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 beautifully 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
tion is perceived in a manner that allows achievement of a
closed interpretation.
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 intellectually is a pleasant and reinforcing experience.
Figure 1 schematically depicts some relationships
among the components involved in intellectual processing.“,21The 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.[31The 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)
882
0 VCH Verlagsgesellschafi mbH, 0-6940 Wernherrn. 1986
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MODIFIED OR
REARRANGED
-----
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POSSIBILITIES
CLOSURE ?
ACTIVITY
INTELLECTUAL
PROCESSING
OPERATIONAL
OPERATIONAL
Fig. I . A schematic representation of normxl i i i l c l l c ~ t u , i l . I L ~ I \ 11) \l~o\\iilg[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
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 counterexamples.[’] 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 extrapolating 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 acknowledge 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 uncomfortable feeling of ambiguity may be relieved and converted into a feeling of excitement by resorting to operations 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 intrinsic 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.
Anyrn
Chem In1 Ed Engl 25 (1986) 882-901
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.[31If so, facile recognition of 3-D geometric forms is genetically embedded into our ability to intellectualize and to understand space. It is easy to appreciate, 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.[61This 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 continuous 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. Topological 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
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 Euclidean 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 threedimensional 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
C
B
ti& 1 < ongruciii ( A .ind 8 ) drid
geometric figurea
\rmiiifr
( A And C , B and C) Euclidean
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
884
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 interested 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.
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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SOLVE
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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
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[xlsuggests 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.
&p
II
CONVENTIONAL
PARADIGMS
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 resolution of the resulting conflicts)? We suggest that the use
of paradigms and geometry can be effective in resolving
these two issues.
See also W. Wieland, Anyew. Chem. 93 (1981 1 627; Angew. Chem. In/. Ed.
Lnyl. 20 (1981) 617.
Anqrii,. Chrm. Int. Ed. Engl. 25 (1986) 882-901
885
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 interrelations 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 disagreements among practitioners concerning legitimate scientific
problems and methods. As a result, practitioners spend little time involved in controversies over fundamentals. For
(1) generates an a priori
example, a mature
(“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
886
occurred over long periods of time and at great expenditure 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 reiterated 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 extraordinary 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 vexation 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 earlier 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. Furthermore, 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 rearticulated (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 diagnostic of paradigm breakdown: they do not renounce the
old paradigm out of hand. They almost always first conAngew. 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 infallible in their predictive ability, or because of their ineluctable 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 exploitable by practitioners, and because they provide the
driving force for fast-developing experimental methodology.
The decision to employ a particular apparatus o r 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 surrendered, resistance guarantees that scientists will not be constantly 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 expensive and disruptive, and is reserved only for the special occasions 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 addressing 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 replicate 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 concluded that problems are solved by remembering past incarnations. The protocol of scientific paradigms as a
method of puzzle solving is somewhat akin to Pluto’s reincarnations.
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
Angew. Chent. Int. Ed. Engl. 25 (1986) 882-901
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-dependent 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. Topological 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.
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 fopological 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 numbered 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-
02
H
H
V = { C , H}
€ < {(l,2),(1,3), (1,4), (1.5))
{
€ = C-H
]
t i f 5 A \ i \ u a l i L e d 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).
888
dean space, R” ( R ’ , R2, and R 3 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.
LJ
0 - o y
3
I’
O 4
R3(V,E,M)
Dimensionless
space
3-D s p a c e
3 - 0 metric
space
i.
?
n--CdH
$
H
t o p o l o g i ~ ~( ~
I ( lI..
. 1,). ~opologic,il gt.oiiii.tric. K I I,1:).
Euclidean geometric, R’( Y. E. M), representations of methane The rnathematical 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).
I-ig. h Gi.iphic.il
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 relinquish 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, portAnyew. 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.
U p 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 o u r
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 observables.
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 predicting 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
F
I
G1V.E)
SPACES
Visualirable
F]
Elastic
R"(V,EI
FORMS
FORMS
Concrete
1-1
M b 4 )
FIGURES
YxY
MOLECULAR
STRUCTURE
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.
Ailyen. C'hem. Inr
Ed Enyl. 25 (19861 882-901
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 historical 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).ii51The historical pathway is summarized in Figure 8 for cyclo-
Abstract
Visualizable
Concrete
-]CONFICURPITIOVI(numberr.
Allowable shapes
of t h i t h r e e dimensional
reprerentatton
of bonding of
nlemtlnti
I
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~ooi i~i ,l i g t i ~ ~ i i ~ i i i ,
and conformation i n 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 eyample.
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-
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 categorization 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.[’61The 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 reduction 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 mixture 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 spatial 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 (conformations) 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 regarded 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. Finally, 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 teAngew. 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 nuclei 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
A n g e h . Chem. Int. Ed. Engl. 25 11986) 882-901
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. Geometric 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 pejorative connotation in the sciences. There is a tendency to assume that scientists are qualitative only when they are incapable of being quantitative; that is, qualitative is merely
second best to quantitative. However, the history and incredible 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. According 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
representations to ... phenomena ... we can now present qualitative results in a rigorous way, thanks to recent progress in topology ... for we know how to define
a form and can determine whether two functions have
o r 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 microscopic 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 mathematically 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 intellectualization, that is, of interpreting geometrically a
given process, either he will seek to create, despite everything, 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 topologi892
cal features of the model are enjoyed by the phenomena
and vice versa. This potential relationship is extremely
powerful when it exists because it 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 diagrams.[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 noninterconvertibility 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
CH4.
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
H
I
n-c--H
I
H
I
n
I
n-c--H--H
H-H-C-H-H
I1
Fig. 10 I’ohbible grdph:. tor the rtpre\entalion 01 methane Class I c o n b i b t b 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.
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 b e
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 3D 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 monosubstitution. 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 pyramid. Thus, the experimental isomer count based on topological and geometric thinking reduces the choices to only
three molecular structures (Fig. 1 l)!
square
square
pyramidal
planar
tetrahedral
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
I
H-C-H
I
-
Y
I
X-C-H
Y
I
or
I
H-C-H
I
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 Euclidean 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-dependent 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 timeaveraged 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 ti ) n c t i Atom h i a >ubstitueiit X in square-planar or
I I; I 2 i < i ~ p l , i ~ c i i i ~ill
\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-disubstiAngew Chem. Inr. Ed. Lngl. 25 (1986) 882-901
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
893
topologically equivalent chemical structures tend. The
chemical nature of methane does not change for sufficiently small deformations of its structure.
An experiment might be possible on a such short time
that most methane
be
“caught” in a nontetrahedral shape. Whatever the technical 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 l o - ’ * to
s. Many conformational systems in
organic chemistry that are normally considered as being in
dynamic equilibria are “static” on such a time scale.
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 anticipating 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 multidimensionality 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 photochemist 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,[271there are four components or states, corresponding to the four possible electron occupancies of the
HO and LU orbital of any R and P: So, S I , S2, and TI
(Fig. 15). The energetic ordering of these states is invariably Str< T i < S < Sz for ordinary organic molecules.f281
894
R
-
-
4-
-4-
+t
44-
+
t
-
SO
TI
S1
s2
Fig. IS. The relationship between the HO and L U orbitals for a reactant
structure R and i t s four lowest-lying electronic states: S,),T,, s,,and s?.
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 Euclidean space, and which we can eventually transform into a
Euclidean geometric figure with a metric, namely, the
quantitative relationship of structure to energy. Before discussing 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 repeat the process to determine the connections between
D/Z and P.
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?
-
LU
-
HO
-
-L U
HO-
DIZ
R
- - -4-+
DI Z
4 - t
%-
'D
21
-%
22
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
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 described by four electronic D / Z states. Twisting about the
-
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 comparable energy for the diradicaloid structure D/Z. Below: The four electronic 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 hydrocarbon 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 3 D states are
expected to be similar, as are the energies of the Z1and Z2
states.lzY1The 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 at 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
-?-#R
tt
D'
-
O/Z
Fig. 17. Keaciioii grdph l o r the transformation R-D/Z
Angew Chem Inr
Ed. Engl. 25 (1986)882-901
(see text).
&
900
00
H
H
180°
H
H
Fig. 18. State correlation diagram or reaction graph for the c f . \ . ~ r u n ,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)]
895
+
or the suprafacial-suprafacial [2 21 cycloaddition of two
molecules of ethylene to give cyclobutane 3 [Eq. (b)].
n -
L
J
L
J
1
2
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. Although 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, groundstate ethylene; in terms of the p orbitals on the two carbon
atoms, p i and pz, the TI’ wave function has the form
p l ( l ) p z ( 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 is purely covalent and has no ionic character; its
wave function has the form p l ( t ) p 2 ( 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 func-
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,
T I , 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 ojumps near the diradicaloid
tions.~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 S2, SI, and TI surfaces
near the D / Z structure.
2. A maximum in the So surface near the D / Z structure.
3. The close approach of the So and TI surfaces near the
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
896
tip. 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
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 considered.’32- -3-31
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:
1. At diradicaloid geometries, So-+TI and T I + & 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-+TIjump 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.
Ph
Ph
Ph Ph
An
An
An
Fig. 20. Left: The prototype redcilun grdph 01 a giuund-ai‘it~-lurhIddt‘nre‘rction. 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- o r trans-1-alkyl-2-anthrylethyleneresults only in for-
mation of trans product. This result is rationalized by postulating the lack of an effective attractor minimum at the
D/Z structure on the S , o r 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 r e a c 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.
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-anthrylethyIenes‘”’ (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 temperature 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 cisor fron.7-stilbene on either the S, or T I surface are consistent with the results of direct photoexcitation and tripletsensitized 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 photoisomerizations are “one way” on both the s, and Ti surfaces ; that is, direct or triplet-photosensitized excitation of
Anyen Chem I n ! Ed Engl. 25 (1986) 882-901
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 T I 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
897
R
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 Coulombic 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
S I surface should occur on the N side of the ground-state
maximum.
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, predicts 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
- b-$
&-&I,
t
t
t
h g 12. Ut.lationships ol~norhornadienednd quadricyclane structures to zwltterion 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 depending 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-
The rearrangement of Dewar benzene to benzene may
be viewed as a forbidden ground-state disrotatory electrocyclic reaction.[381The 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 Dewar benzene and benzene are used to construct the energysurface 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.
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,.
898
Experimentally, chemiluminescence is observed upon
thermolysis of Dewar benzene and triplet benzene is produced, albeit in very low yield.[381The reason for the low
yield is readily explained by the paradigm in one of two
ways: (1) The diradicaloid structure does not allow sufficiently strong spin-orbit coupling (which is expected to be
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 twodimensional 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.[2x1Thus, 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.[391According to the
paradigm (Fig. 24), an efficient adiabatic reaction along
the T I surface is highly probable for the same reason that
the S o - + T jump
I
is inefficient (i.e., the absence of an efficient spin-flipping mechanism). Experimentally, this prediction 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.[381This
corresponds to a different process that is in competition
with and evidently faster than the process leading to excited singlet benzene.
This paradigm has nothing to say directly about the formation o f prismane 8. The paradigm does, however, provide a possible explanation for the formation of groundstate benzene, since an attractor minimum on the s, surface is suggested as a possibility in the prototype surface
Angew. Chem. Int. Ed. Engl. 2s (1986) 882-901
(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
I1 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 IZ
Fig. 25. Prototype reaction graph for the intramolecular hydrogen abstraction of photoexcited ketones. For A, B, and C, see text.
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
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 exci 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,
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. Nonetheless, the possibility of a jump from T I to So is suggested by
the reaction graph. Experimentally, it appears that only the
T , - + 3 Dpathway is followed (Fig. 26), since the quantum
yield for product formation from TI can approach unity,
that is, each molecule in the T I 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 magnetic 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 ratelimiting step in reactions of triplet d i r a d i ~ a l s . [ ~ ' ~
7. Geometry, Intuition, and Imagination
2
'D
-
Norrish type I1 products
SO -
In conclusion, we consider the possible relationships between geometry, intuition, and imagination. Intuition may
be defined as an instantaneous comprehension or apprehension 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 apparently 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 efforts in organic syntheses. Of particular note is the role of
the Platonic solids such as the tetrahedron,["I the
and the dodecahedron.["51The 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 distinguished from ordinary everyday thinking by a willingness
to accept vaguely defined statements and to structure
them, by a persistent preoccupation with puzzles before
T1--sL
s1-
/-
Z
3D
Norrish type I1 p r o d u c t s
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 S 1 + A + ' D pathways is available from the
observation that an optically active ketone in the S , excited
state, 9(S,), undergoes Norrish type I1 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
900
10
Angew. Chem. I n 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 similarities 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 renewed and becomes dynamic again in other scientists
and guides them to new discoveries.
P ~ l a n y i lfurther
~ ~ ~ notes that the vision is driven by the
strength of imagination guided by intuition, which is
homeomorphic with the view that the strengths of creativity 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 discussions 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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901