Chemical Bonding and Bonding Models of Main-Group Compounds

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Cite This: Chem. Rev. XXXX, XXX, XXX−XXX

Chemical Bonding and Bonding Models of Main-Group Compounds Lili Zhao,† Sudip Pan,† Nicole Holzmann,‡ Peter Schwerdtfeger,*,§ and Gernot Frenking*,†,∥,⊥ †

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Institute of Advanced Synthesis, School of Chemistry and Molecular Engineering, Jiangsu National Synergetic Innovation Center for Advanced Materials, Nanjing Tech University, Nanjing 211816, China ‡ Scientific Computing Department, STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, United Kingdom § The New Zealand Institute for Advanced Study, Massey University (Albany), 0632 Auckland, New Zealand ∥ Fachbereich Chemie, Philipps-Universität Marburg, Hans-Meerwein-Strasse, D-35043 Marburg, Germany ⊥ Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain ABSTRACT: The focus of this review is the presentation of the most important aspects of chemical bonding in molecules of the main group atoms according to the current state of knowledge. Special attention is given to the difference between the physical mechanism of covalent bond formation and its description with chemical bonding models, which are often confused. This is partly due to historical reasons, since until the development of quantum theory there was no physical basis for understanding the chemical bond. In the absence of such a basis, chemists developed heuristic models that proved extremely valuable for understanding and predicting experimental studies. The great success of these simple models and the associated rules led to the fact that the model conceptions were regarded as real images of physical reality. The complicated world of quantum theory, which eludes human imagination, made it difficult to link heuristic models of chemical bonding with quantum chemical knowledge. In the early days of quantum chemistry, some suggestions were made which have since proved untenable. In recent decades, there has been a stormy development of quantum chemical methods, which are not limited to the quantitative accuracy of the calculated properties. Also, methods have been developed where the experimentally developed models can be quantitatively expressed and visually represented using mathematically well-defined terms that are derived from quantum chemical calculations. The calculated numbers may however not be measurable values. Nevertheless, as orientation data for the interpretation and classification of experimental findings as well as a guideline for new experiments, they form a coordinate system that defines the multidimensional world of chemistry, which corresponds to the Hilbert space formalism of physics. The nonmeasurability of model values is not a weakness of chemistry but a characteristic by which the infinite complexity of the material world becomes scientifically accessible and very useful for chemical research. This review examines the basis of the commonly used quantum chemical methods for calculating molecules and for analyzing their electronic structure. The bonding situation in selected representative molecules of main-group atoms is discussed. The results are compared with textbook knowledge of common chemistry.

CONTENTS 1. Introduction 2. The Physical Nature of the Chemical Bond 3. Historical Development and Present Situation of Bonding Models for Main-Group Compounds: The Lewis Paradigm 4. Quantum Chemical Methods for Calculating Molecular Structures and Properties 4.1. Molecular Orbital (MO) Theory 4.2. Density Functional Theory (DFT) 4.3. Valence-Bond (VB) Theory 5. Quantum Chemical Methods for Analyzing the Chemical Bond in Molecules 5.1. Natural Bond Orbital Method (NBO) 5.2. Quantum Theory of Atoms in Molecules (QTAIM) 5.3. Energy Decomposition Analysis and Natural Orbitals for Chemical Valence (EDA-NOCV) © XXXX American Chemical Society

6. Physical Reality and Chemical Bonding Models 7. Selected Examples of Chemical Bonds in MainGroup Compounds 7.1. H2+, H2, Li2+, Li2 7.2. H2, N2, CO, and BF 7.3. First Octal-Row Sweep Li2−F2 and Related Molecules 7.3.1. Li2, Be2 and Related Molecules 7.3.2. B2 and Related Molecules 7.3.3. C2 and Related Molecules 7.3.4. N2 and Related Molecules 7.3.5. O2, F2 and Related Molecules 7.4. Chemical Bonding in Heavier Main-Group Compounds

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Special Issue: Frontiers in Main Group Chemistry

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Received: December 2, 2018

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DOI: 10.1021/acs.chemrev.8b00722 Chem. Rev. XXXX, XXX, XXX−XXX

Chemical Reviews 7.4.1. Hypervalent Molecules: Valence, Orbital Symmetry, and 3-Center, 4-Electron Bonding 7.4.2. Hypervalent Molecules: SF6 and PF5 7.4.3. Multiple Bonds of Heavy Diatomic Molecules Na2−Cl2 and N2−Bi2 and Related Molecules 7.4.4. Multiple Bonds of Heavy Main-Group Atoms: N2/N4 vs P2/P4 7.4.5. Multiple Bonds of Heavy Main-Group Atoms: X2CCX2 vs X2EEX2 (E = Si − Pb; X = H, F) 7.4.6. Multiple Bonds of Heavy Main-Group Atoms: HCCH vs HEEH (E = Si − Pb) 8. Dative Bonding in Main-Group Compounds 9. The Octet Rule and the Atomic Valence Space of Main-Group Atoms 10. Relativistic Effects in Main-Group Compounds 11. Concluding Remarks Author Information Corresponding Authors ORCID Notes Biographies Acknowledgments References

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the term “covalent bonding” at a time when modern quantum theory was not developed yet. Over the course of time it was modified and refined like a neural network, using the increase of experimental observations as feedstock for improvement. The electron-pair model of a covalent bond may be considered as a stroke of genius, where the yet unknown quantum chemical nature of the chemical bond is miraculously converted to and hidden in a very simple representation. It is unsurpassed in its convenience to depict and represent molecular structures and reactivities. The model has been somewhat modified over the years and the rules for using Lewis structures slightly changed in the course of time, but the essential features remain the same until today. This is astonishing, because the origin of covalent bonding is a quantum theoretical effect. This was shown for the first time by Heitler and London in 1927,8 when they analyzed the chemical bond in H2 using the newly developed quantum theory by Heisenberg9 and Schrödinger.10 Linus Pauling was the first who presented a bridge between the heuristic electron-pair bonding model of Lewis and quantum theory in his epochal book “The Nature of the Chemical Bond”, which quickly became the reference standard in the community for describing chemical bonding.11 This was done at a time when powerful computers were not yet available and quantum chemistry was in its infant stage. Later studies showed that the nature of the chemical bond is far more complicated than initially thought and that the connection between the Lewis model and the physical nature of chemical bonding is quite intricate. This does not come as a surprise. Considering the enormous complexity of the physical universe and the rather small numbers of chemical elements and particles12 from which it is constructed, it requires a sheer infinite variety of chemical bonding to create such a richness of the material phenomena in our universe including the complex biochemical nature of animate beings. A fundamental approach of chemistry deals with the classification and arrangement of the molecular structures and reactivities using simple models, rules, and guidelines, which serve as classification schemes for the vast amount of compounds and reactions. Chemistry as it is currently applied and understood can be considered as the science of fuzzy concepts, because many models are poorly defined and somewhat arbitrarily used and, yet, they have proven to be very helpful as an orientation for experimental studies. Chemical models were termed as “unicorns”,13 because every chemist seems to know what they mean although a precise definition is often not available. For a scientist who is trained with exact physical quantities, chemical models may appear as unscientific chimera. But if we only would focus on measurable quantities (so-called observables in quantum mechanical terms), disregarding the many beautiful concepts developed over the past centuries, chemistry would be deprived of its inner beauty and many major discoveries would not have been made. The term “chemical bond” is not an observable in a strict quantum mechanical sense but refers to the extensive symbolism used in chemistry to represent the attractive force between atoms or molecules. A puristic viewpoint ignores their enormous usefulness for breaking down the ever increasing complexity and richness of chemical findings into well-ordered classification schemes. Chemical bonding models are not right or wrong, they are more or less usef ul. The usefulness of models comes from their help as an ordering scheme of experimental observations and their use as guidance for future experiments and interpretations. But it is a grave mistake to confuse a bonding model with the

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1. INTRODUCTION In the year 2000, one of the present authors wrote a review for the centennial issue of this journal entitled “The Nature of the Bonding in Transition-Metal Compounds”.1 The introduction describes the peculiar dichotomy of using heuristic bonding models along with quantum chemical calculations for the description of molecular structures, which is characteristic for present-day chemical research. This situation holds even more for main-group compounds, where the depiction of the bonding situation in terms of “resonating” Lewis structures and molecular orbitals (MOs) in combination with quantum chemical calculations along with charge- and energy-partitioning methods is frequently presented side-by-side. The connection between the experimental observations and numerical results of such calculations is usually guided by the attempt to grasp the complicate numerical information, which is provided by quantum chemical calculations, with a simple bonding model. This can clearly be considered a Hercules (and sometimes Sisyphus) task, which tries to connect two extremes: The quantum theoretical description of the electronic structure, which is governed by the Schrö dinger equation in the nonrelativistic case and the Dirac equation in the relativistic case, is given in mathematical terms that are elusive objects for the human imagination trained with classical objects. It is a genuine chemical approach to illustrate this with symbols and figures that are as simple as possible, appealing, and useful to chemists. The most important model for chemical bonding is clearly the heuristic assignment of an electron pair to a chemical bond first suggested by Gilbert Lewis in 1916.2 It was developed by inspection of chemical structures and stabilities and by evaluating similarities mainly of main-group compounds of the first octal row of the periodic system. The model was further elaborated by Langmuir in 1919−1921,3−6 who introduced the octet rule7 along with the 18- and 32-electron rules and coined B


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wave functions, which may either have a positive (+) or a negative (−) sign. The alternative signs yield two possible options for the new wave function, which describe the bonding and antibonding combinations of the interacting electrons. The positive interference leads to an accumulation of electronic charge in the interatomic region whereas the negative interference induces charge depletion. It is only at the level of the wave function that the information, which is provided by the leading sign, becomes operative. The electronic charge is derived from squaring the wave function, which annihilates the sign information given by the wave function. Mulliken,17 Hund,18 Lennard-Jones,19 and Hü ckel20 recognized the gain in information that is provided by the sign of the wave functions. Many years later Fukui21 developed the frontier molecular orbital (FMO) theory and Woodward and Hoffmann22 introduced the orbital symmetry rules for pericyclic reactions, which are based on the information that is provided by the orbitals. Nevertheless, chemists still have difficulties in accepting the more fundamental nature of the wave function as the origin of the chemical bond compared to a simple particle character of the electrons. The most important conclusion that stems from the Heitler− London work is the finding that the physical origin of the covalent bond is not due to the formation of an electron pair. Covalent bonding comes from the interference of the wave functions of the interacting species.23 Chemical bonding occurs already when only one electron occupies the new wave function, such as in H2+. The second electron usually strengthens the chemical bond further such as in H2, but there are molecules where the second electron even weakens the bond. This is the case in Li2, which has lower bond dissociation energy than Li2+.24 The frequent occurrence of molecules with an even number of electrons is due to the Pauli principle, which allows a maximum of two electrons occupying the same spatial orbital. This is a quantum theoretical postulate originating from the more fundamental spin statistics theorem, which requests that electrons are different by at least one quantum number, but it is not the physical origin per se of the chemical bond. The finding that electron pairing is not the physical origin of the chemical bond becomes obvious by the stability of numerous molecules, which have unpaired electrons such as dioxygen O2 and the large number of paramagnetic transition metal complexes. The specific e−e pairing interaction is repulsive in most cases, and it can be energetically more favorable that the two electrons occupy spatially different orbitals, as for example in the ground state of O2. The stabilization of the second electron in a doubly occupied MO stems mainly from the interference of the wave function. We wish to emphasize that Pauli repulsion does not introduce a new type of physical force, which may be overcome by other forces. All interatomic forces, which are relevant for chemical bonding, come from Coulombic interactions. However, the mathematical description of Coulomb forces between atoms requires quantum theoretical postulates. The Pauli (exchange) forces are due to the requirement that the wave function must be antisymmetric, which requests that two electron may not have the same quantum numbers. The large difference between the classical and the quantum theoretical description of an electron comes to the fore when calculating the electron−electron repulsion as a function of the e−e distance either way. Figure 1 shows the 1/r12 curve for the repulsion between two point charges q calculated following the Coulomb law for classical particles (eq 1) and the repulsion

physical nature of the interatomic interactions that give rise to what we call a chemical bond. This has been the source of numerous controversies and misunderstandings in the past. This review article has two goals. One goal is to present the current understanding of the physical nature of covalent bonding in main-group compounds. This shall be done without digging deeply into the mathematical details of the quantum chemical description of the interatomic interactions. This is not an easy task, because the quantum theoretical description of the chemical bond is quite complicated and has many facets, which cannot all be accounted for in the present review article. We strive for a concise presentation of the most important results and conclusions that came out of the numerous studies, which were done since the ground-breaking work by Heitler and London. For a more detailed discussion of the quantum theoretical nature of the chemical bond, we refer to pertinent review articles and monographs.14−16 The second goal of this account concerns the use of modern chemical bonding models and the Lewis electron-pair approximation for describing molecular structures and reactivities. The central aspect of this work aims at bridging bonding models with the quantum theoretical nature of the chemical bond. The focus is the connection between the results of standard quantum chemical calculations, which are mostly based on molecular orbital (MO) theory or density functional theory (DFT), and the presentation of molecules with a bonding model. There are various methods available that convert the results of theoretical calculations into models that aim at describing the essential chemical features of the investigated species. The most important methods will be critically discussed, and the results for archetypical molecules will be presented. It is not possible to review all kinds of chemical bonds of main-group compounds in this work. The present account is restricted to selected classes of archetypical molecules and bonds, which represent not all but the most important types of chemical bonds in molecules of the sp-block elements of the periodic table.

2. THE PHYSICAL NATURE OF THE CHEMICAL BOND The physical origin of the covalent bond was for a long time a mystery not only for chemists but for science in general. It was clear that from the four elementary forces in physics only electrostatic forces could account for the strong interatomic interactions that yield a chemical bond. But the attraction takes place between neutral species, and the application of the classical Coulomb law does not give results that agree with experimental findings. Gilbert Lewis knew about the problem when he suggested in 1916 that the chemical bond should be identified with an electron pair. He speculated, “Electric forces between particles which are very close together do not obey the simple law of inverse squares which holds at greater distances”.2 When the puzzle about the physical origin of the chemical bond was finally solved by Heitler and London in 1927,8 who employed the newly developed quantum theory by Heisenberg9 and Schrödinger10 to the interactions between two hydrogen atoms as well as between two helium atoms, it came as a shock not only to chemists but to natural sciences altogether. The solution was in conflict with all ideas that had been envisaged by scientists for explaining the chemical bond. According to the quantum theoretical approach by Heitler and London, the electrostatic interactions between atoms have to be described in terms of mathematical functions describing quantum objects rather than classical particles. Chemical bonding comes from the interference of spatially extended C


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Figure 2. Copy of the figure by Heitler and London from their 1927 paper which shows the potential energy curves of dihydrogen calculated classically (E11) and using the wave function Ψ (Eα and Eβ). Reprinted with permission from ref 8. Copyright 1927 Springer Nature.

Figure 1. Calculated interaction energies between two charged particles as a function of their distance r12. Repulsive interactions between two electrons calculated classically ΔEelstat(classical) = q1q2/r12 (dashed line); quasiclassical repulsion between two electrons in 1s orbitals ΔEelstat = ∫ ρ1ρ2/r12 dτ1 dτ2 (dotted line); exchange repulsion between two electrons with the same spin in 1s orbitals ΔEPauli (solid line). Reprinted with permission from ref 28. Copyright 2006 John Wiley and Sons.

Figure 2 shows the original curves by Heitler and London8 for the interaction between two hydrogen atoms. The energy curve E11, which is based on the quasi-classical Coulomb interactions between the superimposed atomic charge distributions, has only a weak energy minimum of ∼0.5 eV at a rather long distance of ∼1.7 atomic units, while the curve Eα for the attractive interactions that are calculated using the wave functions has a much deeper potential of ∼2.5 eV at a more reasonable bond distance of 1.5 atomic units. The curve Eβ refers to the repulsive interactions between two hydrogen atoms where the electrons have the same spin. The occurrence of two energy curves for the interactions between the hydrogen atoms is a quantum theoretical phenomenon that cannot be explained by classical physics. This must be kept in mind when bonding models like the Lewis electron pair, which are the results of human interpretation of experiment and thus based on classical objects, are used to explain chemical findings. The essential finding of the Heitler−London work, which features the very core of covalent bonding, might be presented in a simplified way. We skip some mathematical details, in order to extract the central message. Scheme 1 shows the quasiclassical approach (eqs 3−5) for chemical bonding in H2, which employs the electronic charge distribution ρ(H) of the hydrogen atoms as starting point for the interatomic interactions. The calculation using Coulomb’s law results in E11, which has only a shallow energy minimum (Figure 2). The bottom part of Scheme 1 delineates the quantum theoretical approach, where the wave function Ψ(H) is used as elementary physical quantity rather than ρ(H) for the description of the electron. Since ρ(H) is the square29 of Ψ(H) (eq 6), both ±Ψ(H) may be used for the generation of the wave function Ψ(H2) (eq 7) in a simple oneparticle model. The electronic charge distribution ρ(H2) is then

between the electronic charges ρ of two electrons with opposite spin in H2 obtained by the square of the wave function Ψ (eqs 2a): ΔEelstat(classical) = q1q2 /r12 ρ = |Ψ|2 ΔEelstat(qc) =

(1) (2a)

∫ dτ1dτ2ρ1ρ2/r12

(2b)

The two curves are nearly indistinguishable at distances >2 Å where the wave function overlap is negligible. At shorter distances the electrostatic repulsion ΔEelstat(qc) (qc = quasi classical) given by eq 2b is much less than the classical repulsion between two point charges and approaches a finite value at r = 0 whereas the classical eq 1 leads to an infinite Coulomb repulsion. The third curve in Figure 1 depicts the Pauli (exchange) repulsion between two electrons having the same spin. It becomes obvious that the onset of the Pauli repulsion, depending on the overlap of the wave functions, takes place at shorter distances but has a much steeper slope than the Coulomb repulsion. The Pauli repulsion is the physical origin of the loosely defined concept of steric repulsion between bulky substituents. It also provides a quantum theoretical basis for the VSEPR (Valence Shell Electron-Pair Repulsion) model of Nyholm and Gillespie.25−27 The total Coulombic interactions between neutral species are usually attractive, because of the electrostatic attraction between atomic nuclei and electrons.28 This is discussed in more detail below. D


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alteration of E, V, and T for H2 as a function of the H−H distance.

Scheme 1. Schematic Description of the Interatomic Interactions between Two Hydrogen Atoms Using (a) a Classical Approach and (b) a Quantum Theoretical Approach

Figure 3. Energy curves of the total energy E, potential energy V, and kinetic energy T as a function of the H−H distance. Reprinted with permission from ref 42. Copyright 2019 Springer Nature.

When the hydrogen atoms approach each other, the kinetic energy along with the total energy is at first lowered whereas the potential energy rises. This is because the electronic charge experiences a larger volume in the overlapping valence space. The potential energy rises, because the “electron distance” to the nuclei is longer than in the atoms. This is the region where the actual bond formation mainly takes place. At shorter distances, the effect of electronic charge depletion at the nuclei dominates leading to shrinkage of the effective atomic radii. As a result, the depleted charge in the core region experiences a stronger attraction by the nucleus, which lowers V and increases T. The latter process eventually overcompensates the reverse change of T and V in the bonding region and yields an overall increase of kinetic energy and lowering of potential energy at the equilibrium distance. The conclusion is that the driving force for the accumulation of electronic charge in the bonding region associated with the formation of the chemical bond is the lowering of the kinetic energy. This was shown for the first time by Hellmann.30 After some controversy,14 it was finally confirmed by Ruedenberg.31−36 The finding that the origin of covalent bonding is due to the interference of the wave functions of the bonding atoms is not restricted to dihydrogen, and it is not confined to nonpolar bonds. It will be shown below that H2 is in many ways untypical for covalent bonding, but the physical origin for covalence is the same for all atoms. Ruedenberg and others have demonstrated this in detailed theoretical studies.31−34 Electrostatic attraction, exchange (Pauli) repulsion, and further factors contribute to the intricate combination that yields a chemical bond, but the interference of the wave function remains at the origin of covalent bonding. The following conclusions arise from this section: • The physical origin of the covalent bond between atoms is the interference of the atomic wave functions but not the formation of an electron pair. Chemical bonding comes f rom electron-sharing but not f rom electron-pairing between atoms.

given by squaring |Ψ(H2)|2 (eq 8). The associated binomial in eq 9 gives the term ±2[c1c2 Ψ(Ha)Ψ(Hb)], which describes the interference of the atomic wave functions that leads to attraction (+ sign) or repulsion (− sign). The respective attractive and repulsive energy terms are shown in eqs 10a and 10b. This result comes from a quantum theoretical description of the interatomic interactions leading to the curves Eα and Eβ in Figure 2. The interference term Ψ(Ha)Ψ(Hb) is a pure quantum theoretical expression, which does not have a classical analogue. Chemical bonding is therefore a quantum chemical effect and can be seen as caused by the interference of atomic/fragmental wave functions. Chemical bonding is due to electron-sharing but not due to electron pairing. There is a second aspect of covalent bonding that is often misstated in chemistry textbooks and refers to the contribution of kinetic and potential energies to the energy lowering due to the formation of a chemical bond. According to the virial theorem at equilibrium distances E = 0.5 V = −T, it follows that the change in the total energy E has the same sign as the potential energy V, whereas the kinetic energy T has the reverse sign. For the bond energy ΔEb thus holds ΔEb = 0.5ΔVb = −ΔTb. This agrees with the popular statement that the bonding electrons, due to charge accumulation in the interatomic region, are energetically stabilized, because they are attracted by two nuclei. A close examination of the change in the electronic structure of two atoms along the bond path reveals that the formation of the chemical bond is rather complex and exhibits a paradoxical behavior concerning the contribution of the kinetic energy T and potential energy V. Figure 3 shows the E


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• The driving force for the formation of a chemical bond is the lowering of the kinetic energy of the accumulated charge in the bonding region. • The strength of the e−e repulsion as a function of the distance exhibits a drastically different behavior when electrons are considered as point charges compared to orbitals. The Pauli repulsion between electrons that possess the same spin is much stronger than Coulombic repulsion in regions where the orbital overlap becomes important. • The frequent appearance of an electron pair in chemical bonds is due to the Pauli principle. It may be reasonably used as a model for depicting molecular structures and chemical bonds (and particularly useful in explaining organic reaction mechanisms), but it must not be confused with the physical origin of the chemical bond.

electrostatic interaction between charged particles providing the mathematical framework for the puzzling behavior of matter at an atomic scale were introduced ten years later by Heisenberg9 and Schrödinger.10 The suggestions of Lewis were picked up by Langmuir, who elaborated on the electron-pair model in a series of four papers from 1919−1921.3−6 Langmuir coined the term “covalence” for the chemical bond, and he formulated the octet rule for maingroup atoms along with the related 18-electron rule for transition metals and the 32-electron rule for lanthanides. The latter rules were later rationalized by quantum theory with the occupation of the valence shells of the atoms. In 1923, Lewis published the book Valence and the Structure of Atoms and Molecules, where he extended his viewpoint on the electron-pair bonding model.39 The book was partly written as a response to the Langmuir papers, which received much attention and led some chemists to articulate the “Langmuir-Lewis model”.40 Lewis stressed the importance of the electron pairing for chemical bonding, saying “Whether we are dealing with organic or inorganic compounds, the chemical bond is always such a pair of chemical bonds.”39 The dogmatic adherence to the electron pair as the origin of the covalent bond and a deeply rooted aversion against quantum theory, which was at that time in an embryonic state, may have contributed to his unwillingness to open his mind for the upcoming important developments. In his book, he called quantum theory “the entering wedge of scientific bolshevism” but then he concedes “Quantum theory has been criticized for f urnishing no adequate mechanism, but presumably the root of our present problem lies deeper than this, and it is hardly likely that any mechanism based on our existing modes of thought will suf f ice for the explanation of the many new phenomena which the study of the atom is disclosing”.39 It appears that Lewis was neither willing nor prepared to accept the findings of quantum theory, which finally came in 1927. His latest study solely devoted to the chemical bond appeared in 1933,41 six years after the Heitler−London paper about the quantum chemical nature of covalent bonding. The very long paper, which does not have a single reference, reads like a defense of the electron-pair model in spite of its apparent deficiencies for describing some molecules. It seems, however, that Gilbert Lewis himself recognized the limitations of his approach. In the final statement of the 1923 book, he pointed out that his model may have to be modified in the future and that he wants “...to emphasize the necessity of maintaining an opening of mind; so that, when the solution of these problems, which now seem so baff ling, is ultimately of fered, its acceptance will not be retarded by the conventions and the inadequate mental abstractions of the past”.39 His request, which was perhaps inspired by recognizing his own limitations, has not lost its timeliness until today. An important contribution to the manifestation of the electron-pair bond as fundamental model for chemical bonding is the work of Linus Pauling. Unlike Gilbert Lewis, Pauling had an open mind for quantum theory, which he studied during his research stays in Europe in 1926 in the laboratories of Niels Bohr, Arnold Sommerfeld, and Erwin Schrödinger. Like Gilbert Lewis, he had an enormous knowledge of chemistry and he was also acquainted with the newly emerging technique of X-ray crystallography. Pauling blended the information from experimental chemistry, quantum theory, and structural chemistry into a far-sighted but also biased vision of molecular structure and bonding. He published the results in a series of papers that finally culminated in the epochal book The Nature of the Chemical Bond, which was first published in 1939.11 The book clearly shaped the vision of the chemical bond for generations of

3. HISTORICAL DEVELOPMENT AND PRESENT SITUATION OF BONDING MODELS FOR MAIN-GROUP COMPOUNDS: THE LEWIS PARADIGM The development of theoretical concepts and models to understand the physical world is a fascinating chapter in the history of mankind.37 Chemistry played a prominent role in the endeavor to explain observations in nature and in experiments and to formulate conclusions that may be used for future predictions. Based on the corpuscular theory of matter, which was not undisputed until the early 20th century, and following the introduction of the periodic system of elements by Mendeleev and Meyer, chemists designed symbols and formulas as synonyms for characterizing substances that are composed of atoms. The Scottish chemist Archibald Couper was the first who in 1858 suggested a dash connecting two atoms as a symbol for a chemical bond without any physical meaning assigned to it.38 The 19th century witnessed numerous suggestions for explaining the ever-increasing number of experimental observations in terms of bonding models. The wealth of information resulting from chemical research was used as source for a neural network to produce helpful concepts that were improved or dismissed by experimental results. Chemical research was additionally fuelled by the fact that its products became the foundation of the chemical industry that arose as an important new commercial branch. But the concepts and models were considered to have mere symbolic meaning rather than physical relevance. Three-dimensional structures were only later assigned to the chemical formula, most prominently by van’t Hoff, who established together with Le Bel the tetrahedral model for carbon compounds to explain the occurrence of enantiomers and the phenomenon of optical activity. However, the question about the physical foundation behind these models could not be answered. In 1916 Gilbert Lewis published his seminal work in which he suggested that the symbolic dash between two atoms should be identified with an electron pair.2 He also recognized that stable molecules have mostly atoms that possess eight electrons in their valence shell. Lewis introduced the cubic model for an atom where the electrons are located at the corners of the cube. He realized that the cubic atom may describe single and double bonds but not triple bonds, for which the tetrahedron was a better model. As mentioned above, Lewis was aware of the fact that his electron-pair model violates the laws of classical physics for electrostatic forces. The quantum theoretical laws for the F


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suggested an arrow as symbol for a dative bond A→B, as it is called now.45 Sidgwick also showed that the peculiar dipole moments of some molecules such as carbon monoxide, which has its negative end at carbon, can be explained with the direction of the donor−acceptor bond, which he sketched with the formula .46 This explanation was later supported by quantum chemical calculations.47 Pauling dismissed the concept of dative bonds, because he found it inconvenient,11 which may be the reason that the relevance of dative bonding in main-group compounds has not been recognized for a very long time. Haaland stressed the model of dative bonding in main-group compounds in a review article that appeared in 1989, which was, however, restricted to classical donor−acceptor complexes mainly of group 13−15 adducts.48 The full potential of the model of dative bonding in main-group chemistry came to the fore in systematic theoretical studies of low-valent compounds in the recent decade49 that led to the discovery of divalent carbon(0) compounds (“carbones”)50−54 and related systems, and contributed to the amazing development of subvalent maingroup chemistry.55,56 This is discussed in a dedicated section below. The renaissance of the model of dative bonding was not undisputed,57−59 but recent developments show that it is now generally accepted as a valid description for the bonding situation in main-group compounds that was already envisaged by Lewis and Sidgwick.49,60−62 Finally, we want to comment on molecules that are considered as species, which exhibit “unusual bonds”. The origin of the unusual bonding situation is not always the peculiar electronic structure but the difficulty to describe it with the Lewis electron-pair model. A striking example is dioxygen O2, which has an electronic triplet (X3Σg−) ground state. The bonding situation can easily be understood with an orbital interaction diagram. “Unusual bonding situations” may sometimes only indicate that the Lewis model is not suited for the electronic structure, which may easily be understood by another model such as the MO correlation diagram.63 The following conclusions arise from this section: • The shared electron-pair bonding model is unsurpassed in its simplicity for describing molecular structures, but it must not be confused with the physical origin of chemical bonding. The Lewis bonding model lacks the quantum theoretical information about the symmetry of the wave functions, which leads to problems particularly for molecules with delocalized bonds and hypervalent species.

chemists until today. Pauling succeeded in the difficult task to combine the classical viewpoint of the structure of molecules familiar to experimental chemists with the information that was provided by the rather exotic perspective of quantum theory. He recognized a resemblance of the electron-pair model of Lewis to the Heitler−London ansatz for the quantum chemical treatment of H2, which uses an electron-pair function for the H−H bond, and he developed his theory of resonating structures for the description of molecules. The localized picture of valence bond (VB) theory, which was introduced by Heitler and London, appeared to him as far more suitable for chemistry as the delocalized results of molecular orbital (MO) theory, which he strongly opposed. The opposition of Pauling against MO theory is in retrospect difficult to understand for rational reasons, because the advantages of the MO approach over the VB method for explaining chemical phenomena soon became obvious. The electronic triplet ground state of O219 and the peculiar stability of aromatic compounds20 could easily be explained with the symmetry of the MO wave function. Pauling did not recognize the information that is provided by the symmetry of the wave function, which later became an important reason for the breakthrough of MO theory explaining molecular structure and reactivity. It was mainly due to the pioneering FMO work by Fukui21 and to the orbital symmetry rules by Woodward and Hoffmann22 that MO theory became the leading source for bonding models. A second reason for the present dominance of MO theory is the development of computers and efficient algorithms for solving the Schrödinger equation, where the orthogonality of the orbitals make MO calculations substantially faster than VB calculations. More important for the present topic is the fact that MO theory is neither in conflict with nor does it invalidate the electron-pair model of Lewis. Quite on the contrary, the MO method is even better suited for explaining molecular structures in terms of Lewis structures than VB calculations, and it offers a quantum theoretical foundation for writing molecules possessing unusual bonds with the electron-pair model.42 At the same time, it clearly shows the limitations of simple Lewis structures for describing chemical bonding. The diatomic molecules Be2 and O2, which are discussed below, are prominent examples for where the straightforward application of the Lewis paradigm fails without using the symmetry of the underlying wave function. Without knowledge of the orbital symmetry, there is no way to understand why Be2 is essentially unbound with a rather tiny dissociation energy of 2.3 kcal/mol and why O2 has a triplet ground state. Dioxygen is also an example for a molecule where the electronic structure may not reasonably be expressed by Lewis structures. There is no Lewis formula that satisfies the octet rule and the triplet state along with the double bond in O2. According to the Lewis bonding model and the octet rule, dioxygen should be a singlet with a classical double bond OO. Lewis introduced a particular type of electron-pair bond, which he used for a general definition of acids and bases that now carry his name. He wrote that “a basic substance is one which has a lone pair of electrons which may be used to complete the stable group of other atoms and (...) an acid substance is one which can employ a lone pair f rom another”.39 He elaborated this model in a later publication in 1938, which focuses on acid and bases.43 Sidgwick recognized the value of the dative bond for the description of coordinative compounds, which he described in 1923.44 It was further elaborated in his monograph published in 1927, where he introduced the notion of donor and acceptor bonds and

• The choice of the best Lewis structure for a given molecule with unusual bonds should be made in conjunction with quantum chemical calculations. The evaluation of the symmetry of the wave function, which is provided by MO calculations, is very helpful for finding the best description of the bonding situation. • The distinction between electron-sharing bonding and dative bonding, which was already introduced by Lewis, then recognized by Sidgwick and later by Haaland, is a very useful tool to characterize the nature of the chemical bond.

4. QUANTUM CHEMICAL METHODS FOR CALCULATING MOLECULAR STRUCTURES AND PROPERTIES It has become a standard procedure in chemical research to supplement experimental results with quantum chemical G


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E0 MO = E0 0 + E01 + E0 2 + E0 3 + ...

calculations in order to elucidate the bonding situation in molecules. This is often done with a mere presentation of the calculated data without critical examination of the theoretical method used. It is regrettable that the information which is provided by the calculations remains often obscure and questionable, because it is frequently given without knowledge of the relevance of the calculated data and the approximations used. In the following, we will sketch the most important methods that are presently used in chemical research. We will not discuss the details of the mathematical methods. Our focus is not on the technical features of the quite sophisticated approaches used by the quantum chemistry community; they can be found elsewhere in the literature.64−66 We rather emphasize the fundamental aspects of the three most important methods, where we highlight the information that is given by the calculations with respect to the nature of the chemical bond and the relationship to the bonding models. The sheer presentation of calculated numbers in publications is rather useless without a critical examination of the relevance of the data obtained, which requires knowledge about the approximations that are inherent in the respective method. The historically eldest quantum theoretical method is Valence Bond (VB) theory, which was used by Heitler and London in their pioneering study in 1927.8 It was the prevailing approach for some decades, which is mainly due to the strong support by Pauling.11 The advent of computers and the development of numerical algorithms and codes showed that Molecular Orbital (MO) theory is far better suited for quantum chemical calculations, basically rendering VB methods obsolete for numerical studies. Wave function based MO methods clearly became the dominant approaches for quantum chemical calculations during the 1970s. They were gradually replaced in the past two decades by Density Functional Theory (DFT), which is presently, in numerous different variations, the most important tool for quantum chemical studies of larger molecules.67 Examination of the basis of the Kohn−Sham (KS) DFT calculations reveals that they are essentially parametrized variants of the MO method, which is the reason that some authors call them Density Functional Approximation (DFA). We will use the more common term DFT for this method. Because of the close resemblance of the fundamental equations in wave function based MO and KS-DFT calculations, we describe the two methods first before we introduce the basis of VB calculations. The latter has lost its relevance not only because of its computational drawback as already mentioned but also because the apparent advance for the interpretation of chemical bonding in terms of the electron-pair model of Lewis has been lost with the development of the Natural Bond Orbital (NBO) method by Weinhold introduced below.68−71

The associated energies are given by eq 12, where E0MO refers to the energy of the electronic ground state, E00 is the Hartree− Fock energy, and the remaining terms are the correlation contributions of Nth order arising from a step-by-step variational treatment. The first step in determining the MOs consists of a linear combination of the atomic orbitals (LCAO) χa (eq 13) which gives, via Hartree−Fock calculations (eq 14), the molecular orbitals φi that are used to construct the Slater determinant Φ0 as a product wave function (eq 15): φi =

∑ cα χia α

(13)

Fiφi = εiφi

(14)

Φ0 = |φ1φ2φ3 ... φN|

(15)

The order of eqs 13 and 15 indicates the basic difference between MO theory and VB theory (described below): LCAOMO theory is a product of sums while VB theory rests on a sum of products. MO calculations use orthogonal functions, which, compared to VB theory, are much easier to implement into computer programs leading to significantly faster algorithms. This is the reason why quantum chemical calculations that strive for numerical accuracy are nowadays mostly based on MO theory; that is, much larger molecules can be calculated with higher precision than with the VB method. We want to point out that the molecular orbitals φi in the Slater determinant Φ0 (eq 15), the so-called canonical MOs, are a particular set of orbitals that are most commonly used, as these have properties that make them well suited for chemical bonding models. The symmetry of the orbitals was soon recognized as important information that indicates the stability and reactivity of molecules. Hückel realized that the peculiar stability of aromatic compounds can be explained with the symmetry of the π MOs, which was later formulated with the famous Hückel 4n + 2 rule. The delocalized picture of the canonical MOs was for some time an obstacle for accepting it as a valid description of chemical bonding for chemists, because it did not agree with the localized picture of the electron-pair bond of Lewis. The breakthrough came with the FMO model of Fukui,21 the advent of the orbital symmetry rules by Woodward and Hoffmann,22 and the publication of several textbooks on MO theory after 1960.72−74 MO theory does not automatically yield delocalized MOs. The canonical set of orbitals in the Slater determinant is only a particular choice, because they provide the information that is given by the symmetry of the orbitals. The orbitals can be transformed via unitary transformations into equivalent MOs that are subject to the chosen conditions. Thus, the delocalized canonical MOs can be transformed to localized orbitals, which closely resemble the electron-pair model of Lewis. It is not correct to say that MO calculations necessarily yield delocalized orbitals, which are difficult to associate with the Lewis electronpair model. On the contrary, the localized valence MOs exhibit spatial extensions of the electron pairs that may be associated with electron-pair bonds and lone pairs. However, the symmetry information of the canonical MOs is lost and localized MOs do not have an eigenvalue attached. Unlike localized orbitals, canonical orbitals provide symmetry information and they possess energy eigenvalues, which lead to the situation that usually only the latter are used as models.

4.1. Molecular Orbital (MO) Theory

MO theory expresses the electronic wave function for the electronic ground state of a molecule Ψ0MO as a sum of Slater determinants ΦN (eq 11) where the first term Φ0 is usually already a very close approximation to Ψ0MO. The sum of the remaining terms runs over the singly (c1), doubly (c2), and higher excited configurations of Φ0. The theoretical fundament of MO theory can be grasped by understanding the elementary steps for the formation of Ψ0MO (configuration interaction procedure). Ψ0 MO = c0 Φ0 +

(12)

∑ c1Φ1 + ∑ c2Φ2 + ∑ c3Φ3+... (11) H


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correlation through a functional. The expression for the Pauli repulsion in FKS is however different from that in FHF. There are further modifications in various different forms of the KS operator FKS that have been developed over the past decades.78 There is now a rather large and still growing number of DFT approximations available with different KS operators FKS that strive at the same holy grail: To calculate the KS orbitals φiKS via eq 16, which, after quadrature and summation, give a total density ρ(r) that is as close as possible to the exact density. Once this is achieved, all physical properties such as the ground state energy can be obtained. The computational costs of such DFT calculations are comparable to ab initio calculations at the HF level, with the additional bonus that electron correlation is already considered. The enormous progress which has been made in the past two decades in developing DFT led to the present situation where quantum chemical calculations are now mainly performed using DFT with various functionals.79,80 One important aspect that distinguishes DFT from MO theory must be stressed, however. The quality and reliability of MO calculations can stepwise be improved toward the exact solution by using higher correlated methods and larger basis sets. This leads eventually to the correct physical properties in agreement with experiment, albeit at steeply increasing computing costs. DFT calculations at a certain level can, at the moment, not be improved in this fashion.81 We do not wish to discuss the pros and cons of various DFT approximations here. We rather focus on aspects relevant for the present topic, which is the nature of the chemical bond. The Kohn−Sham orbitals φiKS were originally considered a mere technical detail that has accessory character for obtaining the exact electron density ρ(r). Inspection of the shape and the energy levels of φiKS along with their atomic orbital composition showed that they are closely related to HF orbitals φiHF, with the additional bonus that the φiKS are associated with the exact wave function Ψ0 while the φiHF are associated with the HF wave function Φ0 (eq 11). This led to the present situation that KS orbitals φiKS are routinely used for the analysis of the electronic structure and bonding situation in molecules in the framework of qualitative molecular orbital theory, which originally employed semiempirical methods or Hartree−Fock orbitals. Most of the works, which are discussed in this review, are therefore based on DFT calculations and the analysis of Kohn− Sham orbitals φiKS.78−84 There is one important aspect of DFT that is very important for the present topic. There are molecules which possess unusual electronic structures in the sense that they are poorly described with MO methods that use only one determinant as leading configuration (static correlation). Such molecules usually pose problems for traditional DFT methods, which so far are restricted in the wave function picture to single determinant variants. Fractional occupation numbers can be, however, used and, similar to wave function based methods, multideterminant DFT methods are being developed to solve problems such as double-counting electron correlation.85−88 These are interesting developments, but they have not yet reached the stage of routine methods. Accurate wave function based MO methods still play an important role for checking the reliability of DFT calculations for physical properties, particularly in molecules that are not well described with one Lewis structure. Approximate DFT methods as we use them today cover electron correlation in an unspecified way as a consequence of their semiempirical nature. Therefore, they cannot be used as a black box, and the results should always be checked for a smaller test set against high

However, the use of canonical MOs that construct the Slater determinant Φ0 (eq 15) has a drawback: Only the construction of the first component of the wave function Ψ0MO (eq 11) is taken into account. Although the energy contribution of Φ0 is in most cases >99% of the total energy, it may not be sufficient for the bond energy, and thus, the use of Hartree−Fock orbitals φi for chemical bonding may be questioned. It turned out, however, that in most cases the MOs that arise from Hartree−Fock calculations (eq 14) are sufficient for a qualitative discussion of the bonding situation except for van der Waals interactions, which are dominated by electron correlation. Nevertheless, they have been replaced in recent years by the Kohn−Sham orbitals φiKS that arise from DFT calculations, which are briefly introduced in the next section. 4.2. Density Functional Theory (DFT)

The basis of density functional theory is the Hohenberg−Kohn Theorem,75 which states that the ground state properties of a many-electron system are uniquely determined by the electron density ρ(r), depending only on the three spatial coordinates and the external potential; hence, the total energy of the ground state is a unique functional of the electron density. Unlike the wave function based MO and VB theories, DFT takes the density as the starting point of the calculations where the total density of an N-electron system is given by the sum of the oneelectron densities. Thus, the result of a DFT calculation is (in principle) not an approximate energy E0N as in MO calculations but the correct total energy E0 (compare eq 12), provided that the correct density ρ(r) and functional are known. It must be realized that DFT does not focus on the formation of the electron density ρ(r) but rather on the connection between the density and the associated energy. The density ρ(r) is considered as the starting point of DFT. But the work by Heitler and London8 has shown that the chemical bonds in a molecule can be described by the interference of wave functions. The exact density ρ(r) can of course be obtained from the exact wave function Ψ, but it can also be constructed from the Kohn− Sham (KS)76 functions shown in eq 16, which are the working horse of present DFT methods. Within KS theory the total density of an N-electron system is simply the sum of oneelectron densities. The KS equations look similar to the Hartree−Fock equations (eq 14), which are shown again in eq 17, and we use the superscript HF in order to distinguish them from the Kohn−Sham equations: F KSiφi KS = εi KSφi KS

(16)

F HFiφi HF = εi HFφi HF

(17)

The crucial difference between the Kohn−Sham (KS) equations (eq 16) and the Hartree−Fock (HF) equations (eq 17) lies in the expressions for the operators FKS and FHF.77,78 The HF operator FHF contains well-defined terms for the kinetic and potential energy of an electron and its average electrostatic repulsion with the other electrons as well as the exchange (Pauli) interaction with other electrons that have the same spin. This leads to the HF energy E00, which is the first term in eq 12. The individual correlation with the other electrons is neglected but can be considered by the higher order terms in eq 12, which, when included in a CI procedure, is quite computer timeconsuming. This is the well-known bottleneck of correlated ab initio calculations. DFT calculations try to rectify this problem by adding a further term to the KS operator FKS that considers exchangeI


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quality wave function results. We warn against an indiscriminate use of DFT calculations for analyzing the bonding situation in a molecule.

should always be checked for a smaller test set against high quality wave function results. • VB calculations are more cumbersome compared to MO and particularly KS-DFT calculations. The identification of the leading terms of VB calculations with covalent and ionic bonding has no physical fundament.

4.3. Valence-Bond (VB) Theory

Unlike the MO method, where the wave function Ψ0MO is assembled as the product of one-electron functions φi yielding the Slater determinant Φ0 (eqs 11 and 15), the VB wave function Ψ0VB is expressed as the sum of localized two-center productfunctions (λaλb).89−91 The occupation of the product-functions λaλb by two electrons of opposite spins gives the electron-sharing covalent term (λa−λb) (“Heitler−London” term) and the two ionic terms (λa|− λb+) and (λa+ λb|−). The basic VB expansion, which considers all possible products in the molecule, is given by eq 18, where the summation runs over all electron pairs in the molecule:90 Ψ0 VB =

5. QUANTUM CHEMICAL METHODS FOR ANALYZING THE CHEMICAL BOND IN MOLECULES It has become a standard part, also in experimental work, to present the results of quantum chemical calculations in conjunction with a breakdown of the calculated numbers in terms of chemically meaningful information about the bonding situation and electronic structure in the molecules. There are several methods available, which serve as a bridge between the computed numbers of the quantum chemical calculations and the heuristic bonding schemes such as the Lewis model. In the following, we present a short summary of the approaches and basic features of the most common methods, which should be understood in order to prevent misinterpretation of the results. Reasonable conclusions about the bonding situation in a molecule are only possible when the most important details of the sophisticated algorithms and the inherent assumptions, which are intrinsically coded in the algorithms, are known. A mere presentation of the calculated numbers without knowledge of how they were obtained is meaningless. Numerous charge- and energy-partitioning methods have been developed over the course of time. The five conditions given in Scheme 2 should be met by a partitioning procedure, in order to provide a meaningful interpretation of the vast complexity of chemical observations in terms of a bonding model.

∑ c1(λa−λ b) + ∑ c2(λa|−λ b+) + ∑ c3(λa+λ b|−) (18)

As mentioned above, the use of orthogonal orbitals in MO calculations leads nowadays to much faster algorithms in computer programs than VB methods, where overlapping hybrid orbitals are employed. VB methods do not play a role in theoretical research of larger molecules, which strives for numerical accuracy. They were, however, favored over MO theory by Pauling, because they appear at first sight to be more closely related to the electron-pair model of Lewis. The first term in eq 18 could be identified with the covalent electron-pair bond, whereas the second and third terms give the ionic bonding contribution. The coefficients c1 and c2/c3 appear as ideally suited to indicate the covalent and ionic character of a bond. This was suggested by Pauling in his book The Nature of the Chemical Bond, which shaped the understanding of chemical bonding of many chemists for decades.11 The development of computers and algorithms for carrying out VB calculations, however, made it possible to check the validity of this qualitative model. It turned out that there are molecules such as F2 where neither the covalent term (λa−λb) nor the ionic terms (λa|− λb+) and (λa+ λb|−) alone yield the correct bonding energy. It is the mixing of the two terms that leads to a quantitative account of the bond energy. This makes the interpretation of the VB terms in eq 18 for indicating covalent and ionic bonding questionable. The Lewis electron-pair bonding model refers to the total bond while the covalent term (λa−λb) covers only part of the total energy of a VB calculation.92 The nature of the chemical bond in F2 and other diatomic molecules is discussed further below. The conclusion is that VB methods are not necessarily better suited for the interpretation of the chemical bond in terms of the Lewis bonding model. The following conclusions arise from this section:

Scheme 2. Five Conditions That Should Be Fulfilled by a Model for a Chemical Bond

5.1. Natural Bond Orbital Method (NBO)

The presently most popular tool for a bonding analysis of molecules is the NBO (Natural Bond Orbital) method by Weinhold and co-workers.68−71 It was introduced in the 1980s and has been further developed in the course of time during which it underwent significant changes.93 Thus, the results of an early variant of the NBO program may be quite different from those of a newer version. The goal of the NBO method is to provide the most reasonable Lewis structure for a calculated molecule along with information about the polarity of the bond as well as the hybridization of the atoms and their partial charges. The problem of determining the best Lewis structure is that subjective criteria are taken into account when selecting the mathematical steps and computer algorithms. This is true for any chemical bond model. It is adamantly important that the subjective criteria must be known when the results are interpreted and compared with the values of other methods. The NBO method takes the one-electron density matrix, which is given by a quantum chemical calculation, as starting point for the partitioning procedure. Because the NBO method uses the density ρ instead of the wave function Ψ as initial point,

• Molecular orbital calculations provide a reliable framework for the identification of the electronic structure of a molecule, but they may become quite expensive when a single determinant calculation is not adequate for the situation. • The results of MO calculations can easily by expressed in terms of Lewis structures by using localized MOs. • Present DFT calculations can be considered as parametrized MO calculations. They are much more efficient, and they often give very good results at lower costs than Hartree−Fock based MO calculations. However, they should not be used as black box methods, and the results J


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but not to provide information about the electronic structure of a molecule. The NBO method was further developed toward the NRT (Natural Resonance Theory) method, where energy expressions for the intramolecular orbital interaction were suggested.104−106 The NRT was met with immediate scepticism,107 and the energy values that are provided should be taken with some caution. The NBO method focuses exclusively on orbital interactions in a molecule. Electrostatic interactions and Pauli repulsion are only indirectly considered. This may lead to conflicting interpretations about chemical bonding when other methods are employed that include all physical forces.108−110 A strong point of the NBO method is the finding that the numerical results are usually rather insensitive to the level of theory, which is of great advantage compared with the rather outdated Mulliken population analysis whose results are plagued by large variations when different basis sets are used. An extension of the NBO method to molecules that exhibit strongly delocalized bonds, such as electron deficient boron compounds, has been developed by Boldyrev and co-workers.111 The Adaptive Natural Density Partitioning (AdNDP) method describes N-center-two-electron bonds where N gives the number of atoms to which the two-electron bond should be assigned. The AdNDP method has been found to be particularly useful for the description of organic and inorganic molecules and clusters where the standard NBO method is less suitable. A reformulation of the AdNDP approach has recently been introduced by Pendás and Francesco, which provides Lewis structures from real space analyses of general wave functions.112 A warning shall be addressed when the results of a bonding analysis are judged by their agreement with “chemical intuition”. The vaguely defined concept of chemical intuition, which refers to the wealth of experimental experience in chemistry, may be helpful as a guideline for future experiments and for the design of a specific bonding model, but it is very questionable when it comes to a physical interpretation of chemical bonding. As discussed above, chemical bonding can be regarded as a quantum theoretical phenomenon due to the interference of wave functions. Quantum theory and chemical intuition often do not fit together well, since human sensory perception, like classical physics, is limited in space and time. Both quantum theory and relativity are often at odds with the human mind and intuition and may perhaps only be accessible through mathematics. The difficulty of getting along with quantum theory was suitably expressed by Niels Bohr: “Those who are not shocked when they f irst come across quantum theory cannot possibly have understood it”.113

it may be used in conjunction with any correlated ab initio method or DFT calculation. This is a strong point of the method. But there are two little known aspects of the NBO method, which may lead to debatable results. One aspect concerns the choice of the atomic valence orbitals, which are considered for the formation of the molecular orbitals. The NBO algorithm makes a preselection of those atomic functions that are considered as genuine valence orbitals and those who are termed Rydberg functions. The two sets are differently treated in the subsequent algorithms, where the Rydberg orbitals are technically treated with lower priority, which leads to biased results. The division of the atomic orbitals into the two sets of atomic functions is the subjective choice of the authors. Thus, the outcome of NBO calculations is not “natural”, it is rather the consequence of what the authors consider to be “reasonable”. The NBO method considers only those outermost atomic functions as genuine valence orbitals, which are occupied in the electronic ground state of the atom. Thus, the NBO algorithm considers the alkali and alkaline earth atoms having a (n)s valence shell and the group 13−18 atoms to possess a (n)s(n)p valence shell. The question whether the octet rule is valid for main group atoms cannot be addressed in NBO calculations, because the answer is already coded into the algorithms.94 The above criterion leads to a critical situation for transition metals, which have only an (n)s(n − 1)d valence shell. The latter choice leads to a 12-electron rule rather than 18-electron rule for transition metal complexes, because the (n)p AOs are only treated as Rydberg orbitals.95−97 This has been criticized by several authors, because the 18-electron rule is well established in transition metal chemistry.98−100 Very recently, the transition metal compounds [(CO)3Ni-E]− (E = Li − Cs) were reported, where the covalent Ni−E bond at the nickel end comes mainly from the Ni(4p) AO.101 The NBO calculations using version 6.0 do not give a faithful description of the Ni−E bonds. But even for main group atoms, the preselection of valence orbitals according to the NBO method can sometimes become questionable. It was lately shown that the heavier earth alkaline atoms M = Ca, Sr, Ba form strongly bonded octa-carbonyls M(CO)8 using their (n − 1)d orbitals, which challenge the restriction of the valence orbitals to (n)s functions.102 The second critical aspect concerns the algorithms of the current eight orthogonalization steps, which have changed during the different versions of NBO and sometimes lead to inconsistent results when different versions are used. Early versions of the program showed that the 3d participation of sulfur in the bonding in SF6 is rather small, which agrees with the general notion that d-AOs of the heavier main-group atoms are polarization functions and that the model of hypervalence in such compounds should be discarded.103 However, the recent version NBO 6.0 surprisingly gives Lewis structures for SF6 with perfect sp3d2 hybridization. A similar result is obtained for PF5, which, according to NBO 6.0, has sp3d hybridized P−F bonds. This is likely due to the projection operator in the NBO version 6.0, which yields an inappropriate hybridization for SF6 and PF5. This has been fixed in the latest version NBO 7.0, which was distributed while this review was written. The problem remains that the biased treatment of the outermost atomic orbitals and the technical details of the orthogonalization steps may lead to results which should be checked in comparison with other methods that are based on different partitioning schemes. Whatever method is used, it holds that one should know the details of a method in order to make a reasonable statement. The aim of the NBO method is to propose a suitable Lewis structure

5.2. Quantum Theory of Atoms in Molecules (QTAIM)

The NBO method is a charge-partitioning procedure that aims at the division of the wave function Ψ into atomic regions. In contrast, the QTAIM (Quantum Theory of Atoms in Molecules) developed by Bader is based on a topological analysis of the electron density ρ(r) into atomic basins.114 The mathematically well-defined QTAIM method avoids some shortcomings and problems of wave function based partitioning schemes. Using the first derivatives (gradient field) ∇ρ(r) and second derivatives (Laplacian) ∇2ρ(r) of the electron density, physically well-defined atomic regions and interatomic bond paths may be identified, which typify a molecular structure by the zeroflux surfaces that separate the atomic basins. The zero-flux surfaces are defined as the gradient vector field whose trajectories of ∇ρ(r) do not vanish at the atomic nuclei but at K


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5.3. Energy Decomposition Analysis and Natural Orbitals for Chemical Valence (EDA-NOCV)

the bond critical point rb, which is a signature of a bond path. A bond critical point rb has two negative and one positive eigenvalues of the second derivatives of ρ(rb). Other critical points at which the gradient of the electron density ∇ρ(r) vanishes define rings and cages. The bond path is the trajectory which belongs to the positive eigenvalue connecting the bond critical point rb and the bonded atomic nuclei. Thus, the QTAIM method gives a physically sound description of the skeletal structure of the molecule in terms of atomic nuclei and chemical bonds, which has been proven useful for a variety of chemical systems.115 The QTAIM method is possibly the physically best founded theory for chemical bonding, providing a rigorous quantum theoretical proof that the model of discussing the chemical behavior of a molecule in terms of atomic properties is justified. Problems arise when the QTAIM results are connected to the manifold of chemical observations, which are often more meaningfully described with heuristic orbital models. As mentioned above, the wave function and the orbitals exhibit important information through their symmetry, which is lost when only the electron density ρ is used instead of Ψ. This makes the QTAIM method perhaps less powerful for making predictions of chemical reactions. Another problem is the use of the bond critical point rb as an indicator of a genuine chemical bond. Diatomic He2 is a weakly bonded van der Waals complex with negligible interference of the atomic wave functions, and thus, it has no covalent bond. QTAIM calculations indicate a bond critical point and a bond path just like in H2. This was recognized by Bader, who clearly stated that “bond paths are not chemical bonds”.116 On the other hand, there are molecules where by visual inspection of the electronic structure one would ascribe a covalent bond between two atoms but there is no bond path.117,118 There has been theoretical work to distinguish different types of chemical bonds that possess a bond critical point using QTAIM parameters. The most successful one goes back to the investigations by Cremer and Kraka, who suggested that the energy density at the bond critical point H(rb) is a useful criterion for defining the nature of a chemical bond.119 According to the Cremer−Kraka criterion, a covalent bond between two atoms A and B is defined by the existence of a zeroflux surface and bond critical point rb between the atoms (necessary condition) and a negative and thereby stabilizing local energy density H(rb) (suf f icient condition). H(rb) will be close to zero or positive if the interaction between A and B comes from the electrostatic or dispersion forces. This approach has been found very helpful for the bonding analysis between electronegative atoms.120,121 But the numerical absolute values may become very small when bonds between heavy atoms are involved and the degree of covalent bonding is not directly available from the values of H(rb). In spite of the caveats, it may be stated that the QTAIM method has been established as an important method to elucidate the electronic structure not only in molecules but also in solids.122 Inspection of the Laplacian distribution of the electron density ∇2ρ(r) provides valuable information about the areas of charge depletion and charge accumulation in a molecule, which come from the interatomic interaction. The combination of QTAIM calculations with the results of an MO analysis is a very powerful tool, which helps to express the bonding situation in a molecule with an appropriate model.

The NBO and QTAIM methods are charge-partitioning techniques, which divide the electronic structure of molecules into atomic regions. There are complementary methods that first break up the total energy of a molecule into fragments, which are then reassembled in a stepwise fashion where each step is assigned to a particular type of interaction. The choice of the fragments is determined by the question of interest. Trivially, in a diatomic molecule X2 the two atoms are chosen as interacting fragments, but already when a heteroatomic species such as LiF shall be analyzed, it may be more useful to choose the ionic fragments Li+ and F− for the bonding analysis. This seems to make the method somewhat arbitrary and therefore questionable, but the possibility of using different fragments for the bonding analysis provides more flexibility that may be used to address different questions about the bonding nature. Thus, the choice of neutral Li and F as interacting fragments includes all changes along the bond formation between the isolated atoms toward LiF, whereas the choice of the ions Li+ and F− addresses the question about the nature of the eventually formed bond. These are two different questions, which have two different answers that can be addressed by choosing different fragments. The EDA-NOCV method123,124 is a combination of the EDA (Energy Decomposition Analysis) method, which goes back to initial ideas by Morokuma and Kitaura125 and later by Ziegler and Rauk,126 and the original charge-partitioning method NOCV (Natural Orbitals for Chemical Valence) by Mitoraj and Michalak.127,128 It has been proven as a very powerful tool for bonding analyses, because it considers all types of physical interactions, which eventually sum up to the experimentally observable bond dissociation energy (BDE). In particular, the further partitioning of the orbital (covalent) term into pairwise orbital interactions has been proven very helpful, because it provides a quantitative expression of the FMO model of Fukui21 and the orbital symmetry rules of Woodward and Hoffman.22 A nice feature of the EDA-NOCV model is that the orbital interactions are not only given in terms of numbers, which indicate their strength, but the associated charge migration can visually be shown. In the following, we present the most important aspects of the EDA-NOCV method. A more detailed discussion of the method and its application has been given in recent review articles.129,130 At the present time, EDA-NOCV analyses can only be carried out using DFT or Hartree−Fock calculations. Since correlation energy is missing in the latter method, more meaningful results are obtained with DFT approaches. The results do not change very much when different functionals are employed. The focus of the EDA approach is the instantaneous interaction energy ΔEint of a bond A−B between two (or more) fragments A and B in the particular electronic reference state and in the frozen geometry of AB.131,132 It is very important to recognize the correct electronic reference states of the fragments in order to provide a meaningful analysis of the interactions A−B. In doubtful cases, one may carry out EDA calculations with different electronic states of the fragments and compare the results. Those fragments, which give the smallest energy change during the bond formation step, are the most useful species for the bonding analysis. The interaction energy ΔEint is divided into three main components, which are sometimes augmented by a fourth expression for dispersion (van der Waals) interactions: L


Chemical Reviews ΔEint = ΔEelstat + ΔEpauli + ΔEorb + (ΔEdisp)

Review

There is one important aspect of the ΔEelstat term in the above EDA method that deserves to be discussed. The partitioning scheme uses overlapping spheres of the atomic charge distributions, which means that some electronic charge in the molecule that is close to the nucleus of atom α is assigned as coming from atom β. This may appear odd for human imagination that is shaped by senses, which are trained by classical hard objects. If one assumes the atoms are spheres with impenetrable surfaces it seems more “reasonable” to cut a molecule into atomic pieces by sharp boundaries as for instance in the IQA (Interacting Quantum Atoms) model,147−150 which is based on Bader’s QTAIM definition of separated atomic basins.114 However, in quantum theory, an electron is described by a wave function, which has an exponentially decreasing but finite value even at larger distances and so has its square, which gives the charge density of the electron (see below). Thus, for example in N2, an electron of Nα has a finite density also close to the nucleus of Nβ. The estimate of Coulombic interactions using interpenetrating charge distributions in the EDA method thus appears as a physically reasonable approach consistent with quantum theory, which is commonly used in many theoretical approaches, such as Mulliken analysis, Natural Bond Orbital analysis, Hirshfeld partitioning,151 etc. On the other hand, the wave function of an electron of Nα in N2 is subject to interference with the wave functions of the electrons of Nβ. This leads to some ambiguity in assigning the charge in a molecule that comes from the interfered wave functions to a particular atom. In conclusion, the many models of interpenetrating charges as well as strictly separated charges may all appear more or less physically reasonable. The NOCV scheme127 divides the orbital interaction term ΔEorb in the EDA-NOCV approach into pairwise contributions of interacting orbitals of the two fragments. The starting point is the deformation density Δρ(r), which is the difference between the densities of the fragments before and after bond formation. The deformation density Δρ(r) can be expressed in terms of pairs of complementary eigenfunctions (ψk ψ−k) with the eigenvalues υk and υ−k that possess the same absolute value but opposite sign. The absolute numbers of the eigenvalues υk are a measure for the magnitude of the charge transfer:

(19)

In the first step of the EDA procedure, the fragments A and B are superimposed with their frozen densities at the geometry of the molecule AB, which gives the quasi-classical electrostatic interaction between the unperturbed charge distributions of the prepared fragments ΔEelstat. The term is usually attractive, because the frozen charge densities of the fragments overlap and thus penetrate each other, which induces attraction with the atomic nuclei of the other fragment.28 The overlapping densities may have electrons that have the same spin, which according to the Pauli principle is not allowed. This is rectified in the second step of the EDA procedure, where the superposition of the unperturbed electron densities of the isolated atoms is subject to antisymmetrization (operator Â ) and renormalization (constant N), which gives the product wave function Ψ0 = N Â [ΨA(α) ΨB(β)] still without interference of the fragment wave functions ΨA(α) and ΨB(β). The energy term ΔEPauli, which is calculated in the second step, accounts for the electron−electron repulsion due to the orthogonality requirement of the orbitals. The final term ΔEorb accounts for the formation of covalent bonding via interfragment mixing of the orbitals but also for the polarization within the fragments via intrafragment orbital mixing. Polarization and charge transfer between the fragments are not separated in the EDA approach. If the two fragments of the chemical bond are in an electronically excited state or if they have more than one atom, there is an electronic and possibly a geometric relaxation of the fragments during bond rupture into the equilibrium geometry of the fragments, which corresponds to the preparation energy ΔEprep. If the energy value is added to ΔEint, one obtains the bond dissociation energy De, which is by definition the negative value of the total bond energy ΔE: ΔE ( = −De) = ΔEint + ΔEprep

(20)

Although the individual terms in eqs 19 and 20 are not observable quantities, the sum of them gives the experimentally measurable bond dissociation energy. The absolute value of ΔEint is a faithful expression for the intrinsic bond strength, whereas the De values are affected by the geometrical and possibly the electronic relaxation of the fragments. There are molecules where the fragments A and B are lower in energy than the molecule AB, which makes the De values useless for estimating the bond strength A−B. This holds in particular for high-energy materials and explosives where bond breaking is a highly exothermic process. It is thus advisible to use the interaction energy ΔEint for comparing the bond strength of molecules. There are cases where the electronic state of the fragments used in the bonding analysis is not obvious. For example, Noxides R3NO might be described with a dative bond R3N→O or with an electron-sharing single bond R3N+−O− or even a double bond R3NO.133 The EDA method can be used to suggest the best description according to the electronic structure, by using the fragments in the respective electronic state in the actual calculation. Those fragments, whose energy changes the least during bond formation, appear the best choice for the representation of the bonding situation. The smallest absolute value of the orbital term ΔEorb may be used as a measure for the best Lewis description of a molecule. This procedure has been proven very helpful in a variety of compounds where the bonding situation is not directly clear.49,134−146

Δρ(r) =

∑ υk[−ψ 2−k(r) + ψ 2k(r)] = ∑ Δρk(r) k

k

(21)

The NOCVs ψk and the associated eigenvalues υk are obtained through diagonalization of the difference density matrix ΔPμν of the system. Equation 21 facilitates the expression of the total charge deformation Δρ(r) that goes along with the bond formation in terms of pairwise charge contributions Δρk(r) which come from particular pairs of (NOCV) orbitals. The total orbital interaction ΔEorb may likewise be derived from pairwise orbital interaction energies ΔEkorb which are associated with Δρk(r): ΔEorb =

∑ ΔEk orb = ∑ υk[−FTS−k ,−k + FTSk ,k ] k

k

(22)

Experience has shown that the ΔEorb term of the EDA-NOCV approach has usually only a very small number of significant contributions of ΔEkorb, which makes it possible to identify specific orbital interactions that lead to a chemical bond. This connects the results of the EDA-NOCV method with classical models of orbital interactions63 and with the Lewis electron-pair model.152 Numerous examples where the EDA-NOCV method proved helpful for a variety of main-group compounds134−146 M


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and transition metal complexes153−163 have been reported in recent years. For a fruitful and enlightening discussion about different results of the NBO method and EDA-NOCV calculations on the nature of bonding in transition metal compounds, we refer to two recent papers by Weinhold and coworkers.164,165 The following conclusions arise from this section: • There are several powerful analytical tools available which extract information from quantum chemical methods aiming at an understanding of the numerical results in terms of a bonding model. But the user must be aware of the details of the methods in order to provide a meaningful interpretation of the results of the chargeand energy-partitioning schemes.

each other. It would be much better to rely only on measurable quantities and on physical observables. We think that such criticism stems from a misunderstanding of the role and the relevance of bonding models and physical theory in chemistry. It is also based on a questionable definition of observables and physical reality, whose understanding has profoundly changed with the advent of quantum theory. The charge and mass of an electron are fundamental constants which enter the Schrödinger (or Dirac) equation, which then are used to calculate properties to very high accuracy in perfect agreement with experiment. It is often said that the wave function does not correspond to a quantity which can be measured; it is a purely mathematical construct to describe the electronic charge distribution. This led to a debate between Heisenberg and Einstein: “One cannot observe the electron orbits inside the atom. [...] but since it is reasonable to consider only those quantities in a theory that can be measured, it seemed natural to me to introduce them only as entities, as representatives of electron orbits, so to speak” (Heisenberg). “But you do not seriously believe that only observable quantities should be considered in a physical theory?” (Einstein).166 But the one-particle density ρ obtained from the wave function is also a mathematical construct, which is used to describe experimental results. In this case, however, the charge density can be probed experimentally as it is done for example in X-ray diffraction. Since the outcome of a Diels−Alder reaction can only be explained with the help of the symmetry of the wave function Ψ, which uniquely identifies the interacting species, it acquires the quality and the status of a physically relevant entity that becomes evident by the reaction product. Note that a complete wave function of a particle is a valid expression containing all information about its physical properties. We want to make a reference to a recent perspective on quantum mechanics and chemical concepts by Clark, Murray, and Politzer (CMP).167 Although the focus of their work is on noncovalent bonding, the central topic of their work has much in common with the present article. The authors cite in the beginning a statement by Ivanic, Atchity, and Ruedenberg whose final words nicely sum up the central theme “...the resolution of molecules in terms of atoms is not f undamental to rigorous physical theory”.168 CMP rightfully point out that “the distinction between what is mathematical modeling and what is physical reality may become blurred; the model may be taken to represent reality. This can be misleading and conf using.”167 We agree, but the situation becomes a bit complicated when the status of the wave function is concerned. The authors argue that the electron density represents physical reality but not the wave function. They quote Schrödinger, who warned against using the wave function for the direct interpretation of physical observations.169 The pioneers of quantum theory are known not only for their ground breaking contributions to physics but also for their limited acceptance of the outcome of their theories. Einstein never accepted quantum mechanics as a valid physical theory, and Dirac wrongly suggested that relativistic effects are negligible for chemistry. The direct relevance of the symmetry of the wave functions for chemical reactions was not obvious when Schrödinger made his statement in 1926.169 The consideration of the wave function or electron density as the basis of the physical reality of matter can be seen as representing the continuous versus atomistic theory of the universe. They are rather complementary and not mutually exclusive. It is often said that the present generation of scientists is standing on the shoulders of giants. But this enables us to see further than they

• The NBO method has been designed to extract the best Lewis electron-pair description of the molecular electronic structure. The atomic orbitals are not identically treated; the results are biased toward a localized electronpair bonding model favoring preselected atomic orbitals, which is considered by the program designers as reasonable. The NBO method explicitely focuses only on orbital interactions; other interatomic forces such as Coulomb forces are not directly taken into account. • The QTAIM method takes the electron density as starting point for the analysis of the electronic structure. The information that comes from the symmetry of the orbitals is not considered. It provides a direct connection between the physical description of interatomic interactions and the electronic structure. • The EDA-NOCV method is a combination of charge- and energy-partitioning scheme. It provides a quantitative interpretation of the chemical bond in terms of physically meaningful contributions. The NOCV method is a bridge between quantum chemical calculations and the FMO model of Fukui and the orbital symmetry rules by Woodward and Hoffmann. The strength of the orbital interactions can be expressed numerically, and the associated charge deformation can be visualized with the help of the deformation densities. • All analytical methods have their strengths and weaknesses. It is advisible to compare the results of several methods in order to give a reasonable interpretation of the bonding situation. The results of the bonding analysis suggest a model description, which helps to understand the experimental results in the framework of an ordering scheme that is based on quantum chemical calculations. But bonding models are not right or wrong; they are more or less useful.

6. PHYSICAL REALITY AND CHEMICAL BONDING MODELS A frequent topic of controversial discussions among chemists is the relationship between physical reality and chemical bonding models. This holds in particular for models that are derived from quantum chemical calculations such as the NBO and EDANOCV methods, which quantitatively define expressions related to the chemical bond that are given in terms of calculated numbers. It is sometimes criticized that these numbers cannot be measured by experiment and, therefore, they would not have any physical meaning. Moreover, different definitions of equivalent terms such as atomic partial charges or bond orders may lead to very different values and sometimes even contradict N


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to faciliate the comparison of molecules and reactions. Different tools can be applied to achieve a better understanding; their choice may depend on personal preferences and training. Different results and different views from different methods should be analyzed in the light of the particular decomposition scheme, whose suppositions and technical implementations must be known for a meaningful interpretation of the results.

could, and we should not be afraid of challenging traditional viewpoints. If one only repeats earlier findings, this means that nothing new has been learned. The numbers, which come from chemical bonding models such as NBO or EDA-NOCV, arise from a breakdown of Ψ into fragments, which are defined via the particular charge- and energy-partitioning method. They are clearly not observables in a quantum mechanical sense. Their relevance comes exclusively from their ability to construct an ordering scheme, which helps to divide the enormous complexity of chemical reactions and molecular structures into specific classes and groups. Historically, this was done by the finding that certain molecules exhibit similar behavior. Heuristic notions were suggested such as aromaticity, conjugation, nucleophilicity, formal charge, etc. Although their definition is vague, these models are crucial for bringing order into the pandemonium of experimental results. Bonding models are vital for chemical research, because otherwise there would not be any systematic ordering of the infinite number of molecules and reactions. If one would explain chemical results only in terms of physical observables (including the wave function) without bonding models, it would result in a huge table with data that do not provide any intuitive understanding. Chemistry is by its very nature a fuzzy science, because the jungle of boundless number of molecular structures and reactions request methodical schemes that serve as guidance for future directions of chemical research. The quantum theoretical postulates are the physical framework for chemistry, but the inner substance of chemistry is guided by models. Unlike the earlier heuristic models, which were based on assumptions that are rooted in classical physics, modern bonding models come from well-defined though chemicalintuition guided partitioning schemes of accurate quantum chemical methods. They should fulfill the conditions that are shown in Scheme 2. The advantage is that the comparison and trend of different molecules can now be made with the help of numbers rather than hand-waving arguments. The finding that different partitioning schemes may lead to different and sometimes conflicting views should not be taken as argument against their use. Instead, the assumptions of a model and the technical details of their implementation should be considered before a conclusion is made. This requests the knowledge about the most important aspects of the theoretical methods that are employed, just as is done with experimental spectroscopic methods. The following conclusions arise from this section:

7. SELECTED EXAMPLES OF CHEMICAL BONDS IN MAIN-GROUP COMPOUNDS In the following, we discuss the chemical bond in selected maingroup compounds, which serve as examples for the wide variety of molecules that possess different types of bonds. We try to cover a wide range of bonding, which may be used as templates for other species. The goal of this section is to show how modern methods of bonding analysis may be used to (a) elucidate the nature of the interatomic interactions and (b) express the bond in terms of a model that agrees with the physical mechanism of the bond. The latter part may lead to conflicting results with earlier work presented in textbooks, because previous studies were often based on ad-hoc assumptions that are not justified. We present results of quantum chemical calculations mostly using DFT methods, which are well suited for a bonding analysis. The focus is not on high numerical accuracy but on the nature of the interactions. On the other hand, the calculated observables of the molecules given in this work such as the equilibrium geometry and the bond dissociation energy are assured to be in reasonable agreement with experiment, in accordance with condition 1 in Scheme 2. 7.1. H2+, H2, Li2+, Li2

Table 1 shows the calculated energy components for the interactions in H2+, H2, Li2+, and Li2 according to the EDATable 1. EDA-NOCV Results of H2, H2+, Li2, and Li2+ Species at the RPBE/TZ2P//BP86/def2-TZVPP Levela Molecule Fragments ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔEorb(σ)c

• In the quantum world, the physical reality of an electron is not entirely assigned to its electronic charge distribution ρ, which represents only a projection onto a space of lower information content; its completeness is only provided by its wave function Ψ. The wave function Ψ contains more information about the behavior of the electron than ρ. In chemistry, this comes to the fore for example in the outcome of pericyclic reactions, or in any spectroscopic investigation, which can only be explained when the symmetry and sign pattern of Ψ is considered.

H2 [H] + [H] −106.6 0.0 5.5 −112.1 (100%) −111.5 (99.5%)

H2+ +

[H] + [H] −68.7 0.0 17.6 −86.3 (100%) −84.9 (98.4%)

Li2

Li2+

[Li] + [Li]

[Li]+ + [Li]

−20.8 1.9 −8.3 (36.7%) −14.3 (63.3%) −12.9 (90.2%)

−28.5 2.1 1.6 −32.2 (100%) −32.2 (100%)

a

Energies are in kcal/mol. bThe values within the parentheses show the contribution to the total attractive interactions ΔEelstat + ΔEorb. For those cases where ΔEelstat is positive, it is excluded. cThe remaining small contributions come from polarization functions.

NOCV method. The intrinsic interaction energy ΔEint is in these cases identical to the bond dissociation energy De, because there is no electronic or geometric relaxation of the fragments. The one-electron bond in H2+ has a bond energy of De = 68.5 kcal/ mol, which is nearly two-thirds of the bond strength in H2. The bonding in H2+ comes exclusively from the interference of the wave function, and the orbital interactions ΔEorb provide 100% of the attractive contribution to ΔEint. The electrostatic interactions ΔEelstat at the equilibrium distance are repulsive. A naı̈ve view might anticipate some Coulombic attraction of the electron of the hydrogen atom by H+. This is misleading, because the electronic charge would be withdrawn from the

• Models, in particular bonding models, are an integral part of chemical research. Chemical research depends on the existence of bonding models, which provide an ordering scheme for the infinite number of molecules and reactions. • The numerical outcome of modern partitioning schemes is not measurable quantities, but they are very useful tools O


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in H2+ originating from the hydrogen atom ecompasses the proton and the bonding area. The bond formation of the valence isoelectronic species Li2+ and Li2 may superficially be expected to exhibit similar features, but there are important differences between the valence electrons of hydrogen and lithium. Below the 1s shell of hydrogen there is the bare nucleus, whereas below the 2s shell of lithium there is a polarizable 1s core. Charge flow of the valence electrons of H during bond formation of H2 leaves the remaining electronic charge under the Coulombic attraction of the nucleus, which induces a rather large contraction of the 1s AO.173 In contrast, the remaining 2s valence electrons of lithium in Li2 encounter Pauli repulsion by the 1s shell, which prevents penetration toward the nucleus. This leads to significant differences of the physical mechanisms of the bond formation between the lithium and the hydrogen dimers. This becomes obvious by inspection of the EDA-NOCV results in Table 1 and the associated deformation densities in Figure 4. The bond in Li2 is much weaker (De = 20.5 kcal/mol) than in H2 (De = 112.9 kcal/mol). This can be mainly attributed to the more diffuse 2s AO of lithium and the less weakly bonded 2s electron in Li compared with the 1s electron in hydrogen. The ionization energy (IE) of hydrogen atom is 13.60 eV whereas the IE of Li is only 5.39 eV. This induces a significantly smaller energy lowering through the interference of the wave functions.173 The most surprising feature is the stronger oneelectron bond in Li2+ compared with the two-electron bond in Li2. The rather soft singly occupied 2s AO of lithium becomes strongly polarized by the positive charge and it mixes with the 2p(σ) orbital, which leads to enhanced interference of the atomic wave functions compared with Li2. The deformation density in Li2+ shown in Figure 4 nicely vizualizes the polarization of the electronic charge upon bond formation. This effect is so strong that the final energetic stabilization in Li2+ is stronger than in Li2. The second electron weakens particularly the interference of the wave functions, because the interelectronic interactions reduce the polarization, which effectively weakens the bond. The results for H2+, H2, Li2+, and Li2 are striking evidence that chemical bonding does not originate from spin coupling of the electrons, which is often wrongly stated.174−176 The following conclusions arise from this section:

vicinity of the nucleus toward the internuclear region. It can easily be shown that the electrostatic attraction in the internuclear region is weaker than in the core region, because the distance to the nuclei becomes larger. The stabilization comes from the lowering of the kinetic energy due to the interference of the wave function.170,171 This is the physical origin of ΔEorb.172 The second electron in H2 enhances the bond energy to a smaller extent than the first electron in H2+. The electrostatic interactions ΔEelstat at the equilibrium geometry of H2 are repulsive, but only to a minor extent that does not significantly influence the overall internal energy. They become weakly attractive only at longer distances. The deformation densities, which are associated with the bond formation in H2+ and H2, are shown in Figure 4. The direction of the charge flow from the

Figure 4. Plot of deformation densities Δρ of H2, H2+, Li2, and Li2+ species at the RPBE/TZP//BP86/def2-TZVPP level. Energy values are in kcal/mol. The direction of the charge flow is red → blue.

donating to the accepting areas has the color code red → blue. It becomes obvious that there is a charge accumulation in H2 from the hydrogen atoms to the interatomic region. The charge flow

Table 2. Energy Partitioning Analysis of the H−H, N−N, C−O, and B−F Bonds of Diatomics A−B at BP86/TZ2P+a A−B

H2

N2

CO

BF

ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔEσc ΔEπc R(A−B)e P(A−B) μ⃗b,d q(A) De D0e

−113.0 0.0 +5.8 −118.7 (100%) −118.7 (100%) 0.0 0.750 (0.741) 1.0 0 0 −111.6 −105.0 (−103.3)

−240.2 802.2 −312.8 (30.0%) −729.6 (70.0%) −478.7 (65.6%) −250.9 (34.4%) 1.102 (1.098) 3.03 0 0 −237.6 −234.2 (−225.0)

−267.6 582.7 −240.1 (28.2%) −610.1 (71.8%) −310.6 (50.9%) −299.4 (49.1%) 1.136 (1.128) 2.30 −0.19 (−0.11) +0.46 −265.6 −262.6 (−255.7 ± 1)

−186.0 484.4 −214.0 (31.9%) −456.4 (68.1%) −404.8 (88.7%) −51.8 (11.3%) 1.275 (1.262) 0.86 −1.03 +0.54 −184.3 −182.4 (179.9 ± 3)

a The interacting fragments of N2, CO, and BF are shown in Figure 6. Energy values in kcal/mol, bond lengths R in Å, dipole moments μ⃗ in Debye. NBO atomic partial charge q(A), and Wiberg bond order P(A−B). Bond dissociation energies De and ZPE corrected values D0. bThe values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe values in parentheses give the percentage contribution to the total orbital interactions ΔEorb. dA negative value indicates that the first atom of A−B is the negative end of the dipole moment. e Experimental values from ref 24 are given in parentheses.

P


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• The covalent bond in Li2 is much weaker than in H2, because the less strongly bonded 2s electron of Li, shielded by the 1s core electrons, is much more diffuse than the 1s electron of H and encounters weaker stabilization through interference. • The one-electron bond in Li2+ is stronger than the twoelectron bond in Li2, because the induction enhances the interference of the wave functions. This is striking evidence that covalent bonding does not come from electron pairing but from electron-sharing. • The charge flow associated with the covalent bond formation can nicely be visualized using the deformation density from EDA-NOCV calculations. 7.2. H2, N2, CO, and BF

The isoelectronic molecules N2, CO, and BF are well suited to discuss the results of a quantum chemical bonding analysis in comparison with traditional bonding models. Table 2 shows the numerical NBO and EDA-NOCV results of the three diatomic species along with H2, which demonstrates the exceptional features of dihydrogen. The chemical bond of H2 is the standard model for introducing covalent bonding, because the occurrence of bonding in terms of interfering wave functions can best be studied by choosing the simplest example. The data in Table 2 also show the calculated and experimental bond lengths and BDEs, which are in good agreement. There are three major qualitative differences when one goes from H2 to N2. One difference is the occurrence of Pauli repulsion between electrons with the same spin. The very large value of ΔEPauli = 802.2 kcal/mol in N2, which is the largest absolute value of all energy components, demonstrates the relevance of the Pauli repulsion for chemical bonding. The second difference is the large attractive Coulombic attraction of ΔEelstat = −312.8 kcal/mol, which contributes 30% to the total interatomic attraction in N2. This comes from the overlap of the electron density of one nitrogen atom with the nucleus of the other nitrogen atom, which carries a much larger positive charge than hydrogen. A systematic study of 224 homo- and heterodiatomic molecules by Spackman and Maslen in 1986 showed that the quasiclassical Coulomb attraction in nearly all of them is quite strong.177 The rather weak electrostatic attraction in H2 at longer bond lengths, which becomes repulsive at equilibrium distances, comes from the small nuclear charge of hydrogen. A mathematical explanation for this finding was given by Kutzelnigg178 and later by Bickelhaupt and Baerends.131 The third major difference concerns the occurrence of π bonding in N2, which is not surprising. The breakdown of the orbital interactions ΔEorb suggests that two-thirds of the covalent bonding comes from σ orbitals and the remaining one-third from π orbitals. This agrees with the general understanding that π bonds are weaker than σ bonds.179 The relevance of Pauli repulsion for chemical bonding comes clearly to the fore when the factors determining the equilibrium length of a chemical bond are considered. A popular explanation from the early days of quantum chemistry suggests that the bond length of a σ bond, re, is mainly determined by the maximum overlap of the orbitals. The bond becomes further shortened when there is an additional π bond, because the maximum overlap of a π orbital is at r = 0. This explanation is, however, not valid as the strong effect of Pauli repulsion is ignored. Figure 5a shows the overlap of the σ and π orbitals of N2 at different interatomic distances r. It becomes obvious that the maximum overlap of the σ orbitals is not yet reached at equilibrium

Figure 5. (a) Overlap integrals of the atomic 2s and 2p orbitals of N2 as a function of the interatomic distance. (b) Calculated EDA values for N2 as a function of the interatomic distance. The reference value re = 0.0 is the calculated equilibrium bond length 1.102 Å. Reprinted with permission from ref 28. Copyright 2006 John Wiley and Sons.

distance and is rather found at a bond length that is ∼0.2 Å shorter than the equilibrium value. The additional σ bonds should further shorten the bond. The explanation for the equilibrium bond length, which is clearly longer than what one would expect using the maximum overlap guideline, is given in Figure 5b. It shows the variation of the different energy components as a function of the interatomic distance. All attractive components of ΔEint still become stronger when the interatomic distance becomes shorter than the equilibrium value. It is the Pauli repulsion ΔEPauli which prevents further shortening of re below the equilibrium distance. It is only in H2 that the equilibrium bond length is determined by the maximum overlap, because there is no Pauli repulsion in dihydrogen.180 The electronic structure of CO exhibits peculiar features, which can be explained with the help of the EDA-NOCV results. CO has a stronger bond than N2; the BDE of CO (De = 265.6 kcal/mol) is larger than that for N2 (De = 237.6 kcal/mol). The inspection of the energy terms shows that the attractive Q


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Figure 6. Schematic representation of the most important Lewis structures of N2, CO, and BF and the electron configurations of the atoms which lead to electron-sharing bonds A−B or dative bonds A→B. Reprinted with permission from ref 152. Copyright 2019 Springer Nature.

Figure 7. Contour line diagram showing the Laplacian ∇2ρ(r) of (a) CO and (b) BF at BP86/def2-TZVPP. Blue solid lines indicate areas of charge depletion (∇2ρ(r) > 0), and red solid lines indicate areas of charge accumulation (∇2ρ(r) < 0). The bond critical point is shown in black. Solid lines connecting the atomic nuclei are the bond paths, while the solid lines crossing the bond paths indicate the zero-flux lines which separate the donor and acceptor moieties. The thick green arrow indicates the atomic dipole component at C in CO and B in BF. Reprinted with permission from ref 47. Copyright 2006 John Wiley and Sons. .

components in CO, i.e., electrostatic attraction ΔEelstat and orbital interaction ΔEorb, are both weaker than in N2. The stronger bond in CO comes from the significantly weaker Pauli repulsion ΔEPauli than in N2. The strength of ΔEPauli strongly depends on the overlap of the orbitals, even more than ΔEorb (see Figure 1).28 The overlap of the unequal σ AOs in CO is smaller than the overlap of the equal σ AOs in N2, which leads to weaker Pauli repulsion that even compensates for the weaker Coulomb attraction in the former molecule. The breakdown of ΔEorb of CO into the σ and π components suggests that the latter contribution provides 49% of the orbital interactions; that is, σ and π bonding have nearly the same strength. This is in contrast to N2, where π bonding contributes only 34% of ΔEorb. This can be explained with the different types of covalent interactions in CO and N2. Figure 6 shows schematically the interacting orbitals in the two molecules. The σ and π bonds in N2 come from electron-sharing interactions between the nitrogen atoms in the 4S ground state. In contrast, the interactions between carbon and oxygen in the 3P ground states lead to a dative σ bond C←O and two electron-sharing π bonds CO, which agrees with the original description suggested by Sidgwick.46 Since dative bonds are always weaker than electron-sharing bonds between the same atoms (see the section below), the percentage contribution of π bonding in CO is higher than in N2. At the same time, the bond order of CO becomes smaller (2.30) than that in N2 (3.03), because the overlap of the uneven orbitals is smaller than those of even orbitals. The answer to the question concerning the multiple bond character of a bond depends on the criterion

which one has in mind; this may be the overlap population or the energetic stabilization of the σ and π bonds. There is a third property of CO that has puzzled chemists for some time and which has led to controversies: The dipole moment of CO is rather small and has its negative end at the less electronegative carbon atom (−0.11 D).181 A simple consideration in terms of electronegativities would suggest a sizable dipole moment with the negative end at oxygen. The calculation of the atomic partial charges with the NBO method gives indeed a negative charge at oxygen and a positive charge at carbon (Table 2). The apparent contradiction is easily resolved when one recalls that dipole moments are vector properties whereas partial charges are scalar properties. The crucial quantity here is the topography of the charge distribution. This becomes obvious by inspection of the Laplacian distribution ∇2ρ(r) of CO, which is shown in Figure 7a. The electronic charge in the internuclear region is clearly polarized toward the more electronegative oxygen atom. This gives a bond charge component to the dipole moment that is directed toward oxygen. The key components are the charges around the nuclei. The charge concentration around the oxygen nucleus is spherically symmetric, but the charge concentration around the carbon atom has a local center of charge concentration, which is rather far away from the nucleus. This is indicated by the green arrow. This local charge can be associated with the lone-pair electrons at carbon, which strongly influence the chemical reactivity of CO. The aspherical charge distribution at carbon introduces a large dipole component toward the carbon end, which overcompensates the effect of the polarity on the dipole moment. A breakdown of R


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the total dipole moment into individual orbital components showed that all valence orbitals have a dipole component C → O except the HOMO (Highest Occupied Molecular Orbital), which has a large component in the opposite direction C←O that overcompensates the former (Table 3).47 The unexpected direction of the dipole moment of CO comes from the topography of the charge distribution. Table 3. Orbital Components to the Total Dipole Moment of CO at BP86/6-311++G(3df, 3pd)a Orbital

Bond Moment

MO 1 σ MO 2 σ MO 3 σ MO 4 σ MO 5 π MO 6 π MO 7 σ (HOMO) ∑

4. 67 −6. 23 1. 57 4. 52 1. 75 1. 75 −8. 21 −0.18 (exp. −0.11)

a

Values in Debye. Negative values indicate that the negative end is at the carbon atom. The origin of the CO12+ ion is chosen at the position where the dipole moment of the nuclei becomes zero. The orbital components were calculated with nuclear charges which are scaled by 6/7 (carbon) and 8/7 (oxygen). This leads to orbital components of the dipole which after summation give the total dipole moment of the neutral molecule, which is origin independent. The data are taken from ref 47.

Figure 8. MO correlation diagram of the 2s and 2p AOs of first octalrow atoms. Reprinted with permission from ref 28. Copyright 2006 John Wiley and Sons.

Table 2 shows that the BDE of BF is much smaller (De = 184.3 kcal/mol) than that of isoelectronic N2 and CO. All three energy terms of the EDA-NOCV are weaker in BF than in N2 and CO, and therefore, they do not explain the overall reason for the bond weakening. A possible rationalization of the bond weakening comes from the strength of the σ and π bonds in BF. The absolute and relative contribution of π bonding in BF is much smaller than in the other isoelectronic dimers. Inspection of the interacting orbitals explains the finding. Figure 6 shows that the π interactions in BF come from dative bonding from the very electronegative fluorine atom to the electropositive boron, whereas the σ bond is due to electron-sharing interactions in B− F. Since the former provide only 11% to the covalent bond, it seems reasonable to describe the molecule with a Lewis structure that has only a single bond. The nature of the σ and π bonding components in BF are the opposite of CO. The electronic structure of BF nicely supports the above explanation for the dipole moment in CO. The Laplacian distribution ∇2ρ(r) of BF shown in Figure 7b shows that the charge distribution in the bonding region is even more polarized toward fluorine due to the larger difference in electronegativities compared to CO. However, the local charge concentration at boron in BF has a larger distance from the nucleus than the charge concentration at carbon in CO, because the lone-pair orbital at boron is more diffuse than the carbon lone-pair in CO. As a result, the total dipole moment of BF is even larger (−1.03 D) than that of CO with the negative end at boron. The following conclusions arise from this section: • The appearance of valence electrons that possess the same spin introduces strong Pauli repulsion in N2, which is missing in H2. • The equilibrium bond lengths in molecules that have more than two electrons is determined by the equilibrium between Coulombic forces, attractive orbital interactions,

and Pauli repulsion, but not by the maximum orbital overlap. • CO has a triple bond, which consists of a dative σ bond and two electron-sharing π bonds . The stronger bond in CO than in N2 can be explained with the weaker Pauli repulsion in carbon monoxide. • BF has an electron-sharing B−F σ bond, which is supported by two relatively weak dative π bonds . The weak π bonds explain why BF is more weakly bonded than N2 and CO. • The dipole moments of CO (−0.11 D) and BF (−1.03 D), which have their negative ends at the respectively more electropositive atoms C and B, are due to the anisotropic charge distribution in the molecules. The σ lone-pair electrons induce a local vector component, which leads to the surprising direction of the dipole moments. 7.3. First Octal-Row Sweep Li2−F2 and Related Molecules

The physical nature of the chemical bond can be best understood by analyzing the interatomic interactions in diatomic molecules. Polyatomic molecules introduce a plethora of variations, which are the infinite playground of chemistry, but they can significantly change the essential features of chemical bonding. Particular classes of compounds, which exhibit specific types of chemical bonds such as clusters and aromatic systems, are discussed in other articles of this thematic issue. We focus mainly on diatomic molecules and we extend our above presented studies to a systematic first-row sweep of Li2−F2, where we try to connect classical bonding models with a quantum chemical analysis. We also discuss recent theoretical studies of low-valent main-group compounds, which possess S


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Table 4. Energy Partitioning Analysis of the First Row Dimers E2 (E = Li−F) in C2v at BP86/TZ2Pa E el. State ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔΕ(σ)c ΔΕ(π)c ΔEcorr.f De d E−Ed

Li

Be

B

C

N

O

F

Σg+ −20.7 1.8 −8.3 (36.9%) −14.2 (63.1%) −14.2 0.00 0.3 20.4 (24.6) 2.731 (2.673)

Σg+ −7.9 41.6 −17.9 (36.1%) −31.6 (63.9%) −31.6 0.00 0.00 7.9 (2.2; 2.7[calc.])e 2.442 (2.45)e

Σg − −74.7 135.0 −33.1 (15.8%) −176.5 (84.2%) −104.5 (59.2%) −72.0 (40.8%) 1.9 72.8 (71.2) 1.617 (1.590)

Σg+ −140.8 252.2 −3.2 (0.8%) −389.8 (99.2%) −201.7 (51.8%) −188.0 (48.2%) 3.3 137.5 (145.9) 1.253 (1.243)

Σg+ −240.2 802.4 −312.9 (30.0%) −729.8 (70.0%) −478.8 (65.6%) −251.0 (34.4%) 4.2 236.1 (228.4) 1.102 (1.098)

Σg− −141.9 464.9 −159.7 (26.3%) −447.1 (73.7%) −319.5 (71.5%) −127.6 (28.5%) 4.8 137.1 (120.2) 1.224 (1.208)

Σg+ −52.9 146.1 −41.0 (20.7%) −157.8 (79.3%) −151.5 (96.0%) −6.2 (4.0%) 2.7 50.2 (38.3) 1.420 (1.412)

1

1

3

1

1

3

1

a

Energies in kcal/mol, distances E−E in Å. Values are taken from ref 28. bValues in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cValues in parentheses give the percentage contribution to the total orbital interactions ΔEorb. dExperimental values in parentheses from ref 24, unless otherwise specified. eExperimental value for De and E−E; Bondybey, V. E. Chem. Phys. Lett. 1984, 109, 436; calculated value for De; Martin, J. M. L. Chem. Phys. Lett. 1999, 303, 399. fCorrection for the spin polarization.

Figure 9. Schematic representation of the electron configuration and the orientation of the atoms with the chosen orbital populations for the EDA. Note that for the 3Σg− states of Si2, O2, and S2 a spin change of one electron in a singly occupied p(π) orbital takes place in the EDA calculation. Reprinted with permission from ref 28. Copyright 2006 John Wiley and Sons.

elements becomes slightly higher in energy than the 1πu MO. This explains why B2 has a X3Σg− ground state and C2 has a X1Σg+ ground state.24 The hybridization lowers the antibonding character of the 2σu+ MO and raises the antibonding strength of the 3σu+ MO. Thus, the strengths of the bonding 2σg+ MO and antibonding 2σu+ MO do not fully cancel. However, the energy difference between the 3σg+ MO and the 1πu MO is very small and the ordering depends on the nature of the atoms. The analogous valence 4σg+ MO and 2πu MO of Si2 exhibit a reverse

chemical bonds that can be understood by using the undistorted E2 fragments as reference. Figure 8 shows a correlation diagram of the (n)s(n)p valence AOs of main-group atoms of the first octal row Li−F with the MOs of the diatomic molecules. The shape and the splitting of the MOs consider the energy levels of the AOs and the symmetry-allowed mixing (hybridization) of the orbitals. The hybridization lowers the energy level of the bonding 2σg+ MO and raises the energy of the bonding 3σg+ MO, which for some T


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order; that is, the 4σg+ is below the 2πu. Thus, the ground state of Si2 is the X3Σg− triplet state. On the other hand, Al2 has the same X3Σg− ground state as B2.24 7.3.1. Li2, Be2 and Related Molecules. The MO diagram in Figure 8 and the aufbau principle are a useful starting point for the discussion of chemical bonds in diatomic molecules Li2−F2. But they are not sufficient for understanding all features of the bonds, as was shown above for the comparison of Li2 with Li2+. It must be complemented by a quantum chemical analysis, which considers all types of interatomic interactions. Table 4 gives the results of EDA calculations of Li2−F2.28 The comparison of the calculated bond lengths and bond dissociation energies shows the same trends, which makes the data a reasonbale fundament for the bonding analysis. It is important to consider the geometrical arrangement of the atomic configurations for the interatomic interactions, which are shown in Figure 9, in order to understand the calculated numbers. The construction of the AO arrangements corresponds to the MO correlation diagram. The EDA results for Li2 were already given in Table 1, albeit with a different DFT functional. A comparison with the numbers in Table 4 shows that the results change very little when a different functional is used. The EDA values for Be2 suggest very weak bonding, which is due to the strong Pauli repulsion compensating most of the electrostatic attraction and covalent contribution. The net attration of ΔEa1(σ) = −31.6 kcal/mol in Be2 reveals that the hybridization in the bonding 2σg+ MO and antibonding 2σu+ MO yields a net stabilization, which is annihilated by the Pauli repulsion. The naı̈ve view of a bond order of zero for Be2 is only supported when the Pauli repulsion is considered. Inspection of the occupied valence MOs in Be2 gives a hint toward attempts to stabilize molecules that posssess a Be−Be bond. Substituents which withdraw electronic charge from the antibonding 2σu+ MO but donate electronic charge into the bonding 2σg+ MO should enhance the Be−Be bond. This is indeed achieved in Be2F2, which has a theoretically predicted bond dissociation energy of D0298 = 65.0 kcal/mol at the CCSD(T)/cc-VTZ level of theory.182 Figure 10 shows the valence MOs of the molecule. The antibonding inner lobes of Be in the HOMO−4 have nearly completely disappeared whereas the HOMO is a bonding orbital. This explains the surprisingly large strength of the Be−Be bond in Be2F2, which is also predicted to be thermodynamically stable relative to Be + BeF2 in the gas phase. The isolation of Be2F2 in a bulk is energetically unfavorable, because the large atomization energy of beryllium from the metal (ΔHf = 77.4 kcal/mol)183 and the heat of sublimation of BeF2 (ΔHsub = 55.6 kcal/mol)184 make it unlikely that pure beryllium subfluoride can be prepared. But Be2F2 should be a stable molecule in the gas phase, and it may also be isolated in an inert matrix.182 7.3.2. B2 and Related Molecules. According to the simple MO correlation diagram, most of the covalent bonding in the X3Σg− ground state of B2 comes from π bonding, because the contributions from the bonding 2σg+ MO and antibonding 2σu+ MO cancel to some extent. The EDA calculation of B2 suggests that the overall bonding comes mainly from the ΔEorb term, which comprises 60% σ bonding and 40% π bonding. The π interactions come from the two singly occupied degenerate π orbitals. Thus, the hybridization of the bonding 2σg+ MO and antibonding 2σu+ MO in B2 is very effective and leads to a predominantly σ bonded species, if only the stabilizing orbital interactions are considered. A different interpretation arises when the Pauli repulsion is taken into account. Since Pauli

Figure 10. Shape and eigenvalues of the lowest unoccupied Hartree− Fock MO (LUMO) and the relevant occupied valence orbitals of Be2F2 using a cc-pVTZ basis set.182 Reprinted with permission from ref 182. Copyright 2016 John Wiley and Sons.

repulsion in B2 (X3Σg−) comes only from electrons in the σ space, the total σ interactions, i.e., attraction due to interference of the wave functions (ΔEorb) plus repulsion due to spatial exclusion of electrons with the same spin (ΔEPauli), become destabilizing. The simple bonding model of the MO correlation diagram in terms of only π bonding in B2 is thus supported when the Pauli repulsion is considered. The analysis of the orbital interactions in B2 was the starting point for the eventually successful synthesis of a molecule with a genuine BB triple bond. Calculations of group-13 complexes NHCMe→E2←NHCMe (NHCMe = N-heterocyclic carbene with methyl groups at nitrogen) with E = B, Al, Ga, In predicted that the boron species B2(NHCMe)2 has a linear coordinated geometry with a very short B−B distance.185 A bonding analysis suggested that the B2 species in B2(NHCMe)2 is in the electronically excited (3)1Σg+ reference state, which has two vacant σ MOs with suitable symmetry that may serve as acceptor orbitals for donor−acceptor interactions with the NHCMe ligands (Figure 11). The energy calculations predicted that B2(NHCMe)2 is thermodynamically stable with regard to dissociation into B2 in the electronic X3Σg− ground state and two NHCMe ligands by >180 kcal/mol. The bond strength of the intrinsic interactions NHCMe→E2←NHCMe clearly compensates for the excitation energy of B2 for the process X3Σg−→ (3)1Σg+, which amounts to 106.4 kcal/mol. The theoretically predicted stability of B2(NHCMe)2 was verified by the synthesis and X-ray structural characterization of B 2 (NHC R ) 2 carrying bulky groups R at nitrogen by Braunschweig and co-workers.186,187 The experimental value for the B−B distance of 1.45 Å was in very good agreement with U


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Figure 11. Schematic diagram of the valence orbitals and orbital occupation of B2 in (a) the X3Σg− ground state and (b) (3)1Σg+ excited state. (c) Schematic representation of the charge donation from the out-of-phase (+,−) and in-phase (+,+) combinations of the ligand lone-pair σ orbitals into the vacant orbitals of B2 in the 1Σg+ excited state. (d) Calculated energies of the excitation energy of B2 from the ground state to the reference state and bond dissociation energy De of B2(NHCMe)2 at BP86/def2-TZVPP.189 Reprinted with permission from ref 189. Copyright 2015 Royal Society of Chemistry.

detailed insight into the change of the electronic structure of the fragments that comes from the formation of the bonds. The strongest interaction ΔEσ1 (−112.7 kcal/mol) comes from the in-phase (+,+) NHCMe→B2← NHCMe σ-donation into the LUMO+1 of (3)1Σg+ B2 while the contribution of the out-ofphase (+,−) donation ΔEσ2 (−86.9 kcal/mol) into the lowerlying LUMO (Lowest Unoccupied Molecular Orbital) is a bit weaker due to the better overlap of the former (+,+) donation. The other two contributions ΔEπ1 and ΔEπ2 arise from NHCMe←B2→NHCMe π-backdonation, which is smaller than the σ-donation. They are not degenerate, because the planes of the NHC ligands are in a twisted conformation. The shape of the deformation densities shows not only the formation of the boron−ligand bonds but also the alteration of the boron−boron bond. The shapes of Δρ1 and Δρ2 and the

the calculated data of 1.47 Å for the model compound. But the notation of a genuine BB triple bond in B2(NHCR)2 was challenged on the basis of experimental results, where the boron−boron distance in OBBO was taken as reference for a single bond.188 This led to a detailed bonding analysis of B2(NHCMe)2 with the EDA-NOCV method, which convincingly demonstrated that the molecules B2(NHCR)2 have a boron−boron triple bond.189 It was shown that there is a significant π-bonding contribution in the boron−boron bond of OBBO which makes it unsuitable as reference for a single bond. Table 5 gives the numerical EDA-NOCV results for B2(NHCMe)2. Figure 12 displays the deformation densities of the four most important pairwise orbital interactions Δρ1−4 along with the interacting orbitals of the fragments and the resulting MOs of the complex. The combined pictures provide V


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amino carbene) ligand enhances the π backdonation to the ligand, which leads to a longer B−B and shorter B-L bonds.190,191 We point out that the compound B2(NHCR)2 that was isolated by Braunschweig in 2012 was not the first L→B2←L species experimentally observed. Zhou and co-workers reported in 2002 the synthesis of the dicarbonyl complex B2(CO)2 in a low-temperature argon matrix.192 The experimental and theoretical vibrational spectra suggest that the molecule has a linear structure. The authors proposed that B2(CO)2 exhibits some boron−boron triple bond character, and they suggested that the molecule should be drawn as OCBBCO. A similar conclusion was drawn by Li and co-workers, who observed the boron compound [B2(BO)2]− in the gas phase using photoelectron spectroscopy.193 The authors reported quantum chemical calculations of the series [B2(BO)2]0,−,2−, with the dianion being isoelectronic with B2(CO)2. Subsequent theoretical studies of B2L2 with various ligands supported the description of the bonding situation in terms of dative bonds L→B2←L. EDA calculations of B2(CO)2 showed substantial π backdonation OC←B2→CO, which are nearly as strong as the σ donation OC→B2← CO.194 Table 6 shows the EDA results of B2L2 species with isoelectronic ligands L = CO, N2, BO−, not only for the dative bonds between the ligands and B2 but also for the boron−boron bonds. It becomes obvious that all bonds have a significant triple-bond character, which let the authors propose that the LBBL systems are molecules with all-triple bonds. 7.3.3. C2 and Related Molecules. The EDA results for the X1Σg+ ground state of C2 suggest that the attractive σ and π orbital interactions have nearly the same strength (Table 4). The attraction by the σ orbitals is compensated by the Pauli repulsion of the doubly occupied σ MOs, which leaves the π contribution as the sole component of the stabilizing orbital interactions. But

Table 5. Results of the EDA-NOCV calculations for NHCMe → B2 ← NHCMe at BP86/TZ2P Using the Fragments B2[(3)1Σg+] and (NHCMe)2 as Interacting Speciesa B2(NHCMe)2 Interacting fragments

B2[(3) Σg+] and (NHCMe)2

ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔEσ1 L→(B2)←L (+,+) donationc ΔEσ2 L→(B2)←L (+,-) donationc ΔEπ1 L←(B2)→L π backdonationc ΔEπ2 L←(B2)→L π′ backdonationc ΔErest q(B2)

−307.5 259.0 −252.3 (44.5%) −314.2 (55.5%) −112.7 (35.9%) −86.9 (27.7%) −48.0 (15.3%) −42.4 (13.5%) −24.2 (7.7%) −0.36

1

a All energy values in kcal/mol. Calculated NBO partial charge of B2 in e. bThe value in parentheses gives the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe value in parentheses gives the percentage contribution to the total orbital interactions ΔEorb.

associated σ orbitals HOMO−6 and HOMO−13 of the complex (Figure 12a, b) suggest that the NHCMe →B interactions concurrently exert an opposite effect on the boron−boron bonding. The HOMO−6 is bonding and HOMO−13 is antibonding with respect to the B−B bonds. Thus, the effect of the σ interactions NHCMe→B2←NHCMe on the B−B σ bond partially cancel each other. In contrast, the effect of the π backdonation NHCMe←B2→NHCMe clearly weakens the B−B bond, which becomes visible by inspection of Δρ3 and Δρ4 (Figure 12c, d). This has been utilized to fine-tune the B−B bond by variation of the π-acceptor strength of ligands L in L→B2← L. Replacement of the unsaturated NHC ligand by the saturated homologue NHCsat or the CAAC (cyclic alkyl

Figure 12. Plot of the interacting donor and acceptor orbitals and calculated eigenvalues ε of (NHCMe)2 and (1Σg+) B2 (right two columns) and matching MOs of the complex NHCMe→BB←NHCMe (second column from the left). Plot of the deformation densities Δρ with connected stabilization energies ΔE of the four most important orbital interactions in B2(NHCMe)2 which indicate the associated charge flow red → blue.189 Reprinted with permission from ref 189. Copyright 2015 Royal Society of Chemistry. W


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Table 6. EDA Results of LBBL at the BP86/TZ2P Levelba OCBBCO Interacting fragments ΔEint ΔEPauli ΔEelstatc ΔEorbc ΔEσd ΔEπd ΔEprep ΔE(=-De)

4 −

N2BBN2 4 −

[OBBBBO]2− −

4 −

N2BBN2

OCBBCO

[OBBBBO]2−

2 BCO ( Σ )

2 BNN ( Σ )

2 [BBO] ( Σ )

B2 [(3) Σg ] + 2 CO ( Σ )

B2 [(3) Σg ] + 2 N2 ( Σg )

B2 [(3)1Σg+] + 2 BO− (1Σ+)

−155.2 103.9 −108.7 (42.0%) −150.4 (58.0%) −92.1 (61.2%) −58.3 (38.8%) 5.5 149.7

−155.3 93.6 −97.4 (39.1%)

−89.7 144.9 −71.0 (30.2%)

−267.7 203.6 −119.1 (25.3%)

−223.1 221.3 −94.5 (21.3%)

−326.1 195.0 −209.0 (40.1%)

−151.5 (60.9%) −91.3 (60.3%) −60.2 (39.7%) 10.2 145.1

−163.6 (69.8%)

−352.2 (74.7%)

−349.9 (78.7%)

−312.2 (59.9%)

−102.2 (62.5%) −61.4 (37.5%) 6.7 83.0

−193.7 (55.0%) −158.5 (45.0%) 6.1 (112.4)e 261.6 (155.3)e

−168.3 (48.1%) −181.6 (51.9%) 9.8 (123.1)e 213.3 (107.0)e

−233.1 (74.7%) −79.1 (25.3%) 7.6 (180.7)e 318.5 (145.4)e

1

+

1 +

1

+

1

+

a

Values are taken from ref 194. bEnergy values in kcal/mol. cThe percentage values in parentheses give the contribution to total attractive interactions ΔEorb + ΔEelstat. dThe percentage values in parentheses give the contribution to total orbital interactions ΔEorb. eEnergy with respect to B2 (X3Σ g−) ground state.

Figure 13. Coefficients and weights of the four most important electron configurations a−d in the full-valence CAS(8/8)SCF/cc-pVTZ calculation of (X1Σg+) C2.195 Reprinted with permission from ref 195. Copyright 2016 John Wiley and Sons.

CH. According to their VB calculations, the strength of the σ interactions in C2 is 156.6 kcal/mol while σ bonding in acetylene amounts to 138.7 kcal/mol.206−208 Following the reasoning of Shaik et al., the carbon−carbon triple bond in HCCH stays essentially intact when the hydrogen atoms are removed and the remaining unpaired electrons couple with their inward lobe, which leads to a second σ bond that is weaker than the first one but strong enough to establish a fourth bond. The alternative view, which is based on a CASSCF wave function, identifies a significant change in the σ space of the remaining C2 fragment of acetylene after removing the hydrogen atoms.195 The σ interactions comprise indeed two rather than one component, but they are degenerate and very weak. The overall σ interactions in C2 are much weaker than in acetylene and the bonding situation may be described like a Be2 species that is held together by a “π-clip”.209 Table 7 shows some relevant properties of C2 and HCCH for comparison. The carbon−carbon bond in C2 is longer and has a smaller force constant and lower vibrational frequency than in acetylene. Dicarbon has also a significantly lower bond dissociation energy (BDE). The BDE may be questioned as a measure of the bond strength, because the electronic reference state of the dissociated fragments is not the same as in the molecule. However, Figure 14 shows that the energy curve for variation of the C−C distance in HnCCHn (n = 0−3) places C2 between ethylene and acetylene. Thus, all observable quantities suggest that the carbon−carbon bond in C2 is weaker than in

the bonding in C2 is not well described with only one configuration, because the occupied 1πu MO and the vacant 3σg+ MO are close in energy, which requests a multiconfigurational calculation of C2 in order to reliably account for the bonding situation. A (8/8) CASSCF (Complete Active Space) calculation, where the 8 valence electrons occupy the complete space of the 2s2p valence orbitals of C2, showed that the leading configuration (2σg+)2(2σu+)2(1πu)2(1πu′)2(3σg+)0(1πg)0 (Figure 13a) has a weight of 71% of the total wave function.195 The next important configuration, where only bonding σ and π MOs are occupied, contributes 13.6% to the wave function (Figure 13b). The remaining configurations are less important. The formal bond order of 2, which comes from the π bonds in the leading configuration, becomes slightly higher by the second important configuration that has a formal bond order of 4. The nature and strength of the bond in C2 has been a topic of an intensive and controversial debate in recent years, following the suggestion by Shaik and co-workers that dicarbon has a quadruple bond comprising two σ and two π bonds.196−199 The assignment of a quadruple bond in C2 came from an analysis of VB calculations. Several groups, who employed various MO and VB methods, questioned in subsequent studies the suggestion of a quadruple bond in C2.200−205 Calculations of bond orders using different approaches gave values between 2−4, and thus, they were not at all conclusive. The situation became substantiated when Shaik et al. quantified the strength of the intrinsic σ interactions in C2 and triply bonded acetylene HC X


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finally led to the isolation of the compounds C2(CAACR)2 with different substituents R by two groups.214,215 The X-ray structure analysis showed that the molecule slightly deviates from a linear arrangement of the ligands along the central C2 axis. The compounds may thus either be described as cumulene CAACCCCAAC or as complexes CAAC→CC← CAAC. EDA-NOCV calculations were carried out with the fragments C2 and (CAAC)2 in the electronic singlet states for dative bonding and quintet states for electron-sharing bonding in the cumulene. Figure 15 shows schematically the electronic reference states of C2. The (2)1Δg state, which has two vacant σ orbitals, is the reference state for dative bonding L→CC←L while the 5Δg state with four unpaired electrons in two σ and two π orbitals is the reference state for electron-sharing double bonding.216 As mentioned in the section above about the EDANOCV method, the absolute values of the ΔEorb term indicate the more favorable description of the bonding situation. Calculations were also carried out for C2(CAACMe)2 and for the related molecule C2(NHCMe)2. The numerical results of the calculations are shown in Table 8.217 The data in Table 8 suggest that the bonding situation in C2(NHCMe)2 may be reasonably well described with either bonding model. The ΔEorb value for dative interactions (−686.0 kcal/mol) is only slightly smaller than the data for electronsharing double bonding (−707.9 kcal/mol). In contrast, the ΔEorb value for dative interactions in C2(CAACMe)2 is much larger (−720.5 kcal/mol) than that for electron-sharing double bonding (−601.9 kcal/mol). This implies that the latter

Table 7. Calculated Properties of C2 and HCCHa Re(C−C) [Å] kCC [mdyn/Å] υCC [cm−1] De(C−C) [kcal/mol]

C2

HCCH

1.256 11.9 1837 144.0

1.216 15.8 1995 228.2

a

Equilibrium distance Re(C−C), force constant kcc, vibrational frequencies υCC, and bond dissociation energy De. Values are taken from refs 195 and 200.

HCCH. A recent theoretical study of the 13C isotropic magnetic shieldings came to the conclusion that the bond in C2 has a higher multiplicity but it is weaker than in HCCH.210 The suggestion that C2 has a quadruple bond that is stronger than the carbon−carbon triple bond in acetylene appears therefore questionable in the light of the properties of the two molecules. The stabilization of C2 with two donor ligands in compounds C2L2 analogous to B2L2 has been a topic of theoretical and experimental studies. The PPh3 bound derivative has been generated at −78 °C but was reported to be unstable at temperatures above −40 °C.211 Theoretical studies suggest that the stabilization by NHC ligands is stronger than that by phosphanes and that compounds C2(NHCR) might be stable at room temperature.212 Deprotonation of the dication [(C2H2)(NHCR)]2+ with strong bases failed and gave the reduced product instead.213 But the use of the CAAC ligand instead

Figure 14. (a) Calculated stretching force constants kCC at the full-valence CASSCF/cc-pVTZ level of HnCCHn (n = 3−0). (b) Potential energy curves at CASSCF/cc-pVTZ for changing the C−C distances of HnCCHn (n = 3−0) from the equilibrium distance by ±0.15 Å in steps of 0.01 Å.195 Reprinted with permission from ref 195. Copyright 2016 John Wiley and Sons. Y


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16. Note that the deformation densities and the associated energies are the sum of the unequal contributions of the α and β electrons in opposite directions, with the differences between them being less important for the present work. Figure 16 shows only the net deformation densities and the sum of the strength of the orbital interactions. The two strongest orbital interactions ΔEσ(+,+) and ΔEσ(+,−) come from the (+,+) in-phase and (+,−) out-of-phase combinations of the singly occupied σ orbitals, which lead to charge accumulation (blue) in the C−C bonding region between the (CAAC)2 and C2 fragments. The two σ contributions have similar strength. In contrast, the π contribution, ΔEπ(+,−), is much larger than ΔEπ(+,+). This is because the former interaction releases the antibonding π interaction of the central C2 fragment and introduces π bonding between C2 and the CAAC ligands, whereas the latter weakens the π bonding in the C2 fragment. Note that the energy levels of the interacting singly occupied orbitals related to Δρ3 are much closer than the SOMOs that are associated to Δρ4. 7.3.4. N2 and Related Molecules. The chemical bond in N2 has been discussed in the above section 7.2. Dinitrogen is a prototype for an electron-sharing triple bond, which consists of a σ and two degenerate π bonds. The EDA results in Table 4 suggest that two-thirds of the covalent interactions come from σ bonding while one-third is due to π bonding, which nicely agrees with chemical intuition. Dinitrogen is considered a weak donor ligand, which usually binds in complexes in the X1Σg+ electronic ground state through its σ lone-pair orbitals in an end-on fashion.219 The electronic excitation energies of N2 are rather high, and thermochemical reactions of N2 normally involve only the electronic ground state. Dinitrogen as acceptor species was virtually unknown until recently. One decade ago, a series of complexes L→E2←L where E2 are diatomic main-group compounds of groups 14 and 15 stabilized by NHC ligands L were isolated and structurally characterized by X-ray crystallography.220−223 The starting point was the synthesis of Si2(NHCR)2 by Robinson in 2008,224 which was followed by the germanium and tin homologues Ge2(NHCR)2225 and Sn2(NHCR)2226 in 2009 and 2012. The group-15 adducts P2(NHCR)2 and As2(NHCR)2 were isolated in 2008 and 2010.227,228 A theoretical paper by Wilson and coworkers reported in 2012 the calculated bond dissociation energies of the group-14 and group-15 complexes E2(L)2 with L = NHC, PPh3, which showed that the group-15 adducts are clearly less stable than the group-14 homologues.212 Most species should in principle be isolable, except for the dinitrogen complexes N2L2, which were calculated to be thermodynamically unstable toward dissociation of the ligands (Table 9, numbers in bold). Inspection of the experimental literature reveals surprising findings. The compounds N2(NHCR)2 and N2(PPh3)2 had earlier been reported as stable species that can easily be prepared.229,230 The experimental study of the synthesis and properties of N2(PPh3)2 was in conflict with the theoretical result that the molecule is unstable toward loss of the PPh3 ligands by more than 80 kcal/mol, but the compound was reported to be stable up to 215 °C when it slowly decomposes after melting.230 The synthesis was reported in 1964, and the compound was poorly characterized by giving only its high melting point of 184 °C. It was sketched with the formula Ph3PNNPPh3, which is obviously wrong, giving rise to questioning the identity of the compound. A joint experimental and theoretical study by Holzmann et al. reinvestigated the

Figure 15. Schematic diagram of the electronic reference states of C2 for dative bonding and electron-sharing bonding in C2L2. (a) (2)1Δg state; (b) 5Δg state.

Table 8. EDA-NOCV Results at BP86/TZ2P of the Donor− Acceptor and Electron-Sharing Interactions in C2(NHCMe)2 and C2(CAACMe)2 Using Singlet (S) and Quintet (Q) Fragmentsa C2(NHCMe)2 Molecule Fragments ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔE1 L-C2-L σ bonding (+,+)c ΔE2 L-C2-L σ bonding (+,-)c ΔE3 L-C2-L π*bondingc,d ΔE4 L-C2-L πbondingc,d ΔErestc

C2(CAACMe)2

C2 (S)

C2 (Q)

C2 (S)

C2 (Q)

(NHCMe)2 (S)

(NHCMe)2 (Q)

(CAACMe)2 (S)

(CAACMe)2 (Q)

−647.8 361.5 −323.4 (32.0%) −686.0 (68.0%) −205.1 (29.9%) −192.5 (28.1%) −200.2 (29.2%) −42.5 (6.2%) −45.7 (6.6%)

−485.8 622.5 −400.5 (36.1%) −707.9 (63.9%) −228.0 (32.2%) −164.5 (23.2%) −148.1 (20.9%) −102.0 (14.4%) −65.3 (9.3%)

−712.0 344.0 −335.5 (31.8%) −720.5 (68.2%) −220.8 (30.6%) −191.3 (26.4%) −213.2 (29.6%) −50.7 (7.0%) −44.5 (6.4%)

−486.7 480.2 −347.0 (36.6% −601.9 (63.4%) −185.3 (30.8%) −163.6 (27.2%) −146.7 (24.4%) −52.1 (8.7%) −54.2 (8.9%)

a

All energies in kcal/mol.217. bThe value in parentheses gives the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe value in parentheses gives the percentage contribution to the total orbital interactions ΔEorb. dThe notation π and π* refer to the π MO of the C2 fragment.

molecule is best described as cumulene CAACCC CAAC, which agrees with the finding that the singlet−triplet split of the CAAC carbene is significantly smaller than that for NHC.218 The values of ΔEorb in Table 8 are marked in red to highlight the particular relevance of the orbital term, which gives the best description of the type of orbital interaction. It is instructive to compare the deformation densities, which are associated with the formation of the electron-sharing σ and π bonds in C2(CAACMe)2 with the dative σ and π bonds in B2(NHCMe)2 (Figure 12). Table 8 shows that the major components of ΔEorb in C2(CAACMe)2 come from two σ and two π orbital interactions. The associated deformation densities Δρ1−4 and the singly occupied orbitals of the fragments along with the resulting MOs of the molecule are displayed in Figure Z


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Figure 16. Plot of the deformation densities Δρ of the pairwise orbital interactions for electron-sharing bonding between C2 in the excited 5Δg state (see Figure 15) and two CAACMe ligands in C2(CAAC)2. (a) (CAACMe)−C2−(CAACMe) σ-interaction between the the 2σg+ MO of C2 and the SOMO-3 of (CAACMe)2. (b) (CAACMe)−C2−(CAACMe) σ-interaction between the 1σu+ MO of C2 and the SOMO-2 of (CAACMe)2. (c) (CAACMe)−C2−(CAACMe) π-interaction between the 1πg MO of C2 and the SOMO-1 of (CAACMe)2. (d) (CAACMe)−C2−(CAACMe) π-interaction between the 1πu MO of C2 and the SOMO of (CAACMe)2. See ref 217.

Table 9. Calculated Free Energies (kcal/mol) at MP2/ TZVPP of the Metal−Ligand Bond Strengths in E2L2a E2 + L → L-EE-L E C Si Ge Sn Pb N P As Sb Bi

L = NHC Group 14 −158.2 −84.2 −71.9 −66.8 −57.4 Group 15 22.3 −27.4 −21.3 −25.9 −16.0

L = PPh3 −83.6 −55.0 −49.4 −48.0 −44.4 87.8 −16.2 6.4 Diss. Diss.

a

The data are taken from ref 212.

structure and properties of the molecule.231 Following the original simple synthetic protocol followed by an X-ray measurement showed that the molecule is indeed N2(PPh3)2, which has, however, a trans bent geometry of the core unit (Figure 17a). The inspection of the occupied orbitals (Figure 17b−e) revealed a surprising electronic structure and bonding situation of N2(PPh3)2. The out-of-plane π and π* orbitals of the N2 moiety are both occupied (HOMO−14 and HOMO), which makes two in-plane MOs of the dinitrogen fragment available as acceptor orbitals. The HOMO−1 and HOMO−15 are the inphase (+,+) and out-of-phase (+,−) combination of the σ donation Ph3P→N2←PPh3. The electronic reference state of N2 is the highly excited (1)1Γg state which has the valence configuration (1σg)2(1σu)2(1πu)2(2σg)2(1πg)2, where the 1πu′ and 1πg′ are vacant (Figure 18b). There is a formal excitation of two electrons 1πu′ → 1πg from the X1Σg+ ground state (Figure 18a) to the (1)1Γg excited state. The bonding 1πu′ MO is the acceptor orbital for donation from the out-of-phase (+,−)

Figure 17. (a) X-ray crystal structure of N2(PPh3)2. Selected bond lengths [Å] and angles [deg] with calculated values at RI-BP86/def2TZVPP in brackets: N(1)−N(1) 1.497(2) [1.441], P(1)−N(1) 1.5819(12) [1.606], P(1)−N(1)−N(1a) 107.10(11) [112.6], P(1)− N(1)−N(1a)−P(1a) 180.0 [180.0]. (b−e) Plot of relevant occupied orbitals of N2(PPh3)2.231 Reprinted with permission from ref 231. Copyright 2013 John Wiley and Sons.

combination of the σ lone-pair MOs of the PPh3 ligands. The 1πu′ MO of N2 is much lower in energy than the antibonding 1πg′ orbital, which makes the associated HOMO−15 orbital of AA


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Figure 18. Schematic view of the most important valence orbitals of N2 and the orbital occupation in (a) the X1Σg+ ground state and (b) 1Γg excited state. Schematic view of (c) out-of-phase (+,−) donation of the ligand σ orbitals into the vacant 1πu′ orbital of N2 and (d) in-phase (+, +) donation of the ligand σ orbitals into the vacant 1πg′ orbital of N2.231 Reprinted with permission from ref 231. Copyright 2013 John Wiley and Sons.

Figure 19. Plot of the deformation densities Δρ which are associated with most important orbital interactions in N 2 (PPh 3 ) 2 . (a) Deformation density Δρ1 due to donation Ph3P→N2←PPh3 into the vacant in-plane π orbital of N2(1Γg). (b) Deformation density Δρ2 due to donation Ph3P→N2←PPh3 into the vacant in-plane π* orbital of N2(1Γg). (c) Deformation density Δρ3 due to π backdonation Ph3P← N2→PPh3 from the occupied out-of-plane π orbital of N2(1Γg).231 Reprinted with permission from ref 231. Copyright 2013 John Wiley and Sons.

the complex more strongly stabilizing than the HOMO−1. Figure 18c shows the Lewis structure of N2(PPh3)2, which depicts the bonding situation in the molecule. Table 10 shows the numerical results of the EDA-NOCV calculation of N2(PPh3)2 with N2 in the (1)1Γg reference state as

not compensate for the geometrical and electronic relaxation of the fragments, which amounts to ΔEprep = 349.3 kcal/mol. The authors suggested that N2(PPh3)2 is a kinetically unusually stable molecule where the dissociation reaction for loss of the PPh3 ligands may be a symmetry forbidden reaction.231 The activation barrier could not be calculated, because it requires sophisticated approaches for a large molecule that were not available to them. They estimated from an approximated transition state that the barrier was higher than 25.1 kcal/mol but lower than 67.6 kcal/mol, which is the calculated bond dissociation energy for breaking the N−N bond. 7.3.5. O2, F2 and Related Molecules. The peculiarity of O2, which possesses a triplet (3Σg−) ground state, was already mentioned. It arises comprehensibly from the MO diagram in Figure 8, which indicates that dioxygen has a double bond. The EDA results for O2 in Table 4 must be taken with some caution. It is not possible to assemble O2 in the (3Σg−) triplet state from two oxygen atoms in the 3P state without a concomitant spin change of one π electron (see Figure 9). This is how the EDA calculation of O2 (3Σg−) was done.28 Table 4 shows that the total π bonding contribution to the orbital interactions is 28.5%, a little less than in N2, where π bonding makes up for 34.4% of ΔEorb. According to the EDA results shown in Table 4, the largest degree of π bonding contributions of the diatomic species are found in B2 (40.8%) and C2 (48.2%). This is, because the σ orbital interactions in B2 and C2 come from dative bonding while the σ bonds in N2 and O2 are due to stronger electron-sharing bonds. The chemical bond in F2 comprises a single electron-sharing σ bond, which is relatively weak. Difluorine is one of the rare exceptions where the bond of a first octal-row atom is weaker than in the heavier homologues. The BDE of Cl2 is calculated at the same level of theory with De = 62.0 kcal/mol.28 The weak

Table 10. EDA-NOCV Results of N2(PPh3)2 Using the Fragments N2 in the (1)1Γg Reference State and 2 PPh3a Energy terms

N2 + 2 PPh3

ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔEσ1c,d ΔEσ2c,d ΔEπ1c,d ΔErestc

−300.1 853.4 −377.4 (32.7%) −776.1 (67.3%) −357.9 (46.1%) −289.8 (37.3%) −46.9 (6.0%) −81.5 (10.6%)

a Energy values in kcal/mol. The data are taken from ref 231. bThe percentage values in parentheses give the contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe percentage values in parentheses give the contribution to the total orbital interactions ΔEorb. dThe notation σ1, σ2, π1 refers to the orbital pairs which are associated with the deformation densities Δρ1, Δρ2, Δρ3 that are shown in Figure 19.

acceptor fragment. The orbital term ΔEorb comprises two-thirds of the total attraction between the donor and acceptor moieties. There are three major components of ΔEorb, which can easily be identified with the help of the associated deformation densities Δρ (Figure 19) as in-phase (+,+) and out-of-phase (+,−) σ donation Ph3P→N2←PPh3 into the unoccupied 1π′ orbitals of N2 and the weaker Ph3P←N2→PPh3 π backdonation. Table 10 shows that the intrinsic bond strength between N2 in the (1)1Γg reference state and the PPh3 ligands is very large (−300.1 kcal/mol). Thus, dinitrogen in the (1)1Γg state is a very strong Lewis acid. This is due to the rather high electronegativity of nitrogen and because the nitrogen atoms in have only an electron sextet. However, the large interaction energy does AB


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equilibrium distance and the frozen distance of N2. Table 11 shows that the Pauli repulsion in O22+ at the shorter equilibrium bond length (1.051 Å) is stronger than at the frozen N−N distance (1.102 Å), but this is overcompensated by the stronger covalent interactions. The Coulomb repulsion at the shorter equilibrium distance is weaker than at the longer distance, because the increase of the electron−nucleus attraction is larger than the electron−electron and nucleus−nucleus repulsion. All this is somewhat counterintuitive for a classical viewpoint, but it becomes understandable using quantum theoretical arguments.28 Table 11 shows that the Coulomb contribution in O22+ is repulsive, which makes the overall interaction energy ΔEint and the bond dissociation energy De positive. It means that the bond dissociation energy is not always an adequate indicator for the strength of a chemical bond. This holds for all energy-rich molecules, where bond-breaking reactions are exergonic. It can be argued that the activation energy for breaking the bond, ΔE⧧, may be used instead, which would involve the calculation of the activation barrier. However, bond breaking involves a change in the electronic structure when one goes from the molecule to the fragments, which may not be related to the intrinsic binding interactions at equilibrium. A case example is the CC double bond in C2F4, which is discussed below in section 8.237 A direct probe for the intrinsic strength of a chemical bond is the local mode force constant, which was introduced by Cremer and coworkers as a universal measure of the bond strength.200,238−241 The force constant is the second derivative of the energy with respect to the equilibrium coordinates, which does not require the energy of the fragments or the transition state as a reference for bond strength. It may thus be used for stable and metastable species alike. Table 11 also shows the EDA results for the difluorine dication F22+ in the triplet (X3Σg−) ground state.242,243 The F−F bond in the dication (1.266 Å) is much shorter than in neutral F2 (1.420 Å), but it is not shorter than in isoelectronic O2 (1.224 Å). The data for the energy values show that the Pauli repulsion in F22+ is much weaker (ΔEPauli = 210.2 kcal/mol) than in neutral O2 (ΔEPauli = 464.9 kcal/mol, Table 4) but the orbital interactions in F22+ (ΔEorb = −313.9 kcal/mol) are also weaker than in O2 (ΔEorb = −447.1 kcal/mol). It appears that the interactions in the isoelectronic pair F22+/O2 are somewhat different from the system O22+/N2, which may be due to the occurrence of Pauli repulsion between π electrons in the latter pair. But the conclusion remains that the removal of electrons from antibonding orbitals is able to strongly enhance the covalent interactions, which may lead to significantly shorter bonds overcoming strong Coulomb repulsion. The following conclusions arise from this section:

bond in difluorine is usually explained with the repulsion between the lone-pair electrons at short distances. But the strongest repulsion comes from the Pauli exchange interactions between the electrons in the σ orbitals preventing shorter bonds. Table 4 shows also that the Coulombic attraction in F2 is much weaker than in O2 and particularly in N2. The rather long F−F distance compared to O2 and N2 prevents an effective overlap of the p(σ) electrons with the nucleus of the other fluorine. The small stabilization energy of −6.2 kcal/mol of the π orbitals in F2 comes from the change in the mixing of the p(π) fluorine AOs of the TZ2P basis set; it is an intraorbital polarization. The highest lying occupied orbitals of F2 and O2 are antibonding. Removal of two electrons in the dications should enhance the covalent bonding, which might overcome the Coulomb repulsion between the positively charged ions. In fact, the covalent attraction in (X1Σg+) O22+ not only overcomes the intrinsic Coulomb repulsion, it even enforces a shorter O−O bond length in the dication (1.051 Å) than in neutral isoelectronic N2 (1.102 Å). This paradoxical finding has been noted before.232 The dioxygen dication O22+ has been experimentally observed and was the subject of several theoretical studies.232−236 It is a metastable species, which is thermodynamically unstable toward dissociation in two O+ cations by 84.6 kcal/mol,232 but it lies in a deep potential well of about 4.7 eV.233 Why does that O22+ have a shorter bond than N2? Table 11 shows the EDA results for O22+, which may be Table 11. EDA Results of Singlet O22+ and Triplet F22+ Species at the BP86/TZ2P-ZORA//BP86/def2-TZVPP Levela ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔE(σ)c ΔE(π)c r(E-E) De

O22+d

O22+e

F22+

56.7 708.9 126.5 −778.8 (100%) −484.7 (62.2%) −294.0 (37.8%) 1.051 56.7

59.2 579.2 144.0 −664.0 (100%) −418.4 (63.0%) −245.6 (37.0%) 1.102 59.2

129.4 210.2 233.0 −313.9 (100%) −216.5 (69.0%) −97.4 (31.0%) 1.266 129.4

a

Energies are in kcal/mol and bond lengths r(E-E) in Å. bThe percentage values in parentheses give the contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe percentage values in parentheses give the contribution to the total orbital interactions ΔEorb. dCalculated at the equilibrium distance. eCalculated using the frozen equilibrium distance of isoelectronic N2.

compared with the data for N2 (Table 4). The energy values indicate that the Pauli repulsion in the dication (ΔEPauli = 708.9 kcal/mol) is weaker than in N2 (ΔEPauli = 802.4 kcal/mol) whereas the stabilizing orbital interactions in O22+ (ΔEorb = −778.8 kcal/mol) are stronger than in N2 (ΔEorb = −729.8 kcal/ mol). The orbital contraction in the dication due to the positive charge has obviously a much stronger effect on the Pauli repulsion than on the orbital interactions. Although the interatomic distance in O22+ is smaller than in N2, the Pauli repulsion becomes weaker, because the overlap of the occupied orbitals in the dication becomes smaller due to the charge induced contraction. The overlap between occupied and vacant orbitals in O22+ is reduced to a lesser extent, and therefore, the orbital interactions in the dication become stronger than in N2. The effect of the different energy terms on the bond length of O22+ becomes obvious by comparing the calculated values at

• The trend of the bond strength of the first octal-row diatomic molecules can conclusively be explained with the results of the EDA-NOCV calculations, which provide further insight into the nature of the interactions in terms of Coulombic attraction/repulsion, Pauli repulsion, and orbital (covalent) bonding. Accordingly, the very weak attraction in Be2 is due to Pauli repulsion but not to mutual energy compensation of the occupied bonding and antibonding valence orbitals. Likewise, the rather weak bond in F2 is caused by the Pauli repulsion of the electrons in the σ orbitals rather than repulsion of the π lone-pair electrons. AC


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• The covalent bonding in the dications (X1Σg+) O22+ and (X3Σg−) F22+ is stronger and the bonds are shorter than in the neutral parent molecules because the Pauli repulsion is weaker in the charged species due to the orbital contraction. The covalent bonding in O22+ is even stronger than in N2. The Coulomb repulsion makes the dications thermodynamically unstable. The force constant is more suitable as measure for the intrinsic strength of a bond than the bond dissociation energy. • The consideration of the occupied and vacant orbitals in the ground and excited states of E2 provides an understanding of complexes where the diatomic species are stabilized by donor ligands L→E2←L. This led to the synthesis of the first compound with a boron−boron triple bond NHCR→B2←NHCR that is stable under ambient temperatures. • The donor strength in complexes L→E2←L may overcompensate the electronic excitation energy of E2 from the ground to the excited reference state in the adduct, which leads to thermodynamically stable compounds. However, there are molecules like L→ N2←L (L = NHC, PPh3) where N2 is in the high lying (1)1Γg state, which are calculated to be unstable toward dissociation of N2, and yet, they can easily be isolated. This is likely because the extrusion reaction is a symmetry forbidden process.

molecules. There is one particular feature of the atoms, which can be considered as essential for the different chemistry of the two groups. It concerns the ratio of the radii of the (n)s and (n)p valence AOs of the atoms, which significantly changes from the first to the higher octal-row atoms. The 2p orbitals of the first octal-row atoms may penetrate rather deeply into the core, because there are no lower lying occupied p functions, whereas the 2s orbitals are constrained to be orthogonal to the 1s AO. The radii of 2s and 2p orbitals are thus very similar, which leads to effective sp hybridization in chemical bonds of the lighter atoms. In contrast, the occupied (n)p AOs (n > 2) of the heavier atoms are antisymmetric to the p core orbitals, which causes larger radii for the (n)p than for the respective ns AOs. The spatial regions of (n)s and (n)p AOs (n > 2) are more separate than those of 2s and 2p AOs, which leads to less effective hybridization of the heavier main-group atoms.246 Looking from an aufbau principle of atomic structure, chemical bonding of the heavier main-group atoms exhibits “normal” behavior, whereas the f irst octal-row atoms are “anormal”, because the valence p AOs are not affected by lower lying p orbitals. The difference between the relative size of the light and heavier main group atoms is schematically shown in Figure 20.

7.4. Chemical Bonding in Heavier Main-Group Compounds

There is ubiquituous chemical evidence that the structures and reactivities of the first octal-row atoms exhibit siginificant differences from the heavier homologues. This is particularly evident when one compares the organic chemistry of carbon with silicon chemistry. The statement “Silicon is weird” clearly characterizes this finding from an organic chemist point of view.244 The difficulties to understand the large differences between molecules of the first octal row and higher homologues stem from the fact that the heuristic bonding models, which were developed during the last century, received their input mainly from observations which were made with compounds of the first octal-row atoms, henceforth also called lighter elements, for which experimental data were available. The neural network of chemical information, which was used to develop such models, was mainly fed with results that came from first octal-row molecules. Compounds of the heavier elements, which exhibit different properties, were initially considered as exceptions, for which sometimes peculiar explanations were made. This led to difficulties when the information about molecules of the heavier elements increased. The unequal structures and reactivities of heavier main-group compounds could no longer be considered as anomalies. But the bonding models that were suggested to rationalize their behavior were still mainly based on extensions of the existing schemes with some ad-hoc additions. Some of them were simple prescriptions for writing formulas that appear to address the peculiar bonding situations of the heavier molecules without which a sound theoretical basis for the differences between lighter and heavier main-group compounds was given. The nature of the chemical bond in heavier main-group compounds has been discussed from a quantum chemical point of view in a review article by Kutzelnigg in 1984.245 The author presents the fundamental differences between the valence shells of the lighter and the heavier main group atoms, and he discusses their effects on the structures and bonding situations of the

Figure 20. Schematic representation of the relative size of the (n)s and (n)p AOs in atoms (a) of the first octal row and (b) in heavier maingroup atoms.

The relative size of the (n)s and (n)p AOs of the main-group elements is a significant factor for the nature of the chemical bonds and the reactivities of main-group compounds. An approaching atom B “sees” the (n)p AOs earlier than the (n)s AOs (n > 2) of A. In less colloquial wording, it means that the interference of the wave function between B and the valence p AO of heavy atom A occurs earlier and is larger than with the valence s AO. Kutzelnigg analyzed the Mulliken gross population at the central atoms in EH4 (E = C, Si) and found that the hybridization at carbon is sp3 while it is ∼sp2 in SiH4, although both molecules have Td symmetry.245 Table 12 shows the hybridization of the E-X localized orbitals in tetrahedral (Td) EX4 (E = C−Pb; X = H, Cl) given by the NLMO (Natural Localized MO) method.68−71 Only CH4 shows a perfect sp3 hybridization, while the E−H bonds in the heavier EH4 homologues possess ∼sp2 hybridized orbitals at atom E. The chlorine systems ECl4 deviate even more strongly from sp3 hybridization. The percentage p-character of the E-Cl bonds at atom E is much higher than one would expect from the sp3 model. The Pb−Cl bonds have even nearly equal s and p orbital contribution at Pb. Note that the correct hybridization is only given by the NLMOs, which include the nondiagonal elements of the Fock matrix. The standard NBOs give perfect sp3 hybridization for all EX4 species. AD


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Table 12. NLMO Analysis of EX4 (E = C−Pb; X = H, Cl) at the HF/def2-TZVPP Level Using NBO 6.0a q

Pol.

than for the respective (n)s AOs of the heavier maingroup atoms.

Hybrid.

Molecule

E

X

P(E−X)

E%

X%

ns:np:(n−1)d

CH4 SiH4 GeH4 SnH4 PbH4 CCl4 SiCl4 GeCl4 SnCl4 PbCl4

−0.74 0.85 0.74 0.96 0.74 −0.17 1.51 1.64 1.87 1.74

0.19 −0.21 −0.18 −0.24 −0.18 0.04 −0.38 −0.41 −0.47 −0.44

0.97 0.93 0.94 0.91 0.92 1.02 0.88 0.91 0.77 0.76

59 39 41 38 41 49 26 25 23 25

41 61 59 62 59 51 74 75 77 75

25:75:0 33:66:1 33:67:0 35:65:0 36:64:0 30:70:0 37:60:3 43:56:1 42:57:1 49:51:0

• Effective spx hybridization is only possible for the first octal-row atoms, because the 2s and 2p AOs have similar radii. Chemical bonds of the heavier atoms have a rising degree of higher %p character, and the (n)s orbitals become gradually less prone to be engaged in chemical bonding. This stabilizes low-valent compounds of the heavier main-group atoms. 7.4.1. Hypervalent Molecules: Valence, Orbital Symmetry, and 3-Center, 4-Electron Bonding. In order to address the topic of hypervalence, one must first clarify what is meant by the term “valence”. According to the IUPAC, valence is defined as “The maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a f ragment, or for which an atom of this element can be substituted.”251 Accordingly, molecules such as PF5 and SF6 are hypervalent, because they doubtlessly possess five and six bonds at phosphorus/sulfur. However, quantum chemistry suggests that chemical bonding and thus valence are associated with the molecular orbitals, which are used to describe the chemical bonds. Then the question arises, which atomic orbitals are effectively engaged in the interference of the orbitals building the wave function. Since the s and p AOs can be occupied by a maximum of eight electrons, it seems difficult to assign more than four bonds to an atom, which uses only its valence (n)s and (n)p AOs for covalent bonding. A simple explanation for the observation of stable molecules with more than four bonds such as PF5 and SF6 considers the (n)d orbitals, which are available for heavier main-group atoms where n ≥ 3. Early work suggested that hypervalent molecules possess (n)sx(n)py(n)dz hybridized bonds. Accordingly, hypervalent molecules would not obey the octet rule, which states that a maximum of eight electrons are used for covalent bonding. An alternative suggestion, which uses only (n)s and (n)p AOs for covalent bonding in hypervalent molecules, was suggested independently by Rundle252−254 and Pimentel.255 It is now known as the 3-center, 4-electron bonding model, and it was adapted by Coulson to VB theory.256 It is worthwhile to read the original paper, because it already uses the symmetry of the molecular orbitals for explaining chemical bonding in molecules that possess an excess number of electron pairs than required by the octet rule. The authors discuss the bonding in various molecules such as I5−, XeF4, XeF2, HF2−, and X3−.252−255 It is shown that the covalent bonds in the molecules may be described without invoking the use of higher energy basis functions, as it was previously advocated by Pauling11 and by Kimball.257 The 3-center, 4-electron model can also be applied to the π bonds in the allyl anion C3H5−. The basic feature is displayed in Figure 21, which uses a model system of three atoms A3 possessing one AO each. The three atomic basis functions lead to three MOs φ1−3, which are shown in ascending energy order. Covalent bonding comes from the occupation of φ1, which is totally symmetric. There is no covalent bonding between the central and the terminal atoms in φ2, which has one node. But there is still a stabilizing contribution from electrons that occupy φ2, because of the Coulomb attraction between the nucleus of the central atom and the electrons occupying the MO. The energy lowering due to occupation of φ2 is less than that of the electrons in φ1, but the total system is further stabilized. The occupation of φ2 in

a

Atomic partial charges q, Wiberg bond order P(E−X), polarization, and hybridization of atom E in the E−X bond.

The differences in the structures of molecules between first and higher octal-row elements can thus be traced back to the radii of the valence s and p AOs. This shall be demonstrated with selected examples below. We also want to mention relevant theoretical studies by Kaupp, who analyzed the bonding situation in heavier main-group compounds paricularly of the sixth row of the periodic system.247−250 There are two prominent observations that distinguish heavier main-group compounds from compounds of the lighter homologues which shall be analyzed and discussed in this section. One phenomenon concerns the number of electron-pair bonds and the coordination number of the atoms. Heavier maingroup atoms can have more than four bonds in stable molecules. This is essentially unknown for first octal-row atoms. Classical examples are SF6 and PF5, which are discussed below. The question if such molecules should be considered as hypervalent and if they obey the octet rule will also be considered. The second topic concerns the observation that compounds with multiple bonds between heavier main-group atoms are much more difficult to isolate and that they often possess very different equilibrium geometries than the first octal-row analogues. Classical examples are ethylene and acetylene and their heavier group-14 homologues. The standard textbook explanation suggests that the overlap of the p(π) AOs of the heavier atoms is much smaller than for lighter atoms, because of the longer bonds of the former. It has been known for a long time that this suggestion is not valid, because the p(π) AOs of the heavier atoms are more diffuse than for lighter atoms, which largely compensates for the longer bond. Furthermore, the alleged weakening of the π bonds as reason for the experimental findings is questionable, because the overlap of the σ AOs of the heavier atoms is also affected by the longer distance. We shall present an explanation for the observations, which is based on the valence shell structure of the atoms. The following conclusions arise from this section: • The most important difference between atoms of the first and higher octal rows of the periodic system, which determines the experimentally observed variance of their structures and reactivities, is the relative size of the valence s and p orbitals and not the availability of d orbitals for the heavier elements. The (n)p AOs for n > 2 are, as determined by the Pauli principle, antisymmetric to the p core orbitals, unlike the 2p AOs, which experience no lower lying p orbitals. This causes larger radii for the (n)p AE


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are stable only for the heavier atoms lies in the atomic size. First octal row atoms A are too small and cannot accommodate more than four atoms in AXn because the X−X Pauli repulsion prevents the formation of stable species with n > 4. The 3-center 4-electron bonding model is an early version of the MO derived models, which uses the symmetry of the orbitals for the explanation of structures and reactivity that became so prominent in the 1970s. 7.4.2. Hypervalent Molecules: SF6 and PF5. The molecules SF6 and PF5 may be considered as archetypical examples for the validity problem of the octet rule and the chemical bonds in hypervalent compounds. There are definitely six and five bonds in the respective molecules, which pose the question about the correct representation of the bonding situation with Lewis structures in accordance with the octet rule. There have been numerous discussions and suggestions about the usefulness of Lewis structures and the octet rule for hypervalent molecules, which are perhaps better termed as hypercoordinated species. The problem can straightforwardly be addressed with the help of the symmetry of the molecular orbitals describing the valence electrons in the molecule. Figure 22a schematically shows the σ MOs of model compound EX6 where the central atom E has (n)s and (n)p valence functions. In the octahedral (Oh) ligand field in EX6, the (n)s and (n)p atomic orbitals of atom E split into an a1g and a triply degenerate set of t1u AOs, whereas the six σ orbitals of the ligand cage X6 are divided into a1g, t1u, and eg MOs. The molecule SF6 has 12 valence electrons that occupy six σ MOs.

Figure 21. Schematic representation of the molecular orbitals φ1 − φ3 of a 3-center, 4-electron bond.

A3 plays a similar role as the occupation of the 1eg MO in SF6, which is discussed below (Figure 22). The important result of these early works252−255 is the finding that molecules with covalent bonds, which possess an excess number of electron pairs as required by the octet rule, can be described using only their (n)s and (n)p orbitals for the interference of the wave function. When accurate quantum chemical calculations became available later, it was shown that the (n)d AOs of heavy main-group atoms are not genuine valence orbitals in hypervalent molecules but rather serve as polarization functions. The reason why hypervalent molecules

Figure 22. Splitting of the (n)s and (n)p AOs of a main-group atom E (a) in the octahedral field of six ligands X in EX6 and (b) in the trigonal pyramidal field of five ligands X in EX5. AF


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Figure 22a shows that only four MOs in EX6 (a1g and t1u) have the right symmetry to enable interference between the AOs of atom E and the ligands cages. The degenerate eg MO does not induce covalent E−X bonding. This does not mean that the electrons in the eg MO do not contribute to the stabilization of the molecule. The electronic charge of the ligand cage X6 may be strongly stabilized through Coulomb interactions with the nucleus of atom E.258 This also shows the limitation of the model of orbital interactions (covalent bonding) for describing chemical bonding. There are other forces such as Pauli repulsion and Coulombic interactions, which strongly influence the chemical bond. It is a widely held belief that electrostatic interactions between neutral atoms are weak or may even be repulsive, which for most molecules is not correct.28,177,259 Thus, there are actually only four electron-pair bonds in octahedral EX6 that come from the interference of the wave functions of the atoms, which fully agrees with the octet rule. This information is only available by inspection of the symmetry of the MO wave function. But the orbitals are delocalized over more than two atoms (seven atoms in the 1a1g MO and three atoms in each 1t1u components) and the problem is to convert the delocalized picture of the MO calculation into the model of localized Lewis structures. Much of the difficulties and controversies about chemical bonding in molecules in the literature can be traced back to the discrepancy between the viewpoints of delocalized and localized bonds. The adherence to two-center bonds in the works of Gilbert Lewis and Linus Pauling can be identified as a severe limitation of the viewpoint on chemical bonding, and it is the source of much confusion still today. This is surprising, because the ground-breaking study by Erich Hückel on aromaticity showed in the 1930s already the advantages if the description of chemical bonding is not restricted to a localized model.20 The connection of the octet rule to the bonding situation in PF5 is similar to SF6 but with an additional facet. Figure 22b shows that the AOs of E and σ MOs of X5 split in the D3h symmetric model compound EX5 and give rise to σ orbitals possessing a1′, a2″, and e′ symmetry, which are occupied by ten electrons. The 1a1′, 1a2″, and the degenerate 1e′ MOs are bonding orbitals that give four delocalized electron-pair bonds, which result from the interference of the wave functions. The 2a1′ orbital is the HOMO in EX5, but the coefficient of the (n)s valence AO is very small, which makes it effectively a nonbonding MO. There is negligible interference between the atomic orbitals of phosphorus and fluorine in the 2a1′ MO. Covalent bonding refers to the number of bonds, which are due to the interference of the wave functions. There are four such doubly occupied MOs in the molecules. The bonding situation in PF5 is thus in agreement with the octet rule.63,260 How may the information gained by the MO model be expressed by Lewis structures? The most common sketches of Lewis formulas for SF6 and PF5 obeying the octet rule depict one of 14 “resonance forms” for SF6 and one of five for PF5 as shown in Figure 23, which require formally charged fragments. It becomes obvious that the presentation of the chemical bonds is clumsy and the use of charged species may give rise to a misleading interpretation of the nature of the bonding. The “ionic−covalent” resonance forms in Figure 23 must not be interpreted as indication for the appearance of ionic interactions. The charges that show up in the Lewis structures are only formal, and they have no physical relevance. The sometimes accidental agreement between formal charges and actual charge distribu-

Figure 23. Lewis structures for (a) SF6 and (b) PF5 that obey the octet rule.

tion in a molecule should not be used as evidence for the correctness of the model. The following conclusions arise from this section: • The question of hypervalence can only be addressed when valence is defined. • Covalent bonding of main-group compounds usually involves the atomic valence s and p orbitals. Molecular orbitals can be constructed so that they are filled with more than eight electrons, but the individual atoms do not receive more than eight electrons in their valence shell. • The Lewis model of localized electron pairs is not well suited for describing hypervalent molecules. • The 3-center, 4-electron MO model of Rundle and Pimentel provides a simple rationale for the bonding in hypervalent molecules. It shows that the octet rule is fulfilled in molecules where a main-group atom has more than four bonds. The stabilization of the excessive electrons comes from the electrostatic stabilization, which is an important factor in all covalent bonds. • The bonding in SF6 and PF5 is an archetypical example for formally hypervalent molecules that still obey the octet rule. There are only four doubly occupied valence MOs in SF6, which have contributions from the s and p AOs of sulfur. The symmetry of the MOs is the key for understanding the interplay between the octet rule and the number of bonds. The excess electrons in PF5 occupy an antibonding MO, which has a negligible contribution from the s and p AOs of phosphorus. 7.4.3. Multiple Bonds of Heavy Diatomic Molecules Na2−Cl2 and N2−Bi2 and Related Molecules. A commonly made statement which is found in many textbooks of chemistry claims that the π bonds between heavy main-group atoms are weaker than between first octal-row elements. Table 13 gives the EDA results for the diatomic molecues Na2−Cl2 (second-row sweep), which may be compared with the data for the first-row species Li2−F2 (Table 4).28 The BDEs of the heavier molecules are smaller than those of the lighter systems, except for the dihalogens F2 and Cl2, which agrees with the experimental findings. The reason for the exceptionally weak F−F bond has been discussed in section 7.3.5. Note that Si2 has a X3Σg− ground state whereas the valence isoelectronic C2 has a X1Σg+ ground state. A comparison of the nature of the bonding between the different atoms should be done for the same spin state. The trend of weaker bonds for heavier atoms, which is a general feature for the elements of the periodic system, comes from the more weakly bonded valence electrons. The first AG


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Table 13. Energy Partitioning Analysis of the Second Row Dimers E2 (E = Na−Cl) in C2v at BP86/TZ2Pa E ΔEint ΔEPauli ΔEelstatb ΔEorbb ΔΕ(σ)c ΔΕ(π)c ΔEcorre De d E−Ed

Na

Mg

Σg+

1

1

State

−16.3 5.1 −10.7 (50.1%) −10.7 (49.9%) −10.7 (100.0%) 0.0 ( Sb2 > Bi2. The relative Coulombic attraction to the overall binding shows the opposite trend, which is due to the increase of the positive charge of the nuclei. The contribution of π bonding to the overall orbital interaction increases from nitrogen to phosphorus and then becomes AH


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only stretched by 16.2% relative to P2 (1.911 Å). This is a hint toward a possible explanation. More detailed information is available from EDA calculations of the diatomic and tetraatomic molecules. Table 15 shows the EDA results for E4 (E = N, P). Note that the N−N and P−P bonds have a similarly small %s contribution;

particularly weak and, therefore, heavier main-group compounds with multiple bonds are much less stable than homologues of the lighter elements. Since chemical evidence for the latter finding is ubiquitous, other factors rather than weaker π bonding must be responsible for the observation and different explanations have to be provided. This is done in the following sections. 7.4.4. Multiple Bonds of Heavy Main-Group Atoms: N2/N4 vs P2/P4. The most common allotropes of nitrogen and phosphorus are gaseous N2 and tetrahedral P4 (white phosphorus), whose structures and relative energies may be used as prototype to analyze the different chemical bonds of atoms of the first and second row of the periodic system. Tetrahedral N4 (Td) is an energetically high-lying minimum species, which is experimentally unknown so far but has been the subject of quantum chemical studies.265 Experimental data for N2, P2, and P4 are available from the literature.24,266 Figure 24 shows the calculated geometries of the diatomic and tetraatomic molecules.267,268 The computed reaction energies

Table 15. Energy Partitioning Analysis of E4 (E = N, P) (BP86/TZ2P) and Comparison with E2a E4

1/4 E4

1/2 E2

E=N ΔEint ΔEPauli ΔEelstatb

−322.6 1636.6 −705.4 (36.0%)

ΔEorbb

−1253.7 (64.0%) −1253.7 (100.0%) −156.7

−80.7 409.2 −176.4 (36.0%) −313.4 (64.0%) −313.4 (100.0%) −104.5

R(N−N)

s: 7.0% p: 92.5% 1.464

s: 7.0% p: 92.5% 1.464

ΔEint ΔEPauli ΔEelstatb

E=P −293.1 972.3 −545.7 (43.1%)

ΔEorbb

−719.8 (56.9%)

ΔEσc

−719.8 (100.0%) −120.0

−73.3 243.1 −136.4 (43.1%) −180.0 (56.9%) −180.0 (100.0%) −60.0

s: 5.2% p: 93.2% 2.221 (2.223)d

s: 5.2% p: 93.2% 2.221

ΔEσc ΔEσ/nf ΔEπc Hybridizatione of the σ bond

ΔEσ/nf ΔEπc Hybridizatione of the σ bond R(P−P)

−120.1 401.1 −156.4 (30.0%) −364.8 (70.0%) −239.4 (65.6%) −239.4 −125.5 (34.4%) s: 37.5% p: 61.9% 1.102 (1.098)g −59.3 158.8 −93.2 (42.7%) −124.9 (57.3%) −74.4 (59.5%) −74.4 −50.6 (40.5%) s: 20.8% p: 77.8% 1.911 (1.893)g

a

The interacting fragments are four atoms E in the 4S ground state (s2px1py1pz1). Bond lengths are given in Å. Energy values are given in kcal/mol. The data are taken from refs 261. bThe values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe values in parentheses give the percentage contribution to the total orbital interactions ΔEorb. d Experimental value, ref 266. eThere is a small contribution from the d polarization functions. fn = Number of bonds. gExperimental value, ref 24.

Figure 24. Calculated bond lengths [Å] of E2 and (Td) E4 (E = N, P) and dimerization energies De of E2 at BP86/TZ2P. Experimental bond lengths are given in parentheses. Reprinted with permission from ref 267. Copyright 2014 Springer Nature.

for formation of the tetrahedral species are striking evidence for the drastic variation of nitrogen and phosphorus bonding: 2N2 → N4(Td) 2P2 → P4 (Td)

+158.9 kcal/mol −59.7 kcal/mol

the change in the hybridization compared with the diatomic molecules (Table 14) is particularly large for N2. Table 15 gives the values for E4 divided by four and the data for E2 divided by two for the direct comparison of the isomers. The values in the two final columns refer to the energy values per atom. It becomes obvious that the nitrogen atoms in N2 are stabilized with respect to N4 whereas the phosphorus atoms in P4 are lower in energy than in P2. Inspection of the three energy components indicates that the reverse in stability is also found for the orbital term ΔEorb but not for ΔEelstat and ΔEPauli. The covalent interactions stabilize the atoms in N2 much stronger than in N4, whereas the orbital interactions stabilize phosphorus in P4 more than in P2. The diatomic molecules E2 have one σ and two degenerate π bonds, whereas E4 has six σ bonds. Thus, there are three bonds per atom in E2 (one σ, two π) and in E4 (three σ). Table 15 shows that the σ bond strength per atom in E2 is higher than the

(23) (24)

The enormous energy difference of 218.6 kcal/mol between reactions 23 and 24 suggests a fundamental difference between the strengths of the nitrogen and phosphorus bonds in the diatomic and tetratomic isomers. The above results show that different π bond strength in the diatomic species cannot be the reason for the diverse stability order. Inspection of Figure 24 shows that tetrahedral isomers of both elements have longer bonds than the diatomic species, but the bond lengthening of nitrogen is more pronounced than that of phosphorus. The calculated N−N bond length in N4 (1.464 Å) is 32.8% longer than in N2 (1.102 Å), whereas the P−P bond in P4 (2.221 Å) is AI


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Figure 25. Variation of the energy terms of the EDA at different bond lengths of (a) N2 and (b) P2 at BP86/TZ2P. Reprinted with permission from ref 267. Copyright 2014 Springer Nature.

π bond strength for both N and P. The crucial difference between nitrogen and phosphorus is the change of the σ bond strength from E2 to E4. The σ bond strength per atom in N4 (−104.5 kcal/mol) is drastically less than in N2 (−239.4 kcal/ mol) and becomes even lower than the strength of the π bond in N2 (−125.5 kcal/mol). The σ bond strength in P4 (−60.0 kcal/ mol) is also lower than in diatomic P2 (−74.4 kcal/mol), but the weakening is clearly less than for nitrogen and it remains stronger than the π bond strength in P2 (−50.6 kcal/mol). The dramatic weakening of the chemical bonds in N4, which leads to the overall much lower stability of the nitrogen tetramer and the reverse energy shif t of the phosphorus homologues, is related to the σ bonds and not to π bonding.

But what is the driving force for the drastically weaker N−N σ bonds in the tetramer than in the dimer? The bonds in E4 are longer than in E2. As illustrated in section 7.2, the equilibrium bond length in N2 is determined by the Pauli repulsion and not by the maximum overlap of the orbitals. Each nitrogen atom in N4 encounters Pauli repulsion from three other atoms, but only from one atom in N2. The same holds for the phosphorus systems, but the Pauli repulsion in P2 has a different gradient than in N2 with respect to the interatomic distance. Pauli repulsion appears between electrons of the same spin, and the similar radii of the 2s and 2p AOs of nitrogen induce strong Pauli repulsion at the equilibrium distance of N2, which sharply increases at shorter distances. The Pauli repulsion in P2 at AJ


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Figure 26. (a) and (b) Schematic representation of electronic 3B1 and 1A1 states of EX2. Schematic representation of the orbital interactions and bonding models in various isomers of E2X4. (c) Electron-sharing σ and π bonding in planar E2X4 (model A). (d) Lone-pair donation in lp-bridged structures E2X4 (model B). (e) Lone-pair donation in lp-bridged structures E2X4 (model C). (f) Bond pair donation in E-X-bridged structures E2X4 (model D).

possess strongly bent nonlinear geometries.270 The latter aspect is discussed in the following section. Here we investigate the bonding situation in the parent systems H2EEH2 (E = Si − Pb) and in the fluorine compounds F2EEF2. Deviations of the geometries of compounds R2EER2, where R is a bulky group, are caused by steric effects. We focus on the underlying electronic differences between the carbon−carbon bond in ethylene and the E−E bonds in the heavier homologues. The key for understanding the different E-E bonding and the resulting equilibrium structures of X2EEX2 between carbon and the heavier group-14 atoms was provided 30 years ago by Trinquier and Malrieu298,299 and by Carter and Goddard,300 who suggested that the electronic states of the EX2 fragments are crucial for the nature of the E−E bond.301 Figure 26a shows schematically the (3B1) triplet state and Figure 26b the (1A1) singlet state of EX2 and their relative energies. Methylene CH2 has a triplet ground state, which is perfectly suited to build an electron-sharing double bond X2EEX2 (model A) consisting of a σ and a π bond (Figure 26c). CF2 and the heavier EH2 and EF2 species have a singlet (1A1) ground state, and they must be formally exited to the triplet (3B1) state in order to engage in electron-sharing bonding. Some ylidenes EH2 and EF2 with E = Si−Pb bind through their singlet (1A1) ground state via donor− (model B) where the lone-pair acceptor interactions electrons of E interact with the vacant p(π) orbital of the other fragment as shown in Figure 26d. This leads to nonplanar C2h structures of the respective X2EEX2 molecules where atom E has a pyrimidal ligand arrangement. This is a straightforward explanation for the experimentally observed and calculated nonplanar geometries of many heavier group-14 homologues of alkenes. Model C refers to a hypothetical situation with mutual dative bonds in a planar structure X2EEX2. It is used for the discussion of electron-sharing and dative bonding in C2H4 and C2F4 in section 8. There is an alternative method of dative bonding between EX2 fragments in their (1A1) singlet state where the donation occurs from the E−X bonding orbital rather than the lone-pair electrons of E. This is shown in Figure 26e (model D). Textbook knowledge suggests that lone-pair electrons are energetically higher lying and are therefore better donors than bonding electrons, but this is strictly valid only for atoms of the

equilibrium is weaker and increases less steeply, because the singly occupied 3p AOs have a larger radius than the doubly occupied 3s AO. This is shown in Figure 25, which displays the energy terms of the EDA calculations of N2 and P2. The interaction of one nitrogen atom with three N in N4 induces stronger Pauli repulsion than the interaction of one P with the other three phosphorus atoms in P4. This is the reason why the N−N bonds in tetrahedral N4 are stretched by 32.8% relative to N2 whereas the P−P bonds in the tetramer are stretched by only 16.2% compared to P2. The driving force for the particularly long and weak N−N bonds in N4 is thus the Pauli repulsion, which is due to the similar s/p ratio of the valence AOs. The following conclusions arise from this section: • The frequently made statement that π bonds between heavier main-group atoms are particularly weak is not justified. The relative contribution of π orbital interaction to covalent bonding in heavier main-group compounds is even larger than in molecules of the first octal row. • The difficulty to isolate molecules of the heavier maingroup atoms with multiple bonds is not due to weak π bonding. It is rather caused by the change of both the σ and π bond strength in addition reactions where π bonds are converted to σ bonds. P4 (Td) is significantly more stable than 2 P2, but N4 (Td) is much higher in energy than 2 N2. This is mainly due to the very long and weak N−N bonds in N4 (Td), which are caused by the strong Pauli repulsion that originates from the similar size of the 2s and 2p orbitals. 7.4.5. Multiple Bonds of Heavy Main-Group Atoms: X2CCX2 vs X2EEX2 (E = Si − Pb; X = H, F). The history of experimental attempts to synthesize compounds that are heavier group-14 homologues of alkenes and alkynes is a fascinating chapter of chemical research. The various approaches, which finally suceeded in an albeit unexpected way, have been described in several review articles.269−297 They shall not be discussed in this account, which focuses on the nature of chemical bonding. Experiments showed that heavy alkene species X2EEX2 (E = Si − Pb) possess nonplanar equilibrium structures that are significantly different from the planar D2h arrangement of ethylene.269 Similarly, the experimentally observed geometries of the heavier alkyne compounds XEEX AK


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Figure 27. Calculated geometries and relative energies of E2H4 and E2F4 isomers (E = C−Pb) at BP86/def2-TZVPP. Number of imaginary frequencies i. Bond lengths in Å, angles in deg, energies in kcal/mol. The red numbers indicate the lowest energy structures.

Table 16. Calculated Bond Dissociation Energies De of Planar X2EEX2 into 2 EX2 in the 3B1 Triplet State and (in Parentheses) BDE of the Lowest Energy Structure of E2X4 to 2 EX2 in the 1A1 Singlet Statea E2H4 C Si Ge Sn Pb

E2F4

De

ΔE(S→T)

De − 2ΔE(S→T)

De

ΔE(S→T)

De − 2ΔE(S→T)

178.7 (210.3)c 100.0 (67.8)c 93.7 (51.3)c 74.4 (40.5)c 60.4 (38.5)c

−15.8b 16.5 23.3 27.3 35.5

178.7 67.0 47.1 19.8 −10.6

176.7 (74.3)c 111.5 (8.3)c 91.0 (18.8)c 65.9 (31.9)c -d (37.8)c

51.2 69.7 81.5 80.2 89.9

(74.3) −27.9 −72.0 −94.5 -

Excitation energies ΔE(S → T) of EX2 at BP86/def2-TZVPP. All values in kcal/mol. bThe triplet state is the ground state. cDissociation energy of the lowest energy structure of E2X4 to 2 EX2 in the singlet state. dPlanar Pb2F4 dissociates during the geometry optimization into 2 PbF2. a

first octal row. As mentioned above, the (n)p AOs of higher main-group atoms have larger radii than the respective (n)s AO, which leads to an increasingly higher %p character of the heavier atoms in their bonding orbitals. As a consequence, the lone-pair orbitals of the heavier atoms have a higher %s character which lowers their energy. At some point the lone-pair orbitals are lower in energy and less prone to charge donation than the bonding electron pairs of E−X bonds. This is the reason why many low-valent maingroup compounds of the heavier elements possess bridging structures. The rich and seemingly exotic chemistry of the cluster compounds of the heavier main group elements is also based on the stronger donor capabilities of the bonding orbitals than of the lone pairs. This is elaborated and discussed in the paper by Dehnen and co-workers in this issue.302 Figure 27 shows the calculated geometries and the relative energies of the planar D2h form of E2X4 (E = C−Pb), which are not energy minima, and the isomeric equilibrium structures with trans-bent geometry and doubly X-bridging isomers, all of C2h symmetry. The most stable isomer of the E2H4 compounds for E = Si, Ge is the trans-bent form whereas the hydrogen-bridged trans isomer is the energetically lowest lying species for E = Sn, Pb. The most stable isomer for E2F4 is the trans-bent form for E = Si and the F-bridged isomer for E = Ge, Sn, Pb. There are also cis isomers of the bent structures and the bridged forms of E2X4, which are slightly higher in energy than the respective trans isomers. They are not discussed here, because their bonding situation is similar to those in the trans isomers. The stability

order suggests that the E−X bonds become better donors than the lone-pair electrons for heavier E and that the turning point depends on the nature of X. This occurs in the case of E2F4 already for the germanium system, because fluorine is more electronegative than hydrogen. This leads to a higher %s character of the lone-pair orbitals. The strength of the electron-sharing double bonds X2EEX2 in the hypothetical planar forms of the molecules may be estimated from the calculated bond dissociation energies De of the reaction X2EEX2 (D2h) → 2 EX2 (3B1), which refers to EX2 in the electronic reference state. The comparison with the BDEs of the most stable nonplanar structures reveals the different stabilization energies that come from electron-sharing and donor−acceptor interactions. Table 16 gives the computed data together with the singlet−triplet excitation energies. Subtracting the requested promotion energies ΔE(S → T) for obtaining the 3 B1 state of EX2 from the De values indicates the net stabilization of the electron-sharing interactions. It becomes obvious that the energy preference for the nonplanar structures of Si2H4 and Ge2H4 surpasses only slightly the stronger electron-sharing bonding in the planar forms, which requires excitation to the 3B1 state of EH2. The difference of the binding interactions becomes much larger for Sn2H4 and Pb2H4, where the dative bonding in the lowest lying trans-bridged form occurs via E-H donation. The situation in fluorine-substituted systems E2F4 reveals significant differences to the E2H4 compounds. The BDE of planar C2F4 (74.3 kcal/mol), which is even less than the AL


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The comparison of the EDA-NOCV results for the lp-bridged form of Sn2H4(B) with the H-bridged isomer Sn2H4(C) reveals the origin of the change in the relative energy of the two isomers. The lp-bridged form Sn2H4(B) encounters identical Pauli repulsion and stronger electrostatic attraction than Sn2H4(C) (Table 17). However, the orbital interactions in Sn2H4(C) are stronger than in Sn2H4(B), and they overcompensate the former two effects. Thus, the EDA-NOCV results suggest that the higher stability of Sn2H4(C) compared with Sn2H4(B) is due to the stronger charge donation of the Sn−H bonding orbital than the lone-pair MO. The orbital interactions in Sn2H4(C) also comprise two major components ΔEorb(1) and ΔEorb(2), which are the (+,+) and the (+,−) combinations of the Sn−H donor orbitals. This becomes obvious by inspection of the associated deformation densities, which are displayed in Figure 28b. The shape of Δρ and the connected HOMO and LUMO orbitals show that the (+,+) combination of the Sn−H donor orbitals provides, in this case, more stabilization than the (+,−) combination. The fluorine-bridged compounds differ significantly from the hydrogen-bridged molecules, because fluorine carries lone-pair electrons. Table 18 shows the EDA-NOCV results for the energetically lowest lying bridged isomers of E2F4. The turning point from lp-bridging to more favorable F-bridging occurs already for E = Ge. The stabilizing interaction energy in the Fbridged isomer Ge2F4(C) is substantially larger (ΔEint = −41.7 kcal/mol) than in the lp-bridged species Ge2F4(B), which is barely stabilized (ΔEint = −2.3 kcal/mol). Table 18 shows that the Pauli repulsion in Ge2F4(C) is much larger than in Ge2F4(B), but the electrostatic stabilization and the stronger orbital interaction compensate for the increase of ΔEPauli. The largest contribution to the orbital interactions comes from the donation of the σ lone-pair MOs of the bridging fluorine (Figure 29a). There are several smaller orbital contributions, which are associated with a significant polarization of the electronic charge. This becomes visible by inspection of the deformation density Δρ2 (Figure 29b). The red area of charge depletion is localized, but the blue area of charge accumulation is distributed over the whole molecule. 7.4.6. Multiple Bonds of Heavy Main-Group Atoms: HCCH vs HEEH (E = Si − Pb). The peculiar equilibrium structures of the heavier group-14 homologues of actylene can be explained in the same fashion as the nonplanar bridged homologues of ethylene. Since the structures and bonding situation in E2H2 (E = Si − Pb) have been discussed in recent review articles,261,303 only the essential points are presented here. More details can be found there and in the original work.304 Figure 30 summarizes the most important features of the E2H2 species. The nonplanar doubly hydrogen-bridged structure A is the global energy minimum for all E2H2 molecules. The next energetically stable isomer is the singly bridged form B, which exhibits a peculiar inward bending of the terminal hydrogen atom. Species B has all three atoms on the same side of the triply coordinated atom E. This is a highly unusual bonding situation for a group-14 atom. Both E2H2 isomers A and B have been observed for all elements E = Si − Pb in low-temperature matrix studies.305−310 The vinylidene species C is the next energetically higher isomer. It is the only energy minimum structure that is common for carbon and the heavier group-14 atoms Si−Pb. Finally, there are two trans-bent structures D1 and D2, which possess significantly different E−E bond lengths and E−E−H angles. Substituted homologues E2R2 with bulky groups R

calculated BDE of the F−F single bond in C2F6 (87.3 kcal/ mol),237 is much smaller than the intrinsic interaction energy (176.7 kcal/mol) due to the rather large ΔE(S → T) value for CF2 (51.2 kcal/mol). The bonding situation in C2F4 and the change in the electronic structure during bond rupture are discussed in the section on dative bonding below. The electronsharing interactions in the planar structures of the heavier systems E2F4 have a similar magnitude as in the respective hydrogen systems E2H4, but the singlet → triplet excitation energies of EF2 are much higher than for EH2. This makes the planar structures of the former much less stable than the bridged species. As mentioned above, the E−X bonds at some point become stronger donors than the lone-pair orbitals. The gradual change of the relative strength of the two types of interactions can be monitored by EDA-NOCV calculations. Table 17 shows the numerical results of the EDA-NOCV calculations of the energetically lowest lying bridged isomers of Table 17. EDA-NOCV Results of the Most Stable Structures of E2H4 at BP86-D3/TZ2P+//BP86/def2-TZVPP

Isomer

Lp bridged (C2h) Si2H4(B)

Lp bridged (C2h) Ge2H4(B)

Lpbridged (C2h) Sn2H4(B)

H-bridged (C2h) Sn2H4(C)

H-bridged (C2h) Pb2H4(C)

fragments

2 SiH2 (1A1)

2 GeH2 (1A1)

2 SnH2 (1A1)

2 SnH2 (1A1)

2 PbH2 (1A1)

−81.3 294.8 −3.3 −181.2 (48.6%) −191.6 (51.4%) −149.9 (78.2%) −37.6 (19.6%) −4.2 (2.2%) 14.1 −67.2

−60.5 202.6 −2.8 −138.2 (53.1%) −122.2 (46.9%) −81.0 (66.3%) −36.8 (30.1%) −4.4 (3.6%) 9.7 −50.8

−40.7 124.3 −3.2 −89.2 (55.2%) −72.5 (44.8%) −39.8 (54.8%) −29.3 (40.5%) −3.4 (4.7%) 4.7 −35.9

−49.5 124.3 −3.7 −86.1 (50.6%) −84.1 (49.4%) −47.4 (56.4%) −26.8 (31.9%) −9.9 (11.7%) 5.5 −44.0

−48.0 125.4 −3.9 −91.7 (54.1%) −77.8 (45.9%) −45.2 (58.1%) −25.1 (32.3%) −7.5 (9.7%) 5.3 −42.6

ΔEint ΔEPauli ΔEdisp ΔEelstata ΔEorba ΔEorb(1)b ΔEorb(2)b ΔEorb(rest)b ΔEprep -De a

The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. bThe values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.

E2H4. The tin compound Sn2H4 is the turning point where the H-bridged form C becomes lower in energy than the lp-bridged isomer B (Figure 27). Therefore, we present for comparison the results for both species Sn2H4(B) and Sn2H4(C). The calculated energy terms for the lp-bridged isomers B of E2H4 (E = Si − Sn) suggest that the attractive E−E interactions arise with similar strength from electrostatic forces and from orbital interactions. The ΔEorb term has two major components, which contribute >95% to the dative bonding. Figure 28a shows the deformation densities Δρ of Sn2H4, which are associated with the individual orbital interaction. The strongest contribution ΔEorb(1) comes from the mutual donation of the (+,−) combination of the lonepair orbitals whereas ΔEorb(2) is due to the charge donation from the (+,+) lp combination. The associated interacting orbitals HOMO and LUMO are also shown. It becomes obvious that the interfragment interactions lead also to a significant charge alteration (polarization) at each fragment. AM


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Figure 28. Plot of deformation densities Δρ (isovalue = 0.005) of the pairwise orbital interactions ΔEorb(1) and ΔEorb(2) and the associated fragments orbitals in the (a) Lp-bridged isomer of Sn2H4 and (b) H-bridged structure of Sn2H4. The energy values are given in kcal/mol. The direction of the charge flow is red → blue.

EH requires formally the a4Σ− excited state of EH for the formation of an electron-sharing triple bond. The excitation energy X2Π → a4Σ− of the heavier EH species is significantly larger than for CH. Table 19 shows the calculated bond dissociation energies De of the linear structures HEEH yielding (a4Σ−) EH. Subtracting the excitation energies for two EH fragments to the a4Σ− state ΔEexc from the De values gives the net stabilization energy of the electron-sharing triple bond. This value of 240.0 kcal/mol for CH is rather high; it clearly surpasses the stabilization of an electron-sharing single bond, which could be provided by coupling the unpaired electrons of the EH fragments in the X2Π ground state. The net stabilization energy De − 2ΔEexc for Si and Ge is much smaller (Table 19). It does not compensate for the energy gain of a E−E single bond. The excitation energies of two SnH and PbH fragments are even higher than the bond dissociation energy of linear HEEH. It follows that the bonding interactions between two EH species for E = Si − Pb occur from the X2Π ground state. The isomer C is not considered, because it has a different atomic connectivity than the other species. Figure 32 shows three possible arrangements of the EH fragments in their X2Π ground state where the unpaired electrons are coupled to an electron-sharing σ bond. The assignments (a) and (c) are unfavorable, because they leave the unoccupied p(π) orbitals in the resulting species vacant. Consequently, the structures are not energy minima but transition states (i = 1). The arrangement (c) leads to structure D2, which becomes an energy minimum for E = Pb when bulky substituents prevent the formation of the electronically more favored isomers.316 The arrangement (b) in Figure 32 places the EH bonds in perfect position to donate the associated electron pairs into the respectively vacant p(π) orbitals. This leads to the global energy minimum form A of the heavier E2H2 molecules. It becomes clear that the structure A has a triple bond, which

Table 18. EDA-NOCV Results of the Most Stable Structures of E2F4 at BP86-D3/TZ2P+//BP86/def2-TZVPP

Isomer

Lp bridged (C2h) Si2F4(B)

Lpbridged (C2h) Ge2F4(B)

F-bridged (C2h) Ge2F4(C)

F-bridged (C2h) Sn2F4(C)

F-bridged (C2h) Pb2F4(C)

fragments

2 SiF2 (1A1)

2 GeF2 (1A1)

2 GeF2 (1A1)

2 SnF2 (1A1)

2 PbF2 (1A1)

−9.7 68.6 −3.4 −29.4 (39.3%) −45.5 (60.7%) −26.0 (57.3%) −17.9 (39.4%) −1.5 (3.4%) 0.3 −9.4

−2.3 21.7 −2.9 −5.0 (23.8%) −16.1 (76.2%) −11.1 (69.3%) −4.2 (26.2%) −0.7 (4.5%) 0.2 −2.1

−41.7 161.7 −3.3 −112.0 (56.0%) −88.0 (44.0%) −52.0 (59.0%) −13.7 (15.6%) −22.3 (25.4%) 20.3 −21.4

−49.7 135.4 −3.9 −109.6 (60.5%) −71.6 (39.5%) −39.5 (55.2%) −11.1 (15.5%) −21.0 (29.3%) 14.7 −35.0

−53.3 113.6 −4.2 −99.4 (61.1%) −63.3 (38.9%) −34.9 (55.1%) −10.1 (15.9%) −18.3 (29.0%) 12.1 −41.1

ΔEint ΔEPauli ΔEdisp ΔEelstata ΔEorba ΔEorb(1)b ΔEorb(2)b ΔEorb(rest)b ΔEprep −De a

The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. bThe values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.

featuring D1 have been synthesized and structurally characterized for E = Si, Ge, Sn.311−314 The lead analogue Pb2R2 exhibits structure D2.315 The linear form E is an energetically high lying second order saddle point (number of imaginary frquencies i = 2) for all heavy group-14 species E2H2. Figure 31 schematically shows the X2Π ground and the a4Σ− excited state of EH. It is obvious that the linear structure HE AN


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Figure 29. Plot of deformation densities Δρ (isovalue = 0.005) of the pairwise orbital interactions ΔEorb(1) and ΔEorb(2) and the associated fragments orbitals in the (a) Lp-bridged isomer of Ge2F4 and (b) F-bridged structure of Ge2F4. The energy values are given in kcal/mol. The direction of the charge flow is red → blue.

consists of one electron-sharing σ component and a degenerate EH donor component. Figure 33 shows three different placements of the (X2Π) EH fragments where the unpaired electrons yield a π bond. Placement (a) shows that the electron-sharing π bond is supported by a lone-pair donor bond and a EH donor bond. The optimal EH donation requires a tilting of the vacant p(π) orbital of the right EH fragment, which leads to the inward bending of the associated E−H bond. The formation of the EH donor bond nicely explains the unusual equilibrium structure of the isomer B, which has a triple bond that includes an electron-sharing π bond, a lone-pair donor bond, and a EH donor bond. Rotation of one EH fragment leads to structure G, which is not an energy minimum, although it is lower in energy than B. Structure G possesses two EH donor bonds, which are stronger than one EH and one lone-pair donor bond in B. But G is the transition state for the degenerate flip−flap rearrangement of the global energy minimum A. Finally, there is the energetically highest lying energy minimum D1, which has a triple bond consisting of an electron-sharing π bond and two lone-pair donor bonds. Substitution of hydrogen by bulky groups in E2H2 stabilizes D1 so much that it can be isolated. When E = Pb, the isomeric form D2 becomes lower in energy than D1, because the electron-sharing π bond is replaced by a stronger σ bond and the lone-pair donation of lead is rather weak. The model presented here straightforwardly explains the experimentally observed structures of E2H2, which possess equilibrium geometries that are not directly accessible by the Lewis bonding model. They can be straightforwardly explained with an extension of the model by Trinquier and Malrieu298,299 and by Carter and Goddard,300 that was suggested for heavier homologues of ethylene. With the information that is gained from the analysis of the orbital interactions in E2H2 it is possible to display the bonding situation in the isomers in terms of

modified Lewis type structures for the isomers A, B, D1, and D2 as shown in Figure 34. It turns out that the Lewis model should not be discarded or replaced, but it should be used in conjunction with quantum chemical calculations. The following conclusions arise from these sections: • The nonplanar equilibrium geometries of the heavier ethylene homologues E2X4 (E = Si − Pb) can be understood in terms of dative interactions between ylidene fragments EX2 in the electronic (X1A1) ground state. The excitation energy from the (X1A1) ground state to the 3B1 excited state is too high to make the planar species with electron-sharing double bonds X2EEX2 energy minima. • The dative bonds in E2X4 can be formed either via the lone-pair electrons or by the E−X bond pairs as donors, which become favorable when E is a heavier atom. The turning point from lone-pair donation to E−X donation depends also on the nature of X. • The unusual equilibrium structures of the heavier acetylene homologues E2H2 (E = Si − Pb) can be explained analogously to the heavier ethylene systems with the interaction of the EH fragments in the (2Π) doublet ground state. The E−E bonds in the lowest lying energy minima A, B, and D1 possess electron-sharing σ or π single bonds, which are completed by lone-pair or E−H dative bonds. • The trans-bent equilibrium geometries of the isolated systems E2R2 are enforced by steric repulsion of the bulky groups R. In the case of E = Si, Ge, Sn, they have the structure D1 with a triple bond whereas Pb2R2 features structure D2 with a single bond. AO


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Figure 30. Optimized structures of E2H2 isomers A−E at BP86/QZ4P. Bond lengths are given in Å, angles in deg. The values for Θ give for structure A the dihedral angle between the E2H and E2H′ planes. The relative energies with respect to A are given at the bottom of each entry in kcal/mol together with the number of imaginary frequencies i. Reproduced from ref 304. Copyright 2005 American Chemical Society.

8. DATIVE BONDING IN MAIN-GROUP COMPOUNDS

decade following the suggestion that the chemical bonds and the reactivity of carbodiphosphorane C(PPh3)2 can best be understood in terms of donor−acceptor interactions Ph3P→ C←PPh3.50 The proposal not only led to an understanding of the sometimes unusual geometries of related molecules such as the bent equilibrium geometry of carbon suboxid C(CO)2;52,53 it was also the starting point for the synthesis of new molecules with uncommon features.49,55,56 The term “carbone” was suggested for divalent carbon(0) compounds CL2, which exhibit a distinctly different chemical behavior than the carbenes CR2, which are carbon(II) compounds.54 Carbones possess two lone electron pairs whereas carbenes have only one. A particular class of carbones CL2 are carbodicarbenes C(NHC)2, which possess carbon−carbon dative bonds (NHC)→C←(NHC). Carbodi-

In the examples of chemical bonding that were discussed so far, a distinction was often made between electron-sharing bonding and dative bonding. As mentioned above, the concept of dative bonding and the associated notation with an arrow A→B in contrast to an electron-sharing bond A−B is quite old, but it was neglected by the strong influence of Pauling on the general view of chemical bonding in the following period. Arne Haaland was a lonely caller in the desert when he published a review article on donor−acceptor interactions in main-group chemistry in 1989.48 It received little attention, which is probably due to the fact that the work was restricted to well-known standard molecules. The model of dative bonding was revived in the past AP


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guidance for synthesizing one-center complexes EL2, it can also be applied to multicenter complexes. Several examples where diatomic species E2 are stabilized by ligands L in stable complexes L→E2←L have been discussed in section 7.3 above. The basic principle of the stabilizing interactions in the twocenter adducts E2L2 is the same as in the one-center complexes EL2.49 The mono- or diatomic fragment E or E2 is formally excited to an electronic reference state, which is usually an excited state and which makes the fragment a good acceptor. The dative interactions L→En←L are strong enough to stabilize the complex so it can be experimentally observed or even isolated in bulk form. Recently, the concept was extended to systems with triatomic cyclic systems E3L3 (Figure 36). The triatomic Si(0) cluster Si3(CAAC)3, which has three Si⇆CAAC donor−acceptor bonds, could be isolated and structurally characterized by X-ray analysis.334 The triboron complexes [B3(NN) 3] + and [B 3(CO)3 ]+ have been observed and spectroscopically identified in the gas phase.335 The molecules feature the smallest π-aromatic system B3+, which is stabilized by three dative bonds with the ligands N2 and CO. The usefulness to distinguish between electron-sharing and dative bonding for understanding and also explaining chemical reactions can be demonstrated with a recent example, in which very similar substrates led to completely different products. Roesky and co-workers reported in 2009 that the reaction of NHC with HSiCl3 leads to the synthesis of the complex NHC→ SiCl2 (A), which was the first ligand stabilized Si(II) compound that was isolated and structurally characterized by X-ray analysis (Reaction 25, Figure 37).336 In a subsequent work, the authors reacted A with CAAC, which according to quantum chemical calculations binds more strongly to SiCl2 than NHC. Surprisingly, the reaction gave the compound (SiCl2)(CAAC)2 (B) where silicon is four-coordinated, rather than the expected complex (CAAC)→SiCl2 via 1:1 ligand exchange (Reaction 26, Figure 37)337 The calculations showed that the Si−C bonds in (SiCl2)(CAAC)2 are significantly shorter than in (CAAC)→ SiCl2.

Figure 31. Schematic representation of the electron configuration of the 2Π electronic ground state and the a4Σ− excited state of EH (E = C − Pb). The experimental24 and calculated (BP86/QZ4P) excitation energies are given in kcal/mol.

Table 19. Calculated Bond Dissociation Energies De (kcal/ mol) of Linear HEEH → 2 EH (a4Σ−) and X2Π → a4Σ− Excitation Energies ΔEexc (kcal/mol) of EH at BP86/QZ4P E

De

ΔEexc

De − 2xΔEexc

C Si Ge Sn Pb

270.9 121.6 113.3 89.4 69.0

15.44 38.56 47.09 45.87 52.01

240.0 44.5 19.0 −2.3 −35.0

carbenes were theoretically predicted as stable compounds in 2007,51 and they were synthesized for the first time in 2008.317,318 It was recently found that carbodicarbenes may be used as catalysts in a variety of reactions, which are a topic of intense experimental investigations.319−324 Following the discovery of carbones, heavier group-14 homologues EL2 (E = Si − Pb) were proposed as synthetically achievable targets.325,326 Stable silylones SiL2, germylones GeL2, and stannylones SnL2 have been isolated in the meantime.327−329 Perhaps the most amazing outcome of the carbone model CL2 was the synthesis of the isoelectronic borylene complex (BH)L2 with L = CAAC, which has a three-coordinated boron atom that possesses an electron pair CAAC→(BH)← CAAC.330 Substituted homologues (BR)L2 with different ligands L have also been isolated in the meantime, of which the dicarbonyl complex (BR)(CO)2 is the most surprising example.331,332 It is the first stable dicarbonyl complex of boron and the only dicarbonyl complex of any main-group atom besides carbon suboxide C(CO)2 and the isoelectronic nitronium cation N+(CO)2.333 The examples demonstrate the value of a bonding model, which is based on a quantum chemical analysis of the electronic structure of the molecules. Figure 35 gives an overview of the calculated and experimental structures of carbones CL2 and isoelectronic boron and nitrogen homologues (BH)L2 and N+L2 with various ligands. The trend in the bond angles can be explained with the relative strength of the π backdonation from the ligands to the central atom.49,58 The model of dative bonding is not only useful as a means of interpretation of the bonding situation and as predictive

HSiCl3 + 2NHC → NHC→SiCl 2(A) + NHC(H+)(Cl−) (25)

NHC→SiCl 2(A) + 3CAAC → (SiCl 2)(CAAC)2 (B) + NHC‐CAAC

(26)

The unexpected formation of four-coordinated (SiCl2)(CAAC)2 rather than (CAAC)→SiCl2 can straightforwardly be explained with the presence of Si−C electron-sharing bonds in the former compound in contrast to the latter adduct, which has a Si←C dative bond. The formation of electron-sharing Si− C in (SiCl2)(CAAC)2 (B) requires open-shell fragments in the electronic triplet state, which leaves one unpaired electron at each CAAC ligand (Figure 37). According to Hund’s rule, the overall electronic state of (SiCl2)(CAAC)2 should be a triplet. The EPR spectrum actually suggests a triplet state for the compound. Interestingly, compound B possesses two polymorphic forms, where only one exhibits the EPR spectrum.337 It appears that the unpaired electrons in one isomer are weakly coupled via intermolecular interactions, making it diamagnetic. There remains the question why SiCl2 binds two CAAC ligands with covalent bonds while it only binds one NHC ligand with a dative bond. The associated energies of bond formation and electronic excitation provide a straightforward answer. AQ


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Figure 32. Qualitative model for the orbital interactions between two EH molecules in different orientations where the unpaired electrons yield a σ bond. Reproduced from ref 304. Copyright 2005 American Chemical Society.

Figure 34. Suggested Lewis structures for isomers A, B, D1, and D2 of E2H2 isomers.

gives only a net stabilization of 7.3 kcal/mol, because the singlet → triplet excitation energy of NHC (88.9 kcal(mol) is much higher than for CAAC (49.9 kcal/mol). The energies calculated on the basis of the two bond types provide not only an explanation for the experimental observation. They serve also as a guideline for predicting the experimental outcome of further reactions with different carbene ligands, which mainly depends on the energy difference between their singlet and triplet state. How can a dative bond, A→B, be distinguished from an electron-sharing one, A−B? The question seems to be easily answered by looking at the fragments that arise after breaking the bond. Dative bonds tend to dissociate heterolytically while electron-sharing bonds usually break up in a homolytic fashion. However, the situation is more complicated, because the nature of the bonding interactions may change during the bond breaking process. A good example is the carbon−carbon double bond F2CCF2, which is a classical electron-sharing bond that

Figure 33. Qualitative model for the orbital interactions between two EH molecules in different orientations where the unpaired electrons yield a π bond. Reproduced from ref 304. Copyright 2005 American Chemical Society.

Figure 37 gives the calculated bond dissociation energies of the mono and bis adducts of SiCl2 with NHC and CAAC ligands and the singlet → triplet excitation energies.337 The bottom end shows the energy balance for electron-sharing bonding of SiCl2 with NHC and CAAC. The sum of energy gained due to bond formation and singlet → triplet excitation energies for the CAAC ligand results in a net stabilization of 67.3 kcal/mol for (SiCl2)(CAAC)2, which surpasses the BDE of (SiCl2)(CAAC) (42.5 kcal/mol). In contrast, the formation of (SiCl2)(NHC)2 AR


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Figure 35. List of isoelectronic molecules EL2 with calculated bond angles and partial charges Δq of the central fragments E = BH, C, N+. Experimental bond angles of isolated molecules are given in parentheses. Reproduced from ref 49. Copyright 2017 Elsevier.

revealed by the values of the orbital interactions ΔEorb values. As mentioned above, those fragments, whose electronic energies change least during the covalent bond formation, provide the best description of the chemical bond. This criterion was used for a variety of chemical bonds where the description in terms of dative or electron-sharing interactions was not clear.133−146 Table 21 shows that the ΔEorb values using triplet fragments are smaller than with singlet fragments for H2CCH2 and F2C CF2, which supports the notion that also the latter compound has an electron-sharing double bond. The change from electronsharing bonding at equilibrium to dative bonding during carbon−carbon bond breaking can be monitored by EDA calculations along the dissociation profile. Figure 38 shows the calculated reaction pathway for the carbon−carbon bond breaking/bond formation reaction of C2F4 at different levels of theory.237 The calculations suggest a smooth reaction profile where the CF2 fragments initially approach each other in a sideway orientation. The molecule becomes planar only at the final stadium of the bond formation. There is no activation barrier for the reaction; the small hump at the CASSCF(8,8) level is due to the lack of dynamic correlation. Table 22 shows the calculated ΔEorb values at different C−C distances using the singlet and triplet electronic states of CF2, which refer to the three different bonding models A, B, and C. The electron-sharing model A provides the best description for shorter distances as suggested by the ΔEorb values, but for longer distances the dative model B becomes better than A. Model C is not relevant at any carbon−carbon distance. The trend of the ΔEorb values shows that the change from electron-sharing to dative bonding can be illustrated with the EDA-NOCV method. It should be kept in mind that the transition involves a change of the model and not a change in the nature of the interatomic interactions.

consists of a σ and a π component. But the fragments, which are eventually formed during the bond-breaking process, are two CF2 species in the singlet (1A1) state. Since bond breaking is the reverse process of bond formation, the initial interaction must have a dative nature (see model B in Figure 26d). A different situation exists for ethylene H2CCH2, because the interacting methylene fragments have a triplet (3B1) ground state, which straightforwardly leads to electron-sharing σ and π bonding in the molecule (model A, Figure 26c). The different electronic states of the carbene fragments have a drastic effect on the BDE of the alkenes. Table 20 shows the calculated values of the alkenes C2X4 and alkanes C2X6 (X = H, F). The De values for the hydrogen bonded parent systems show the expected order; that is, the H2CCH2 double bond is much stronger than the H3C− CH3 single bond. In contrast, the F2CCF2 double bond is weaker than the F3C−CF3 single bond, although the former bond is even shorter than in H2CCH2. It becomes obvious that the bond dissociation energy is not a reliable measure for the strength of a chemical bond. The carbon−carbon double bonds in H2CCH2 and F2C CF2 have recently been studied in terms of dative and electronsharing bonds with the EDA-NOCV method.237 Table 21 gives the numerical results of the calculations, where the carbene fragments were arranged in the singlet and triplet states according to bonding models A and C (Figures 26c and 26e). It becomes obvious that the calculated interaction energies ΔEint between the triplet fragments in H2CCH2 and F2CCF2 are a more faithful description than the bond dissociation energies De. The ΔEint values give the strength of the stabilizing interactions in the eventually formed bond while the De values for C2F4 include the electronic relaxation of the CF2 fragments from the triplet to the singlet state. The preference for describing the double bonds in terms of electron-sharing interactions rather than dative bonds is AS


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Figure 36. (a) Geometry and bonding situation of the isolated complex Si3(CAACR)3 with experimental and (in parentheses) calculated bond lengths (Å) and angles (deg). Reprinted with permission from ref 334. Copyright 2016 John Wiley and Sons. (b) Calculated geometry and bonding situation of the experimentally observed aromatic cations B3L3+ (L = CO, N2). Reprinted with permission from ref 335. Copyright 2016 John Wiley and Sons.

for compound 2, where the dative interactions bz-NHCMe→ C2H2←bz-NHCMe (2b) lead to a smaller ΔEorb value than the electron-sharing model 2a, which refers to a tetraaminobutadiene. There is chemical evidence that 2b is indeed a more faithful representation of the chemical bonds than 2a. When the dication 22+ is reacted with silver or gold compounds, the in situ generated 2 reacts and gives a major product the metal complexes bz-[NHCMe→M←bz-NHCMe]+ (M = Ag, Au) where the metal cation replaces the central C2H2 moiety.140 Such reaction is unknown for tetraaminobutadiene compounds. The EDA-NOCV calculations for Si2H2(CAACDip) (Dip = 2,6diisopropylphenyl) suggest that the description in terms of dative and electron-sharing bond have essentially equal weight. Consequently, the synthesis of 3 was reported with the title “The Structure of the Carbene Stabilized Si2H2 May Be Equally Well Described with Coordinate Bonds as with Classical Double Bond”.136 The following conclusions arise from this section: • The distinction of the covalent electron-pair bonding into electron-sharing and dative interactions is a helpful model for understanding the structures and reactivities of molecules. It is also a powerful tool for predicting stable molecules with unusual chemical bonds.

Finally, we would like to discuss three systems LX2L with valence isoelectronic species X2 = N2, C2H2, and Si2H2, where the dichotomy of dative and electron-sharing bonds is a challenge for chemical intuition. Figure 39 shows three molecules 1−3, which may be written in either of the two depicted ways. Compound 1 (E = N) with a bz-NHCMe ligand (bz-NHCMe denotes a benzoannelated NHCMe ligand with methyl groups at nitrogen) was synthesized in 2012, where it was termed as a guanidine compound, which was sketched with the classical Lewis structure 1a.338 The above-described bonding analysis of N2(PPh3)2, which was recognized as donor−acceptor complex Ph3P→N2←PPh3, let it seem possible that compound 1 is better described with dative bonds bz-NHCMe→N2←bzNHCMe (1b). The choice of the bz-NHCMe ligands was made, because the molecule was supposed to be compared with the isoelectronic species C2H2(bz-NHCMe) (2), which was reported to have some surprising chemical behavior.140 Very recently, the silicon system 3 was reported, which may be considered as ligand stabilized disilaacetylene complex 3b or as substituted 2,3-disilabutadiene 3a.136 Table 23 shows the results of the EDA-NOCV calculations of 1−3. The data suggest that the chemical bonds at the central N2 fragment in 1 are indeed better described in terms of dative interactions bz-NHCMe→N2←bz-NHCMe (1b) than with electron-sharing bonds (1a). The same conclusion is drawn

• Dative bonds A→B usually tend to dissociate heterolytically and electron-sharing bonds A−B normally break up AT


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Figure 37. Reactions of SiCl2 with NHC and CAAC and schematic view of the bonding situation in the complex NHC→SiCl2 (A) and the molecule SiCl2(CAAC)2 (B) in the triplet state. Below are the calculated bond dissociation energies at M05-2X/def2-TZVPP of the complexes SiCl2(NHC) and SiCl2(CAAC) and the compounds in the triplet state SiCl2(NHC)2 and SiCl2(CAAC)2 as well as the singlet−triplet gaps of the fragments. The bottom lines give the net stabilization energies ΔE for the formation of the electron-sharing bonds in SiCl2(NHC)2 and SiCl2(CAAC)2 in comparison with the dative bond. Reprinted with permission from ref 337. Copyright 2013 John Wiley and Sons.

Table 20. Calculated (Experimental) C−C Bond Lengths Re [Å] and Calculated (Experimental) Bond Dissociation Energies De [kcal/mol]a H3CCH3 H2CCH2 F3CCF3 F2CCF2

Re

De

1.532 (1.522) 1.333 (1.336) 1.567 (1.545) 1.329 (1.311)

93.1 (89.7) 178.2 (172.1) 87.3 (96.4) 73.3 (70.3)

Table 21. EDA Calculations of C2F4 and C2H4 at the M06-L/ TZ2P Level Using Triplet and Singlet Fragments According to Models A and C (Figure 26)a C2F4

a

Calculated values were obtained at the M06-L/TZ2P level of theory. Taken from ref 237.

Fragments

Triplet (A)

Singlet (C)

Triplet (A)

Singlet (C)

ΔEint ΔEMetaGGA ΔEPauli ΔEelstatb

−197.7 6.2 305.0 −182.7 (35.9%) −326.1 (64.1%) −209.2 (64.2%) −87.0 (26.7%) −29.9 (9.2%) 122.2 −75.5

−279.9 −4.8 292.9 −175.6 (30.9%) −392.5 (69.1%) −221.5 (56.4%) −142.0 (36.2%) −29.0 (7.4%) 204.4 −75.5

−196.7 7.9 291.4 −183.6 (37.0%) −312.4 (63.0%) −218.0 (69.8%) −79.4 (25.4%) −15.0 (4.8%) 20.7 −176.0

−278.1 7.6 282.8 −181.2 (31.9%) −387.2 (68.1%) −241.2 (62.3%) −129.9 (33.5%) −16.1 (4.2%) 102.1 −176.0

ΔEorbb

homolytically, but the dissociation products are not always reliable probes for the type of the bonding. This is because the nature of the interatomic interactions may strongly change during the bond formation in a way that the initially dative interactions eventually turn into electron-sharing bonding. • Dative and electron-sharing bonds are models, which are helpful tools for the description and prediction of the structures and reactivities of molecules. There may be cases where both descriptions have similar validity.

C2H4

ΔEorb(σ)c ΔEorb(π)c ΔEorb restc ΔEprep ΔE = −De

a Energy values in kcal/mol. Taken from ref 237. bThe values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.

9. THE OCTET RULE AND THE ATOMIC VALENCE SPACE OF MAIN-GROUP ATOMS There is ample evidence that the atoms of groups 1, 2, and 13− 18 of the periodic system called main-group elements use only their s and p valence orbitals for chemical bonding. Since the exchange (Pauli) principle allows at most two electrons in one orbitals, the four valence orbitals can accommodate a maximum of eight electrons. This is the quantum theoretical basis for the octet rule, which was suggested by Langmuir already in 1921.6 The extension of the valence space to d-orbitals, which was

earlier suggested to explain the stabilty of so-called hypervalent molecules, was discarded on the basis of quantum chemical calculations. Atomic orbitals with higher angular momentum such as d and f functions only serve as polarization functions for the sp space, but they are not genuine valence orbitals in maingroup compounds.339,340 The evaluation of the symmetry of the molecular orbitals shows that molecular wave functions may be constructed that assign more than four bonds to an atom although they only use its four atomic valence s and p functions for chemical bonding. This has been discussed in section 7.4. AU


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Figure 38. Calculated reaction pathway for rupture of the carbon− carbon bond of C2F4 with different theoretical methods. The energy is given with respect to the dissociation product 2 CF2 (1A1). Reprinted with permission from ref 237. Copyright 2018 John Wiley and Sons. Figure 39. Schematic representation of alternative bonding situations with dative or electron-sharing bonds for molecules with valence isoelectronic fragments. (a) N2(bz-NHCMe)2; (b) (C2H2)(bzNHCMe)2; (c) (Si2H2)(CAACDip)2 (Dip = 2,6-diisopropylphenyl).

Recently, joint experimental/theoretical studies reported the observation and identification of the earth alkaline carbonyl complexes M(CO)8 (M = Ca, Sr, Ba) in low-temperature matrices, which possess chemical bonds that are typical for transition metals.102 The molecules fulfill the 18-electron rule, and the metal−carbonyl bonds were explained with the Dewar− Chatt−Duncanson (DCD) model1,341,342 in terms of M ← CO σ donation and M → CO π backdonation. Surprisingly, the latter interactions involve the valence d orbitals of the earth alkaline atoms, which are vacant in the electronic ground state. The electronic reference state of the metal atoms is the excited triplet (3F) state with the electron configuration (n)s0(n−1)d2, which is stabilized via strong M(d) → CO π backdonation. In a forerunner of the work, the authors had found that the barium carbonyl cation Ba(CO)+ exhibits a strong red shift of the C−O stretching mode, which was explained with large Ba(5d)+ → CO π backdonation.343 It is highly unusual that a cation serves as a donor to a neutral species. The possible involvement of d orbitals in barium compounds was earlier suggested by Pyykkö, who coined the term “honorary transition metal” for barium.344 But it seems that transition metal-like bonding may not be restricted to barium.

Figure 40 shows the calculated equilibrium geometries of neutral M(CO)8 and the cations M(CO)8+ (M = Ca, Sr, Ba), which were observed in the gas phase.102 The molecules were identified by comparing experimental infrared spectra with calculated wave numbers. The complexes are stable with respect to loss of one CO ligand by 8.4−11.5 kal/mol. The neutral species exhibit strong red shifts of the single IR active C−O stretching mode by 130−160 cm−1. Also the cations show a red shift by 30−50 cm−1, which can be explained with one and two occupied d-orbitals in the cations and neutral molecules, respectively. Figure 41 shows the correlation diagram of a metal with a (n)s, (n)p, (n−1)d shell and eight CO in a cubic (Oh) field. The qualitative model suggests that a total of 14 electrons are donated from the CO lone-pair MOs to the metal into (a) the triply degenerate t2g (n−1)d-AOs, (b) the triply degenerate t1u (n)p-AOs, and (c) the (n)s AO. One σ electron pair of the ligand cage has a2u symmetry, which does not match

Table 22. Calculated EDA Values at M06-L/TZ2P of the Orbital Term ΔEorb [kcal/mol] for the Interactions between CF2 with Different Electronic States and Different C−C Distances dC−C [Å]a

dC−C

1.30

1.326a

1.40

1.50

1.60

ΔEorb

−339.5

−326.1

−290.9

−245.6

−217.3

ΔEorb

−856.4

−802.0

−665.8

−420.9

−275.1

ΔEorb

−406.0

−392.5

−357.1

−359.4

−365.5

1.70 Model A −195.0 Model B −188.0 Model C −349.3

1.80

1.90

2.00

2.20

2.40

3.00

−178.4

−165.9

−156.5

−144.3

−137.7

−131.4

−133.3

−96.4

−70.4

−38.1

−20.7

−3.9

−326.1

−303.3

−283.7

−255.0

−237.1

−215.6

The bold values depict the smallest ΔEorb value at the respective C−C distance. Data taken from ref 237.

a

AV


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Table 23. EDA-NOCV Calculations at the BP86-D3(BJ)/TZ2P Level of Theory of N2(bz-NHCMe)2, C2H2(bz-NHCMe)2, and Si2H2(CAACDip)2 and Using the Interacting Fragments in the Singlet (S) or Quintet (Q) Electronic Statea N2(bz-NHCMe)2 2 Fragments

bz-NHCMe (S) + N2 (S)c

C2H2(bz-NHCMe)2 Me

2 bz-NHC (Q) + N2 (Q)

Me

2 bz-NHC (S) + C2H2 (S)c

Si2H2(CAACDip)2 Me

2 bz-NHC (Q) + C2H2 (Q)

Me

2 CAACMe (Q) + Si2H2 (Q)

2 CAAC (S) + Si2H2 (S)c

Bonding

dative

electron-sharing

Dative

electron-sharing

dative

electron-sharing

ΔEint ΔEPauli ΔEdisp ΔEelstatb ΔEorbb

−393.5 1275.3 −7.2 −547.8 (33.0%) −1113.9 (67.0%)

−633.5 1083.2 −7.2 −487.4 (28.5%) −1222.1 (71.5%)

−465.3 636.5 −10.4 −385.4 (36.8%) −706.1 (63.2%)

−436.0 924.8 −10.4 −503.9 (37.3%) −846.5 (62.7%)

−193.0 665.2 −33.4 (3.9%) −423.9 (49.4%) −401.0 (46.7%)

−323.5 436.0 −33.4 (4.4%) −328.6 (43.3%) −397.5 (52.3%)

a

Energy values are given in kcal/mol. The data are taken from refs 49, 136, and 140. bThe values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb. cThe π and π* MOs in the electronically excited reference state are occupied.

Figure 40. Calculated equilibrium geometries of alkaline earth octacarbonyls. (A) M(CO)8 (M = Ca, Sr, or Ba), (B) [M(CO)8]+ (M = Ca or Sr), and (C) [Ba(CO)8]+. Bond lengths are in angstroms. The D0 values in roman type are the ZPE-corrected bond dissociation energies for loss of one CO ligand; the italicized values are the corresponding energies for the loss of eight CO ligands and M/M+ in the ground state. The values without parentheses are from M06-2X-D3/def2-TZVPP calculations; the values in parentheses are from CCSD(T)/def2-TZVPP using M06-2X-D3/def2TZVPP optimized geometries. Reprinted with permission from ref 102. Copyright 2018 American Association for the Advancement of Science.

characteristic for transition metal compounds. The possible extension of this concept to other atoms than the alkaline earth elements and further ligands has yet to be explored. A guideline for the search of other stable molecules is the excitation energy from the occupied sp shell to the d orbitals and the strength of the bonds to a ligand. This concept proved to be useful for finding the class of carbones and related compounds, which are mentioned in section 8. It might also be helpful for the search of molecules where classical main-group atoms bind like transition metals. The following conclusions arise from this section:

any valence orbital of the metal. The earth alkaline atoms have therefore only 16-electrons in their valence shells. Although the complexes are formally 18-electron species, they are effectively 16-electron complexes. The HOMO is a degenerate orbital, and the molecules are therefore in agreement with Hund’s rule triplets. The singly occupied MO of the cations induces due to Jahn−Teller distortion a lower symmetry of the equilibrium structure. Table 24 shows the results of EDA-NOCV calculations on M(CO)8, which give among others a quantitative account of the individual orbital contributions that are sketched in Figure 41. The data show that the most important valence orbitals of the metals are the (n−1)d functions, which contribute 85%−95% to the total orbital interactions. The dominant contribution is due to the (n−1)d → (CO)8 π backdonation. Therefore, the alkaline earth atoms calcium, strontium, and barium bind the CO ligands in M(CO)8 like typical transition metals. The intrinsic interaction energies ΔEint between the metal atoms in the excited 3F state and the (CO)8 ligand cage is very strong; it overcompensates the (n)s → (n−1)d excitation energies of the metals. The bonding analysis of the recently reported 20electron systems M(CO)8− (M = Sc, Y, La), which also have cubic (Oh) symmetry, showed a very similar bonding pattern.345 The HOMO in Figure 41 is fully occupied in the latter compounds, which are thus effectively 18-electron complexes. The above results indicate that some atoms of the sp shell are able to use their d orbitals, which are empty in the electronic ground state, for chemical bonds in stable molecules that are

• The valence space of the main group atoms usually comprises the outermost s and p orbitals, which is the basis for the octet rule. However, main-group elements can also form stable molecules with covalent bonds using their d-orbitals when the excitation energy to the d-shell is compensated by the binding energies. This has recently been demonstrated by the synthesis of the octacarbonyl complexes M(CO)8 (M = Ca, Sr, Ba).

10. RELATIVISTIC EFFECTS IN MAIN-GROUP COMPOUNDS When one moves to the heaviest elements in the periodic table, relativistic effects need to be considered in chemical bonding. Since relativistic quantum theory is relatively a newcomer to the field of quantum chemistry, and from the theoretical point of AW


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Figure 41. Bonding scheme and shape of the occupied valence orbitals of M(CO)8 (M = Ca, Sr, or Ba). Splitting of the spd valence orbitals of an atom M with the configuration (n−1)d2(n)s0(n)p0 in the octacoordinate cubic (Oh) field of eight CO ligands is also given. Only the occupied valence orbitals that are relevant for the M−CO interactions are shown. Up and down arrows indicate electrons with opposite spin. Reprinted with permission from ref 102. 2018 American Association for the Advancement of Science.

Loosely speaking, the two large components ΨLα and ΨLß of the relativistic wave function ΨD refer to electrons with α and ß spin, and the remaining two small components ΨSα and ΨSß are required to allow for the existence of positrons (the antiparticles of the electrons with the same mass and spin but opposite charge), which were not yet known in 1928 when Dirac developed his theory and whose existence was predicted by him.355 Positrons were only identified in 1932.356 Thus, the Dirac equation directly yields the electron (and positron) spin, whereas spin (and its resulting spin−orbit coupling) must be introduced ad-hoc into the nonrelativistic Schrödinger equation. Calculations involving the Dirac equation with the full fourcomponent wave function ΨD are significantly more complicated and demanding than those of the nonrelativistic Schrödinger equation, mainly due to the fact of the appearing negative energy continuum and increasing computational costs of the matrix form of the Dirac operator. Nevertheless, a number of four-component methods have been developed over the past 30 years, applicable routinely to smaller systems.357 For larger systems, approximate two-component methods have been developed, which involve the separation of the large from the small component in the Dirac wave function and a transformation to a two-component formalism.358−360 Twocomponent methods such as the recently introduced X2C approximation are less computer time-consuming than fourcomponent procedures but are still computationally expensive especially when electron correlation is included.346−354,361,362 Therefore, the presently most commonly used quantum chemical approaches including relativistic effects in chemistry are based on one-component (scalar relativistic) approximations, such as the Douglas−Kroll method modified by Hess363 or the Zeroth Order Regular Approximation (ZORA).364 A very popular time-efficient and quite accurate alternative is the use of relativistic effective core potentials (ECPs), also called

view quite demanding for chemists, we briefly outline the most important qualitative view points of this field. The Schrödinger equation is usually presented in quantum chemistry in its time-independent form (eq 27). The kinetic energy requests the second derivatives with regard to the spatial coordinates (x,y,z). The time-dependent Schrödinger equation (eq 28) involves the derivative with respect to the time originating from the energy quantization, but the time differential is only of first order. Relativistic theory demands that time and space are equivalent coordinates in the four dimensional space-time continuum, and thus, they must be treated equally in physics. The Schrödinger equation is therefore not compatible with relativity, but it naturally emerges from the nonrelativistic limit (velocity of light c → ∞). [−h2 /2m(∂ 2/∂x 2 + ∂ 2/∂y 2 + ∂ 2/∂z 2) + V (x , y , z)] Ψ(x , y , z) = E Ψ(x , y , z)

(27)

[−ℏ2 /2m(∂ 2/∂x 2 + ∂ 2/∂y 2 + ∂ 2/∂z 2) + V (x , y , z , t )] Ψ(x , y , z , t ) = ih∂/∂t Ψ(x , y , z , t )

(28)

The problem to bring the Schrödinger equation in harmony with special relativity was solved by Dirac. He introduced a (4 × 4) matrix operator, which brings the spatial and time coordinates on equal footing. Mathematical details on the Dirac operator D shall not be presented in this work; they are discussed in several review articles and monographs.346−354 The most important result of the Dirac eq 29 is the manifestation of ΨD = (ΨLαΨLßΨSαΨSß) being a four-component wave function (spinor). DΨD(x , y , z , t ) = E ΨD(x , y , z , t )

(29) AX


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This is because valence electrons can have substantial density in the core region where relativistic perturbation operators act.375,376 This gives rise to three major effects of relativity on the valence electrons, which are important for the electronic structure and bonding situation in molecules:368 (a) The lowest lying s orbitals (and often p orbitals but to a lesser extend) stabilize and contract; that is, their shell radii become smaller (direct relativistic effect). (b) Because of this direct relativistic effect the nucleus becomes more screened lowering the effective nuclear charge for each orbital, and most notably the higher angular momentum (more diffuse) functions expand and the orbital energies increase. This effect is called the indirect relativistic ef fect and is more important for the d and higher angular momenta shells. (c) The third effect is the coupling of the electron spin with its angular momentum. This effect is called spin−orbit coupling (correctly derived only from the Dirac equation), which, for example, leads to a splitting of the p orbitals into a p1/2 and p3/2 component. We note that the contraction of the p1/2 component of the orbital is much larger than for the p3/2 component. Figure 42 shows the variation of the orbitals under the influence of relativistic effects.377 In many-electron systems the underlying shell structure has significant influence on the magnitude of relativistic effects.378 For example, in neutral Tl the 6p1/2 orbital contracts (−10.4%) while the 6p3/2 orbital expands slightly (+2.2%). Spin−orbit splitting in the p-levels can be substantial; for example, for the heavy p-block row we get for the spin−orbit splitting (in kcal/ mol) in the spectroscopic levels379 of Tl 22.3 (2P1/2/2P3/2), Pb 30.4 (3P0/3P2), Bi+ 48.7 (3P0/3P2), Po 48.1 (3P2/3P1) and At 60.3 (2P3/2/2P1/2).380 It becomes obvious that such large energy differences due to relativity become manifest also in chemical bonding. Interestingly, subtle shell structure effects378 can lead to a much larger percentage contraction of the valence s-orbital compared to the core−shell s-orbitals. For example, for the neutral atom Tl we get the following s-contractions/orbital stabilizations:381 1s −12.7/10.7%, 2s −13.4/17.4%, 3s −10.6/ 18.1%, 4s −9.4/20.1%, 5s −9.5/21.7%, 6s −13.1/24.4%. Moreover, relativistic effects in the s-levels of the p-block elements are particularly large from filling the lower lying core dshell.382 This rather strong relativistic valence s-shell contraction leads to a large separation between the ns/np levels and to a reduced mixing (hybridization) between the two, thus altering chemical bonding in the p-block heavy-element section substantially. This comes on top of the reduced s/p mixing from the first to the second octal-row atoms discussed above. Interestingly, these perhaps unexpected large relativistic effects and their consequence for chemical and physical properties have been accepted far earlier by the chemistry than by the physics community. Spin−orbit effects can be large in heavy elements but are suppressed in strong ligand fields, which is best described by ionic bonding models. This is obvious as we formally move electron density from orbitals undergoing spin−orbit splitting at the central atom to the electronegative ligand. For the p-block elements in high oxidation states with strong ionic bonding partners (such as fluorine) spin−orbit effects become rather small. However, for covalent bonding spin−orbit effects can also be substantially reduced by mixing of p1/2 with p3/2 orbitals. These relativistic orbitals are complex spin α/β mixtures; that is, two p1/2 combinations at different centers give 1/3σ and 2/3π* bonding or a 2/3π and 1/3σ* bonding, while two p3/2 combinations at different centers give one π or a 2/3π and 1/3σ* bonding.383 Thus, s-p hybridization and the formation of

Table 24. EDA-NOCV Results for Triplet State M(CO)8 (M = Ca, Sr, Ba) Complexes at the M06-2X/TZ2P-ZORA//M062X-D3/def2-TZVPP Level Taking (CO)8 in Singlet Ground State and M in Triplet Excited State with a (n)s0(n−1)d2 Valence Electronic Configuration as Interacting Fragmentsa Energy term ΔEint ΔEhybridb ΔEPauli ΔEelstatc ΔEorbc ΔEorb(1)d,e (eg) ΔEorb(2)d,e (t2g) ΔEorb(3)d (a1g) ΔEorb(4)d,e (t1u) ΔEorb(5)d (a2u)

[M(d)]→(CO)8 π backdonation [M(d)]←(CO)8 σ donation [M(s)]←(CO)8 σ donation [M(p)]←(CO)8 σ donation (CO)8 polarization

ΔEorb(rest)d ΔEprep(a) ΔEprep(b)f ΔE (ΔEint + ΔEprep) = −De

8 CO → (CO)8 M, (n)s2 → (n) s0(n−1)d2 (T) M(CO)8 → M (1S) + 8 CO

Ca + (CO)8

Sr + (CO)8

Ba + (CO)8

−243.1 41.8 19.5 −65.3 (21.5%) −239.1 (78.5%) −206.2 (86.2%) −21.3 (9.0%) −2.4 (1.0%) −0.9 (0.3%) −0.6 (0.3%) −7.7 (3.2%) 13.9 159.5

−224.1 46.9 30.5 −61.7 (20.5%) −239.7 (79.5%) −206.4 (86.2%) −20.7 (8.7%) −2.9 (1.2%) −0.9 (0.3%) −1.0 (0.4%) −7.8 (3.3%) 7.7 150.9

−145.2 37.4 30.8 −78.5 (36.8%) −135.0 (63.2%) −95.0 (69.8%) −22.8 (16.8%) −3.2 (2.4%) −2.7 (2.1%) −2.3 (1.7%) −9.0 (6.7%) 3.3 68.2

−69.7

−65.5

−73.7

a

Energy values are given in kcal/mol. bContribution of the metahybrid term in M06-2X. cThe values within the parentheses show the contribution to the total attractive interactions ΔEelstat + ΔEorb. dThe values within the parentheses show the contribution to the total orbital interaction, ΔEorb. eThe sum of the two (eg) or three (t2g, t1u) components is given. fThe EDA calculations give a triplet state with spherically symmetrical distribution of the d electrons. The experimental values for excitation into the energetically lowest lying 3 F state with (n)s0(n−1)d2 configuration are 124.2 and 59.8 kcal/mol for Ca and Ba. There is no experimental value for the relevant 3F state of Sr. The data are taken from ref 102.

pseudopotentials.365,366 The basic idea behind the pseudopotential approximation goes back to Hellmann, who introduced the concept already in 1933.367 Here, the chemically inactive core electrons of the atom are replaced by an effective potential, which is optimized in relativistic atomic calculations mimicing the influence of relativistic effects on the valence electrons. Thus, ECP calculations do not only consider relativistic effects but also reduce the number of electrons in quantum chemical calculations. Small-core ECPs explicitely retain the outer core−electrons in the calculations, whereas the less accurate large-core ECPs consider only the immediate valence shell. The relativistic ECP approximation transfers relativistic effects extremely well from the atomic into the molecular environment; thus, the change of the relativistic influence on the electronic structure during the interatomic interactions is accurately described. For a very long time it was thought that relativistic effects are unimportant for chemistry, because the relativistic increase in kinetic energy of the electrons in heavy atoms takes place mainly at the core electrons close to the nucleus. It was only in the 1970s when it was discovered that the valence electrons in heavier atoms are also affected by relativity and to the extend where the chemistry of heavy elements is changed significantly.354,368−374 AY


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Figure 42. Radial densities D(r) = 4πr2ρ(r) (r in atomic units) for the 1s, 2s, 3s, and 2p states of a hydrogen-like atom with Z = 80. The dashed curves are nonrelativistic (NR) and the full curves relativistic. The contractions for 1s, 2s, 2p1/2, and 3s are of the same order of magnitude while that for 2p3/2 is much smaller. Reproduced with permission from ref 377. Copyright 1967 Elsevier.

strong σ bonding can be severely hampered if we do not correctly mix p1/2 with p3/2.383 This is especially the case when spin−orbit splitting becomes very large, as for the superheavy pblock elements. For example, the 2P3/2/2P1/2 splitting in the heaviest rare-gas element known, Og+, is 229 kcal/mol,384 larger than most bond dissociation energies. Spin−orbit effects have major influence in systems, which are sensitive to small changes in energy such as in (first or second-order) Jahn−Teller distortion385,386 or in dispersive types of interactions. All the important bonding models for molecules or the solid state developed in nonrelativistic quantum chemistry can be transferred and applied to the relativistic domain, even if spin− orbit coupling is considered. Furthermore, like electroncorrelation effects which can be switched on and off in calculations, relativistic effects are extremely useful in discussing anomalies in chemical bonding situations and periodic trends.387,388 Concerning relativistic effects in the group 1 and 2 elements of the periodic table,389,390 which are dominated by the direct relativistic s-shell contraction/stabilization, these are rather small compared to the corresponding group 11 and 12 atoms (except for the superheavy elements with nuclear charge 119 and 120).391−393 This was already recognized early on by Pyykkö and Desclaux.394 In these groups we expect for example small bond contractions and bond stabilizations for the dimers, which however will not significantly alter the bonding picture. The situation completely changes when the electropositive group 1 and 2 elements bind to atoms such as gold, which exhibit very

strong relativistic effects. For example, in the compound CsAu the gold atom acts like an electronegative ligand inducing ionic bonding best described by Cs+Au−,395−397 which is due to the relativistically increased electronegativity of Au (from 1.9 to 2.4).398 This leads to the interesting situation where solid Cs+Au− adopts a rocksalt structure rather than being an intermetallic compound as one would expect399 (two metals do not necessarily give a metallic bond!). CsAu can be dissolved in liquid ammonia like several other ionic alkali or alkaline earth halides,400 and related to this, the interesting coordination compound AuCs·NH3 has been isolated by Jansen and coworkers in 2002.401 A similar interesting example is the closedshell interaction between Ba and Au− with an unprecedented high dissociation energy of 34 kcal/mol due to relativistic effects.402 Far more interesting for relativistic effects in chemical bonding are the p-block elements. The direct relativistic sshell contraction, enhanced by filling the core d-shell, results in the valence (n)s AOs of the heavy atoms becoming less prone to chemical bonding; that is, the (n)s electrons retain as lone-pairs rather than engaging in chemical bonding (inert-pair effect, originally proposed by Sidgwick403−405). This reduces s−p mixing in heavy main group compounds and, as a consequence, destabilizes high oxidation states.406 For example, the nonrelativistic 6s population for TlCl3 was calculated to be 0.73, while the relativistic one gave 0.96.406 It is well-known that low valencies in heavy p-block element compounds arise naturally from the periodic trend down the group in the periodic table AZ


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distance by 0.17 Å and reduce the bond energy by about 19 kcal/ mol in Tl2.417 This feature of weak Tl−Tl bonding propagates into the solid state with Tl having a rather small cohesive energy of 43 kcal/mol. Further, Tl−Tl interactions in coordination compounds (termed thallophilic interactions) are rare and extremely weak, and mostly counterion-mediated, requiring the anions to provide favorable electrostatics,418 or due to crystal packing effects.419,420 Similar large spin−orbit coupling effects are observed for lead. For example for solid lead, nonrelativistic PW91 results give a cohesive energy of 74 kcal/mol, scalar relativistic effects reduces it slightly to 70 kcal/mol changing the crystal structure from diamond to face center cubic, and finally spin−orbit effects lead to a substantial reduction to reach a final calculated value of 46 kcal/mol (exp. 47 kcal/mol) and a change in crystal structure to the hexagonal close packing arrangement.421 Kraka et al. studied the energetics and mechanism for the hydrogenation reaction XHn + H2 → XHn+1 + H with X belonging to group 4, 5, 6, or 7 of the periodic table, and found that spin−orbit coupling becomes very important for the heavier elements.422,423 The chemistry of short-lived isotopes of superheavy elements can be studied using atom-at-a-time experiments.424 To prepare for such experiments, theoretical predictions about their chemical and physical behavior are required.425 Here, however, relativistic effects in the bonding of the superheavy elements become so dominant that they drastically change their chemical and physical behavior.391−393 The inert-pair effect becomes so strong for the higher oxidation state that even the fluorides NhF3 and FlF4 decompose easily.426,427 For example, the 7s population in NhF3 increases dramatically from 0.3 to 1.2 due to relativistic effects. In fact, NhF3 adopts a T-shape instead of the expected trigonal-planar structure. Because of strong spin− orbit splitting in Fl (the 3P0/3P2 splitting is 83 kcal/mol428), Fl has a closed-shell (7p1/2)2 configuration, and p3/2 mixing into the Fl−Fl bond to form strong σ- and π- bonding becomes unlikely, predicting only very weak van der Waals type of interactions. Indeed, solid-state calculations gave a cohesive energy of only 12 kcal/mol for Fl.421 Pitzer already suggested in 1975 that Fl should be chemically inert and could even be a gas at room temperature.429 Moving to the elements where the spin−orbit destabilized 7p3/2 level becomes occupied, B3LYP ZORA calculations by Mitin and Wüllen show a very large spin−orbit destabilization for the dimer of tennessine (eka-At), Ts2, from 37 to 11 kcal/ mol due to the inaccessability of the 7p1/2 electrons to form proper σ-bonds.430 This might be termed as a spin−orbit caused 7p1/2 inert pair ef fect. The rare-gas element oganesson (eka-Rn) is even more spectacular. The extremely large spin−orbit splitting results in a rather diffuse and polarizable 7p3/2 shell and increases the dispersive type of interactions. Hence, Og2 has the highest dissociation energy of any rare-gas dimer predicted at 1.8 kcal/mol.431 In fact, the electron localization function (ELF) exhibits a uniform electron gas-like behavior in the valence region not seen for any of the lighter elements shown in Figure 43.384 The ELF clearly shows the shell structure for the heavier rare gases Xe and Rn, but for Og the density becomes smeared out over the whole atom. Moreover, Og has a positive electron affinity of 1.5 kcal/mol due to the very large relativistic 8s shell contraction, while all the other rare-gas elements do not bind an extra electron.432,433 One might therefore speculate that solid Og has a small band gap and becomes semiconducting or even metallic due to relativistic effects.434 Further, the diffuse four 7p3/2 electrons can easily be removed to form compounds like

because of smaller overlap between the more diffuse higher principal quantum number orbitals; however, relativistic effects significantly enhance this trend. For example, the TlCl3 → TlCl + Cl2 decomposition energy reduces substantially by 42 kcal/ mol (Table 25) due to relativistic effects to a final experimental Table 25. Relativistic and Nonrelativistic Reaction Energies (in kcal/mol) of TlX3 (X = F−I) for the Process TlX3(D3h) → X2 + TlX at the QCI Level Using the Pseudopotential Method (from Ref 407) Reaction

Nonrelativistic ΔE

Relativistic ΔE

TlF3 → F2 + TlF TlCl3 → Cl2 + TlCl TlBr3 → Br2 + TlBr TlI3 → I2 + TlI

154.9 97.0 82.0 63.7

105.4 54.6 45.2 33.5

value of 50 kcal/mol.407 Solid-state effects shift the equilibrium even more toward the lower oxidation state for thallium; that is, solid TlCl3 is rather unstable and disproportionates at 40 °C, loosing chlorine. This originates from the large relativistic increase of the TlCl dipole moment by more than 1 D,407 leading to stronger ionic interactions in the solid. Concerning TlI3, it only exists as thallium(I)triiodide in solid form.408 Similarly large relativistic effects are obtained for the halides of lead.406 For example, Sn(II) and Pb(II) compounds are quite stable, whereas C(II) and Si(II) require electronic stabilization in order to be isolated. The influence of relativity on the stability of group-14 compounds in the oxidation states E(II) and E(IV) becomes obvious by the calculated reaction energies for the decomposition ECl4 → Cl2 + ECl2 (E = C − Pb) shown in Table 26. It is clear that there is already a sizable difference between Table 26. Relativistic and Nonrelativistic Free Reaction Energies (in kcal/mol) of ECl4 (E = C−Pb) for the Process ECl4 → Cl2 + ECl2 at the BP86-D3(BJ)/TZ2P Level Using the Scalar ZORA Approach Reaction

Nonrelativistic ΔG298

Relativistic ΔG298

CCl4 → Cl2 + CCl2 SiCl4 → Cl2 + SiCl2 GeCl4 → Cl2 + GeCl2 SnCl4 → Cl2 + SnCl2 PbCl4 → Cl2 + PbCl2

60.9 98.0 61.4 65.2 72.4

60.8 97.0 55.9 49.4 14.4

nonrelativistic and relativistic values for Ge, which becomes much larger for Sn and particularly large for Pb. For example, while CCl4 is a stable compound, PbCl4 decomposes quickly at 50 °C into PbCl2 and Cl2 due to relativistic effects (the gas phase decomposition energy of PbCl4 has been calculated to be 24 kcal/mol at the CCSD(T) level of theory).409 A nice peculiarity concerns the lead acid battery, for which we have the overall reaction Pb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(s) + 2H2O(l). The relativistic 6s stabilization destabilizes PbO2, thus raising the voltage from 0.4 V to the standard value of 2.1 V.410 Thus, lead batteries only work because of relativistic effects. To illustrate the role of spin−orbit effects, we choose two prime examples, the dimers Tl2 and Pb2.411−414 Tl2 has an unusually small dissociation energy of 0.43 ± 0.04 eV.415 Christiansen and Pitzer stated in 1981 that “Tl2 is only weakly bound... The cause is undoubtedly related to the large spin-orbit splitting between the 6p1/2 and 6p3/2 thallium spinors”.416 Han and Hirao calculated that spin−orbit effects increase the bond BA


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Spin−orbit coupling becomes very important for the fifth and higher octal rows. • Quantum chemical methods are now available, which consider relativistic effects with high accuracy. The presently most common methods are the ZORA approximation, Douglas−Kroll−Hess approach, X2C, and relativistic pseudopotentials. Spin−orbit coupling can be either included at the Hartree−Fock or Kohn− Sham level or at a later stage at the correlated level (e.g., multireference configuration interaction).

11. CONCLUDING REMARKS The approach and the relationship of many chemists to the quantum theoretical explanation of chemical bonding may be seen as example of Plato’s cave allegory, which is described in his work Republica.444 It is written as a fictitious dialogue where Socrates pictures a group of imprisoned people who spent all their life chained in a cave and facing a blank wall. The people watch moving shadows projected on the wall in front of them from a fire behind them. Since these are the only processes accessible to their senses, the shadows are seen as real objects. They develop a science of shadows and try to determine laws in their occurrence and movements and to derive prognoses from them. They give praise and honor to the one who makes the best predictions. One day a prisoner manages to break his chains and he escapes from the cave. He discovers the sun and realizes that their reality was not what they thought it was. But the sun “would hurt his eyes, and he would escape by turning away to the things which he was able to look at, and these he would believe to be clearer than what was being shown to him.”444 Slowly his eyes are adjusting to the light, and the freed prisoner understands that the real world outside the cave is superior to the world he experienced in the cave. He is happy about the change, and he wants to free his fellow cave dwellers out of the cave and into the sunlight. But returning to the cave would again hurt his eyes, which have become accustomed to the sunlight, and the prisoners left behind would infer from the returning man’s pain that leaving the cave had harmed him and that they should not undertake a similar journey. Thus, anyone who attempted to drag them out of the cave would therefore be attacked and cursed.444−448 The resistance and even hostility of experimental chemists toward quantum chemistry is a painful experience of many theoretical chemists. This work is an attempt to connect the present understanding of the nature of the chemical bond in terms of quantum theory with the heuristic bonding models, which were derived from experimental observations. We do not consider the empirical models as irrelevant or unimportant shadows on the wall. On the contrary, they are central ingredients and key variables in chemical research, which serve as indispensable guidelines for further development. However, the models must not be identified with the physical reality of chemical bonding and interatomic interactions. They should be used on the basis of and in conjunction with accurate quantum chemical calculations. The sometimes ambiguous interpretation of experimental results can be better focused in favor of more targeted planning of further experiments. The laws of physics are only the framework within which chemical research, with the help of human creativity and fuzzy concepts, always creates something new, often with different viewpoints about the interpretation of chemical bonding. This is not a weakness of chemistry but shows the limitation of the human mind, which has been expressed by

Figure 43. Electron localization functions from nonrelativistic (NR, left) and Dirac−Hartree−Fock calculations (R, right) for the heavy rare gas atoms Xe (top), Rn (middle), and Og (bottom). Reprinted with permission from ref 384. Copyright 2018 American Physical Society.

OgF4. This structure, however, does not adopt the expected D4h symmetry but becomes tetrahedral due to strong spin−orbit coupling.435 This indicates that we expect rather unusual bonding features in the superheavy element region due to relativistic effects, which need yet to be explored. As already mentioned for the Group 1 and 2 elements, strong relativistic effects in molecular properties also appear when elements are bound to “relativistic” atoms such as platinum, gold or mercury. To mention a few examples for the p-block elements, the strong Pt−C-bond (∼110 kcal/mol) in [Pt(CH2)]+ profits from a 50 kcal/mol relativistic stabilization with the two Pt configurations d8s1 and d9 being involved in the formation of the Pt−C double bond.436,437 That relativity can dictate enzymatic reaction was nicely demonstrated by Kozlowski et al., who showed that mercury methylation by cobalt corrinoids is dictated by spin−orbit splitting of the Hg 6p orbitals.438 In this respect, (CH3)2Hg, a highly volatile compound and a well-known toxin, shows very large relativistic effects in the Hg−C bond strength.439,440 For many further examples on relativistic effects including main-group elements, we refer to a number of review articles.368,370−374,394,441−443 The following conclusions arise from this section: • Relativistic effects have a strong influence on the structures and reactivities of heavy main-group atoms. They become already important for atoms of the fourth octal row of the periodic system, and they have a dominant effect on atoms of the fifth and higher rows. • Relativistic effects make the s-orbitals less inclined to form chemical bonds. This leads, among other things, to the stabilization of compounds in low oxidation states and enhances the inert pair effect for the p-block elements. BB


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Niels Bohr with the saying “We must be clear that when it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections.”113

Foundation to join Auckland University and the Australian National University. He earned many awards, amongst them the Prince & Princess of Wales Science Award, the Hector medal, Fellowships of the Royal Society New Zealand and the International Academy of Quantum Molecular Science, a James Cook Fellowship and most recently a Humboldt Research Prize, the Fukui Medal and the Rutherford medal. His research interests are in fundamental chemistry and physics.

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected].

Gernot Frenking, FRSC, studied chemistry at the universities Aachen, Kyoto, and TU Berlin where he received his doctoral degree in 1979. After obtaining his habilitation in theoretical organic chemistry at the TU Berlin in 1984, he moved to the USA. Following one year as a visiting scientist with Fritz Schaefer at UC Berkeley he worked at the Stanford Research Institute in Menlo Park, California. In 1989 he returned to Germany as Associate Professor for Computational Chemistry at the Philipps University Marburg, where he was appointed Full Professor for Theoretical Chemistry 1998 and became HansHellmann-Professor from 2011−2014. He is also Visiting Research Professor at the DIPC (Donostia International Physics Center) at San Sebastian, Spain. Awards include the Elhuyar-Goldschmidt Prize of the Royal Society of Chemistry of Spain (2007) and the Schrödinger Medal of the WATOC (2009). In 2019 he received the International Solvay Chair of Chemistry of the International Solvay Institutes. His research interests lie in the field of Chemical Bonding Theory and Theoretical Inorganic Chemistry. Major topics of his work are the nature of the chemical bond and molecules with unusual electronic structures.

ORCID

Lili Zhao: 0000-0003-2580-6919 Sudip Pan: 0000-0003-3172-926X Gernot Frenking: 0000-0003-1689-1197 Notes

The authors declare no competing financial interest. Biographies Lili Zhao received her Ph.D. degree at the Graduate University of Chinese Academy of Sciences in 2012. She then worked in the IHPC, A*STAR in Singapore as a Scientist during 2012−2014, modeling homogeneous catalysis. From 2014 to 2016, she held a Humboldt research fellowship at Philipps-Universität Marburg in Germany. She joined Nanjing Tech University (China) in the end of 2016 and worked independently as a full professor. Her research interest include the homogeneous catalysis, bonding analysis by using state-of-the-art methods (i.e., EDA, EDA-NOCV, QTAIM), as well as carbone chemistry.

ACKNOWLEDGMENTS G.F. wants to express his gratitude to Dr. Alfred Paulus for inspiring suggestions and Prof. Eugen Schwarz as well as Klaus Ruedenberg for enlightening discussions about the nature of the chemical bond. He also expresses his gratitude for financial support by the Alexander von Humboldt foundation and the Deutsche Forschungsgemeinschaft. L.Z and G.F. acknowledge financial support from Nanjing Tech University (grant numbers 39837123 and 39837123) and a SICAM Fellowship from Jiangsu National Synergetic Innovation Center for Advanced Materials, National Natural Science Foundation of China (grant no. 21703099), and Natural Science Foundation of Jiangsu Province for Youth (grant no. BK20170964). S.P. thanks Nanjing Tech University for a postdoctoral fellowship and the High Performance Computing Center of Nanjing Tech University for providing computational resources. P.S. thanks the Royal Society of New Zealand for financial support in terms of a Marsden Fund (17-MAU-021) and Prof. Helmut Schwarz for useful discussions.

Sudip Pan obtained his Ph.D. degree from the Indian Institute of Technology Kharagpur (India) in 2016 under the supervision of Prof. Pratim K. Chattaraj. In the same year, he moved to work as a Postdoctoral Fellow at Cinvestav Merida (Mexico) under Prof. Gabriel Merino. Toward the end of 2017, he moved to Nanjing Tech University (China) for another postdoctoral stay under Prof. Gernot Frenking and Prof. Lili Zhao, where he is currently working. His research interests include the theoretical predictions of viable noble gas compounds, effective hydrogen storage material, clusters with unusual bonding, nanomachinery, compounds with boron−boron triple bonds and reactivity, catalysis and reaction mechanism, planar hypercoordinate carbon and boron systems, and ligand stabilized species and reactivity. Nicole Holzmann studied chemistry at the universities Darmstadt, Bristol/UK, and Marburg. In 2013 she received her doctoral degree in Theorerical Chemistry at the Philipps-Universität Marburg in the group of Prof. Dr. Gernot Frenking, where she was working on bonding analyses of donor−acceptor complexes. From 2014 to 2015 she was a postdoc with the CNRS (Centre National de la Recherche Scientifique) at the University of Lorraine with Prof. Chris Chipot and Dr. François Dehez. During this time she carried out MD simulations of the membrane proteins. In her current position in the group of Dr. Gilberto Teobaldi at the STFC (Science and Technology Facilities Council), she works in close collaboration with user groups and instrument scientists at the Rutherford Appleton Laboratory facilities and in the area of ab inito molecular dynamics simulations and TD-DFT.

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Peter Schwerdtfeger currently holds a chair in Theoretical Chemistry at Massey University and serves as Director of the Center of Theoretical Chemistry and Physics within the Institute for Advanced Study. He took his first degree in Chemical Engineering (Aalen) and studied both chemistry and mathematics at Stuttgart University, where he received his Ph.D. in theoretical chemistry, at habilitated at Marburg University. He held a position as a software analyst at Stuttgart University before receiving a Feodor-Lynen fellowship of the Alexander von Humboldt BC


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