A 21st Century View of Allowed and Forbidden Electrocyclic Reactions.

In 1965, Woodward and Hoffmann proposed a theory to predict the stereochemistry of electrocyclic reactions, which, after expansion and generalization, became known as the Woodward-Hoffmann Rules. Subsequently, Longuet-Higgins and Abrahamson used correlation diagrams to propose that the stereoselectivity of electrocyclizations could be explained by the correlation of reactant and product orbitals with the same symmetry. Immediately thereafter, Hoffmann and Woodward applied correlation diagrams to explain the mechanism of cycloadditions. We describe these discoveries and their evolution. We now report an investigation of various electrocyclic reactions using DFT and CASSCF. We track the frontier molecular orbitals along the intrinsic reaction coordinate and modeled trajectories and examine the correlation between HOMO and LUMO for thermally forbidden systems. We also investigate the electrocyclizations of several highly polarized systems for which the Houk group had predicted that donor-acceptor substitution can induce zwitterionic character, thereby providing low-energy pathways for formally forbidden reactions. We conclude with perspectives on the field of pericyclic reactions, including a refinement as the meaning of Woodward and Hoffmann's "Violations. There are none!" Lastly, we comment on the burgeoning influence of computations on all fields of chemistry.

Variations on the Bergman Cyclization Theme: Electrocyclizations of Ionic Penta-, Hepta-, and Octadiynes.

The Bergman cyclization of (Z)-hexa-3-ene-1,5-diyne to form the aromatic diradical p-benzyne has garnered attention as a potential antitumor agent due to its relatively low cyclization barrier and the stability of the resulting diradical. Here, we present a theoretical investigation of several ionic extensions of the fundamental Bergman cyclization: electrocyclizations of the penta-1,4-diyne anion, hepta-1,6-diyne cation, and octa-1,7-diyne dication, leveraging the spin-flip formulation of the equation-of-motion coupled cluster theory with single and double substitutions (EOM-SF-CCSD). Though the penta-1,4-diyne anion exhibits a large cyclization barrier of +66 kcal mol-1, cyclization of both the hepta-1,6-diyne cation and octa-1,7-diyne dication along a previously unreported triplet pathway requires relatively low energy. We also identified the presence of significant aromaticity in the triplet diradical products of these two cationic cyclizations.

Theory of Borazine-Derived Nanothreads: Enumeration, Reaction Pathways, and Piezoelectricity.

Nanothreads are one-dimensional macromolecules formed by pressure-induced polymerization along stacks of multiply unsaturated (or highly strained) molecules such as benzene (or cubane). Borazine is isoelectronic to benzene yet with substantial bond polarity, thus motivating a theoretical examination of borazine-derived nanothreads with degrees of saturation of 2, 4, and 6 (defined as the number of four-coordinated boron and nitrogen atoms per borazine formula unit). The energy increases upon going from molecular borazine to degree-2 borazine-derived threads and then decreases for degree-4 and degree-6 nanothreads as more σ bonds are formed. With the constraint of no more than two borazine formula units within the repeat unit of the framework's bonding topology, there are only 13 fully saturated (i.e., degree-6) borazine-derived nanothreads that avoid energetically costly homopolar bonds (as compared to more than 50 such candidates for benzene-derived threads). Only two of these are more stable than borazine. Hypothetical pathways from molecular borazine to these two degree-6 borazine-derived nanothreads are discussed. This relative paucity of outcomes may assist in kinetic control of reaction products. Beyond the high mechanical strength also predicted for carbon-based threads, properties such as piezoelectricity and flexoelectricity may be accessible to the polar lattice of borazine-derived nanothreads, with intriguing prospects for expression in these extremely thin yet rigid objects.

Rearrangements in the multiverse.

A brief and personal history of the binding interactions between Hans-Beat Bürgi and Roald Hoffmann is given, and a potential rearrangement of their molecules is considered.

Theoretical Studies of Furan and Thiophene Nanothreads: Structures, Cycloaddition Barriers, and Activation Volumes.

This theoretical study examines the formation, structure, and stability of two of the most ordered nanothreads produced yet, those derived from furan and thiophene. The energetic consequences and activation barriers of the first two steps of oligomerization via a Diels-Alder mechanism were examined. The ca. 20 GPa difference in the synthetic pressures (lower for furan) is explainable in terms of the greater loss of aromaticity by the thiophene. The effects of pressure on the reaction profiles, operating through a volume decrease along the reaction coordinate, are illustrated. The interesting option of polymerization proceeding in one or two directions opens up the possibility of polymers with opposing, cumulative dipole moments. The computed activation volumes are consistently more negative for furan, in accordance with the lower onset pressure of furan polymerization. The energetics of three ordered polymer structures were examined. The syn polymer, with all O/S atoms on the same side, if not allowed to distort, is at a high energy relative to the other two due to the O/S lone pair repulsion, understandably greater for S than for O at the 2.8/2.6 Å separation. Set free, the syn isomers curve or arch in two- or three-dimensional (helical) ways, whose energetics are traced in detail. The syn polymer can also stabilize itself by twisting into zig-zag or helical energy minima. The release of strain in a linear thread as the pressure is relaxed to 1 atm, with consequent thread curving, is a likely mechanism for the observed loss of the crystalline order in the polymer as it is returned to ambient pressure.

Varying Electronic Configurations in Compressed Atoms: From the Role of the Spatial Extension of Atomic Orbitals to the Change of Electronic Configuration as an Isobaric Transformation.

A quantum chemical model for the study of the electronic structure of compressed atoms lends itself to a perturbation-theoretic analysis. It is shown, both analytically and numerically, that the increase of the electronic energy with increasing compression depends on the electronic configuration, as a result of the variable spatial extent of the atomic orbitals involved. The different destabilization of the electronic states may lead to an isobaric change of the ground-state electronic configuration, and the same first-order model paves the way to a simple thermodynamical interpretation of this process.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part C. Toward Consilience.

In the last part of our Essay, we outline a future of consilience, with a role both for fact-seekers, and for searchers for understanding. We begin by looking at theory and simulation, surrounded as they are by and interacting with experiment, especially in Chemistry. Experimenters ask questions both conceptual and numerical, and so draw the communities together. Two case studies show what brings the theoretician authors joy in this playground, and two more detailed ones make it clear that computation/simulation is anyway deeply intertwined with theory-building in what we do, or for that matter, anywhere in the profession. From a definition of science we try to foresee how simulation and theory will interact in the AI-dominated future. We posit that Chemistry's streak of creation provides in that conjoined future a link to Art, and a passage to a renewed vision of the sacred in science.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part B. The March of Simulation, for Better or Worse.

In the second part of this Essay, we leave philosophy, and begin by describing Roald's being trashed by simulation. This leads us to a general sketch of artificial intelligence (AI), Searle's Chinese room, and Strevens' account of what a go-playing program knows. Back to our terrain-we ask "Quantum Chemistry, † ca. 2020?" Then we move to examples of Big Data, machine learning and neural networks in action, first in chemistry and then affecting social matters, trivial to scary. We argue that moral decisions are hardly to be left to a computer. And that posited causes, even if recognized as provisional, represent a much deeper level of understanding than correlations. At this point, we try to pull the reader up, giving voice to the opposing view of an optimistic, limitless future. But we don't do justice to that view-how could we, older mammals on the way to extinction that we are? We try. But then we return to fuss, questioning the ascetic dimension of scientists, their romance with black boxes. And argue for a science of many tongues.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part A. Stage Setting.

We begin our tripartite Essay with a triangle of understanding, theory and simulation. Sketching the intimate tie between explanation and teaching, we also point to the emotional impact of understanding. As we trace the development of theory in chemistry, Dirac's characterization of what is known and what is needed for theoretical chemistry comes up, as does the role of prediction, and Thom's phrase "To predict is not to explain." We give a typology of models, and then describe, no doubt inadequately, machine learning and neural networks. In the second part, we leave philosophy, beginning by describing Roald's being beaten by simulation. This leads us to artificial intelligence (AI), Searle's Chinese room, and Strevens' account of what a go-playing program knows. Back to our terrain-we ask "Quantum Chemistry, † ca. 2020?" Then move to examples of AI affecting social matters, ranging from trivial to scary. We argue that moral decisions are hardly to be left to a computer. At this point, we try to pull the reader up, giving the opposing view of an optimistic, limitless future a voice. But we don't do justice to that view-how could we? We return to questioning the ascetic dimension of scientists, their romance with black boxes. Onward: In the 3rd part of this Essay, we work our way up from pessimism. We trace (another triangle!) the special interests of experimentalists, who want the theory we love, and reliable numbers as well. We detail in our own science instances where theory gave us real joy. Two more examples-on magnetic coupling in inorganic diradicals, and the way to think about alkali metal halides, show us the way to integrate simulation with theory. Back and forth is how it should be-between painfully-obtained, intriguing numbers, begging for interpretation, in turn requiring new concepts, new models, new theoretically grounded tools of computation. Through such iterations understanding is formed. As our tripartite Essay ends, we outline a future of consilience, with a role both for fact-seekers, and searchers for understanding. Chemistry's streak of creation provides in that conjoined future a passage to art and to perceiving, as we argue we must, the sacred in science.

Expanding the Frontiers of Higher-Order Cycloadditions.

The concept of pericyclic reactions and the explanation of their specificity through orbital symmetries introduced a new way of understanding reactions and looking for new ones. One of the 1965 Woodward-Hoffmann communications described "the (as yet unobserved) symmetry-allowed 6 + 4 combination", the prediction of a new field of "higher-order" cycloadditions, involving more than six electrons. Later these authors predicted exo-stereoselectivity for the [6 + 4]-cycloaddition. Chemists rushed to test this prediction (for the most part successfully). For more than half a century, chemists have hunted for additional higher-order cycloadditions. The application of catalysis within organic chemistry allows the accomplishment of previously unattainable reactions, including higher-order cycloadditions. The many examples of [8 + 2], [6 + 4], and cycloadditions of even higher electron-counts discovered since the Woodward-Hoffmann rules were introduced illustrate the difficulty in predicting which of these transformations will occur when two highly unsaturated molecules react. Periselectivity has been a challenge, and the development of enantioselective variants has been elusive. While progress was made, the rise of organocatalysis in asymmetric synthesis has led to a surge of interest in stereoselective versions of higher-order cycloadditions. Through organocatalytic activation of conjugated cyclic polyenes and heteroaromatic compounds, asymmetric [8 + 2]-, [6 + 4]-, and [10 + 4]-cycloadditions have been realized by our groups. In this century, [6 + 4]-cycloadditions have been found also to occur in enzyme-catalyzed reactions for the biosynthesis of spinosyn A, heronamide, and streptoseomycin natural products. A whole new class of enzymes, the pericyclases that catalyze pericyclic reactions, has been discovered. A remarkable aspect of these recent developments is the cross-disciplinary research involved: from organic synthesis to computational studies integrated with experimental studies of reaction mechanisms, intermediates, and dynamics, to understanding mechanisms of enzyme catalysis and engineering of enzymes. This Account describes how our groups have been involved in the expansion of the higher-order cycloaddition frontiers. We describe both the history and recent progress in higher-order cycloadditions, and how these advances have been made by our collaborative experimental and computational studies. Progress in asymmetric organocatalysis, incorporating enantioselective higher-order cycloadditions in organic synthesis, and the stereoselective synthesis of important scaffolds will be highlighted. Experimental progress and computational modeling with density functional theory (DFT) has identified ambimodal cycloaddition pathways and led to the realization that multiple products of pericyclic reactions are linked by common transition states. Molecular dynamic simulations have provided fundamental understanding of factors controlling periselectivity and have led to discoveries of a group of enzymes, the pericyclases, which catalyze pericyclic reactions such as [6 + 4]-cycloadditions.

Do Diradicals Behave Like Radicals?

This review sets out to understand the reactivity of diradicals and how that may differ from monoradicals. In the first part of the review, we delineate the electronic structure of a diradical with its two degenerate or nearly degenerate molecular orbitals, occupied by two electrons. A classification of diradicals based on whether or not the two SOMOs can be located on different sites of the molecule is useful in determining the ground state spin. Important is a delocalized to localized orbital transformation that interchanges "closed-shell" to "open-shell" descriptions. The resulting duality is useful in understanding the dual reactivity of singlet diradicals. In the second part of the review, we examine, with a consistent level of theory, activation energies of prototypical radical reactions (dimerization, hydrogen abstraction, and addition to ethylene) for representative organic diradicals and diradicaloids in their two lowest spin states. Differences and similarities in reactivity of diradicals vs monoradicals, based on either a localized or delocalized view, whichever is suitable, are then discussed. The last part of this review begins with an extensive, comparative, and critical survey of available measures of diradical character and ends with an analysis of the consequences of diradical character for selected diradicaloids.

Squeezing All Elements in the Periodic Table: Electron Configuration and Electronegativity of the Atoms under Compression.

We present a quantum mechanical model capable of describing isotropic compression of single atoms in a non-reactive neon-like environment. Studies of 93 atoms predict drastic changes to ground-state electronic configurations and electronegativity in the pressure range of 0-300 GPa. This extension of atomic reference data assists in the working of chemical intuition at extreme pressure and can act as a guide to both experiments and computational efforts. For example, we can speculate on the existence of pressure-induced polarity (red-ox) inversions in various alloys. Our study confirms that the filling of energy levels in compressed atoms more closely follows the hydrogenic aufbau principle, where the ordering is determined by the principal quantum number. In contrast, the Madelung energy ordering rule is not predictive for atoms under compression. Magnetism may increase or decrease with pressure, depending on which atom is considered. However, Hund's rule is never violated for single atoms in the considered pressure range. Important (and understandable) electron shifts, s→p, s→d, s→f, and d→f are essential chemical and physical consequences of compression. Among the specific intriguing changes predicted are an increase in the range between the most and least electronegative elements with compression; a rearrangement of electronegativities of the alkali metals with pressure, with Na becoming the most electropositive s1 element (while Li becomes a p group element and K and heavier become transition metals); phase transitions in Ca, Sr, and Ba correlating well with s→d transitions; spin-reduction in all d-block atoms for which the valence d-shell occupation is d n (4 ≤ n ≤ 8); d→f transitions in Ce, Dy, and Cm causing Ce to become the most electropositive element of the f-block; f→d transitions in Ho, Dy, and Tb and a s→f transition in Pu. At high pressure Sc and Ti become the most electropositive elements, while Ne, He, and F remain the most electronegative ones.

Cross Conjugation in Polyenes and Related Hydrocarbons: What Can Be Learned from Valence Bond Theory about Single-Molecule Conductance?

This study examined the nature of the electronic structure of representative cross-conjugated polyenes from a valence bond (VB) perspective. Our VBSCF calculations on a prototypical dendralene model reveal a remarkable inhibition of the delocalization compared to linear polyenes. Especially along the C-C backbone, the delocalization is virtually quenched so that these compounds can essentially be considered as sets of isolated butadiene units. In direct contrast to the dendralene chains, quinodimethane compounds exhibit an enhancement in their delocalization compared to linear polyenes. We demonstrate that this quenching/enhancement of the delocalization is inherently connected to the relative weights of specific types of long-bond VB structures. From our ab initio treatment, many localization/delocalization-related concepts and phenomena, central to both organic chemistry and single-molecule electronics, emerge. Not only do we find direct insight into the relation between topology and the occurrence of quantum interference (QI), but we also find a phenomenological justification of the recently proposed diradical character-based rule for the estimation of the magnitude of molecular conductance. Generally, our results can be conceptualized using the "arrow-pushing" concept, originating from resonance theory.

Electronegativity Seen as the Ground-State Average Valence Electron Binding Energy.

We introduce a new electronegativity scale for atoms, based consistently on ground-state energies of valence electrons. The scale is closely related to (yet different from) L. C. Allen's, which is based on configuration energies. Using a combination of literature experimental values for ground-state energies and ab initio-calculated energies where experimental data are missing, we are able to provide electronegativities for elements 1-96. The values are slightly smaller than Allen's original scale, but correlate well with Allen's and others. Outliers in agreement with other scales are oxygen and fluorine, now somewhat less electronegative, but in better agreement with their chemistry with the noble gas elements. Group 11 and 12 electronegativities emerge as high, although Au less so than in other scales. Our scale also gives relatively high electronegativities for Mn, Co, Ni, Zn, Tc, Cd, Hg (affected by choice of valence state), and Gd. The new electronegativities provide hints for new alloy/compound design, and a framework is in place to analyze those energy changes in reactions in which electronegativity changes may not be controlling.

Alkyl Isosteres.

By substituting an ER3- unit (E = Group 13 element) or E'R3+ (E' = Group 15 element) for CR3 one gets to methyl isosteres, compounds analogous to alkyls and isoelectronic or iso-valence-electronic to them. The substituent charge can be used to stabilize countercharged aromatic systems; some compounds of this type are known. Nature makes available all kinds of escape routes to such formally zwitterionic species. Strategies for impeding the often facile reaction channels that open up can be designed. We construct what we believe are viable further examples of zwitterionic methyl isosteres based on 3-, 5-, 7-, and 8-membered rings. A similar strategy is laid out for dicationic and dianionic xylene isosteres.

Mirrors of Bonding in Metal Halide Perovskites.

We explore the chemical bonding and band gap in the metal halide perovskites ABX3 (where A is a cation, B a metal dication, and X a halide) through detailed calculations and a qualitative, symmetry-based bonding analysis that moves between chemical and physical viewpoints, covering every aspect of bonding over a range of 15 eV around the band gap. We show how the gap is controlled by metal-halide orbital interactions that give rise to a characteristic mirror of bands, a bonding signpost which first shows up in turning on and off the scalar relativistic effects in computation of the band structure of CsPbBr3. The mirror is made up by a Pb 6s and Br 4p combination that moves in an understandable way through the Brillouin zone, setting the valence band maximum. The mirror is also there when the A cation is changed to an organocation and is robust enough to persist through moderate distortions of the lattice. The analysis predicts how a modification of Pb2+ to Sn2+ and Ge2+ and a variation of the halide X influence the band gap. In describing in equal detail the lowest three conduction bands, a second mirror of bonding emerges. For CsPbBr3, this mirror is made up by Pb 6p and Br 4p combinations. An understanding of the way these combinations move in reciprocal space to set the conduction band minimum allows us to see why the band gap is direct. The orbital analysis provides a chemical and intuitive picture of band gap engineering in this popular class of materials.

Coarctate and Möbius: The Helical Orbitals of Allene and Other Cumulenes.

As brought to the attention of the community by Hendon et al. and noted by previous workers, the π orbitals of the equilibrium geometry odd-carbon (even number of double bonds = n) [n]cumulenes may be written in either rectilinear or helical form. We trace the origins and detailed composition of the helical orbitals of cumulenes, which emerge in the simplest Hückel model and are not much modified in advanced computations. For the α,ω-disubstituted even [n]cumulenes, the helical representation is obligatory as the symmetry is reduced from D2d to C2. A relationship is apparent between these helical orbitals of the even [n]cumulenes, seen as a Herges coarctate system, and the corresponding Möbius cyclic polyene orbitals. The twist of the orbitals varies in interesting ways along the helix, and so does the contribution of the component atomic orbitals. Though the electronic structures of even [n]cumulenes and Möbius cyclopolyenes are closely related, they differ for higher n in intriguing ways; these are linked to the constrained rotation of the basis orbitals along the helical twist itinerary. Relations are constructed between the level patterns of the π-systems of even [n]cumulenes and ideas of Hückel and Möbius aromaticity.

Quantum Interference, Graphs, Walks, and Polynomials.

In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.