The nature of the chemical bond

The nature of the chemical bond has long been a topic of much interest, beginning with the speculations of Isaac Newton about the “forces” that hold atoms together to the famous book by Pauling with this title.¹ For chemists, the existence of bonds between the atoms in a molecule is without question, but what are they, how can they be categorized, and what accounts for their existence? Early theories focused on the attraction of electrical charges of opposite sign, which could account for the binding in many inorganic molecules. However, what would account for the binding in electrically neutral organic molecules? Slightly more than a century ago, Lewis noted that the chemical bond appeared to be associated with electron pairs.² Shortly after the formulation of the wave equation by Irwin Schrödinger in 1925–1926, Heitler and London showed that quantum mechanics, in the form of what would eventually be called valence bond (VB) theory, accounted for the binding in that most improbable of molecules, the simple H₂ molecule.³ 

With the advent of increasingly powerful electronic computers and the development of increasingly sophisticated chemical theories and computational techniques, chemists have gained new insights into the nature of the various types of chemical bonds, including those associated with weak interactions. With these developments, the traditional VB theory of the chemical bond introduced by Heitler and London has been replaced by increasingly sophisticated and much more accurate versions of the theory, including Valence Bond Self-Consistent Field (VBSCF),⁴ Breathing Orbital Valence Bond (BOVB),³ and Spin-Coupled Generalized Valence Bond (SCGVB)⁵ theories. These theories optimize the orbitals in the wave function, thus freeing the results from any preconceived notions about the nature of the atomic orbitals. Molecular orbital (MO) theory⁶ and its computational counterpart, restricted Hartree–Fock theory,⁷ offered another view of the electronic structure of molecules and were extensively used to describe the ground and excited states of molecules, e.g., O₂. Although the MO theory, with orbitals delocalized over the entire molecule, appears to offer a very different view of bonding in molecules, Edmiston and Ruedenberg⁸ showed that VB and MO theories were very closely linked—they showed that doubly occupied orbitals representing two-electron bonds and lone pairs were just a unitary transformation away from the delocalized molecular orbitals of the traditional MO theory (for early thoughts on this topic, see Refs. 9–12). Finally, Ruedenberg and co-workers and Roos developed multiconfiguration wave functions, the full optimized reaction space¹³ (FORS) and the complete active space self-consistent field¹⁴ (CASSCF) wave functions, that corrected the major deficiencies of the RHF wave function and laid the foundation for additional insights into the nature of the chemical bond and the electronic structure of molecules.¹⁵ These new multiconfiguration approaches are closely related to the VBSCF and SCGVB approaches, especially the latter (see the discussion in Ref. 5).

In the past decade, a new generation of petascale and exascale computers was deployed in the U.S., the European Union, Japan, and China. These extraordinarily powerful computers are based on new computing technologies, which have led to a concerted effort to adapt the existing suites of electronic structure methods for these new technologies, with the development of exascale versions of NWChem, GAMESS, EXAALT, and QMCPACK being funded by the Exascale Computing Project in the U.S. Department of Energy.¹⁶ These developments will dramatically increase the number of correlated electrons and active orbital spaces that can be accurately described via highly correlated methods^17–21 as well as the overall capabilities of these suites of electronic structure codes. We look forward to the advances in our understanding of the nature of the chemical bond resulting from computational studies with these new modeling capabilities.

Over the past half century, theoretical and computational chemists have developed not only the above-mentioned theories of the chemical bond, but also numerous techniques for analyzing increasingly sophisticated molecular wave functions, including atoms-in-molecules analysis,²² natural bond orbital analysis,²³ quasi-atomic orbital analysis,^24–26 bond orders and multicenter bond indices,^27–29 domain-averaged Fermi hole analysis,^30–33 and AO Projection techniques,^34,35 to obtain invaluable insights into the nature of the chemical bond (for the various ways of analyzing the SCGVB wave function, see Ref. 5). This work belies the comment made by Coulson:

We can, in fact, now “see” the chemical bond in unprecedented detail despite the ambiguities imposed by its quantum mechanical nature.^14,24,36,37 The papers included in this Special Issue of the Journal of Chemical Physics devoted to the “Nature of the chemical bond” are further proof of these advancements.

This collection of articles begins with two perspectives, one written by Professor Klaus Ruendenberg and one by Professor Sason Shaik. In the early 1960s, Professor Ruedenberg upended our understanding of the physical origin of the covalent bond in the H₂ molecule by showing that the energy lowering associated with the formation of the bond was due to a lowering of the (interference) kinetic energy of the electrons, not to an increase in the magnitude of the potential energy.^14,38 Since that time, his research has continued to provide valuable insights into the nature of the chemical bond. Although the MO theory essentially replaced the VB theory as the “theory of choice” in the early 1960s, largely because of the computational efficiencies associated with the use of orthogonal molecular orbitals, Professor Shaik has been instrumental in reviving the role of VB theory in chemistry.³⁹ This is, in part, due to increasing computational power and the development of efficient techniques for performing calculations with non-orthogonal orbitals. However, it is also due to the natural affinity that many chemists have for the concepts associated with VB theory and (equivalently) the localized orbitals in MO theory (hybrid orbitals, bond pairs, lone pairs, resonance, etc.). One of the most striking aspects of these new ab initio VB theories (VBSCF, BOVB, and SCGVB) is the development of new concepts that will enable chemists to make sense of their increasingly detailed exploration of the molecular world.

The articles included in the Special Issue of the Journal of Chemical Physics on the “Nature of the chemical bond” can be roughly divided into six categories: physical nature of the chemical bond, characteristics of the chemical bond, analysis of the chemical bond, chemical reactions and the chemical bond, molecular interactions, and computational approaches and benchmarks. There are also a few articles that are not easily categorized. Below we briefly describe the articles in each of these categories.

Ruedenberg contributed two articles to the issue. The first expressed the electronic wave function in quasi-atomic orbitals, resulting in a global energy expression that provides a unified description of various modes of chemical bonding in terms of physical interactions. He then examined the bonding in several molecules.⁴⁰ In the second article, Ruedenberg used a series of intrinsic orbital and configurational transformations of the wave function to partition the energy of formation of a molecule in terms of intra-atomic energy changes, interference energies, quasi-classical, non-classical, and charge-transfer Coulombic interactions, thereby providing an algorithm for the quantitative breakdown of the bond energy.⁴¹ Shaik et al. provided a panoramic description of the nature of the chemical bond according to valence bond theory, discussing single and multiple bonds as well as charge-shift bonds.⁴² Along the way, they outline an effective methodology of reducing the description of the electronic structure of unusual molecules, such as C₂ and O₂, to a minimum. Subsequently, they describe the roles of π vs σ in the geometric preferences in unsaturated molecules and their Si-based analogs. They discuss bonding in clusters of univalent metal-atoms, which possess only parallel spins, but are, nevertheless, bonded due to multiple resonance interactions. Finally, they discuss the singlet excited states of ethylene, ozone, and sulfur dioxide, offering new insights into the electronic structure of these species.

Arasaki and Takatsuka discuss the physical nature of the chemical bond in terms of energy natural orbitals, analyzing the nature of chemical bonding in three low-lying electronic states of the hydrogen molecule.⁴³ Bacskay quantified the intra- and inter-fragment changes in the kinetic energy, which largely represent the effects of orbital contraction, for a wide range of diatomic and polyatomic molecules.⁴⁴ See also the articles by Ruedenberg and Shaik.

Ariyarathna et al. examined the bonding in the ground and low-lying electronic states of HfO and HfB to obtain insights into a range of catalysts and materials that contain Hf–O or Hf–B moieties, comparing the DFT errors to coupled cluster reference values for the dissociation energies, excitation energies, and ionization energies.⁴⁵ Bressanini showed that a positron can attach to PsH₂ to form a locally stable species with three positrons whose potential energy curve shows an equilibrium structure at about 8 bohr and a binding energy of 11.5(5) mhartree with respect to the dissociation into PsH + e⁺PsH.⁴⁶ Chen and Yang characterized the bonding in cerium oxides, finding that the increasing overlap of the (Ce4f, O2p) orbitals with increasing coordination of Ce atoms enhances Ce–O bond covalency, which, in return, leads to a given molecular geometry.⁴⁷ Depastas et al. determined the bonding, dissociation energies, and spectroscopic parameters of seven states of the Mo₂ molecule that correlate with ground state atoms; they reported that the ground state has a sextuple chemical bond and each of the calculated excited states has one less bond than the previous one.⁴⁸ Fletcher et al. presented a theoretical basis for modeling the correlation effects between specific electron pairs by incorporating inter-electronic coordinates in the variational subspace valence bond (VSVB) wave function.⁴⁹ Florez et al. studied the closed-shell flerovium in detail to predict its solid-state properties, including its melting point, from a decomposition of the total energy into many-body forces.⁵⁰ They found that flerovium does not behave like a typical noble gas element despite its closed-shell configuration. Jackson and Miliordos examined beryllium–ammonia dimers coupled together by linear hydrocarbon chains and showed that the electrons on each of the beryllium–ammonia dimers occupy diffuse, singly occupied s-type orbitals.⁵¹ The spins of the electrons in these orbitals can be coupled into a triplet or singlet state just as in H₂. For long hydrocarbon chains, the ground state is an open-shell singlet state, which is nearly degenerate with the triplet state. For n = 1, the molecule becomes a closed-shell singlet, thereby imitating the σ-covalent bond of H₂. Kalemos examined the archetypal CH₄ species, examining the formation routes CH_n + (4 − n)H → CH₄ (n = 0, 1, 2) from both diabatic and adiabatic viewpoints.⁵² Lucci et al. combined experimental and computational methods to determine the bond energies of the AuSc, AuTi, and AuFe diatomic molecules.⁵³ Mato et al. examined the many-body expansion of the energy for alkaline earth metal clusters Be_n, Mg_n, and Ca_n (n = 4, 5, 6) at several levels of theory, finding that the behavior of the expansion for these clusters depends strongly on the geometrical arrangement and, to a lesser extent, on the level of theory.⁵⁴ McCutcheon et al. investigated the bonding features in 15 ruthenium-ligand complexes, providing a modern computational tool set for the investigation of bonding features in these molecules.⁵⁵ Montero-Campillo et al. examined why the weak acid properties of triazoles and tetrazoles are enhanced by bonding to beryllium complexes, finding that the formation of the complexes between the N basic sites of the azoles and the Be center of the BeF₂ molecule and the (BeF₂)₂ dimer leads to a significant perturbation in the bonding of both interacting subunits.⁵⁶ Rask and Zimmerman examined Fe(II)–porphyrin complexes finding that, for two low-lying triplet states, strong metal d–d and macrocycle π–π orbital interactions preferentially stabilize the ³A_2g state and d–π interactions preferentially stabilize the ³E_g state.⁵⁷ Ren et al. assessed the use of compact, yet accurate ab initio valence bond wave functions for describing electron transfer in the classic but challenging covalent–ionic interaction in LiF.⁵⁸ Simons considered the possibility that a neutral Rydberg radical could form a covalent bond either to another Rydberg radical or to a conventional valence orbital.⁵⁹ He found that any covalent bonds formed by these radicals are weak and the molecules they form are susceptible to exothermic fragmentations that involve small activation barriers. Shui et al. used multiple bonding analysis methods to characterize the bond between the metal atoms in Lu₂(C_2n) (2n = 76–88) fullerenes, finding that there is one (two-center, two-electron) σ covalent bond between the two Lu ions in these fullerenes.⁶⁰ Tzeli and Xantheas presented a novel implementation of the many-body expansion method to account for the breaking of covalent bonds, greatly extending the applicability of this approach.⁶¹ In two articles, Xu and Dunning examined the dependence of the dynamical correlation energy on the internuclear distance for the covalent bonds in the AH molecules (A = B–F) and the recoupled pair (two-center, three-electron) bonds in the CH and CF molecules.^62,63 They found the changes in the dynamical correlation energy to be non-monotonic, reflecting the changes in the underlying orbital structure of the molecule. Zhao et al. used both experimental and computational methods to characterize the uranium nitride-oxide cations and their complexes with equatorial N₂ ligands, finding that the bond distances and vibrational modes could be rationalized as due to cooperative covalent and dative triple bonds.⁶⁴ 

Hagebaum-Reignier et al. studied valence bond wave functions from the density point of view, with the density being given as a difference with a quasi-state built on the same orbitals.⁶⁵ Menéndez-Herrero et al. examined the spatial position of the N electrons of an atom at the maximum of the square of the wavefunction, the so-called Born maximum, as a shell structure descriptor for ground state atoms with Z = 1–36, comparing it to other available indices.⁶⁶ Nakatani et al. presented a methodology for analyzing chemical bonds embedded in the electronic wavefunction of molecules in terms of spin correlations or the so-called “local spin.”⁶⁷ This enabled them to clarify the relationship between spin correlations and traditional chemical concepts such as resonance structures. Otero-de-la-Roza tackled the problem of locating the critical points in Bader’s QTAIM when the density is only known on a three-dimensional uniform grid.⁶⁸ Reuter et al. used probability density analysis to describe a range of multicenter bonds, comparing the results with valence bond calculations.⁶⁹ Scemama and Savin discussed the effect of uncertainty on the probability of finding a chosen number of electrons in specified domains of space, putting forward a tool to assist in choosing the domains.⁷⁰ Silvi and Alikhani presented a method for partitioning atomic and molecular charge densities in non-overlapping chemically significant regions, providing a tool to determine “good boundaries” with the help of elementary statistical methods and information theory.⁷¹ Sousa and Nascimento used interference energy analysis to examine (three-center, two-electron) bonds and concluded that in A–B–C bonds, the A–C interactions are as important as the A–B/B–C ones.⁷² Weinhold used the natural bond orbital and natural resonance theory tools to analyze the enigmatic properties of the C_2v-symmetric isomer of chlorine dioxide radical ClO₂, finding that the molecule exhibits a bonding pattern based on “different Lewis structures for different spins.”⁷³ Xu et al. proposed a general tight-binding based energy decomposition analysis scheme for intermolecular interactions that can be used to analyze the interactions from all self-consistent charge type density functional-tight binding methods.⁷⁴ Zhao et al. examined the nature of the interatomic interactions in LiF, BeO, and BN with the EDA-NOCV method, critically discussing the results and comparing the results of this analysis with that from QTAIM, NBO, and Mayer approaches.⁷⁵ 

Danilov et al. used the quantum Ehrenfest method to study the nuclear and electron dynamics in the glycine cation starting from localized hole states and then used the Fourier transform of the spin densities to identify the normal modes of the electron dynamics.⁷⁶ David et al. examined the C_4n cyclacenes, which exhibit strong bond alternation in their equilibrium geometries, and discussed the physical factors involved in the energy differences between the isomers as well as the intervening barrier heights.⁷⁷ Ebisawa et al. used a new orbital analysis method, the natural reaction orbital method, which automatically extracts orbital pairs to characterize electron transfer in reactions, finding that electron transfer occurs mainly in the vicinity of transition states and in regions where the energy profile along the intrinsic reaction coordinate shows shoulder features.⁷⁸ Molina-Aguirre et al. found that the mechanism of the gas-phase halogen exchange reaction between boron- and aluminum-halides is a two-step mechanism with the intermediacy of a diamond-core structure analogous to diborane.⁷⁹ Sandoval-Pauker and Pinter characterized the electronic structure changes along the oxidative and reductive quenching cycles of a photoredox catalyst using quasi-restricted orbitals, electron density differences, and spin densities.⁸⁰ Sigmund et al. calculated the potential energy surfaces of 15 tetrahedral p-block element hydrides to determine whether stereoinversion competes against other reactions such as reductive H₂-elimination or hydride loss, and if so, along which pathway the stereomutation occurs.⁸¹ 

Fan et al. proposed an inter-anion chalcogen bond and analyzed the chalcogen bonds found in a series of complexes formed by negatively charged bidentate chalcogen donors with a chloride anion.⁸² Harville and Gordon used the quasi-atomic orbital bonding analysis method to study the intramolecular hydrogen bonding (IMHB) in salicylic acid and an intermediate that is crucial to the synthesis of aspirin and found that each intramolecular hydrogen bond is a (three-center, four-electron) bond.⁸³ Herman et al. present a classical electrostatic induction model to evaluate the three-body ion–water–water and water–water–water interactions in aqueous ionic systems, benchmarking the classical model against an accurate dataset of three-body terms for 13 different monatomic and polyatomic cation and anion systems.⁸⁴ Nottoli et al. used charge displacement analysis to determine the profile of the charge redistribution along a given interaction axis for several intermolecular complexes, including that between dimethyl sulfide and sulfur dioxide.⁸⁵ Sirianni et al. report a quantum chemical study of π–π interactions in an aqueous solution, as exemplified by a T-shaped benzene dimer and a cationic pyridinium–benzene dimer surrounded by 28 or 50 water molecules with the goal of understanding how solvent molecules can lead to changes in the π–π interactions of the benzene dimer.⁸⁶ They found that nearby solvent molecules cause very little change in the solute–solute interactions in the benzene dimer but had a significant effect on the interaction in the cationic pyridinium–benzene dimer.

Almeida et al. used an f-block ab initio correlation-consistent composite approach to predict the dissociation energies of lanthanide sulfides and selenides.⁸⁷ Aldossary and Head-Gordon presented a non-iterative method for constructing valence antibonding molecular orbitals and a molecule-adapted minimum basis set.⁸⁸ Chakraborty et al. benchmarked the semi-stochastic CC(P,Q) approach for singlet–triplet gaps in biradicals.⁸⁹ Feller et al. calculated the benchmark quality isotropic hyperfine properties for first row elements (B–F) using a systematic composite approach.⁹⁰ Pradhan et al. presented a unified one-electron Hamiltonian formalism for spin–orbit vibronic interactions for molecules in all tetrahedral and octahedral symmetries, covering all spin–orbit Jahn–Teller and pseudo-Jahn–Teller problems in these symmetries with arbitrary types and arbitrary numbers of vibrational modes.⁹¹ Vila et al. discussed newly developed coupled-cluster methods that enable simulations of ionization potentials and spectral functions of molecular systems for a wide range of energy scales ranging from core-binding to valence.⁹² Zheng et al. put forward a hybrid density functional valence bond method, which is a combination of the valence bond self-consistent field method and the Kohn–Sham density functional theory.⁹³ 

The articles in this Special Issue of the Journal of Chemical Physics illustrate the advances that have been made in understanding one of the most fundamental concepts in chemistry—the chemical bond. A little over a century ago, what held the atoms together in a molecule was largely a speculation, especially for molecules that are bound covalently. Initial applications of quantum theory to understanding the chemical bond in the late 1920s and 1930s, while very promising, were limited by daunting computational problems. Fortunately, with the advent of digital computers, that is no longer the case. Our current understanding of the chemical bond is now founded on rigorous theoretical principles. These advances are a result of advances in both theoretical and computational chemistry. We now have advanced theoretical and computational methods to explore the bonding in a wide range of molecules, analyze the electronic density in those molecules, and understand the various contributions to the bond energy. We expect to see additional advances in our understanding of the chemical bond as we and our colleagues continue our explorations in the future.

The editors of this Special Issue of the Journal of Chemical Physics thank all the contributors to this issue—their work illustrates the advances that theory and computing has made to advancing our understanding of the chemical bond. We would also like to thank Professor C. David Sherrill, the Associate Editor of the Journal of Chemical Physics responsible for the Special Issue, as well as the staff at AIP Publishing, in particular, Ms. Olivia Zarzycki and Dr. Jenny Stein. Finally, we thank all the reviewers whose diligent reviews of the articles maintained the high quality of the article published in the Special Issue.

Finally, we wish to acknowledge our sponsors. T.H.D., Jr. and S.S.X. were supported by the Center for Scalable Predictive Methods for Excitations and Correlated Phenomena (SPEC), which is funded by the U.S. Department of Energy (DoE), Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences, as part of the Computational Chemical Sciences (CCS) program at Pacific Northwest National Laboratory (PNNL) under Grant No. FWP 70942. PNNL is a multi-program national laboratory operated by Battelle Memorial Institute for the U.S. DoE. M.S.G. was supported by a Department of Energy Exascale Computing Project to the Ames National Laboratory, Project No. 17-SC-20-SC. The Ames National Laboratory is operated by Iowa State University under Contract No. DE-AC02-07CH11338.