzenes, metallabenzynes, heterometallaaromatics, dianion metalloles, metallapentalenes and metallapentalynes (also termed carbolongs), and spiro metalloles, have been reported and extensively studied. Their parent organic analogues may be aromatic, non-aromatic, or even anti-aromatic. These unique systems not only enrich the large family of aromatics, but they also broaden our understanding and extend the concept of aromaticity. This review provides a comprehensive overview of metallaaromatic chemistry. We have focused on not only the six major classes of metallaaromatics, including the main-group-metal-based metallaaromatics, but also other types, such as metallacyclobutadienes and metallacyclopropenes. The structures, synthetic methods, and reactivities are described, their applications are covered, and the challenges and future prospects of the area are discussed. The criteria commonly used to judge the aromaticity of metallaaromatics are presented.

NBO/NRT Two-State Theory of Bond-Shift Spectral Excitation

We show that natural bond orbital (NBO) and natural resonance theory (NRT) analysis methods provide both optimized Lewis-structural bonding descriptors for ground-state electronic properties as well as suitable building blocks for idealized "diabatic" two-state models of the associated spectroscopic excitations. Specifically, in the framework of single-determinant Hartree-Fock or density functional methods for a resonance-stabilized molecule or supramolecular complex, we employ NBO/NRT descriptors of the ground-state determinant to develop a qualitative picture of the associated charge-transfer excitation that dominates the valence region of the electronic spectrum. We illustrate the procedure for the elementary bond shifts of SN2-type halide exchange reaction as well as the more complex bond shifts in a series of conjugated cyanine dyes. In each case, we show how NBO-based descriptors of resonance-type 3-center, 4-electron (3c/4e) interactions provide simple estimates of spectroscopic excitation energy, bond orders, and other vibronic details of the excited-state PES that anticipate important features of the full multi-configuration description. The deep 3c/4e connections to measurable spectral properties also provide evidence for NBO-based estimates of ground-state donor-acceptor stabilization energies (sometimes criticized as "too large" compared to alternative analysis methods) that are also found to be of proper magnitude to provide useful estimates of excitation energies and structure-dependent spectral shifts.

Effect of substitution on the bonding in He dimer confined within dodecahedrane: A computational study

The effect of substitution in the dodecahedrane (C₂₀ H₂₀ ) cage on bonding in the confined He dimer is analyzed. The HeHe distances inside the halogenated dodecahedrane C₂₀ X₂₀ (X = FBr) cages are found to be less than half of that in the free He dimer. Comparing the equilibrium structure of He₂ @C₂₀ H₂₀ with He₂ @C₂₀ X₂₀ at ωB97XD/def2-TZVPP level, it is found that the He-He distances are relatively larger in the latter cases indicating the influence of halogen groups on the interaction between the cage and the trapped He pair. The viability of the He₂ @C₂₀ X₂₀ complexes is reflected in the presence of a very high activation energy barrier against the thermochemically feasible dissociation process producing free He₂ and C₂₀ X₂₀ . Quantum theory of atoms in molecules (QTAIM) approach reveals a partial covalent interaction between He pair.

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.

The Atomic Partial Charges Arboretum: Trying to See the Forest for the Trees

Atomic partial charges are among the most commonly used interpretive tools in quantum chemistry. Dozens of different 'population analyses' are in use, which are best seen as proxies (indirect gauges) rather than measurements of a 'general ionicity'. For the GMTKN55 benchmark of nearly 2,500 main-group molecules, which span a broad swathe of chemical space, some two dozen different charge distributions were evaluated at the PBE0 level near the 1-particle basis set limit. The correlation matrix between the different charge distributions exhibits a block structure; blocking is, broadly speaking, by charge distribution class. A principal component analysis on the entire dataset suggests that nearly all variation can be accounted for by just two 'principal components of ionicity': one has all the distributions going in sync, while the second corresponds mainly to Bader QTAIM vs. all others. A weaker third component corresponds to electrostatic charge models in opposition to the orbital-based ones. The single charge distributions that have the greatest statistical similarity to the first principal component are iterated Hirshfeld (Hirshfeld-I) and a minimal-basis projected modification of Bickelhaupt charges. If three individual variables, rather than three principal components, are to be identified that contain most of the information in the whole dataset, one representative for each of the three classes of Corminboeuf et al. is needed: one based on partitioning of the density (such as QTAIM), a second based on orbital partitioning (such as NPA), and a third based on the molecular electrostatic potential (such as HLY or CHELPG).

Revealing the Real Size of a Porphyrin Molecule with Quantum Confinement Probing via Temperature-Dependent Photoluminescence Spectroscopy

The confinement energy of electrons in an aromatic molecule was studied by indirect and direct methods, namely, temperature-dependent photoluminescence (TDPL) spectroscopy and scanning tunneling microscopy (STM). We observed a decrease in the tetraphenylporphyrin (H₂TPP) PL intensity with increasing temperature. The increase in temperature provides kinetic energy for the electrons to overcome the confinement energy barrier, making recombination via nonradiative pathways more favorable. The results of fitting the integrated TDPL intensity with a modified Arrhenius equation suggest two confinement energy values. We propose that these energy values are related to the size of the delocalized electron cloud along the plane and thickness of the H₂TPP ring. These values quantitatively express an abstract form of the size of the aromatic ring system. These results are in good agreement with the topography images of single H₂TPP molecules and monolayer H₂TPP obtained by a direct probing method using STM. These results are also supported by the porphyrin ring orientation relative to the excited crystal face during the TDPL measurements.

Coupling of two curved polyaromatic radical-anions: stabilization of dimers by counterions

In this study, a comprehensive theoretical investigation of both kinetic and thermodynamic stabilities was performed for dimeric dianionic systems (C₂₀H₁₀)₂^2- and (C₂₈H₁₄)₂^2-, neutralized by two alkali metal cations. The influence of the counterions was of primary interest. The impact of the additional/spectator ligand(s) was elucidated by considering adducts with four molecules of diglyme or two molecules of 18-crown-6 ether. Importantly, both types of systems - in the form of contact-ion pair (CIP) and solvent-separated ion pair (SSIP) - were considered. The SSIP set was augmented by the adduct, in which the dimeric dianionic species were neutralized with purely organic cations N(CH₃)₄⁺ and P(CH₃)₄⁺. Detailed analysis of the bonding revealed that the presence of the counterions made these systems thermodynamically stable. This finding is in sharp contrast with results obtained for isolated (PAH)₂^2- systems, which were previously found to be thermodynamically unstable, but kinetically persistent. The introduction of the alkali metal cations to the system significantly increases the ionic term (ΔE_elstat), whereas the repulsive ΔE_Pauli one was found to be substantially reduced. Considering that the orbital component (ΔE_orb) exhibited only a moderate decrease and the preparation energy (ΔE_prep) showed no changes, the above-mentioned changes in ΔE_elstat and ΔE_Pauli provided a clear explanation for the increase of the thermodynamic stability of the target species. Importantly, a clear correlation between the size of the alkali metal cation and stability of the target dimeric product was established. Thermodynamic stability of the system rises with a decrease in the size of M⁺ due to enlargement of the ΔE_orb. Evaluated energy barriers (as spin-crossing points between singlet and triplet energy surfaces) were found to be equal to +15.85 kcal mol^-1 and +18.5 kcal mol^-1 for [(Cs⁺)₂{(C₂₀H₁₀)₂^2-}] and [(Cs⁺)₂{(C₂₈H₁₄)₂^2-}], respectively, which is substantially higher than those calculated for isolated (PAH)₂^2- systems (+10.00 kcal mol^-1 for (C₂₀H₁₀)₂^2- and +12.35 kcal mol^-1 for (C₂₈H₁₄)₂^2-). Thus, this study identified the presence of counterions as the key factor, which have a dramatic influence on the thermodynamic and kinetic stabilities of the aimed dianionic dimeric systems, which are formed by two curved polyaromatic monoanion-radicals.

Where Is the Fluoro Wall?: A Quantum Chemical Investigation

Despite their isoelectronic properties, fluoro and oxo ligands exhibit completely different chemical behavior. Formally speaking, the first is known to exclusively form single bonds, while the latter is generally observed to form double (or even triple) bonds. The biggest difference, however, lies in what is known among inorganic chemists as the Oxo Wall: the fact that six-coordinate tetragonal transition metal oxo complexes are not observed beyond group 7 elements. While the Oxo Wall was explained a few decades ago, some questions regarding the nature of the Oxo Wall remain unanswered. For example, why do group 8 oxo complexes with high oxidation states not violate the Oxo Wall? Moreover, why are transition metal fluoro complexes observed through the whole transition metal series? In order to understand how the small difference between these two isoelectronic ligands can give rise to such different chemical behaviors, we conducted an extensive computational analysis of the geometric and electronic properties of model fluoro and oxo complexes with metals around the Oxo Wall. Among many insights into the details of the Oxo Wall, we mostly learned that the oxygen 2p orbitals are prone to meaningfully interact with transition metal d orbitals, because they match not only spatially but also energetically, while for fluorine the p orbital energies are lower to an extent that interaction with transition metal d orbitals is much reduced. This in turn implies that in those instances where the metal d orbitals principally accessible for interaction are occupied, the oxygen 2p orbitals are too exposed to be stable.

A computational analysis was conducted in order to understand how a subtle difference between the isoelectronic fluoro and oxo ligands can give rise to such different chemical behaviors, including the presence of an Oxo Wall but not a Fluoro Wall. It seems that the small energetic difference of the ligands’ 2p orbitals is enough to inhibit meaningful interaction with transition metal d orbitals for the fluoro ligand, which eventually leads to the observed chemical behaviors.

Homolytic Versus Heterolytic Bond Breaking in Functionalized [R-C20 H10 ]+ Systems

The comprehensive theoretical investigation of stability of functionalized corannulene cations [R-C₂₀ H₁₀ ]⁺ with respect to two alternative bond-breaking mechanisms, namely, homolytic or radical ([R-C₂₀ H₁₀ ]⁺  → R^•  + C₂₀ H₁₀ ^+• ) and heterolytic or cationic ([R-C₂₀ H₁₀ ]⁺  → R⁺  + C₂₀ H₁₀ ), was accomplished. The special focus was on the influence of the nature of R-group on the energetics of the bond cleavage. Detailed study of energetics of both mechanisms has revealed that the systems with small alkyl groups such as methyl tend to undergo bond breaking in accordance with homolytic mechanism. Subsequent elongation of the chain of the R-group resulted in shifting the paradigm, making heterolytic path more energetically favorable. Subsequent analysis of different components of the bonding between R-group and corannulene polyaromatic core helped to shed light on trends observed. In both mechanisms, the covalent contribution was found to be dominating, whereas ionic part contributes ~25-27%. Two leading components of ΔE_orb , C₂₀ H₁₀ → R and R → C₂₀ H₁₀ , were identified with NOCV-EDA approach. While the homolytic pathway is best described as R → C₂₀ H₁₀ process, the heterolytic mechanism shows domination of the C₂₀ H₁₀ → R term. Surprisingly, the preparation energy (ΔE_prep ) was identified as a key player in stability tendencies found. In other words, the relative stability of corresponding molecular fragments (here R-groups as the corannulene fragment remains the same for all systems) in their cationic or radical forms determine the preference given to a specific bond breaking path and, as consequence, the total stability of target functionalized cations. These conclusions were further confirmed by extending a set of R-groups to conjugated (allyl, phenyl), bulky (iPr, tBu), β-silyl (CH₂ SiH₃ , CH₂ SiMe₃ ), and benzyl (CH₂ Ph) groups. © 2019 Wiley Periodicals, Inc.

Charge Displacement Analysis-A Tool to Theoretically Characterize the Charge Transfer Contribution of Halogen Bonds

Theoretical bonding analysis is of prime importance for the deep understanding of the various chemical interactions, covalent or not. Among the various methods that have been developed in the last decades, the analysis of the Charge Displacement function (CD) demonstrated to be useful to reveal the charge transfer effects in many contexts, from weak hydrogen bonds, to the characterization of σ hole interactions, as halogen, chalcogen and pnictogen bonding or even in the decomposition of the metal-ligand bond. Quite often, the CD analysis has also been coupled with experimental techniques, in order to give a complete description of the system under study. In this review, we focus on the use of CD analysis on halogen bonded systems, describing the most relevant literature examples about gas phase and condensed phase systems. Chemical insights will be drawn about the nature of halogen bond, its cooperativity and its influence on metal-ligand bond components.

Bonding rearrangements in organometallic reactions: from orbitals to curly arrows

Reaction mechanisms include a description of both the nuclear and electronic rearrangements along the energetically most favourable pathway. Extracting the nuclear rearrangements from the outcome of quantum chemical calculations is straightforward, while it is more intricate for the electronic rearrangements. This is particularly true when changes in the bonding pattern are of interest, just as in the arrow-pushing formalism used in chemical schemes. Here, we report on our use of a simple and highly visual procedure to recover the bonding rearrangements along a reaction pathway from DFT calculations and to draw curly arrows. We show that the procedure allows us discern among mechanistic proposals in the context of organometallic reactions featuring the forming and breaking of bonds.

Tungsten-Ligand Bond Strengths for 2p Elements Including σ- and π-Bond Strength Components, A Density Functional Theory and ab Initio Study

Three W^VI crystal structures with multifarious metal-ligand bond types are used to theoretically predict homolytic metal-element bond enthalpies with 11 popular DFT functionals, MP2 wave function methods, and four common valence basis set/pseudopotentials in order to evaluate the accuracy and precision of the resultant bond enthalpy data. To our knowledge, for the first time, estimates of component metal-ligand σ- and π-bond strengths are computed. The WE (E = C, N, O) bond enthalpies have the consistent trend σ > second π > first π. In contrast, the element-element BDE trend for the 2p homologues is second π > first π > σ for nitrogen and oxygen, and σ > first π > second π for carbon. These differences may underpin the differences in stability trends and thus reactivity behavior for metal-element multiple bonds as compared to the element-element multiple bonds, and metal-element triple bonds versus their corresponding double bonded counterparts. For example, Odom et al. show that MeI nucleophilically attacks at the imide (M═N) rather than the nitride (M ≡ N) ligand; the relative π-bond strengths derived herein provide a thermodynamic rationalization for this site preference. In this study, it is deduced from the calculated thermodynamics that the W-oxo ligand is more congruous with a triple bond than a double bond, consistent with the bonding model set forth in the seminal 1961 Ballhausen-Gray paper.

Unveiling the role of intra and interatomic interactions in the energetics of reaction schemes: a quantum chemical topology analysis

In this work we present a detailed analysis of selected reaction schemes in terms of the atomic components of the electronic energy defined by the quantum theory of atoms in molecules and the interacting quantum atoms method. The aim is to provide an interpretation tool for the energy change involved in a chemical reaction by means of the atomic and interaction contributions to the energies of the molecules involved. Ring strain in cyclic alkanes, the resonance energy of aromatic and antiaromatic molecules, local aromaticity in polycyclic aromatic hydrocarbons, intermolecular bonding in hydrogen fluoride clusters, and hydration of d-block metal dications were selected for the study. It was found that in addition to the changes in the strong C-C interactions in the carbon skeleton of the organic molecular rings, other contributions not usually considered to be important such as those between C and H atoms (either bonded or not) need to be considered in order to account for the net energy changes. The analysis unveils the role of the ionic and covalent contributions to the hydrogen bonding in HF clusters and the energetic origin and extent of cooperative effects involved. Moreover, the "double-hump" behavior observed for the hydration energy trend of [M(H2O)6]2+ complexes is explained in terms of the deformation energy of the metal cation and the increasingly covalent metal-water interactions. In addition, proper comparisons with the description provided by other methodologies are briefly discussed. The topological approach proposed in this contribution proves to be useful for the description of energy changes of apposite reaction schemes in chemically meaningful terms.

Comprehensive Theoretical Study of Interactions between Ag+ and Polycyclic Aromatic Hydrocarbons

The first comprehensive and systematic theoretical exploration of the bonding nature and energetics of the interactions between Ag(I) cation and a wide set of π-ligands was accomplished. This set ranges from simple ethylene and aromatic benzene to planar and curved polyaromatic molecules and to closed-cage C₆₀ -fullerene. Simultaneous application of two energy decomposition schemes based on different ideas, namely, NBO-NEDA and EDA-NOCV, allowed shedding light on the nature of the bonding and its energetics. Importantly, our results unambiguously indicate that reliable results can be obtained only if using more than one theoretical approach. All methods clearly revealed the importance and even domination of the ionic contribution of the bonding in all adducts, except for those of C₆₀ -fullerene, in which the covalent component was found to be the largest. Subsequent decomposition of the orbital term onto components showed that it consists of two major parts: (i) ligand-to-metal (π(C=C)→s(Ag), L→M) and (ii) metal-to-ligand (M→L) terms, with significant domination of the former. Interestingly, while the L→M component is essentially the same for all systems considered, the nature of the M→L one depends on the coordination site of the polycyclic aromatic hydrocarbons (PAH). In most of adducts, the M→L can be described as d_xy (Ag)→π^* (C=C) donation, whereas for systems [Ag-spoke-C₁₂ H₈ ]⁺ and [Ag-spoke-C₂₀ H₁₀ ]⁺ it corresponds to the d_z ² (Ag)→π^* (C=C) type of interaction. As a result, the coordination mode in such complexes is switched from η² -type to η¹ . Thus, the nature of the bonding, its energetics and even coordination mode in adducts of unsaturated hydrocarbons with late transition metal cations should be considered as a function of many components, which primarily includes the topology and aromaticity of the (poly)aromatic molecules.