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 bond[’] 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- [*I 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 experience. 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. 703 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. I I I I I I 1 I 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. 704 /678511 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 satisfy: (a,b) = ( c , d ) if and only if a = c and b = d. 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 structure. Injection lone-one into1 11\78521 Surjection lontoi Fig 2 Various types of mapping of sets A Bijection Ione -oneonto1 5B 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: EI: A 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.). VUE.A:URU “t/ (‘a (Reflexive) The relation R cortains the diagonal ‘) a Aa E II: (aR b) 3 (6 R a ) (a, b) 6 (6, a) R R (Symmetric) The relation K i s Symmetric about the diagonal (Transitive) 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) 705 Each metric D induces a topology. For example, by using the <,y} ~ ]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,, 1 1 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). 1 3 G I: 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 . , (&*) (49) If the condition f(Q.a') = f(4 .f(a') 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 entries f (49) GUn,R) - 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. AA = AB = BB BA = = A B 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 1 -1 +l I +1 -1 +I -1 -1 -1 The concept of a symmetry group is illustrated by Fig. 4. b 2 3 i:)I - __-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 conditions: [7] E . Ruch and I. Ugi, Theor. Chim. Acta 4 , 287 (966); Top. in Stereochem. 4, 99 (1969). 706 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 A-F 6: E A E C D F the multiplication table obtained is that of the group symmetry operations. 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}, cnh: {(2nxoh}> 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 matrix allx + a12y + a31x + 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' a32 = 1 M 11: allazl + al2aZ2 aI3az3= 0 + + + + + + a12a32 + a13a33 a21a31 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 HR, == {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. G = ( 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: AAT = E, 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-'A 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 = +1} 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) invariance. The symmetric group S, of n letters or symbols consists of the n ! mappings of a set of n letters on itself. etc. 1 2 3 ..... n 1 2 3 . . . . . 707 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 . SP”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. CsH5 H \ C6H5 (3b) x X”) --- = S” kOzH 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. c1 c1 c1 The three isomeric dichlorobenzenes (la)- ( I c ) are permutational isomers having skeleton VI (see Section 2.4.2), but are not stereoisomers. P Br\ Cl’ H C1 Br, c=c= Br\ c=c, H/ H H/ H c=c= c1 [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. 708 H 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 species. 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., 1967. 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. 1~ 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. + txsi = i-{x.QI A chiroid with two elements of chirality x and x’occurs in four stereoisomeric forms. {xibxkl (xs3xb) { X R I X J i t X S > X k I 709 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 with Y x V‘ = (( + I ) ( + I)‘,( - 1)( - I)‘,( + I ) ( - I)’,( - I ) ( + 1)‘) and \\ isahomomorphismonto:Y=( , (+I) \ / ). (-1) 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 products. The nomenclatural designation (2(S),3(R),4(R)J)-tetrafor D-arabinose (6) is based upon the hydroxypentanal 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. + + CHO I HO-C-H I H-C-OH I H-C-OH I C H2OH (4) 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 ,H ,I’ csH( C‘ (74 H, c l 0 H 9 F’e C‘ f 7b) H3 H. CH,/ -..,NH2 ‘c~H~ 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 1969. 710 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” object. 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. :IF(3 I b.8 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. H H 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 cyclooctane. 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, SP8h.16 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 Hu H H H 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). 15 13 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 a1 Xi: 2 L 16 15 bl 1p7856j 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 3 n7,is possible. To accomplish this, sets must be used at 1 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 II 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 = (1: x Y x . . . x Y") H:. 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 1178571 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. 712 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 ps 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 I 2 -_ J_ )_ n . c) 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. c), 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 ,z 1 \ 5 3 4, 5, :j t; r: 1(" ; '.----__ __- -_- _ , _-\ I , \=----- 1 / / gives the operator P, operator = ( 1 4 2 5 ) . When used as a descriptor, the 12345 generates P, = ( I 4 2 5 ) (1). { I1 22 33 4 5 } according to (;) = = Ps( b)E. 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. 713 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 XXVII). S V : If the observer looks along a principal C , axis [Zbl the assigned skeletal numbers increase in a clockwise manner. I 1 ; I n 6 8 10 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). I Y 2 I' 6+3 2' XXIII (Y) xxv (D5d) Angew. Chem. internat. Edit. XXIV ( Y ) XXVI (Td) / Vol. 9 (1970)1 No. 9 2 CD I 1 I I +--- - _ , XXVII (Td) XXIX XXVIII (0,) 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 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 1 2 3 . . . . . 4 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 . . . . . n Il/s 0 0 0 c)E . . . . . 1 0 0 . . . . 0 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 follows: 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 matrix. 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 : 0 . . . . . . . . . . . . . . . . . . . . . . . 0 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. c),,, 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 &. (Oh) 2.4.3. L i g a n d s E):( 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 c):,,,, 715 c) 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 matrix 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. c)pre), c),,,, oder: (’34) [ j(9;[:R : 0001 0 1 0 0 1 0 0 0 0 1 0 01 0 l o o 0 [n R :81 1 0 0 0 =(Id) J(9b, = ( I I ) E4 (9c) 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,) and (1) ={ n, n, n,.. . 1 . . . 2 . . . 3 . . . n, I 2 3 . . . n , . . . n, . . . n L . . n (2) (3) 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 2.4.7.1. 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) 716 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 a B Y 1 I 3 4 (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 (3. Angew. Chem. internat. Edit. Vol. 9 (1970)1 No. 9 - - x (1357) (2 4 6 8 ) 2 - x z z x 5 x 2 2 z x z x I - - x x z @ x x z H xo z H @ = o o x - z x x z x (1 8 ) ( 3 5 ) o x x x x z z @ x A fa x x 0. z c1 - - 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 ~ CI 1 For comparison, try naming the truxinic acids (3c) and (3d) by any other nomenclature, and then reconstruct their steric formulas! NH ( C H Z C H ~ N H ~ ) ~ 3 2 5.6 4 6 B a Y 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 g r H OCH, 3 CH30 : a O C H 3 H3C H OCH, OH c1 H H (Ila) 4 CH, H3C = OCH, M 3 1 2 (IIb) 9' 2' 5 3,4 3,4 1 2 7?.12' 2' a 1 2 3 4 1' 4 3 2 1 9' (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 12' CH3 H H (Ilc) = Mopt 11' 1' 6' ' 6 ' 7 5 1 2 ' 7112' 1' 7' 8' 16' 7'12' 7'12' 3' = E 10' 4' 3' 4' 5' 7' 8' 9' 10' 11' 12' 10' 11' 12' 6' 5' 8' 7' 1' 2' 3 4' 9' 6' I 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: 717 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 9 n :I{ 1 2 3-5 3-5 1 2 3 4 3-51 = (.I - 2.4.7.2. 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 skeletons. 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. 2.4.7.3. 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 7 H3C-&-Si(CH3)3 6 I { f 3' 3'-5' 1' '' 5' 4' 3'-5' 3'-5' 1' 2' 3' 4' 3' 6' 5' 1' 2' G(1' <2J t' 2' 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). 718 4') (--3c c jp178591 Fig. 9. Formal dissection of cyclic and polycyclic compounds with multiple bonds (a) o r bridges (b,c), the resulting free valencies being represented bynils. 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. H KPt) (ISh) ( = M 5 ( l5a) (ISCi ( = E) 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,, skeleton 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 3 4 11 6 7 8 9 10 3 5 5 13 20 21 16 17 18 12 14 15 2 2 6 7 8 9 10 11 12 13 14 15 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 B2i+2 zzi-1 zzi+i : Y2i-1 > Y2i+1 > z2i : Y21 221+2 :Y2i+2 f164 (166) 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 i 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 results. 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 ' 59 l l T l ' 34 7 I H H H H H H I H H I H H I 26 (17a) ( = E) p-y@-H-, 12 (176) ( = Mopt) 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). 719 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). I (19~) (19bl 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 H The (19c) H methyl-3,4-0,O'-isopropylidene-~-~-arabinopyranoside 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 14 [21] H. Maehr and C . Schaffner, J. Amer. chem. SOC. 92, 1697 (1970). 720 15 16 10 18 17 18 I [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). NH,-cH,-cH,, I 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. H’ I I (236) 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) LxL’ 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. H3CTCH2S03H 12 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. CH,COO OH I I 5=5 = < ( h , . . . . . h,) (4) Q CH3 dcH~COOH IH3I (24 c ) (244 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 = 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 722 I kOCH, 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 ligands. 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,. P,F, (A, . . . . . k”) = F, (1,. . . . . h,) with 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), , i, 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) - 17) 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. c),, c),pt (14)(23)(36)(45)(58)(67)(78)(81) - (18)(27) (37)(48)(78)(54)(63)(32)(41)1 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 = P~,xiihih4= h7h, + h,h, = &(hlh4 = i{(XI - + h 5 ) (h4 - + . . . . . + h4h5+ h,A,} = (8) 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 (9) 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). (3’ = = 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 (15) 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. 9 723 reverse; for instance if P, = ( I 2 3 4), then P, = (4 3 2 1) = (1 4 3 2). Multiplying equation (14a) by P, and equation (14b) by 6 produces equation (16) I P from which equation (17) is obtained by multiplication with Pi. (6)’ = Example 22 P;xq:) (17) Combining equations (16) and (13) yields equation (18). Ho @ H = HO H W OH H OH 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). D npl (1% = (e) OH H (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 interconversions. 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 ~ CH3 Example 21 H ( 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 ;& H CH,’ (27b) (1 4.) ( 2 7 ) It 11 Ib.= (1 564) I,, = ( 1 2 3 5 ) lob = ( I 46)(1 5 ) = ( I 465) I,, = (253)(15) = (1532) I,, = (2 5 3) (1 6 4) Icb = (146)(235) follow from the description according to Section 2.4.6. = H*cH3 F H :*F CH3 ( 2 . 5 ~ () 1 5 ) F = (1 (25b) (1 6 4.) 5) H3c$F : H ( 2 . 5 ~ )(2 3 5 ) 724 E (2 5 3) 8 (1 4 6) )q$ 6 4 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 3.1.3.1. G e n e r a l R e m a r k s L1+L3 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. 3.1.3.2. (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 instead. 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. + + 725 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. + F + + + 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 726 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 atoms. 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 principles. 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 d~211~ 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 adequate. 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 reaction. 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 r1 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. [**I C f . c g . , Chem. Eng. News 48, No. 22, 12 (1970). Angew. Chem. internat. Edit. Vol. 9 (1970)J No. 9 727 Fig. 12. Reaction (32a) -t (326) ordinate system. - (32c) in a topology oriented co- (334 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: i? (33fl R\ “C 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 3.1.3.2) 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). n - Ph (33a) @’ [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). 728 I + I“ O-p---Ph \P h ,’C\ R“ H (33i) 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: R; R ”, -__ [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 H ‘c- C / H (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. (34fl 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- 729 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). 730 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 (1970) [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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