On the Predictive Power of Chemical Concepts
OOn the Predictive Power of
Chemical Concepts
Stephanie A. Grimmel † and Markus Reiher (cid:63) Laboratory of Physical Chemistry, ETH Zurich, Vladimir-Prelog-Weg 2, 8093 Zurich,Switzerland † Winner of the Best Poster Presentation Award in Computational Chemistry at the SCSFall Meeting 2020 (cid:63)
Corresponding author: [email protected]
Many chemical concepts can be well defined in the context of quantumchemical theories. Examples are the electronegativity scale of Mullikenand Jaffe and the hard and soft acids and bases concept of Pearson. Thesound theoretical basis allows for a systematic definition of such concepts.However, while they are often used to describe and compare chemical pro-cesses in terms of reactivity, their predictive power remains unclear. In thiswork, we elaborate on the predictive potential of chemical reactivity con-cepts, which can be crucial for autonomous reaction exploration protocolsto guide them by first-principles heuristics that expoit these concepts.
Keywords: chemical concepts, conceptual DFT, reactivity prediction, automated mech-anism exploration
Understanding chemical processes often starts with structure elucidation of the rele-vant reactive species and their connection in terms of a reaction network. Quantumchemical calculations allow us to detail the reaction mechanism by mapping it to anetwork of elementary reaction steps characterized by reactants, stable intermediates,and products that are connected by transition states. In the most basic formulationof transition state theory, it is necessary to identify the stationary points, i.e. , localminima and first-order saddle points, on a potential energy surface to arrive at sucha reaction network at full conformational resolution. The mathematical definition ofthe potential energy surface may be taken from the Born-Oppenheimer approximationwhich freezes the nuclear motion compared to that of the electrons and introduces the1 a r X i v : . [ phy s i c s . c h e m - ph ] F e b lectronic energy. The electronic energy depends on the positions of the nuclei frozenin space. It can serve as the hypersurface for the description of reactive as well asdiffusive processes. Quantum chemistry has provided routine tools to calculate andexplore such surfaces, up to the point where this is possible in an automated man-ner in order to provide the full detail of a chemical process in terms of thousandsof interconnected elementary steps. With subsequent inclusion of nuclear motion toarrive at free-energy differences for the elementary reaction steps, we have all infor-mation, obtained in a first-principles way with no experimental information required,to study the kinetics in the network. While this procedure provides us with a quantitative description of chemical reac-tions in terms of relevant species involved, barrier heights, and concentration fluxes,it does not necessarily allow us to describe and understand what is actually going on.In particular, common reactivity patterns need to be identified from the quantitativedata that is available from experiment and from detailed calculations. In order toaccomplish this, chemical concepts have been developed, many of them introduced ina rather ad hoc fashion to describe changes in electronic structure upon a chemical re-action. Examples are the electronegativity concept as defined by Pauling and thehard and soft acids and bases (HSAB) concept by Pearson.
In the past decades,it has become possible to root these concepts in quantum chemical theories. Whilethis is natural for electronic structure related concepts such as partial charge and bondorder, it is also possible for electronegativity and HSAB.These concepts allow us to relate and discuss data obtained for different moleculesand reactive systems. As such, they have a categorizing value that can hardly be over-estimated. However, their predictive power remains obscure. Even though rigorousmathematical definitions for many concepts exist, most concepts are not unique in asense that they can be introduced in different ways ( e.g. , there exist various ways to de-fine atomic and hence partial charge as an atom is not rigorously defined in a moleculewithout ad hoc assumptions — even Bader’s elegant atoms in molecules theory isdifficult to develop consistently in a fundamental relativistic framework without addi-tional assumptions; loosely speaking, due to the lack of second derivatives in theDirac equation ). Moreover, the concepts are typically evaluated for the unreactive,isolated reactants, i.e. , in their equilibrium structures. It is not at all clear how theycould possibly point the way toward the minimum-energy reaction path and hence tothe transition state and barrier height. Therefore, we may anticipate that it will bedifficult to define a predictive concept that offers a rationale or better a quantitativerelation between the (molecular and/or electronic) structure of reactants and reactiv-ity measures such as activation barriers (that eventually relate to rate constants, andhence, also to equilibrium constants).Reactivity predictions should allow one to answer questions such as these: Willcertain reactants react with one another under given reaction conditions? Whichreaction paths and/or products are to be expected, i.e. , how do the reactants reactwith each other? From which direction is a given reactant attacked preferably? Wefocus on the prediction of elementary steps as opposed to more complex multistepreactions to which our discussion can be generalized in a straightforward manner.For planning chemical synthesis, it is important to know whether and which of these2oncepts can be reliably employed. Whereas one might argue that traditional chemicalsynthesis planning has rather been based on chemical intuition that is nourished bythe vast amount of experimental knowledge ( e.g. , by name reactions) — even whenexploited in a data-driven manner, recent developments in automated reaction ex-ploration schemes based on extensive quantum chemical calculations would benefitfrom knowledge about the predictive value of reactivity concepts.We have put forward the idea of first-principles heuristics, i.e. , the idea of exploit-ing features of the electronic wave function to cut the deadwood from the reactionspace, which for bimolecular reactions alone formally grows with the number ofpairs of atoms in both reactants. This can only be fruitful if reactivity predictionsbased on concepts are reliable (at least to a certain degree). Therefore, we elicit onwhat is known about the predictive power of chemical reactivity concepts in this work.For a concept to have predictive power, it should enable us to judge reliably on thereactivity of a given set of reactants without any a priori information about possiblereaction products and paths. Note, however, that it can be equally important — espe-cially in automated mechanism exploration — to determine what atoms or functionalgroups in a molecule are unreactive.In the following section, we review mathematical definitions of four chemical con-cepts. As examples we select the electronegativity and the chemical hardness, be-cause they belong to the core of chemical concepts taught to chemistry students fromhigh school education onwards – however, typically, without discussing their quantumchemical foundations. Furthermore, we consider the Fukui functions and the dual de-scriptor for the identification of nucleophilic and electrophilic sites of reactants. Weoutline limitations of specific descriptors as well as overall constraints. Finally, weexpand on how to gather further insights upon the predictive capabilities of chemicalconcepts. We elaborate on why, besides the value of chemical concepts for describingand understanding chemical processes, massively automated reaction exploration al-gorithms will be important to assess actual reactive behavior in full depth, sheddingalso light on the application range of reactivity concepts. Electronegativity, hard and soft acids and bases, Fukui functions, and the dual de-scriptor, all refer to electronic structure evaluated for a frozen molecular structure,typically the equilibrium structure of a reactant. Hence, we limit ourselves to the stan-dard electronic-energy based discussion for specific individual structures on the Born-Oppenheimer surface, typically those that mark local minima. Accordingly, thermaland entropic effects are left aside so that one can expect predictions for contributionsto enthalpic, but not to entropic contributions. Naturally, these can only be sensi-ble if the elementary step under consideration is governed by electronic effects. Wenote, however, that there exist generalizations of quantum theory as well as chemicalconcepts towards thermodynamic ensembles (see, e.g. , Refs. 36,37).3 .1 Electronegativity
Electronegativity is among the oldest and most widely used chemical concepts. Theterm dates back to Berzelius’ work in the early 19 th century and the concept evenfurther. It denotes the tendency of atoms to attract electrons towards themselves. Among the many quantitative definitions that have been put forward, the first one byPauling remains to be the most popular one, which is why we briefly consider it here.Pauling proposed a thermochemical recipe driven by the availability of such data atthe time. His definition relates differences in the electronegativity χ of two elements A and B to the deviation of the dissociation energy of the heteroatomic diatomicmolecule, E d ( AB ), from the average of the dissociation energies of the correspondinghomoatomic molecules, E d ( AA ) and E d ( BB ): (cid:12)(cid:12) χ P A − χ P B (cid:12)(cid:12) := (cid:114) E d ( AB ) − E d ( AA ) + E d ( BB )2 · (eV − / ) (1)With regard to the predictive capabilities of the electronegativity it should be pointedout that by rearranging Eq. (1) one obtains an expression for the heteroatomic disso-ciation energy in terms of Pauling electronegativities χ P and homoatomic dissociationenergies. Considering the reliability of a heteroatomic dissociation energy calculatedin this manner already sheds doubt on how reliable the electronegativities can poten-tially be. Hence, examples of poor predictions can be easily found, e.g. , in the case ofalkali fluorides and chlorides. Indeed, the definition of χ P through Eq. (1) has profound limitations, some of whichare:1. If the evaluation of bond energies relies on enthalpies of formation (as in Pauling’swork ), it depends on macroscopic environmental parameters such as tempera-ture.2. The definition marries electronic with nuclear dynamics because experimentaldata was taken for the parametrization. As a consequence, the data cannot beeasily calculated with sufficient accuracy.3. It is not obvious whether only data on diatomics may be used for the definitionor whether bond breaking processes in any molecule can be considered. I.e. , thedefinition ignores the fact that atoms in different bonding situations may not becomparable. In other words, one will be forced to find a categorizing scheme thatgroups atoms in similar binding situations, for example, by introducing differentvalence states (and we refrain here from a discussion of how that could possiblybe accomplished in a rigorous way).4. Moreover, Pauling’s definition only allows for the calculation of relative elec-tronegativity values, which requires an arbitrary definition of an absolute refer-ence. In his original work, he chose 0.0 for hydrogen, which was later replacedby 2.1 and today is typically set to 2.2.
5. The parametrization results in an overdetermined system of equations.4. Depending on which element pairs are employed the resulting electronegativitiesmay differ.From this list, it becomes clear that Pauling’s scheme is ambiguous for a couple of rea-sons. Most of these issues are not present in the absolute electronegativity definitionby Mulliken, which may be rigorously traced back to quantum chemical quantitiesthat can be routinely calculated on modern computer hardware. The Mulliken elec-tronegativity χ M of some atom is the average of its ionization energy I and electronaffinity A , χ M := I + A , (2)which is a natural definition in the sense that it connects the concept to the two quanti-ties that measure the energy required or liberated when an electron is removed or addedto the system. In our setting here, we may identify I and A as the vertical electronicenergy differences calculated between the system’s ground state and the correspondingstates at the same structure with one electron removed and added, respectively (withproper sign conventions). With this definition, the concept of electronegativity is nolonger limited to atoms but can be generalized to entire molecules.An at first sight different approach to define electronegativity is taken in the contextof conceptual density functional theory (cDFT). For an extensive review of thefoundations of cDFT see Ref. 47 and for a recent perspective Ref. 48. cDFT providesmathematical definitions of numerous chemical concepts (see also below). Its basis isthe standard procedure of physical modeling and that is to expand a quantity thatdepends on a set of parameters in terms of a Taylor series. In this case, it is theelectronic energy E el [ N, v ; { R k } ] that depends on the number of electrons N andthe external potential v . The latter is typically given as the Coulomb potential of theunderlying nuclear framework in Born-Oppenheimer theory, and hence, it also encodesthe specific molecular structure given by the set of nuclear Cartesian coordinates ofall nuclei, { R k } .Although this consideration is typically done in the framework of conceptual DFT,we emphasize that such a Taylor expansion of the electronic energy may be writtenfor any electronic structure model. However, DFT may offer specific advantages overother models ( e.g. , as Koopmans’ theorem becomes exact; however, see our discussionbelow). The identification of reactivity concepts as responses of E el towards changesin N and v is in line with the understanding that they vary upon interactions betweenreactants during chemical reactions.In 1961 – and hence even before cDFT evolved as a field – Iczkowski and Mar-grave identified the electronegativity with the negative response of the energy towardschanges in the number of electrons at constant external potential: χ := − (cid:18) ∂E el ∂N (cid:19) v = − µ (3)Later on, this expression was recognized as the negative of the chemical potential µ byParr. The minus sign is introduced to have a high (positive) electronegativity flag astrong tendency to attract electrons, and hence, it should be associated with a decrease5n energy upon an increase in N (note that for sufficiently large molecular systems,the electronic energy can be considered to decrease upon reduction of the system byone electron). Despite appearing mathematically concise, Eq. (3) suffers from the factthat the number of electrons N is restricted to integer numbers when treating isolatedatoms or molecules. This issue is discussed in great detail elsewhere. In practical applications, Eq. (3) is most often evaluated within a finite differenceapproach. The electronegativity is then assumed to be equal to the average of the leftand right derivatives of Eq. (3), χ − and χ + : χ − ≈ E el ( N − − E el ( N ) = I (4) χ + ≈ E el ( N ) − E el ( N + 1) = A (5) χ ≈ χ − + χ + ≈ I + A χ M (6)Even though derived differently, this expression is equal to the Mulliken electronega-tivity χ M given in Eq. (2). Therefore, finite differences clearly link both approaches.Rather than calculating the electronic energy of the system with different electronnumbers ( i.e. , N , N + 1, and N − (cid:15) HOMO and (cid:15)
LUMO , respectively, of thesystem with N electrons by virtue of Koopmans’ theorem, to obtain χ M ≈ − (cid:15) HOMO + (cid:15) LUMO . (7)Whereas Koopmans’ theorem relies on error cancellation in the case of Hartree-Fockcalculations (owing to the neglected orbital relaxation upon oxidation vs. the missingelectron correlation in the independent particle model), it can be considered exact forionization energies in exact Kohn-Sham DFT. However, the LUMO Kohn-Shamenergy cannot be exploited that easily.
It refers to an excited Kohn-Sham electronrather than to an added Kohn-Sham electron so that one might consider theHOMO energy of the N + 1 Kohn-Sham system and reverse its sign to obtain theelectron affinity of the N Kohn-Sham system. Mulliken pointed out that the electronegativity is not only an atom’s ground stateproperty, but instead depends on its valence state. This notion was formalized byHinze and Jaffe by introduction of orbital electronegativities. The electronega-tivity χ i of orbital i with N i being the number of electrons in that orbital is givenby χ i := − (cid:18) ∂E el ∂N i (cid:19) v . (8)Dramatically different electronegativity values were found, for example, for differentvalence states of carbon. Although there is a decent correlation between Mulliken and Pauling electronega-tivities, the theoretical basis of Mulliken-Jaffe electronegativities χ M has thesalient feature that it links directly to quantities of electronic structure theory, in6hich any (fixed) molecular scaffold represents a point on the Born-Oppenheimer sur-face given by the electronic energy, as opposed to the profound problems associatedwith Pauling’s definition.Electronegativities of atoms are widely used to identify reactive centers withinmolecules. For example, the fact that tetrahedral carbon has a lower electronegativitythan chlorine (8.07 vs. 10.05 eV according to the Mulliken-Jaffe values from Ref. 68)allows us to predict that a nucleophilic attack on chloromethane should preferablyoccur at the carbon center. The question whether electronegativity can be used topredict reaction barriers in a quantitative manner was tackled, e.g. , in the contextof hydrogen abstraction reactions: Nguyen et al. established a correlation betweenthe reaction barriers of the abstraction of a given hydrogen atom from propene andmethane and the electronegativity of attacking radicals. A larger share of chemicalspace was investigated in a study of the relation between different cDFT propertiesand 216 electrophilicity values from the Mayr database by Lee et al. They ob-served only a very poor correlation ( R = 0 . R = 0 . ω , which is a jointproperty calculated from the electronegativity χ and the chemical hardness η : ω := χ η (9)The concept of chemical hardness will be considered in the following section. Thepoor correlation coefficients may be taken as a clear indication that kinetic predictionsbased on Eqns. (6) and (9) are unreliable. However, it should be pointed out thatwith Mayr’s electrophilicities being obtained from experimental data, deviations maynot be exclusively attributed to shortcomings of the concepts’ definitions: Measuringinaccuracies and the protocol for the calculation of electrophilicities from experimentalrate constants (see Ref. 73) as well as limitations of the applied electronic structuremethod can also contribute to the discrepancies. This issue is considered in some moredetail in Section 4. Based on experimental observations Pearson formulated the hard and soft acids andbases (HSAB) principle as follows: “Hard acids prefer to bind to hard bases and softacids prefer to bind to soft bases.”
This means, that with A denoting an acid, Ba base, and the subscripts h and s hard and soft, respectively, the exchange reactionA h B s + A s B h −− (cid:42)(cid:41) −− A h B h + A s B s should be exergonic. In Pearson’s early work, soft acids and bases were defined to bethose that are easily polarizable, while hard ones are hard to polarize. It is rathersurprising that such a phenomenological concept that was proposed as an attempt tocategorize a wealth of experimental observations can actually be based on rigorousmathematical definitions in a quantum chemical framework. cDFT provided such a7ramework, in which hardness η appears as the resistance of the chemical potential µ towards changes in the number of electrons N . Accordingly, a derivative term inthe aforementioned Taylor expansion of the electronic energy is interpreted as thehardness: ˜ η := 12 (cid:18) ∂ E el ∂N (cid:19) v = 12 (cid:18) ∂µ∂N (cid:19) v (10)However, the prefactor 1/2 is frequently omitted: η := (cid:18) ∂ E el ∂N (cid:19) v = (cid:18) ∂µ∂N (cid:19) v (11) I.e. , this prefactor emerging in front of any second-derivative term of a Taylor seriesexpansion is to be understood as included in the definition, implying a scaling of thehardness. In the following, we continue to apply the scaled hardness.In analogy to electronegativity, the chemical hardness is often calculated within afinite difference approach, which eventually reads as η ≈ I − A. (12)The application of Koopmans’ theorem then yields η ≈ (cid:15) LUMO − (cid:15) HOMO . (13)The global softness, S , is identified as the inverse of the hardness, S := 1 η . (14)A prototypical example of Pearson’s HSAB principle is the reaction between LiI andCsF: LiI(g) + CsF(g) −− (cid:42)(cid:41) −− LiF(g) + CsI(g)With I – being softer than F – and Cs + softer than Li + , the HSAB principle correctlypredicts that the conversion should be exothermic. This is especially notable consid-ering that a prediction based on Pauling electronegativities and a rearranged form ofEq. (1) fails for this example. Nevertheless, the HSAB principle should by no means be applied blindly. Mostimportantly, as Pearson himself pointed out, it only constitutes one amongst manyeffects determining the interaction between two reactants. Considering the terms inthe Taylor series expansion of the electronic energy, this means that an exact corre-spondence between chemical reactivity and predictions based on the HSAB principlecan only be expected if all other derivatives are negligible – a condition that is hardlyever met in chemical reactions. For example, it is known that the HSAB principle tendsto be dominated by the preference of strong acids to recombine with strong bases. Applications of the HSAB principle to ambident reactants were heavily criticized dueto notable failures occurring even for prototypical systems.
Although its now established roots in quantum chemistry allow for explicit calcu-lations, the HSAB principle should not be mistaken as a universal tool for reactivitypredictions. It may be regarded as one measure for reactivity in the context of all suchconcepts evaluated for a reactant. 8 .3 Fukui Indices and the Dual Descriptor
To obtain information about where a reactant is attacked, local reactivity descriptorsare required. One prominent example is the Fukui function f ( r ) with the Cartesiancoordinate r denoting some position in space. It was introduced by Parr and Yangas a density-functional generalization of Fukui’s Frontier Molecular Orbital (FMO)theory. The Fukui function is the mixed second derivative of the electronic energy withrespect to v and N at a given position r . It can be represented either as the responseof the chemical potential to a change in the external potential at position r or as theresponse of the electron density ρ ( r ) towards a change in the number of electrons, i.e. , f ( r ) := (cid:18) δδv ( r ) ∂E el ∂N (cid:19) = (cid:18) δµδv ( r ) (cid:19) = (cid:18) ∂∂N δE el δv ( r ) (cid:19) = (cid:18) ∂ρ ( r ) ∂N (cid:19) . (15)Regions where f ( r ) is large are supposed to be those from which reactive attacks arethe most favorable. Due to the derivative discontinuities with respect to N , leftand right derivatives are not equal, and hence, two different quantities are typicallyreported, f − ( r ) and f + ( r ), which can be approximated by finite differences: f − ( r ) := (cid:18) ∂ρ ( r ) ∂N (cid:19) − v ( r ) ≈ ρ N ( r ) − ρ N − ( r ) (16) f + ( r ) := (cid:18) ∂ρ ( r ) ∂N (cid:19) + v ( r ) ≈ ρ N +1 ( r ) − ρ N ( r ) (17)Hence, f − ( r ) describes the response of the system towards a decrease of N , i.e. ,towards electrophilic attacks and f + ( r ) the response towards an increase of N , i.e. , thereactivity with respect to nucleophilic attacks. To allow for an easier interpretation,the Fukui functions are often not analyzed in full spatial resolution, but instead ina condensed-to-atom representation. Staying in the finite difference approximation,these Fukui indices f ± k can be obtained for any atom k from the atomic charges q k and are given in atomic units as f − k ≈ q k,N − − q k,N (18) f + k ≈ q k,N − q k,N +1 (19)However, atomic charges are yet another concept without a unique definition. Numer-ous mathematical definitions exist because of the difficulty to define spatial boundariesof an atom in a quantum system such as a molecule (see above). Therefore, the choiceamong these can significantly affect the values of the resulting Fukui indices. Hirsh-feld charges are a commonly recommended choice. If frozen orbitals are assumed, f − ( r ) and f + ( r ) can be expressed in terms of theremoved or incoming orbital upon change of the number of electrons, f − ( r ) ≈ ρ HOMO ( r ) (20) f + ( r ) ≈ ρ LUMO ( r ) , (21)9 .e. , in terms of HOMO and LUMO densities, ρ HOMO ( r ) and ρ LUMO ( r ), respectively,calculated as the absolute square of the respective orbital, which manifests the connec-tion to FMO theory. Naturally, this interpretation is prone to fail for quasi-degeneratefrontier orbitals. For example, it cannot reliably predict the regioselectivity of elec-trophilic aromatic substitutions. In orbital-weighted Fukui functions this behavioris cured by the inclusion of further orbitals — however, at the cost of introducing anempirically motivated weighting factor. Instead of considering f − ( r ) and f + ( r ) separately to identify nucleophilic and elec-trophilic sites, Morell et al. suggested the use of the dual descriptor f (2) ( r ) definedas the derivative of the chemical hardness with respect to the external potential or,equivalently, the response of the Fukui function towards changes in the number ofelectrons: f (2) ( r ) := (cid:18) δηδv ( r ) (cid:19) N = (cid:18) ∂f ( r ) ∂N (cid:19) v ( r ) (22)Within the finite difference approximation, f (2) ( r ) can be written as the differencebetween the electrophilic and nucleophilic Fukui functions, f (2) ( r ) ≈ f + ( r ) − f − ( r ) . (23) f (2) ( r ) takes values between − The dualdescriptor was reported to be more robust than Fukui functions, especially in thefrozen-orbital approximation. As a demonstration for the identification of reactive sites through the inspectionof f (2) ( r ), we consider propanone in the finite-difference approximation (see Fig. 1).We obtained the underlying electronic densities from a DFT/PBE /def2-SVP calculation with the program Orca . The condensed-to-atom values given in Table 1were determined from Hirshfeld charges. The fact that they turned out to be neg-ative at the oxygen atom and positive at the adjacent carbon atom agrees with thenotion that these are the primary attack sites for electrophilic and nucleophilic attacks,respectively.Figure 1: f (2) ( r ) = ± .
01 a − isosurface. Blue encodes negative values, orange positiveones. 10able 1: Condensed dual descriptor indices for propanone.Atom f (2)k O CO − CO α refuting the common belief that one requires an or-bital model to predict pericyclic reactions.Unfortunately, the applicability of Fukui indices and the dual descriptor for predict-ing nucleophilicity and electrophilicity is limited to orbital-controlled soft–soft inter-actions. The most reactive sites for hard–hard interactions can be better predictedusing atomic charges or with descriptors extracted from the molecular electrostaticpotential.
For example, Fukui functions are known to poorly predict protona-tion sites.
Another level of complexity arises due to the fact that for many chemicalreactions neither charge control nor orbital effects are highly dominating, but bothhave to be considered.
As we outlined for the examples above, the predictive power of a single chemicalconcept is very limited. Following the Taylor series expansion of the electronic energy,which explicitly states that various partial derivatives, now understood as differentchemical concepts, are to be added up, it is not sufficient to limit oneself to onechemical concept when trying to understand the change in energy upon reaction, butseveral must be considered simultaneously. They all need to be incorporated into areactivity prediction protocol. The fact that machine-learned models based on severalindicators clearly outperform schemes based on a single selected descriptor underlinesthis necessity.
The key difficulty of all reactivity descriptors defined in terms of a Taylor seriesexpansion of the electronic energy around a reference point is that this is alwayslocal information in the sense that it is valid for the unreactive and non-activatedequilibrium structure of a reactant, i.e. , for the reference point: The local slope andcurvature of the energy functional is supposed to correlate with the barrier heightof eventual transition states. As formulated in Klopman’s non-crossing rule differentreaction curves shall not cross between reactants and transition states.
In practice,this means that the behavior at the onset of reactions has to be decisive for the overallcourse of the reaction. This is why Geerlings et al. state that cDFT descriptorsare expected to be unreliable for late transition states. Making direct use of thisstatement for reactivity predictions, however, is impossible because transition statesare obviously unknown prior to the analysis.11his issue may be considered more severe for all but intramolecular reactions dueto the fact that to project towards reaction paths and energy barriers the interactionsbetween all reactants have to be considered, which will disturb the information gath-ered at the reference point. cDFT descriptors can be evaluated for associated reactivecomplexes and even along entire reaction paths, but the relevant complex geometryand especially the path are, obviously, not known a priori in prediction tasks. For alist of further approaches which incorporate the effect of a second reactant see Ref. 48.In principle, it would be possible to evaluate derivatives of E el [ N, v ; { R k } ] up toarbitrary order. In practice, however, there are mathematical and technical difficultiesassociated with taking such derivatives. Moreover, even if these derivatives wereevaluated, there is no reason to believe that the radius of convergence of the seriesexpansion is such that it encodes information up to the transition state structure. Inother words, the series expansion is not expected to converge for structures fartheraway from the reference point on the Born-Oppenheimer surface.Due to these arguments, a perfect matching between reactivity descriptors, whichare evaluated at equilibrium structures, and activation barriers (and even less so forreaction energies) cannot be expected. They might, however, still be a valuable toolin order to (de-)prioritize more or less promising reaction guesses for further analyses.
I.e. , if one can tell (for instance, in an automated mechanism exploration algorithm),which sites are very likely to be unreactive then this knowledge accelerates the ex-ploration as unreactive paths can be omitted by the exploration protocol without thenecessity to consider them explicitly.
To verify the usefulness of a reactivity prediction scheme and gather profound knowl-edge about the associated uncertainties detailed statistical analyses have to be carriedout. Automated reaction space exploration schemes cannot only benefit from the ex-ploitation of chemical concepts in the context of first-principles heuristics, theyalso offer the potential to help validating these. For statistically rigorous analyses largeand chemically diverse test sets have to be investigated. Automated algorithms allowto carry out and analyze vast numbers of quantum chemical calculations, inaccessibleto manual work with quantum chemistry programs, and can therefore deliver the datarequired.The compilation of such a test set is non-trivial: Typically researchers hand-pickreference reactions based on their prior knowledge or the literature or make use ofdatabases such as the aforementioned Mayr database. However, this approach bearsthe risk of assembling biased test sets: Chemical concepts are integral parts of chem-istry education. That is why it appears reasonable to assume that reactions conformingto these are overrepresented in the literature and, hence, in the test sets. This shouldnot affect the comparison of different evaluation schemes for a given concept. How-ever, there might be a tendency to overrate the predictive power of chemical concepts12s such in a circular-argument situation. Automated reaction network explorationalgorithms allow one to enumerate large numbers of different reaction coordinates —including those that appear to be unpromising from a chemist’s perspective. Hence,there is hope that the bias in favor of known chemical concepts is less pronounced.Furthermore, reaction databases typically only contain information on successfulreactions, but no explicit information about unproductive reaction coordinates. Thepossibility to sort out clearly unreactive reaction guesses is, however, one of the keytasks a useful reactivity predictor must fulfill, which is why this information should beincluded in a test set. A brute-force enumeration of reaction paths by automated ex-ploration algorithms will certainly also include unreactive scans. Finally, the fact thatthe target quantity of the reactivity prediction ( e.g. , reaction barriers) is calculatedwith the same quantum mechanical method as the reactivity descriptor itself, opposedto experimental data, offers the possibility to study the reliability of the reactivityscheme separated from possible shortcomings of the underlying method compared toexperiment.So far, such extensive data has not been produced yet, but it can be anticipatedthat this will change.
Once available, the machine learning evaluation of reactivity descriptors faced withexplicit knowledge of vast reaction networks will provide us with means to assessthe extent to which we may prune quantum chemical reactivity explorations by first-principles heuristics in order to dramatically accelerate the automated, but com-putationally costly exploration process.
In this work, we discussed why chemical concepts are not only valuable to explain,describe, and categorize reactions a posteriori , but also bear hope for reactivity pre-diction. Mathematical definitions allow for the quantification of well-known chemicalconcepts by means of quantum chemistry. In particular, partial derivatives of the elec-tronic energy with respect to the decisive parameters of a system (such as the numberof its electrons) allow us to clearly define reactivity concepts relying on electronicstructure information.However, ambiguities may arise due to competing definitions of the same conceptas well as different approximation and localization schemes. An exact correspondencebetween chemical concepts and reactivity (measured, e.g. , in terms of the activationbarrier associated with that process) cannot be expected because of the fact that theconcepts are evaluated for a specific molecular structure, whereas barriers require thecomparison of energies of two sets of structures, i.e. , reactants and transition statestructures. The usefulness of chemical concepts critically depends on whether, despitethese expected uncertainties, they still allow for discriminating between reactive andunreactive pathways with a high degree of confidence.Especially automated quantum chemical reactivity exploration can effectively ex-ploit this situation. If, due to the aforementioned uncertainties, we do not have clearlyseparable clusters of reactive and unreactive pairs of atoms, this will be a sign that our13ethodology cannot cut deadwood in reaction space with sufficient precision. How-ever, if we can define a trust margin that will allow us to include all of the potentiallyreactive sites, even at the cost of including some unreactive ones, then that will stillrepresent a reliable basis for an efficient heuristic pruning scheme.Statistically rigorous analyses are required to quantify the reliability of reactivityprediction schemes and to discover instances in which knowledge of reactant propertiescan, in fact, allow us to make useful predictions about paths and barrier heights.Automated reaction exploration algorithms are a suitable tool to provide the vastamounts of data required for such analyses. They bear the promise to reduce the biastowards elementary steps that align with known concepts in the test sets and allow forefficient high-throughput analyses with modern statistical techniques.Notorious machine learning techniques provide efficient algorithms to crawl throughhuge data sets produced in automated mechanism explorations. They are a means tonavigate reactivity data efficiently to highlight potentially important correlations thatthen make the data accessible to our chemical understanding. As such, there is thenno contradiction between the production and handling of huge amounts of (calculated)reactivity data and the desire to understand the chemistry in terms of concepts thatare simple, rigorous, and effective. Concerns raised in the literature are likely to beseen resolved in a fruitful interplay of numerical data and conceptual understanding.
This work has been financially supported by the Swiss National Science Foundation(Project No. 200021 182400).
References [1] Warren, J. H.; Radom, L.; von R. Schleyer, P.; Pople, J.
Ab Initio MolecularOrbital Theory ; Wiley: New York, 1986.[2] Cramer, C. J.
Essentials of Computational Chemistry: Theories and Models , 2nded.; Wiley: Chichester, 2006.[3] Jensen, F.
Introduction to Computational Chemistry , 3rd ed.; Wiley: HobokenNew Jersey, 2017.[4] Sameera, W. M. C.; Maeda, S.; Morokuma, K. Computational Catalysis Usingthe Artificial Force Induced Reaction Method.
Acc. Chem. Res. , , 763–773, DOI: .[5] Dewyer, A. L.; Arg¨uelles, A. J.; Zimmerman, P. M. Methods for ExploringReaction Space in Molecular Systems: Exploring Reaction Space in Molecu-lar Systems. Wiley Interdiscip. Rev. Comput. Mol. Sci. , , e1354, DOI: . 146] Simm, G. N.; Vaucher, A. C.; Reiher, M. Exploration of Reaction Pathwaysand Chemical Transformation Networks. J. Phys. Chem. A , , 385–399,DOI: .[7] Kee, R. J.; Miller, J. A.; Jefferson, T. H. CHEMKIN: A General-Purpose,Problem-Independent, Transportable, FORTRAN Chemical Kinetics Code Pack-age ; 1980.[8] Kee, R. J.; Rupley, F. M.; Miller, J. A.
CHEMKIN-II: A Fortran ChemicalKinetics Package for the Analysis of Gas-Phase Chemical Kinetics ; 1989.[9] Kee, R. J.; Rupley, F. M.; Meeks, E.; Miller, J. A.
CHEMKIN-III: A FORTRANChemical Kinetics Package for the Analysis of Gas-Phase Chemical and PlasmaKinetics ; 1996; DOI: .[10] Hoops, S.; Sahle, S.; Gauges, R.; Lee, C.; Pahle, J.; Simus, N.; Singhal, M.;Xu, L.; Mendes, P.; Kummer, U. COPASI—a Complex Pathway Simulator.
Bioinformatics , , 3067–3074, DOI: .[11] Goodwin, D. G.; Moffat, H. K.; Speth, R. L. Cantera: An Object-Oriented Soft-ware Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes.2017.[12] Glowacki, D. R.; Liang, C.-H.; Morley, C.; Pilling, M. J.; Robertson, S. H. MES-MER: An Open-Source Master Equation Solver for Multi-Energy Well Reactions. J. Phys. Chem. A , , 9545–9560, DOI: .[13] Proppe, J.; Husch, T.; Simm, G. N.; Reiher, M. Uncertainty Quantification forQuantum Chemical Models of Complex Reaction Networks. Faraday Discuss. , , 497–520, DOI: .[14] Shannon, R.; Glowacki, D. R. A Simple “Boxed Molecular Kinetics” ApproachTo Accelerate Rare Events in the Stochastic Kinetic Master Equation. J. Phys.Chem. A , , 1531–1541, DOI: .[15] Proppe, J.; Reiher, M. Mechanism Deduction from Noisy Chemical Re-action Networks. J. Chem. Theory Comput. , , 357–370, DOI: .[16] Pauling, L. The Nature of The Chemical Bond. IV. The Energy of Single Bondsand the Relative Electronegativity of Atoms. J. Am. Chem. Soc. , , 3570–3582, DOI: .[17] Pauling, L. The Nature of the Chemical Bond , 3rd ed.; Cornell University Press:Ithaca, New York, 1980.[18] Pearson, R. G. Hard and Soft Acids and Bases.
J. Am. Chem. Soc. , ,3533–3539, DOI: .1519] Pearson, R. G. Hard and Soft Acids and Bases, HSAB, Part 1: FundamentalPrinciples. J. Chem. Educ. , , 581, DOI: .[20] Pearson, R. G. Hard and Soft Acids and Bases, HSAB, Part II: UnderlyingTheories. J. Chem. Educ. , , 643, DOI: .[21] Bader, R. F. W. Atoms in Molecules: A Quantum Theory ; The InternationalSeries of Monographs on Chemistry; Clarendon Press: Oxford, 1990; Vol. 22.[22] Cioslowski, J.; Karwowski, J. In
Fundamentals of Molecular Similarity ; Carb´o-Dorca, R., Giron´es, X., Mezey, P. G., Eds.; Mathematical and Compu-tational Chemistry; Springer US: Boston, MA, 2001; pp 101–112, DOI: .[23] Anderson, J. S. M.; Ayers, P. W. Quantum Theory of Atoms in Molecules:Results for the SR-ZORA Hamiltonian.
J. Phys. Chem. A , , 13001–13006, DOI: .[24] Reiher, M.; Wolf, A. Relativistic Quantum Chemistry: The Fundamental Theoryof Molecular Science , 2nd ed.; Wiley: Weinheim, 2015.[25] Kowalik, M.; Gothard, C. M.; Drews, A. M.; Gothard, N. A.; Weckiewicz, A.;Fuller, P. E.; Grzybowski, B. A.; Bishop, K. J. M. Parallel Optimization ofSynthetic Pathways within the Network of Organic Chemistry.
Angew. Chem.Int. Ed. , , 7928–7932, DOI: .[26] Klucznik, T. et al. Efficient Syntheses of Diverse, Medicinally Relevant TargetsPlanned by Computer and Executed in the Laboratory. Chem , , 522–532,DOI: .[27] Szymku´c, S.; Gajewska, E. P.; Klucznik, T.; Molga, K.; Dittwald, P.; Startek, M.;Bajczyk, M.; Grzybowski, B. A. Computer-Assisted Synthetic Planning: TheEnd of the Beginning. Angew. Chem. Int. Ed. , , 5904–5937, DOI: .[28] Schwaller, P.; Laino, T. Machine Learning in Chemistry: Data-Driven Algo-rithms, Learning Systems, and Predictions ; ACS Symp. Ser.; Am. Chem. Soc.,2019; Vol. 1326; Chapter 4, pp 61–79, DOI: .[29] Schwaller, P.; Laino, T.; Gaudin, T.; Bolgar, P.; Hunter, C. A.; Bekas, C.;Lee, A. A. Molecular Transformer: A Model for Uncertainty-CalibratedChemical Reaction Prediction.
ACS Cent. Sci. , , 1572–1583, DOI: .[30] IBM RXN for Chemistry. https://rxn.res.ibm.com/.[31] Segler, M. H. S.; Waller, M. P. Neural-Symbolic Machine Learning for Ret-rosynthesis and Reaction Prediction. Chem. Eur. J. , , 5966–5971, DOI: . 1632] Segler, M. H. S.; Waller, M. P. Modelling Chemical Reasoning to Pre-dict and Invent Reactions. Chem. Eur. J. , , 6118–6128, DOI: .[33] Segler, M. H. S.; Preuss, M.; Waller, M. P. Planning Chemical Syntheses withDeep Neural Networks and Symbolic AI. Nature , , 604–610, DOI: .[34] Bergeler, M.; Simm, G. N.; Proppe, J.; Reiher, M. Heuristics-Guided Explorationof Reaction Mechanisms. J. Chem. Theory Comput. , , 5712–5722, DOI: .[35] Grimmel, S. A.; Reiher, M. The Electrostatic Potential as a Descriptor forthe Protonation Propensity in Automated Exploration of Reaction Mechanisms. Faraday Discuss. , , 443–463, DOI: .[36] Miranda-Quintana, R. A.; Franco-P´erez, M.; G´azquez, J. L.; Ayers, P. W.;Vela, A. Chemical Hardness: Temperature Dependent Definitions and ReactivityPrinciples. J. Chem. Phys. , , 124110, DOI: .[37] G´azquez, J. L.; Franco-P´erez, M.; Ayers, P. W.; Vela, A. Temperature-Dependent Approach to Chemical Reactivity Concepts in Density FunctionalTheory. Int. J. Quantum Chem. , , e25797, DOI: .[38] Berzelius, J. J. Essai sur la Nomenclature Chimique. J. Phys. Chim. Hist. Nat.Arts , 253–286.[39] Jensen, W. B. Electronegativity from Avogadro to Pauling: Part 1: Ori-gins of the Electronegativity Concept.
J. Chem. Educ. , , 11, DOI: .[40] Pritchard, H. O.; Skinner, H. A. The Concept Of Electronegativity. Chem. Rev. , , 745–786, DOI: .[41] Huggins, M. L. Bond Energies and Polarities. J. Am. Chem. Soc. , ,4123–4126, DOI: .[42] Allred, A. L. Electronegativity Values from Thermochemical Data. J. Inorg.Nucl. Chem. , , 215–221, DOI: .[43] Mulliken, R. S. A New Electroaffinity Scale; Together with Data on ValenceStates and on Valence Ionization Potentials and Electron Affinities. J. Chem.Phys. , , 782–793, DOI: .[44] Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. Electronegativity: TheDensity Functional Viewpoint. J. Chem. Phys. , , 3801–3807, DOI: . 1745] Parr, R. G.; Yang, W. Density-Functional Theory of the Electronic Struc-ture of Molecules. Annu. Rev. Phys. Chem. , , 701–728, DOI: .[46] Geerlings, P.; Proft, F. D. Conceptual DFT: The Chemical Relevance of HigherResponse Functions. Phys. Chem. Chem. Phys. , , 3028–3042, DOI: .[47] Geerlings, P.; De Proft, F.; Langenaeker, W. Conceptual Density FunctionalTheory. Chem. Rev. , , 1793–1874, DOI: .[48] Geerlings, P.; Chamorro, E.; Chattaraj, P. K.; De Proft, F.; G´azquez, J. L.;Liu, S.; Morell, C.; Toro-Labb´e, A.; Vela, A.; Ayers, P. Conceptual DensityFunctional Theory: Status, Prospects, Issues. Theor. Chem. Acc. , , 36,DOI: .[49] Iczkowski, R. P.; Margrave, J. L. Electronegativity. J. Am. Chem. Soc. , , 3547–3551, DOI: .[50] Parr, R. G. Density Functional Theory of Atoms and Molecules.Horizons of Quantum Chemistry. Dordrecht, 1980; pp 5–15, DOI: .[51] Lieb, E. H. Density Functionals for Coulomb Systems. Int. J. Quantum Chem. , , 243–277, DOI: .[52] Dreizler, R. M.; Gross, E. K. U. Density Functional Theory: An Approach to theQuantum Many-Body Problem ; Springer-Verlag: Berlin Heidelberg, 1990; DOI: .[53] Chermette, H. Chemical Reactivity Indexes in Density Func-tional Theory.
J. Comput. Chem. , , 129–154, DOI: .[54] Kvaal, S.; Ekstr¨om, U.; Teale, A. M.; Helgaker, T. Differentiable but ExactFormulation of Density-Functional Theory. J. Chem. Phys. , , 18A518,DOI: .[55] Koopmans, T. ¨Uber die Zuordnung von Wellenfunktionen und Eigenwertenzu den einzelnen Elektronen eines Atoms. Physica , , 104–113, DOI: .[56] Mulliken, R. S. Quelques aspects de la th´eorie des orbitales mol´eculaires. J.Chim. Phys. , , 497–542, DOI: .[57] Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Density-Functional Theoryfor Fractional Particle Number: Derivative Discontinuities of the Energy. Phys.Rev. Lett. , , 1691–1694, DOI: .1858] Almbladh, C.-O.; von Barth, U. Exact Results for the Charge and Spin Den-sities, Exchange-Correlation Potentials, and Density-Functional Eigenvalues. Phys. Rev. B , , 3231–3244, DOI: .[59] Perdew, J. P.; Levy, M. Comment on “Significance of the Highest Occu-pied Kohn-Sham Eigenvalue”. Phys. Rev. B , , 16021–16028, DOI: .[60] Baerends, E. J.; Gritsenko, O. V.; van Meer, R. The Kohn–Sham Gap, theFundamental Gap and the Optical Gap: The Physical Meaning of Occupiedand Virtual Kohn–Sham Orbital Energies. Phys. Chem. Chem. Phys. , ,16408–16425, DOI: .[61] Baerends, E. J. Density Functional Approximations for Orbital Energies andTotal Energies of Molecules and Solids. J. Chem. Phys. , , 054105, DOI: .[62] van Meer, R.; Gritsenko, O. V.; Baerends, E. J. Physical Meaning of VirtualKohn–Sham Orbitals and Orbital Energies: An Ideal Basis for the Descriptionof Molecular Excitations. J. Chem. Theory Comput. , , 4432–4441, DOI: .[63] Amati, M.; Stoia, S.; Baerends, E. J. The Electron Affinity as the Highest Oc-cupied Anion Orbital Energy with a Sufficiently Accurate Approximation ofthe Exact Kohn–Sham Potential. J. Chem. Theory Comput. , , 443–452,DOI: .[64] Hinze, J.; Jaffe, H. H. Electronegativity. I. Orbital Electronegativity of NeutralAtoms. J. Am. Chem. Soc. , , 540–546, DOI: .[65] Hinze, J.; Whitehead, M. A.; Jaffe, H. H. Electronegativity. II. Bond andOrbital Electronegativities. J. Am. Chem. Soc. , , 148–154, DOI: .[66] Hinze, J.; Jaffe, H. Electronegativity. III. Orbital Electronegativities and Elec-tron Affinities of Transition Metals. Can. J. Chem. , , 1315, DOI: .[67] Hinze, J.; Jaffe, H. H. Electronegativity. IV. Orbital Electronegativity of the Neu-tral Atoms of the Periods Three and Four A and of Positive Ions of Periods Oneand Two. J. Phys. Chem. , , 1501–1506, DOI: .[68] Bergmann, D.; Hinze, J. Electronegativity and Molecular Properties. Angew.Chem. Int. Ed. , , 150–163, DOI: .[69] Skinner, H. A.; Pritchard, H. O. The Measure of Electronegativity. Trans. Fara-day Soc. , , 1254–1262, DOI: .1970] Bratsch, S. G. Revised Mulliken Electronegativities: I. Calculation and Conver-sion to Pauling Units. J. Chem. Educ. , , 34, DOI: .[71] Nguyen, H. M. T.; Peeters, J.; Nguyen, M. T.; Chandra, A. K. Use of DFT-BasedReactivity Descriptors for Rationalizing Radical Reactions: A Critical Analysis. J. Phys. Chem. A , , 484–489, DOI: .[72] Mayr, H.; Ofial, A. R. Kinetics of Electrophile-Nucleophile Combinations: AGeneral Approach to Polar Organic Reactivity. Pure Appl. Chem. , ,1807–1821, DOI: .[73] Mayr, H.; Ofial, A. R. Do General Nucleophilicity Scales Exist? J. Phys. Org.Chem. , , 584–595, DOI: .[74] Mayr, H.; Breugst, M.; Ofial, A. R. Farewell to the HSAB Treatment ofAmbident Reactivity. Angew. Chem. Int. Ed. , , 6470–6505, DOI: .[75] Mayr, H. Reactivity Scales for Quantifying Polar Organic Reactivity:The Benzhydrylium Methodology. Tetrahedron , , 5095–5111, DOI: .[76] Lee, S.; Goodman, J. M. Rapid Route-Finding for Bifurcating Organic Reactions. J. Am. Chem. Soc. , , 9210–9219, DOI: .[77] Parr, R. G.; v. Szentp´aly, L.; Liu, S. Electrophilicity Index. J. Am. Chem. Soc. , , 1922–1924, DOI: .[78] Szentp´aly, L. V. Modeling the Charge Dependence of Total Energy and Its Rel-evance to Electrophilicity. Int. J. Quantum Chem. , , 222–234, DOI: .[79] Parr, R. G.; Pearson, R. G. Absolute Hardness: Companion Parameter toAbsolute Electronegativity. J. Am. Chem. Soc. , , 7512–7516, DOI: .[80] Pearson, R. G. Chemical Hardness and Density Functional Theory. J. Chem.Sci. , , 369–377, DOI: .[81] Pearson, R. G. Chemical Hardness: Applications from Molecules to Solids , 1sted.; Wiley-VCH: Weinheim; New York, 1997.[82] Pearson, R. G. The HSAB Principle — More Quantitative Aspects.
Inorg. Chim.Acta , , 93–98, DOI: .[83] C´ardenas, C.; W. Ayers, P. How Reliable Is the Hard–Soft Acid–Base Principle?An Assessment from Numerical Simulations of Electron Transfer Energies. Phys.Chem. Chem. Phys. , , 13959–13968, DOI: .2084] Bettens, T.; Alonso, M.; Proft, F. D.; Hamlin, T. A.; Bickelhaupt, F. M. Am-bident Nucleophilic Substitution: Understanding Non-HSAB Behavior throughActivation Strain and Conceptual DFT Analyses. Chem. Eur. J. , 3884–3893, DOI: .[85] Parr, R. G.; Yang, W. Density Functional Approach to the Frontier-ElectronTheory of Chemical Reactivity.
J. Am. Chem. Soc. , , 4049–4050, DOI: .[86] Bultinck, P.; Fias, S.; Van Alsenoy, C.; Ayers, P. W.; Carb´o-Dorca, R. CriticalThoughts on Computing Atom Condensed Fukui Functions. J. Chem. Phys. , , 034102, DOI: .[87] Hirshfeld, F. L. Bonded-Atom Fragments for Describing Molecular Charge Den-sities. Theoret. Chim. Acta , , 129–138, DOI: .[88] Proft, F. D.; Alsenoy, C. V.; Peeters, A.; Langenaeker, W.; Geerlings, P. AtomicCharges, Dipole Moments, and Fukui Functions Using the Hirshfeld Partition-ing of the Electron Density. J. Comput. Chem. , , 1198–1209, DOI: .[89] Dewar, M. J. S. A Critique of Frontier Orbital Theory. J. Mol. Struct.THEOCHEM , , 301–323, DOI: .[90] Brown, J. J.; Cockroft, S. L. Aromatic Reactivity Revealed: Beyond Reso-nance Theory and Frontier Orbitals. Chem. Sci. , , 1772–1780, DOI: .[91] Pino-Rios, R.; Ya˜nez, O.; Inostroza, D.; Ruiz, L.; Cardenas, C.; Fuentealba, P.;Tiznado, W. Proposal of a Simple and Effective Local Reactivity Descriptorthrough a Topological Analysis of an Orbital-Weighted Fukui Function. J. Com-put. Chem. , , 481–488, DOI: .[92] Morell, C.; Grand, A.; Toro-Labb´e, A. New Dual Descriptor for Chemical Reac-tivity. J. Phys. Chem. A , , 205–212, DOI: .[93] Morell, C.; Grand, A.; Toro-Labb´e, A. Theoretical Support for Us-ing the ∆f(r) Descriptor. Chem. Phys. Lett. , , 342–346, DOI: .[94] Mart´ınez-Araya, J. I. Why Is the Dual Descriptor a More Accurate Local Re-activity Descriptor than Fukui Functions? J. Math. Chem. , , 451–465,DOI: .[95] Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Ap-proximation Made Simple. Phys. Rev. Lett. , , 3865–3868, DOI: . 2196] Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient ApproximationMade Simple [Phys. Rev. Lett. 77, 3865 (1996)]. Phys. Rev. Lett. , ,1396–1396, DOI: .[97] Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple ZetaValence and Quadruple Zeta Valence Quality for H to Rn: Design and As-sessment of Accuracy. Phys. Chem. Chem. Phys. , , 3297–3305, DOI: .[98] Weigend, F. Accurate Coulomb-Fitting Basis Sets for H to Rn. Phys. Chem.Chem. Phys. , , 1057–1065, DOI: .[99] Neese, F. The ORCA Program System. Wiley Interdiscip. Rev.: Comput. Mol.Sci. , , 73–78, DOI: .[100] Ayers, P. W.; Morell, C.; De Proft, F.; Geerlings, P. Understanding the Wood-ward–Hoffmann Rules by Using Changes in Electron Density. Chem. Eur. J. , , 8240–8247, DOI: .[101] Geerlings, P.; Ayers, P. W.; Toro-Labb´e, A.; Chattaraj, P. K.; De Proft, F. TheWoodward–Hoffmann Rules Reinterpreted by Conceptual Density FunctionalTheory. Acc. Chem. Res. , , 683–695, DOI: .[102] Melin, J.; Aparicio, F.; Subramanian, V.; Galv´an, M.; Chattaraj, P. K. Is theFukui Function a Right Descriptor of Hard-Hard Interactions? J. Phys. Chem.A , , 2487–2491, DOI: .[103] Stuyver, T.; Shaik, S. Unifying Conceptual Density Functional and Valence BondTheory: The Hardness–Softness Conundrum Associated with Protonation Re-actions and Uncovering Complementary Reactivity Modes. J. Am. Chem. Soc. , , 20002–20013, DOI: .[104] Anderson, J. S. M.; Melin, J.; Ayers, P. W. Conceptual Density-FunctionalTheory for General Chemical Reactions, Including Those That Are NeitherCharge- nor Frontier-Orbital-Controlled. 1. Theory and Derivation of a General-Purpose Reactivity Indicator. J. Chem. Theory Comput. , , 358–374, DOI: .[105] Anderson, J. S. M.; Melin, J.; Ayers, P. W. Conceptual Density-Functional The-ory for General Chemical Reactions, Including Those That Are Neither Charge-nor Frontier-Orbital-Controlled. 2. Application to Molecules Where FrontierMolecular Orbital Theory Fails. J. Chem. Theory Comput. , , 375–389,DOI: .[106] Lee, B.; Yoo, J.; Kang, K. Predicting the Chemical Reactivity of Organic Ma-terials Using a Machine-Learning Approach. Chem. Sci. , , 7813–7822,DOI: . 22107] Hoffmann, G.; Balcilar, M.; Tognetti, V.; H´eroux, P.; Ga¨uz`ere, B.; Adam, S.;Joubert, L. Predicting Experimental Electrophilicities from Quantum and Topo-logical Descriptors: A Machine Learning Approach. J. Comput. Chem. , ,2124–2136, DOI: .[108] Klopman, G. Chemical Reactivity and Reaction Paths ; Wiley, 1974.[109] Toro-Labb´e, A. Characterization of Chemical Reactions from the Profiles of En-ergy, Chemical Potential, and Hardness.
J. Phys. Chem. A , , 4398–4403, DOI: .[110] Unsleber, J. P.; Reiher, M. The Exploration of Chemical Reac-tion Networks. Annu. Rev. Phys. Chem. , , 121–142, DOI: .[111] Hoffmann, R.; Malrieu, J.-P. Simulation vs. Understanding: A Tension, inQuantum Chemistry and Beyond. Part B. The March of Simulation, forBetter or Worse. Angew. Chem. Int. Ed. , , 13156–13178, DOI:10.1002/anie.201910283