Curly Arrows, Electron Flow, and Reaction Mechanisms from the Perspective of the Bonding Evolution Theory

Despite the usefulness of curly arrows in chemistry, their relationship with real electron density flows is still imprecise, and even their direct connection to quantum chemistry is still controversial. The paradigmatic description – from first principles – of the mechanistic aspects of a given chemical process is based mainly on the relative energies and geometrical changes at the stationary points of the potential energy surface along the reaction pathway; however, it is not sufficient to describe chemical systems in terms of bonding aspects. Probing the electron density distribution during a chemical reaction can provide important insights, enabling us to understand and control chemical reactions. This aim has required an extension of the relationships between the concepts of traditional chemistry and those of quantum mechanics. Bonding evolution theory (BET), which combines the topological analysis of the electron localization function (ELF) and Thom’s catastrophe theory (CT), provides a powerful method that offers insight into the molecular mechanism of chemical rearrangements. In agreement with the laws of physical and aspects of quantum theory, BET can be considered an appropriate tool to tackle chemical reactivity with a wide range of possible applications. In this work, BET is applied to address a long-standing problem: the ability to monitor the flow of electron density. BET analysis shows a connection between quantum mechanics and bond making/forming processes. Likewise, the present approach retrieves the classical curly arrows used to describe the rearrangements of chemical bonds and provides detailed physical grounds for this type of representation. We demonstrate this procedure using the test set of prototypical examples of thermal ring apertures, and the degenerated Cope rearrangement of semibullvalene.

distribution during a chemical reaction can provide important insights, enabling us to understand and control chemical reactions. This aim has required an extension of the relationships between the concepts of traditional chemistry and those of quantum mechanics. Bonding evolution theory (BET), which combines the topological analysis of the electron localization function (ELF) and Thom's catastrophe theory (CT), provides a powerful method that offers insight into the molecular mechanism of chemical rearrangements. In agreement with the laws of physical and aspects of quantum theory, BET can be considered an appropriate tool to tackle chemical reactivity with a wide range of possible applications.
In this work, BET is applied to address a long-standing problem: the ability to monitor the flow of electron density. BET analysis shows a connection between quantum mechanics and bond making/forming processes. Likewise, the present approach retrieves the classical curly arrows used to describe the rearrangements of chemical bonds and provides detailed physical grounds for this type of representation. We demonstrate this procedure using the test set of prototypical examples of thermal ring apertures, and the degenerated Cope rearrangement of semibullvalene.

Concept of Chemical Bond
Chemistry is the science of atoms, molecules, and matter. Chemists are capable of obtaining a wide range of molecules and determining their structures and transformations, even at extraordinary levels of complexity, using sophisticated experimental techniques as well as accurate first-principle calculations. This type of work makes it possible to elucidate how atoms, as constitutive entities, hold together by chemical bonds in order to understand the 3D rearrangements of atoms in molecules and solids. In general terms, a molecule possesses one stable structure with a defined geometry under a given condition that is determined by its chemical bond.
In chemistry, theoretical concepts play an important role and useful qualitative concepts such as the chemical bond appeared very early in the chemical literature. The chemical bond is one of the most prominent concepts in chemistry to describe and understand the structure of molecules and their chemical reactivity. It is employed in all fields of chemistry and on all levels, from school education to research. However, the concept of chemical bond is a legacy from the development of structural chemistry during the 19 th century, and it is ultimately determined by the representation of the matter one can have. As chemists, we are primarily concerned with composition in terms of elemental atoms (in a molecule). To consider the structural formula as graphics, atoms are shown at vertices, linked by straight lines (bonds). Removal of a bond increases the number of independent species by one or opens at least one cycle. The possible numbers of bonds around a center are determined by the group of the elemental atom. The resulting representation is that of elemental atoms linked by bonds. Building rules have been proposed by Lewis based on a partition of the electrons between kernels and valence shells, and magic numbers rules, namely the duet and octet rules, which have to be fulfilled by bonding and non-bonding electron pairs. Lewis's original approach has been further complemented by expended octet rules, 1-4 mesomery, 5, 6 by methods providing the spatial arrangement and the size of electron pairs, [7][8][9][10][11] and so on. A closer look, however, often reveals that attempting to quantify the concept of chemical bond is futile and it is still awaiting a solution. 12 Owing to the still ambiguous definition of the concept of chemical bond, like other concepts in chemistry, 13 a great deal of effort has been devoted to clarifying its intrinsic nature. However, this concept is not directly associated with experimentally observable values, which lack unique quantitative assessments. 14 In the special issue of the Journal of Computational Chemistry in 2007 entitled "90 Years of Chemical Bonding", published with this aim, 15 Frenking and Krapp 16 compared the chemical bond to a "unicorn", as a "mythical but useful creature which brings law and order in an otherwise chaotic and disordered world" where everyone knows what they look like despite nobody ever having seen one. 17,18 This line of reasoning is similar to Coulson's comment: "Sometimes it seems to me that a bond between two atoms has become so real, so tangible, so friendly, that I can almost see it. Then I awake with a little shock, for a chemical bond is not a real thing. It does not exist. No one has ever seen one. No one ever can. It is a figment of our own imagination". 19 Likewise, chemical bonds have been described as "noumenon" rather than as "phenomenon". [20][21][22] In chemistry as in other scientific disciplines, there is no place for ambiguity. The definition of a concept such as chemical bond must be clear and unequivocal; building those definitions is the raison d'être for the existence of the IUPAC. Unfortunately, like many other fundamental concepts in chemistry, the concept of chemical bond lacks a precise and unique definition and, even worse, no quantum mechanical "bond operator" can provide a conventional expectation value. In contrast, as quoted in Ritter's paper, 23 Nobel Laureate Roald Hoffmann says: "Push the chemical concept to its limits. Be aware of the different experimental and theoretical measures out there. Accept that (at the limits) a bond will be a bond by some criteria, maybe not others. Respect chemical tradition, relax, and instead of wringing your hands about how terrible it is that this concept cannot be unambiguously defined, have fun with the fuzzy richness of the idea".
In many fields, advanced theories are supported by two important milestones: a mathematical structure/formalism disclosing the basic entities of theory together with their mathematical relationships, and an interpretative recipe of the basic entities of the theory. Moreover, it is important to recognize that the connection between the mathematical formalism of a theory and its interpretation is always subtle. This problem can be traced back to the lack of a clear and unambiguous definition of a bond in quantum mechanics, and a plethora of interpretations have been introduced with various "meanings" of the "mathematical symbols/entities" of the theory. 24 From this quandary, two opposing attitudes can be envisaged. On the one hand there is the old and negative statement of the French mathematician R. Thom: "Il me faut cependant avouer que la chimie proprement dite ne m'a jamais beaucoup intéressée. Pourquoi? Peut-être parce que des notions telles que celles de valence, de liaisons chimique etc., ont toujours semblées peu claires du point de vue conceptuel" (I should admit that chemistry never really interested me. Why? Perhaps because notions such as that of valence, chemical bond, etc., always appeared unclear from the conceptual point of view). On the other hand, there is the more actual and pragmatic comment by Álvarez et al.: "Chemistry has done more than well in a universe of structure and function on the molecular level with just this imperfectly defined concept of a chemical bond. Or maybe it has done so well precisely because the concept is flexible and fuzzy". 25 However, it is important to note that scientific arguments, debates, and controversy are at the heart of chemistry. This situation has been clearly stated in the very recent paper entitled "The Nature of the Fourth Bond in the Ground State of C 2 : The Quadruple Bond Conundrum" by Shaik et al. 26 , where the authors recognize that they find themselves before a "Rashomon effect" (see en.wikipedia.org/wiki/Rashomon effect), in which the bonding picture risks becoming too fuzzy to be constructive anymore.
To describe the matter at the molecular level, we can make a first universal statement: "matter is made of nuclei and electrons electrostatically interacting". To address this statement, quantum mechanics allows the Hamiltonian to be written out in order to calculate energies and other observables, and to bring a large part of chemistry in line with Dirac's research program. Such a theory is predictive and is able to provide chemical explanations. Thanks to the density functional theory formalism, chemistry can account for external potential, which expresses the interaction between the nuclei and the electrons.

Quantum Mechanics in Chemical Interpretation
Quantum mechanics constitutes the fundamental framework underlying the theoretical study of molecular systems. The quantum-mechanical approach to chemical phenomena usually deals with very complex numerical algorithms for the approximate solution of the many-body Schrödinger equation. From a theoretical perspective, molecules are quantum matter for which structure is defined by a full specification of quantum numbers for the correlated electron and nuclear motion. Similarly, chemical reactions are transformations of the eigenstates of the reactant into those of the reaction products. In the context of quantum mechanics, a molecular process is described completely by the time-dependent state vector |Ψ(t)>, which evolves according to Schrödinger's equation governing the simultaneous coupled motions of electrons and nuclei. In the most common instance, the process is taken to be electronically adiabatic (i.e., the light, fast electrons adjust instantaneously to the movements of the heavy, slow nuclei). To describe such a process, one typically invokes the Born-Oppenheimer approximation (BOA). [27][28][29] From this approximation, the concept of potential energy surface (PES) is derived, which plays a central role in the theory and computational simulation of chemical structure and reactivity. This representation of the reaction mechanism is intuitively apprehended as a progression of states from reactants to products, where the electronic energy of the different states is described as a function of the positions of the nuclei in the coordinate-energy space 30 Quantum chemical methods developed since the 1960s have made tremendous progress so that it is now possible to compute PES with a number of varying approximations. These range from semiempirical methods, ab initio Hartree-Fock approaches and density functional theory to higher computational levels such as multiconfigurational self-consistent field or complete active space second-order perturbation, thus yielding efficient predictions and accurate energies (better than fractions of a kcal mol -1 ) for the thermodynamic and kinetic parameters of a sufficiently large number of chemical reactions. In the above context, chemical reactivity is determined by the potential energy landscape of the reacting species, and the course of a chemical reaction can be considered as an evolution of atoms on a PES.
In this energy-based picture, a given chemical reaction can be viewed as a sequence of snapshots of a complex process in space and time. This type of modeling is without a doubt a very important tool for a deeper understanding of chemical reactions, and thus a key for building the knowledge base we need to be able to design chemical processes. However, as energy is a global quantity that contains all energy changes, such an energy-based partitioning of a chemical reaction misses mechanistic details, i.e., bond making/forming processes that are the essence of chemical reactivity. Particular interest has been focused on extracting information about the stationary points of the energy surface. In the BOA framework, minima on the N-dimensional PES for the nuclei can be identified with the classical picture of equilibrium structures of molecules, while saddle points can be related to transition states (TSs) and reaction rates. Within this approach, minima and saddle points have been fully characterized through the first and second derivatives of the energy (gradient and Hessian) over the positions of the nuclei.
A large number of computational studies of chemical reactions are based on the concept of the PES, while the main parameters in chemical kinetic models (equilibrium constants and rate coefficients) are increasingly derived from quantum mechanical calculations on the stationary points along the PES. Local minima, i.e., reactants, intermediates, and products, are generally easy to characterize due to simple bonding and also because the negative of the gradient along the PES always points downhill. Hence, reaction mechanisms can be modelled as minimum energy paths between stable configurations on a 3N-6 multidimensional PES. [31][32][33][34][35]

Chemical Reactivity and Electron Density
The nature of the bond determines the intrinsic chemical reactivity. However, the complexity of the electronic structure in the transient regime of emerging or breaking chemical bonds cannot be unambiguously defined in pure quantum theory, thus hindering the understanding of how atoms or molecules bond at the most fundamental level. Likewise, there is no unambiguous relationship between kinetics and reaction mechanism that can be used to predict the conditions for the favorable evolution of a given chemical process. Quantitatively, characterizing the fundamental basis of reaction pathway preference remains an elusive ideal to be reached in chemical reactivity, although recent advances in electron microscopy techniques have allowed direct observation of reactive intermediates despite their short lifetimes and high reactivities. 36,37 Thus, a better understanding of reaction mechanisms and product distributions can be achieved by molecular dynamics of reactions at or in the immediate vicinity of transition states, which can be complemented with experiments such as those performed by Polany 38 and Brooks, 39 the timeresolved pump-probe "femtochemistry" experiments pioneered by Zewail,40 or the negative ion photodetachment experiments of Neumark 41 and Lineberger. 42 Thus, these experimental observations can be compared with predictions based on molecular simulations and/or transition state theory.
Electronic structure calculations have been used to help understand the chemical bond.
Some of them were computed directly from calculated N-electron wave-functions, such as valence bond theory 43 or natural bond orbitals, 44 while frontier molecular orbital theory is a well established tool for rationalizing and predicting chemical reactivity. 45 In 1990, Bader published a book that changed the mindsets of many chemists. In Bader's Atoms in Molecules theory, 46 the chosen function is the one electron density ρ(r), and the basins are associated with each of the atoms in the molecule. 47 Electron density ρ(r) rather than an orbital approach is a better choice for the description of chemical processes since it is a local function defined within the exact many body theory; moreover, it corresponds to an experimentally and accessible scalar field. Its paramount role in the description of many-body problems is supported by the Hohenberg-Kohn theorem (HKT), 48 while density functional theory (DFT) 48,49 asserts that the single particle density ρ(r) contains all the information of a system, and the total energy attains the minimum value for the true density.
Conceptual DFT focuses on understanding chemical reactivity through various reactivity descriptors based on derivatives of ρ(r) and its energy, such as electronegativity, hardness, and so on. 50,51 These reactivity indices are well established concepts in the chemical language and help in the interpretation of chemical reactivity. But we need to recognize, as was recently remarked by Neese, 52 that "a corollary is a statement commonly attributed to Max Planck that 'experiment is the only source of knowledge, the rest is poetry and imagination'. This candidly formulated sentence is, on one hand, a strong reminder to ground theory in experimental reality and, on the other hand, to carefully distinguish physical observables from unobservable quantities. The latter, for example molecular orbitals, various energy decomposition schemes, or reactivity indices, are 'interpretation aids'. They are of vital importance in creating a chemical language, guiding chemical thinking, and eventually inspiring new experiments. There is, however, no objective truth to these quantities, and it is largely a personal matter which interpretation aid provides the greatest inspiration to a given individual. However, physical observables have well-defined values and definitions that provide an unambiguous meeting point between theory and experiment".
One further step toward gaining a deeper insight into chemical reactivity is achieved by a successive detection of the electron density change throughout the course of a chemical reaction, in which continuous redistribution of ρ(r) occurs, thereby providing valuable information about how and where the bond forming/breaking processes take place. Therefore, a chemical process cannot be understood in terms of just a simple redistribution of the atoms, but also as a dynamic process of evolution of the electron density along the reaction coordinate, where it is possible to identify changes in the electronic structure. In this sense, Solano-Altamirano and Hernández-Pérez have developed DensToolKit as a cross-platform suite for analyzing the molecular electron density (ρ) and several fields derived from it, including the electron localization function (ELF). 53

Curly Arrow and Reaction Mechanism
Every chemist is familiar with these drawings, which make this new selection rule very intuitive and accessible, while it bridges the gap between traditional reactivity theory and molecular electronics. In the essay entitled "Chemistry: A Panoply of Arrows", 54 Prof. Álvarez presents an overview on the historical use of arrows in chemistry highlighting the variety of meanings that a simple symbol such as an arrow may have. Their uses have varied over time: the alchemical symbols representing elements or compounds; in chemical equations to show the reversibility of a given chemical process; double-headed arrows to represent resonance structures or even tautomerism associated with the interconversion of two isomers through a simultaneous shift of a double bond and a proton; in orbital energy diagrams; in Jablonski diagrams indicating radioactive (straight arrows) and non-radioactive (wave arrows) transitions; in the stimulated emission of radiation that takes place in lasers; and up-and down-pointing arrows to depict the positive and negative spin of an electron, respectively. Very recently "bond" arrows have even been used to describe dative bonds in main-group compounds. 55 Likewise, the description of chemical processes has been classically represented as sequences of elementary steps where transformations of formally double to simple bonds (or vice versa), electron pair rearrangements, and bond breaking/forming processes are clearly symbolized. The prototypical representation is provided by electron-pushing formalisms where the electron flow is represented with curved arrows. 56 This type of representation for the electron flow movements is displayed to symbolize chemical reactivity in much of organic chemistry.
All these processes are still imagined and represented in chemistry textbooks that employ drawings involving curly arrows models. The history of curly arrows goes back about ninety years, when they were introduced in the seminal papers by Robinson and Ingold [56][57][58] . In general, the tails and heads of the curly arrows indicate chemical bonds that are weakened and strengthened due to loss or gain of valence electron density during the reaction, respectively, while the curve is not meant to describe a trajectory: it is only one way of connecting the departing and ending points of the electron displacement. In organic chemistry, transformations were simply represented by arrows that would result in an electron pair and atom placement consistent with the products in a meaningful way, providing mechanisms for transformations. Mechanistic reactions can be drawn as "arrow-pushing" diagrams 59 showing the concerted electron movements. This representation appears to be a consequence of chemical intuition and is a fundamental part of the chemist's activity, although there is no experimental support for these curly arrows. In this sense, an elementary reaction can be associated with a single electron movement (e.g., radical reactions), movement of a single pair of electrons (e.g., simple addition or bond dissociation reactions), or the complex concerted movement of many electrons, e.g., S N 2 or S N 1 substitution, and pericyclic (Diels-Alder) reactions. For example, the usual reaction mechanism for these reactions that can be found in many organic chemistry textbooks is written as: These curly arrows indicate how the bonds have to be rearranged in order to go from the reactants to the products. Generally speaking, there are many different possible ways to draw these arrows. These examples tend to explain the corresponding reaction mechanism, connecting the reactants to the reaction products, and it is associated with the electronic structure change during a chemical rearrangement. This concept is essential in chemical education as a fundamental tool enabling the comprehensive representation of the reaction mechanism associated with a chemical reaction. Interpretations of organic reactions by means of curly arrows [60][61][62]

Electron Flow
The attempts made so far to extract the flow and electron transfer processes along the reaction pathway associated with a chemical reaction from quantum chemical calculations have been based either on wave function-based and orbital-based methods. These methods generate orbital representations of the wave-functions, and the derived based-energy properties that, in turn, assist the qualitative interpretation of chemical structure and reactivity. In this sense molecular orbitals make it possible to define a chemical bond, assigned to a pair of electrons shared by two or more nuclei, as put forward by Lewis. 76 However, valence bond theory developed by Pauling, [77][78][79][80] in which the superposition of resonating Lewis  At the present time, a great deal of effort has been focused on characterizing and quantifying the electron flow in chemical processes. [83][84][85][86][87][88][89][90][91][92][93][94] In this sense, it is now becoming possible to generate tunable, intense, ultrashort X-rays, 95

Quantum Chemical Topology
The topology of the scalar fields associated with the electron density distribution is, in principle, independent of the approximations performed to calculate the approximate wave function.
Theoretical studies on chemical reactivity are usually carried out within a Hilbert space, where the Hernández-Trujillo 163 investigated the topology of a vector field, i.e., the Ehrenfest force density, which is the electrostatic force acting on any point in the electron density due to all the other particles in the molecule, while Dillen 164 has analyzed the topology of the Ehrenfest force density using basis sets based on Slater-type orbitals. In this context, the best known approaches are the "atoms in molecules" theory (QTAIM), which relies on the properties of the empirically observable electron density ρ(r) 46,156,159,165 and the ELF methods. 166 This was emphasized by the founder of QTAIM, Richard Bader, who stated, "further study of the gradient vector field of the electron density leads to a complete theory of structure and structural stability". 167 The QTAIM has been widely used to study molecules, 168,169 solids, [170][171][172] complexes, [173][174][175][176] , and chemical reactions. [177][178][179] Although electron density in principle contains all the chemical information about the corresponding molecule, it does not give a clear picture of the electron distribution. Therefore, it is often preferable to partition such total electron density into different regions, which can be viewed as the more conventional chemical bonds, lone pairs, etc. For this purpose and using the topology of the electron density, the atoms inside a molecule are defined by regions in space called atomic basins, which are bounded by zero-flux surfaces. Atomic properties (e.g., electron population and atomic volume) are calculated using the volume integral of the appropriate variable over the atomic

basin. ELF was introduced by Becke and Edgecombe 166 for Hartree-Fock theory and extended to
Kohn-Sham DFT via an alternative interpretation put forward by Savin and Silvi, 144 and has enjoyed enormous success as a tool for understanding and visualizing chemical bonds. This analysis is based on the topology of the ELF. 166,[180][181][182] The ELF has been originally designed by Becke and Edgecombe to identify "localized electronic groups in atomic and molecular systems". 166 187,188 and by one of us, 189 with the spin pair composition, cπ (r), in order to generalize ELF to correlated wave functions. 190,191 The partition of the molecular space provided by the ELF gradient field yield non-overlapping volumes which minimize the variance of its population with respect to the variation of their boundaries. 192 This assumption is supported by numerical experiments on atoms 192 and hydrogen bonded complexes 193 as well as by theoretical arguments. 194 It is therefore widely used to characterize localized electrons.
ELF basins are related to pairs or groups of electrons, such as core and valence basins. Lone pairs and bonds involving hydrogen atoms are associated with monosynaptic basins, whereas covalent and polar bonds usually exhibit disynaptic basins. 192,195 The electron population and the shape of the ELF basins are commonly used to feature bond interactions. 144,196  Nine years ago, we showed that the formation or dissociation of diatomic molecules is not accompanied by any change in the topology of the electron density. 198 To overcome this drawback, the changes in the structure throughout the progress of a bond making/forming process can be achieved by analyzing the topology of the Laplacian of ρ(r). This strategy has been used by Cortés-Guzman et al., 199 based on the valence shell charge concentration, 200 to describe the changes in electron density concentrations and depletions around the bonding area of an atom. In addition, Quirante et al. 201 followed the evolution of the topology of the Laplacian of the electronic charge density and its gradient vector field for rationalizing the catalytic activity of copper in the cycloaddition of azide and alkynes.

Bonding Evolution Theory
The analysis of the electronic structure, at the corresponding stationary points of the PESs, represents the most frequent and relevant application of modern computational chemistry. Although the accuracy of the prediction of molecular structures, energies, and physical observables is not always guaranteed, the description of chemical processes has been carried out to a great extent using quantitative concepts derived from first-principle calculations. In this sense, interpretative tools are necessary to retrieve chemical structure reactivity, and more specifically to understand the process of bond formation and bond breaking during chemical reactions. Although the HKT 48 guarantees that all the molecular information is encoded in the electron density, the physical catastrophic points at which at least one critical point is non-hyperbolic. Therefore, during a chemical rearrangement, the chemical system goes from a given SSD to another by means of bifurcation catastrophes occurring at the turning points (TPs). These TPs are identified according to Thom's classification. 206 In this way, a chemical reaction is viewed as a sequence of elementary chemical processes characterized by a catastrophe. These chemical processes are classified according to the variation of the number of basins µ and/or of the synaptic order σ of at least one basin. Thom's classification in chemical reactions has been described in detail elsewhere. 148 Several research groups have used the electron density, 207-212 its laplacian, 213 ELF, 148,[214][215][216] to study a wide range of reaction mechanisms. 148,198,[214][215][216][217][218][219][220][221][222][223][224][225][226][227][228][229][230] This combined method that we use herein has previously been described in great detail to establish electron density redistribution in the course of structural rearrangements. 177,198,218,231 On the other hand, Silvi et al. 232 have developed a cross ELF/noncovalent interaction (NCI) analysis to offer an alternative look at chemical mechanisms for prototypical organic reactions, while a combination of QTAIM and the NCI index has been employed to describe the molecular mechanism for the NH 3 + LiH → LiNH 2 + H 2 reaction. 233 Very recently, BET has been coupled with the quantum mechanics/molecular mechanics (QM/MM) method in order to study biochemical reaction paths. The evolution of the bond breaking/forming processes and electron pair rearrangements in an inhomogeneous dynamic environment provided by the enzyme has been elucidated, 234 while Piquemal et al. 235  bond. Interestingly, the breaking/forming processes of C2−C8 and C4−C6 (strictly from the ELFtopological point of view) neither take place at the TS structure nor occur simultaneously. Thus, the ELF topological description accounts for the asynchronicity of the electron density flow in the course of thermal ring aperture. This behavior is predicted in good agreement with previous studies. 88,93,94,242 According to the above findings, the reaction mechanism can be illustrated as Yet, the reorganization of electron density under the framework of BET corresponding to the thermal ring apertures for the cyclobutene and cyclohexa-1,3-diene giving rise to 1,3 butadiene and (Z)-hexa-1,3,5-triene, respectively, have also been studied in detail. 218 In particular, for the ring aperture in cyclobutene, the process is predicted to be exothermic by 12.0 kcal mol -1 and associated with an energy barrier of 35.6 kcal mol -1 . The full ELF-topological analysis predicts five different SSDs for the process (see Figure 2). The topological description is described in detail elsewhere. 218 The Additionally, the activation energy for the thermal ring aperture of the cyclohexa-1,3-diene is predicted to be 45.0 kcal mol -1 , which is considerably higher in energy than the thermal ring aperture for cyclobutene; likewise, the process is calculated to be endothermic by 25 kcal mol -1 (see The last example analyzed concerns the reaction mechanism of the ring closure process in (3Z,5Z)-octa-1,3,5,7-tetraene to yield (1Z,3Z,5Z)-cycloocta-1,3,5-triene. 222 Nevertheless, for the sake of clarity, here this process will be analyzed in a reverse way in order to compare this chemical rearrangement with the previous ring aperture cases (see Figure 4). The process represents complex and coupled rearrangements of both formal single and double carbon-carbon chemical bonds. The global process involves the formation of one single carbon-carbon bond C7-C8, a transformation of three double bonds C1=C2, C3=C4, C5=C6 into single ones, and transformation of four single bonds C1-C8, C2-C3, C4-C5, and C6-C7 into double ones. The ring aperture takes place via a onestep mechanism and is calculated to be endothermic by 19.6 kcal mol -1 , while its activation energy barrier is predicted to be 22.1 kcal mol -1 . The BET analysis along the IRC path reveals six SSDs distributed throughout the thermal process (see Figure 4).

Conclusions
One of the most fundamental concepts within chemistry, i.e., the chemical bond, is still a matter of lively debate among chemists and physicists. The nature of chemical bonds is still ambiguous and the lack of a unique and precise definition makes this concept very controversial. In the above context, new and more robust models of chemical bonds (structure) and chemical reactivity are necessary to further our understanding of chemical phenomena. Probing the electron density distribution during a chemical reaction can provide important insights, making it possible to understand and control chemical reactions.
Understanding chemical structure and unraveling reactivity patterns, i.e., mechanisms that govern the making and breaking of bonds in terms of the electron density distribution and their changes along the reaction pathway, is a major goal of chemistry. Electron density distribution is an observable and therefore it can also be determined experimentally. 155 This representation can be considered as a step ahead with respect to the interpretation based on the concept of molecular orbitals (MOs) 75,244 or the valence bond theory (VB), 245 or reactivity indices derived from conceptual DFT. 50,51 Actually, the current electronic structure theory of molecules, i.e., quantum chemistry, can provide the accurate snapshots of electronic distribution associated with geometrical changes of even very large molecules. Although the HKT guarantees that all the molecular information is encoded in the electron density, the physical description of chemical systems requires additional postulates for extracting observable information in terms of atomic contributions. The combination of the ELF and Thom's catastrophe theory has been consolidated as a powerful tool to analyze the course of a given chemical rearrangement, and allows us to identify how electronic flow in a molecule occurs as a function of reaction progress, which constitutes the motivation of the present work. In the present study, we have used the ELF and CT to analyze and monitor the progress of chemical events, that is, bond breaking/forming process, lone pair rearrangements, and so forth, throughout a given reaction mechanism. However, a direct relationship between concepts such as curly arrows, aromaticity, and electron redistribution patterns along a given reaction pathway is not trivial. To properly justify any such connection, it is necessary to employ a theoretical framework that directly relates the experimental observable with the hypothesis being investigated. Taking into account this theoretical background, our results intend to clarify the evolution of atomic interactions. Furthermore, curly arrows can be based on physical grounds and show how BET can directly afford fresh and richer insights and more correct descriptions of reaction mechanisms in intuitive terms and in unprecedented detail.
The development of BET, as a quantum mechanical treatment of chemical structure and reactivity, provides an understanding based on the analysis of ELF topology and the CT of electron flow as the reaction proceeds. The authors apologize to the reader for the often intentional but sometimes unintentional simplification of the presentation. The purpose of this work is to demonstrate the need to allow a degree of chemical complexity to enter into the rigorous world of structure and chemical reactivity. This work should show that bringing together electron flow and the reaction mechanism is a central paradigm in quantum mechanics to unify concepts in chemical reactivity. We think that this work brings concepts of two worlds of chemistry together (curly arrows and electron flow from BET) through the formulation of a practical scheme to represent the reaction mechanism of a given chemical rearrangement on a quantum mechanics framework. The BET method allows us to search for the degree of fitness of the Lewis hypothesis of chemical bonding as an electron pairing phenomenon, while acquiring a quantum chemical support. On this basis, we should be able to predict quantitatively the reaction mechanics and outcome of the physical processes that lead from reactants to products via the corresponding transition structure and possible intermediates.
With respect to other quantum chemical techniques, which explain the reactivity from the electronic structures of the reactants and of the transition state at given static geometries, BET considers the classical nuclear trajectories as the driving force of the electronic rearrangements occurring along the reaction pathway. It is therefore consistent with the DFT, in which the chemistry is contained in the external potential, in other words the nuclear potential. In a thermal chemical reaction, the conversion of translational kinetic energy into vibrational energy due to the inelastic collisions of the molecules induce excitations of the vibrational modes which promote the system to the transition state. The deformation of the nuclear geometry is at the origin of a stress of the electron density, which undergoes rearrangements under relaxation. The BET quantitatively describes these rearrangements, which can be qualitatively predicted by the reciprocal VSEPR rules. 237 Our long-term objective is to improve our ability to predict chemical reactivity using BET, and here we have shown how this end is achieved without performing (often costly) high-level energy calculations. We believe that this kind of study may serve to provide more specific information with which to nourish chemical reactivity theories. Furthermore, it is also our contention that these observations might be amenable to experimental verification through selective laser pulses, and monitoring the electronic fluxes that accompany the breaking and making of chemical bonds during chemical rearrangements.
In sum, we try to study chemical structure and reactivity from the strict point of view of quantum mechanics and, by extension, of quantum chemistry. Such a situation is clearly defined by using BET. We hope that these examples demonstrate the power of relatively simple ideas applied to understanding chemical structure and reactivity.