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Chemistry and Logical Structures.

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Chemistry and Logical Structures
By Zvar Ugi, Dieter Marquarding, H a n s Klusacek, George Gokel, and Paul Gillespier*l
The utility of the basic structures of modern mathematics ,for chemisfry is discussed;
general set theory, topology, and group theory are shown to pervade almost all static
and dynamic aspects of chemistry. Chemical analogy, the systematic classification of
molecules and a corresponding nomenclature system, conformational transformations,
polyhedral rearrangements, and the relations between starting ma ferials, fransition
complexes, and final products of chemical reactions are examples of where we apply
the elements of modern mathematics to the solution of chemical problems.
“Die Moral dieser Geschichte ist ofenkundig: man nehme
solche vorZau$gen kombinatorischen Schemata wie die Valenzdiagramme nicht zu wortlich, so niitzlich sie auch als erste
Fiihrung in einer scheinbar zusammenhanglosen Masse von
Tatsachen sind. Von einem mit ein paar scharfen Strichen
entworfenen Bild der Wirklichkeit kann nicht erwartet werden, dap es der Vielfalt all ihrer Schattierungen adaquat sei.
Gleichwohl muJ gerade der Zeichner den Mut haben, die
Linien kraytig ZU ziehen.” - Hermann Weyl, on the chemical
1. Introduction and General Considerations
1.1. Chemistry and Mathematics
Mathematics has recently undergone substantial reforms.
A Paris group of mathematicians formalized the new approach to mathematics (see Section 1.3.1.) in their book
“Elkments de Mathkmatique” published under the pseudonym Nicolaus Bourbaki
The organization of classical
pathernatics, which was strongly influenced by its applica-
Prof. Dr. lvar Ugi, George Gokel, B.Sc.,
and Paul Gillespie, M.Sc.
Department of Chemistry
University of Southern California
Los Angeles, California, 90007 (USA)
Dr. Hans Klusacek and Dr. Dieter Marquarding
Wissenschaftliches Hauptlaboratorium
der Farbenfabriken Bayer A.G.
509 Leverkusen (Germany)
Currently Postdoctoral Research Associates
at the Department of Chemistry
University of Southern California
Los Angeles, California, 90007 (USA)
[ l ] H . Weyl: Philosophie der Mathematik u n d Naturwissenschaft,
Oldcnbourg. Munchen. 1966. p. 352
[ 2 ] See, e.g. : a) M . Humermesh: Group Theory and its Application to Physical Problems. Addison- Wesley, Reading, Mass
1962: b) R. McWeeney: Symmetry. Pergamon, New Yorh 1963;
c) S . Lipschutz: Set Theory. McGraw-Hill, New York 1964;
d) S . Lipschutz: General Topology. McGraw-Hill, New York
1965; e) H . J . Kowalski: Topological Spaces. Academic Press,
New York 1965; f) H . Meschkowskic Mathematisches Begriffsworterbuch. Bibliographisches Institut, Mannheim 1965; g) J . Dugundji: Topology. Allyn and Bacon, Boston 1966; h) J . Schmidt.
Mengenlehre. Bibliographisches Institut, Mannheim 1966; i) N .
Bourbaki: Elements of Mathematics. General Topology. AddisonWesley, Don Mills, Ontario 1966; j) A . J . Coleman in E. M . Loebf .
Group Theory and its Applications. Academic Press, New York
1968, p. 61 ; k) R. H . Fox and R. Crowell: Introduction to Knot
Theory. Blaisdell, Waltham, Mass., 1963.
Angew. Chem. internal. Edit. 1 Vol. 9 (1970) / No. 9
tions, is now being oriented towards generating mathematics from the “basic structures”, i.e. the algebraic structures, the ordered sets, and the topological spaces. The ideas
of the Bourbaki group can be considered a consequential
extension of the formalistic approach initiated by Hilbert.
Pure mathematics, from the contemporary viewpoint, is a
general, hypothetical, deductive formalism of relations.
This formalism, which may serve an objective, is created
without any consideration of the objective. It provides us
with a theory of logical structures which can be used as
models for some characteristic properties of reality. It is
our intention to develop a mathematical concept of molecules, confined and prejudiced as little as possible by our
The relations between objects, rather than the objects
themselves, are the primary concern of science. Usually,
spatial relations are p r i m facie defined on the basis of the
Euclidean space of our experience. It is, however, as physics illustrates, often advantageous for the solution of certain problems to use purely mathematical concepts of
space, which have in common with Euclidean space only
some formalistic characteristics.
The question of whether or not a certain mathematical concept provides the basis for the most suitable model of a
phenomenon is consciously asked in physics. In chemistry,
however, three-dimensional Euclidean space and classical
mathematics have been so strongly favored by tradition
that the use of the more abstract mathematical structures
was confined to those who considered chemistry an extremely complicated special case of physics, and accordingly treated chemical problems with the advanced and
sophisticated methods of theoretical physics. The use of
mathematics in chemistry has been oriented almost exclusively towards generating numerical data in order to test
the validity of some mathematical model of an experimentally observed phenomenon. Little use has generally
been made of the fact that mathematics is formalized logical
thought and can be used directly to gain insight into the intrinsic logical structure behind chemical problems.
The coming generation of scientistswill have grown up with
modem mathematics and it is safe to predict that this
generation will use these fundamental concepts to an extent which we cannot even imagine. The present paper attempts to outline some aspects of chemistry where structure-oriented mathematics offers a promising approach.
1.2. Chemical Topology
Among the mathematical structures, the group and topological space have the greatest potential for chemistry.
The striking success of group theory in solving problems
which are related to symmetry is well documented. Group
theory is also a powerful mathematical tool for dealing
with such deviations from geometrical symmetry which
can be represented by algebraic symmetry [ 7 . l5].
Topology has not found extensive application to the solution of chemical problems. It was pointed out during
various pertinent discussions[31 that stereochemistry ought
to be discussed in terms of topology, because some o f the
metric features of molecules are not essential for stereochemistry.
PreZogc4l proposed the term “chemical topology” t o be
used for the description of stereochemical entities in topological terms :
“Chemical topology deals with properties of geometrical figures
that are either isomorphous or homomorphous with the
momentary or time average topography of chemical particles.”
“When I use a word”, Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean - neither more
nor less”. Lewis Carroll, Through the Looking-Glass 151.
It is difficult t o predict whether this concept o f chemical
topology will ever provide a basis for an operational formalism in chemistry or lead to new insights.
Nonetheless, Prelog’s definition of chemical topology has
initiated and stimulated the use of mathematical structures
in chemistry, as discussed in the present paper.
The main difficulty in generating a topological theory of
stereochemistry arises from chirality; it is not possible to
distinguish between a geometrical system and its nonsuperimposable mirror image by the representation of
molecules as topological spaces. Yet, it is possible to
generate a framework of thought for the treatment of
chemical problems by a combined application of basic
concepts from topology and group theory.
1.3. Elements of Modern Mathematics
Sets are specified by giving them a property P which can be used
to test the claim of membership to a set by any object in the universe, the set of all objects having the property P is defined by
{xlx has the property P}.In the following specific applications
of set theory, all sets under investigation are subsets of a fixed
set. We call this set the universal set, U . The empty or nullset, p, is
the set which contains no elements; it is a subset of every other
set. Important additional concepts are intersections and unions.
The intersection of two sets A and E, symbolized A n E, is the
set of elements which belong to both A and B, i.e.
A n B = { x l x e A and x t B }
The union of two sets A and B, symbolized A u E, is the set of
elements which belong to at least one, A or B, i.e.,
A uB
{ x l x t A or x e B }
Here, “or” is used in the sense of “and/or”.
Two more important set operations are best introduced at this
point. The complement of set E with respect to set A or simply the
dzference between A and E, denoted by A\E, is the set of elements which belong to A but not to E, i.e.,
A\B = { x l x ~ A , x $ B }
The absolute complement of set A , symbolized by A‘, is the set of
elements which do not belong to A , i.e.,
AC = {xlx.
u, x $ A }
Or, A‘ is the difference between the universal set Uand the set A .
These four set operations can be further illustrated with the use
of Venn-Euler diagrams. Here, the sets A and E are symbolized
by circles and the universal set U shown as a rectangle. The
indicated operations are shaded.
a1 A f l B
bi A U B
%61 I*]
1.3.1. Sets [Zc,f-hl
A fundamental concept in all branches is that of a set. A set is
any well defined collection or class of distinguishable objects of
our experience or thought. These objects are called elements or
members of the set. The theory of sets contains only one basic
relation, usually denoted by E and expressed as belongs to. Examples pertinent to our discussion include not only sets of
individual elements, numbers, and points, but sets (classes,
families) of sets. A set A is a subset of a set E, or conversely E
is a superset of A ;
A c B
or B I A
if each element in A also belongs to E.
[3] Euchem Conferences on Stereochemistry, Biirgenstock
141 V . Prelog. Abstract of t h e Roger Adams Award Lecture on
l i t n e 17. 1969. 111 the A . C S Meeting. Salt Lake City, Utah.
[S] L. Carroll: The Annotated Alice (edited and annotated by
M . Gardner). World Publishing Company, New York 1969.
p. 269; see also M . Gardner: The Arnbldextrous Universe. Mentor
Books, New York 1969, pp. 74, 118.
[6] For a review, see Mathematical Aspects of Chemical Reactions
- R . Aris, Ind Eng. Chem. 61, No. 6, p. 17 (1969), especially footnotes 17, 18, 27. 34, 35, 216.
[*I We suggest that readers familiar with elementary topology
and group theory omit Section 1.3.
cl A\ B
d l A‘
Fig. 1. Venn-Euler diagrams of set operations.
To each two objects, a and b, there corresponds a new object
( a , b ) called their ordered pair. Ordered pairs are required to
( c , d ) if and only if a = c and b
The product set in set theory, written A x B and called the Cartesian product, is the set whose members are all ordered pairs
( a , b ) where a E A and b e E , i.e.,
A x B
{ ( a , b ) l a G A ,b t B }
The product of a set with itself, i.e., A x A is symbolized by A’
and plays an important part in the definition of group structure
(see Section 1.3.3.).
A rulefthat assigns to each element of a set A a unique element
of a set E is called a mapping f of A into E : A + E (or A 2 E).
The specific element 6 E E assigned to a E A byfis called the image
f ( a ) of a underf(or the value off at a). Figure 2 illustrates the
various types of mapping.
Angew. Chem. internat. Edit.
Vol. 9 (1970)1 No. 9
mathematical structure has at its foundation a set, but one usually imposes an extra structure. Concepts of neighborhood, open
ser, limit, and continuif!. necessary for the formalisms that follow.
can be applied to a set A only if it has been supplied with a topology. A topological space is a set equipped with a topological
lone-one into1
Fig 2 Various types of mapping of sets A
Ione -oneonto1
A propositional function P ( x , y ) on the Cartesian product
A x B has the property that P(a,b),a and b being substitutes for
x and y in P ( x , y ) , is true or false for any ordered pair
(a,b) E A x B. If A is the set of all chemists and B is the set of all
known chemical compounds, then P(x,y) = “ x discovered y” is
a propositional function on A x B. For example, P( Waksman,
streptomycin) = “ Wuksman discovered streptomycin”, P(Fischer, streptomycin) = “Fischer discovered streptomycin” are
true and false, respectively.
A binary relation R in the set A is, intuitively, a proposition such
that for each ordered couple (a,a’) of elements of A one can determine whether aRa’ (a is a relation. R. to u’) is or is not true.
Any subset R of A x B is called a binary relation, R, from A to B.
Formally stated in terms of the set concept, the binary relation R
ofA x Bis:
R c A x B
( a , b ) E R = aRb
A binary relation R in A is called an equivalence relarion if:
If we let A be any non-empty set, a class T of subsets of A is
called a topology on A if T satisfies the following axioms:
T I: A and 9 belong to T.
T I1 : The union of any number of sets in T belongs to T.
T 111: The intersection of any two sets in T belongs to T.
The pair (A,T) is called a topological space and the members
of T are called the open sets of the topological space (A,T). If
only one topology on A is considered we refer to the topological
space A. Sometimes it is necessary to use different topologies on
the same set to produce the formalism test suited for the solution of a problem. Here a comment by Co/eman[2j1is appropriate. ‘‘. . . the chief skill needed is the ability to switch adroitly
from one topology to another as rapidly as a quick change artist
exchanges personalities. In such situations the old notations
and habits of mind are very inhibiting. . .”
If an open set T contains a point p, the open set T is called a
neighborhood of p .
A family, (V,[txt T f , of sets is called a basis for a topology if
the family of arbitrary unions of these sets forms a topology.
To illustrate the defined concept of topological space, one can
see that a topological structure on the set of real numbers R is
defined by using the open intervals and their unions as the open
sets T.(Rcan be represented by a real line whose points are paired
with the real numbers in a natural manner.) In this example,
the neighborhood concept is obvious. The neighborhood of a
point on a real line is any open interval which contains this
point (compare the sequence rules discussed in Section 2.3.).
The relation R
cortains the diagonal
a Aa
(aR b) 3 (6 R a )
(a, b) 6
(6, a)
The relation K
i s Symmetric
about the diagonal
The equivalence class [a]of any element a E A is the set of elements to which a is related. If the equivalence relation R is “ x
is congruent t o y modulo 2” in the set Nof integers, these are two
distinctive equivalence classes, the even and the odd numbers.
In chemistry most reasoning is done by analogy, i.e. by classification according to equivalence relations. For this to be meaningful, the pertinent equivalence relation must always be defined.
1.3.2. Topology[Zd-87’zj’
In modem mathematics the notion of a set is an undefined term
whose use is governed by various axioms. This notion is used
only as a framework from which mathematical theories are developed by defining axioms and applying logical deduction. Any
Angew. Chem. internat. Edit.
Vol. 9 11970) 1 No. 9
Open and interior are best illustrated with the concept of the open
interval of real numbers.
Open interval from a to b : ]a,b[ = { x [ a< x < b }
Closed interval from a to b : [a,b] = { x l a 6 x 6 b}
We can further define our topology to include the Hausdorff
separation property:
T IV: Any two points of a space A have disjoint neighborhoods M and N whose intersection is the empty (null)
set, i.e., M n N = 9.
A set A is called a metric set if there is given for any two points
a and b a real number D(a,b), the distance between a and b.
A metric in a set A is a function D : A x A + R such that:
T V : D(u,u) = 0
T VI: D(a,b) = D(b,a) > 0, for a # b
T VII: D(a,b) 5 D(a,c) + D(c,b)
Each metric D induces a topology. For example, by using the
~ ]for all y > 0, U E A ,the balls and all
balls B(,,,,= { X I D ( ~
their unions are the topology on A .
Let (Ai,Ti)be a finite family of topological spaces where i c I
with I being a finite set of indices. The Cartesian prodrtct topology
n T i o f n A i , is that topology having the open sets U,x . . . . x U,,
where Vi E Ti, as a basis.
The mutual neighborhood relations within families of intersecting sets can be described graphically by the nerve of the family.
The nerve of an n-fold non-empty intersection is given by an
n - 1 dimensional graph (simplex); the non-empty intersection
of two sets is represented by a line, a non-empty intersection
of three sets by a triangle (see Fig. 3).
The associative law is obeyed, i e . , (a.a').a'' = a.(a'-a'')
for all a,a',a"e A .
G 11: There exists an element e e A (called the unit element)
such that e . a = a for all a e A .
G 111: For each a e A there exists an element a-' (called the
inverse element) such that u-' .a = e.
The group G = (A,cp) is characterized by the set A and the
mappingcp. Two groups are compared by mappings of the underlying sets that comprise the structures, i e . ,
If the condition
is satisfied, such mappings are called homomorphous. A special
case is a homomorphism which is a one to one mapping of a
group onto another group, a bijection (see Fig. 2). Such a map is
called an isomorphism. In this case the two groups have the same
structure. Any homomorphism of a given group into the general
linear group GL(n,R) of non-singular ( n x n ) matrices with real
is a representation of the group (A,cp). If this mapping is an isomorphism into GL(n,R) we have a faithful representation.
A group is characterized by its multiplication table, a square
array of group elements arranged to show the results of the combination of group elements. If we have a group of order two
with elements A and Band let the combination of these elements
be defined such that
Fig. 3. Graph and matrix representation of neighborhood relations of a
family of intersecting sets (nerve).
Such graphs can also be represented by matrices, using zero and
one to indicate whether two sets are disjoint or have a nonempty intersection respectively. If matrix element aij = 0, this
indicates that sets i and j are disjoint, and aij= 1 means their
intersection is not empty (cf. Fig. 3). The secular determinants
of the HMO theory are based upon such topological matrices.
By their very definitions, these matrices are symmetrical across
the main diagonal. If one replaces the connecting lines between
pairs of points by arrows (as in category theory) the direction of
the connection can be expressed by the indices of the matrix
elements; a12= 1, azl = 0 means point one is connected with
point two by an arrow 1 -+ 2. A formalism for the representation
of molecules including all the steric features could be based upon
category theory. However, such a treatment would afford a
formalism which, except for computer use, is too complicated
for practical purposes (see Section 4).
For the description of complicated reaction mechanisms with
consecutive and parallel reactions (see ref. [6,7]),the category
theory offers interesting possibilities. If we represent the reactants, products and all the observed and hypothetically assumed
intermediates of a reaction by points and represent the observed
and assumed reaction by arrows, the mapping of the sets of
arrows onto the sets of points describes the reaction kinetic
scheme. Here, the recent discussion 16] of topological (homology theoretical) concepts of "chemical reaction" and "reaction mechanism" is of particular interest. In this context.
thermodynamic equilibria can be viewed as equivalence relations.
this statement could be represented by the multiplication table
If we take the real number + 1 as the element A and the real number -1 as element B, and make multiplication the group
operation, we have a particular realization of the above group
with the multiplication table
The concept of a symmetry group is illustrated by Fig. 4.
- __-2
1.3.3. Groups L2a*b*g*j1
Another structure that can be imposed upon a set is the group
structure. A set is called a group G = (A,cp) if there is a mapping
cp : A x A + A (we write cp(a. a') = a . a') satisfying the following
[7] E . Ruch and I. Ugi, Theor. Chim. Acta 4 , 287 (966); Top. in
Stereochem. 4, 99 (1969).
Fig. 4. Rotations C. and reflections 0 .
Angew. Chem. infernat. Edit.
/ Vol. 9 (1970)/ No. 9
The operations that bring this triangle in Fig. 4 into a position indistinguishable from its original position, i.e. self-coincidence, are called symmetry operations. Symmetry operations may be performed sequentially. We define, for any set
of symmetry operations R and S, a law of combination cp symbolized by RS, meaning first apply S, then R. Sequentially performed symmetry operations obeying the combination law cp
form groups G = (A,cp).
If we map the symmetry operations G, of Fig. 4 on the letters
the multiplication table obtained is that of the group symmetry
Three dimensional point groups are characterized by the following generators:
C, = positive rotations moving 360"/n (C, axis normal to the
plane of the paper).
o, = reflection across a vertical plane.
oh = reflection across horizontal plane.
o, = reflection across a vertical plane bisecting the angle between two C; axes.
C; = rotation through 180" about axes normal to the principal
C , axis.
S, = iniproper positive rotations through 360"/n.
The Schoenflies notation is used to denote the point group symmetries in terms of their generators. C,: {C"}, Cnv:{Cn,ov},
Sn: {SnI, D n : {CnLG}, Dmi: ~Cn,C;,~d},
D,,,,: {C,, C;,o,}, T : {C;,C;yz}, T,: {S&C;yz),T,: {C;. Ciyz,i},
0: {c:,C;YZ}, 0,: {c;,c;y'}, Y: {C3,C5}.
The trajectory followed by a given point resulting from sequentially applied group operations is called its orbit. One can use this
concept for numbering a symmetrical set of points (see Fig. 6 ) .
An informative mapping of one three-dimensional Euclidean
space E3 onto another 8, can be represented by an orthogonal
+ a12y +
+ a32y + a332
Each row in the resulting matrix is the new coordinate of the
original point. The distance relation between the points as well
as the origin is maintained. A has the following properties:
M I: a l l Z+ at,' + a12 = 1
aZl2 a,?
a,,' = 1
a32 = 1
M 11: allazl + al2aZ2 aI3az3= 0
+ a12a32 + a13a33
Notice that the group {( + 1)( - I)} also satisfies the same multiplication table, but that the latter group is homomorphic to the
first two.
Order, subgroup, and coset are further basic concepts of group
theory essential for our discussion. In a group Gcontaininggelements, g is called the order of the group. Any collection of the
elements of G which by themselves form a group H i s a subgroup
of G. Every group contains two trivial or improper subgroups,
the unit element, E, and the whole group itself, the rest of the
subgroups considered proper. The set of subgroups does not
correspond to a partition of the original group as E must be a
member of every subgroup.
Let H = { A , , A , . . . Ah] and R,, R, . . . . be elements of G not
contained in H ; then the collection defined by
{AIRk,A,R,. . . A,R,)
is called the right coset of H with respect to R,. Analogously,
the double coset HRJT (R,EG, R i g H , R,#H') is obtained from
the two subgroups H and H' and their relative complement.
Notice that the coset HR, is not a group as it does not contain
the identity element E, the same also applying to H R , H .
The symmetry group of a regular polygon with n vertices is of the
order 2n. It contains n rotations and n reflections.
( E = Ct,C,,C2, - - - Ct-' = C.,O,GC, - - - o C t - ' }
Since this group G can be generated from the rotations C, and
the reflections o,these are called the generators of G .
Symmetry operations that leave one point of a figure unmoved
form a point group.
Angew. Chem. internat. Edit. / Vol. 9 (1970) 1 No. 9
+ a22a32 + a23a33
M I and M I1 state: The product of the matrix A and its transpose
AT (the 'mirror image" of A across its main diagonal) is equal
to the unit matrix E3 symbolized:
The pr3duct of any two orthogonal matrices is an orthogonal
matrix and the inverse of such a matrix is its transpose:
A A - ' = E,
A-' = AT
A set of all orthogonal matrices is a group O(n). We define the
group operation as multiplication.
If we dcfine:
SO(n) = {AeO(n)[DetA
as that group, resulting from the transformation, whose determinmts are equal to + 1, it can be shown that a three-dimensional figure retains its "handedness" as a result of the operation
(see Sections 2.2. and 2.5.) and the result can be defined as a
rotation. If the Det A = - I a three-dimensional figure reverses
its "handedness" upon transformation, this result corresponding to a reflection.
We h a w a choice as to the number of dimensions allowed in the
representation space E" of a molecule. An EZspace would allow
some molecuies to be chiral [e.g.. (2aJ - ( Z r) ]when they would
not be SO in E3. However, since all E' and E2 can be embedded
into E', we use the E3 space to describe chemical systems.
Further, we will consider achirality as equivalent to O(3)
The symmetric group S, of n letters or symbols consists of the
n ! mappings of a set of n letters on itself.
1 2 3 ..... n
1 2 3 . . . . .
These mappings are transformations from one arrangement of
the numbers 1 . . . n to another and are called permutations. The
symmetric group S,,the group of all permutations on n letters
is of great general importance in mathematics and science.
Permutations play a central role in this paper.
cp is an action, cp:
g x X - X , such that the transformation g (elements of the
group G ) on the set X reproduce the set X where g (all g e G )
acting on x = (gx) and g[g‘x] = [gg’lx. Each g e G is a bijection
of X onto X.A family of such bijections is called a symmetry.
A Cartesian product (X, x X, x . . . x X,) which is symmetrized
with respect to the permutational transformation (permutation
of the indices) of the group S, is called the symmetrized Cartesian
product of X = S P X .
( X I x X, x
All the bromochloroethylenes (2a) - (2c) are permutational isomers because they have the same skeleton I, and
they differ only by permutations of the ligand set. (2a)
differs in chemical constitution from (2b) and (2c), which
are stereoisomers.
x X”)
2. Classification of Molecules
2.1. Isomers
An equivalence relation in a set divides that set into pairwise disjoint classes. The members of an equivalence class
are “the same” with reference to the defining relation.
Equivalence relations are the basis for many useful classifications in chemistry whereby order is achieved. This
order provides for the possibility of a systematicnomenclature for chemical systems. A clear example of the utility
of a classification for chemical systems according to equivalence relations is the periodic system of elements.
The set of all chemical compounds can be divided into those
classes which have the same empirical formula, i.e., the
isomers. Those compounds classified according to the criterion of molecular skeletal simiiarity form a further class,
i e . , topologically similar compounds. We can now define
the concept of permutational isomers, chemical compounds
which have in common the same molecular skeleton and set
of ligands, differing only by the distribution of the ligands
on the skeletal positions.
Chemical constitution can be defined as a set of neighborhood relations between the atoms of a molecule *’]. This
can be visualized in terms of bonds and bonded neighbors.
Those permutational isomers which differ only by ligand
permutations at constitutionally equivalent positions are
the subclass of stereoisomers [*I.
The three isomeric dichlorobenzenes (la)- ( I c ) are permutational isomers having skeleton VI (see Section 2.4.2),
but are not stereoisomers.
Br\ c=c,
[8] a) E. L. Eliel: The Stereochemistry of Carbon Compounds.
McGraw-Hill, New York 1962; Stereochemie der Kohlenstoffverbindungen. Verlag Chemie, Weinheim 1966; b) K . Mislow:
Introduction to Stereochemistry. Benjamin, New York 1965;
Einfiihrung in die Stereochemie. Verlag Chemie, Weinheim 1967.
The isomers ( 3 a ) [= ( 3 b ) ] , (3c) - (3e) are all’permutational isomers; (3a) is identical with (36) and can be
brought into coincidence with (36) by rotation and translation. Of these permutational isomers, the truxinic acids
(3a) and (3c) are mutual stereoisomers,whereas a-truxillic
acid 19] (3d) and the reference isomer (3e) (see Section
2.4.4.) differ in chemical constitution from the other
The determination of whether (4a) and (4b) are stereoisomers or non-stereoisomeric permutational isomers is
relevant only to some problems, and involves an arbitrary
decision as to whether the two “axial” chloro ligands distort the 0, symmetry of the skeleton sufficiently to create a
distinction between “axial” and “equatorial” skeletal positions [lol. If such a distortion occurred, the constitutional
equivalence of the ligand positions would be destroyed,
violating a precondition for (4a) and ( 4 b ) being stereoisomers.
The non-stereoisomeric permutational isomers arise from
molecular skeletons whose equivalent skeletal positions
become non-equivalent on attachment of a ligand. In fact,
those skeletons with a symmetry that precludes the nonequivalence of skeletal positions regardless of the ligand
[9] A . Mustufa, Chem. Rev. 57, 1 (1952).
[lo] E.g.: J . Lewis and R. G. Wilkins: Modern Coordination Chemistry. Interscience, New York 1960; F. A . Cotton and G . Wilkinson,
Advanced Inorganic Chemistry, Interscience, New York 1966;
Anorganische Chemie. Verlag Chemie, Weinheim 1967; C. I(.J#rgensen: Inorganic Complexes, Academic Press, New York 1963;
R. F. Gould, Werner Centennial, A.C.S. Publ. Washington, D.C.,
Angew. Chem. internat. Edit.
1 Vol. 9 (1970) No. 9
distribution (like XXVII) are rather the exception than the
rule. Likewise, those skeletons with a symmetry (like VI)
which allow only constitutionally different, non-stereoisomeric permutational isomers, are also in the minority.
A consistent nomenclature for permutational isomers
would simultaneously be a solution to the remaining unsolved problems of stereochemical nomenclature.
Any ‘somer nomenclature must be so devised that there is
a :I1 defined and clear correspondence between names and
iwmers One must be able to determine, unequivocally and,
if possiblc, easily, all constitutional and stereochemical
features of a given isomer from its name, and be able to
classify a given isomeric species according to unambiguous
criteria. It would be a convenient feature of such a nomenclature if it contained both the (R),(S)[‘I1 and the
nomenclature [12] and related nomenclatures [ I 3 ] as special
cases (subsets).
The simplest type of unequivocal assignment of nomenclature names to molecules with a given skeleton and set of
ligands is the kind of nomenclature which is used, for
instance, for benzene derivatives. Thus an adequate description of a given molecule is achieved by numbering the
skeletal positions and including the position numbers with
the ligands (which substitute hydrogen) in the nomenclature name. For instance, ( 5 ) would be l-chloro-2-nitro-4amino-5-methylbenzene.
Achiral objects have symmetries which include reflections
and improper rotations. Whether an object is chiral in
Euclidean space or not depends also upon its dimensionality and the dimensions of the space in which it is observed
(see Section I .3.3.). For example, a two-dimensional chiral
object, such as an idealized, suitably substituted ethylene
[eg., (Za)- (2c)I is chiral in two dimensional space E2, but
achiral in three dimensional space, E3.
Lord Kelvin’s intuitive definition of chirality serves most
purposes of stereochemistry well, although in certain exceptional cases (see below) it does not provide an unambiguous basis for the mathematical treatment of chirality problems.
The mathematical description of chirality is related to
(@,(a- representations of orientation-preserving transformations
A variety of other names could be found for ( 5 ) , however,
by the same procedure (if no further definitions were included in the nomenclature) depending upon which ligand
occupied skeleton position number one, and whether the
numbering was carried out clockwise o r counterclockwise.
However, no matter how the numbering is selected, the
resulting name can always be used to reconstruct (S).
Such simple nomenclatural procedures cannot be applied
to chiroids and molecules with highly symmetrical threedimensional molecular skeletons, e . g . , XXII - XXIX, as
no simple unequivocal correlations of molecules and names
can be achieved.
2.2. Chirality Problems
In order to establish a basis for such a universal nomenclature some implications of chirality must be discussed. Lord
Kelvin [ l 4 ] introduced the concept of chirality as follows:
“I call any geometrical figure or group of points chiral
and say it has chirality if its image in a plane mirror ideally
realized, cannot be brought to coincide with itself.”
[ I l l R. S . Cahn, C. K. Ingold, and V . Prelog, Angew. Chem. 78,
413 (1966); Angew. Chem. internat. Edit. 5, 385 (1966), and preceding papers cited there.
[12] J . E. Blackwood, C. L . Gladys, K . L. Loening, A . E. Petrarca,
and J. E. Rush, J. Amer. chem. SOC.90, 509 (1968).
[I31 I. Ugi, Jahrb. 1964 Akad. Wiss., Vandenhoeckand Rupprecht,
Gottingen 1965, p. 21; Z. Naturforsch. ZOb, 405 (1965); I . Ugi,
K . Offermann, and H . Herlinger, Chimia (Aarau) 18, 278 (1964).
[14] Lord Kelvin, Baltimore Lectures, 1884 and 1893. C. J. Clay
and Sons, London 1904, pp. 436, 619.
Angew. Chem. internat. Edit. / Vol. 9 (1970)/ No. 9
by orthogonal matrices (see Section 1.3.3.).
Along with the objects classified as chiral and achiral, there
is a class of objects called a m p h Z ~ h i r a l [ These
~ ~ l . are characterized by the fact that they are chiral according to Kelvin’s definition but achiral according to orientation-preserving transformations from O(n).
We call an object K amphichiral if there exists an orientation-preserving mapping, h, such that, h : E3 E3 with
h(K) = p(K), i.e., K is amphichiral if there is an orientationpreserving homomorphism of E3 taking K into its mirror
image. For example, the twisted loop and the trefoil knot
are amphichiral objects.
Amphichiral systems might be of significant importance
in chemistry and biochemistry. The discovery of cyclic and
catenated nucleic acid structures leads one to suspect that
amphichiral superstructures of nucleic acids are responsible
for neoplastic processes by interfering with the transfer of
information on chirality.
We propose to make the distinction between chiroids and
achiroids only with respect to the action of orthogonal
matrices rather than the criterion of the superimposibility
of an object and its mirror image by reflection in a plane
mirror and rigid motions.
The set of operators A = { e = “leave the object as it is”,
i = “convert the object into its mirror image”} with the
combination law cp = “apply the operation to the result
of the preceding operation”, forms a group Z=(A,cp).
Converting the mirror image of an object into its mirror
image leads back to the original object (i i = e ) .
The inversion group Z is isomorphic to, and is represented
by, the group V = ({( + I),( - l)}, multiplication) (see below). With regard to achiral objects it is isomorphic to
({( l)}, multiplication). In stereochemistry a chiroid xR
with an (R)-configuration as the standard situation can
produce the (S)-chiroid xs as the result of the operator i
acting upon an (R)-chiroid.
= i-{x.QI
A chiroid with two elements of chirality x and x’occurs in
four stereoisomeric forms.
(xs3xb) { X R I X J i t X S > X k I
This class of stereoisomers is generated from ( x R , x k } by
the direct product group Z x I’ where Z (see above) acts
upon x and I’ acts upon x‘. The above set is isomorphic
Y x V‘ = (( + I ) ( + I)‘,( - 1)( - I)‘,( + I ) ( - I)’,( - I ) ( + 1)‘) and
This relation can be used for the classification of diastereoisomers [131.
The (R),(S)-nomenclature of chiroids with many elements
of chirality essentially corresponds to a mapping of the direct products of the group Z onto the group Vor its direct
The nomenclatural designation (2(S),3(R),4(R)J)-tetrafor D-arabinose (6) is based upon the
direct product group I, x Z3 x Z4 and corresponds to
( - 1)( 1)( 1). The classification of sugars into D- and
L-sugars makes use of a mapping of the set of sugars into
V rather than an ordered product of the Vs.
+ +
The (R),(S)-nomenclature is currently accepted for chiral
stereoisomers. It is far superior to the classical D,L-nomenclature [*l which is not based upon rigorous definitions.
The (R),(S)-nomenclature was conceived in order to discriminate between enantiomers and to classify the absolute
configurations of chiral molecules or elements of chirality
unambiguously according to the criterion of either homochirality or heterochirality, concepts originated by Lord
Kelvin and further refined by Ruth""' for chiroids of
category a (see below).
Two chiral objects x and xi are termed homochiral if it
is possible to define a criterion by which x and x’ are similar,
whereas, by the same criterion, x’ and z, the antipode of x,
are dissimilar. If one finds, by the same criterion, that x’
and are similar (and x and x’ are not) 2 and x’are heterochiral. As it is used here, the term “chiral” does not refer
to the action of an orthogonal matrix. It is used only to
discuss a suitably defined series of compounds in a manner
some chemists find useful. In the following figure
(7a) - (7c) the criteria for comparison is that three ligands
must be identical and distributed in a like manner. As
C,H, and C,,H,Fe are different yet on equivalent skeletal
positions, (7a) and (76) are homochiral, whereas (7c)
is heterochiral to (7a) and ( 7 6 ) .
NHz \
NHz \ ,,’3
csH( C‘
c l 0 H 9 F’e C‘
f 7b)
f 7c)
[15] a) E. Ruch, A. Schonhofer, and I. Ugi,Theor. Chim. Acta 7,
420 (1967); b) E. Ruch and A . Schonhofer, Theor. Chim. Acta 10,
91 (1968); c) E. Ruch, Theor. Chim. Acta I I , 183 (1968); d) E. Ruch
and A. Schonhofer, Theor. Chim. Acta, in press; e) E. Ruch, personal communication; f ) E. Ruch, W. Hasselbarth, and B. Rich-
ter, Theor. Chim. Acta, in press; see also G. Polya, Acta Math.
68, 145 (1937); g) E. Ruch, Lectures at Zurich and Los Angeles
Accordingly, the classification of a chiral object with regard
to homochirality is possible only by comparison with a
chiral reference system.
It is possible to communicate the fact that some object is
“right handed”, only by making the statement that it is
homochiral to some other comparable “right handed”
Despite its obvious merits, the (R),(S)-nomenclature is not
the ultimate solution to the problem of stereochemical
nomenclature for the following reasons:
First, it is limited to configurational assignments.
Second, in the case of molecules with complex molecular
skeletons, its underlying definitions do not provide a basis
for dealing with the molecule as an entity, but require consideration of the molecule by (tetrahedral) subunits and
thus sometimes makes additional definitions necessary.
Third, and most serious, the (R),(S)-nomenclature is confor chiroids of class
fined to chiroids of class
b [15b,c1 its definitions become meaningless unless they are
arbitrarily extended. The (R),(S)-nomenclature does not
therefore allow for the consequence of two different
chirality classes a and b. This nomenclature was conceived
before Ruch [”I showed by the group theoretical analysis of
chemical chirality that there are two different classes, u
and b, of chiroids.
A class of chiroids for which there is a non-arbitrary rightleft classification is termed a class of category a, otherwise
a class of category b. A molecule belongs to category a if
and only if it has either only two skeletal positions, or if
each skeletal plane of reflection contains n - 2 skeletal
positions, n > 2 being the number of skeletal positions
Let us now consider Ruch’s [15brc’ classification:
By definition, any transformation of a chiroid x, of class u
into its antipode must pass through some state in which
all differences between x, and ii. vanish. This can be
visualized in terms of a sharp achiral border without
loopholes between x, and L,such that all compounds on
the x,-side of the border are homochiral to x,, and that
beyond that achiral border any chiroid is homochiral
to L.
For chiroids of class b there is no such sharp border between the antipodes % and zb (class b,) or the border has
loopholes (class be). Accordingly, chiral molecules of
class u are subject to classification by statements of homochirality, like right- and left-handedness o r the D,L or
(R),(S)-nomenclature, and chiroids of class b are not.
Let a chiral molecule be represented by an achiral skeleton
with n ligand bearing positions and n ligands (groups) attached to the framework. Then, the following applies:
Chiroids of class u do not contain any skeletal planes of
reflection or improper axes of rotation where more than
two non-equivalent ligands are attached to skeletal positions not coincident with the stated fixes of symmetry. For
these chiroids of class u there are no reflections or improper
rotations of the skeletal symmetry group which exchange
more than two non-equivalent ligands when the chiroid is
transformed into its antipode. For the molecules in class u
to become achiral, it is necessary and sufficient to remove
or to make equal two non-equivalent ligands which do not
Angew. Chem. internat. Edit.
Vol. 9 (1970) NO.9
occupy skeletal positions not coinciding with skeletal
planes of reflection or improper axes of rotation (see Section 1.3.3.).
For chiroids of class b, there are only those skeletal planes
of reflection or improper axes of rotation that allow more
than two non-equivalent ligands to occupy skeletal positions outside any of these skeletal planes or improper axes
of symmetry. It is not possible to transform a chiroid of
class b, into its mirror image by permuting only two differing ligands; more than two non-equivalent ligands must
be exchanged in order to generate the antipode. More than
two ligands must be removed or equalized in order to
destroy chirality.
Chiroids of class b, have at least one skeletal plane of reflection or improper axis of symmetry of type u and at least
one of type b,. This classification is illustrated by the twodimensional examples shown below.
Fig 5 Reflection across the lines ~or - - - - - - transforms the laminae
into their mirror images. Two of the distinguishable vertices (, . . ., 4 lie
outside the lines --,
more than two lie outside the lines - - - - ~ -
Class b contains various types of molecules. For some,
the concept of handedness has no meaning. However, it is
possible to dissect others into elements of chirality of class
u. For example, the cyclopropane (XI) and prismane
derivatives (with D,, skeleton and six ligands) can be dissected into tetrahedral elements of chirality which can be
described in terms of the (R),(S)-nomenclature. Conversely,
it is not possible to apply the (R),(S)-nomenclature meaningfully to a molecule whose skeleton consists of a central
atom with valencies directed towards the vertices of a
trigonal prism with six different ligands.
2.3. The Nerve and Beyond
Now the various types of mathematical structures can be
combined to represent molecules in a detailed manner.
We will describe a cyclooctane molecule (8a) first in topological terms.
bors and can be described by an indexed family of 24 intersecting sets, e.g. a family of open spheres (8c). The nerve
of this family (8d) represents the chemical constitution of
No information beyond the neighborhood relations is included in the nerve of the molecule. All stereochemical
features of a molecule are lost when it is represented by its
nerve. Cyclooctane's behavior is that of a molecule possessing D,, symmetry. This is usually described by stating
that cyclooctane has a D,, time average symmetry. While
this notion serves some purposes well it is not at all well
founded. Geometrical properties cannot be averaged like
scalar properties. We will make use of the highest symmetry
which a molecule or its skeleton can be imagined to attain
within its given constraints. The constraints of a given
molecule and its skeleton can depend upon the time scale
inherent to the method of observation. Symmetrical molecules suffering distortion, as well as those having none of
the symmetry described as time average symmetry in any
of its vibrational, internal rotational or conformational
states can all be represented without auxiliary hypotheses
using the highest available symmetry concept.
The highest attainable D,, symmetry of the cyclooctane
molecule can be described by a permutation group. The
indexed open spheres of (8c) are employed as the coordinates of the carbons and hydrogens of cyclooctane. Then
the permutational transformation properties of the coordinate system represent not only the skeletal symmetry,
considering the equivalence of the hydrogens, but the symmetry of the whole molecule. When the necessity arises to
differentiate between the hydrogen atoms of the various
molecular ligands in substituted cyclooctanes one must
further define a mapping process. This mapping must correlate the various parts of the molecule with suitably
chosen coordinates. The coordinates of the eight equivalent
carbon atoms of cyclooctane are not needed for the description of the steric features of the molecule and its derivatives. Therefore, the partially symmetrized product,
X, of sixteen indexed copies of one set is
adequate as a coordinate system for the idealized D,,
skeleton of cyclooctane. The subgroup H16 of the symmetric group S,, is used as a partial symmetrizerof the product
set {X16xX, x . . . x X"}.The indexed set @ I , . . . , q6 }
of points with disjoint neighborhoods, piE X,,can be used
X2,. . .,X,, }. This is obtained from one
as the set {X,,
point p2 by sequentially applying the generators C, and
q, to pl and assigning odd indices, 2s - 1 (s = 1, . . . ,8), to
the points on the orbit of C,. We then reflect this set of
points (Fig. 6a) across a plane parallel to and above the
Neighborhood relations provide the chemical constitution
of cyclooctane. They are given by bonds and bonded neighAngew. Chem. internat. Edit.
1 VoI. 9 (1970) 1 No. 9
71 1
plane of the paper. Now, assign the even indices 2s to oh
mirror images of the points with 2s- 1 indices (Fig. 6b).
Representation of cyclooctanes substituted by a numbered
set of sixteen distinguishable ligands L = { Ll, L,, . . .,L16}
is achieved through the bijective mapping of the ligands
on the skeleton and thus assigning the skeletal coordinate
of each ligand. By using the partially symmetrized set of
coordinates SP8W6 X the fact that some skeletal coordinates are symmetry equivalent is accounted for and one
is allowed to select that one among the equivalent bijections which is the best according to some criterion. The
latter may be defined as that mapping where the highest
number of ligand and coordinate indices match.
Li: i
Fig. 6. Sequential application of the point group symmetry operations
(generators) C , and mb to a point (odd indices 2s - 1 (s = 1, . ., 8) assigned to orbit of C8).
We may now formally consider the carbon atoms. The carbon atom with the index one belongs to a distinguished
point, the cross-section of the neighborhoods of those
atoms to which the carbon is directly connected. Thus, the
assignment is in accord with skeletal symmetry.
In principle, as is illustrated by Fig. 7, the simultaneous
use of product sets and the descriptions of neighborhood
relations by non-empty intersections of q , s,, and ' T ~of
n7,is possible. To accomplish this, sets must be used at
different "levels" and symbolized by double indexing.
p3 E Tlr3,3i
1 2 3
_ _ _16
1 1 1 . . . ' 1-
i = I 2 3 _ _ . 16
Since the ligands of molecules are not per se indexed, a
procedure must be established for assigning the indices to
the n ligands of a molecule. This produces, for example, a
one to one mapping of a set of n ligands onto the numbers
1,2, . . .,n that we call the indices of the coordinates Y,of
the ligands such that L / E ,L, E 5 , . . . L, E 7 . . . L,E Yn.
This is equivalent to the statement that the set of ligands
belongs to the product of the coordinates. We propose to
assign the ligand coordinates 1 according to the sequence
rule of Cahn, Ingold, and Prelog["], which permits the
ordering of a set of distinguishable ligands. The coordinate
index 1 = 1 is assigned to the ligand with the highest sequential priority, 1 = 2 to the second highest priority
ligand, . . . and 1 = n to the ligand with the lowest priority.
If there are several equivalent ligands then the corresponding index interval is assigned as illustrated by the following
example :
Let L be the sequentially ordered set of ligands for a given
molecule [like (3a)J.The coordinate indices are
L , : H0,C-, H0,C-, C,Hy, CeHy, H-,H-, H-, HY; : [ 1 I
21 [3 I:1 I
41 [S I
1 5 81
The equivalence of ligands (see the example above) is
taken into account, just as the symmetry equivalence of
skeletal positions is, by partially or fully symmetrizing the
Cartesian product {yI x U, x . . . x Y,) of the ligand coordinates Y. The ligand coordinate spaces I: correspond
to open intervals in a line of sequential points. This line of
points represents the sequence of all ligands that can occur in permutational isomers. Sequential intervals are the
open sets of the topology on the linear sequential manifold.
The partially symmetrized Cartesian product.
sp"'"y =
x Y x . . . x Y")
refers to the subgroup Hk of permutations of non-distinguishable ligands, a subgroup of S,, and becomes equal
to S,, if all ligands are equals as in the case of cyclooctane.
p,c T 4 . 7 1
Fig. 7 Example of combination of a product set with the description of
neighborhood relations by non-empty intersections.
A physical feeling for these operations can best be visualized by looking upon neighborhood relation operations as
vertical projections and product topological and group
theoretical operations as horizontal projections. This approach avoids violation of the product of sets definition by
the intersections.
The bijective mapping of the index set of the partially
(or fully) symmetrized ligand coordinates Yionto the
index set of partially (or fully) symmetrized skeletal coordinates X,permits unequivocal classification of topological isomers and is introduced by us as the basis of a permutational nomenclature.
There is a bijective mapping from the set of these mappings
onto the given class of isomers. According to Ruch and
Schiinhofer [''dl the total number of isomers in such a class
Angew. Chem. internat. Edit. / Yol. 9 (1970)1 No. 9
2.4. Describing Molecules by Permutationdl61
can be calculated from the number of double cosets of H,,,
and H,,
in S,,
The use of the permutational nomenclature system is illustrated by the flow sheet shown in Fig. 8.
2.4.1. D e s c r i p t o r
The permutational isomers are described by mapping the
indexed set of ligands I., onto the indexed set of permutational coordinates X,, the equivalent of the molecular
skeleton. This mapping is represented by a (2 x n) matrix
notation using the (3 matrices of the indices 1 and .r.
An effective nomenclature for permutational isomers
results from this matrix representation. That permutational transformation, Pa, of the permutational coordinates s which generate the matrix (:), of a given molecule M from the matrix (:), of a reference molecule E according to
SPm'" Y
s Prn" x
is called the descriptor of the molecule M. The descriptor
is an unambiguous symbolism for the nomenclatural
classification of the permutational isomers [I6].
2.4.2. Skeleton
Fig. 8. Flow sheet for generation of the descriptor of permutation isomers.
The permutational descriptors are based upon the following definitions:
S I: A molecule M is represented by an achiral skeleton
Here, L + M + S symbolizes the division of a molecule
into a skeleton and a set of ligands. Coordinate assignment
and numbering is represented by { Yl} and {X,}.
The n ligand and skeletal indices 1 and s of the set { Y,}and
(X,}are correlated by mapping in a (2 x n) unit matrix.
the (3, matrix. We use the term matrix for any rectangular
array of mathematical symbols. If these mappings are
represented by permutations, this corresponds to the
identity permutation e (see 2.4).
I 2 -_ J_ )_ n .
Among the
matrices of equivalent mappings
SP'"Y + S F X that matrix &,, which is obtained from
by the most simple s permutation, Ps, is chosen as the
descriptor of a given molecule.
In the context of their solution of the Polya problem, Ruch
et al. [15f,*1 demonstrated that the number of distinguishable isomers of a given class of isomers corresponds to the
number of double cosets PxG ( x E S,);P and G are the subgroups which correspond to the skeletal symmetry of the
skeleton with n positions and the permutation ofequivalent
ligands. The various isomers correspond to systems of
representatives of the double cosets PxG. Therefore, a
nomenclature system for permutational isomers can only
be adequate and non-arbitrary if it corresponds to a
system of the above double coset representatives.
Ruch et al. will soon report on a proof that the descriptions
of the present paper (which are here derived according to a
mapping formalism - which is closer to chemistry - rather
than by a formalism that is in immediate relation to the
double cosets) are indeed equivalent to a representation
system of the PxG double cosets.
Angew. Chem. internat. Edit.
Vol. 9 (1970)/ No. 9
S, with n ligand positions and a corresponding set of n
ligands L, . . . . . L,.
S I1 : A permutational coordinate system is introduced to
represent the molecular skeleton, such that the permutational transformation properties of the coordinates correspond to the symmetries of the skeleton.
S I11 : A set of skeletal position numbers s = I , 2,3, . . . ,n
is assigned to the n skeletal positions of S,,by a convention
which is defined for each type of skeleton, and is characteristic of the given class of permutational isomers according to the rules S I - S IX. The result is a standard
skeleton (see I - XXIX) whose skeletal numbers correspond to the indices of the skeletal coordinates.
[16] The following may serve to illustrate permutations and descriptors: The permutation P, = (1 4 2 5 ) represents the cycle
5 t 2
1 t and means I replaces 4 (or is the image of 4 ) and similarly,
1 4 4
4 - 2 , 2 + 5 , 5 + I . The ordered set A = { I 2 3 4 5 1 is transformed into and mapped onto B = ( 5 4 3 1 2 ) by the permutation
P, = ( 1 4 2 5 ) . Given that permutation P,,which represents the
mapping of set A onto set B, is found from the matrix
(As) { :::::)
; following the arrows
t; r:
1(" ;
'.----__ __- -_- _ ,
I ,
gives the operator P,
( 1 4 2 5 ) . When used as a descriptor, the
P, = ( I 4 2 5 )
(1). { I1 22 33 4 5 } according to (;)
Now, the descrip-
tor ( I 4 2 5 ) expresses the following mapping: Skeletal position I
carries the ligand with the sequential number 4, position 4 carries
ligand 2, position 2 carries ligand 5, and position 5 carries ligand 1 ;
at position 3, unaffected by Ps,the ligand number 3 matches the
skeletal number.
S IV : The skeletal numbers are assigned such that, whenever possible, the configurations of the reference isomers E
of class a have the (R)-configuration (see XVII and
S V : If the observer looks along a principal C , axis [Zbl the
assigned skeletal numbers increase in a clockwise manner.
S VI : If there are several C, cycles, belonging to the same
principal C,-axis, the cycle which is closer to the observer
has the lower numbers s (see XVI). The portion of the cycle
with the lowest s lies within a plane that includes the principal axis and is the same for each one. If this is not possible,
the lowest s lies in a clockwise direction, less than 90' from
this plane.
S VII : Pairs of skeletal positions which belong to the same
atom have consecutive s (see XI).
S VIII: If there are several non-equivalent sets of symmetry equivalent skeletal positions, separate sets of skeletal
position numbers s = J . . . . . n, s' = 1' . . . . . n',
s" = I" . . . . . n", etc., are assigned. Similarly separate sets
of ligand numbers 1 = 1 . . . . . n, 1' = 1' . . . . . n', 1" =
1" . . . . . n" are associated with these.
S IX: Skeletal positions and ligands on a principal symmetry axis of the skeleton are described by primed or multiply primed l and s (see VII).
Angew. Chem. internat. Edit.
XXIV ( Y )
/ Vol. 9 (1970)1 No. 9
+--- - _ ,
L 11: If there are ni ligands (L,= L,+, = Li+2.
... Li+",,) of the same kind in the set, they are given thecorresponding n, ligand numbers 1 = i, i + 1, . . . . . , i ni - 1.
2.4.4. Reference Isomer
E I: The reference isomer E is that permutation isomer in
which the ligands L, . . . . . L, of M are attached to the positions of the standard skeleton (see I-XXIX) so that the
ligand numbers 1 and the skeletal position numbers s
match. We represent this mapping by the unit matrix
(d), of the set of isomers.
1 2 3 4 . . . . .n
I 2 3 4 . . . . .n
(3, =(
E 11: The reference matrix with more than one set of
equivalent skeletal positions is that matrix in which the
ligand numbers 1, l', l", etc. of all sets are superimposed
upon all corresponding skeletal numbers s, s', s", etc.
1 _ _ _ n. _1' _ . _ _
n' _ 1" _ _ . .n"
. .....
1 . . . . .n 1 ' . . . . .n' 1". . . . . n " . . . . .
The mapping of 1-+s of a set of n indexed ligands onto
an indexed set of n skeletal coordinates can be particularly
well represented by (2 x n) matrices. An (n x n) matrix
will also represent this mapping. In the latter case, all
matrix elements a,, are zero except for a one in each row
and column. a,, = 1 expresses that s and 1 belong together,
and a,, = 0 means s and 1 are not mapped onto each other.
The ( n x n ) matrices of the reference isomers are unit
matrices Em.
1: : 1
. . . . .
. . . . .
. . . .
0 11: We consider that mapping of M to be most similar
to 0, whose matrix
can be transformed into (3, by
the simplest permutation of the skeletal numbers, s. The
s-permutations correspond to permutations of the rows
of the (n x n) matrices. The simplest permutation is the
permutation which involves the smallest number of s, if
possible in small cycles. Accordingly, Mop,is that arrangement of M in whose matrix there is a maximum set of superimposable ligand and skeletal numbers 1 = s.
In terms of an (n x n) matrix representation the optimum
arrangement is that one in which as many ones as possible
are matrix elements of the main diagonal, and those ones
which are off the main diagonal can be brought into it by
the simplest permutation of rows.
0 111: If there is more than one such matrix the alphanumerically lowest set of non-matching 1 and s is preferred.
If this is not sufficient to identify the optimum matrix
(d), , then that matrix is preferred for which the difference 1 - s is the smallest. This corresponds to a (n x n)
matrix where the diagonal zeros are as close to the upper
left corner of the matrix as possible.
0 IV: The optimum matrix
of an isomer with nonequivalent sets of equivalent skeletal positions i s that
matrix which is generated from
by the simplest, alphanumerically lowest permutation of the unprimed s.
If M does not happen incidentally to be given in its optimum arrangement, its matrix ,():
can be determined as
0 V: First, the ligand and skeleton numbers, 1 and s are
determined according to L I - S IX for the particular arrangement of M, and are then represented by the (3
0 VI: If M contains sets of equal ligands (like n, 2 2
equal ligands Li = L,+, . . . . + Li+nj-J, then, at first,
only a corresponding range i 5 1 5 i + ni of the ligand
numbers is "reserved" for these ligands :
. . . . .
. . . . .
. . . . .
. . . .
. . . .
belong together in different ways. This is due to the equivalence of some skeletal positions S (topological coordinates) and, in certain cases, the equivalence of some of the
ligands L, . . . . . L,. Some of these different mappings correspond to different spatial orientations of M. These different arrangements can be compared to the different Fischer
projectionsI8"I of a molecule. For the comparison of M
with its reference isomer E the optimum mapping, Mop,
of M is chosen.
L I : A set of ligand numbers 1 = 1,2,3, . . . . . n is assigned
to the ligands L, . . . . . L, of M according to the sequence
rule[*."], in the order of the sequential priorities of the
ligands. The number 1 is assigned to the ligand which has
the highest sequential priority and n to the ligand with the
lowest sequential priority 1"'.
A molecule M can generally be described by many equivalent mappings (see Section 2.4.7.) in which the 1 and s
0 I: The optimum mapping Mop,is that mapping of M
whose matrix (,:),
is most similar to &.
2.4.3. L i g a n d s
2.4.5. O p t i m u m P r o j e c t i o n
Angew. Chem. internat. Edit. 1 Vol. 9 (1970) 1 No. 9
In the corresponding (n x n) representation the matrix
element a , , = x = {h is used for i s 1 5 i + n i - l if the
skeletal position s carries one of the equal ligands
Li= Li+"j-,.
0 VII: The ligand and skeleton numbers, 1 and s of M
are adjusted to an optimum matrix,
by permuting
the skeletal numbers s in the matrix or the rows in the
(n x n) matrix. Only those permutations of s are permitted
which correspond to transformations of the permutational
coordinates into equivalent coordinates, i.e., permutations which correspond to the rotational symmetry
operations (of the subgroup H) of the skeletal symmetry
group G.
0 VIII: If not all of the ligands L, . . . . . L, are different,
i.e., if there is at least one set of ni 2 2 of equal ligands
(L,= L,+l . . . . . = Li+,,-,), and some of the ligand
numbers 1 had only been pre-assigned by 0 VI (for final
assignment after establishing
a pre-optimum matrix
(rj)pre is obtained by applying 0 VII. The pre-optimum
is transformed into
by assigning the ni
individual numbers (i I
1 Ini) of the ligands
L i . . . L j . . . Li+,,-l in such a manner that as many as
possible of them match the skeletal numbers of (&,re. In
the (n x n) representations this is done by simply replacing
that x in each row which is on the main diagonal or that is
permutationally “closest” to it by a one, and all other x’s
by zero.
[ j(9;[:R :
0 1 0 0
1 0 0 0
0 1
0 01 0
l o o 0
[n R :81
1 0 0 0
= ( I I ) E4
2.4.6. Assigning D e s c r i p t o r s
(see Section
The conversion of the reference matrix
2.4.4.) into the c)-matrix of a given permutational isomer
M is given by a permutation Pa, the descriptor of M, according to equation (l),
with P, = ( . . . n, 1 n, . . . n, 2 n,. . . nk 3 n, . . . n n,)
(1) ={
n, n, n,.. . 1 . . . 2 . . . 3 . . . n,
I 2 3 . . . n , . . . n, . . . n L . . n
We propose to use the descriptor P, of M as the nomenclature name of the isomer M, because the set of permutational isomers is in an unequivocal relation to the set of
these operators Pa. The operator P, gives unequivocal
instructions as to how to construct its corresponding isomer from a given molecular skeleton S, (see I-XXIX)
and a set of ligands L, . . . . L,. The identity operator
P,= (e) represents the reference isomer E, according to
The set of descriptors of a set of isomers is not a group, but
the set of permutations which interconvert a11 the isomers
is a group G. Those permutations which interconvert the
stereoisomers among the permutational isomers, belong
to a subgroup of G.
2.4.7. Examples S i m p l e S k e l e t o n s
One might infer from the preceeding definitions that use of
this nomenclature involves lengthy and complicated
operations. This is however, not the case, as one can
readily demonstrate with the examples of Section 2.4.7.
Any systematic nomenclature which appears less complicated is bound to contain arbitrary rules and oversimplifications which will severely limit its generality and accuracy (see Section 2.3 and footnotes 115d-15g1 >.
Example I
The spatial arrangement of ( 9 ) , (s = I , . . .,4, see XXVII,
1 = 1 , ...,4 w i t h O K = l , C H O = 2 , C H , O H = 3 , a n d H = 4 )
is represented by (4)(9,,
which is converted into (:)(96) by the
s-permutation ( I 3 4 ) or the equivalent I-permutation ( 1 4 3)
The I-permutation (1 4 3) corresponds to a counterclockwise 120”
rotation (C,) of the ligands, L,,L,,L,.
The permutation ( I 2) converts (9c) into (9b) and is therefore
the descriptor of (9b) [ - ( 9 a ) ] . The steric formula of (9b) is
reconstructed from the descriptor ( I 2) as follows: The ligands
OH, CHO, CH,OH, and H are written in this sequence. When
these ligands are attached to the ( I Z)-transform of the standard
framework in their sequential order, OH 42, CHO + 1,
CHZOH+4, the formula (9b) results.
The matrix
(I 2)
1 44 .1 22 13 13 1
contains an abbreviated formalism for deriving the descriptor
( I 2) for ( 9 a ) . The rows are written in the order a, p, y. The
ct-row contains the s in their natural order. The sequential numbers of the ligands are mapped upon the numbers of the skeletal
positions to which the individual ligands are attached (row p).
The y-row is obtained from the a-row by the permutation ( 1 3 4);
this permutation (which corresponds to a 120” rotation around
2) brings 3 and 4 into coincidence with 3 and 4. The p- and y-rows
which is transformed into (i),by the permucorrespond to (i),,,
tation ( I 2).
The descriptor ( I 2) is obtained by permuting row y such that
it matches row p. Since the pairs 3 - 3 and 4-4 match already,
only 1 and 2 must be permuted in order to transfer those mappings 1-2,2-2 into l - l , 2 - 2 .
Example 2 (see XII) [’I
The molecule of (3a) (=M) contains three sets of equivalent
ligands for which the corresponding intervals of 1 are “reserved”
(ie.,CO,H = 1,2; C,H, = 3,4; H = 5 - 8 ) , according to Section 2.4.3. Then, if (1) is mapped onto {s} (see above) the preoptimum matrix (i),,,(see Section 2.4.5) is generated by the
permutation ( 1 3 5 7) (2468) which represents a 90” rotation of
the skeleton. A comparison of the s in y, and the 1 in p make the
assignments of all the 1 except 5 and 8 obvious. The aim is simply
to match as many 1 and s as possible. The assignment of 5 and 8 is
not so simple. Since an optimum assignment of an isomer should
be, whenever possible, that assignment which leads to the reference isomer via the most simple permutation of s, 8 is assigned
to I and 5 to 3 so that these interchanges may be made directly.
Then, the (~),,, is
Angew. Chem. internat. Edit.
Vol. 9 (1970)1 No. 9
(2 4 6 8 )
= o
o x
(1 8 ) ( 3 5 )
Example 5
For the purpose of assigning ligand numbers, the coordinating
moieties of the multidentate ligands of ( l o ) are treated as separate ligands.
Accordingly, the ligand numbers of (10) are,
Example 3 (see XII)
The steric formulas (3c) and (3d) of the ( I 7) (3 5)- and ( I 5 ) (3 7)dicarboxy-diphenyl-cyciobutanesare derived from the descriptors as follows:
The permutations (1 7)(3 5) and ( 1 5)(3 7) of the ligands 1
(C0,H) and 7 (H) as well as 3 (C,H,) and 5 (H) of the reference
isomer (3e) generate (3c) and (3d). In this case the 1- and the
s-permutations have the same effect.
OO2C - C H Z - N H ~
For comparison, try naming the truxinic acids (3c) and (3d)
by any other nomenclature, and then reconstruct their steric
NH ( C H Z C H ~ N H ~ ) ~
Example 4
The steric formula (4a) of the (2 4)-dichloro-aquo-triamminocobalt(1rr) ion is obtained by first writing down the ligands in
and the descriptor of the chiral complex (10) is (45)-chloroaminoacetato-diethylenetriamine-cobalt.
Example 6
3 1 2
2' 5
3,4 3,4 1 2 7?.12' 2'
2 1
(7' 8 ' )
sequential order (C1 = 1,2; H,O = 3; NH, = 4,5,6) and then
attaching them to the (24)-transform of the skeleton XXIX to
form ( 4 a ) . The reference isomer is ( 4 b j .
Angew. Chem. internat. Edit. / Vol. 9 (19701 / No. 9
= Mopt
1' 6'
' 6 ' 7 5 1 2 ' 7112' 1'
16' 7'12'
10' 11'
6' 5' 8'
3' 7112' 416'
The unprimed and primed skeletal numbers of XXVI are arranged in one line (ct), and the (assigned or preassigned) sequential numbers of the ligands (1: C1; 2: OCH,; 3,4: H ; 1 ' : C1:
2': OCH,; 3': OH;4'-6': CH,; 7'- 12': H)arepaired with their
positions (p). The unprimed part of the matrix is optimized by
a permutation which corresponds to a C,-rotation [- ( 7' 8' ) ] .
The pre-optimal matrix is converted into the optimal matrix ( 6 )
by assigning the final ligand numbers I. The upper and the lower
rows are the rows of (J),,, of ( l l b ) . The permutation ( 1 2 )
( I ' 8' 6' 11' 5 ' ) (2' 3' 10') is therefore the descriptor of
( I l a ) = ( I l b ) , a chiroid of class b. Admittedly, it involves
some work to find the descriptor of ( Z l a ) , but it takes, even in
this complicated case, very little effort to reconstruct ( l l b )
[ = ( I l a ) = MI from its descriptor and the reference isomer
( I 1 ) (=E).The attempt to treat a compound like ( l l a ) by any
other nomenclature would lead to a rather frustrating experience.
Example 9
- C o m p o s i t e S k e l e t o n s
There are molecules whose skeletons have internal rotational degrees of freedom. These skeletons can be treated
as if they were combinations of independent simpler
In example 9, chiral side chains are attached to the ferrocene
system. In order to describe the permutation of the ligands of
those side chains in the description of the whole molecule and
in order to differentiate that part of the descriptor which belongs
to the side chain, we used starred ligand numbers and starred
skeletal numbers for the side chain. If there is more than one side
chain which belongs to the system, we use double starred ligands
and skeleton numbers. The priorities follow the sequence rule.
Accordingly, in (14), the unstarred ligands belong to the ferrocene skeleton; I, s with one star belong to the benzyl alcohol
group and those 1 and s which are doubly starred belong to the
dimethylamino group.
Example 7
The metallocenes [17'1 with their internal rotation are considered
to be a combination of two C,, skeletons (V).
According to the usual procedure, we find the descriptor to be
(cf. example 1) (e)( I 2)* (I 2)**. This is equivalent to the designations which have been used in the past, and corresponds to
(S)(S)(R)[17bl. T h e Use of N i l
In the case of permutational isomers with these composite
skeletons containing independent equivalent parts, the unprimed skeletal numbers s are assigned to that moiety which
carries the greatest number of different ligands.
The assignment of the skeletal numbers s is done as shown in
formula (12) i.e., the metal is coordinated to those sides of the
rings from which the skeletal numbers appear to be numbered
in a counter-clockwise sequence. This definition is intended to
have general validity for metal complexes which contain'symmetrical subskeletons whose skeletal positions lie in a plane or
whose idealized representations lie in a plane.
Cyclic molecules which contain multiple bonds, annellations, or bridges can be included in the present concept of
permutational nomenclature by dissecting bonds and
representing the resulting free valencies by nils (= 1).
By this dissection procedure (see Fig. 9), a cycloalkane
derivative results with a D,, skeleton which carries 2 z nils
if z bonds were dissected. The positions of the nils are determined by the stereochemistry of the dissected molecule.
Example 8
f 3'
<2J t'
The primed 1 and s are assigned to the upper ring which has
three equal ligands (H). Both rings are represented by their
numbers 1 are asskeletal number s and s' (a), and the linand
signed and the 1' are preassigned to their corresponding s- and
s'(p). Then the optimum matrices of (13) (i),,, (&,, are generated
(via the pre-optimum of (&,)
by cyclic permutations of the s
and s'(y). The resulting descriptor of (13) is ( 2 3 ) ( 1 ' 4 ' ).
[I71 a) K . Schlogl, Top. in Stereochem. 1, 39 (1967); b) D . Marguarding, H . Klusacek, G . Gokel, P . Hoffmann, and I. Ugi, Angew.
Chem. 82, 360 (1970); Angew. Chem. internat. Edit. 9, 371 (1970).
Fig. 9. Formal dissection of cyclic and polycyclic compounds with multiple bonds (a) o r bridges (b,c), the resulting free valencies being represented
Angew. Chem. internat. Edit.
1 Vol. 9 (1970) No. 9
Whether a given skeletal position number of a D,, (cycloalkane) skeletal representation is odd or even is usually obvious. In complicated cases, the solution is obtained unequivocally as shown in Fig. 10.
(ISh) ( = M 5
( l5a)
(ISCi (
The reference isomer of (15a) [= (15b)] is ( I S c ) and the descriptor is (2 16). Therefore (150) is (2 16)-cyclooctene-(2- 15).
Example 11 [I9]
Fig. 10 Unequivocal assignment of skeletal position numbers to a D,,
An idealized Cartesian coordinate system is drawn so
that two neighboring members, C iand Ci+l of a ring
belong to the -x axis and the +x axis. The ring
1 2
1 2
19 4
11 6 7 8 9 10 3 5
16 17 18
12 14 15 2 2
6 7 8 9 10 11 12 13
16 17 18
19 20 21 22
belongs to the space with y < 0. Ligand positions Azi- and
B2i+z belong to y 2 0. Azi-, and
have odd s (are
above the clockwise ring) Bzi and BZi+*have even s (are
below) if
A2,-1, Bzi
: Y2i-1 >
: Y21
Longifolene (16a) is dissected, as shown by (16h), into a cycloundecane derivative with the ligands CH2 = : 1.2 ; CH, : 3 - 5 ;
H : 6-18; 1 : 19-22, and the matrix is
From this follows the descriptor ( 3 19 I2 5 11) (1420) (1521)methylene-trimethyl-cycloundecane-dicon-(3- IS,14 - 22).
Example 12IZo1
The steric formula of ( I 9 28 17 30 19 2 10 3) (4 I2 29 18) (26 32)hydroxy - [ ( I ' 2') -a,&-dimethyl- hexyl]-dimethyl-cycloheptadecene-(31-33)-tricon-(9-34, 12-27, 17-26) is derived by the
following steps.
The descriptors of unsaturated and/or polycyclic compounds are assigned according to these definitions. Whenever the stereochemistry is unknown or purposely neglected, the topological coordinates of the nils are chosen such
that the most simple, alphanumerically lowest descriptor
a) MuItipIe and Transannular Bonds
For unsaturated compounds and a wide variety of polycyclic compounds the permutational nomenclature in combination with the concept of nils provides a satisfactory
solution to the nomenclature problem, even if chiral
skeletons are involved.
The bonds which are considered to be formed from the
nils are designated by attaching the suffixes ene,yne,and con,
respectively, to the name of M for double and triple bonds
and transannular connections and, in brackets ( ), the
numbers s of the skeletal positions in MAptwhich are connected by these bonds.
Example I0 [ l 8 I
The chiral trans-cyclooctane (15a) is, for descriptor purposes,
considered as a cyclooctane (with an idealized D,, symmetry,
see Section 2.3.); the two nils in (15b) in adjacent skeletal positions characterize the formation of a trans double bond.
[I81 A . C. Cope, C. R. Ganellin, and H . W. Johnson, Jr., J. Amer.
chem. SOC. 84, 3191 (1962); A . C. Cope, C. R. Ganellin, H. W.
Johnson, Jr., T. V. Vun Auken, and H . J. S . Winkler, ibid. 85, 3276
(1963); A. C.Cope and A . S. Mehtu, ibid. 86, 5626 (1964).
Angew. Chem. internat. Edit. / Vol. 9 (1970)/ No. 9
l l T l
(17a) ( = E)
(176) ( =
1191 P. Nuffu and G. Ourisson, Chem. Ind. (London) 1953, 917;
R. H. Moffett and D . Rogers, ibid. 1953, 916.
[20] J. W . Cornforth, F. Youhotsky, and G. Popluk, Nature 173,
536 (1954); B. Riniger, D.Arigoni, and 0. Jeger, Helv. Chim. Acta
37, 546 (1954).
c) Bridged Molecules
The nil method can be extended to bridged systems, by
simply considering the bridges as connections between nils,
and describing them by attaching to the name of MAptthe
suffix pon (Iat.: pons = bridge), and in bracketso the
numbers s of the skeletal positions which are connected
and (between those numbers) the name of the bridge.
E.rample 14 [221
The formula of /19cj, (I 9) (312) (513) (716)-cyclooctadiene(I -15, 5 - 7)-con-(IO- I4)-pon-(3-carbonylo-11) follows
from (19aj via (196) by generating (19c) with the permutation
operator (I 9) (312) (513)(7 16) and then connecting the nil
ligands as indicated by diem-( 1- 15, 5- 7)-con-(IO- I4)-pon(3-carbonylo-11). The result is barbaralone (19cj.
The molecule is based upon the skeleton of cycloheptadecane
with an idealized DIThsymmetry. The 34 skeletal positions are
numbered according to the rules of Section 2.4.2. and the reference isomer (17a) is generated by attaching the ligands (1 : OH,
2: C8HI7,3,4: CH,, 5-26: H, 27-34: nils). The descriptor
( I 928Z730192103)(4 122918)(2632)permutesthecoordinate
indices of (17a) and transforms it into (17bj. Then the nils of
the skeletal positions 31 and 33 are used to make a double
bond, and the pairs of nils at 9 and 34,12 and 27,17 and 26 are
connected to produce the tetracyclic framework of (17cj,
which is cholesterol.
b) HeterocycIes
The concept of nils is useful for the description of the aza-,
oxa-, and thia-derivatives of the carbocyclic permutational
isomers. If the covalency of the heteroatom is lower than
four (4- x), this is taken into account by attaching x nils
to the heteroatom. Further, the positions of heteroatoms
must be specified. For this purpose the (carbon) atoms of
the skeleton are numbered by skeletal nuclear numbers
s, = I N . . . n,, differentiated from the skeletal position
numbers s by the subscript N. The nuclear number I , is
assigned to the skeletal nucleus with the skeletal position
number 1 (or numbers I and 2, respectively); number 2,
belongs to the nucleus with the position number 2 (or 3
and 4, respectively), etc. The nature and position of a
heteroatom is indicated in the descriptor (see example 13).
The concept of bridged skeletal positions can, in a slightly
modified form, be extended to the chelated metal complexes. As a rule, it suffices to indicate the skeletal positions
of the central atoms that are connected by chelating ligands.
Permutation of the skeletal positions in a complex having
chelating ligands might easily lead to confusion regarding
the original skeletal position-ligand relationship. In order
to retain the information provided by the original (3
matrix, those columns of skeletal positions which belong
together because of chelation are indicated by brackets
and these brackets are retained during s-permutations.
Example 13c21]
Example 15
f18a). an intermediate in the synthesis of gentosamine is treated
as a trioxa-cyclononane.
The optimal matrix (i),,, of (18h) yields the descriptor
(I 5 13 3 6 I4 4) (1017)-con-(l0- 18)-3,,6,,8,-trioxa-methoxydimethyl-cyclononanol for (18aj.
4 1 5 16
[21] H. Maehr and C . Schaffner, J. Amer. chem. SOC. 92, 1697
10 18
[22] W . v. E. Doering, B. M . Ferrier, E. T . Fossel, J . H . Hartenstein, M . Jones, Jr., G. Klumpp, R . M . Rubm, and M . Saunders,
Tetrahedron 23, 3943 (1967).
Angew. Chem. internat. Edit. 1 Val. 9 (1970)
1 No. 9
According to
the descriptor of the chiral complex (20) is
dioxalato-(1 - 3,2- 4)-ethylenediamine-(5 -6)-cobalt, and the
( 1 -2,3-4)-(5-6)-isomer
is the antipode of (20).
The descriptors of multinuclear metal complexes indicate
the connecting bridges and the numbers of the nils of the
nuclei which are used for bridging. Ligands inserted into
faces of complexes as bridges are symbolized by pon and
in ( ) brackets the numbers of those ligands between which
the additional ligand is inserted (see some of the heptacoordinate complexes).
Example 17 (see XXIX)
The (2 4 3)-azido-{(lO’ 14’)-se-(Ik 7 , ) - f N
2,, , S,-triaza-bis-[Paminoethyll-cyc1oheptane)-cobaltion is (23d).
A wide variety of polycyclic molecules could not be adequately represented by descriptors unless further operations were defined for the treatment of complications arising from the bridges of certain polycyclic systems.
A bridge of a polycyclic molecule that is capable of existing
in various isomeric forms requires a nomenclatural procedure which allows one to describe unequivocally the topology of this bridge in relation to the rest of the molecule.
A one-membered bridge of a molecule M such as (21)
can be adequately represented by a tetrahedral skeleton
XXVII. Positions 1 and 2 carry the ligands L, and L2 and
positions 3 and 4 are empty, ie., occupied by nils, and are
used for attaching this bridge to the rest of the molecule
M, in a manner indicated by the descriptor.
Example 16
The camphor x-sulfonic acid (22a) is an example for compounds
with bridges capable of existing in different “epimeric” forms.
The trenen ligand (23a) of the complex (23d) is obtained
from its reference isomer E’= (I’ 2’)-bis-(P-aminoethyl)-14‘-nz/cycloheptane by applying its descriptor and attaching a nil to
the skeletal position 10’ according to (10’ 14‘), and replacing the
carbon atoms at the nuclear positions l k , 2k, and 5; by nitrogen.
The chirality of the asymmetric nitrogen at 10’ is stabilized by
complexation in (23a) [= (23b)l in the configuration indicated.
For nomenclatural purposes (22a) is “dissected’ into (22b) and
(22c), which leads through the C)-matrices of (226) and (22c)
to the descriptor (35 IZ)-pon-(3’-5,4‘ - I 1)-[sulfonylmethylmethyl-methanol-methyl-cyclohexanonefor (22a).
The azido group has the highest sequential priority (1) among
the ligands of the complex (23d), and the chelating nitrogen
atoms of the amine groups are sequentially numbered as indicated in (23b). Hence the reference isomer for the octahedral
complex is (23c), a structure that is only conceptual and could
not exist for ring strain reasons. The permutation (243) transforms (23c) into (23d).
Two-membered bridges are derived from I by attaching this
system in a specified manner (see above), with its clockwise
side toward the positions which must be connected. The
most convenient way to describe bridges with three or
more members is to describe a corresponding cyclic
system and open the ring between two of its members. The
resulting chain is fully described by the descriptor of the
cyclic system and further, by specifying which bond was
transformed into two nils by the prefix se (lat. : seeare) and
finally, in brackets ( ),the numbers of skeletal nuclei whose
bond was broken.
[24] D . A . Buckinghum, P . A . Marzilli, and A . M . Surgeson, Inorg.
Chern. 8, 1595 (1969).
[23] F. S. Kipping and W. F. Pope, J . chern. SOC. 63, 549 (1893);
67,351 (1895); A. M . T. Finch and W. R . Vaughun, J. Amer. chem.
SOC. PI, 1416 (1969).
[25] W . Oppolzer, V . Prelog, and P . S . Suns, Experientia 20,336
(1964); M. Burfoni. W. Fedeli, C. Gincornelto, and A . Vuciago,
ibid. 20, 339 (1964); J . Leitlich, W. Oppolzer, and V. Prelog, ibzd.
20, 343 (1964).
Angew. Chern. internat. Edit. / Yo[. 9 (1970) 1 No. 9
Example 18 1251
Rifamycin B (24a) is treated as a naphthalene derivative with
a se-azacyclononadecane bridge and an additional oxa-bridge.
The combination of the description of the aliphatic bridge (24d)
and the naphthalene moiety (24c) leads to the descriptor
( I 29 26 12 38 2 16 7 17 10 19 8 36 1324 5 9 37) (I1 21) (6 32 27)se-( I , - 19N)-l,-aza-l 7,-oxa-dioxo-acetoxy-methoxy-dihydro-
72 1
xy-hexamethyl-nonadecatriene-(6- 8, 10 - 12, 30 -31>-dipon(IN-1k,19N-3i,18,-oxa-3~)-(1’2”)
( I ” 2‘) (3’4‘‘)-carbomethoxYdihydroxy-methyl-naphthalene.On reconstructing Rifamycin B
the ‘product ratios of stereoselective syntheses, we use
approximations 4 of chirality functions F, [151.
with Dreiding models, according to the descriptor, one obtains
a model which is strikingly similar to the model of the molecule
as determined by X-ray methods.
< ( h , . . . . . h,)
(24 c )
2.5. Chiral Descriptors
An observation of chirality relates to a measurement
of some property of a chiral system, the numerical value of
which will (by definition) be the same for the two antipodes
but each will have the opposite algebraic sign 115b,c1.
With respect to a class of molecules, we may call a function
F, a chirality function 17,13*151 if it is capable of expressing
these numerical values in terms of the nature of the particular molecules, i.e., in terms of the ligands by which the
molecules of the given class differ. For chiral permutational isomers with n places for the ligands and a skeletal
symmetry S the chirality function F, is of the form
Fz ( L , . . . . . L”)
The operations of the group S produce permutations Ps,
of the L,. Here the L, are used as symbols for the ligands,
and the indices s refer to the skeletal positions to which
L, is attached.
We use the symbol H’for the subgroups of all permutations
belonging to rotations H of S (group order = 8).The
function F, is invariant towards the operations of H and
changes its algebraic sign on permutations, s’,which correspond to the improper rotations and reflections of S.
Accordingly, F, vanishes for achiral molecules.
In order to describe observations of chirality, such as
In the approximated (hl . . . . . h,) the variables hirepresent those properties of Li which are relevant for a giver,
observation of chirality. The value of the ligand parameter hiof Li differs for the description of different types
of chirality observations. The approximation of F, by
F, is achieved by two methods [15b1; 1) the method of the
polynomials of the lowest degree, and 2) the method of
linear combinations of functions of a minimum set of
The approximated chirality function F, (h, . . . . . h,) is
such that F, is invariant towards those permutations P of
the variables hiwhich belong to the subgroup Hand changes its algebraic sign by permutations representing the
coset C.Accordingly, these chirality functions F, are invariant towards the projection operator PA of the antimetric representation of the permutation group s‘ of the
variables h, . . . . . h,.
(A, . . . . . k”) = F, (1,. . . . . h,)
This antimetric projection operator (6) can also be used
to generate the approximation functions F, from monomials (1) or functions of a minimum set of ligands
(2) [15b’.
Example 19
The subgroup H of the skeletal symmetry group S of XI1 is
Angew. Chem. internat. Edit. / Vol. 9 (1970)
No. 9
and its coset is
= 1 u ( 1 2 ~ , a ( ~ 3 ~&s),
u ( 1 7 ~ , u(371,
sa4), sp}.
Therefore, projection operator Pa,xllof the standard skeleton
XI1 is
PA = & { E
+ (1357)(2468) + (7531)(8642) + (12)(38)(4 7)(56)
+ (15) (26)(37)(48) + (16)( 2 5 )( 3 4 )(78)
( 1 4 )(23)(76)(85)
(18)(27)(36)(45)- (12)(34)(56)(78)- (13)(24)(75)(86)
(15)(26) - ( I 7 ) (28)(35)(46) - (16)(25)(38)(47) -
Another way of determining whether a descriptor P, describes a chiral or an achiral molecule, without drawing
a steric formula, is to generate the matrix
= P,&
and then apply an s-permutation from the coset C‘of H’
to C),,t. If the resulting C)-matrix can be brought into
“coincidence” with
by s-permutations of H’, the
molecule that is described by the descriptor P, is achiral,
otherwise it is chiral.
- (18)(27)
The lowest degree, homogeneous polynomial F,
is an
approximation chirality function of compounds with the
skeleton XII. It is generated by applying PA,,,, to the
monomial h,h, (or an equivalent monomial).
4,xir =
h7h, + h,h,
h 5 ) (h4
+ . . . . . + h4h5+ h,A,}
b)- (h2 - h.6) (h3
This chirality function, based upon the skeletal numbers of
XII, can also be obtained from the previousIy published function F;,.xll,[L5b1
by applying the permutation P‘ = (2’ 5’ 3’)
(6‘ 7 ’ 4 7 , which converts the original skeletal numbers s’ (indices of the XII‘) into the standard skeletal numbers of XII.
P‘Fi,xll= (2‘5’3‘)(4’6’7’)
( ( h ;- h;)(h; - k8)
( h ‘ 2- h;)(h; - h i ) ) += fi , X I 1
In the reference matrix C), the ligand numbers 1 and the
skeletal numbers s are superimposed. Therefore, in the
standard chirality function, the skeletal indices i of the hi
can be replaced by ligand indices i (hi+ hi), by which the
standard chirality functions F, is transformed into the
chirality function fi,”of the reference isomer. The descriptor P, of a given permutational isomer generates the
specific chirality function F,,n of this particular isomer
from the reference chirality function @,n
The isomer functions Fx,“ are easier to use than the original
chirality functions, because the indices i of the hiin the isomer functions are the sequential ligand numbers of the
ligands of the given set of permutation isomers. With the
isomer functions there is no need to draw the particular
isomer with numbered skeletal positions in order to assign
the indices of the variables hi.
Without knowledge of the particular numerical values of
the ligand parameters hithe isomer functions can be used
to determine whether a descriptor belongs to a chiral or
an achiral molecule.
3. Dynamic Chemistry
3.1. Permutational Isomerization
3.1.1. G e n e r a l F o r m a l i s m
Redistribution of ligands on a polytopal framework is
called a polytopal rearrangement
The mutual transformations of permutational isomers, called polytopal
rearrangements, involve a permutation of at least two
skeletal sites, or ligands, respectively.
We wish to differentiate between two types of polytopal
isomerizations, regular and irregular. The regular polytopal isomerizations are those that occur without breaking and reforming bonds, conserving neighborhood relations, as in the closely related process of interconversion
of conformers. The irregular polytopal rearrangements OCcur with bond fission and reformation. While irregular
rearrangements can be easily treated with this formalism,
we restrict ourselves, in this paper, to regular processes.
Polytopal rearrangements, M e M’, can be described by
transforming the matrix (El into (i)’or the reverse by the
operation of the ligand or coordinate isomerizer [eq. (13)]
on the appropriate matrix.
The s permutations are used as descriptors. In order to
differentiate those permutations used to describe isomerization reactions from the descriptors, we will use I-permutations, Zlof equation (13) to represent the isomerizations. Ziwill be called the isomerizer. The isomerizer which
describes the transformation of a molecule M to M’ is
derived from their descriptors P, and Pi as follows: M
and M’ are generated from their reference isomer E according to equations (14a) and (14b).
In these equations the equivalent s and 1 permutations P,
and PIas we11 as P: and Pi are numerically reversed. For example, one obtains P, from the inverse P, of Ps,
by replacing
its s with 1, i.e., by “deitalicizing” its numbers. The inverse
p, of P,is defined by equation (1 5).
P, x P, = e
Equation (15) states that if you first apply the permutation
P,and then Ps upon the elements of a set, the result is the
identity operation, e. Inverse permutations are numerically
[26] We use the term polytopal rearrangements in a more general
sense than it has been used by previous authors (cf. [27]).
Angew. Chem. internat. Edit. / Vol. 9 (1970) / No. 9
reverse; for instance if P, = ( I 2 3 4),
P, = (4 3 2 1) = (1 4 3 2). Multiplying equation (14a) by
P, and equation (14b) by 6 produces equation (16)
from which equation (17) is obtained by multiplication
with Pi.
Example 22
Combining equations (16) and (13) yields equation (18).
The isomerizer I, can be directly obtained from the descriptor,
by reversing the sequence of and “de-italicizing”
[P,x Pi].The product of the isomerizers of a closed cycle
of reactions corresponds to the identity operation (e).
= (e)
(26b)‘ (2 8 ) (4 10) (6 1 2 )
3.1.2. C o n f o r m a t i o n s
The energetically preferred geometrical configurations of
molecules which can be interconverted by movements of
some of their parts without breaking bonds are the conformers of a given molecular species (see below) and can
be treated by the following methods. The skeletal coordinate systems and their transformations must be chosen
taking into account the peculiarities of the conformational
The isomerizer (1 7) (2 8) (3 9)(4 10) (5 11) (6 12) of the conforma) (26b) of neoinositol is derived in a
tional equilibrium ( 2 6 ~S
straight forward manner on the basis of skeleton XIV.
Example 23
Example 21
( 2 7 4 (2 6)
The conformational interconversions of (25a) - (25c) are treated on the basis of the (D3d)skeleton (25d) and the reference
isomer (25e) ; the isomerizers
(27b) (1 4.) ( 2 7 )
Ib.= (1 564)
I,, = ( 1 2 3 5 )
= ( I 46)(1 5 ) = ( I 465)
I,, = (253)(15) = (1532)
I,, = (2 5 3) (1 6 4)
= (146)(235)
follow from the description according to Section 2.4.6.
( 2 . 5 ~ () 1 5 )
= (1
(25b) (1 6 4.)
( 2 . 5 ~ )(2 3 5 )
(2 5 3)
(1 4 6)
The conformation independent descriptor (26)-dimethykyclopentane refers to the skeleton XI11 whereas the descriptors of
(27a) - (27d) and the isomerizers of their interconversions
(e.g., Inb
= (1 4) (2 7) (2 6) = (1 4) (276)) are derived from (27e)
when CH; = 1, CH, = 2.
Angew. Chem. internat. Edit. 1 Vol. 9 (1970)J No. 9
3.1.3. T r i go na 1 B i p y r a mid a 1 M o lecu le s [ 2 i , 2 8 1 [*I G e n e r a l R e m a r k s
A polyhedral skeleton is that molecular skeleton with
ligand positions equivalent to the n vertices of a polyhedron (see e.g. VII-XIV). There have been several recent
attempts to determine the number of interconvertible
polyhedral isomers (Muetterties
calls these polytopal
stereoisomers) and the number of steps needed for the
interconversions of these isomers by graph theoretical
methods 127b*c1 or some matrix equivalent thereof [27f1.In
all of these contributions particular attention was focused
upon the isomerizations of the pentacoordinate phosphorus compounds by the generally accepted pseudorotation rz81mechanism. (2+ 3 ) - T u r n s t i l e P r o c e s s e s [ z 9 1
Of the formal possibilities invoked to explain regular
pentacoordinate isomerization, the one alternative to the
(1 +4)-pseudorotation mechanism that satisfies all experimental data is the (2 3)-internal rotation process
shown in example 24. We call this process the (2 3)-turnstile mechanism.
Isornerizations of polyhedral 'molecules are treated according to the general procedure of Section 3.1.1. If an isomerization process leads to an exchange of ligands between
non-equivalent skeletal positions, it must be represented
by transformations involving skeletal coordinates of different, non-equivalent sets. The separate treatment of the
ligands of non-equivalent coordinates, suitable for nomenclature purposes. can be given up, and the combination of the unprimed and the primed sets of ligands used
Application of this permutational formalism to the rearrangements of pentacoordinate compounds (28b) with
the skeleton (28a) and skeletal coordinates 1 . . . 5 shows,
in a straightforward manner, which mechanisms are conceivable ['I'.
If the ligands of (28b) are indexed in sequential order
(286) is the reference isomer E and is represented by the
descriptor (e).
[27] a) E. L . Muetterties, J. Amer. chem. SOC.90, 5097 (1968);
b) J. D. Dunitz and V . Prelog, Angew. Chem. 80, 700 (1968).
Angew. Chem. internat. Edit. 7,725 (1968); c) P . C . Lauterbur and
F. Ramirez, J. Amer. chem. SOC.90, 6722 (1968); d) E . L. Muetterties, ibid. 91, 1636 (1969); e) E. L. Muetterties and A . T . Storr,
ibid. 91, 3098 (1969); f) M . Gielen, M . DeClerq, and J . Nasielski,
J. Organometal. Chem. 18, 217 (1969); M . Gielen and J . Nasielski,
Bull. SOC.Chim. Belges, in press; M . Gielen, C. Depasse-Delit, and
J. Nasielski, ibid., in press; g ) R . G . Pearson, J. Amer. chem. SOC.
91,4947 (1969); h) F. N . Tebbe, P . Meakin, J . P . Jesson, and E . L .
Muetterties, J. Amer. chem. SOC.92, 1068 (1970).
[28] a) R . S. Berry, J. Chem. Phys. 32, 933 (1960); b) see also:
L. H. Somrner: Stereochemistry, Mechanism, and Silicon.
McGraw-Hill, New York 1964; c) R . F. Hudson: Structure and
Mechanism in Organophosphorus Chemistry. Academic Press,
New York 1965; d) R . Schmutzler, Advan. Fluorine Chem. 5, 31
(1965); e) E. L. Muetterties, Inorg. Chem. 6 , 635 (1967); f) G . Witf i g , Bull. SOC.Chim. France 1966, 1162; g ) D . Hellwinkel, Chem.
Ber. 99, 3668 (1966); h) E. L . Muetterties and R . A . Schunn, Quart.
Rev. (London) 20,245 (1 966); i) J . H . Letcher and J. R . van Wazer,
J. Chem. Phys. 45, 2926 (1966); j) D . Gorenstein and F. H . Westheimer, Proc. Nat. Acad. Sci. U.S. 58, 1747 (1967); k) R . R . Holmes and R . M . Deiters, J. Amer. chem. SOC.90, 5021 (1968);
I) F. Ramirez, Accounts Chem. Res. I , 168 (1968); m) F. Ramirez,
J . F. Pilot, 0.P . Madan, and C . P . Smith, J. Amer. chem. SOC.90,
1275 (1968); n) F. Ramirez, Trans. N.Y. Acad. Sci. 30, 410
(1968); 0) M . Sanchez, R . Wolf, R . Burgada, and F. Mathis, Bull.
SOC.Chim. France, 1968, 7338; p) F. H . Westheimer, Accounts
Chem. Res. I , 70 (1968); q) G . M . Whitesides and H. L. Mitchell,
J . Amer. chem. SOC.91, 5384 (1969).
[*I F. Ramirez is co-author of Section 3.1.3
[29] a) I. Ugi, D . Marquarding, H . Klusacek, P . Gillespie, and
F. Ramirez. Accounts Chem. Res., in press; b) F. Ramirez, S . Pfohi,
E. A . Tsolis I. Ugi D . Marquarding P . Gillespie and P . Hoffman,
J. Amer. chem. SOC.,in press.
Angew. Chem. internat. Edit. / Vol. 9 (1970)/ No. 9
Example 24
i-This scheme can be viewed as an attempt to illustrate the
topological concept of the (2 3)-turnstile process (1 2 4)
(3 5 ) in geometrical terms [*I by a 120 & 180" relative internal
[*] Note added in proof: The mechanistic alternatives for the
positional exchange of pentacoordinate molecules belong to different classes of the permutation group s,. Pseudorotation of D,,
trigonal bipyramidal molecules correspond to the alternating
permutations (a e a'e') of the apical a and a ' and equatorial e,
e" and e"' ligand indices whereas turnstile processes by permutations (a e) (e" e' a') from the class of (2 3) cycles. As a consequence
of the skeletal symmetry of V I I the permutations (ae) (e"e'a').
(a'e) (e"e'a), (a'e') (e"ea) and ( a e a ' e ' ) are equivalent. For instance, the pseudorotation isomerizer (2435) as well as the four
different but equivalent turnstile isomerizers (24) ( 1 35), ( 2 5 ) ( 1 34).
(34) (125) and (35) (124) all transform ( 2 8 6 ) ( - E ) into the same
isomer. During the equivalent turnstile process, the pivot L, of
the pseudorotation mechanism replaces one of the equatorial
ligands (L2 or L3). It must be noted that during a turnstile process.
the pair has an approximate local C, skeletal symmetry and undergoes a corresponding rotation about the local C, axis while the
trio has an approximate local C, skeletal symmetry and simultaneously undergoes a rotation about the local C, axis with opposite
angular momentum.
The BPR (Berry pseudorotation or (1 +4)-pseudorotation) and
TR ((2 3)-turnstile rotation) differ fundamentally by the pathways by which they occur and by the intermediate species involved.
The BPR, as it is defined, is the combination of two synchronized
bending motions of a pair of equatorial and a pair of apical bonds
respectively, and does not involve any internal rotation. The TR
process, by contrast, involves internal rotation.
In cyclic pentacoordinate compounds, enforcing BPR upon a
molecule invariably involves internal rotation of the cyclic moiety
YS. the remaining ligands and thus becomes a TR unless the BPR
involves the interconversion of a diequatorial ring and an apicalequatorial or a diapical ring respectively which is quite unlikely
and only reasonably conceivable for sufficiently large rings.
rotation of the ligands 1, 2, and 4 vs. 3 and 5. We propose
that the regular pentacoordinate isomerizations occur by a
(2 + 3)-internal rotation through a (2 + 3)-intermediate or
an interconvertible set of (2 + 3)-intermediates rather than
by the generally accepted (1 4)-pseudorotation mechanism. Both processes can, in the idealized case, occur with
conservation of angular momentum. While there is experimental evidence which contradicts the (1 + 4)-pseudorotation mechanism, all data are in accordance with the
(2 + 3)-turnstile concept.
Intermediate species of the type (296) or (29d) in the
(2 3)-internal rotation concept can only be characterized
in terms of the permutational symmetry of the skeletal
coordinates and not by a particular geometry. Species
(29b) for instance, is characterized only by the fact that it can
be represented by five skeletal coordinates of which a pair
(3’ and 5 ’ ) and a triple (l‘, 2’, and 4’) are equivalent and
permutable. If all of the possible (2 3)- intermediates of a
given set of 20 pentacoordinate topological isomers are
interconvertible, this is represented by complete equivalence of the five skeletal coordinates which are interconverted by 5 ! permutations. Such a state cannot be described by geometry or the conventional concept of symmetry because it is not possible to place five points mutually
equidistant on the surface of a sphere. Since it can be assumed that the (2 3)-intermediates are short-lived species,
a geometrical description loses its meaning in any case, due
to the uncertainty principle.
by the authors as proof that none of the published (“nonBerry”) alternatives to the (1 +4)-pseudorotation mechanism of pentatopal rearrangements could be responsible
for the F-exchange of (31). However, the (2+3)-process
corresponds exactly, like the (1 4)-pseudorotation process, to all the assumptions that entered Whitesides’ and
Mitchell‘s evaluation of their experimental data.
The ‘H- and ”F-NMR data of (30) [29b1 indicate rapid positional exchange (permutational isomerization) of the
ligands of the pentacoordinate phosphorus. This can only
be interpreted by a (2 + 3)-turnstile process; the ( 1 + 4)pseudorotation is not possible as a consequence of the
structure and the strain it imposes. Even if one forces (30)
into the pseudorotation pathway by overcoming the structural limitations, the result would involve relative internal
rotation of the adamantanoid moiety vs. the five membered
ring, asuperpositionoftheturnstileuponpseudorotation.
Whitesides and Mitchell [”ql analyzed the temperature dependence of the 31P-NMR spectrum of (31), and were
able to show that at low temperatures ( T I - 100 “C)
structure (31) is frozen, i.e., the dimethylamino group and
two of the fluorines occupy equatorial positions, and two
of the P - F bonds are apical, without any exchange of the
equatorial and apical fluorines. Above -50°C all four
fluorines become completely equivalent, but there is an
intermediate temperature range in which the equatorial
pair of fluorines replaces the apical pair of fluorines and
vice versa in a single concerted step. This result was invoked
If such a (2+3)-internal rotation is terminated with the
dimethylamino group in the equatorial position, the result
is an exchange of the equatorial and apical pairs of fluorine
3.2. Chemical Reactions and Transition States
3.2.1. G e n e r a l C o n s i d e r a t i o n s
The mechanisms of chemical reactions can be described
on various levels and in a variety of terms [301. In any detailed discussion of the mechanisms of a chemical reaction
the transition complex, which is often called the transition
state, plays a central role 13’]. The transition state corresponds to a complex which contains all participants in a
reaction such that the reactants have already lost some of
their characteristics, and in which the characteristics of
the products are not yet fully developed. The transition
complex is a short-lived state, or better, a statistical ensemble ofshort-lived states,which corresponds to the peak
of an energy barrier between starting materials and products. These transition complexes, as statistical entities, cannot be characterized in geometrical terms (as has been attempted, often with admirable imagination), since they do
not represent a uniform long-lived species, but a collection
of different, short-lived states.
Even if we could observe one single, isolated transition
complex of an “ultimate kinetic unit” [31b1, a geometrical
description would not be possible. As a consequence of the
uncertffinty principle, the geometrical features (distance)
of short-lived species are not subject to an accurate observation 13*1.
The reader of current chemical literature could be led to
believe that there exists a wealth of detailed knowledge on
transition complexes. However, a critical analysis of the
statements often reveals assertions which go far beyond the
evidence or, in some cases, even contain violations of basic
There seems to be a tradition, among mechanistically
oriented chemists, to describe reaction mechanisms ac(301 A . Srreitwieser, J r . : Molecular Orbital Theory. Wiley, New
York 1962, p. 310.
[31] a) H . Eyring, J. Chem. Phys. 3, 107 (1935); W . F. K . WynneJones and H Eyring, ibid. 3, 492 (1935); S. Glasstone, K . J . Laidler, and H . E y i n g : The Theory of Rate Processes. McGraw-Hill,
New York 1942; b) H . Eyring, D . Henderson, B. J . Stover, and
E. M . Eyring: Statistical Mechanics and Dynamics. Wiley,
New York 1964, p. 29.
[32] 0.K . Rice, J. Phys. Chem. 65, 1588 (1961).
Angew. Chem. internat. Edit.
Vol. 9 (1970)/ NO.9
cording to accepted terminologies and formalisms, without
ascertaining the pertinence of the proceedings. The occurrence of a chemical reaction is frequently expounded
in detailed geometrical terms. Conclusions are drawn about
the transition complex from the so-called “reaction coordinate diagram”, in which the only thing less certain than
the ordinate is the abscissa.
To interpret and predict the steric course of stereoselective
reactions, often one arbitrarily chosen conformation of the
transition complex is discussed. The geometrical distribution of nuclei and electrons of this choice, of which at
best, only its relative free energy, origin, and destination
are known, is quite often the basis of interpretation and
prediction of the steric course of stereoselective reactions.
10021, 12001, 10201, and 12221. These eight points are also
the vertices of a cube (0,) or two interlocking tetrahedra,
the di-dodecahedron (T,). The six points 101 11, I l O l j ,
~ 1 1 0 ~ , ~ 1 1 2 ~ , j 1 2 1 j , a nlieinthecentersofthefacesof
that cube and at the vertices of an octahedron (0,).There
are also sets of points which correspond to the vertices of a
trigonal bipyramid (D3,), e.g. 10001, 12221 (the apical
points), 10211, 11021, and 12101; if the points 10121,
11201, and 12011 are added to the latter set, we have a
hexagonal bipyramid (D6J.
3.2.2. R e l a t i o n s between Reacting E n t i t i e s
These current extrapolations of the transition state
theory [311result from the fact, that there are many things
which can and should be said about transition states, but
no formalized framework of thought has been available to
support these statements. The quantum mechanical description of reacting systems must be interpreted with
caution as the assumptions one requires are not made on
the basis of a complete understanding of the phenomenon [**I. Further, with the proper choice of assumptions,
one can produce the result one intuitively feels is the most
A description of chemical reactions and their states employing mathematical structures, offers less to visualize
but substantially avoids unjustifiable statements. It is
beyond the scope of this article to discuss, in a detailed
manner, the major objectives of mechanistic chemistry.
We wish to generate a partially symmetrized topological
coordinate system in order to discuss some examples of
reactions whose mechanisms have previously been critically analyzed on the basis of sound experimental evidence.
A topological description yields some information on re-
action mechanisms because the coordinates of the relevant
particles, including the phase of the wave functions
of the reacting species can be chosen in such a manner that there is conservation of coordinates during the
It is possible to choose the coordinates of certain parts of
the reacting species in such a manner that the reactions
can be described by permutation operators in analogy to
the “isomerizer” description of isomerizations. Relative
to the description of isomerization mechanisms, by virtue
of the partitioning of the permutation groups S, into their
distinct classes[29a1of conjugate elements, it will also be
possible to classify the mechanisms of the reaction between two (and more) reactants.
On this basis, it will be possible to establish rules concerning mechanistic alternatives in certain types of reactions.
Fig. I I . Sets of points in a cubic lattice
Finite sets of points belonging to the cubic point lattice and
lattices which are contained within the cubic lattice are
useful for the construction of coordinate systems with
suitable permutational transformation properties and
neighborhood relations to be applied to chemical reactions.
A topology whose open sets contain the above mentioned
finite sets of points as internal points can be chosen as a
topological coordinate system for chemical reactions such
that the coordinates of all relevant parts of the molecule
are preserved throughout the reaction, from the starting
material through the transition state to the final products.
Example 26
The following brief discussion of certain symmetrical sets
of lattice points will also be useful for the latter discussion
of the topology of chemical reactions.
The classical configur; ional correlation [331 of (+)-or-phenylethyl chloride (32a) and (-)-or-phenylethyl azide (32c) is
chosen to illustrate the topological approach to a chemical
reaction and its transition state.
The lattice points lOOO], 10221, 12021, and 12201 are the
vertices of a regular tetrahedron (TJ, as are the points
[33] E. D . Hughes, C. K . Ingold, and C . S. Parel, J. Chem. SOC.
1933, 526; see also: E. D . Hughes and C. K . Ingold, ibid. 1933,
1571; 1935, 244; J . L. Gleave, E. D . Hughes, and C. K . Ingold.
ibid. 1935, 255.
C f . c g . , Chem. Eng. News 48, No. 22, 12 (1970).
Angew. Chem. internat. Edit. Vol. 9 (1970)J No. 9
Fig. 12. Reaction (32a) -t (326)
ordinate system.
(32c) in a topology oriented co-
The process whereby (32a) + (32b) + (32c) can be described
in terms of a topology oriented skeletal coordinate system that
applies to all participating species is pictured using the cubic
lattice point set as follows:
The central atom C, is represented by a distinguished point at
the cross-section of the coordinates z1-z5. Conservation of
these coordinates throughout the process (32a) + (326) + (32c)
is characteristic of the S,2 mechanism and is equivalent to a
correlation of the configuration of (32a) and (32c).
Example 27
A trigonal bipyramidal adduct of the ylide (33a) and aldehyde
(33b) has been suggested as an intermediate in the Wittig olefin
synthesis [zsf*341. This concept was recently invoked in an attempt to explain the preferential formation of cis isomers (33i)
by Wittig’s kinetically controlled synthesis of 1,2-disubstituted
olefins [34bl.
Here it was assumed that the nucleophilic oxygen of an aldehyde
(336) attacks the electrophilic phosphorus of the ylide (33a)
such that a trigonal bipyramidal adduct (33c) with an apical
P - 0 bond and an equatorial P-carbanion bond was formed
directly. It was further assumed that the erythro-intermediate
(33d) was formed from (33c) by “a small anticlockwise rotation about the C - 0 bond” [’sf]. A corresponding clockwise
rotation would yield the threo-isomer of (33c) and thus the
trans-olefin. Employment of the (2 3)-turnstile process (see
Section provides an alternate interpretation for the formation of these cis-olefins (33i) by the Wittig reaction in salt
free, non-polar solvents at low temperatures [351.
If the hypothesis of the “open-pyramid mechanism” is invoked
at all, it is reasonable to assume that the nucleophilic oxygen of
the aldehyde (33b) attacks the electrophilic phosphorus of the
ylide (33a) on the side opposite the P-carbanion bond to give
the trigonal bipyramidal intermediate (33d).
[34] a) L. D. Bergelson and M . M . Shemyakin, Pure and Appl.
Chern. 9, 271 (1964); Angew.Chern. 76, 113 (1964); Angew. Chern.
internat. Edit. 3, 250 (1964); b) W. P.Schneider, Accounts Chern.
Res. 2, 785 (1969).
[35] M . Schlosser and K. F. Christmann, Liebigs Ann. Chem. 708,
1 (1967).
\P h
This is analogous to formation of an S,2 transition state. The
primary adduct (33d) is transformed by the (2 3)-turnstile
mechanism into the intermediate species (33f) via (33e). It is
reasonable to assume that the conformations of (33e) and (33f)
occur such that the (2 3)-turnstile process suffers minimal
steric hindrance from the bulky groups R and R’. The stereochemical transformation of (33f) into (33g) arises from the
fact that the carbonium ion moiety of (33f) can approach its
carbanion moiety only through a rotation about the P- 0 bond.
This is explained by observing that while resonance stability of the carbonium ion moiety of (33f) exists, groups
H, C , R’, 0, and P must occupy a coplanar set of positions. A
clockwise rotation about the P - 0 bond of (33f) leads to (33g),
the counterclockwise rotation generates the antipode of (33g).
Both (33g) and its antipode are intermediates of the cis-olefin
formation. As long as there is no rotation about the C - 0 bond
of the carbonium ion moiety, there can be no formation of the
trans-olefin via the threo-isomer of (33g) or (33h) respectively.
The isocyanide-nitrile rearrangements 1361(and the SNireactions)
probably also occur through an intermediate state which resembles topologically the intermediate states of the (2 + 3)-turnstile rearrangement. It is even conceivable that
of the
reactions which have heretofore been considered to occur by
an SN2 mechanism do involve a (2 + 3)-turnstile type process.
The topological consideration of this section may also be useful
for the discussion of the pentacoordinate carbon systems [381.
We will refer to Dewar’s PMO formulation of the more
than 30 year old Evans’ rule as the Dewar-Evans’ rule 139*401
of electrocyclic and related reactions:
[36] F. W. Schneider and B. S. Rabinovitch. J. Amer. chern. SOC.
84,4215 (1962); 85, 2365 (1963); K.M . Maloney and B. S . Rabinovitch in I. Ugic Isonitrile Chemistry. Academic Press, New York
1970, in press.
[37] See e.g.: J . L. Fry, C. J. Lancelot, L. K. M . Lam, J . M . Harris,
R. C. Bingham, D. J . Raber, R. E. Hall, and P. v. R. Schleyer,
J. Arner. chern. SOC.92, 2540 (1970).
[38] G. A . Olah, G. Klopman, and R. H. Schlosberg,J. Arner. chern.
SOC. 91, 3261 (1969); J . J. C. Mulder a n d J. S . Wright, Chem.
Phys. Lett. 5 , 445 (1970).
[39] M . G. Evans, Trans. Faraday SOC.35, 824 (1939).
[40] M . J . S. Dewar: The Molecular Orbital Theory of Organic
Chemistry. McGraw-Hilk, New York 1969 a) p. 319; b) p. 323;
c) p. 335, 339:
Angew. Chem. internat. Edit.
Vol. 9 (1970) / No. 9
“Thermal electrocyclic reactions take place via aromatic
transition states. Photochemical electrocyclic reactions
take place through excited forms of aromatic transition
states” [40cl.
The thermal electrocyclic reactions can be considered as
that class of reactions whose transition states are topologically equivalent to the aromatic hydrocarbons. Although
Dewar did not use a topological formalism as the basis of
his treatment of electrocyclic reactions, he was fully aware
of the topological nature of the problem. He describes
the transition states of the Diels-Alder reactions as “delocalized systems - topologically equivalent to, or isoconjugate with the 7c system in benzene”
The DewarEvans’ Rule can be applied to an individual case by selecting such a topological coordinate system for the transition
state that this system also applies to the isoconjugate
hydrocarbon. By conserving the topology and suitably
choosing coordinates throughout a chemical reaction one
can correlate the features of interest (e.g. the stereochemistry) in the starting materials and products of the electrocyclic process.
Both in their essential content and consequences, the Dewar-Evans’ rules are roughly an equivalent to the frontierorbital 1411 or correlation diagram I4’I based WoodwardHoffmann rules of orbital symmetry conservation [431.
(34 e i
Yet, in some aspects, these rules differ. There are cases in
which the application of the essentially topological DewarEvans’ rules seem to be more simple and allow one to
analyze a chemical situation more clearly, as in the case
of the Cope rearrangement [40b1 and the Diels-Alder synthesis 140,43J. Recently, Mulder and Uosterhoff[44J announced rules concerning the conservation of permutational symmetry which seem to be particularly clear,
general, and theoretically well founded in their application
to electrocyclic reactions and related processes.
The Woodward-Hoffmann rules [431 are primarily oriented
towards geometrical symmetry. They can also be stated
1411 K . Fukui, T . Yonezawa, and H . Shingu, J. Chem. Phys. 20,
722 (1952); K . Fukui, T. Yonezawa, C. Nagata, and H . Shingu,
rbid.22, 1433 (1954); K . Fukuiand 0. Sinanoglu: Modern Quantum
Chemistry. Academic Press, New York 1965, Vol. 1, p. 49.
1421 R . Hoffmann and R. B. Woodward, J. Amer. chem. SOC.87,
2046 (1965); H . C. Longuet-Higgins and E. W . Abrahamson, ibid.
87, 2045 (1965).
[43] R. Hoffmann and R . B. Woodward, Science 167, 825 (1970);
R. B. Woodward and R. Hoffmann, Angew. Chem. 81,797 [ 1969),
Angew. Chem. internat. Edit. 8, 781 (1969), and preceding papers
cited there; see also: J . J. Vollmer and K . L. Servis, J. Chem. Educ.
45. 214 (1968); 47, 491 (1970).
[44] J . J . C. Mulder and L. J. Oosterhoff, Chem. Commun. 1970,
305, 307.
Angew. Chem. internat. Edit. / Voi.9 (1970)/ No. 9
in the form of topological skeletal coordinate conservation.
Example 28
(34 4
(34 c)
The prediction of the stereochemistry of the process (34a) to
(34dj by the Woodward-Hoffmann frontier orbital treatment
is symbolized by the disrotatory opening (34c) --t ( 3 4 d ) .
The topological equivalent of the process (34aj + (34d) results
from the conservation of the topological coordinates during the
entire reaction.
As can be seen immediately, the set of coordinates of (34e) apply
to the transition state (34c) of the disrotatory process as well as
the initial and final states (3413) and (34d). This treatment could
also be represented algebraically by mapping the set of indices
to those parts of the involved species of interest onto the set of
their coordinates but in this case the simple graph suffices.
There is no set of coordinates conserved during the contrarotatory analog of the process (34a) + (34d) if one forbids
the overlap of orbitals which are not in phase. For example, if
one takes the coordinates of (34f) of a conrotatory transition
state, the in-phase overlap of the C - C G bond of (34a) that is
being opened has as a consequence an out-of-phase K overlap
of (34dj and vice wrsu.
4. Conclusion
The authors’ intention was clearly not to expound and exemplify all the aspects and applications of modem mathematics to chemistry. Neither is this an effort to present a
complete and perfect discussion of the utility of mathematical structures to chemistry. In many areas, our treatment is open to profound improvement. Often, only the surface is scratched, hopefully deeply enough to stimulate interest and to point in a promising direction for further develop-
ment. It should be noted that this paper is a compromise between imperative mathematical rigor and the desire to present a document that will be readable and comprehensible.
The apparent overemphasis of classification problems
and nomenclature resulted from the obvious application
of elementary topological and group theoretical concepts
to these problems, and the desire to illustrate the direct
applications of basic mathematical structures. Further,
the definition of a nomenclature is only useful if it is complete with regard to its whole scope and its limitations,
which, for our case, is all of chemistry. Macromolecules
could have been included by some additional definitions,
but the authors feel that this would over-extend the introduction of the present concept.
Many different systems ofchemical nomenclature have been
peopoaed, some specifically designed for convenient handling of rather limited sets of very similar compounds, such
as the cholestane derivatives, whereas others are defined to
serve as a universal nomenclature. Some of these proposed
nomenclatures have found general acceptance, while others
were never mentioned again, quite often undeservedly.
The present permutational nomenclature differs considerably in some essential features from any previous attempt
to create a universal chemical nomenclature which does
not break down in “pathological cases”, i.e. for molecules
with complicated skeletons and involved stereochemistry.
In contrast to other systems, the permutation nomenclature deals with the molecule as an entity, and dissects it
only in exceptional cases, which are described separately,
as is done with the (R),(S)-qomenclature of molecules
with more than one element of chirality. In contrast to
the systems of nomenclature in current use, the descriptor
does not tell “what a molecule looks like”, but it is an
instruction on how to draw or let a computer draw the
steric formula of the molecule, or (which is often more
easily done) how to construct a steric model of the molecule,
e.g. a Dreiding model. The descriptors of molecules are
based upon the representation of the molecule by an algebraic formalism, which in fact, is a general principle of
order in chemistry and provides a framework of thought
which helps to understand the steric aspects of chemistry.
There are many cases in which the conventional nomenclatures are fully adequate and more convenient to use than
this descriptor system. For open chain chiral compounds
(of class a) with only a few elements of chirality (e.g.
“asymmetric carbons”) the (R),(S)-nomenclature works
well and is easier to use than the descriptors. On the other
hand, the (R),(S)-nomenclature is very cumbersome to
use or fails completely with compounds like ( 3 ) , for which,
to our knowledge, there is no other adequate system of
nomenclature than the present. The metal complexes with
a coordination number higher than four are compounds
for which the permutational nomenclature seems to be the
only acceptable solution to the nomenclature problem 14’].
For molecules with a polycyclic skeleton and complicated
stereochemistry the permutational nomenclature is superior and often the only satisfactory description of the
molecule, with the exception of some classes of very similar
compounds which have their own specific reference system
and nomenclature, like the steroids.
1451 Cf. K . A . Jensen, Inorg. Chem. 9, 1 (1970).
Some of the computer-oriented nomenclature systems
which have been developed recently are called topological
nomenclature systems. These systems do not use the concept of topological spaces for the representation of steric
features, but represent the chemical constitution of molecules in terms of graph theory.
Since the treatment of enantiotopicity l4’],diastereotopicity 1471, and prochirality 14*1 according to the present formalism follows in a straightforward manner from the principles
presented, in addition to the definitions and procedures included herein, no chapter on these subjects will be found.
Within the framework of thought, as presented here,
chirality appears only as a secondary aspect of permutational isomerism. Chirality is, however, a phenomenon
which is relevant to chemistry, and a test for any theoretical
treatment of the steric aspects of chemistry. A formalism
which fails to deal adequately with chirality is severely
limited in its application to stereochemistry.
The present formalism does not represent the mirror
image of a chiral object as the mirror image “as reflected
by a plane mirror”; it does, however, represent molecules
in such a manner that the algebraic representations of
mirror images differ, and it deals with chiroids of both
classes, a and b, adequately.
The discussion of an alternate mechanism for pentacoordinate isomerization was introduced in an effort to
show how the extension of the algebraic formalisms led to
the reinvestigation of a reaction. The mechanism proposed
for the Wittig reaction on the basis of the turnstile discussions illustrates the use of the formalism further.
The utility of topological concepts for the discussion of
reaction mechanisms has already been recognized by
several authors. Evans [391 and Dewar
applied the topological concept of isoconjugation to transition complexes,
a particularly important contribution in this context.
The present authors will discuss the relation of mathematical structures and chemical reactions in greater detail in the near future. A promising field of application is
biochemistry. Hydrophobic interactions, quaternary structures of proteins, the nucleic acid structures, and enzymatic processes are some of the problems to which the
present concept could be applied.
We gratefully acknowledge the contributions of Professor
J. Dugundji whose advice, many suggestions, and discussions guided us substantially and of Professor E. Ruch who
stimulated us with his lectures and lengthy discussions during
his visit to Los Angeles in October-November 1969. The
authors are further indebted to Professors M . Eigen, F.
Hawthorne, K . Kirschner, G. A . Segal, and W . K . Wilmarth as well as Dr. G. Kaujhold for helpful hints and
discussions, and to Mrs. P. Gillespie, Mrs. D. Marquarding,
and Mrs. V. Schneider for their valuable help in preparing
the manuscript.
Received: April 16, 1970
[ A 785 IE]
German version: Angew. Chem. 82, 741
[46] a) J. Lederberg: The Mathematical Sciences. The M.I.T.
Press, Cambridge, Mass. 1969, p. 37; b) J. E. Dubois, Entropie 27,
1 (1969); Lecture at the Twelfth International Symposium on
Combustion, at Poitiers, France, July 14-20, 1968.
[47] K. Mislow and M . Raban, Top. in Stereochem. I, 1 (1967).
[48] K. R. Hanson, J. Amer. chem. SOC. 88,2731 (1966); P . Corradini,
G. Maglio, A . Musro, and G. Paiaro, Chem. Commun. 1966, 618.
Angew. Chem. internat. Edit. 1 Vol. 9 (1970)1 No. 9
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