Geometric and Topological Thinking - Semantic Scholar · Geometric and Topological Thinking in Organic Chemistry ... young children often may fail to acknowl- ... figures in space.L71

Geometric and Topological Thinking in Organic Chemistry

By Nicholas J. Turro*

The beginning student of organic chemistry is often bewildered by what appears to be an enormous maze of random structural variations and reactions that can be mastered only by tedious memorization. To the organic chemist, however, the same subject is often a beauti- fully ordered discipline of elegant simplicity. An important value of learning organic chemistry is the mastering of “organic thinking,” an approach to intellectual processing whereby the “sameness” of many families of structures and reactions is revealed. This article offers the author’s personal views of organic thinking and explores the intellectual and scientific foundations of organic chemistry and of the powerful methods that provide the field with a platform for making rapid conceptual and experimental advances. It is proposed that these methods involve a geometric and topological approach to scientific reasoning within the framework of scientific paradigms that guide experimental design and execution. The basis of this approach is considered in relation to day-to-day thinking, problem solving, and the psychological drive for intellectual closure. The power of the approach is illustrated by the analysis of several photochemical and chemiluminescent reactions.

1. Introduction - Intellectual Processing

1.1. Intellectual Processing, Problem Solving, and Closure

Normal day-to-day mental activity and learning involve a high degree of intellectual processing directed toward problem solving. Difficulties arise with the recognition of ambiguities that result from the existence of many possible solutions. These ambiguities and the recognition of potential deficiencies or conflicts associated with the selection of a single solution can give rise to anxiety and tension. On the other hand, the process of “closing out” a problem in- tellectually is a pleasant and reinforcing experience.

Figure 1 schematically depicts some relationships among the components involved in intellectual processing.“,21 The phenomena (facts or events) in the world around us generate a set of beliefs. Intellectual processing involves preservation, rearrangement, or modification of these beliefs. Successful problem solving and concomitant closure reinforce beliefs and breed satisfaction. Failure to resolve problems breeds tension and conflict because of the implication that beliefs may be incomplete or incorrect. One may hypothesize that the dominant principle driving intellectual processes is a Law of Intellectual CIosure:121 intellectual processing is naturally driven toward as closed a mental state as circumstances permit. The origin of such a law could be traced to its survival value and the associated evolution of an appropriate genetic composition and constitution of the human brain.[31 The intellectual process of closure involves the goal of achieving a complete, stable, and self-consistent interpretation of a phenomenon or event. For example, an unfinished act tends to be mentally completed, a n ambiguous object tends to be interpreted in terms of its familiar aspects, a word, an object, or a situa-

[*I Prof. N. J . Turro Department of Chemistry, Columbia University New York, NY 10027 (USA)

tion is perceived in a manner that allows achievement of a closed interpretation.

BELIEFS MODIFIED OR - - - - - r------ it: REARRANGED

I I I I I

POSSIBILITIES CLOSURE ? ACTIVITY

INTELLECTUAL

PROCESSING

OPERATIONAL OPERATIONAL

Fig. I . A schematic representation of normxl i i i l c l l c~ tu , i l . IL~I\ 11) \ l~o\ \ i i lg [he interplay of beliefs, possibilities, closure, and intellectual processing (cf. Fig. 4).

Much of the learning experience consists of the process of resolving conflicts among related beliefs. Learning is fa cilitated and even made enjoyable by artificially creating intellectual conflicts that can be resolved to a high degree of closure. Students who learn to tolerate the tension that normally accompanies the process of resolving such intellectual conflicts often feel an excitement that is stimulating and rewarding in itself. Indeed, the development of an ability to tolerate tension during the activity of problem solving is important to the learning process. Intellectual processing that involves the achievement of problem definition and problem solution with a high degree of intellectual closure is naturally attractive and readily adopted. A

882 0 VCH Verlagsgesellschafi mbH, 0-6940 Wernherrn. 1986 0570-0833/86/1010-0882 $ 02.50/0 Angew. Chem. In(. Ed. Engl. 25 (1986) 882-901

major aim of many motivational programs for learning is to promote the visibility of “living examples” of successful behavior o r accomplishment and to suppress that of coun- terexamples.[’] This theme will reappear in the discussion of scientific paradigms (Section 3.3). Let us now consider two general and important classifications of intellectual processing that are employed in day-to-day thinking.

1.2. Concrete and Formal Operational Intellectual Processing

According to Piaget,l4] two distinct stages of intellectual development may be characterized, namely, concrete operational (typical of children from 6 to 12 years of age) and

formal operafional (typical of children from 1 1 to 16 years of age). I n Figure 1, these stages may be viewed as alternative mechanisms for intellectual processing. During the concrete operational period, intellectual processing involving the logic of real (concrete) objects is mastered. This period is characterized by accepting concrete objects or events in the immediate present with little extrapolation to what hypothetically is possible in the future or was possible in the past. During the formal operational period, intellectual processing involving the generation and manipulation of possible combinations of objects and events is mastered. I n this period, the mind becomes capable of extra- polating from objects and events in the immediate present to objects and events that are hypothetically possible.

At the concrete operational level, reality is accepted as it is found, perhaps without recognition that alternatives exist (for example, young children often may fail to acknowl- edge another’s point of view because they d o not consider the possibility that other points of view exist!). At the formal operational level, on the other hand, the recognition of competing possibilities often brings with it an ambiguity that may be psychologically unsettling. However, this un- comfortable feeling of ambiguity may be relieved and converted into a feeling of excitement by resorting to opera- tions that solve problems effectively and thereby achieve intellectual closure. Let us now consider how geometry can assist thinking at either the concrete operational or formal operational levels.

1.3. The Role of Geometry in Intellectual Processing

Many of the beliefs that are employed in intellectual processing may be viewed as containing information, which, in turn, can be represented by concrete or abstract

forms. According to Thorn,[’“] there is a natural tendency of the mind to give the form or shape of an object some in- trinsic meaning:

. . . our perceptual organs are genetically developed as to detect the living beings that play a large role, as prey or predators, in our survival and in the maintenance of our psychological equilibrium. It is clear that some forms have special value for us or are biologically important, for example, the shapes of foods, of animals, of tools. These forms are genetically imprinted into our understanding of space, and ... are narrowly and strictly adapted to them.

During the process of evolution of the brain, it is plausible that, since forms of information (beliefs) were generated from the stimuli provided by the environment in which we are embedded, the recognition of forms in terms of a three-dimensional (3-D) geometry had survival value and led to the development of perceptual receptors in the brain that are particularly suited to embrace and process 3-D geometric forms.[31 If so, facile recognition of 3-D geometric forms is genetically embedded into our ability to in- tellectualize and to understand space. It is easy to appre- ciate, therefore, how the use of geometry, with its powerful methods for processing geometric structures and its logical and internally consistent mathematical basis, both of which facilitate closure, can take on enormous importance as a vehicle for intellectual processing.

The notion that geometric thinking involves only 3-D forms, which correspond to figures in Euclidean geometry, is too restrictive, however. The geometric thinking that the author has in mind is much more “elastic” and general than that allowed by the rules of Euclidean geometry. An appreciation of what is meant by this elastic geometry may be obtained by considering some aspects of a branch of mathematics termed topology.[”

2. Euclidean and Topological Geometry

2.1. Topology and Topological Geometry

Topology is a branch of mathematics concerned with the “sameness” of mathematical forms. Topology provides a basis for determining whether two mathematical forms are the same or not via a mapping procedure that attempts to place the topologically relevant properties of one form onto a second form.

Topology may be described as “rubber sheet” geometry.[61 This definition emphasizes the elasticity of the concepts of topology, which is concerned, in general, only with very fundamental geometric properties. Topological properties may be visualized as those geometric properties of a figure on a rubber sheet that are conserved upon twisting and stretching, such as the connectivity, the sequence, and the continuity of points. It is easy to visualize the con- tinuous mapping of a figure on a rubber sheet onto the new image of the figure generated as a result of elastic twisting or stretching. No topological features of the figure are created or destroyed by the twisting or stretching process. However, tearing or joining parts of the original figure is not allowed, since this creates new topological properties and does not conserve the initial topological properties.

The model of the geometric figure on a rubber sheet provides the flavor of topological geometry, which emphasizes the possible rather than the concrete, an important feature of formal operational thinking. Euclidean geometry, on the other hand, emphasizes figures exactly as they are perceived, the essence of concrete operational thinking. Topo- logical geometry considers geometric figures as they might be transformed by mapping procedures that conserve fundamental topological properties. To obtain a further appreciation for topological geometry, let us further compare

Anyrn Chem In1 Ed Engl 25 (1986) 882-901 883

its essential features to the more familiar Euclidean geometry.

2.2. Comparison of Euclidean and Topological Geometry

Euclidean geometry is the study of certain properties of figures in space.L71 To a geometrician not all properties of a figure are of interest, only the geometric properties. What is a pertinent geometric property? In Euclidean geometry the key idea is that of geometrically equivalent or congruent Jigures. Two figures are called congruent if an intellectual transformation or mapping allows one figure to be “placed on the other” so that the two figures exactly coincide in all geometric properties. A geometric property of a figure is a property shared by every congruent figure.

Topological geometry also involves the study of certain properties of figures in space. Not all properties of a figure are of interest, only the topological properties. What is a pertinent topological property? In topology the key idea is that of topologically equivalent or homeomorphic figures. Two figures are called homeomorphic if an intellectual transformation or mapping allows one figure to be “placed on the other” so that the two figures exactly coincide in all topological properties. A topological property is one shared by every homeomorphic figure.

In Euclidean geometry, how d o we “place one figure on another”? How can we move a figure? How can we conserve its geometric properties during the movement? The sameness or identity of geometric figures is established by transformations termed isometries. An isometry is an intellectual transformation that conserves the size and shape of a geometric figure. The three common isometries of Eucli- dean geometry are rotation, translation, and reflection. Shapes that are recognized to be the same by isometric transformations are termed congruent. In Euclidean geometry, we are allowed to move a figure only by applying motions that d o not change the distance and angle relationships between any two points of the figure. Thus, the geometric properties of interest are those that are invariant during such motions. Euclidean geometric figures are characterized by rigidity owing to their characteristic metric (measurable) properties, such as the lengths and areas of sides and the angles between sides. A requirement of Euclidean figures i s that motion of the figure in three- dimensional space cannot change its metric properties.

Euclidean figures may be similar but not identical, i.e., two figures may have the same shape but different sizes.

A B C ti& 1 < ongruciii ( A .ind 8 ) drid \rmiiifr ( A And C , B and C) Euclidean geometric figurea

For example, A (Fig. 2) is a square whose sides are the same length as those of B. A and B are the same figure in

Euclidean geometry, because they are congruent. C is a smaller square which is similar but not congruent to A and B .

Topological figures may be of different size and shape yet topologically equivalent (i.e., homeomorphic). For example, D, E, F, and G (Fig. 3) are all homeomorphic, because the points of any one figure may be continuously mapped onto those of the others by elastic motions that conserve sequence and connectivity. The differing size and shape of the figures is of no topological importance. Thus, D, E, F, and G are the same figures in topological geometry, because they are homeomorphic. Topological geometric figures are characterized by elasticity and complete lack of fixed metric properties such as lengths, areas, and angles. In terms of the rubber sheet analogy, elastic distortions of such figures do not change topological geometric properties such as the connectivity of points or the existence of an inside and an outside. Thus, in spite of the wide variation in appearance, D , E, F, and G are the same figures topologically in that they are closed simple curves, having an inside, an outside, and a boundary.

D E F G

Fig. 3. Homeomorphic iopological geometric figures

The search for sameness in structural representations of objects is a crucial feature of geometric thinking. Given two structural representations of objects, how is it established whether the objects are identical or different? As may be inferred from the previous discussion, the terms identical and different need to be qualified. If we are inter- ested in sameness in geometry, we must specify the level of sameness (topological or Euclidean, congruent or similar). In each case, however, the intellectual process of analyzing sameness can be viewed as a mapping procedure. In mathematics, the term transformation is used for such mapping. A mathematical transformation R- P (or its inverse R+P) may conserve certain topological properties or Euclidean properties. Sameness implies that the transformation (and its inverse) involves a one-to-one correspondence of the pertinent features of the structures. This observation is important because it implies, in turn, that if membership of a test structure within a topological family of homeomorphic structures can be established, then the test structure will share all the topological geometric features of the family.

The relationship of topological geometry and the elasticity of topological figures to formal operational thinking, that is, the extrapolation from the given to the possible, and the relationship of Euclidean geometry and the rigidity of Euclidean figures to concrete operational thinking are readily apparent.

884 Angew. Chem. I n f . Ed. Engl. 25 119861 882-901

3. The Scientific Method, Strong Inference, and Paradigms

3.1. The Scientific Method'*'

The scientific method is usually defined as a n iterative inferential process involving the formulation of a hypothesis, followed by the devising of crucial experiments capable of a clean confirmation or rejection of the hypothesis and recycling as required by the outcome of the experiments. Platt'" suggests that certain systematic methods of scientific thinking produce more rapid progress than others. He uses the term strong inference to describe a version of the scientific method that emphasizes the formal, explicit, and regular use of alternative hypotheses and alternative crucial experiments. The scientific method may be viewed as a specific form of inteIlectual processing.

Kuhn")' takes the position that it is not the scientific method per se that leads to rapid progress, but rather the development of mature and effective paradigms, which are universally recognized scientific achievements that provide guidance to define scientific puzzles and to provide clues for their solution by a community of practitioners. Figure 4 is a transformation of the schematic description of normal mental activity (cf. Fig. 1) to that of normal science. In science, paradigms replace beliefs in the intellectual processing cycle. The role of geometric thinking in the scientific method is highlighted by replacing formal operational and concrete operational thinking with topological and Euclidean geometric thinking, respectively.

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r RESISTANT &p 1 1

GENERATE SOLVE

1 CONVENTIONAL

SCIENCE PARADIGMS PUZZLES PUZZLES ?

I t

J \ EUCLIDEAN TOPOLOGICAL

I

Fig 4. A schematic representation of the acti\ilq 0 1 norindl hcrence showing the interplay of paradigms, puzzles, solutions, and the scientific method (cf. Fig. I ) . Concrete and formal operational thinking "topologically map" onto Euclidean and topological geometry, respectively.

Thorn['] emphasizes the role of geometry in scientific analyses by suggesting that the human mind cannot be

[*I See also W. Wieland, Anyew. Chem. 93 (1981 1 627; Angew. Chem. In/ . Ed. Lnyl. 20 (1981) 617.

comfortable in a universe in which phenomena are governed by mathematic formulations that are coherent and quantitative but so abstract as to be impossible to visualize, that is, to interpret geometrically. The geometric interpretation allows the closure that drives intellectual processing. Furthermore, Euclidean geometry and topological geometry play an important role in structuring the thought processes involved in the scientific method.

The PIatt[" proposal of a strong inference methodology for solving puzzles scientifically raises two interesting questions: ( I ) Are there systematic and effective ways of generating alternative hypotheses and alternative crucial experiments?; (2) How does one best satisfy the Law of Closure in applying the strong inference approach (which inherently creates psychological tension by demanding the generation and consideration of possibilities and the reso- lution of the resulting conflicts)? We suggest that the use of paradigms and geometry can be effective in resolving these two issues.

3.2. Strong Inference

Some scientific fields appear to be capable of making more rapid advances than others. Clearly, development of instrumentation, funding, quality of practitioners, etc., all contribute to the speed of scientific advances in a field. Platt[xl suggests that an intellectual factor may also be important. He proposes that fields that systematically use and teach strong inference (the formal, regular, and explicit application of alternative hypotheses) are inherently better positioned for making rapid scientific advances.

The strong inference (scientific) method consists of the following steps: (1) creation of several alternative hypotheses; (2) devising of crucial experiments that exclude one or more of the hypotheses; (3) execution of experiments to allow "clean" exclusion of some hypotheses; (4) repetition of the cycle after refining the remaining hypotheses.

It is clear that observational knowledge must guide scientific ideas and that laboratory experiments must challenge and test the validity of the ideas. However, Platt181 puts particular emphasis on the regular and explicit use of multiple hypotheses or possibilities. By embracing multiple hypotheses at the beginning of an inquiry, the scientist will exhibit less tendency to become attached to a single hypothesis (which may become a sort of intellectual offspring from the moment it is proposed as an original and satisfactory explanation of a phenomenon). The intellectual and experimental attempts to exclude hypotheses should provide a n area of conflict, not between scientists, but between ideas. Excitement can be derived from the puzzles generated by alternative hypotheses. Which one will be right? Zeal and passion for experimental work surely can be derived from viewing the scientific method as requiring clever detective work in addition to experimental skill.

However, the question arises how scientists generate the formal schemata that guide them through the strong inference process without getting them bogged down in irrelevancies. The use of scientific paradigms provides such guidance.

885 Anqrii,. Chrm. Int. Ed. Engl. 25 (1986) 882-901

3.3. Scientific Paradigms

The exact meaning of the term is a bit “fuzzy.” A paradigm may be a constellation of beliefs, values, techniques, and methods that are shared universally by a community of practitioners. A paradigm may also be a smaller subset or a single element in such a constellation. I n either case, the paradigms serve as models or examples that can replace explicit rules for generating and solving scientific puzzles. There is an implicit process of intellectual mapping of features of the paradigm onto the puzzle under analysis. Since topology is a branch of mathematics concerned with the identification of equivalences by mapping processes, we shall examine the possible interrela- tions of scientific paradigms and geometric methods, first considering the overall role of paradigms in scientific methodology.

According to Kuhn,‘’”] paradigms provide a “mind set” for the execution of normal science. When scientific research is approached, questions of the following type arise commonly: How does one generate experiments to be performed? Of the many conceivable experiments that might be performed, how are priorities set? What aspects of a phenomenon are relevant for scientific examination? What questions may be legitimately asked and what techniques may be legitimately employed in seeking a scientific solution to a puzzle?

Effective paradigms, once established and accepted, provide a vehicle for the definition and solution of scientific puzzles. Successful paradigms prevent overt disagree- ments among practitioners concerning legitimate scientific problems and methods. As a result, practitioners spend little time involved in controversies over fundamentals. For example, a mature (1) generates an a priori (“intuitive”) backdrop of expectations for the community by providing a common body of beliefs, (2) defines legitimate puzzles that may be addressed by the community of practitioners and legitimate methods for the solution of these puzzles, ( 3 ) guarantees that legitimate puzzles have solutions that are limited only by the cleverness and skill of the practitioner and the experimental technique used, and (4) allows the search for solutions to proceed rapidly, because irrelevancies and constant justification are avoided, since standards are universally accepted by practitioners.

The successful solution of a scientific puzzle within the protocol of the paradigm both demonstrates the skill of the practitioner and strengthens the authority of the paradigm. The iterative process of puzzle generation and puzzle solution also provides a high degree of closure to the intellectual processing involved in the scientific method and in this respect can be stimulating and fun for the successful practitioner.

Of course, the more powerful a paradigm becomes as the result of continuing successes, the deeper and more profound becomes its grip on scientific minds. Attempts to question an enshrined paradigm are typically dismissed out of hand. The scientist may become oblivious to even the existence of other possibilities. On the darker side, truly new paradigms are sometimes resisted because they challenge whole systems of scientific development which

occurred over long periods of time and at great expendi- ture of energy and finances. A scientific revolution, as viewed by K ~ h n , [ ~ “ ] is the process of replacing an entrenched paradigm with a truly different one.

The course of a scientific revolution may be described by consideration of Figure 4. In normal science the conventional paradigms are employed to formulate legitimate puzzles which are solved by legitimate methods to reinforce the paradigm. However, according to Kuhn,[”]

. . . sometimes a normal problem, one that ought to be solvable by known rules and procedures, resists the re- iterated onslaught of the ablest members of the group within whose competence it falls ... when the profession can no longer evade anomolies that subvert the existing tradition of scientific practise-then begin the ex- traordinary investigations that lead the profession at last to a new set of commitments . . .

Such resistant puzzles are typically viewed initially by practitioners as anomalies which will be resolved eventually within the framework of normal science. There is often an implicit suppression of such anomalies, because their existence reflects poorly on the operating paradigm. However, the accumulation of anomalies can lead to a vex- ation among practitioners because this raises doubts concerning the rules governing the prior practice of normal science and the scientific work already completed, that is, the occurrence of an anomaly implies that an error in the paradigm may have been propagated unknowingly at ear- lier times. If so, puzzles were solved incorrectly in the past. Eventually a crisis may develop among practitioners, who, in the face of paradigm collapse, are forced to seriously consider a tradition-shattering commitment to the new paradigm, that is, a scientific revolution is underway. Once a successful new paradigm has emerged, the practitioners are mercifully reinstated in the normal science loop (see Fig. 4). Candidates for new paradigms must resolve some outstanding and generally recognized puzzle that cannot be solved by the protocol of the existing paradigm. Fur- thermore, the new paradigm must preserve or replace the ability of the conventional paradigm to define and solve puzzles.

A true revolution (change of a paradigm) is much less common than an “articulation” of a paradigm (topologically speaking, an elastic distortion of the local rules within the global paradigm). Both, however, begin with the awareness of an anomaly that results from a disparity between paradigm-generated expectations and experimental or theoretical observations. This “opens” the paradigm and leads to a feeling of discomfort among the practitioners. Closure is sought and is only achieved when the paradigm has been replaced (rare) or the rules have been re- articulated (more common). The re-articulation of a paradigm often corresponds to merely a realization of some aspect of the conventional paradigm that had been previously ignored or overlooked.

It is important to note what organic chemists never d o when confronted by a sudden and severe anomaly diag- nostic of paradigm breakdown: they do not renounce the old paradigm out of hand. They almost always first con-

886 Angew. Chem. Int . Ed. Engl. 25 (1986) 882-901

sider the anomaly to be the result of an artifact or to be interpretable in terms of a new articulation of the old paradigm. The old paradigm will survive until a viable alternative candidate is found. The decision to reject one paradigm is almost always coincident with the decision to accept another after comparison of both with the pertinent phenomena and with each other. Ad hoc modifications are usually a more preferable response than tossing out the conventional paradigm without a suitable replacement. Many of the paradigms of organic chemistry are retained, not because they are free of fault, or because they are infal- lible in their predictive ability, or because of their inelucta- ble logic. They are more often retained because they are useful for the understanding of observational knowledge at a qualitative level, because they are portable“’] and ex- ploitable by practitioners, and because they provide the driving force for fast-developing experimental methodology.

The decision to employ a particular apparatus or a particular method to solve a chemical puzzle carries an assumption that certain chemical phenomena exist and can be investigated by means of the protocol set forth by the paradigm. Novelty and anomaly emerge against the backdrop of the expectation (intuition) provided by the paradigm. Since accepted paradigms are not readily surren- dered, resistance guarantees that scientists will not be con- stantly distracted with irrelevancies and artifacts. So Iong as a paradigm continues to prove capable of identifying solvable problems, science moves fastest and penetrates most deeply through employment of the tools approved by the paradigm. In science as in industry, retooling is expen- sive and disruptive, and is reserved only for the special oc- casions that demand it.[9a1

It is the study of paradigms that prepares the student for admission to the scientific community. The practitioners share paradigms and are committed to the same rules and standards for scientific practice. By doing so, they can proceed rapidly in their research without having to start each time from first principles and to justify each basic concept that is used in a n argument. Indeed, when we discuss research within a field, we usually assume that we are ad- dressing colleagues who possess knowledge of a shared paradigm. The student is trained to attempt to “see” a new problem in the form of a n old, solved problem and to re- plicate a solution by mapping the features of the “understood” problem onto the new problem. At the beginning of a scientific inquiry, the student is sometimes only vaguely aware of what he or she is expected to discover and may lack confidence in knowing what to look for. Plutoi“’ understood this dilemma when in the Meno he noted that if we know the solution to a puzzle there is no puzzle, but if we don’t know the solution we d o not know what we are looking for and can not expect to find anything. H e con- cluded that problems are solved by remembering past in- carnations. The protocol of scientific paradigms as a method of puzzle solving is somewhat akin to Pluto’s rein- carnations.

In organic chemistry there are numerous examples of revolutions in thinking that occurred as chemistry developed into a molecular science. First, the notion of composition (number and kinds of elements), then the notion of

connectivity (the bonding between elements) and the concept of a three-dimensional representation (the stereochemical aspects of the array of bonded elements), and finally the concept of dynamic stereochemistry (the time-de- pendent conformational aspects of the array of bonded elements) were developed. At each of these stages in the development and refinements of the ideas of molecular structure a scientific revolution occurred because chemists changed entrenched paradigms which had previously determined the definition and solution of chemical puzzles.

4. The Relation of EucIidean and Topological Geometry to Chemical Structures

4.1. Mathematical Graphs, Forms, Figures, and Structures’*’

The scientific method often involves consideration of the information content of various mathematical objects. For the purposes of this article we wish to distinguish between mathematical graphs, forms, figures, and structures, which are used to model chemical objects. By a graph we shall mean a dimensionless but visualizable topological object that contains information concerning the components and connectivity relations of the chemical object to be modeled. By a form we shall mean an elastic topological geometric object that is embedded in Euclidean space. A figure shall mean a rigid Euclidean geometric object. We shall reserve the term structure for the mathematical object that models a chemical object (i.e., a molecule).

4.2. Transformation from Topological Objects to Euclidean Objects

We are all accustomed to analysis of concrete, readily visualized geometric figures embedded in Euclidean spaces of one, two, or three dimensions. We shall discuss briefly how objects in an abstract topological space can be transformed into objects in a concrete Euclidean space. In topology, sets (defined collections or families of objects) are employed for mathematical analysis. When relationships exist between members of the set, the objects of the set are said to constitute a topological space, T. How can one proceed from an abstract topological space to a concrete Euclidean space, R ? In this section we suggest a pathway that uses topological structures, namely, graphs, as the bridge from T-space to R-space.

Topology involves the study of those properties of spaces that depend only on the “nearness” of the elements of the space and that are independent of geometric aspects such as distances and angles. A topological space,[61 T(X, Y), is usually defined as a set, X , together with a family of subsets, Y, which satisfy certain conditions. Topolog- ical spaces have rich generality, but are completely abstract mathematical objects devoid of a visualizable geometric form. A finite topological space is said to have structure in the sets X and Y if there are some properties of the elements (members of the sets) that can be brought into

[*I See also 1. Ugi, D. Marquarding, H. Klusacek, G. Gokel, P. Gillespie, Angew. Chem. 82 (1970) 741; Angew. Chem. Int. Ed. Engl. 9 (1970) 703.

Angew. Chent. Int. Ed. Engl. 25 (1986) 882-901 887

correspondence. In order to “visualize” and eventually interpret a topological space geometrically, it is natural to speak of “points” rather than elements or members of the sets X and Y. The topological space can be visualized by imagining connections between the points of T ( X , Y). A fo-

pological graph is thereby produced. Among the most important fundamental properties of a topological space are neighborhood relations between points of the space. For an eventual chemical interpretation, it is natural to view the topological points in terms of atoms and the neighborhood relations in terms of chemical bonds. A finite topological space and its graph have the same number of components. A connected graph corresponds to a connected topological space.

A specific method of visualization, with powerful implications for chemistry, is to cast the topological space into the form of a connected graph, G , which is defined as an ordered pair, ( V , E) , where V is a set of points in a topological space and E defines the binary relations between the points. The points V are the vertices of the graph and the binary relations E are the edges connecting the vertices. The binary relations E can be viewed as the rules for a transformation that maps lines connecting elements of the abstract topological space T(X, Y) to produce a visualizable graphical space, G( V, E ) . Visualization is possible, for example, when the vertices are represented as small num- bered circles and the connections are represented as lines. It is important to recognize that C( V, E> is still a topological object, having no geometric parameters such as distances and angles. Chemists come into close contact with topological concepts when they use graphs to represent molecular structures. Indeed, in the usual mathematical sense, a graph, although it can be visualized, does not have dimensionality in the Euclidean sense.

As a concrete example of the conversion from a topological space to a connected graph let us consider the graph representing methane (Fig. 5). The members of the topological space T(X , Y) represent carbon atoms and hydrogen atoms ( X = (C, H)) and connections between carbon atoms and hydrogen atoms (Y={C-H)). The degree of a vertex of a graph is the number of edges connected to the vertex. A connected graph of degree four is employed to represent the normal valence of carbon. In an abstract formulation of methane there are two different kinds of points in the topological space and one kind of connectivity relation. Embedding of the graph C ( V , E ) into Eucli-

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t i f 5 A \ i \ua l iLed connected graph (let”) and a inolecular graph (right) of methane The molecular graph, which represents the molecular topology of methane, is produced by embedding the connected graph into the chemical paradigm for molecular structure. represented by the mapping process C. V is the set of points (left) or vertices (right); E is the set of binary relations between the points (left) or edges connecting the vertices (right).

€ = { C-H ]

dean space, R” (R’ , R2, and R3 correspond to one-dimensional, two-dimensional, and three-dimensional spaces, respectively), produces an elastic object R”( v, E), which is a topological form having the flexible geometric properties of distance and angle.

In Figure 6, the visualized graph G( V , E ) , which represents methane, is embedded in a dimensionless topological space to give a concrete Euclidean geometric figure. R’( V , E ) represents any elastic distortion of the geometric form that conserves the constitutional properties of the figure. Application of a metric, M (corresponding to measurable distances and angles), to R”( V, E ) produces a concrete object, R”( V , E , M ) , which is a rigid geometric figure that obeys the rules of ordinary Euclidean geometry.

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I-ig. h Gi.iphic.il t opo log i~~~ l . ( I ( I . . 1,) . ~opologic,il gt.oiiii.tric. K I I, 1:). Euclidean geometric, R’( Y. E. M), representations of methane The rnathe- matical objects G ( V , E), R’( Y. E) , and R‘( Y. E. M ) become models for the chemical object methane by embedding in the chemical paradigm, represented by the mapping process C. This produces a structural model at the giaphicd! (left), topological (middle), or Euclidean (right) level of representation. V, points (vertices); E, lines (edges); M . metric (distances, angles).

4.3. Molecular Models

We have seen how mathematical objects may be used as models for molecular structures. The meaning of the term model, like that of the term paradigm, is a bit fuzzy. For the present purposes, a model is an alternative form of an object or concept, which is created with the expectation that it will provide some insight into the nature of the object or concept. The mind is a model builder par excellence as it seeks, with only fragmentary information, a closed interpretation of the surrounding world. To many organic chemists, models are the “pictures” on which imagination may be exercised and which enhance an intuitive understanding of the object or concept under consideration.

According to Trindle,~’2’‘1 in building a model we relin- quish any claim to perfect truthfulness. Models are always different from the objects or concepts that are being represented. In science models are temporary aids, always subject to revision and usually destined for an intellectual junkyard. The best and most useful models have certain characteristic properties :[‘“I they are memorizable, port-

888 Anyew. Chem. Inr. Ed. Engl. 25 (1986) 882-901

able, simple, self-consistent, elastic, and widely applicable. Models are particularly delightful when they produce surprises. Since it is now.taken for granted that molecular objects in the real world can be faithfully represented by geometric objects, the wonderment of molecular models as representations of chemical reality is often no longer appreciated by the practitioner.

Up to this point we have considered purely mathematical (topological and geometric) objects, which can exist in intellectual processing but which d o not require any relation to an ultimate reality. We now need a bridge from these objects to molecular structures, which are models"*] used to represent the reality of chemical systems. For our purposes we shall assume that the molecules we wish to model exist as real objects. Mislow["' has provided a proper perspective for the approach we wish to embrace:

The (geometric) figures which can be developed from molecules and which become their models are mathematical objects and intangible abstractions of the reality they are intended to represent ... Model and molecule are thus separate and distinct entities, one abstract. the other concrete . . . By contrast, molecules and molecular ensemble exist entirely in the realm of observa- bles.

The concepts of molecules and molecular structure, the so-called intellectual units of the chemist, are remarkably successful models of chemical reality. However, we should keep in mind the caution of Hummond et al.:L'41

It is . . . important to remember the fine distinction between ... models and physical reality. A good model can be so successful in ordering our thoughts and pre- dicting the behavior of systems that we come to regard the model as real . . . Trouble may arise when two scientists, using ... different models engage in a bitter conflict in which they try to prove, or disprove, the reality of their models ... We must remember that scientific thinking is really a branch of symbolic logic.

In Figure 7, the relationship of topological spaces, graphical forms, geometric forms, and geometric figures to molecular structure is shown schematically. A mapping process, C, which corresponds to the chemical paradigm, transforms mathematical objects such as spaces, forms, and figures into chemical objects such as molecular struc-

Abstract Visualirable Elastic Concrete FI G1V.E) F] R"(V,EI M b 4 ) 1-1 SPACES FORMS FORMS F I G U R E S

M O L E C U L A R YxY S T R U C T U R E

Fig. 7. A whematic reprc\ciil.iii~iii 01 ihe rnicrrclationship of topological spaces. graphical forms, geometric forms, and geometric figures. The embedding of any of these objects in conventional chemical paradigms produces a model 01' molecular structure. R , Euclidean space (n= 1. 2, 3). For other abbreviations, see Fig. 6 and text.

tures. We shall now discuss the nature of this mapping process.

4.4. Transformation from Mathematical Objects to Chemical Objects

The transformation from mathematical objects (graphs, forms, figures) of topological geometry and Euclidean geometry may be viewed as the embedding of the mathematical objects into a chemical paradigm (see Fig. 7) to produce models for molecular structures. The paradigm provides the basis for the mapping process C that transforms mathematical objects into chemical objects. The his- torical pathway to modern chemical structure is related to that shown in Figure 7 except that Euclidean geometric figures (configurational stereochemistry) were accepted as the models for molecular structure before topological geometric forms (conformational stereochemistry).ii51 The his- torical pathway is summarized in Figure 8 for cyclo-

Abstract Visualizable

(numberr . reprerentatton of bonding of nlemtlnt i

I

Concrete

-]CONFICURPITIOVI-

Allowable shapes of t h i t h r e e - dimensional

1

\ /

I OF C Y C L O H E X A N E I

Fig. 8. The relationship hc.i\reeii c~)iiipo~itioii, ~ o i i ~ t i ~ t i i i o i ~ , ~ o i i l i g t i ~ ~ i i ~ i i i , and conformation in the mathematical and chemical sense. Molecular i t r u c ~ ture may be modeled at any of these levels. Cyclohexane is given as an ey- ample.

hexane as an example. Starting from the notion of composition (number and kinds of elements possessed by the chemical object), the notion of constitution"" (bonding relations between elements) was introduced and visualization was achieved by creation of a molecular graph. The graph was then embedded in 3-D Euclidean space to produce a rigid configurational geometric model of the molecular structure of the chemical ~bject .~" . Finally, the molecular structure was allowed to be conformationally active by giving it an elasticity consistent with conventional chemical paradigms. The configurational level of representation may be viewed as corresponding to concrete operational thought and the conformational level of representation to formal operational thought.

4.5. Molecular Graphs

Since chemists come into closest contact with topological ideas in the use of graphs to represent molecular struc-

Ailyen. C'hem. Inr Ed Enyl. 25 (19861 882-901 889

tures, we now consider how graphs may be used to represent molecular topology and how these graphs are converted into geometric forms which model molecular structure.

The graphical method for representing molecules, which is nowadays taken for granted by organic chemists, represented a major intellectual breakthrough, since it provided a conceptual framework for further progress in organic chemistry. The informational content of a molecular graph and its ease of visualization are important reasons for its success among chemists. A special attraction of graphs is their power in handling combinational problems in chemistry. From a practical standpoint, the use of graphs and simple rules with pencil and paper, without need for lengthy calculation, allows representation and categoriza- tion of a large number of chemical systems. Let us analyze the basis for why graphs are so valuable in analyzing chemical systems and why these graphs appear to have such a broad and robust informational content.

In a molecular graph the vertices represent atoms (or groups of atoms considered as a unit) and the lines represent bonds. The points and the connections between points are topologically significant. The chemist would term the points the composition of the molecular graph, and the network of connections the constitution of the graph. To a topological chemist the composition and constitution of the graph are essentially global topological properties of the graph. The graph of a molecule may be viewed as an expression of the topological concept of special neighborhood relationships, which are chemically described by the valence of a n atom (A). Only n atoms of the entire set of atoms in a molecule have a valency relationship with A, i.e., A has a valency of n. This topological notion of valency or bond order was the foundation of the first formulations of structural theory.[’61 The actual arrangement of atoms in space, their bond lengths and bond angles, need not be specified. Indeed, at this level, bonded atoms may be drawn further apart than nonbonded atoms without re- duction of the topological information, which is concerned only with the property of connectedness of atoms in the set with A.

Two graphs that have identical numbers of vertices connected in the same way are said to be isomorphic. The chemist recognizes two isomorphic molecular graphs as having identical composition and constitution. The type of line used to make the connection or the symbols used to represent the graph are not of topological significance.

H I

H-C-H I

P’

40*

H 1

Hyp big. Y. .Tho iaomorphic (mathematical) graphs of degree four (left) and two isomorphic molecular graphs representing methane (right).

890

Thus, a symmetrical, simple mathematical graph of degree four (see Section 4.2) is the same if it is written with a mix- ture of wiggly, straight, long and short lines (Fig. 9, left). The same holds true for the molecular graph of, for example, methane (Fig. 9, right). Because the points of the graph are abstract entities, the same molecular graph may represent many different molecules.

4.6. Stereochemistry in Euclidean and Topological Molecular Structures

Stereochemistry examines and categorizes relative spa- tial relationships between atoms and groups of atoms within a molecule and between different m o l e c ~ l e s . ~ ~ ~ ~ From everyday experience, the chemist recognizes that, although the relative arrangements of atoms (configurations) and the relative shapes of networks of atoms (conforma- tions) are of enormous importance, quantitative metric quantities such as bond lengths and bond angles may remain completely unspecified for many types of analyses. Within this framework, it is natural to employ an extension of topological ideas (which ignore metric relations) in stereochemical analysis.[’9]

We have seen that graphs allow a topological representation of molecules in terms of the idealized entities that constitute geometry and that a molecular graph can be re- garded as a representation of molecules in topological geometry. Examination of the stereochemical features of molecules requires the embedding of the molecular graph into Euclidean space, which produces geometric forms. This embedding is required because constraints, which are not fun’damental to topology, are placed on the physical characteristics of molecular entities.

The step from geometric forms (topological) to geometric figures (Euclidean) involves the application of a metric. The introduction of distances and angles into topological molecular structures creates geometric molecular structures. The transition from topological to geometric structures is a profound one from the standpoint of intellectual processing; indeed, it is analogous to the distinction between formal operational and concrete operational thought! Thus, a geometric figure represents a concrete chemical object with fixed distances and angles, whereas a topological structure represents the chemical object only under the condition of conservation of topological properties. Final- ly, the labeling of the points of the geometric figure, which results in a fixed orientation of the figure, allows the representation of chemical objects such as enantiomers.[201

The discovery of optical activity revealed a limitation of molecular graphs, which was overcome by introduction of the concept of configurational stereochemistry resulting from the arrangement of atoms in 3-D space. Optical isomerization could be understood in terms of the Euclidean 3-D geometry of n atoms connected to a central atom A. The key notion of the tetrahedral arrangement of valences about carbon is the same whether the configurational molecular figure representing the geometry is regular (van ’t Hoffj[”] or irregular (Le BeI).”*’ The neglect of metric relations, a fundamental feature of topological thinking, is apparent here. The bond lengths and bond angles of the te-

Angew. Chem. Int. Ed. Engl. 25 (1986) 882-901

trahedral structure can vary over a wide range without changing the (nontopological) geometric property of configuration. Of course, in real systems a limit may be reached for which the configuration is not conserved (bond breaking, formation of an achiral structure, etc.).

In the light of scientific paradigms, the step from a graphical, abstract, hypothetical representation of real objects to the acceptance of atoms as real entities that may be modeled by molecular structures in 3-D space represented a scientific revolution because it profoundly changed the way the practitioners designed and interpreted chemical processes. The acceptance of the concept of stereochemistry met with the expected resistance of practitioners doing normal science under the guidance of entrenched paradigms. For example, Kolbe‘”] castigated the new paradigm as a step backward:

I t is typical of the present time, when there is so little criticism and so much hatred of criticism, that two practically unknown chemists, one from a veterinary college, and the other from an agricultural institute, pass judgement on the loftiest problems of chemistry, those which will probably never be solved, particularly the question of the position of atoms in space, and they undertake to answer these problems with an impudence and assurance that absolutely astonish the true scientist.

Such a vitriolic attack on a paradigm that is now universally accepted brings to mind the skeptical but perceptive remark by P l a n ~ k : ’ ~ ~ ]

New scientific truth usually becomes accepted, not because opponents become convinced, but because opponents gradually die, and because the rising generation are familiar with the new truths at the outset.

4.7. Boundaries in Euclidean and Topological Molecular Structures

In mathematics a boundary implies a discontinuity, that is, the qualitative nature of a function changes as a “behavioral point” crosses from one side of the boundary to the other. In organic thinking, the notions of inside/outside and above/below are commonly used with reference to volumes or surfaces, respectively. For example, an organic molecule may be solubilized inside a micellar aggregate, outside the aggregate in the aqueous phase, or at the micellar/aqueous interface. A specific topological form that faithfully represents the inside/outside/boundary aspects of a micelle is a sphere. Of course, any geometric form that is homeomorphic with a sphere will also faithfully represent these topological aspects.

The wave functions of orbitals may or may not change sign when passing through a plane defined by certain nu- clei of the molecule of interest, that is, the qualitative properties of the wave function either change or remain the same on passing through the plane. A specific topological form that faithfully represents the above/below/boundary aspects is a Euclidean plane. Of course, in topological

terms, the plane is elastic and may be bent, twisted, and stretched without loss of the notion of above/below/ boundary. The power and robustness of Huckel molecular orbital theory probably reflects its topological foundations and explains its remarkable ability to faithfully represent the qualitative features of molecules, even those with no apparent Euclidean geometric symmetry.’231

5. The Use of Topological Thinking and Geometric Models to Examine Chemical Structures

5.1. Topological Thinking in Chemistry: A Qualitative but Precise Approach to the Scientific Method

Having given examples of geometric thinking and the use of topological concepts, we can suggest that there is a general procedure for structured intellectual processing that can be termed topological thinking. This method uses elastic mapping procedures for establishing the correspondence between a model, which can be expressed in topological form, and a phenomenon or observation. Geo- metric thinking emphasizes the concrete aspects of the model that relate to rigid forms and figures in space, whereas topological thinking emphasizes the elastic mapping process. Thus, the intellectual process by which a molecular structure is predicted to have a certain color or to be characterized by a certain NMR spectrum does not involve purely geometric components. The color or the spectrum can be related to a more general topological space in which the properties corresponding to color or the NMR spectrum can be defined.

Another way to view topological thinking is to consider it as an intellectual process that involves the pulling, stretching, and twisting of ideas in a search for sameness or correspondence in a manner reminiscent of the process involved in searching for homeomorphisms in topological geometry. Phenomena are imbued with properties that allow them to be converted into mathematical forms, which, in turn, can be mapped onto topological geometric forms.

Topological thinking is always qualitative, because, like topological geometry, it does not involve quantitative metric constraints. The term qualitative tends to have a pejora- tive connotation in the sciences. There is a tendency to assume that scientists are qualitative only when they are in- capable of being quantitative; that is, qualitative is merely second best to quantitative. However, the history and in- credible pace of progress in organic chemistry over the past two centuries is proof of the power of qualitative thinking in science. In most day-to-day organic research, what appears to be of greatest import is a qualitative result and not the precise value of a quantity; for example, is a rate slower or faster, is a yield higher or lower, does a structure contain a carbonyl or not. It is probable that the poor reputation of qualitative thinking is due to the naive and imprecise appearance of certain qualitative ideas. Ac- cording to such need not be the case:

(We can furnish) . . . by a refinement of our geometric intuition . . . our scientific investigations with a stock of ideas and procedures subtle enough to give satisfactory

89 I A n g e h . Chem. Int. Ed. Engl. 25 11986) 882-901

representations to ... phenomena ... we can now present qualitative results in a rigorous way, thanks to re- cent progress in topology . . . for we know how to define a form and can determine whether two functions have or have not the same form or topological type.

Thorn goes on to suggest that topological thinking is capable of releasing our intuition from the constraint of three-dimensional space and is capable of providing a more general, richer intuition for the examination of mi- croscopic phenomena. Furthermore, topological thinking can be precise if only the topological aspects are considered in a geometric analysis. According to M i ~ l o w : “ ~ ]

As chemists we often employ inexact terms such as fast/slow, strong/weak, concentrated/dilute, hot/cold, etc., yet we are confident that these words suffice to carry the desired message, unburdened by superfluous precision, within the context of the report.

Quantitative thinking, of course, plays a crucial role in the scientific method. However, quantitative, abstract mathematics may result in noncausal thinking that is math- ematically precise, consistent, and rigorous, but chemically or physically irrelevant. Thorn13b1 warns:

The human mind would not be fully satisfied with a universe in which all phenomena are governed by a mathematical process that is completely coherent but totally abstract. Are we not then in wonderland? In the situation where man is deprived of all possibility of in- tellectualization, that is, of interpreting geometrically a given process, either he will seek to create, despite ev- erything, through suitable interpretation, an intuitive justification of the process or he will sink into resigned incomprehension.

Perhaps Plaff[’] best epitomizes the attitude of the topological thinker:

Many-perhaps most-of the great issues of science are qualitative not quantitative, even in physics and chemistry. Equations and measurements are useful when and only when they are related to proof: but proof or disproof comes first and is in fact strongest when it is absolutely convincing without any quantitative measurement ... you can catch phenomena in a logical box or in a mathematical box. The logical box is coarse but strong. The mathematical box is fine-grained but flimsy. The mathematical box is a beautiful way of wrapping up a problem, but it will not hold the phenomena unless they have been caught first in a logical box.

In other words, a phenomenon should be mapped onto the correct topological form (logical box) before it is modeled in a quantitative fashion (mathematical box).

Topological thinking assumes that if the correct topological model for a phenomenon has been devised, then all of the topologically relevant aspects of the model can be “mapped onto” the phenomenon; that is, all the topologi-

cal features of the model are enjoyed by the phenomena and vice versa. This potential relationship is extremely powerful when it exists because i t automatically defines qualitative attributes that can be transferred from a given topological form to a wide range of chemical phenomena.

For example, the isoelectronic principle involves the mapping of the properties of a topological chemical structure having an array of valence electrons onto specific atoms. Thus, all atoms with octets of electrons “telescope down” to a single point in a topological space; the same is true for all atoms with sextets of valence electrons. In topological thinking, carbenes, nitrenes, oxygen atoms, etc., have the same qualitative chemical properties. This means that each reaction or physical property that is related to the topological aspects of a given chemical structure is shared qualitatively by every member or structure in the family. Other important examples of topological thinking in organic chemistry include the Hammond postulate,’241 the Woodward-Hoffmann rules,‘”] and Salem dia- grams.[z6’

5.2. Tetrahedral Carbon: A Triumph of Topological Thinking

The first important step in the development of the structural theory of chemistry was probably the formulation of the concept of valency. This concept when mapped onto the concepts of graph theory naturally leads to the recognition of connectedness, the special chemical relationship that distinguishes the n atoms bound to an atom A from the rest of the set. The connectedness, in turn, is interpreted in terms of chemical bonds. Thus, at the earliest stages of development, the key concepts of bonding were topological and graphical.

Consider the following argument, based on the use of topological and geometric thinking, an experimental isomer count, and the assumption of the noninterconvertibil- ity of isomers modeled by 3-D geometric figures. If the composition of methane is CH, and if it can be represented by a molecular graph and by a 3-D geometric figure, there are two distinct classes of possibilities (Fig. 10): either the vertices of the molecular graph are equivalent (class I ) or they are not equivalent (class 11). These two possibilities may be visualized with a molecular graph showing a C atom with four equivalent bonds to the H atoms, or by any other molecular graph with composition

If 1 is the correct representation, then replacement of a single H atom by any equivalent atom will lead to one and only one isomer. If I1 is the correct representation, then replacement of a single H atom by any equivalent atom

CH4.

H n I

n-c--H I I I1

H-H-C-H-H I

n-c--H--H

H

Fig. 10 I’ohbible grdph:. tor the rtpre\entalion 01 methane Class I conb ib tb o< a graph that shows four equivalent hydrogen atoms (and all isomorphous graphs) and class I 1 consists of all other graphs of composition CHI.

892 Angew. Chem. Int . Ed. Engl. 25 (1986) 882-901

will lead, in principle, to two distinct isomers. Since all experimental examples have been consistent with I but would require an ad hoc explanation in each case to be consistent with 11, strong inference suggests a molecular graph of CH, that has four equivalent H atoms and a C atom with a valency of four. By extrapolation, the valence of any member of the methane family is four. Any geometric figure that represents the molecular graph in 2-D or 3- D space and has four equivalent vertices and any molecular structure that has four equivalent H atoms bound to a C atom is consistent with the experimental finding that only one isomer is produced from methane by monosubsti- tution. In 2-D geometry the only such molecular graph places the H atoms at the corners of a square and the C atom at the center (square planar). In 3-D geometry the only such structures are a tetrahedron and a square pyra- mid. Thus, the experimental isomer count based on topological and geometric thinking reduces the choices to only three molecular structures (Fig. 1 l)!

square square tetrahedral planar pyramidal

Fig. I I I’o\\ible molecular structure5 produced b! embedding t h e topological molecular graph for methane, deduced from the isomer count, into 3-D space ( R ’ ) . Three possible Euclidean molecular structures result.

By employing the isomer count criterion again (Fig. 12), the square-planar and square-pyramidal structures can be ruled out. Replacement of one H atom in a monosubstituted methane produces only one product, as expected for a tetrahedral structure. The square-planar and square-pyramidal structures, on the other hand, would have given two possible products (cis and trans).

Y Y Y I I I

I I I H-C-H - X - C - H or H-C-H

V

tuted benzenes should exist. Since isomerism in 1,2-disubstituted benzenes has not been observed, it is inferred that the bonds in benzene are equivalent.

Fig. 13. Two representations of a 1,2-disuhstituted benzene. Lett: the bonds are equivalent (one possible structure). Right: alternating single and double bonds are present (two possible structures).

5.3. The Role of Time in Topological Thinking: Structural Stability and Chemical Reactivity

At the concrete operational level of thinking or the Eu- clidean geometric level of modeling, structures are rigid and d o not change with time. At the formal operational level of thinking or the topological geometric level of modeling, structures are elastic and capable of changing with time. It is crucial to distinguish two types of time-depend- ent structural changes: those that conserve all topological properties and those that change the topological properties. For example, the former would correspond to the dynamic stereochemistry of a stable chemical structure and the latter to the dynamics of a chemical reaction.

Consider a structural model of a methane molecule. At any instant in time, a “snapshot” of the methane structure would not reveal a perfect tetrahedron, because vibrations will slightly change each bond length and bond angle (Fig. 14). Nevertheless, a chemist would not fail to recognize methane in any of its vibrational guises unless one of the C-H bonds becomes so long that it becomes unclear whether the observed structure corresponds to methane or to a methyl radical plus a hydrogen atom. The notion that methane, under “normal” conditions, can be modeled as a perfect tetrahedron derives from the premise that all experimental observations of methane correspond to a time- averaged structure and that the limiting time-averaged structure is a perfect tetrahedron. The time-averaged tetrahedral model is valid because, in a typical analysis, more than a trillion molecules are sampled.

I I; I 2 i < i ~ p l , i ~ c i i i ~ i i t ill i )nc t i Atom h i a >ubstitueiit X in square-planar or \quare-pyramidal (upper) and in tetrahedral (lower) monosubstituted methane, C H I Y (see text).

The existence of enantiomers follows logically from the nonequivalence in 3-D space of mirror images of tetrahedral molecular figures having four different substituents. Moreover, these general conclusions are also valid for other molecular structures having a central atom of valence four.

A second example of topological thinking in organic chemistry is found in the use of the isomer count method to infer the structure of benzene (Fig. 13). If benzene has equivalent bonds, then only one type of 1,2-disubstituted benzene should exist; if benzene has alternating single bonds and double bonds, then two isomers of 1,2-disubsti-

H

Fig. 14. Ideal tetrahedral represmtdtion of methdne as an “attractor” structure for all homeomorphic representations of methane.

In topological terms the perfect tetrahedral structure represents an “attractor” structure toward which all other

Angew Chem. Inr. Ed. Lngl. 25 (1986) 882-901 893

4- -4- +t

44- + t SO TI S1 s2

- topologically equivalent chemical structures tend. The - chemical nature of methane does not change for suffi- -

- - ciently small deformations of its structure. R An experiment might be possible on a such short time

“caught” in a nontetrahedral shape. Whatever the techni- cal difficulties of such an experiment, it is topologically equivalent to an experiment for detecting cyclohexane conformers. Before fast methods of analysis were available, cyclohexane was considered as a single, static structure. Conformers were not detected because the time scale of the experiment was large relative to the time scale of the conformational change. With modern laser techniques, chemical structures may be examined on time scales of the order of lo-’* to s. Many conformational systems in organic chemistry that are normally considered as being in dynamic equilibria are “static” on such a time scale.

that most methane be Fig. IS. The relationship between the HO and L U orbitals for a reactant structure R and i ts four lowest-lying electronic states: S,), T, , s,, and s?.

6. The Use of Topological Thinking, Geometric Models, and Paradigms to Examine Chemical Reactivity

Are there archetypical graphs, topological forms, and geometric figures for modeling chemical reactivity? If we assume such models exist, we now have a recipe for antic- ipating their structure and the pathways for producing them; that is, we can map the pertinent aspects of the molecular graphs, molecular forms, and molecular figures used to model chemical structures onto reaction graphs, reaction forms, and reaction figures (i.e., a reaction network). It is natural to associate the points of the reaction network with the structures of reactants and products and the lines connecting the points with elementary reaction steps.

Topological thinking allows us to handle the multidi- mensionality of ground- and excited-state reactivity for a given structural transformation with a single reaction network generated by geometric procedures. Chemical paradigms guide the expression of specific geometric features of the reaction network.

6.1. Electronically Excited States and Diradicaloid Structures

Let us consider a chemical transformation of a reactant structure (R) to a product structure (P): R-P. The photo- chemist is concerned not only with the ground-state surface for this reaction but also with the excited-state surfaces associated with the same overall transformation. In constructing a reaction graph we need to establish the components (vertices) and connections (edges) of the graph. What are the components that correspond to the pertinent states of R and P? Using the highest occupied/ lowest unoccupied (HO/LU) orbital paradigm for theoretical guidance,[271 there are four components or states, corresponding to the four possible electron occupancies of the HO and LU orbital of any R and P: So, SI , S2, and TI (Fig. 15). The energetic ordering of these states is invaria- bly Str <Ti < S < Sz for ordinary organic molecules.f281

What are the connections between the components of R and P? If we can answer this question in a general way we will be able (as we are for molecular structures) to generate a visualizable reaction graph, which we can transform into a topological geometric form by embedding it in a Eucli- dean space, and which we can eventually transform into a Euclidean geometric figure with a metric, namely, the quantitative relationship of structure to energy. Before dis- cussing examples of “reaction structures” (i.e., specific chemical reactions), we will discuss an important paradigm for the so-called “diradicaloid” structure that is commonly used for photoreactions.l2“I This diradicaloid structure generally has four electronic states that are related to the So, S,, S2, and Ti states of R and P. Importantly, we shall see that the diradicaloid structure is usually optimal for transformation from one energy surface to another, which occurs at the so-called “behavioral point” (corresponding to the change in atomic arrangement as R is transformed into P). Since getting from an excited-state surface to a ground-state surface is a critical aspect of all photochemical reactions, the diradicaloid geometry plays a crucial role in determining the possible “traffic patterns” at this behavioral point.

6.2. The DiradicaVZwitterion Paradigm

The important and extremely useful diradicaVzwitterion (D/Z) paradigm for analyzing many photoreactions was proposed by Salem[26’ as follows: If a molecular structure (having an even number of electrons) occurs along a reaction pathway for which the highest occupied orbital and the lowest unoccupied orbital are of comparable energy, the structure, termed a diradicaloid or diradicaVzwitterion structure, will have four low-lying electronic states that will determine the chemical pathways leading to and from the structure. The diradical/zwitterion paradigm has the typical fuzziness of topological structures and qualitative organic thinking. The “composition” of the topology, namely, the set of four electronic states, is specified, but the energetic ordering of the states and the connectivity of the surfaces from R to P are not. We can proceed to the “constitutional” level of a reaction graph by assuming a common orbital situation for the reactant structure and then deriving the possibilities for the diradicaloid structure. By inspection, we can attempt to list the most probable state orderings of the diradicaloid structure and note the possible connections, thereby establishing the possible constitutional relationships between R and D/Z. We can then re- peat the process to determine the connections between D/Z and P.

894 Angew. Chem. In,. Ed. Engl. 25 (1986) 882-901

First, we consider a limiting situation in which the HO and LU orbitals have exactly the same energy at the D/Z structure (Fig. 16). Since composition is a topological property, we postulate that it must be conserved in an elementary, chemical step. Thus, there must be a one-to-one correlation between the number of electronic states for R and D/Z. What are the four lowest-lying states of the D/Z structure? How d o they connect with the four lowest states of R?

L U - - HO- - LU

HO - R D I Z

of R. The topological arguments are independent of the detailed chemical structure of R or D/Z. The next critical questions are how to map the reaction graph of Figure 17 onto plausible experimental examples and how, for a given system, to generate crucial experiments to eliminate by strong inference all candidates that cannot be mapped onto the reaction graph.

6.3. Some Archetypical Reaction Graphs Involving D/Z Structure

As examples of topological thinking in the use of reaction graphs, we shall consider two families of reactions involving D/Z structures along the reaction pathways. One family is typified by the cis-trans isomerization of a carbon-carbon double bond and the other family by a hydrogen-abstraction reaction.

The twisting and breaking of a carbon-carbon n bond may be imagined to occur as follows:'271 as an ethylene - - -4-+ 4 - t % - - %

DI Z 'D 21 2 2 molecule is twisted, it eventually arrives at a structure in which the two methvlene ErouDs are mutuallv DerDendicu-

- & - . . lar (Fig. 18). At this diradicaloid geometry the n bond is completely broken. The resulting I ,2-diradical can be de-

Fig. 16. The relationship between HO and LU orbitals and V/Z structures. Above: A chemical reaction results in transformation of the well-separated HO and LU orbitals of the startine structure R into a Dair of orbitals of com- - parable energy for the diradicaloid structure D/Z. Below: The four elec- scribed by four electronic D / Z states. Twisting about the tronic states (ID, -'D, Z,, 2,) of the D/Z structure.

In Figure 16, the states, (labeled ID, 3D, Z, , and Z,) that are possible for the two electronic configurations of the D/Z structure, are shown. Let us now generate the possible connections that would constitute reaction graphs of a hy- drocarbon system for which the state compositions are only So, T I , S,, and S2 for R and ID, 3D, Z, , and Z 2 for D/Z. For hydrocarbons, the energy of the ID and 3D states are expected to be similar, as are the energies of the Z1 and Z2 states.lzY1 The Z , and Z2 states can be considered as a unit because of their similar energies and electronic and spin configurations. The 'D and 3 D states, however, cannot be considered as a unit, because they have different spin configurations. In making connections between the states of R and D/Z, orbital occupancy and electron spin are used as descriptors a t the topological level. In Figure 17, the transformation R-D/Z can be viewed as a reaction graph (termed a Salem diagram) for this structural change starting from any one of the four lowest-lying electronic states

+ + ti-

'0 tt

'D

-?-#- R - O/Z

Fig. 17. Keaciioii grdph lo r the transformation R-D/Z (see text).

00 900 180° H H H H

Fig. 18. State correlation diagram or reaction graph for the cf . \ .~run, isomerization of ethylene.

carbon-carbon bond of excited ethylene sharply relieves electron-electron repulsion, thereby lowering the energy of any ethylene state for which bonding is not important (because electronic excitation has already effectively broken the bond). As a result, the electronic energies of S2, S , , and T, drop rapidly as a function of the twist angle. At the same time, the electronic energy of So increases as the molecule is twisted, because the A bond is being broken without any compensating bond formation. The breaking of a n bond by twisting is thus a prototype of a ground-state-forbidden concerted reaction, such as the disrotatory ring opening of cyclobutene 1 to give 1,3-butadiene 2 [Eq. (a)]

Angew Chem Inr Ed. Engl. 25 (1986) 882-901 895

or the suprafacial-suprafacial [2 + 21 cycloaddition of two molecules of ethylene to give cyclobutane 3 [Eq. (b)].

n - L J

1 2

L J 3

The correlations So+’D, T , - + 3 D , S,-+Z,, and Sz+Zz may be made on the basis of orbital-symmetry consideration~.[~‘.’’~ The symmetry operation that brings the starting planar geometry into the twisted (diradicaloid) geometry is a rotation of one CH, group. The overall state symmetries must be definable in terms of this symmetry operation. Al- though the state correlation is best accomplished by use of group theory and point-group analysis, the following qualitative description indicates the basis of the correlation.

The wave function for the 7c2 configuration (Fig. 18) at the planar geometry is essentially covalent in character, that is, there is very little ionic character to planar, ground- state ethylene; in terms of the p orbitals on the two carbon atoms, p i and pz, the TI’ wave function has the form pl ( l )pz( j ) . This means that at all times there is only one p electron near each carbon atom, and the two electrons have paired spins. For the 3(7c,7c*) configuration at the planar geometry there can never be two electrons on one carbon in the same p orbital since the electrons have parallel spins (violation of the Pauli principle). The triplet (TI) state i s purely covalent and has no ionic character; its wave function has the form pl ( t )p2( f ) .

The wave functions for ’(n,n*) and TI*’ must differ from that for n’, because these two states are much higher in energy. They are best described by zwitterionic wave functions.~26. 271

The twisting of a TI bond thus represents a prototype topological reaction surface. The state correlation diagram (Fig. 17) exhibits the following qualitative features:

1. The occurrence of minima in the S 2 , SI, and TI surfaces

2. A maximum in the So surface near the D / Z structure. 3. The close approach of the So and TI surfaces near the

near the D/Z structure.

D/Z structure.

The important topological features of these surfaces are the existence of four surfaces, their energetic ordering, and their electronic character. The closeness of approach of the ‘D and 3D surfaces and of the Z, and Z 2 surfaces is not topologically significant but is geometrically significant, as shown for the examples discussed below.

6.4. Application of Prototype Surfaces to Experimental Examples

If they are to be useful, prototype reaction surfaces should be “topologically adjustable,” that is, mappable

onto different but related experimental examples. We shall start by considering the prototype surfaces for TI-bond twisting, since we have just shown that the topological form of this surface may be as being equivalent to all thermally forbidden concerted ground-state reactions! This means that the twisting and breaking of the TI bond of ethylene can be topologically mapped onto any 4n-electron concerted electrocyclic reaction or cycloaddition

The mapping procedure can be simplified by invoking Kasha’s which states that in solution only the So, TI , and S, states are involved in organic photoreactions. Kasha’s rule allows us to ignore Sz (or any higher singlet state) and to consider only correlations involving the So, T I , and S, states and the corresponding states of the D/Z structure. The movement of a representative point along each of the three lowest-energy surfaces (So, TI, and S,) in the prototype diagram is shown in Figure 19. Movement of the representative point along the So and TI surfaces is chemically interesting in the region of the diradicaloid structure. At and near this structure, the S , and TI surfaces “come close together,” so that there is a high probability for “jumping” from the So to the TI surface (Fig. 19, left) or from the TI surface to the So surface (Fig. 19, middle). Since jumping from one surface to another implies a topological change (in this case, change in the electron spin of the system), such jumps can only occur if the “topology” of the total experimental system is conserved. Therefore, somewhere in the experimental system a corresponding and compensating spin change must occur. Thus, an So-T, or T,-So jump is possible when the surfaces get close, but is not probable unless certain selection rules are obeyed. Chemically, the pertinent selection rules state that either strong electron spin-electron orbit (spin-orbit) coupling or electron spin-nuclear spin (hyperfine) coupling is needed for efficient So-T, or T l - + S o jumps near the diradicaloid

t ip. 19. Prototype redctioii graph lor d ground-state-forbidden concertrd reaction showing various pathways for motions of a representative point along the reaction surfaces (see text).

The most important feature of the S, surface (Fig. 19, right) is the occurrence of a zwitterionic minimum near the D / Z structure; this minimum corresponds to the geometry at which SI-So transitions will occur. These transitions are spin allowed and can occur as rapidly as a means of “dumping” the electronic energy of the S I -So transition into vibrational or collision energy can be found. The position of the minimum is not a topological property of the system and may occur at the diradicaloid structure or at a

896 Angew. Chem. h i . Ed. Engl. 25 (1986) 882-901

position corresponding to a structure “on either side” of the diradicaloid structure. Furthermore, other minima may occur owing to energetic features not explicitly consid-

We have now developed a paradigm that can be used to generate experimental puzzles and to guide the search for solutions. To the extent that the paradigm is correct, only the experimentalist’s ability to articulate the paradigm experimentally will determine whether the puzzle is solved. The paradigm has the following implications:

ered.’32- -3-31

1. At diradicaloid geometries, So-+TI and TI+& jumps are possible if an adequate mechanism for intersystem crossing is available and can operate. Since intersystem crossing can occur near the minimum of the TI surface, there is generally sufficient time for a spin-orbit or hyperfine interaction, which will promote the TI-+& jump, to occur. A more remarkable possibility suggested by the paradigm (Fig. 19, left) is that a n So-fTl jump is possible, and, since this corresponds to a “chemiexcitation,” it could result in chemiluminescence if phosphorescence can occur from TI of the product. The efficiency of the So-+TI jump will depend on the efficiency of the spin change in the region of the diradicaloid structure.

2. An SI-So jump is possible from the minimum in the S, surface to the So surface ; the corresponding structure will reflect the nuclear geometry of the S, minimum. An intriguing possibility suggested by the paradigm is that the rapid removal of thermal energy will cause the representative point (Fig. 19) to “slide” rapidly toward the minimum on whichever side of the So curve it lands.

We will now present experimental examples of photochemical and chemiluminescent reactions whose qualitative features can be readily interpreted in terms of “topological distortions” of the prototype energy surface.

6.4. I . The cis-trans Isomerization of Alkenes

An example of the archetypical reaction graph for a ground-state-forbidden reaction is shown in Figure 20, left, along with reaction graphs for the cis-trans isomerization of ~ t i l b e n e l ~ ~ ] (Fig. 20, middle) and I-alkyl-2-anthrylethy- Ienes‘”’ (Fig. 20, right). From experimental data the relative (and, in some cases, the absolute) positions of the So, S , , and T, surfaces have been determined. The energy barrier indicated between the trans-stilbene structure and the D/Z structure on the S , and TI surfaces (Fig. 20, middle) is consistent with photochemical quenching data and tem- perature effects. There is no evidence for a barrier between the cL-stilbene structure and the D/Z structure on either the S i or T, surface. The representative behavioral points proceeding to cis and trans product starting from either cis- or fron.7-stilbene on either the S, or TI surface are consistent with the results of direct photoexcitation and triplet- sensitized excitation experiments.[341

In the case of I-alkyl-2-anthrylethylenes, e. g., cis- and trans- 1-(2-anthryl)-3,3-dimethyl-l-butene,[3s1 the photo- isomerizations are “one way” on both the s, and Ti surfaces ; that is, direct or triplet-photosensitized excitation of

Ph Ph Ph Ph An An An R

Fig. 20. Left: The prototype redcilun grdph 0 1 a giuund-ai‘it~-lurhIddt‘n re‘rc- tion. Middle: The “reaction structure” for cis-trans isornerization o l strlbene. Right: The “reaction structure” for &-trans isomerization of I-aikyi-2-anthrylethylenes. An = anthryl. See text for discussion.

cis- or trans-1-alkyl-2-anthrylethylene results only in formation of trans product. This result is rationalized by pos- tulating the lack of an effective attractor minimum at the D/Z structure on the S, or TI surface.

6.4.2. The Interconversion of Norbornadiene and Quadricyelane

The interconversion of norbornadiene ( N ) and quadricyclane (Q) is an example of a thermally forbidden concerted [2 + 21 cycloaddition reac t i~n . ‘ ’~ .~‘~ Accordingly, the prototype energy-surface diagram for thermally forbidden concerted reactions (cf. Fig. 20, left) can be used again as a paradigm to examine N and Q on the SO, TI, and S, surfaces.

Fig. 21. Reaction graph for the norbornadrent: to quddr1c)cldne irorneriza tion. See text for discussion. 0 =representative point.

Mapping of prototype reaction graph for a thermally forbidden concerted reaction onto the N and Q systems produces the reaction graph shown in Figure 21. The prototype paradigm allows arguments to be made concerning the position of the minimum on the S, and on the TI surface relative to the ground-state maximum. Since the S, surface is expected to have zwitterionic character near the diradicaloid structure, a charge-separated zwitterionic

Anyen Chem I n ! Ed Engl. 25 (1986) 882-901 897

structure is expected. Of the two zwitterionic structures conceivably formed via approach from N or from Q, the one having a 1,2 arrangement of positive and negative charges, N,, is expected to be more stable than the one having a 1,3 arrangement, Q,, on account of simple Cou- lombic factors (Fig. 22, top). The structure corresponding to the S I minimum should therefore resemble the 1,2 charge-separated species N,; that is, the minimum on the SI surface should occur on the N side of the ground-state maximum.

&-&I, t - t t b - $ h g 12. Ut.lationships ol~norhornadiene dnd quadricyclane structures to zwlt- terion species (top) and diradical species (bottom). See text for discussion.

A similar line of reasoning for the TI surface suggests that the species with a 1,3 separation of parallel spins, Qv, should be more stable than that with a 1,2 separation, ND (Fig. 22, bottom). The structure corresponding to the minimum on the Ti surface should resemble the 1,3 diradical Qv; that is, the minimum on the TI surface should occur on the Q side of the ground-state maximum.

Even though the above line of reasoning is qualitative, the paradigm leads to experimentally verifiable predictions; for example, either N or Q is the major product de- pending on the surface on which the representative point finds itself. Anytime a representative point is placed on the S, surface, N is the expected product, since the minimum on S, carries the point, via internal conversion, to So on the N side. Anytime a representative point is placed on the Ti surface, Q is the expected product, since the minimum on T I , corresponding to Qo, carries the point, via intersystem crossing, to So on the Q side.

The paradigm has thus produced a framework to generate experimental puzzles and to articulate experimental solutions. How can the point be placed on the S, or TI sur-

Ti QO TI

Fig. 23. A m precursors 4 and 5 for N, ( S , surface) and Q,, (T, surface). Top: On the S , surface both N and Q structures collapse to a zwitterionic structure, N,. Bottom: On the T, surface both N and Q structures collapse to a dirddical structure, Qr,.

faces? Experimentally the point can in principle be placed on the S , surface by direct photoexcitation of N or by direct photoexcitation of a precursor such as the azo compounds 4 and 5 (Fig. 23). The paradigm, if correct, pre- dicts that, irrespective of precursor structure, the major product from the S, surface will be N and the major product from the TI surface will be Q. Thus, eight predictions can be made, each of which has been confirmed experi-

6.4.3. The Chemilumineseent Electroeyelie Rearrangement of Dewar Benzene to Triplet Benzene

The rearrangement of Dewar benzene to benzene may be viewed as a forbidden ground-state disrotatory electrocyclic reaction.[381 The prototype energy-surface diagram for thermally concerted reactions (Fig. 19) can therefore be used as a paradigm to examine the chemistry of Dewar benzene in its So, T,, and S, states.

Initially, the prototype energy-surface diagram must be modified to accomodate any experimental data available for the Dewar benzene-benzene system. For example,[381 Dewar benzene is approximately 60 kcal/mol (AH,) less stable than benzene, and the activation enthalpy, AH', for the rearrangement of Dewar benzene to benzene is about 30 kcal/mol. The state energies for St,, T I , and S, for Dew- ar benzene and benzene are used to construct the energy- surface correlation diagram shown in Figure 24. The paradigm shows that three reactions are energetically conceivable: ( 1 ) the thermal production of triplet benzene; (2) the adiabatic conversion of triplet Dewar benzene to triplet benzene; (3) the adiabatic conversion of excited singlet Dewar benzene to excited singlet benzene.

very weak spin-orbit

coupling

Reaction coordinate(r,) -

Fig. 24. Energy-surface correlation diagram ("reaction structure") for the Dewar benzene to benzene transformation.

Experimentally, chemiluminescence is observed upon thermolysis of Dewar benzene and triplet benzene is produced, albeit in very low yield.[381 The reason for the low yield is readily explained by the paradigm in one of two ways: (1) The diradicaloid structure does not allow suffi- ciently strong spin-orbit coupling (which is expected to be

898 Angew. Chem. Int. Ed. Engl. 25 (1986) 882-901

the only spin-flipping mechanism capable of causing an So-T, jump); (2) the time spent by the representative point in the region of the diradicaloid structure is too small to allow spin-orbit coupling to develop. Of course, a third possibility I S that the paradigm is quantitatively incorrect with respect to the energetics; that is, the simplified two- dimensional representation may be misleading with respect to the energy of the diradicaloid structure.

The paradigm has thus created puzzles for the experimentalist: Is it the near absence of spin-orbit coupling, the too short time scale, or the energy that causes the low yield of triplet benzene? Can one devise a convincing test?

It is well established that certain “heavy atoms” such as halogens enhance spin-orbit coupling.[2x1 Thus, a specific test of the puzzle is to study the thermolysis of a Dewar benzene having a heavy-atom substituent. The paradigm suggests that thermolysis of a halogenated Dewar benzene should produce higher yields of triplet benzene if the near absence of spin-orbit coupling is the major factor in determining the yield of triplets. Experimentally, the thermolysis of 1,4-dichloro Dewar benzene produced a yield of triplets that was five times higher than that produced from Dewar benzene

The triplet state of Dewar benzene may be produced by triplet sensitization of Dewar benzene.[391 According to the paradigm (Fig. 24), an efficient adiabatic reaction along the TI surface is highly probable for the same reason that the S o - + T I jump is inefficient (i.e., the absence of an efficient spin-flipping mechanism). Experimentally, this pre- diction has been confirmed; triplet Dewar benzene 6(Tl) produces triplet benzene 7(T,) [Eq. (c)].[~~]

The paradigm further suggests that an adiabatic reaction along the S, surface is possible because S, of benzene is lower than S, of Dewar benzene and because the S, surface is separated from the So surface all along the reaction coordinate. Experimentally, no evidence has been found for this adiabatic reaction. Instead, prismane 8 and ground-state benzene 7(S0) were produced by excitation of Dewar benzene 6 to the S, excited state [Eq. (d)1.[381 This corresponds to a different process that is in competition with and evidently faster than the process leading to excited singlet benzene.

(Fig. 19). Should the representative point be “captured” by this minimum, and if the minimum occurs on the benzene side of the ground-state maximum, internal conversion would lead to benzene.

6.4.4. Reactions of n ,z* Excited States

The reactions of the n,n* excited states of carbonyl compounds are among the best understood of all photochemical processes. The n,n* state has topological features relative to a symmetry plane that allow the generation of topological reaction graphs that are applicable to a wide range of photoreactions.

R D/Z P

A prototype reaction is intramolecular hydrogen abstraction by an excited carbonyl compound (Norrish type I 1 photoreaction) [Eq. (e)]. The primary photochemical product is a diradicaloid structure, so that the overall transformation corresponds to an R- D/Z structural transformation. Application of the topological reaction graph for such transformations (Fig. 17) gives Figure 25. Starting from the S,(n,n*) state there is an adiabatic connection to ‘ D and starting from the T, (n,n*) state there is an adiabatic connection to 3D. Thus, both reactions are topologically allowed in the sense that the direct connections S , - ’D and TI-.’D exist.

D I Z

Fig. 25. Prototype reaction graph for the intramolecular hydrogen abstraction of photoexcited ketones. For A, B, and C, see text.

This paradigm has nothing to say directly about the formation of prismane 8. The paradigm does, however, provide a possible explanation for the formation of ground- state benzene, since an attractor minimum on the s, surface is suggested as a possibility in the prototype surface

Since these arguments are topological, moreover, they are independent of the specific molecular structure we have selected to exemplify the reaction graph (i.e., So, T I , S,, ‘D, 3D, and Z are merely symbols given to the abstract topological points!). Therefore, any conclusions derived for the specific example may be mapped onto all examples of H-atom abstractions from n p * excited states of carbonyl compounds or n,n* states of compounds containing other functional groups. Furthermore, the reaction graph applies to any reaction that is topologically equivalent to (or homeomorphic with) hydrogen abstraction, that is, any

Angew. Chem. Int. Ed. Engl. 2s (1986) 882-901 899

reaction involving low-lying S, (n,n*) or T,(n,n*) excited states in which the n orbital draws electron density from a substrate. Important examples are additions to alkenes, electron abstraction from amines, and the formation of ex- ci pi e xes.["I

Among the chemically interesting aspects of the reaction graph (Fig. 25) is the occurrence of three regions, A, B, and C, where different states have similar energies for the same structure. In these regions a transition from one state to the other is only expected to be favorable if a mechanism is available and if selection rules are obeyed. For example, the region A corresponds to structures for which the S,- ID surface and So-Z surface have comparable energies. This region is analogous to a traffic intersection. If a representative point moves down the S , - ID surface relatively slowly, it may be driven by thermal collisions in one of three directions when it arrives in region A: toward Z, toward So, or toward 'D. Since Z is higher in energy, this pathway is least likely. Thus, the plausible topological pathways are S,-A-So and S, -A- 'D (Fig. 26). In fact,

2

'D - Norrish type I1 products

SO -

Z s1- /- T1--sL 3D - Norrish type I1 products

t-ig. 26. I'ohaible reaction pathways along the reaction graph for intramolecular hydrogen abstraction of photoexcited ketones.

experimental evidence in favor of the occurrence of both S,-A+So and S1+A+'D pathways is available from the observation that an optically active ketone in the S, excited state, 9(S,), undergoes Norrish type I 1 reaction both to form an optically active cyclobutanol 10 and to regenerate the starting structure 9(So) without loss of optical activity [Eq. (f)].[4*1

The situation is different for region B (Fig. 25) , because the S o - Z surface is close to the T I - t 3 D surface and the surfaces differ in their spin characteristics. Transition from

'D 10

one state to the other in region B requires a mechanism for intersystem crossing. In general, the rates of intersystem crossing in diradicals are relatively slow compared with motion (vibrational relaxation) along a surface. Nonethe- less, the possibility of a jump from TI to So is suggested by the reaction graph. Experimentally, it appears that only the T , - + 3 D pathway is followed (Fig. 26), since the quantum yield for product formation from TI can approach unity, that is, each molecule in the TI state produces products derived from the 'D state.["]

Region C (Fig. 25) is of interest from the standpoint of topology, because at geometries near C the ' D and 'D states are nearly degenerate. This means that weak mag- netic forces whose energies are of the order of the energy difference between ' D and 3D can induce intersystem crossing between these states. There is considerable evidence that intersystem crossing from 3D to ' D is the rate- limiting step in reactions of triplet d i rad i~a ls . [~ '~

7. Geometry, Intuition, and Imagination

In conclusion, we consider the possible relationships between geometry, intuition, and imagination. Intuition may be defined as an instantaneous comprehension or appre- hension of an object or an event in the past, present, or future. In the spirit of the themes put forth in this article, some intuitions might be defined in terms of the instantaneous comprehension of a geometry representing an object or an event. lmagination may be defined as the power of creating or inventing mental images of what is not actually present and of perceiving the resemblances between appar- ently different objects and events. We can therefore view imagination as the manipulation of geometries, for example, in topological mapping.

In organic chemistry the classic figures of Euclidean geometry have provided the driving force for enormous ef- forts in organic syntheses. Of particular note is the role of the Platonic solids such as the tetrahedron,["I the and the dodecahedron.["51 The belief in the reality of chemical objects that can be represented by these geometric figures and the realization of the syntheses of materials whose chemical properties correspond precisely to those expected from conventional scientific paradigms represents, in the author's opinion, a spectacular achievement of the highest order of intellect and scientific skill. Perhaps the future will bring comparable accomplishments involving the use of topological figures, especially those whose geometric properties are not revealed by an intuition that is provided by Euclidean geometry alone: for example, knots and Mobius strips.[461

From time to time chemists create ideas that are judged by the profession to be imaginative, intuitive, and novel. Whether the values discerned in these ideas by the community of practitioners are intellectual, aesthetic, or practical, there is a consensus that such ideas increase the rate at which progress can be made in the field. It has been suggested that imaginative and creative thinking are distin- guished from ordinary everyday thinking by a willingness to accept vaguely defined statements and to structure them, by a persistent preoccupation with puzzles before

900 Angew. Chem. In t . Ed. Engl. 25 (1986) 882-901

solutions are apparent, and by extensive intuitive knowledge of the relevant aspects of The ability to achieve sudden insights into situations by recognizing sim- ilarities between new puzzles and solved puzzles would appear to depend on the ability of the brain to recognize familiar patterns and clues for the mapping of these patterns onto new puzzles.

P ~ l a n y i [ ~ ” has proposed that great scientific discoveries are made by practitioners who believe in the attribution of reality to scientific concepts, and that there can be an a priori knowledge of such underlying realities. He terms this knowledge a vision:

. . . the vision of a hidden reality, which guides a scientist in his quest, is a dynamic force. At the end of the quest the vision is becalmed in the contemplation of the reality revealed by a discovery; but the vision is re- newed and becomes dynamic again in other scientists and guides them to new discoveries.

P ~ l a n y i l ~ ~ ~ further notes that the vision is driven by the strength of imagination guided by intuition, which is homeomorphic with the view that the strengths of creativ- ity in intellectual processing are guided by geometry.

The author thanks the National Science Foundation and the Air Force OfJice of Scientijic Research for their generous support of this investigation. The author is also tremendously indebted to numerous friends and colleagues who over the past decade have participated with him in delightful discus- sions of the geometric and topological approaches to intellectual processing in the chemical sciences. A special thanks goes to Professor Lionel Salem for prouiding. over many years, a stimulating and revealing tutorial on seeking sameness in the solution of chemical puzzles.

Received: Ocrober 14, 1985 [A 592 IE] German version: Angew. Chem. 98 (1986) 872

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