GitBucket
Newer
zeynalig
<?xml version="1.0" ?> <tei> <teiHeader> <fileDesc xml:id="_Ayers_2015_Six_questions_on"/> </teiHeader> <text xml:lang="en"> <front> Six Questions on Topology in Theoretical Chemistry <lb/> Paul L. Ayers a , Russell J. Boyd b , Patrick Bultinck c , Michel Ca↵arel d , Ramon Carbó-Dorca e , Mauro Causá f , Jerzy <lb/>Cioslowski g , Julia Contreras-Garcia h , David L. Cooper i , Philip Coppens j , Carlo Gatti k , Simon Grabowsky l , Paolo <lb/>Lazzeretti m , Piero Macchi n , Ángel Martín Pendás o , Paul L. A. Popelier p,q , Klaus Ruedenberg r , Henry Rzepa s , <lb/>Andreas Savin h , Alexander Sax t , W. H. Eugen Schwarz u , Shant Shahbazian v , Bernard Silvi h,⇤ , Miquel Solà w , <lb/>Vladimir Tsirelson x <lb/> a Department of Chemistry and Chemical Biology, McMaster University, 1280 Main Street West, L8S 4M1, Hamilton, Ontario, Canada <lb/> b Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2 <lb/> c Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281 (S3), B9000 Ghent, Belgium <lb/> d Laboratoire de Chimie et Physique Quantiques, CNRS-IRSAMC Université de Toulouse, France <lb/> e Institute of Computational Chemistry, University of Girona, Campus de Montilivi, 17071 Girona, Spain <lb/> f Dipartimento di Chimica Paolo Corradini, Universitá degli Studi di Napoli " Federico II " , Via Cintia, 80126 Napoli, Italy <lb/> g Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland <lb/> h Sorbonne Universités, UPMC, Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique, case courrier 137, 4 place Jussieu, F-75005 Paris, <lb/>France <lb/> i Department of Chemistry, University of Liverpool, Liverpool L69 7ZD, United Kingdom <lb/> j Chemistry Department, University at Bu↵alo, SUNY, Bu↵alo, New York 14260-3000, United States <lb/> k CNR-ISTM Istituto di Scienze e Tecnologie Molecolari, via Golgi 19, 20133 Milano, Italy <lb/> l School of Chemistry and Biochemistry, Chemistry M310, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia <lb/> m Dipartimento di Scienze Chimiche e Geologiche, Universitá degli Studi di Modena, via G. Campi 183, 41100 Modena, Italy <lb/> n Department of Chemistry and Biochemistry, University of Berne, Freiestrasse 3, CH-3012 Berne, Switzerland <lb/> o Departamento de Química Física y Analítica, Universidad de Oviedo, E-33006 Oviedo, Spain <lb/> p Manchester Institute of Biotechnology (MIB), 131 Princess Street, Manchester, M1 7DN, Great Britain <lb/> q School of Chemistry, University of Manchester, Oxford Road, Manchester, M13 9PL, Great Britain <lb/> r Department of Chemistry and Ames Laboratory USDOE, Iowa State University, Ames, Iowa 50011,USA <lb/> s Department of Chemistry, Imperial College London, South Kensington Campus, Exhibition Road, London SW7 2AY, U.K <lb/> t Department of Chemistry, University of Graz, Heinrichstrasse 28, 8010 Graz, Austria <lb/> u Theoretical Chemistry Groups at Tsinghua University, Beijing, 100084, China, and Physical and Theoretical Chemistry Laboratory, Faculty of <lb/>Science and Engineering, University of Siegen, Siegen, 57068, Germany <lb/> v Faculty of Chemistry, Department of Pure Chemistry, Shahid Beheshti University, G.C., Evin, Tehran, Iran <lb/> w Institut de Química Computacional i Catàlisi (IQCC) and Departament de Química, Universitat de Girona, Campus de Montilivi, E-17071 <lb/>Girona, Catalonia, Spain <lb/> x Department of Quantum Chemistry, Mendeleev University of Chemical Technology, Miusskaya Square 9, 125047 Moscow, Russia <lb/> Abstract <lb/> The paper collects the answers of the authors to the following questions: <lb/> • What is the significance of the topological approach? <lb/> • Can new chemical concepts be found by a topological approach? <lb/> • What is the status of a chemical concept within a topological approach? <lb/> • Should topological approaches provide measurable quantities? <lb/> • Is it possible to predict the outcome of a topological approach without performing a calculation on a computer? <lb/> • What are new domains for which topological approaches would be useful? <lb/></front> <front>⇤ Corresponding author <lb/> Email address: silvi@lct.jussieu.fr (Bernard Silvi) <lb/></front> <front>Preprint submitted to Computational and Theoretical Chemistry <lb/>September 24, 2014 <lb/></front> <body>Andreas Savin and Bernard Silvi. We have been asked by the editors to write an introductory article to this special <lb/>issue of Theoretical and Computational Chemistry. They kindly left us the choice between a short review and anything <lb/>else. We retained the second option and decided to produce a collective paper in which a panel of scientists were <lb/>invited to give their opinions on the topological approaches in Theoretical Chemistry. In March 2014, a letter giving <lb/>the rules of the game was sent to about sixty quantum chemists involved in the development and applications of <lb/>topological methods. We asked the contributors to answer as concisely as possible six questions. A wiki page was <lb/>created on the web site of the Laboratoire de Chimie Théorique enabling each contributor to leave his answers and <lb/>update them in accordance with the answers of the others. The wiki page was closed on May 31. We did not edit the <lb/>answers, as they reflect the diversity of opinion that characterizes scientific discussions. During the editing process <lb/>we received the answers of Piero Macchi that are included in this paper. <lb/> Henry Rzepa. George Whitesides is one of those who have urged the community to make their science more acces-<lb/>sible (his solution was video). I too find experimenting with di↵erent formats refreshing, and part of the process of <lb/>making our science more accessible. I myself took part in a " trialogue " with two others where again a serial format <lb/>illustrating our thoughts over a six month period was captured[1]. I recently attended a celebration of the work of <lb/>the pioneering Jean-Claude Bradley (who died at an absurdly young age) where he introduced the concept of " open <lb/>notebook science " in getting the community to collaborate on new drugs, and to curate a melting point database. <lb/>There are many other experiments of this type out there; some (many?) will not persist, but I think as a community <lb/>it is our obligation to do what Whitesides urges and communicate better. I regard the current " interesting object " as <lb/>yet another of these experiments. In a sense, since it started on a Wiki, that this too was an experiment in notebook <lb/>science (albeit not quite open in the manner JC envisaged). We have done something very similar for > 10 years now <lb/>with our undergraduate students, and they seem to relish it. <lb/> Shant Shahbazian. This contribution is freezing a snapshot of the history of theoretical chemistry. It reveals how peo-<lb/>ple " really " think about this field now. Why not leaving a text for historians of quantum chemistry in next generation? <lb/>I am wondering what will be our answers twenty years in future.... Let not be too sensitive to rigor in this contribution. <lb/>I guess others will also enjoy reading this strange text. It was a bold (and now I think really fruitful) decision made <lb/>by the guest editors and may open the door for such contribution in future. <lb/> What is the significance of topological approach? <lb/> Is it the development of functions of spatial (or momentum) coordinates and the further discussion of their rep-<lb/>resentation? or is it the use of mathematical methods belonging to topology (like singularity theory) in order, for <lb/>instance, to carry out the analysis of a function? <lb/> Paul Ayers. I think that either of the two choices is plausible, but I usually think of the first topic: finding new <lb/>quantities that are amenable to " chemical topology " are (inappropriately? appropriately? I'm not sure) stagnant: once <lb/>one has a new function, the next steps are often clear. Obviously a mathematician (or mathematically-inclined chemist) <lb/>who was well-versed in topology might find new and interesting approaches, but (to me) the simplest approaches seem <lb/>to work well. <lb/> W. H. Eugen Schwarz. The question has two aspects, i) a semantic one and ii) that of purpose and meaning. <lb/>i) Physicists often insist that words have and should have only the meaning that is used by themselves. A nice <lb/>example is " interference " . In physics it means that the superposition of waves results in a new interference pattern; in <lb/>analytical and pharmacological chemistry it means the change of an e↵ect due to the presence of some other substance; <lb/>in chromosomal genetics, in cognition and teaching psychology, in chess, etc. " interference " has other meanings. I <lb/>hold that chemists should stay with their use of the word " topological " as a somewhat vague concept (rather common <lb/>in chemistry) indicating the connection of physical or imagined points or regions of microscopic matter, either in <lb/>the classical sense or represented by some quantal matter field function. Since in Greek (topos) means place, <lb/> (graphein) means write, (legein) means say, both words topological and topographical are similarly <lb/>appropriate for the field under discussion. <lb/>ii) Consequently, the topological approach is the analysis of points or functions in some space, these points or <lb/>functions characterizing the chemical substances. Various tools may be used, including also the concepts and methods <lb/> <page>2 <lb/></page> of mathematical topology. As spaces, one may investigate the position space, the momentum or linear and angular <lb/>momentum space, the classical phase space or one of its quantum analogs, chemical similarity spaces and so on. The <lb/>aim is to extract chemically useful compact indices and more extended information. The aim should not be replacing <lb/>chemistry by sterile numerology. The aim should be to find additional, or more compact or more sharply defined <lb/>descriptors for chemical substances. Since chemistry has two aspects: Many (stationary) Substances, and Changing <lb/>the substances, the topological approach should be applied to the investigation of substances and of reactions. Note <lb/>that in the Chinese language, the word for Chemistry ( f, huà-xué) literally means " Science of Change " . So far <lb/>most of the topological approach has concentrated on the description of stationary substances, i.e. molecules and <lb/>crystals in equilibrium, and be it under pressure. The significance of the topological approach in chemistry could be <lb/>greatly extended in the future, if one succeeds in eciently handling chemical processes. Thereby, two goals could <lb/>be reached. First the description of substances and how they are formed in reactions; and second, it might become <lb/>possible to explain, why this or that isomer is stable, why this or that one is formed. So far the topological approach <lb/>has mainly been restricted to describe " how nature is " , but not yet extended to explain " why nature is as it is " . <lb/>Summarizing, the topological approach in chemistry is a whole set of approaches including tools from mathematical <lb/>topology. <lb/> Mauro Causá. As a first answer I prefer the second. I think the term " topology " is not appropriate for defining <lb/>the present field, so the use of physical observables (or quasi observable) for defining unambiguously the chemical <lb/>phenomenological concepts, even if we make geometrical analysis of physical space functions like ⇢(r), ELF(r) and <lb/> P(n, ⌦) for finding chemistry in physics. I prefer to reserve the term topology to the use of mathematical topology <lb/>analysis to network of " bonds " , a field that seems newly interesting, if applied not only to poly aromatics. <lb/> Paolo Lazzeretti. As regards the first question I would emphasize that topology is a branch of mathematics, born as <lb/> " analysis situ " , and applied nowadays to physics, chemistry and biology. In chemistry, vague or ill-defined concepts, <lb/>say " aromaticity " , can be dealt with allowing for topological indices which provide quantitative information. <lb/> Bernard Silvi. Rather than a topological approach there are topological approaches to definite topics in which a <lb/>mathematical method belonging or related to topology is explicitly or intuitively used to build a model or to analyse <lb/>a set of data. As non exhaustive examples: graph theory has been applied to molecular structure, dynamical system <lb/>theory (enabling to extract qualitative pieces of information from numbers) to the study of di↵erent scalar fields <lb/>(BO energy surfaces, electron density, MESP, ELF), knot theory to the study of chirality, fractal theory to colloïds and <lb/>dendrimers, chaos theory to oscillating reactions. Interesting reviews have been written by Babaev[2] and Cramers[3]. <lb/> Russell Boyd. It is a shame that Richard Bader is no longer a participant in the discussion. He could be relied upon to <lb/>enter all debates and discussions with his unique style and intense passion for science. Life was never dull in Richard <lb/>Bader's presence. The first answer probably comes closer to capturing the view of chemists, while the second is closer <lb/>to the purview of mathematicians. I sometimes think chemists are thinking more in terms of the topography of the <lb/>electron density than topological analysis of the electron density and other quantities. <lb/> Patrick Bultinck. If we consider the first choice a more chemistry oriented one and the second a more mathematical <lb/>one, I still think both are needed. It would be good to have the unbiased view of a mathematician (or anybody that lacks <lb/>the " chemical bias " ) on topological analysis. If I may consider Richard Bader's AIM theory a topology, I think it is <lb/>too often forgotten that this is not a pure topological approach. Essentially, these AIM are based on domains where the <lb/>Schrodinger kinetic energy and the " gradient scalar product " kinetic energies[4] are the same despite the underlying <lb/>property densities being significantly di↵erent (see for example Anderson et al. for some more allowable kinetic <lb/>energy densities[5]). That they share the same expectation value over an AIM domain is thanks to the disappearance <lb/>of the Laplacian of the density over this domain[6]. Now, there is an infinite choice of domains that fulfils this, as is <lb/>immediately clear[7]. However, Bader himself in 1973[8] excluded any gradient line that ends (starts) on a nucleus. <lb/>Moreover, it is an integral that needs to be zero and not an integral of zero values. So from the very start, a topological <lb/>analysis was " guided " in an admittedly brilliant way to lead to chemistry. But what if we do not guide it. Who knows <lb/>what could be found ? Second, in all these approaches we attempt to recover chemistry and in this context, there is <lb/>nothing to be ashamed of to define new functions and then to analyse these. But here also lies a risk. Is a function <lb/>wrong if it does not reveal what we " want to see " ? <lb/> <page>3 <lb/></page> Shant Shahbazian. Topological approach in itself does " not " yield novel mathematical information but simply makes <lb/>it possible to " magnify " and " concentrate " on some special characteristics of a scalar function namely its extrema <lb/>using its gradient vector field. For theories like the QTAIM that are interested in such extrema to grasp their basic <lb/>objects, e.g. topological atom as the basin of attraction of the gradient field of the one-electron density, topological <lb/>approach is a proper tool to reveal the " relevant " points/surfaces, i.e. bond critical points/inter-atomic surfaces. It <lb/>is in principle applicable and useful for any theory that is somehow interested in extrema of functions, e.g. the <lb/>ELF[9]. The Catastrophe theory and other more sophisticate topological approaches, though interesting, have just a <lb/>secondary/auxiliary role. <lb/> Klaus Ruedenberg. The aim in our field is to relate molecular properties to electronic structure, which means elec-<lb/>tronic densities (1st and 2nd order) and energy contributions from various parts of a molecule. Topological reasoning <lb/>is one of the tools that can sometimes be helpful in identifying " macro " -features without being distracted by " micro " -<lb/>details. More sophisticated topological methods are only interesting if they enhance the capabilities of extracting <lb/>basic physical implications that are embedded in i specific electronic wave functions. <lb/> Carlo Gatti. In my view, topological approach has a broader meaning than that covered by either of the two views <lb/>that have been proposed above. Moreover, it is not just represented by their sum. It is rather connected to the <lb/>ancient Greek primary meaning of the words (locus in Latin and in English) and (to speak, to say, to <lb/>tell about), thereby indicating an approach that puts a special emphasis on the local properties of functions of spatial <lb/>(or momentum) coordinates, to reach some, possibly useful, insight on the properties of a system. Functions can be <lb/>selected, for instance, as for their ability to convey chemical information tied to some simplistic chemical model. The <lb/>loci where to study these functions may be suggested by singularity theory. Local properties may be then related <lb/>either to global, " integral " properties of (a part of ) the system, or also to the features of convenient model " objects " <lb/>describing the system. An example of integral property is the energy or the number of electrons of a portion of system, <lb/>one of a convenient model object is the " chemical bond " or its representation in terms of simple diagrams (e.g. Lewis <lb/>structures). <lb/> Julia Contreras-Garcia. As I see it, what we are referring to is chemical topology. So the application of mathematical <lb/>topology concepts to recover, and hopefully expand, classical chemical concepts. Thus, to our community 2) does not <lb/>exist without 1), because we need a function of some kind to apply the mathematical concepts to. Topology is a well <lb/>adapted tool thereto because it enables us to go from point-wise analysis, which scapes our understanding and which <lb/>we do not know how to relate to macrocopic properties of the system to something we can understand, dissect, and <lb/>build -predictive-concepts upon (I tried to avoid mixing questions, but I could not resist). <lb/> Simon Grabowsky. I like the idea of Russel Boyd to distinguish between topography and topology. Every scalar <lb/>function based on spatial or momentum coordinates has a topography. Maybe we could say that topological analysis <lb/>in the sense we mean it in chemistry is the use of mathematical tools and procedures in order to break down the <lb/>three-dimensional topography of a physically meaningful function into interpretable properties in a quantitative way. <lb/>Hence, I think the topological approach is not the development of new functions, but the described way of analysing <lb/>new or old functions – any scalar function -to make them accessible to chemical interpretation. <lb/> Jerzy Cioslowski. Strictly speaking, the term " topological " should be reserved for: 1) analysis of functions in the <lb/>Cartesian or momentum space (such as the electron density or ELF) that is based upon identification of critical <lb/>points and the relations ( " connectivities " ) between them and 2) construction of Hamiltonians that use only atomic <lb/>connectivities as input (as in the Hückel theory a.k.a. chemical graph theory). For example, QTAIM really involves <lb/>two steps: 1) topological analysis of electronic density that cannot be disputed from mathematical standpoint and 2) <lb/>identification of the topological features of electron density with chemical concepts that is subject to rather arbitrary <lb/>interpretation (recall the raging discussion on weak bonds/ " negative bonds " /repulsive bonds in e.g. planar biphenyl). <lb/> Paul Popelier. To me the significance of the " topological approach " has always been its ability to partition a system <lb/>into well-defined regions. This is crucial because many questions in chemistry boil down to a partitioning issue. <lb/>The way the " topological approach " partitions is minimal and I like that; moreover, it has a visual appeal that has <lb/>not be fully exploited (see movies in drug design). The partitioning is minimal because it is parameter-free and <lb/> <page>4 <lb/></page> does not assume anything, such as a reference system, or a priori sphericity. The gradient vector is all one needs <lb/>to go ahead with partitioning. " Minimal " should not be confused with " simple " (see my little story on the ellipse <lb/>versus the circle in solar system mechanics in the preface of reference[10]). I like minimal approaches because they <lb/>generate a lot of information without imposing anything. Putting the idea of gradient vector field upfront is what <lb/>binds the various topological approaches (QTAIM (both the electron density and its Laplacian), ELF, IQA, . . . , See <lb/>Box 8.1 in reference[11]), which is one of the reasons for the overarching name " Quantum Chemical Topology " <lb/>(QCT)[12, 13], which really stands for " the topological approach " . If forced to choose between the two subquestions <lb/>above, I therefore veer towards the second one. The separatrices that QCT discovers are part of a wider language, <lb/>that of dynamical systems, which includes critical points etc. More advanced and related mathematical concepts (e.g. <lb/>Riemannian curvature) should be explored in connection with QCT. <lb/> Ángel Martín Pendás. A point of view which is floating around in many of the answers provided distinguishes be-<lb/>tween the functions that are supposed to convey chemically relevant information (either in real or momentum space) <lb/>and the mathematical tools (borrowed from topology, graph theory, catastrophe theory, etc) used to data mine them. <lb/>To me, the " topological approach " provides a unique way to discover specific points in (real or momentum) space <lb/>-the local stance, with its critical points correlating with all sort of molecular properties-, and at the same time to <lb/>coarse-grain or partition the space into regions. This leads to the global stance, with integrated properties that add to <lb/>the total quantum-mechanical expectation values. In a sense, it is this bimodality that makes the approach so useful, <lb/>since one may switch from one stance to the other at convenience without leaving the general framework. <lb/> Vladimir Tsirelson. I prefer the first answer. I also think that neither " topological approach " nor " Atoms in molecules " <lb/>and " Quantum chemical topology " are not proper names of the field. I would suggest the following: " Quantum <lb/>topological theory of molecular and crystal structure " . <lb/> Michel Ca↵arel. To me the key issue is to construct simple mathematical representations allowing chemists/physicists <lb/>to visualize and rationalize what happen in a complex molecular system described by a highly structured and entangled <lb/>3N-dimensional wavefunction through the use of much more simpler 3D functions and topological analysis. Due to <lb/>this loss of information, the notion of chemical bond in ordinary space, and more generally most chemical concepts, <lb/>appear to me as necessarily and intrinsically not well defined, although useful as any taxonomy. Localization functions <lb/>and topology can help (eventually a lot) but within these strict limits. Hopefully, the problem can be reduced to lower <lb/>dimensions through two-and one-body densities. Localization functions in close relation to the two-body matrix are <lb/>thus attractive. An example is the electron pair localization function (EPLF) introduced in [14, 15] that is directly <lb/>related to the two-body density matrix and that is usable for any type of wavefunctions and excited states. Another <lb/>fundamental aspect related to intricacy and delocalization of the highly multi-dimensional wavefunction is that any <lb/>partition in 3D space into well-defined regions seems hardly compatible with quantum mechanics. Some fuzzy aspects <lb/>in domains taking their source in multi-particle correlations should appear. In that respect I find the idea of Savin of <lb/>defining overlapping domains of maximal probability for a given number of electrons very interesting (see [16], and <lb/>[17] also). <lb/> Alexander Sax. The topological approach helps chemists to feel like mathematicians: Just call everything related <lb/>to geometry (distances, angles) or graph theory (arcs connecting nodes) topology. And you need not care about <lb/>horrifying aspects like filters, compactness, homotopy etc. <lb/> Piero Macchi. In Chemistry, a topological approach is a link between a simple concept (structure, reaction mecha-<lb/>nism, functionality etc.) and an observable. It can be used to understand chemical bonding, to anticipate chemical <lb/>reactions or to rationalize materials properties. Mathematically, a topological approach involves both the analysis of <lb/>a function and its representation in terms of simple elements. <lb/> Can new chemical concepts be found by a topological approach? Can traditional chemical concept definitions <lb/>be improved by topological approaches? <lb/> Henry Rzepa. I will answer this perhaps in an unexpected manner, where I am redefining the topological approach <lb/>to chemistry rather more broadly. Our interest in topological techniques such as QTAIM and ELF arose because we <lb/> <page>5 <lb/></page> wanted to study the properties of aromatic systems. Conventional approaches for assessing aromaticity do not work as <lb/>easily if the aromatics are topologically non planar. We discovered the wonderful topological world of knots and links, <lb/>and the theorems associated with them (see figure 1). The White/C˘ alug˘ areanu/Fuller theorem relates to the geometry <lb/> Figure 1: Torus knots and links of relevance to twisted conjugated cyclic molecules. <lb/> of closed twisted ribbons by a topological invariant known as the Linking number, where Lk = T w + Wr, where Wr <lb/> or Writhe measures the extent to which coiling of the central curve has relieved local twisting T of the ribbon and <lb/> T w is <lb/> R T . The ⇡-density ⇢(r) ⇡-manifold of benzene for example can be described with a linking number of 0, but <lb/>the interesting linking numbers are , 0. For example, a Möbius ⇢(r) ⇡-manifold has a linking number of ±1⇡, and a <lb/>three dimensional lemniscate is ±2. In chemistry, these topological invariants are also chiral descriptors, thus -1 is the <lb/>mirror image/enantiomer of +1. In that sense, the linking number is a manifestation of the Cahn-Ingold-Prelog M/P <lb/>helical notation for higher-order topologies. <lb/>Molecules exhibiting these properties are in fact well known; many extended porphyrins adopt such topologies <lb/>(see C. S. M. Allan and H. S. Rzepa[18] or H. S. Rzepa[19]. It turns out that this allows a general extension of <lb/>the simple Hückel rules for aromaticity: For even values of the linking number (including zero), the ⇡-density ⇢(r) <lb/> manifold will exhibit aromaticity/diatropicity if populated with 4n+2 ⇡-electrons and antiaromaticity/paratropicity for <lb/>values of 4n under thermal conditions. For odd values of the linking number, the ⇡-density ⇢(r) manifold will exhibit <lb/>aromaticity/diatropicity for values of 4n and antiaromaticity/paratropicity if populated with 4n + 2 ⇡-electrons under <lb/>thermal conditions. So to answer the question directly " Can traditional chemical concept definitions be improved by <lb/>topological approaches? " , yes one can use these topological approaches to gain insight into the aromaticity of twisted <lb/>(non-planar) conjugated molecules. And one can achieve a fusion of two classical 19th century concepts of aromatic <lb/>molecules and of chirality since as I noted above, manifolds with non-zero linking numbers are chiral. I close by <lb/>saying that the above also provides an answer to questions 5 and 6. More details can be found at S. M. Rappaport and <lb/>H S. Rzepa[20] <lb/> Paul Ayers. Yes, and Yes. The latter question is most apparent and in my narrow field (conceptual DFT) one often <lb/>takes " traditional " (or at least quasi-traditional) concepts and applies the topological toolbox to them. As for new <lb/>chemical concepts, I hope so, and I have faith that it is true. For example, in our work on using machine-learning <lb/>to identify molecular descriptors, it seems likely that a " topological signature " that is common to some phenomenon <lb/>could be identified. (This is to say, it is possible that the descriptor(s) one finds when trying to organize a broad range <lb/>of chemical phenomena into classes will be topological.) That said, I don't know how many compelling new concepts <lb/>have been identified; it seems to me that the most powerful and useful chemical concepts are half a century old, or <lb/>older. <lb/> Mauro Causá. Yes, The so called topological approaches (QTAIM, ELF and MPD) can be definite answers to the <lb/>problem of understanding the chemical phenomenological concepts within a physical framework. <lb/> Vladimir Tsirelson. Yes. It is really doing very often. <lb/> <page>6 <lb/></page> Paolo Lazzeretti. That's precisely what I expect from my " topological " approach to " chemical ideas " . For instance, <lb/>the interpretation of aromaticity on the magnetic criterion is based on a topological object named " stagnation graph " . <lb/>The singular branching points of this graph are analyzed in connection with the notion of " delocalized electron cloud " . <lb/> Russell Boyd. I would answer yes to both questions, but I am not entirely happy with the current state of a↵airs. Let <lb/>me explain with the example of electronegativity, which is introduced to students at an early stage in their study of <lb/>chemistry. Electronegativity is a chemical concept that is used to rationalize many observations and is a major part <lb/>of the vocabulary of chemistry, but the theoretical basis for the concept is far from being satisfactory. In 1988, Ken <lb/>Edgecombe and I tried to evaluate atomic and group electronegativities from the topological properties of the electron <lb/>density distributions of molecules[21, 22]. Although our manuscript was published in a prominent journal, I have <lb/>always had the feeling that it should be possible to produce a better approach. Electronegativity is only one of many <lb/>chemical concepts for which I think there is a need for a more rigorous theoretical foundation. If that foundation <lb/>is possible, then I think it is mostly likely to come from the application of topological approaches to the results of <lb/>quantum mechanics. <lb/> Patrick Bultinck. Do we really need even more concepts? I'd be happier if we concentrated first on getting these <lb/>right or at least linked to something quantum chemically/mechanically better defined. Topology may help in this but <lb/>I think we need to realize that many of these concepts were never meant to be quantified. Take aromaticity, there is <lb/>discussion on what is more aromatic; e.g. benzene or the pentadienyl anion, but we quite often forget to define closely <lb/>the meaning of aromatic. To me it is the property of having a delocalized system (also to be properly defined first) and <lb/>the property of exhibiting a current density of right nature (read: like in benzene) but to many others it is the property <lb/>of having a such and such NICS value. So, new concepts might make everything even more di↵use. Narrowing down, <lb/>or describing more accurately existing concepts is a good option but remember Mulliken's statement that the more we <lb/>know and compute, the more concepts disappear[23]: are we prepared, willing, capable of accepting that ? I sure do <lb/>hope so but it'll be a hard job to ask " regular chemists " to throw away a tool that is not broken in their view. Even <lb/>R.G. Parr has struggled with this question, given his statement " Accurate calculation is not synonymous with useful <lb/>interpretation. To calculate a molecule is not to understand it " that supports the conceptual approach[24]. Maybe now, <lb/>with even better methods and higher accuracy than in 1965, we could ask this question again instead of coming up <lb/>with more concepts. <lb/> Shant Shahbazian. The general answer to the first question is " no " . Topological approach is indeed a mathematical <lb/>tool to " extract " chemical information from wavefunctions in real space making a " bridge " between quantum me-<lb/>chanics (QM) and chemistry. But, to make a bridge, one needs to the both sides. Chemical concepts are introduced <lb/>in chemical discourse based on their significance in chemistry not emerging from QM. We know there are atoms in <lb/>molecules from chemical experiences (not QM!) then seeking to extract them from wavefunctions " inventing " the <lb/>QTAIM and its topological apparatus. This is also true in the case of ELF methodology that assumes there are lone <lb/>pairs, then seeking them using topological apparatus. The answer to the second question is just a conditional " yes " . <lb/>Yes, if topological procedure is part of a comprehensive theory that " redefines " the relevant chemical concept in a <lb/>rigorous manner like " the " QTAIM. It must be stressed that topological approach is " not " in itself a theory but al-<lb/>ways part of a theory/methodology that aims to make the mentioned bridge. Thus, its success depends on the main <lb/>theory[25]. <lb/> Julia Contreras-Garcia. I would say that, like any other theoretical development, topological approaches should be <lb/>able to 1) recover previous (classical chemical) concepts and 2) expand them (otherwise, we would not need them!). <lb/>The recovery should ideally be done at the qualitative and quantitative level, not only we have to find bonds, but we <lb/>should be able to find a charge for it which fits into classical theories. In this matching, we should concentrate in <lb/>text-book examples. Once this is done to a reasonable (which of course is quite ambiguous) level, the 'extension' <lb/>phase should come in. E.g. It is very good to see lone pairs where we know lone pairs are, but where are the lone <lb/>pairs in the electrides, for example. This is a very neat example that I always use with students: sodium under very <lb/>high pressure loses its metallic properties, i.e. it is an insulator. This escapes classical approaches, however, ELF is <lb/>able to show very clearly that electrons localize in the structural voids as pseudo anions, thus forming an insulating <lb/>pseudo-ionic solid, whereas it shows a very clear metallic structure at ambient pressure (and is thus conducting at <lb/>ambient conditions). <lb/> <page>7 <lb/></page> Simon Grabowsky. The discussion seems to develop into the direction (especially Patrick Bultinck and Shant Shah-<lb/>bazian) that topological analysis should be unrelated to the discovery, confirmation, improvement or refutation of <lb/>chemical concepts because chemical concepts are meant to be broad with fuzzy boundaries. This allows their uni-<lb/>versal applicability and enhances chemical understanding beyond the level of numerical details. Within this train of <lb/>thoughts, topological analysis should rather provide a framework to compare properties of di↵erent compounds and <lb/>enable correlation to their chemical function, i.e. bridging quantum mechanics/experiments with chemical concepts. <lb/>As an example, I must think of the use of pure QTAIM[26] as well as the related Source Function[27] to define di↵er-<lb/>ent classes of hydrogen bonds, i.e. to " improve " the fuzzy concept of hydrogen bonds by introducing sharp boundaries <lb/>between hydrogen bond types. These ideas of classification are certainly very interesting, and maybe they are useful <lb/>for correlations, but do they enhance our understanding of the underlying chemical concept? <lb/> Carlo Gatti. Yes, it occasionally happens that the topological approach discovers new chemical concepts or that <lb/>it simply finds a sharper definition/classification of loosely categorized physico/chemical features. One odd, but <lb/>interesting case, is that of the occurrence of non-nuclear attractors (NNAs) of the electron density gradient field that I, <lb/>as a young " reckless " fellow, introduced in QTAIM, along with Pacchioni and Fantucci, while working on the chemical <lb/>bond features in Li clusters[28]. QTAIM had not to be modified, but simply generalized to systems where, besides the <lb/>usual atomic basins enclosing a nucleus, " strange " basins bounded by the usual zero-flux surfaces but not enclosing <lb/>a nucleus also occur. When I told Richard (Bader) about NNAs, he stared at me clearly disconcerted for a while, but <lb/>after ten minutes or so, he enthusiastically embraced the idea that under some circumstances the electron densities of <lb/>atoms may rearrange upon combination as to form maxima at positions other than the nuclei and I cooperated with <lb/>him and his group in publishing a second paper on NNAs[29]. Occurrence of non-nuclear maxima in the density has <lb/>then been elegantly rationalized by A. Martín Pendás et al.[30]. They have shown that each homonuclear diatomics <lb/>has a stability window of internuclear distances for a midpoint non-nuclear maximum to occur. Therefore, rather than <lb/>being an oddity, non nuclear maxima are a normal step in the chemical bonding of homonuclear groups, if analysed in <lb/>the appropriate range of internuclear distances. For most elements, this range occurs however far away from the stable <lb/>geometry under normal thermodynamic conditions, but such range may easily reached under (high) pressure, where <lb/>the occurrence of NNAs is very frequent (for instance in the insulating structural phase of Na[31]. The NNA concept <lb/>has now been largely recognised and put in correspondence with F centers in crystals[32, 33], or as a mark of solvated <lb/>electrons (e.g. [34]), or to the necessity of concentrating electrons in the interstitial regions of (elemental) solids (e.g. <lb/>[31]) to contrast the application of a large external pressure and to so lower the total energy. Such electron density <lb/>displacement may be also envisaged as a promotion, under pressure, of atomic states of higher angular momentum <lb/>than the ground state. <lb/> Klaus Ruedenberg. Topology is an analysis of relative positions of objects with respect to each other in space. It <lb/>contains no information whatsoever regarding the physics that drives objects to move or be where they are in space. <lb/>Manifestly, topology per se cannot yield physical concepts. On the other hand, the complexity of molecular and, <lb/>in particular, super-molecular structures frequently embodies certain spatial motives that re-occur under a variety of <lb/>conditions. Some of these motives can be characterized by topological identifiers, which some may call chemical <lb/>concepts. Recall in this context that Dirac denounced the " pest of group theory " because it contains no physical <lb/>concepts. Yet the theoretical concepts of point groups have proven very useful, even though they contain no physical <lb/>concepts. Recall also that the basic physical concept of constants of motion is a consequence of the mathematical <lb/>invariance theorem of Emmy Noether. I doubt whether topology can add to elucidating basic bonding concepts. It has <lb/>not done so far. But it may be well positioned to organize the understanding of bonding patterns in complex structures, <lb/>e.g. nano-structures. <lb/> Paul Popelier. Yes to both questions but we need to be brave and follow the full consequences of this approach through <lb/>to the end. With enough work done, the whole of chemistry could be recast into a nice and coherent framework based <lb/>on the topological approach. Space allows me to mention only two examples. In work to be published we show that <lb/>the rotation barrier in biphenyl (from planar to equilibrium at the central torsion angle of ⇠ 45 ) is not due to the text-<lb/>book explanation of steric hindrance between the bay hydrogens H2· · · H2 0 . Instead, a full topological partitioning <lb/>of the molecular energy (using the topology of the electron density) shows that this barrier is due to the destabilised <lb/>intra-atomic energies in the planar conformation of only four atoms: C2,H2 ,C2 0 and H2 0 . In fact, the interaction <lb/> <page>8 <lb/></page> between the two " clashing hydrogens " is actually stabilising, resulting from substantial exchange energy between <lb/>them. As a second example, imagine that you have never seen a hydrogen bond. You would soon see one via its <lb/>topological signature in a plot of interatomic exchange energy versus internuclear distance. This is clear from Figure <lb/>2 in reference[35] where both the O· · · O and H· · · O interaction in the water dimer are anomalously high for their <lb/>internuclear distance. It also shows that one better thinks of a hydrogen bond as a strongly interacting triplet of atoms <lb/>(O-H· · · O) rather than just the H· · · O interaction. Improved insight in hydrogen bonding follows from the nice IQA <lb/>work on hydrogen bonding[36]. <lb/> Ángel Martín Pendás. Nice discussion, indeed. Returning to R. G. Parr's " Accurate calculation is not synonymous <lb/>with useful interpretation " quoted by Patrick Bultinck, are we happy with all the main stream interpretations we <lb/>teach our students? Let us say, to name just a few, hyperconjugation, the anomeric e↵ect, back-donation,... If we <lb/>all agree that we need interpretations to jump from calculation (or experiment) to prediction, has the topological <lb/>approach provided any new chemical insight? I would say yes, and some of the examples in the previous answers <lb/>show it. I would also like to recall how many concepts used today have been influenced by ideas emanating from the <lb/> " topological approach " : core, lone pair, binding regions (the latter going back to Berlin's work); dihydrogen bonds, <lb/>halogen (and pcnide, beryllium, etc) bonds; electron localization and delocalization measures, etc. On the other hand, <lb/>is the present situation idyllic? Here I would say no. I feel, in line with Paul Popelier, that much work needs still to <lb/>be done. <lb/> W.H.Eugen Schwarz. Chemistry is the art and science of less or more complex matter, synthesizable and then really <lb/>existing under " human conditions " , i.e. in the ranges around temperatures of 300 K⇥10 ±1 , pressures of 1 at⇥10 ±3 <lb/> and times of 1 d⇥10 ±6 . This is a rather special selection of conditions. Accordingly, the laws and ordering rules of <lb/>chemistry will not be that simple and that general as those of physics, describing idealized material systems in the <lb/>most comprehensive and general manner. Mathematical chemistry including topological approaches should be useful <lb/>to find characteristic relations of known descriptors, as well as new descriptors of chemical compounds. There exists <lb/>a synergic interaction between the synthetic skills of experimental organic, inorganic, macro-and supra-molecular <lb/>and nano chemists, and the analytic conceptual and computational skills of theoretical chemists of the various breeds. <lb/>Above, Rzepa mentioned an example. <lb/>Many chemical concepts aim at bringing some order into a vast mass of data material. Both the type of com-<lb/>pounds and the interests of the investigating chemists vary largely. Accordingly, many useful chemical concepts have <lb/>been defined in somewhat di↵erent ways by di↵erent researchers. Examples are atomic radii, bond orders, valence <lb/>numbers, iconicity and electronegativity, hard-and softness of acids and bases (HSAB), aromaticity, solvent polarity, <lb/>periodicity of elements, and so on. All these chemical concepts and their definitions are somewhat vague and o↵er <lb/>themselves for criticism. Statistical concept-analyses (main-component, factor, cluster analyses) have shown that the <lb/>various electronegativity and e↵ective atomic charge scales all 'measure' basically the same thing with some noise <lb/>(meaning a small specific contribution for each individual atom and bond in each molecule or crystal), but using dif-<lb/>ferent charge units; i.e. electronegativity is a one-dimensional property[37]. New definitions are still possible, simply <lb/>enlarging the number of basically similar suggestions. The solvent polarity should be described by two parameters, it <lb/>is a two-dimensional property[38, 39]. The hardness and the softness are one-and two-dimensional, respectively[40]. <lb/>The aromaticity concept seems to be three-dimensional[41]. Reducing any of these chemical concepts to a single <lb/>physical noiseless parameter eliminates some part of the richness of chemistry! I remind that it had been suggested <lb/>in the QTAIM[42, 43] that chemists should forget much of their experience about di↵erent bonding types (covalent <lb/>attraction, Pauli repulsion, ionic attraction, ionic repulsion, van der Waals attraction, and a new infant that has ad-<lb/>mittedly not yet shown its chance of survival is the so-called charge-shift bonding) since QTAIM does not allow for <lb/>such a rich description. However, adding various topological approaches to the toolbox of theoretical and conceptual <lb/>chemistry, without forbidding the use of other successful concepts, can enrich chemistry with new concepts and throw <lb/>new light on old ones. <lb/>Since " chemistry is creating its own subjects " , it may happen that indeed new concepts and new notions are <lb/>appropriate for new fields of chemistry, concerning new classes of compounds and reactions, or new applications, <lb/>or new points of view on known compound classes. However, sharpening the definitions in chemistry is somewhat <lb/>dangerous, owing to two aspects. First, chemical concepts should be applicable to a large number of cases which are <lb/>all somewhat di↵erent. If one confines oneself or even the whole community to a single " sharp and clear " definition, <lb/> <page>9 <lb/></page> one refuses all the other, may be more fruitful options. For instance, the concept of general fuzzy atomic radii for <lb/>atoms in a given bond state and coordination multiplicity (coordination number) allows predictions of structures, <lb/>explanations of trends and the discovery of exceptional cases. One may introduce the topological radius of each atom <lb/>in each molecule in each direction as one of several accurate descriptors for cataloging the chemical compounds. <lb/>Yet one still needs the fuzzy concepts both in research and teaching[44]. Pupils and young students like simple <lb/>answers to complex questions. The definition of sharp criteria allows strict qualitative classification of border cases. <lb/>A most welcome rule, found in some textbooks, reads: A bond is covalent if the two bonded atoms di↵er in their <lb/>electronegativities by up to 0.99; for electronegativity di↵erences of 1.00 or more the bond is ionic. I strongly oppose <lb/>against this type of black-white world views, which can be promoted by o↵ering only one sharp criterion for the <lb/>judgment on complex issues. <lb/>Summarizing, new chemical concepts can be found by the topological approaches, provided one follows the <lb/>dicta of chemical usefulness. Traditional chemical concepts can be supported, enriched, elucidated by topological <lb/>approaches, again provided the chemical needs are taken as the maxim. <lb/> Alexander Sax. I am sceptical about a predictive power of a topological approach, but I see that traditional chemical <lb/>concepts can be supported by topological approaches. I just mention the recent paper by Arne Lüchow[45], or A. <lb/>Scemama et al[17], showing that the Lewis electron-pair concept is related to maxima of the squared wave function. <lb/>Or think of the arrangement of like spins in the valence shell as described by Lennard-Jones[46]. This was found <lb/>without topological approaches, but it can be nicely visualized using quantum Monte-Carlo methods. <lb/> Philip Coppens. Topological theory of the electron density as initially developed by Richard Bader constitutes a new <lb/>paradigm with a di↵erent language from that used up to that time. History is full of examples of new paradigms <lb/>encountering fierce opposition. But in this case the theory complements existing interpretive methods by adding new <lb/>concepts, such as the Laplacian, atomic basins bounded by zero flux surfaces, the source function and many others. As <lb/>Richard emphasized many times, the electron density is an observable. Consequently topological analysis has been <lb/>widely applied in X-ray Charge Density Analysis and greatly contributed to its results. Yes, new chemical concepts <lb/>have indeed been found and are utilized by application of the topological approach. They continue to be developed. <lb/> Piero Macchi. It could occur. Because chemical concepts are always produced by a particular theoretical model (one <lb/>for all: "isolobal analogy", coming from fragment molecular orbital theory), a topological approach may also give <lb/>rise to new concepts. However, one should make sure that concepts are really new, not simply the translation of the <lb/>old ones in a new language. <lb/> What is the status of a chemical concept within a topological approach? <lb/> Does the topological approach yield results concerning well-defined chemical concepts? or Is the output of the <lb/>topological approach identified with weakly defined chemical concepts? <lb/> Paul Ayers. Certainly the topological approach helps clarify and elucidate chemical concepts. I'm not sure about <lb/> " well-defined " chemical concepts. For example, Richard Bader's work certainly helped clarify the concept of chem-<lb/>ical bonding but I don't think the chemical bond is a well-defined concept (and perhaps it never shall be). Similarly, <lb/>ELF and related approaches help clarify the nature of " lone pairs " but these concepts, alas, are also perhaps not " well <lb/>defined " (in the sense that not everyone can agree on a definition). So in this sense, I think that the topological ap-<lb/>proach elucidates (and allows one to quantitatively define, at least in some cases) weakly-defined concepts. Perhaps <lb/>this is not always true: polarization and polarizability are concepts that everyone can agree on (I think, perhaps I <lb/>should not be so sanguine). I don't know if polarization has a topological signature but, if it did, then the topological <lb/>approach could be helpful for both weakly-defined and well-defined chemical concepts. Even so, the topological <lb/>approach seems to be mainly applied to weakly-defined concepts. <lb/> Mauro Causá. The topological approach (QTAIM, ELF and MPD) helps in distinquishing between " strong " and <lb/> " weak " chemical concept. I define as " weak " a chemical concept that is based only on chemical phenomenology and <lb/>that poorly resists to a strictly physical analysis. In the next future the topological approach, that is firmly based on <lb/>physical general concepts, could be used to easily eliminate "weak" chemical concepts. <lb/> <page>10 <lb/></page> Vladimir Tsirelson. The topological approach started from well-defined chemical concepts and it now mainly focuses <lb/>on " weakly defined chemical concepts " . That is a true way. Also, I agree with Paul that approach aims to reach the <lb/>quantitative level. Let me cite Richards Bader (private conversation in his home, Burlington, January of 1994): <lb/> " Vladimir, my top achievement is the application of [Schwinger] action principle to bounded atoms. I brought the <lb/>numbers to chemistry " . <lb/> Paolo Lazzeretti. The ill-defined concept of aromaticity can be dealt with on the " magnetic criterion " , allowing for <lb/>topological tools, e.g., the " stagnation graph " of the quantum-mechanical current density induced by an external mag-<lb/>netic field. The analysis of the singular points at which the graph lines branch out provides fundamental quantitative <lb/>information. <lb/> Ramon Carbó-Dorca. The topological approach seems to have provided many ways to look at the connection between <lb/>empirical chemical concepts and theoretical results. However, trying to deepen on such a connection can produce <lb/>more confusion than new insight, if no proper basis for the new point of view is present. Typical examples of this <lb/>situation are the chemical bond and the related concept of aromaticity. Has topology ameliorated the visualization <lb/>of chemical bond by means of Bader analysis, for instance? In many cases one can answer armatively, with some <lb/>exceptions. A new way to look at the density function has arisen from Bader's simple procedure and the resultant <lb/>graphical results have been printed in innumerable papers. But nothing really new has appeared. Has aromaticity <lb/>provided insight over molecular structure, or has been just an excuse to publish more of the same? Now aromaticity <lb/>is like an overdimensioned phantom, which haunts an incredibly vast segment of the literature. The concept has been <lb/>useful to extend benzene particular chemical behavior to many other, even quite dissimilar structures. Yet, has this <lb/>changed chemistry in a way to gain e↵ective information about molecules? <lb/> Patrick Bultinck. To me a well-defined concept would be an observable or at least something that leaves no room <lb/>for debate. A horse is a horse and is biologically well-defined. A unicorn is not and I have the impression too often <lb/>discussions go on and on, discussing whether a unicorn is brown or white. So my opinion is that topology gives <lb/>excellent insights in chemical concepts but do not provide a stronger definition of the concept. Once accepted that a <lb/>concept, however weakly defined from a hardcore point of view, might be of interest, topology allows us to understand <lb/>it better. But when you build a castle on quicksand, even if in the end the castle looks like the strongest one since <lb/>Camelot, it is still built on quicksand and might sink... However, a too strict stance may result in disconnecting with <lb/>the majority of chemists who feel comfortable with concepts while at the same time, this same rigorous attitude could <lb/>also forbid working with present concepts that may in the end result in observables or link very closely to them. <lb/> Shant Shahbazian. " Anything " emerging from topological approach is mathematically rigorous and well-defined <lb/>since the whole procedure has this character. If a concept is ambiguously defined like the " chemical bond " , then it <lb/>has no " unique " analogue in " entities " emerging from topological procedures. In such cases there is no " one-to-one " <lb/>correspondence between the ambiguously/weakly defined chemical concept and topological indicators. However, it <lb/>is feasible seeking for loose connections. For instance, topological indices derived from the topological analysis of <lb/>the QTAIM can be used to classify chemical bonds using " correspondence rules " [47]. Since subjective elements are <lb/>always involved in constructing correspondence rules, the emerging classifications always have vague " borderlines " . <lb/>Thus, the initial floppiness remains even after " quantification " [48]. <lb/> Miquel Solà. There is not a unique way to define atomic partitions and all of them have its own limitations. Conse-<lb/>quently, when we define a chemical concept within a topological approach the result is always at certain point arbitrary <lb/>because the outcome is determined by the partition. The degree of arbitrariness depends also on the definition. If we <lb/>can impose certain physical properties to the quantity defined and the definition proposed obeys these properties, the <lb/>result can be a reasonably defined chemical concept. This is the case of the recent definition of local spin[49] that <lb/>provide a way to describe polyradicals in the framework of quantum mechanics. In other cases, the link between <lb/>the quantities derived from a topological analysis and the chemical definitions is fuzzier and the concept is weakly <lb/>defined. This is the case of the equivalence between the bond critical point and the chemical bond in the QTAIM <lb/>theory of Bader et al. Although this point is still controversial, evidences that such relationship is too stringent have <lb/>been accumulated over the years[50–56]. <lb/> <page>11 <lb/></page> I do not think that aromaticity as a chemical concept is more or less ill-defined that let's say the chemical bond <lb/>or the similarity between molecules. For some reason, the concept of aromaticity has a bad reputation (probably it <lb/>comes from the so many existing di↵erent possible measures), but there is no doubt that it is necessary to understand <lb/>the chemistry of many compounds. If you find an alternative way to understand why C 6 H 6 , [Al 4 ] 2– or [B 12 H 12 ] 2– are <lb/>particularly stable, then maybe we could abandon the aromaticity concept. It seems to me that we have not reached <lb/>this point yet. <lb/> Simon Grabowsky. Why do we need to justify the use of topological analysis with its connection to chemical con-<lb/>cepts? Why is it so important to discuss whether, e.g., the QTAIM bond path and the chemical bond relate to each <lb/>other closely or not or if a monosynaptic ELF/ELI basin really represents a lone pair? I prefer to make use of the <lb/>fact that the topological analysis of the electron density or the ELF/ELI indeed gives us numbers (as Richard Bader <lb/>stated, see Vladimir Tsirelson's comment) for statistical analysis within a physically meaningful framework. If the <lb/>outcomes of these statistical or correlation analyses help us to control, e.g., substituent e↵ects in a chemical system, <lb/>the behaviour of materials in di↵erent compositions, crystal packing e↵ects, reactivity etc. (and there are numerous <lb/>examples for successful applications of topological analysis in these di↵erent fields[47, 57–69]), then it doesn't really <lb/>matter how deep the philosophical connection to well-defined or weak chemical concepts is. In other words, we don't <lb/>need to find justifications for the use of topological analysis if it proves to be helpful for a certain application or <lb/>chemical question. <lb/> Carlo Gatti. To answer this question, we should first ask ourselves what is a " well-defined " and what is a " weakly-<lb/>defined " chemical concept. I do hardly believe to be able to trace a neat border between the two, nor I believe other <lb/>may do, despite their, admittedly useful, broad distinction. In particular, a chemical concept may be strong or weak <lb/>depending on the circumstances. The " chemical bond " is clearly a good example of what I'm trying to say. No <lb/>one will doubt about the existence of a C-C bond in ethane, although it will be described in di↵erent ways by the <lb/>various topological approaches. However, recent literature is crowded of endless discussions about the existence or <lb/>not of bonds in more delicate situations. When the chemical concept becomes fuzzy, the topological approach also <lb/>becomes fuzzy in its answers, but eventually it turns out to be definitely useful and informative. The answer one <lb/>obtains in such cases, commences to depend on the selected topological approach, on the kind of wave-function, on <lb/>the " quality " of the basis set. Or, on the experimental side, if we are dealing for instance with topological descriptors <lb/>based on electron-density derived quantities, the answer starts to depend on how the X-ray data have been sampled, <lb/>treated, modelled. . . .The topological approach may serve, therefore, as a tool to understand how " weakly defined " <lb/>is a chemical concept in given circumstances. It provides in those cases a lot of details, which may be useful to <lb/>quantify, order and classify situations which are by their nature hardly manageable and understandable in other ways. <lb/>Another interesting chemical concept is that of atomic shell structure. We know that there many suitable functions <lb/>able to qualitatively recover this structure and to be more or less close to the " ideal " one in terms of number of shells <lb/>and associated populations. In general, the description of the atomic shell structure by the topological approaches is <lb/>indeed quite robust. Still this concept, which is " strong " in terms of the principal quantum number, may become loose, <lb/>hence " weak " in terms of the various functions and related topological approaches, when the shells and the electronic <lb/>energy levels [typically, (n+1)s and (n)d for transition metal elements] start to be not so clearly di↵erentiated. The <lb/>fuzziness of the topological approach is such cases just reflects a change in the physics of the problem. Chemical <lb/>concepts, especially the " well-defined " ones, are quite often the cumulative result of the observation of the chemical <lb/>behaviour of many classes of systems, but usually under quite " standard " conditions. When applied to more " exotic " <lb/>situations (high external pressure is a very good case) topological approaches yield results that provide insights about <lb/>the robustness or not of a chemical concept or of " standard " chemical behaviour in conditions far out from equilibrium. <lb/>In such cases, new chemical concepts may be revealed by the topological approaches. For, instance, a simple example <lb/>is provided by the unexpected partially ionic behaviour of an element (B) under pressure[70]. <lb/> Paul Popelier. First, a short reaction (rather than answer) to the subquestions. Science should strive for the crispest <lb/>possible definition of a concept, especially when given enough time and especially in Chemistry, which studies a <lb/>complicated universe. In the long term I am therefore uncomfortable with " weakly defined " concepts, which could be <lb/>an excuse to do poor science, if no-one tries to strengthen the definition. I am not trying to oversimplify the chemical <lb/>universe but I defend clear thinking. If one tries hard enough " weakly defined chemical concepts " should eventu-<lb/>ally disappear (see also last part of my answer to Question 6). One of the prime outcomes of QTAIM is a modern, <lb/> <page>12 <lb/></page> computable and practical definition of an atom inside a system. Our work on a protein force field with multipolar <lb/>electrostatics now explores the addition of intra-atomic energies. Tests on oligopeptides show that this atomic energy <lb/>( " self energy " , which includes kinetic, Coulomb and exchange energy) for a carbon atom in tri-alanine, for example, <lb/>changes by only 0.6 kJmol 1 compared to the same carbon energy in penta-alanine ( 0.4 kJmol 1 for isoleucine and <lb/>threonine). This high degree of transferability shows that a topological atom is a useful high-information fragment <lb/>carrying a precise energetic fingerprint of its chemical environment. At the same time, this excellent energetic trans-<lb/>ferability shields the force field from the huge (non-chemical) energy released when an atom is formed from its bare <lb/>nucleus and electrons. As such, the topological atom forms a bridge between the physical Hamiltonian (written in <lb/>terms of non-chemical entities that are nuclei and electrons) and chemical entities (that are atoms). Topological atoms <lb/>are fragments that have unique kinetic energies[71] provided one stays within the " Laplacian family of local kinetic <lb/>energies " [5]. Secondly, the topological approach o↵ers a computable yet intuitive picture of chemical bonding but <lb/>this picture is not yet complete. Historically[11], QTAIM first defined the atomic energy and then, many years later, <lb/>the bond critical point. The meaning of the latter is still not fully established but a promising approach, in my opinion, <lb/>is linking it with energy partitioning, as pioneered[72] in 2007 and further worked out more recently[73]. <lb/> Ángel Martín Pendás. Are there well-defined concepts in chemistry? Maybe we should start with this. An observable <lb/>(Patrick) is well-defined, but is it a chemical concept? How many quantum-mechanical observables may be associated <lb/>to chemical concepts? I am afraid that the number is small, to be optimistic. Ordinary chemical thinking is plagued <lb/>with fuzzy concepts and ideas. From this point of view, attempts to give them a solid foundation should be gauged <lb/>using standard philosophical rules: minimality and closeness to the physical models which we believe rule the world <lb/>(i.e. quantum mechanics). In my opinion, topological schemes approach these rules better than, for instance, orbital <lb/>based reasoning. <lb/> Alexander Sax. Weakly defined chemical concepts are only obtained when the weak topology is used. <lb/> W. H. Eugen Schwarz. In the formative period of the chemical science, say before the late 18th century, chemistry <lb/>was a mainly qualitative science, with hardly any serious conservation law. (I may say so, although the synthetic <lb/>prescriptions contained mass and volume instructions, and although 'recycling' substances such as sulfur or copper <lb/>through what we now call oxidation and reduction were rated as important empirical evidences for the early chemical <lb/>theories.[74]) After the creation of the modern chemical science by the group around Lavoisier in Paris, a positivistic <lb/>period set in putting emphasis on numerical weight relations and strict connectivity relations, i.e. on numbers and <lb/>topology. Developing intuitive explanations with the help of hypotheses had a hard time. During the second half <lb/>of the 19th century, the development of thermodynamics, statistical mechanics and molecular mechanics, and half a <lb/>century later of quantum chemistry gave both, numbers and the first steps towards a causal understanding. Yet large <lb/>parts of the chemical community believe mainly in positive numbers, so that from time to time a minority cry can be <lb/>heard: don't give me (chemical) numbers, give me (physical) insight. <lb/>Concepts are created by scholars, thus the status of chemical concepts is determined by the personality and the <lb/>philosophical (pre)conceptions of these various individuals. Anyhow, typical chemical concepts throughout have two <lb/>di↵erent aspects, a quantitative hard side and a qualitative soft, fuzzy side. Geometric structure may be specified <lb/>by the numerical geometric parameters of the (quantum and thermally smeared) positions of the atomic nuclei or <lb/>qualitatively specifying the connectivity of the atomic core regions. Energetic stability may be accurately specified <lb/>by the numerical thermodynamic parameter set or qualitatively through the bond strengths or bond orders. Chemical <lb/>compounds are assumed to consist of conserved atoms hold more or less permanently together by bonds. The atoms <lb/>can be specified by their rather constant averaged isotopic weights, or by their rather variable spatial and charge dis-<lb/>tributions. Concerning the numerical quantitative aspects of chemical concepts, probably the topological approaches <lb/>can't contribute more. However topological approaches can become relevant for the qualitative weak fuzzy side of <lb/>the chemical concepts. <lb/>If topological approaches can contribute some numerical characterization indices, one may first check correlations <lb/>with other numerical parameters suggested in the same context. Main-component analysis can elucidate whether the <lb/>topological parameter(s) corroborate previous chemical views or add some new aspect. The discussion in the chemical <lb/>community will, hopefully after some years, come to a conclusion, which numerical parameters are most fruitful for <lb/>the chemical concept with a given name, and whether such a concept is linear one-dimensional and may be described <lb/> <page>13 <lb/></page> by a real number, or multidimensional to be describable by a point in a two-or three-dimensional parameter space <lb/>(such as polar solvent, basic ligand, aromatic molecule). <lb/>All chemical concepts are quantitative (numerical in some meaning parameter space), even if sounding qualitative <lb/>(yes-no, more or less: polar – covalent or aromatic – non-aromatic – anti-aromatic). There is a tendency to classify <lb/>chemical substances, chemical groups, chemical reactions etc. This is fine for getting a first overview and for intro-<lb/>ductory teaching, and it works well for the clear cut cases. Indices which are integer or yes-no are good for some <lb/>book-keeping. E.g. whether C is counted as tri-or tetra-valent in the C 2 molecule is not that relevant, what matters <lb/>are its spectroscopic and reactive properties. Topological indices that are of this discrete type are nice for the clear-cut <lb/>cases. Examples are the synapticity of an ELF region, the existence of an internuclear bond path or a non-nuclear <lb/>electron density maximum. Changing the molecular system a little, or changing the computational or experimental <lb/>approach a little, may change the topological index from two to one, or from one to zero. It makes little sense in such <lb/>border cases to define a numerically unstable criterion as a chemically relevant characterization. <lb/>An example is the appearance of the " non-nuclear attractors " in the molecular and crystal electron density dis-<lb/>tribution. Atomic matter follows wave mechanics. Localized states show a wavy exponentially decaying density. If <lb/>only a few states are occupied (as in light or electron-deficient atoms) the wavy nature of the density can be recovered <lb/>more or less easily. Local perturbations of a rather homogeneous electron gas induce the Friedel oscillations. If more <lb/>states with their density maxima at di↵erent positions are superimposed, it becomes more and more dicult to recover <lb/>density wave maxima. This is the case for heavy atoms: light atoms show an electron density shell structure, heavy <lb/>atoms only show the shell structure in the cores, no longer in the valence region. As Martin et al.[30] have shown, <lb/>non-nuclear electron density maxima are nearly ubiquitous though not for all internuclear distances. They appear <lb/>more or less accidentally and seem to have no specific meaning when they just show up at equilibrium bond lengths. <lb/>Similarly, there may occur bond paths in the electron density distribution sometimes rather accidentally, indicating <lb/>the near-formation of a bond, or just a non-bonded contact, looking from various chemical points of view[75, 76]. <lb/>Summarizing , the yardstick for chemical concepts is practical chemistry and the ordering desire for the vast mass <lb/>of phenomena. Chemical concepts can appear in two forms, as physically well defined, quantitative numerical ones, <lb/>and that's it. Or as 'typically chemical', soft, fuzzy ones. Here topological results, both of qualitative and quantitative <lb/>kind, can help on the way of clarifying the meaning, always keeping the chemical purpuses in mind. <lb/> Piero Macchi. The question is dicult to answer. A chemical concept may be fully embedded in the topological <lb/>approach or it may cross two or more theoretical frameworks. Therefore its status may substantially vary. <lb/>The most classical example is the chemical bond. While it is in principle a standalone concept within topological <lb/>methods, it normally embraces several approaches. This makes it not perfectly compatible with all interpretations, but <lb/>more general and universally understood. A concept like the bond order, instead, cannot be defined by the "classical" <lb/>quantum theory of atoms in molecules, although some indices may well reflect this concept. However this is an <lb/>example of the "translation" of a concept defined within molecular orbital theory into the framework of topological <lb/>approaches. <lb/>Sometime, concepts defined by two approaches are quite unrelated or even conflicting. For example, the stability <lb/>of a molecular graph is clearly defined in topology, but it has little analogy with other approaches and it has nothing <lb/>to do with the stability of a molecule, which is what the typical reader of chemical journals would understand. <lb/> Should topological approaches provide measurable quantities? <lb/> Paul Ayers. Yes. Ideally we should base our understanding on quantities that are, at least in principle, measurable. <lb/>Doing so forces a certain intellectual discipline, and keeps one discussions grounded in reality. But this is not always <lb/>practical; I would not wish to discard all of the (sometimes very useful) exceptions to this rule. E.g., the topology of <lb/>Hartree-Fock/Kohn-Sham orbitals is obviously very useful, even though these orbitals are manifestly unobservable. <lb/> Mauro Causá. Yes. Topological analysis will provide a quantitative measure of the chemical hypotheses. This target <lb/>will be reached for example using the ELF critical branching values or the MPD values of an electronic structure. <lb/> Vladimir Tsirelson. Yes, I think so. <lb/> <page>14 <lb/></page> Paolo Lazzeretti. Hopefully, yes. <lb/> Ramon Carbó-Dorca. I'm afraid to slightly disagree with the previous comments, if there is not added some nuance <lb/>into the question. If the basis of the topological approach is quantum mechanical, usually in terms of a density <lb/>function, then the only way to provide measurable, observable information is to define topological results by means <lb/>of a suitable hermitian operator acting on the density. Visualization of density features might be loosely connected by <lb/>such a claim. However aromaticity hardly meets such a condition. <lb/> David L. Cooper. The chemical concepts that continue to prove to be the most useful tend to be those that allow us <lb/>to rationalize important features of e.g. molecular properties and reactivity. Much as we might prefer to concentrate <lb/>on deriving measurable quantities from topological approaches, the reality is that some of the most useful chemical <lb/>concepts that have stood the test of time are far from easy to quantify. As such, it seems to me that it would be <lb/>somewhat limiting simply to reject outputs from topological approaches that are more qualitative, such as pictorial <lb/>representations of molecular electronic structure, for example. <lb/> Patrick Bultinck. As Paul W. Ayers says, we need intellectual discipline. As David L. Cooper implies: if we now <lb/>all of a sudden reject all age-old concepts that have guided and are guiding e.g., organic chemists in synthesis, we'll <lb/>be fools to think this will be accepted anytime soon. I do like the intellectual discipline and for me this also means <lb/>that we keep reminding ourselves that these concepts were never meant to be used as quantitatively as they are now. <lb/>There are no heaven-sent atoms in molecules and it is pointless to claim the atomic charge is too large or too small <lb/>with method x, y or z. More e↵ort should be devoted to at least understanding why the result is what it is. And <lb/>what if we base all our analysis on observables ? What with e.g., the ever so interesting underlying matrices as <lb/>in density/density matrix, property/property density matrix, ... What about interesting functions that -in my limited <lb/>knowledge-are not (yet) truly observable like the current density in an aromatic molecule ? As Paolo Lazzeretti wrote <lb/>in his paper on aromaticity based on observables: Electronic currents are not directly observable, but are related via <lb/>integral formulae to the magnetic properties[77]. This suggests that we need to leave some freedom. Current densities <lb/>are not observables but they are key in the Biot-Savart law that ultimately is key to NMR and thus current densities <lb/>lead to observables in a pretty direct way. So to conclude; maybe not all should be based on strict observables (as <lb/>Ramon Carbó-Dorca says: expectation values from Hermitian operators) but we need to be very cautious when we <lb/>might drift too far away from observables too. <lb/> Shant Shahbazian. The concept of measurable quantity (probably means " observable " here) is more problematic that <lb/>is usually assumed[25, 78]. So, the answer is an explicit " no " . Topological approach is a mathematical procedure that <lb/>can be used to decipher topological characteristics of both 3D scalar functions that are usually assumed to be measur-<lb/>able like charge densities as well as those that are assumed to have purely mathematical origin (though, personally, <lb/>I do not defend this distinction). What is really important is whether a topological procedure is e↵ective enough to <lb/>reveal characteristics that are " correlated " with chemical experiences. If yes, then the proposed topological procedure <lb/>is successful in its mission. <lb/> Miquel Solà. Ideally yes, but the problem is that any chemical concept defined within a topological approach gives <lb/>arbitrary results because the outcome depends on the partition and there is neither a unique way to define atomic <lb/>partitions nor a manner to decide which partition is better. By comparing the results obtained with the " chemically " <lb/>predicted results, in some cases it is possible to conclude that certain partition does a better job than others. For <lb/>instance, calculation of bond orders with the Mayer and fuzzy atom approaches are closer to the formally expected <lb/>bond orders than QTAIM delocalization indices[79]. <lb/> Simon Grabowsky. No, topological approaches definitely do not need to provide measurable quantities. And again, as <lb/>with the previous points, I closely agree with Shant Shahbazian. As an example, the great strength of the topological <lb/>approach is to be able to quantify bonding phenomena, which are inherently not measurable, and yet it provides a <lb/>deeper understanding of how bonding manifests itself on the sub-atomic electronic level than most experiments can. <lb/> <page>15 <lb/></page> Carlo Gatti. The electron density and related functions (e.g. its Laplacian) are observables (as properly defined <lb/>by Dirac and reminded by Ramon) and with some proviso also " measurable " quantities. Topological approaches <lb/>reveal values at and locations of their critical points, hence measurable quantities. But I do agree with David's view <lb/>that topological approaches should not be constrained to only such kind of quantities. For instance, one may use <lb/>topological approaches, applied to measurable quantities like the electron density and related functions, to assign <lb/>approximate values to the relative weights of the resonant forms (VB structures) describing a system and to judge <lb/>how such weights are slightly modified by a change of phase (e.g. from in vacuo to crystal) or of a substituent. <lb/>Although resonant forms and their weights clearly represent an hypothetical and idealized view of a system and all <lb/>but measurable quantities, they still provide tremendous chemical insight. Another example is that related to the value <lb/>of the dipole moment of a molecule in crystalline phase. Topological approaches (QTAIM) may provide this value <lb/>and also its often recovered magnitude increase upon crystallization. The molecular dipole moment in condensed <lb/>phase is not a measurable quantity, but still a very useful one for quantifying the importance of packing e↵ects and, in <lb/>general, the changes of the properties of a molecule when crystallized. <lb/> Jerzy Cioslowski. One has to be careful about the definition of measurable quantities. In principle, anything that is <lb/>derivable from the electronic wavefunction alone (i.e. derivable through a well-defined sequence of mathematical <lb/>operations that does not vary from one molecule to another and does not involve additional input) is measurable as <lb/>experimental measurement of electron density and energy suces to define the electronic Hamiltonian and identifies <lb/>the electronic state. Within this interpretation, even the exact KS orbitals are measurable as they can be obtained <lb/>through a well-defined mathematical manipulation of the wavefunction. On the other hand, any quantity employing <lb/>basis sets (such as the Mulliken charges) is not measurable as it relies on an additional input of basis functions. I guess <lb/>one should not frown upon such generalization as even the " honestly measurable " quantities such as electron density <lb/>are in fact not measured directly but derived from other data such as di↵raction intensities. <lb/> Paul Popelier. Measurement is a concept that can be stretched. Originally, it refers to what our human senses can <lb/>detect directly. Humans were able to expand their observable universe to see, for example, images of UV light or hear <lb/>ultrasonic sounds (which some animals can do naturally), and of course measure much more (see Herschel telescope). <lb/> " Enhanced measurement " can also occur on computed data. For example, a molecular dynamics simulation of a <lb/>crystal nucleation event in solution o↵ers data that no experiment can deliver. The set of system snapshots and their <lb/>properties then become observable. However, a major concern is that this simulation must be (suciently) realistic. In <lb/>other words, it must map onto reality in an accurate way. If not, interpretation will derail and Frenking's unicorns[80] <lb/>will appear. If the mapping to reality fails then one will make false observations. All this also applies to a theory: <lb/>if a theory maps onto reality accurately then the concepts this theory proposes also become part of reality. In that <lb/>sense, Newton's laws are real, and when pushed further, even the imaginary number i is real (no pun intended). So, <lb/>practically, this means that the topological approach needs to prove continuously that it maps onto observed reality <lb/>as accurately as possible. While doing so it creates, within its bosom, other observable quantities (in the enhanced <lb/>way described above). Carlo's example of a molecular dipole moment in the condensed phase nicely illustrates this <lb/>assertion. <lb/> Philip Coppens. Not necessarily. Theory is often used to extend experimental observations to obtain information of <lb/>importance that is not directly accessible to experimentation but provides further insight. With some exotic exceptions <lb/>orbitals can not be directly measured and can be defined in many di↵erent ways. But if an atomic connectivity and <lb/>structure are accurately known much information can be gained by now classical theoretical calculations, information <lb/>which may not be directly accessible otherwise.The same is true for topological analysis. <lb/> W.H.Eugen Schwarz. Yes, topological approaches should provide both numerical and qualitative graphical data, al-<lb/>ways furnished with the caveat that chemical compatibility and usefulness is granted. <lb/> Piero Macchi. Any approach that explains chemistry is intended to justify some observations. Therefore a connection <lb/>with measurements is inherent. <lb/> <page>16 <lb/></page> Is it possible to predict the outcome of a topological approach without performing a calculation on a computer? <lb/> Paul Ayers. I think that most experienced scientists know, before they perform a calculation, about what to expect. So <lb/>I think that often we can predict the answer before we do the calculation, and often a back-of-the-envelope calculation <lb/>suces. However, there are interesting cases where calculations, or even experiments, are helpful: think about using <lb/>accurately computed, or even experimentally measured, (spin-) electron densities to elucidate the bonding pattern in <lb/>complex materials. In those cases, the experiment/computation is done because we don't know what to expect. <lb/> Mauro Causà. A full knowledge of the results of a topological analysis can be reached only performing calculations <lb/>or measurements. QTAIM needs the detailed knowledge of electronic density, that in many cases can be reached. ELF <lb/>is not directly an observable, but is possible to fit at observed density laplacian to a model local kinetic term. MPD at <lb/> n level could be derived in principle from experimental reduced density of order n. On the other hand is possible to <lb/>think about a moderate transferabilty of topological quantities. <lb/> Vladimir Tsirelson. No. Unfortunately no. Of course the experienced worker certainly knows what he/she can expect <lb/>from calculations. However our fixed belief can play a spiteful joke... <lb/> Paolo Lazzeretti. Although a guess is sometimes possible, a thorough computational investigation is frequently <lb/>needed. <lb/> David L. Cooper. As well as helping with the rationalization of important features of e.g. molecular properties and <lb/>reactivity, we should seek outcomes of topological approaches that also allow useful predictions to be made for more <lb/>complex systems, preferably without the need to perform a full calculation for every system of potential interest. <lb/>Often, though, the results that are really interesting are those for which the outcome of the topological approach are <lb/>somewhat di↵erent from those that we anticipated. <lb/> Bernard Silvi. As far as qualitative or semi-quantitative chemically related properties are concerned the answer is <lb/>certainly yes. I see two reasons for that. On the one hand the axioms and the theorems of the topological model <lb/>constrain the output. On the other hand, Chemistry proposes a series of very ecient tools such as the Periodic Table, <lb/>the octet rule, electronegativity scales, Lewis's model, VSEPR rules which enable (together with chemical knowledge) <lb/>to anticipate the trends of the outcomes of a topological approach. Moreover, at least for QTAIM and ELF, a moderate <lb/>transferability is observed justifying cut and paste strategies as first steps of a study. <lb/> Patrick Bultinck. I think there is room for debate here. For a method to be useful, it is best if the outcome can be <lb/>roughly predicted using the back of an envelope. This is what most chemists are used to (and frankly, willing to do). <lb/>For theoreticians, as David L. Cooper says, it is interesting to study cases where the outcome is a but unexpected from <lb/>what was anticipated or appeared from the back of an envelope, but most chemists will not expect such accuracy in a <lb/>prediction. <lb/> Shant Shahbazian. The general answer to this question is " no " . Topological approach is usually used within context <lb/>of theories that aim to extract chemical information from intricate ab initio wavefunctions not from " model " simplified <lb/>wavefunctions derived from " paper and pencil " methods (like Hückel's approach). However, it is probable that for <lb/>an " expert " eye it is feasible to guess (but not predict!) the outcome a topological analysis with a high degree of <lb/>confidence. But, there are always " surprises " even for experts that exotic Molecular Graphs emerging occasionally <lb/>from the topological analysis of the QTAIM are the best examples. If a " paper and pencil " method is capable to <lb/>predict the outcome of a topological procedure, then what would be the motivation to introduce the procedure per se? <lb/> Julia Contreras-Garcia. I think that most of us know the answer to common systems without calculating it. What I <lb/>would say is even more interesting is that we can identify to which systems we know the answer, and for which ones <lb/>our intuition could fail. <lb/> <page>17 <lb/></page> Simon Grabowsky. Let me try to answer the question with a short story. During my PhD work with Prof. Luger in <lb/>Berlin, we investigated the degree of transferability of bond-topological and atomic properties derived from QTAIM <lb/>between di↵erent organic systems. After a while, we started a competition between the PhD students: Can we <lb/>guess the values of the electron density at the bond critical points and the atomic charges of a dipeptide within the <lb/>experimental uncertainties of the measured electron density? Of course, all of us could, because there is no unexpected <lb/>bonding situation in such simple organic systems. So we just needed some experience with the topological analysis of <lb/>similar systems to predict the values as accurately as a measurement can give them. Consequently, we asked ourselves <lb/>why we were actually pursuing these analyses if we don't learn new things about chemistry? The answer was that <lb/>we were developing library methods based on transferability assumptions that can quickly and reliably reconstruct <lb/>an electron density from building blocks in order to get better experimental geometries and derived properties, such <lb/>as interaction energies, that cannot be guessed[81, 82]. However, in my opinion performing topological analysis of <lb/>compounds in usual bonding situations in the ground state is not more than collecting stamps, but the interesting <lb/>applications involve method development and chemical problems. <lb/> Carlo Gatti. I always suggest my students to guess (or roughly " predict " ) the outcome they expect before making a <lb/>calculation. This holds true also when they make use of a topological approach. And I'm clearly obliged to do the <lb/>same, myself! Besides being the only way to easily uncover possible computational mistakes, this attitude helps to <lb/>understand the outcome, to appreciate its smaller or larger deviation from the guessed one, and even more importantly, <lb/>to be able to recognize a possibly unexpected result. Which, as it has been already said by many of you, is clearly <lb/>one of the main reasons motivating the use of topological approaches and of why we keep have fun working with <lb/>them... That is to exploit a multi-facets method which in usual conditions recovers what is already known, boring <lb/>and expected, but that becomes really insightful and sometimes definitely surprising when the standard chemical <lb/>paradigms fail or become uncertain or inapplicable. <lb/> Jerzy Cioslowski. Fortunately, it is usually not possible to predict the outcome of calculations with absolute certainty. <lb/>I write " fortunately " because this fact keeps us employed (and comfortably so...). <lb/> Paul Popelier. My answer is yes but then one should question the value of such a prediction because a back-of-<lb/>the-envelope prediction is only coarse. An example from recent work[35] shows that the exchange energy between <lb/>two hydrogens in a [H-C. . . C-H] system in saturated hydrocarbons is consistently higher if this four-atom system is <lb/>planar. This phenomenon clearly appears in a plot of exchange energy versus H. . . H internuclear distance, in the set <lb/>of ethane, propane, butane and pentane. Without calculating the case of hexane one predicts correctly that, also here, <lb/>the planar configuration will again yield a larger H. . . H exchange energy than in a non-planar configuration, for the <lb/>same internuclear distance. This case study delivered a clear trend and hence a rule. This rule can be applied to cases <lb/>outside the original set of observations and hence make a prediction without computation. However, the topological <lb/>approach should be used more at the cutting edge of Chemistry, where open discussions occur that can decide on the <lb/>performance and even validity of topological versus non-topological methods. <lb/> Ángel Martín Pendás. I think that we all agree that no scientific field acquires that status until it becomes systematized. <lb/>This means that someone was able to find order in the raw data. Chemistry is a successful, predictive science, so <lb/>anyone approaching it from the topological side should be able to " predict " qualitative-or semi-quantitatively the <lb/>outcome of a calculation beforehand in those cases that were used to forge the field. And no one should be able to <lb/>make such a prediction in what we today think as interesting systems. What is important is that the border between <lb/>those two sets moves in time to hold more and more boring (predictable without calculation) molecules as we improve <lb/>the topological machinery at our disposal. <lb/> W. H. Eugen Schwarz. First there are the computational as well as the mathematical and paper-pencil approaches. <lb/>Concerning the second ones, most questions can be directly answered, though so to say at the level of the Hueckel <lb/>approximation. Concerning computational, e.g. QTAIM or ELF investigations, on the other hand, an experienced <lb/>scientist should have a good feeling for the results, and should mistrust the computation if it deviates strongly. Of <lb/>course there are also the 'special cases' and the 'borderline cases', and there the results cannot be guessed but can <lb/>supply arguments, e.g. is there a bond, is there a multiple bond, what is the shape of the lone pair. <lb/> <page>18 <lb/></page> Piero Macchi. Yes, definitely. Calculations may prove or dismiss hypotheses, but they do not create themselves <lb/>hypotheses. <lb/> What are new domains for which topological approaches would be useful? <lb/> Paul Ayers. I feel that topological approaches have been broadly used as a tool for understanding chemical phenom-<lb/>ena, but not as much for actually making predictions. There are a few exceptions. For example, Samantha Jenkins <lb/>has found " missing topologies " for clusters and used that to identify new stable cluster geometries. In the zeolite <lb/>community, they're making an exhaustive (or at least nearly exhaustive) search for structures guided by " what is topo-<lb/>logically possible. " But shouldn't topology also be useful in more practical realms? Can topology be used to design <lb/>an appropriate catalyst? To screen possible drug molecules? I would like to believe that our tools can be used in this <lb/>way, but there is scant evidence to support my faith. <lb/> Mauro Causà. The topological approaches (QTAIM, ELF and MPD) can be useful to completely re-write the general <lb/>chemistry: that will be useful for complete the synthesis between phenomenological chemistry and molecular physics, <lb/>initiated by Linus Pauling. The topological analysis of the excited states is, to my knowledge, a totally unexplored <lb/>field: then topology could define new chemical descriptors, and may be become predictive. The other chemical <lb/>topology, as the topological analysis of bond networks, could be really predictive. <lb/> Vladimir Tsirelson. I can indicate at least two new domains for which topological approach would be useful. These <lb/>are its combinations with i)DFT and ii) information theory. I personally expect the wave of new results along these <lb/>scientific directions. <lb/> Paolo Lazzeretti. I am presently working on a " falsifiable " (in the Popper jargoon) definition of " delocalized current <lb/>density " induced by a uniform magnetic field in cyclic and non cyclic molecules. Tentatively, such a current density <lb/>would be defined as that flowing beyond a topological surface called " separatrix " , which separates the internal set of <lb/>branching points from a nodeless external portion. <lb/> Ramon Carbó-Dorca. My experience is very limited, but I will guess that for the moment these approaches have <lb/>been useful to develop several graphical techniques which might serve to illustrate the molecular electronic entourage, <lb/>sometimes with beautiful pictures. <lb/> Patrick Bultinck. Property predictions (as in some improved QSAR), predicting reaction outcome. Rewriting general <lb/>chemistry would be, indeed, most exciting but at least for me, that requires first sorting out loose ends. As I stated <lb/>earlier: QTAIM is a chemically guided topology that at least in two stages has chemical input. 1) concentrate on <lb/>critical points not associated to nuclei to define AIM and 2) reducing a zero integral condition to a condition of zero <lb/>integrands over the AIM boundary. <lb/> Shant Shahbazian. I am personally working on the extension of the QTAIM beyond the Born-Oppenhiemer paradigm <lb/>as well as for exotic species[83], I must emphasize that concomitant topological analysis is central to decipher " atoms <lb/>in molecules " (AIM) structure in these domains. Traditionally, these domains were not considered to be amenable <lb/>to AIM analysis so the topological approach of the extended QTAIM (termed multi-component QTAIM) is a novel <lb/>application of the topological analysis that unifies the AIM concept of a large (and traditionally irrelevant) classes of <lb/>systems (for a recent example see: [84]). Time-dependent topological analysis is another area that may have a bright <lb/>future when considering molecular dynamics and time-dependent Schrödinger equation for chemical reactions, even <lb/>applications beyond chemistry are foreseeable (for some interesting examples of nonorthodox applications of the ELF <lb/>analysis see: [85, 86]). <lb/> Simon Grabowsky. I believe that the various numerical descriptors and indicators that arise from topological analysis <lb/>can be used to learn more about reactivity properties if analysed along pseudo-reaction pathways. Polo, Silvi et al. <lb/>have done some very inspiring work in this resepct using the ELF[87, 88]. If these ideas can be combined with <lb/>crystallographic experiments in the tradition of Bürgi's structure correlation philosophy[38], we can expect a deeper <lb/>insight into reactive properties of simple molecules. <lb/> <page>19 <lb/></page> Julia Contreras-Garcia. I think there is one vast field in which topological approaches could have important con-<lb/>sequencies: inverse design. One example is materials. Understanding the microscopic reason behind interesting <lb/>properties is a straight way towards imaterial design. Another nice example is drugs. I think that one of the greatest <lb/>potentialities of topology is that it focuses on local properties, so that if we were able to make the link topology-<lb/>property/energy, we would not need to worry for the size of the system since we would just care for the local part of <lb/>the protein-ligand interaction. <lb/> Carlo Gatti. I'm working on the extension of the source function concept to electron spin densities, which, should <lb/>eventually help in elucidating the mechanisms of the transmission of electron spin information (through bond, through <lb/>space, with ferro or anti-ferro " coupling " . . . ) in complex molecular systems and materials. I've presented preliminary <lb/>results at a Gordon Research conference last year and I've shown as an interesting side-product , the rich information <lb/>one obtains from the Laplacian of the spin density. My work is by now only on calculated spin densities, but due to <lb/>recent development of spin-split multipole model refinement procedures[89] it will be hopefully possible to extend <lb/>it also to experimentally measured spin densities. On another side, it is well known that computing electron spin <lb/>densities for some systems, like the transition metal complexes containing non-innocent ligands, may be sometimes <lb/>a dicult task, with results that may largely depend on the adopted computational method[90]. Use of a topological <lb/>approach on spin densities and perhaps also of the source function concept extended to spin densities, might concur <lb/>in shedding light on such a large dependency. Moreover, although it is not a new domain, but just a recent one, I <lb/>believe that topological approaches will be still incredibly useful for understanding the unexpected new compounds, <lb/>along with their exotic bonding patterns that may be formed under high pressure. Rationalizing their observed phase <lb/>changes with pressure using topological approaches is also quite exciting. <lb/> Jerzy Cioslowski. Some time ago, Paul Mezey proposed partitioning of the space of internal coordinates of a molecule <lb/>into catchment regions[91] that is analogous to the partitioning of the Cartesian space into atoms in QTAIM. Within <lb/>the Born-Oppenheimer approximation, each catchment region defines a distinct chemical species. In my opinion, this <lb/>very interesting idea has not been given enough attention by the chemical community. <lb/> Paul Popelier. I can think of two big areas where the topological approach could make a di↵erence that will be <lb/>appreciated by a large community of chemists and material scientists. One big area is understanding in local terms <lb/>how matter assembles, whether a molecular crystal or a protein-ligand interaction. A topological energy partitioning <lb/>can detect and characterise hydrogen bonding in its widest sense but also pinpoint other interactions that have no name <lb/>yet. Much of Chemistry has become an " assembly science " and this is where much systematic work must be done. <lb/>Secondly, the other big area is that the interpretation and rationalisation of traditional chemistry, as a " single-molecule <lb/>science " , which needs to be revisited and " cleaned up " . For example, textbooks do not explain satisfactorily[92] why <lb/>the BF bond length is so short in BF 3 . A minimally constructed topological electronegativity scale (without the <lb/>assumptions[93] of the Electronegativity Equalisation Method or any other framework), is also highly desirable. <lb/>Finally, there are two types of research activity we should see more of. There should be more bottom-up research, <lb/>i.e. apply a topological quantity to a huge number of cases, observe trends and formulate a rule or concept. This <lb/>is how concepts and rules (old or novel) emerge naturally and rigorously connect to an underlying quantum reality. <lb/>In contrast, an example of top-down research is: here is a very old and vague concept called aromaticity that needs <lb/>to be quantified topologically (in spite of its definition changing every 30 years). If aromaticity ultimately boils <lb/>down to mere " benzenicity " then research should focus on a bottom-up strategy, looking for emerging " patterns <lb/>of remarkable delocalisation stability " , without worrying if those patterns coincide with aromaticity. Give these <lb/>patterns/concepts/rules new names and use them with confidence. <lb/>The second type of research activity is falsification. There are only a few comparative studies[36, 94, 95] between <lb/>topological and non-topological approaches but the full consequence of a fundamental di↵erence in interpretation <lb/>between them has not been followed up. For example, in B 2 H 6 , the value of the topological exchange energy between <lb/>the bridging hydrogens is about three times larger than that between B and B. In a private communication Roald <lb/>Ho↵mann states " . . . I noted the HH interaction between the bridging atoms in diborane, this is something new to <lb/>me; that there is some BB bonding is easier to understand. " Here is a clash between a topological method and a <lb/>non-topological one. Which one is right? Or is a falsehood simply complementary a truth? <lb/> <page>20 <lb/></page> Ángel Martín Pendás. I would add to Carlo on pressure. Without any new tools, the topological approaches di↵er <lb/>substantially from other interpretive tools at hand in providing answers wherever preconceptions do not work. Our <lb/>naïve chemical rules do only apply to a very narrow temperature and pressure regime amenable to experimentation. <lb/>The rest is terra incognita. Finding new chemical rules valid in extreme thermodynamic conditions (think of planetary <lb/>science, for instance) may well become a golden territory for topological thinking. <lb/> Philip Coppens. Molecular processes in chemistry and biology are now becoming accessible on timescales down to <lb/>tenths of femtoseconds as a result of the dramatic increasing power and time-resolution of light sources, which is <lb/>continuing. This is a field that to my knowledge has not been addressed by topological examination. How does the <lb/>molecular topology change during the early stages of a chemical reaction? Can dynamic changes in the topology of <lb/>the electron density provide insight in the course of a chemical reaction? How does it change at conical intersections? <lb/>Can time-dependent topology be developed? <lb/> W. H. Eugen Schwarz. The chemical topological approaches have so far largely been limited to the investigation of <lb/>stationary systems. But chemistry is 'changing the substances', by nuclear rearrangements (chemical reactions) or <lb/>by electronic rearrangements (photochemistry, electronic excitation). Therefore one open field of development is the <lb/>investigation of the path from the initial to the final state, with two aims of the research: i) describing the reaction <lb/>path, ii) understanding, i.e. explaining, why the stationary product has this electronic-geometric structure. <lb/> Piero Macchi. Topological approaches are useful not only to rationalize the electron density distribution but in many <lb/>other areas of chemistry, in particular structural chemistry. Among the best applications are the classifications of <lb/>extended solids, the predictions of conversion paths in solid state reactions, the rationalization of bulk properties <lb/>in crystals. It is dicult to anticipate which domain will take more advantage or in which domain the topological <lb/>approaches will produce more success. Often this depends on the ability of the scientists who apply them to commu-<lb/>nicate new findings and how rapidly these are received by the scientific community. <lb/> </body> <back> <div type="acknowledgement">Acknowledgements <lb/> Andreas Savin and Bernard Silvi. thanks the editors and the participants of this strange object. <lb/></div> <listBibl> References <lb/> [1] S. Shaik, H. S. Rzepa, R. Ho↵mann, One molecule, two atoms, three views, four bonds?, Angew. Chem. Int. Ed. Engl. 52 (2013) 3020–3033. <lb/>[2] E. V. Babaev, Intuitive chemical topology concepts, in: D. Bonchev, R. Rouvray (Eds.), Chemical Topology: Introduction and Fundamentals, <lb/>Gordon and Breach, Reading, 1999, pp. 167–264. <lb/>[3] J. A. Cramers, K. S. Booksh, Chaos theory in chemistry and chemometrics: a review, Journal of Chemiometrics 20 (2006) 447–454. <lb/>[4] K. Ruedenberg, The physical nature of the chemical bond, Rev. Mod. Phys. 34 (1962) 326–376. <lb/>[5] J. S. M. Anderson, P. W. Ayers, J. I. Rodriguez Hernandez, How ambiguous is the local kinetic energy?, The Journal of Physical Chemistry <lb/>A 114 (33) (2010) 8884–8895. <lb/>[6] R. F. W. Bader, P. M. Beddall, Virial field relationship for molecular charge distributions and the spatial partitioning of molecular properties, <lb/>J. Chem. Phys. 56 (1972) 3320–3329. <lb/>[7] P. Cassam-Chenaï, D. Jayatilaka, Some fundamental problems with zero flux partitioning of electron densities, Theor. Chim. Acta (Berlin) <lb/>105 (2001) 213–218. <lb/>[8] R. F. W. Bader, P. M. Beddall, J. Peslak, Jr., Theoretical development of a virial relationship for spatially defined fragments of molecular <lb/>systems, J. Chem. Phys. 58 (1973) 557–566. <lb/>[9] M. Causà, A. Savin, B. Silvi, Atoms and bonds in molecules and chemical explanations, Foundation of Chemistry 13 (2014) 2–26. <lb/>[10] P. L. A. Popelier, Solving the Schroedinger Equation: Has Everything Been Tried?, Imperial College Press, London, 2011. <lb/>[11] P. L. A. Popelier, The quantum theory of atoms in molecules, in: G. Frenking, S. Shaik (Eds.), The Nature of the Chemical Bond Revisited, <lb/>Wiley-VCH, Weinheim, 2014, Ch. 8, pp. 271–308. <lb/>[12] P. L. A. Popelier, F. Aicken, Atomic properties of amino acids: Computed atom types as a guide for future force-field design, ChemPhysChem <lb/>4 (2003) 824–829. <lb/>[13] P. L. A. Popelier, Quantum chemical topology: on bonds and potentials. in structure and bonding, in: D.J.Wales (Ed.), Structure and Bonding. <lb/>Intermolecular Forces and Clusters, Vol. 115, Springer, Heidelberg, 2005, pp. 1–56. <lb/>[14] A. Scemama, P. Chaquin, M. Ca↵arel, Electron pair localization function: A practical tool to visualize electron localization in molecules from <lb/>quantum monte carlo data, J. Chem. Phys. 121 (2004) 1725–1735. <lb/>[15] A. Scemama, M. Ca↵arel, R. Chaudret, J.-P. Piquemal, Electron pair localization function (eplf) for density functional theory and ab initio <lb/>wave function-based methods: A new tool for chemical interpretation, J. Chem. Theory Comput. 7 (2011) 618–624. <lb/> <page> 21 <lb/></page> [16] A. Savin, Probability distributions and valence shells in atoms, in: K. D. Sen (Ed.), Reviews of Modern Quantum Chemistry: A Celebration <lb/>of the Contributions of Robert G.Parr, Vol. I, World Scientific, Singapore, 2002, pp. 43–62. <lb/>[17] A. Scemama, M. Ca↵arel, A. Savin, Maximum probability domains from quantum monte carlo calculations, J. Comput. Chem. 28 (2007) <lb/>442. <lb/>[18] C. S. M. Allan, H. S. Rzepa, Chiral aromaticities. AIM and ELF critical point and nics magnetic analyses of mobius-type aromaticity and <lb/>homoaromaticity in lemniscular annulenes and hexaphyrins, J. Org. Chem. 73 (17) (2008) 6615–6622. <lb/>[19] H. S. Rzepa, The chiro-optical properties of a lemniscular octaphyrin, Org. Lett. 11 (2009) 3088–3091. <lb/>[20] S. M. Rappaport, H. S. Rzepa, Intrinsically chiral aromaticity. rules incorporating linking number, twist, and writhe for higher-twist möbius <lb/>annulenes, J. Am. Chem. Soc. 130 (2008) 7613–7619. <lb/>[21] R. J. Boyd, K. E. Edgecombe, Atomic and group electronegativities from the electron-density distributions of molecules, Journal of the <lb/>American Chemical Society 110 (1988) 4182–4186. <lb/>[22] R. J. Boyd, S. L. Boyd, Group electronegativities from the bond critical point model, Journal of the American Chemical Society 114 (5) <lb/>(1992) 1652–1655. <lb/>[23] R. S. Mulliken, Molecular scientists and molecular science: Some reminiscences, J. Chem. Phys. 43 (1965) S2–S11. <lb/>[24] R. G. Parr, Density-functional theory in chemistry, in: R. M. Dreizler, J. da Providencia (Eds.), Density Functional Methods in Physics, <lb/>NATO ASI B Series, Plenum, New York, 1985, p. 141. <lb/>[25] S. Shahbazian, Letter to the editor: Are there " really " atoms in molecules?, Foundations of Chemistry 16 (1) (2014) 77–84. <lb/>[26] U. Koch, P. L. A. Popelier, Characterization of c-h-o hydrogen bonds on the basis of the charge density, J. Phys. Chem. 99 (1995) 9747–9754. <lb/>[27] C. Gatti, F. Cargnoni, L. Bertini, Chemical information from the source function, J. Comput. Chem. 24 (2003) 422–436. <lb/>[28] C. Gatti, P. Fantucci, G. Pacchioni, Charge-density topological study of bonding in lithium clusters.1. planar li n clusters (n=4,5,6), Theor. <lb/>Chim. Acta (Berlin) 72 (1987) 433–458. <lb/>[29] W. L. Cao, C. Gatti, P. J. MacDougall, R. F. W. Bader, On the presence of nonnuclear attractors in the charge distribution of li and na cluster, <lb/>Chem. Phys. Lett. 141 (1987) 380. <lb/>[30] A. Martín Pendás, M. A. Blanco, A. Costales, P. Mori Sánchez, V. Luaña, Non-nuclear maxima of the electron density, Phys. Rev. Lett. 83 <lb/>(1999) 1930–1933. <lb/>[31] Y. Ma, M. Eremets, A. R. Oganov, Y. Xie, I. Trojan, S. Medvedev, A. O. Lyakhov, M. Valle, V. Prakapenka, Transparent dense sodium, <lb/>Nature 458 (2009) 182–185. <lb/>[32] G. K. H. Madsen, C. Gatti, B. B. Iversen, L. Damjanovic, G. D. Stucky, V. I. Srdanov, F center in sodium electrosodalite as a physical <lb/>manifestation of a non-nuclear attractor in the electron density, Phys. Rev. B 59 (1999) 12359–12369. <lb/>[33] P. Mori-Sánchez, J. M. Recio, B. Silvi, C. Sousa, A. Martín Pendás, V. Luaña, F. Illas, The rigorous characterization of mgo f centers as <lb/>pseudoatoms., Phys. Rev. B 66 (2002) 075103. <lb/>[34] Q. K. Timerghazin, G. H. Peslherbe, Electronic structure of the acetonitrile and acetonitrile dimer anions: A topological investigation, The <lb/>Journal of Physical Chemistry B 112 (2) (2008) 520–528. <lb/>[35] M. García-Revilla, E. Francisco, P. L. A. Popelier, A. Martín Pendás, Domain-averaged exchange-correlation energies as a physical under-<lb/>pinning for chemical graphs, ChemPhysChem 14 (6) (2013) 1211–1218. <lb/>[36] A. Martín Pendás, M. A. Blanco, E. Francisco, The nature of the hydrogen bond: A synthesis from the interacting quantum atoms picture, <lb/>The Journal of Chemical Physics 125 (18) (2006) 184112. <lb/>[37] J. Meister, W. H. E. Schwarz, Principal components of ionicity, The Journal of Physical Chemistry 98 (33) (1994) 8245–8252. <lb/>[38] T. Bürgi, J. Dunitz, Structure Correlation, VCH, Weinheim, 1994. <lb/>[39] C. Reichardt, Solvents and Solvent E↵ects in Organic Chemistry, VCH, Weinheim, 1988. <lb/>[40] I. Fleming, Frontier orbitals and organic chemical reactions, John Wiley and sons, New York, 1976. <lb/>[41] J. Neuss, Aromatizitsät: Geschichte und mathematische Analyse eines fundamentalen chemischen Begri↵s, Hyle publications, Karlsruhe, <lb/>2002. <lb/>[42] R. F. W. Bader, On the non-existence of parallel universes in chemistry, Found. Chem. 13 (2011) 11–37. <lb/>[43] R. F. W. Bader, Worlds apart in chemistry: A personal tribute to j. c. slater, J. Phys. Chem. A 115 (2011) 12667–12676. <lb/>[44] J.-B. Liu, W. H. E. Schwarz, J. Li, On two di↵erent objectives of the concepts of ionic radii, Chemistry -A European Journal 19 (44) (2013) <lb/>14758–14767. <lb/>[45] A. Lüchow, R. Petz, Single electron densities: A new tool to analyze molecular wavefunctions, J. Comput. Chem. 32 (2011) 2619–2626. <lb/>[46] J. E. Lennard-Jones, New ideas in chemistry, Advancement of Science 11 (1954) 136–148. <lb/>[47] P. Macchi, A. Sironi, Chemical bonding in transition metal carbonyl clusters: complementary analysis of theoretical and experimental electron <lb/>densities., Coord. Chem. Rev. 238-239 (2003) 383–412. <lb/>[48] C. Foroutan-Nejad, S. Shahbazian, R. Marek, Toward a consistent interpretation of the QTAIM: Tortuous link between chemical bonds, <lb/>interactions, and bond/line paths, Chemistry -A European Journal 20 (2014) 10140–10152. <lb/>[49] E. Ramos-Cordoba, E. Matito, I. Mayer, P. Salvador, Toward a unique definition of the local spin, Journal of Chemical Theory and Computa-<lb/>tion 8 (4) (2012) 1270–1279. <lb/>[50] L. J. Farrugia, C. Evans, M. Tegel, Chemical bonds without " chemical bonding " ? a combined experimental and theoretical charge density <lb/>study on an iron trimethylenemethane complex, The Journal of Physical Chemistry A 110 (25) (2006) 7952–7961. <lb/>[51] J. Poater, M. Solà, F. M. Bickelhaupt, Hydrogen-hydrogen bonding in planar biphenyl, predicted by atoms-in-molecules theory, does not <lb/>exist, Chemistry -A European Journal 12 (10) (2006) 2889–2895. <lb/>[52] J. Poater, M. Solà, F. M. Bickelhaupt, A model of the chemical bond must be rooted in quantum mechanics, provide insight, and possess <lb/>predictive power, Chemistry -A European Journal 12 (10) (2006) 2902–2905. <lb/>[53] T. Strenalyuk, A. Haaland, Chemical bonding in the inclusion complex of he in adamantane (He @ adam): The origin of the barrier to <lb/>dissociation, Chemistry -A European Journal 14 (33) (2008) 10223–10226. <lb/>[54] S. Grimme, C. Mück-Lichtenfeld, G. Erker, G. Kehr, H. Wang, H. Beckers, H. Willner, When do interacting atoms form a chemical bond? <lb/>spectroscopic measurements and theoretical analyses of dideuteriophenanthrene, Angewandte Chemie International Edition 48 (14) (2009) <lb/> <page> 22 <lb/></page> 2592–2595. <lb/> [55] S.-G. Wang, Y.-X. Qiu, W. H. E. Schwarz, Antibond breaking-the formation and decomposition of he @ adamantane: Descriptions, explana-<lb/>tions, and meaning of concepts, Chemistry -A European Journal 16 (30) (2010) 9107–9116. <lb/>[56] J. R. Lane, J. Contreras-García, J.-P. Piquemal, B. J. Miller, H. G. Kjaergaard, Are bond critical points really critical for hydrogen bonding?, <lb/>Journal of Chemical Theory and Computation 9 (8) (2013) 3263–3266. <lb/>[57] G. A. Je↵rey, J. F. Piniella (Eds.), The Application of Charge Density Research to Chemistry and Drug Design, NATO ASI Series, Vol. 250, <lb/>Plenum Press, New York, 1991. <lb/>[58] P. Coppens, X-Ray Charge Densities and Chemical Bonding, International Union of Crystallography Texts on Crystallography, Oxford <lb/>University Press, Oxford, 1997. <lb/>[59] C. F. Matta, R. J. Boyd (Eds.), Wiley-VCH, Weinheim, 2007. <lb/>[60] C. Gatti, P. Macchi (Eds.), Modern Charge Density Analysis, Springer-Verlag, Berlin, 2012. <lb/>[61] D. Stalke (Ed.), Electron Density and Chemical Bonding I, Experimental Charge Density Studies, Structure and Bonding, Vol. 146, Springer-<lb/>Verlag, Berlin, 2012. <lb/>[62] T. S. Koritsanszky, P. Coppens, Chemical applications of x-ray charge-density analysis, Chem. Rev. 101 (2001) 1583–1628. <lb/>[63] W. Scherer, G. S. McGrady, Agostic interactions in d 0 metal alkyl complexes, Angew. Chem. Int. Ed. 43 (2004) 1782–1806. <lb/>[64] C. Gatti, Chemical bonding in crystals: new directions, Zeit. Kristallogr. 220 (2005) 399–457. <lb/>[65] P. Luger, Fast electron density methods in the life sciences-a routine application in the future?, Org. Biomol. Chem. 5 (2007) 2529–2540. <lb/>[66] D. Stalke, Meaningful structural descriptors from charge density, Chem. Eur. J. 17 (2011) 9264–9278. <lb/>[67] S. Grabowsky, D. Jayatilaka, R. F. Fink, T. Schirmeister, B. Engels, Can experimental electron-density studies be used as a tool to predict <lb/>biologically relevant properties of low-molecular weight enzyme ligands?, Z. Anorg. Allg. Chem. 639 (2013) 1905–1921. <lb/>[68] M. S. Schmøkel, J. Overgaard, B. B. Iversen, Experimental electron density studies of inorganic materials, Z. Anorg. Allg. Chem. 639 (2013) <lb/>1922 1932. <lb/>[69] A. Savin, R. Nesper, S. Wengert, T. F. Fässler, ELF: The electron localization function, Angew. Chem. Int. Ed. Engl. 36 (1997) 1809–1832. <lb/>[70] A. R. Oganov1, J. Chen, C. Gatti, Y. Ma, Y. Ma, C. W. Glass, Z. Liu, T. Yu, O. O. Kurakevych, V. L. Solozhenko, Ionic high-pressure form <lb/>of elemental boron, Nature 457 (2009) 863–868. <lb/>[71] T. L. Fletcher, S. M. Kandathil, P. L. A. Popelier, The prediction of atomic kinetic energies from coordinates of surrounding atoms using <lb/>kriging machine learning, Theor. Chem. Acc. 133 (2014) 1499:1–10. <lb/>[72] A. Martín Pendás, E. Francisco, M. A. Blanco, C. Gatti, Bond paths as privileged exchange channels, Chemistry -A European Journal 13 (33) <lb/>(2007) 9362–9371. <lb/>[73] V. Tognetti, L. Joubert, On the physical role of exchange in the formation of an intramolecular bond path between two electronegative atoms, <lb/>The Journal of Chemical Physics 138 (2) (2013) 024102. <lb/>[74] K. Ruedenberg, W. Schwarz, Three millennia of atoms and molecules, in: E. Storm, A. Wilson (Eds.), Pioneers in Quantum Chemistry, ACS <lb/>Symposium Series, Vol. 1122, ACS, Washington DC, 2013, pp. 1–45. <lb/>[75] W. H. E. Schwarz, H. Schmidbaur, Observations and descriptions versus explanations-an example: Does nature, does theory know about <lb/>steric hindrance?, Chemistry -A European Journal 18 (15) (2012) 4470–4479. <lb/>[76] S.-G. Wang, Y.-X. Qiu, W. H. E. Schwarz, Bonding or nonbonding? description or explanation? " confinement bonding " of He @ adamantane, <lb/>Chemistry -A European Journal 15 (24) (2009) 6032–6040. <lb/>[77] P. Lazzeretti, Assessment of aromaticity via molecular response properties, Phys. Chem. Chem.Phys. 6 (2004) 217–223. <lb/>[78] S. Shahbazian, M. Zahedi, The role of observables and non-observables in chemistry: A critique of chemical language, Foundations of <lb/>Chemistry 8 (1) (2006) 37–52. <lb/>[79] E. Matito, J. Poater, M. Solà, M. Duran, P. Salvador, Comparison of the AIM delocalization index and the mayer and fuzzy atom bond orders, <lb/>The Journal of Physical Chemistry A 109 (43) (2005) 9904–9910. <lb/>[80] G. Frenking, A. Krapp, Unicorns in the world of chemical bonding models, Journal of Computational Chemistry 28 (1) (2007) 15–24. <lb/>[81] S. Grabowsky, R. Kalinowski, M. Weber, D. Förster, C. Paulmann, P. Luger, Transferability and reproducibility in electron-density studies <lb/>– bond-topological and atomic properties of tripeptides of the type l-alanyl-X-l-alanine, Acta Crystallographica Section B 65 (4) (2009) <lb/>488–501. <lb/>[82] B. Dittrich, C. B. Hübschle, K. Pröpper, F. Dietrich, T. Stolper, J. J. Holstein, The generalized invariom database (GID), Acta Crystallograph-<lb/>ica Section B 69 (2) (2013) 91–104. <lb/>[83] S. Shahbazian, Beyond the orthodox QTAIM: motivations, current status, prospects and challenges, Foundations of Chemistry 15 (2013) <lb/>287–302. <lb/>[84] M. Goli, S. Shahbazian, Deciphering the "chemical" nature of the exotic isotopes of hydrogen by the MC-QTAIM analysis: the positively <lb/>charged muon and the muonic helium as new members of the periodic table, Phys. Chem. Chem. Phys. 16 (2014) 6602–6613. <lb/>[85] T. Burnus, M. A. L. Marques, E. K. U. Gross, Time-dependent electron localization function, Phys. Rev. A 71 (2005) 010501. <lb/>[86] P.-G. Reinhard, J. A. Maruhn, A. S. Umar, V. E. Oberacker, Localization in light nuclei, Phys. Rev. C 83 (2011) 034312. <lb/>[87] V. Polo, J. Andres, S. Berski, L. R. Domingo, B. Silvi, Understanding reaction mechanisms in organic chemistry from catastrophe theory <lb/>applied to the electron localization function topology, The Journal of Physical Chemistry A 112 (31) (2008) 7128–7136. <lb/>[88] V. Polo, P. Gonzalez-Navarrete, B. Silvi, J. Andrés, An electron localization function and catastrophe theory analysis on the molecular <lb/>mechanism of gas-phase identity s n 2 reactions, Theor. Chem. Acc. 120 (2008) 341–349. <lb/>[89] M. Deutsch, N. Claiser, S. Pillet, Y. Chumakov, P. Becker, J.-M. Gillet, B. Gillon, C. Lecomte, M. Souhassou, Experimental determination of <lb/>spin-dependent electron density by joint refinement of X-ray and polarized neutron di↵raction data, Acta Crystallographica Section A 68 (6) <lb/>(2012) 675–686. <lb/>[90] K. Boguslawski, C. R. Jacob, M. Reiher, Can dft accurately predict spin densities? analysis of discrepancies in iron nitrosyl complexes, <lb/>Journal of Chemical Theory and Computation 7 (9) (2011) 2740–2752. <lb/>[91] P. G. Mezey, Catchment region partitioning of energy hypersurfaces, i, Theor. Chim. Acta (Berlin) 58 (1981) 309–330. <lb/>[92] R. J. Gillespie, Covalent and ionic molecules: Why are BeF 2 and AlF 3 high melting point solids whereas BF 3 and SiF 4 are gases?, Journal <lb/> <page> 23 <lb/></page> of Chemical Education 75 (7) (1998) 923. <lb/>[93] P. Bultinck, R. Vanholme, P. L. A. Popelier, F. De Proft, P. Geerlings, High-speed calculation of AIM charges through the electronegativity <lb/>equalization method, The Journal of Physical Chemistry A 108 (46) (2004) 10359–10366. <lb/>[94] A. Martín Pendás, M. A. Blanco, E. Francisco, Steric repulsions, rotation barriers, and stereoelectronic e↵ects: A real space perspective, <lb/>Journal of Computational Chemistry 30 (1) (2009) 98–109. <lb/>[95] E. Francisco, A. Martín Pendás, M. A. Blanco, A molecular energy decomposition scheme for atoms in molecules, J. Chem. Theory Comput. <lb/>2 (2006) 90–102. <lb/></listBibl> <page> 24 </page> </back> </text> </tei>