ISOlIERISM AND COSFIGURATIOS* BY -1.C. LUNN A N D J. K. SEXIOR
I. Introduction Foiezcord.-The general ideas of isomerism and the configuration of molecules have been treated by so many different investigators and from so many different points of view that limitations of space practically forbid any attempt at a connected review of these subjects. In the present paper, therefore, historical aspects mill be mentioned only in so far as the articles cited bear directly an the theory arid formulae herein developed. Another form of introduction, however, appears to be unavoidable. Since the discussion is to be largely mathematical, it is nrcessary to review certain current chemical terms and to define with adequate precision the sense in wfiich these will be used. .It the same time, there will be introduced certain new expressions which in part refer to new ideas, and in part consist’ only of names applied to old hut’ hitherto unnamed concepts. Puye Sztbntunces.-The terms “atom,“ “molecule” and “valence” are here accepted in the sense in which they are ordinarily used. For the moment at least they require no more precise definition. “Isomerism” however is a relation existing between “pure substances” which are not objects of experiment but ideal limits to series of separation processes. And just TThat i s meant by a pure substance is by no means free from ambiguity. To begin with, it is not assumed that all the molecules of a pure substance are in an identical condition. Modern theories of chemical reaction usually assume that’ in any lot of material certain molecules exist in an “activated state,” and that the activated molecules are in dynamic equilibrium with the unactivated ones. But it is never assumed merely because some of the molecules are activated while others are not that t’he substance is therefore a mixture-or in other words impure. On the other hand, aceto-acetic ester is generally agreed to be a “tautomeric” mixture of two isomeric substances, the enol and the keto forms, which are in dynamic equilibrium with one another. The separation of these two forms is difficult but not impossible. Between such tautomeric mixtures at one extreme and the so-called pure substances containing activated molecules at the other, there may or may not exist a n unbroken series of intermediate cases. It is certain that there are some tautomeric mixtures which are more difficult t o separate into their component pure substances than is aceto-acetic ester. In some of these cases a partial but not a complete separation has been accomplished. Easily racemized pairs of optical isomers fall in this class. Certain other materials in which no separation has ever been effected nevertheless resemble tauto* Contribution from the Kent Chemical Laboratory and the Department of Mathematics of the University of Chcago.
I028
A. C. LUNN A S D J. K. SESIOR
meric mixtures so closely that it seems justifiable to assume that they also consist of two or more isomeric forms in dynamic equilibrium. By contrast, the current theory’in regard to hexahydro-benzene is that this substance consists of at least two different forms also in dynamic equilibrium. With the methods at present available, however, there seems to be almost no chance of separating these components. Again, the Kekul6 formula for benzene calls for two forms of all ortho-disubstituted benzene derivatives, and here also the members of the various pairs are supposed to be in dynamic equilibrium. No one has ever met with any success in attempts to isolate the members of such a pair. Further, the rotation dispersion phenomena in tartaric acid and some of its derivatives have given rise to the hypothesis that in the fluid states these substances are equilibrium mixtures of several distinct forms2 And lastly, the question of whether solutions of certain pentaerythrite derivatives contain only one variety of these substances, or whether some of the molecules may be in a tetrahedral and others in a pyramidal form has been argued at some length.3 There appears to be no record that any attempt has been made in the last two cases to isolate the individual forms supposed t’oconstitute the mixture, but the present prospects of such an isolation are to say the least discouraging. A successful mathematical theory of isomerism should certainly permit the number of isomers in a given class to be calculated, and it seems futile t o attempt to devise such a theory if the activated and unactivated moleculrs in a pure substance be considered isomeric. For it is never certain just how many varieties of activated molecules exist in a sample of such material, and so there is no means of comparing the numerical consequences of any theory with observed facts. Similarly, as long as the object of enumeration is kept in mind, supposed varieties like those of hexahydro-benzene, orthodisubstituted benzene derivatives, tartaric acid or pentaerythrite can hardly be considered as isomers. Present knowledge of these variations in molecular structure and configuration is too small to permit of an accurate counting of the varieties in the so-called equilibrium mixtures. I n other words, the authors of this paper do not consider i t at present advisable to treat the cases intermediate between tautomerism and activation as forming a smooth and unbroken series. The criterion of separability (partial or complete) serves to divide this series with the requisite degree of distinctness, provided it is used under the guidance of practical judgment in a somewhat extended sense. Certain cases where no actual separation has yet been made may be treated as separable on the grounds of analogy with cases where 1 Sachse: Z. physik. Chem., 10, 203 (1892);Ber., 23, 1363 (1890); Rlohr: J. prakt. Chem. (21, 98, 3 1 j (1918). * Arndtsen: Ann. Chim. Phvs., ‘3) 54, 403 (1858); Lowrv: J. Chim. phys.,23, 5 6 j (1926); Longchambon: Compt. rend., “178, 951 (1924); 183, 9 j 8 (1526);Lucas: Ann. Phys., (10) 9, 381, (1928:. 3 Ebert and v. Hartel: Saturivissenschaften, 15, 669; Keissenberg: 662 (1927); Boeseken and Felix: Ber., 61, 787,185j; Kenner: 2470 (1928); Ebert, Eisenschitz and v. Hartel: Z. physik. Chem., [B] 1, 94 (1928).
ISOMERISM AND CONFIGURATION
1029
such a separation is an experimental fact. A criterion which is strictly free from ambiguity in all instances seems at present unattainable. Of course a distinction based on separability assumes the present state of chemical technique. It is probable that substances inseparable today may, in view of improvements in chemical methods, be separated to-morrow. But this prospect of improvement is not subject to any known limitation, and consequently to postpone all attempts a t a mathematical theory of isomerism until the perfection of available methock of separation would be equivalent to the abandonment of such attempts. Nor does there seem to be any reason why the present status of chemistry should be considered particularly illsuited to supply the foundation of such a theory. On the contrary, there is some reason to believe that it may be particularly well adapted to serve this purpose. Connezity and Configuration.-From the percentage elementary composition and molecular weight of any pure substance, its empirical formula is calculated. With such empirical formulae in hand, it has been possible to work out the connexity relations between the component atoms of the molecules of a large number of substances. These connexities are arrived a t principally by interpretation of the synthetic, disruptive, and replacement reactions which form the bulk of struct'ural chemistry. The present paper need not attempt to deal with this side of the subject; the connexity data are assumed in their ordinarily accepted form. I t is important however to note that connexity data do not by themselves determine the space relations of the atoms in the molecule. Saturated unbranched hydrocarbon chains are usually written in somet,hing like the following form :
H
H
H
I l ---c-c-c--I l
l l
H
H
H
(1)
(2)
(3)
But this conventional mode of representation is not supposed to indicate that the hydrogen atoms directly connected with carbon atom ( 2 ) really lie closer to this atom than they do to carbon atom ( 3 ) , or that carbon atoms ( I ) and ( 2 ) really lie closer to one another than do carbon atoms ( I ) and (3). Similarly, the simple statement that the six carbon atoms in benzene are cyclically connected is compatible with a great number of diagrams of which the following may be considered samples.
/"\ c c
c---c
I I c-c
Kc c
1030
A. C . L U S S A S D J. K. SEXIOR
The attempt to rule out from such a list of possibilities all but one form (or failing that a minimum number of forms) is the attempt to establish t,he configuration of the molecule, and all data which aid in this process of elimination are "configuration data" as distinguished from the "connexity data" hitherto referred to. It, is of course possible that in a part,icular case a single observed fact may be part of both bodies of data. And it is obvious that while t,he connexity4data can always be adequately represented in two dimensions, the configuration dat,a usually need three. Skeleton and Univalent Substituents.-If the'structural (connexity) formula of a compound be written out in full, it will be seen that the molecule can be thought of as a skeleton carrying a certain number of univalent substituents. The skeleton may be a single atom, a straight or branched chain, a single or condensed ring, or any complex of such forms. Since radicals as well as atoms may be treated as univalent subst'ituents, the division of the molecule is not unique. For example, in xylene, at least three methods of division a t once suggest themselves. ( I ) The skeleton consists of the eight carbon atoms of the xylene molecule; the univalent substituents are the ten hydrogen a t o m attached to this skeleton. (2) The skeleton consists of the six carbon atoms of the benzene ring; the univalent substituents are the two methyl groups and the four hydrogen atoms attached to this skeleton. ( 3 ) The skeleton consists of the six carbon atonis of the benzene ring and the t\To met,hyl groups; the univalent substituents are the four hydrogen atoms attached to this skeleton. An$ one of these methods of division is admissible. The choice between t'hem depends on the object' in view. But although the choice of a skeleton may in some cases be somewhat arbitrary, each possible choice fixes a certain re tant' set of properties of the system. In the first place, the number d of un lent substituents ialways a positive integer) is thereby uniquely determined.$ In the second place, choosing a skeleton in such a ITay that a particular atom or radical (A) functions as a univalent substituent is equivalent to assuming (at least for the purposes of the present discussion) that (-1)has certain properties. From the moment that (a)is so chosen, it is to be considered as an indivisible unit, and as such it acquires the infinite symmetry usually represented by a geometrical point, and becomes permutable with other univalent substituents supposed to possess like propertice. That is to say, when such univalent atoms or radicals 4 Connexitv is a relation of order independent of considerations of space. The "structural" relations treated by cheniists are relations of just this sort, and it is uniortunate that the v o r d structure as used by engineer?. etc.. should carry with it geometrical connotations which are too special for chemistry. L'hroughout the present artkle "connexity" and "structure" are used as synonyms. and the iormer expression is introduced here merely t o emphasize the fact that the rrlations indicated by both terms are independent of space limitations. except perhaps the vague one implied in the wrrd vicinity. 5 llolecules like CO: and S ? which do not contain anv univalent atom or radical may be considered as cases where d = 0 . 3uch molecules do n& give rise to instances of isomerism of the kind treated in this paper.
ISOMERISM AXD CONFIGURATIOS
103 I
as hydrogen, iodine, methyl or naphthyl are thought of as univalent substit,uents, the differences between them become qualitative, like the differrences between red, blue, and yellow geometrical points. From these considerations arises a certain limitation on the choice of univalent substituents which will be discussed later in connection with enantiomorphism. Partitions.-It is next necessary to define certain purely mathematical terms which at this point assume an essential role in the discussion. If d and dl, d2 . . . . di are positive integers such that dl d:! . . . . dt = d, then t,he set of integers d!, dz . . . . di is called a partition of d, and each such set may be designated as a value of the general symbol (p). Every value of ip) naturally determines the corresponding value of d; and d determines a finite set of values for (p). The number of values of (p) in such a set is q(d)- a single-valued, positire, integral function of d. If d is considered constant, there ail1 be distinguishable among the values of (p) certain pairs which may be called adjacent. When the transfer of a single individual from one class to another converts (pa) into (pb), then these two partitions are adjacent. For example, where d is 6, the partition (4, 2) is adjacent to the partitions ( j , I ) , (4, I , I ) , ( 3 , 3) and 13, 2 , I ) . It is not adjacent to the partitions ( 2 , 2 , z ) , ( 3 , I , I , I ) , etc. Khen any niolecule is divided into a skeleton and univalent substituents, not only is the number d of univalent substituents thereby fixed, but a value for (p) is also determined, since the univalent substituents chosen must consist of a certain number of c1asse.q of like individuals. T y p e Properties a n d Speci$c Propertie.y.--TVith these ideas in mind, the properties of niolecules may be divided into two kinds as follcms. I. Type properties are determined by the nature of the molecular skeleton, the value of ( p j and the distribution of the various sorts of univalent substituents among the unsatisfied valences of the skeleton. 11. Specific properties are not completely determined by the considerations just mentioned, but depend also on the specific nature of the atoms or radicals which function as univalent substituents. An example viill make this general statement somewhat clearer. Suppose that the skeleton be the carbon ring of benzene so that d is 6. In hydroquinone, the value of (p) is (4) 2 ) . SOK it is a property of hydroquinone that if one of t'he hydrogen atoms rhich function as univalent substituents be replaced by some univalent substituent other than a hydrogen atom or a hydroxyl radical (whereby a substance is produced for which (pi is (3, 2 , I ) ) , the compound formed is the same whichever hydrogen atom is replaced. This property of hydroquinone is hoxever a type property. I t is dependent only on the nature of the benzene carbon skeleton, the adjacent partitions (4, 2 ) and (3, z , I ) , and the distribution of the various univalent substituents in the original compound into the para position. Statements perfectly analogous to the one just made for hydroquinone could be made about para-dinitrobenzene, para-dimethyl-tetrachloro-benzene, etc. On the other hand, hydroquinone is a powerful reducing agent. This, however, is a specific property
+
+
1032
A. C. LUNS AND J. K . SENIOR
dependent on the fact that the para pair of like univalent substituents which form one class are hydroxyl groups. No similar reducing property is found in para-dinitro-benzene, para-dimethyl-tetrachloro-benzene, etc. From what has been said, it is clearly of importance to have a complete list of the type properties of molecules. For it may be stated a t once that, although a general theory of isomerism competent to explain type properties seems well within reach, a theory which accounts for specific properties is still some distance in the future. Further discussion of these type and specific properties is however postponed until the subject of isomerism has been considered in greater detail. Isomerism and zts Varzeties.-For treating this subject, the following definitions will be used. ( I ) If any experimental procedure serves to distinguish under like conditions between two lots of pure material of identical empirical formula, then these two lots of material are specimens respectively of two isomeric compounds. Isomerism is the relation existing between them. The use of connexity data makes possible the extension of this idea to include the idea of isomeric radicals.6 Structural isomers are isomeric substances which differ in regard to (2) the connexity relations within their respective molecules; stereo-isomers show no such difference. (3) Physical isomers are substances which as solids show different properties but which are alike in the fluid (that is the gaseous, liquid or dissolved) states. ( 4 ) Diamers are best defined indirectly: diamerism is the relation between any two stereo-isomers which are not enantiomorphs. (See Note A.) Enantzomorphism.-( 5 ) Enantiomorphism or optical isomerism is a special form of stereo-isomerism. If two substances (A) and (B) show the following properties, they are called enantiomorphs or optical isomers. (a) (A) and (B) in the fluid states rotate the plane of polarized light equally but in opposite directions. That is, they are both "optically active," and the optical activity of (A) is the negative of that of (B). In the crystalline state, certain special effects obtained from (A) are equal to those obtained from (B), but of opposite sign; of these the piezo-electric phenomenon is the best known. Usually (but not invariably) the crystals of (A) are not congruent with the crystals of (B), but are congruent with the mirror images of such crystals. (b) (A) and (B) are identical in respect to all ordinary physical properties, and identical in their chemical behavior towards all reagents except those 6 Throughout the prcsent paper, it will be necessary to refer repeatedly to the number N of isomers in a given class. The equation K = I is therefore equivalent to the assertion 'that the number of isomers is one. In spite of the somewhat paradoxical nature of this language, such use of the terminology is indicated by considerations of brevitv and uniformity. And so hereafter, when the number of isomers in a given class is said"to be one, the statement should be understood to mean that there is only one compound in the clam in question. This implies that there is no isomerism a t all. Interpreted in this fashion, the terminology has never been found to lead to confusion.
ISOMERISM A N D COZFIGCRATIOS
I033
which are themselves optically active in the fluid states. It is understood to be typical that they differ in their chemical behavior towards optically active reagents, although the amount of difference may in some cases be too small to measure. Such is the ordinary definition of ‘enantiomorphism. The usual interpretation is that enantiomorphous molecules possess space configurations which are non-congruent mirror images of one another. But it is possible to define enantiomorphism with equal accuracy and in more abstract terms as follows. Let S be a complete class of isomers containing every substance or every radical of given empirical formula. The enantiomorphous relation E is such that I Every class S is uniquely divided into two subclasses S’ and S” such that (a) For every member u of S‘there exists a member v of S such that u E v.’ (b) For every member x of S” there exists no member y of S such that x E y.
I1 (a) u E v implies that u and v are not identical. (b) u E v implies v E u. (c) u E v and u E v’ imply that v and v’ are identical. That is to say, enantiomorphism is the only symmetric, intransitive and constantly dyadic relation found among the isomers of a complete class s. At this point it is probably best to introduce parenthetically the limitation on the choice of univalent substituents already referred to (see p. 1 ~ 3 1 ) . It’ is as follows: Every univalent substituent x must be such that there nowhere exists a univalent substituent y such that x E y. Because if x E y, then x cannot be considered to possess the infinite symmetry representable by a geometrical point, and cannot therefore be permuted with the univalent substitutents considered to possess such symmetry.
Skeletal Isomerism and Univalent Substitution Isomerism.-(6) The various categories of isomers so far mentioned are well recognized by chemists and their names are firmly established in the literature. But there is another easily definable kind of isomerism for which no special name is current. The division of any molecule into skeleton and univalent substituents implies a unique division into two classes of the isomeric relations between compounds of the empirical formula in question. I. Univalent substitution isomerism is the relation existing between compound (A) and the isomeric compound (B) if the formula of (A) can be converted into that of (B) by a permutation of the univalent substituents without disturbing the skeleton. In this case (A) and (B) are univalent substitution isomers. The notation u E v is a customary one to denote that u stands in the relation E to v.
I034
A . C. L E N S A S D J. I ’1-ip) and S “ v ( p ) are all equal to one when ip) = (d), no niatter what J’ may be. That is to say, if the unsatisfied valences of any skeleton V are saturated by d identical univalent substituents, only one compound can be formed. There are therefore just, as many types of compound determined by Y and (p) = (d) as there me determined by Y alone. iSkeietnl Isoinerz’siiz,-If the class of isomers to be enumerated is to coritain skeletal isomers, then the results at present most interesting are obtained by imposing a further restriction on the method of dividing the molecule into a skeleton and univalent substituents. This is to be done in such a way that S o univalent’ substituent is a polyatoniic radical. (I) S o univalrnt atom is part of the skeleton. (2) If the division is performed in this unique manner, then the problem of finding the total number of isomers of given empirical formula resolves itself into two separate problems as follow: (a) The probleni of finding in how many ways the skeletal atoms can he connected in a single skeleton Y with d unsaturated valences. The class of all such skeletons is T ; the number of members of T is IT. (b) The problem of finding the number of univalent substitution isomers derivable from each member of T. Although these two problems are so closely related chemically that the distinction between them seems almost artificial, from the point of view of the mathematical methods required, they shoIy important differences. Problem (b) (which has already been discussed in some detail) is directly a probleni in partitions and permutation groups;’O probleni (a) involves also preliminary consideration of a method of representing connexity between polyvalent atoms. Of the two, (a) is thus distinctly the more complex. The value of S T is a function only of the nature of the skeletal atoms and the value of d. It is independent not only of the specific nature of the univalent substituents but also of ( p ) . But since the value of KT is a type prol o The word “suhstitution” is used bv chemists and mathematicians in conflicting senses. Throughout the present article this word is t o he understood to have its ordinary chemical significance; for the mathematical term “substitution” the expression used is “permutation”-a usage for which there is good mathematical precedent.
ISOMERISM BKD C O S F I G T R I T I O ? ;
1039
perty and the number S v ( p ) of univalent substitution isomers derivable from each member of T is likewise a type property, the total number of isomers of given empirical formula is also a type property. From this and similar reasoning, it is clear that the following three additional defining agreements as to classes of isomers to be enumerated also involve only type properties. The isomers of given empirical forxnula, taking account of isomerism, (A,) structural and steric, both enantiomorphous and diameric. The number in the class r i p ) so defined is Q(p;. ( j ) The isomers of given empirical formula, taking account of structural isomerism and diamerism, but disregarding enantiomorphism by counting each enantiomorphous pair as one individual. The number in the class I-’(p) so defined is Q’(p). (6) The isomers of given empirical formula, taking account only of structural isomerism and disregarding stereo-isomerism by counting each subset of stereo-isomers as one individual. The number in the class C”(p) so defined is Q”(p). On account of the disparity in the mathematical machinery necessary for the solution of problems (a) and (b) (p. 1038) it seems unlikely that the values of ( l i p ) , Q’(p) and Q”(p) can be conveniently given by any single formula. It is easy to show that X r ( p ) , S ’ v i p ) and S”v(p) can be represented as the sums of series, and hence Q(p), Q’ip) and Q”(p) as the sums of series each member of which is itself a series. In certain cases, however, a simple expression exists. If ip) = (d), then only one type of univalent substitution deriT-ative may be formed from each member of T. Hence the number of isomers of given empirical formula is NT. In this case, (b) is vacuous and (a) is the whole problem. On the other hand, where the skeleton consists of less than three atoms, the value of NT is usually one,11and
Here (a) is vacuous and the whole problem lies in (b). In the literature there are a number of attempts to deal with the matter of skeletal isomerism. The two papers of widest scope are those of Sylvester’* and of Gordan and ;Ile~ejeff,’~ in both of which the parallelism between the chemical theory of molecular structure and the algebraic theory of invarients is stressed. I t seems probable that interesting and important results might be attained by exhaustively pursuing this line of attack, but chemists have never devoted to these papers the attention which they deserve. The case where V consists of one carbon and one nitrogen atom and d is one is an exception to this rule. Sylrester: .lm.Jour. Math., 1, 64 f1878). See also Clifford: ibid., p. 126. 13 Gordan and Alexejeff: Z. physik. Chem., 35, 610 (19001.
1040
A. C. L U S S A S D J. K . SESIOR
Cayley14set himself a more restricted problem by considering only open chain skeletons composed entirely of one kind of atom. His work attracted considerable notice and called forth a number of articles purporting to contain improvements on his methods.’j Rut the formulae for the numbers of isomers developed in these papers, although in many cases correct, are never convenient to apply, and some of them involve so large an amount of enumeration and summation that they represent only a moderate advance over the graphic methods which they are intended to replace. It it is further noteworthy that all of these articles (Cayley’s included) are restricted to the field of structural isomerism, and that even within this limited range the various authors do not always agree on the isomer numbers obtained. The intricate nature of the problem is thus clearly indicated. Since the systematic treatment of problem (b) appears to be a natural first stage in the treatment of the more general questions raised by problem (a), this article deals no further with skeletal isomerism. It is hoped to take up this subject a t some future time, when the papers cited will be further discussed.16 For the present, attention will be confined t o the study of the classes Sv(p), S’\,(p) and S”v(p) as well as the determination of Nv(p), S’v(p) and I\;”v(p) which enumerate these classes. Genetzc Relataons.-But before the details of these problems are considered, attention may be called to another type property which has been familiar to cheniists a t least since 187 j, but to which no special name has apparently been applied. Let Sv(pa) and Sy(pb) be two sets of univalent substitution isomers determined by 5’ and the adjacent partitions (pa) and (pb). Since the partitions are adjacent, the two sets of isomers may be called adjacent sets. In favorable cases, each member of &(pa) may be converted by simple subsitution reactions into one or more members of S\,(Pb). That is to say, each member of &(pa) is genetically connected with a certain number of members of &(Pb), and the number of members of &(Pb) genetically connected with each member of Sv(pa) is a type property. If the genetic relations of all members of &(pa) with all the members of SY(Pb) be considered together, then they may be called the genetic ;elations of Sv(p,) and SV(Pb), and these relations are also type properties. A classic example of the use of these genetic relations is found in Korner’s work on the substitution derivatives of benzene. Korner took the carbon ring of benzene as a skeleton V, and investigated the adjacent sets of isomers , Nv(3, 3) are both 3. He found that a Sv(4, 2 ) and Sv(3, 3). S V (2 )~and certain member (A) of S v ( 4 , 2 ) is genetically connected with only one member (D) of Sr(3 3) ; a second member (B) of Sv(4, 2 ) is genetically connected with l1Cayley: Brit. Ass. Adv. Sci., Reports, p. 2 j 7 ( r 8 7 j ) ; Phil. Mag., (4) 47, 44 (1874’8; (5),3,34 (1877). These papers are reprintedin Vol. IX of Cayley’s collected wo,%s. l5 Schiff: Ber., 8, 1542 (1875); Losanitsch: Ber., 30, 1917; Herrmann: 2423 (1897); Delannoy: Bull., (31, 11, 2.39 (1894). A general method of determining in terms of the group theory the number of isomers of arbitrary empirical formula has already been obtained.
ISO\IERIShI A S D CONFIGURATION
1041
two members (D and E) of S d 3 , 3 ) ; and the third member (C) of Sv(4, 2 ) is genetically connected with all three members (D, E and F) of S d j , 3) These relations are illustrated schematically by the following diagram.
They are sufficient to identify each one of the six substances involved and serve as the basis for assigning names as follows. (A) is the para, (B) the ortho and (C) the meta isomer in &(4, 2 ) ; (D) is the asymmetric, (E) the vicinal and (F) the symmetric isomer in Sv(3, 3 ) . Korner actually carried out this investigation on the dibromo- and tribromo-benzenes.” But in choosing these particular sets of isomers to work with, he was guided purely by considerations of convenience. Had he chosen t,he dinitro- and trinitro-benzenes or the dimethyl-tetrachloro- and trimethyltrichloro-derivatives instead, the demonstration might (for experimental reasons) have been much more difficult, but the results would have been the same, because the relations inrestigated are type properties independent of the specific nature of the univalent subst’ituents involved. Reasoning quite analogous to that of Korner has been applied repeatedly to the solution of problems in structural and stereo-chemistry-often with conspicuous success. It needs no further argument to prove that the relations involved are always type properties. In many cases they suffice to identify uniquely each one of the compounds concerned, and even where the identification is not unique, they serve to limit the choice of a formula to be assigned to any one compound. But curiously enough these type properties do not appear to have been named.ls They will be referred to hereafter in this paper as the genetic relations between adjacent sets of univalent subst’itutionisomers. Other Type Properties.-The type properties SO far enumerated do not complete the list of all such properties known. If t’he class of isomers to be enumerated contains skeletal isomers, but the division into skeleton and univalent substituents is not subjected to the restrictions mentioned on p. 1038, certain other isomer numbers are obtained, which do not correspond t o any common method of chemical classification, but which are nevertheless type properties of the molecules involved. Korner: Gam., 4 , 3 0 j (1874). Holleman in his “Textbook of Organic Chemistry,” p. 504 et seq. (1920)speaks of “Korner’s principle” and refers to the “absolute” determination of the ortho. meta and para positions by its means. J. B. Cohen in his “Organic Chemistry,” Part 11, p. 430 et. seq. (1928)uses the same terminology. There is however no explicit statement that the type of reasoning is quite general and by no means restricted in its application to the derivatives of benzene. For instance, of the five truxillic acids, only one should yield two diameric monoanilides, and the isomer in question is thus uniquely identified by this fact. (See Stoermer: Ber., 57, 18 (19241). T h e word “absolute” has been overworked and should be avoided where possible. I’
I8
1042
A . C. L U S S A S D J. I;. SEXIOR
If the molecule is divided into skeleton and substituents in such a way that some Qr all of the substituents are polyvalent, again certain isomer numbers which are type properties are obtained. In part,icular classes of compounds ifor example the spiro compounds and the complex molecules derived from cobalt, chromium, etc.) this matter of polyvalent substitution isomerism is of great importance. It lies outside the scope of the present paper but does not appear t o offer any addit,ional sort of difficulties and is also a problem in permutation groups. Thus it, should prow amenable to treatment by methods similar to those developed in subsequent wctions of this article. Possibly it may be discussed more fully at some later time. Beyond the type properties mentioned, no others are known to the authors of this paper. Those listed have one important characteristic in common. They are all pure numbers; and it will be shown that this characteristic of all these type properties is a fact of great significance. Problems to be Considered.-Since attention is to be confined from now on to univalent substitution isomerism, the five type properties connected n i t h isomerism of this sort are here recalled. They are (I) The value of K\;v(p). (2) The value of S ’ v ( p ) . ( 3 ) The value of I S”&). (4) The number of members of each of t’he X”v(p) structurally isomeric subsets of SyCp). (5) The genetic relations between adjacent S\;v(p)s. The mathematical machinery for dealing with isomers of the three classes and properties of the sorts mentioned is next developed. 11. Determination of ZG(p) T h e Group G and its Induced Groups Hc(p).--dny class consisting of d individuals determines definite classes of certain other entities such as unordered or ordered pairs; unordered, partially ordered, or completely ordered triples;lg ordered pairs of unordered pairs, etc. In fact it is vital to the folloming discussion that to every value of (p) there corresponds a class of such entities. The table lists some examples, together with t’he number of entities in each class and the partit,ion with which each class of entities is associated. But although there is a class of entities corresponding to every value of (p), the converse is not true in the sense intended. For example, the class of unordered pairs of totally distinct unordered pairs, the number of which is As unordered pairs ( a h ) and (ba! are identical; as ordered pairs they are not. The six forms (ahc) (ach (hac) (cab) (cha) I I1 I11 Iv v 1-I are identical as completely unordered triples; as completely ordered triples they are all different. Considered as triples ordered only as to ( a , I and I1 are identical. I11 and IV are identical. V and VI are identical. The three pairs are however different from one another.
ISO>IERISJf A S D COSFIGURATIOS
1013
,
d’ (d - 4) ! 2 l. 2 1. z . ’ is not the class belonging to any partition in the sense of the general definition following. Moreover, the use of special names for the entities is superfluous, since every class of the sort here needed is uniquely defined by a particular value of (p) and may therefore be called simply the class belonging to (p). For an arbitrary partition (dl, d,. . . , d o , the correspondence is fixed as follows. If d things are assigned to d places, there will be d ! distinct assignments, provided the d things are individually distinguishable. This corresponds to the partition ( I , I . . . . I , I ) . But if, of the d things, dl are alike of one kind, d2 alike of a second kind, and so on, the number of distinguishable d! assignments is , and these constitute the class of entities belonging di! d2!. . . .di! to the partition (dl,dz.. . . di). It will be seen however that the placing of the last set of df things is determined by the placing of the other sets-hence the fitness of the special names mentioned above. Entity
Partition
Unordered d-tuples
(d)
d! a -
Individuals
(d-1,1)
d! (d-I)! I ! = d
Unordered pairs
(d-z,z)
Ordered pairs
(d-:
d! - d(d - I ) (d-2)! Z ! 2 d! = d(d-I) (d-2)! I ! I !
L-nordered triples
(d-3,3)
Partially ordered triples
(d - 3 , 2 ,1)
Completely ordered triples
(d- 3,1,1,1)
Ordered pairs of totally distinct unordered pairs
(d - 4 , 2 , 2 )
Conipletely ordered d-tuples
(1,I..
I
I, J )
. . 1,I)
Sumber I
d! d ( d - I ) (d-2) -(d-3)! 3! 6 d! - d ( d - I ) d-2) (d-3)! 2! I ! 2 d! = dkd-1) (d-2) (d-3)! I ! I ! I ! d! (d-2) (d-3) _ d(d-I) _ (d-4)! z ! 2!4 d! l I , = d ( d - ~ ) .. .(2)(1) I! I ! I ! .
.,I.
I . 1.
A permutation of the d places determines a corresponding or “induced” perniutation of the various assignments belonging to any single partition. It is clear, therefore, that any group G of permutations on the places induces a determinate group H G ( ~of) permutations on the class of entities belonging to the partition (p). The following table lists some of these groups which match classes of entities in the preceding table. (See S o t e C.)
I044
A. C . LUSN A S D J. K . SESIOR
Partition
Group
Degree
(d - 2,1,1)
HG(I,I.. . . I , I )
(1,I... .I,I)
d! = d(d- I ) (d-2)! I ! I! d! = d(d-I). . . . (2) I! I ! .
.
,
, I ! I!
(I)
Sets of Transitzvzty.-Every such induced group has a certain number Z,(p) of sets of transitivity. The symbol zG(dl,d2,.. . di) will be used to mean the number of sets of transitivity in the group H G ( d~z,. ,, dr) whichisagroup of d! degree induced by the group G (of degree d) for the partition di! dz! di!
.
(d1,dz.. di). If all the q(d) values of (p) be considered in connection with any group G, there results a set of q(d) numbers which may be called a complete set of Z ~ ( p ) s . One of these values of Z,(p) is, however, of a peculiar nature. The partition (d) induces a group H(d) which is always of degree one, no matter what G may be. So H(d) is always the identity and Z(d) is always unity. Determination of Z&) by Direct Enumeration.-If G and (p) are known, a method of direct enumeration will theoretically determine &(p) in all cases. This method is illustrated by the following example in which (p) is (6,2), and the group G is generated by the operations (I) abcd.ejgh
and
(2)
ae.hh.cg.df
Let a single letter be assigned to each of the of letters in the set a . . . . h as follows:
ah = A
bc
=
cd ad
=
B C
D E F fs = G gh = H = eh = ef =
hj
= I = J = K = L = hI = T\T
cy
=
nc bd eg
fh ae
Then the generators ( I ) and group H ~ ( 6 , zof ) degree 28 (I) (2)
0
dh = P (2)
28
(= 8 X
7/2)
unordered pairs
aj = Q bg = R ch = S de = T ag = U bh = V ce = K dj = X
ah = Y be = Z cf = A dg = B’
in the group G of degree 8 become in the
ABCD.EFGH.hI?;OP.QRST.UF‘WS.YZA’.B’.I J.KL dE.BH.CG.DF.NP.QT.RS.U~~~.YZ.A’B’.IK. JL
ISOMERISM A S D CONFIGURATION
I045
These two operations generate a group H ~ ( 6 , 2 which ) belongs to a class of groups expressible in’the Cayley notation as follows:
{ [ (ABCDEFGH)s(JlN0P.QRST.UVWX.YZA’Bf)s]~.i(I JKL) 4 } 2 : i Every group of this class has 6 sets of transitivity, and therefore Z ~ ( 6 , 2 is ) 6, where G has the significance assigned to it on the previous page. A Formula for Z&).-There is never any theoretical difficulty in determining ZG(p) by such enumeration] but it may readily be seen that for large values of d and the more complex values of (p) the task becomes insufferably tedious. It is therefore important to have a more expeditious way of calcutaing this value. Such a method is to be found in a n adaptation of the well known character theorem of Frobenius, which states that if L be the sum of the letters left invariant by the individual operations of a permutation group G of order g and degree d, then the number of sets of transitivity ZG is equal to L;g. If any partitition of d be written out in full and all the integers equal to unity omitted, the resulting set of integers represents a type (t) of operation which may occur in a permutation group of degree d. Conversely every operation of a permutation group has a determinate type in this sense. For instance where d is 8, the operation abcdef.gh is of type ( 6 , ~ ) the ; operation abc.def is of type (3.3); etc. According to this definition, the identity operation is unique in that its type is represented by a vacuous set of integers no matter what the degree of the group, and that it is the only operation of this type. Let K(t) be the number of letters left invariant in a group G of degree d by a n operation of type (t); and let nt be the number of operations of type (t) in G. Then L =ZntK(t) t
where the summation is in respect to all types it) of operation in G. Hence
ZG
hK(t) =
t
g Where (p) is (d- I , I ) and the group G is the group on the individuals] the formula becomes ZntK(t) ZG(d-I,I) = g which is the Frobenius theorem in its ordinary form. To adapt the formulation of this theorem to any partition (p) other than (d- I,I), it is necessary to have, not the number of individuals left invariant by an operation of type (t), but the number of invariant entities of the kind associated with the partition (p). I n the general case the theorem is ZntK(p,t) ZGb)
E
(A) g
1046
A . C. L C S S AXD J. E[. S K S I O R
whenever the induced group G(p) is simply isomorphous with the group G(d- I , I ) . It is probable though still unproven that (except where (p) = (d) and G(p) is the identity) this simple isomorphism always occurs, inashave been compared in a large much as the results calculated by formula (-4) number of cases with those obtained by the process of direct enumeration, and no discrepancy has been found. Clearly the heart of the problem of determining Z ~ l p for ) any G and for any value of (p, lies in the determination of the sets of values of K. By induction, an empirical method of computing these has been found. I t i s as follows. If the sum of the integers in (p) is d, and the sun1 of the integers in (t) is e, then d-e is either o or a positive integer 5 d. If the integers in (t) are subtracted from those in (p) in any particular way, there is obtained ti set of remainders 11, = a1,ap. . . . . . . .ai such that al a2....... a, = d-e If the subtraction is carried out in all possible ways, a number of sets of rp. . . R I are obtained. And, if the following definitions are adopted R,! = al! as!. . . . ai! Rb! = bl! bz! . . . . bi! etc. (d-e)! (d-e)! (d-e)! then
+
+
K=-
R,!
+- H b ! + . . . . . . + _Rr!_
The first special cases under this general formula which need particular attention are those where a t least one integer in the set R, has a negative value. If a1 is a negative integer, al! = oc
K,! = and
cr)
(d-e)! =
R,! Hence if (for any given value of (t) and (pj) every R contains a t least one negative integer, then K is 0 . If only certain Rs contain negative integers, then it is sufficient, to consider only those which do not, because those containing negative integers have no effect on the value of I(. The special case of the identity comes under the general scheme, provider1 it is understood that a vacuous set of integers can be subtracted in only one d! way. In this case, t'he only R is dl,d2 . . . df and I< = _____ dl! d p ! . . . .di! Where d-e = 0 , every R which contains no negative integers is of the O! form o,o.., . . , . . o , and since olo!. . . . . . . o!
o!
o!
O!
= I
K=-+-+,,..+-=f R,! Rb! lif! This method of computing I< is very nearly automatic. The main difficulty is to be sure that the subtraction is carried out just once in every way
1047
ISOXERISM A S D CONFIGURATION
which leads to no negative remainders. One means of attaining this end is to assign a fixed order to the integers in the minuend, and to index the identical integers in the subtrahend. Two examples illustrate the proper procedure. d-e
=
3
d-e 4
2
21
22
4-21
2-22
2
4-21
2
0
2
2
2
2
4
2
21
2
4 22
__
2
4
2 21
-
__
4-22
2
2-z1
2
2
3 4
0
22
22
21
4-22
2-21
2
0
2
2
21
23
5
2 21
-
__
__
2-22
4
2-22
2-21
0 2
2 22
__
0
1
2
_______ 2
2
+ _ _ _ _ - 21
= 4 2
0
2-21
2
4
2-22
- ___
2
3
2
- __
--
3
3
4
0
0
2
22
4-21-22 0
2
2
2
2
By this method, the full set of . d u e s of I< which correspond to any given d may be computed. Khen such a set has been completely evaluated, all the values of Z,(p) according to formula (A) may be determined for any group G of degree d. Table I gives all the values of E; for d < I I ; Table I1 gives all the values of ZG(P) for d
1
is a complete set of Sv(p)s where the skeleton J’ consists of two carbon atoms united by a double bond.
J
Ethylene, C2H1,is one example of a substance derived from this skeleton when (p) is ( 4 ) ; tetrachloro-ethylene, C?C1,, is another. The substances derived from this skeleton when (p) has any other value may therefore be thought, of as substitution derivatives of ethylene (or tetrachloro-ethylene) . Examination of the table containing all groups of degree four shows that the two equations S v ( 3 , 1 ) = I and S T ~ Z=~3Ztaken ) simultaneouslyagree respectively with the values of Z G ( 3 , 1 ) and ZG(Z,Z)for no other group beside ( a b e d ) + The fact that for the skeleton in question SV(P,I,I) is 3 agrees with the value of Z G ( Z , I , I ) for the same group. But if, as matter of experiment, Xv(z,l,l) for the skeleton under consideration were 2 , this fact would still be compatible with the group requirement though no longer in agreement with it. Such a state of the experimental data might mean that one of the theoretically possible substitution derivatives of ethylene had never been prepared-due to inadvertence, lack of interest, or experimental difficulties. I , Ifound ) experimentally to be 4, But if, for the same skeleton, S ~ ( Z ,were this fact, taken in connection with the previously established values of Sr(3,1)and S v ( z , z ) would not be compatible with any group of degree four. If Z G ( 3 , I ) is I , then & ( 2 , 1 , 1 ) cannot be 4 and vice versa. As already stated, throughout the range of chemistry no such data are known. It is always possible to find a group which is a t least compatible with the experimental facts. The Determination o j Isomer -Yunzbers and Xolecular Group-The association of at least one permutation group with every molecular skeleton in the manner just described is attended with very wide consequences. It gives in terms of a permutation group HG(P)the mathematical analogue of any set Sv(p) of isomers determined by the skeleton J’ and the partition (p). ) ,the same as Furthermore ZG(P),the number of sets of transitivity in H G ( ~ is S,(p), the number of isomers in Sv(p); and so any method of calculating ZG(P) becomes a method of calculating S v ( p ) . This means that formula (A) z0 As a matter of fact, no case where the six isomers required bv the partition ( I , I ! I , I ) have all been prepared has been found in the literature. For the case where the four different substituents are the hydrogen and bromine atoms and the methyl and carboxyl radicals, Beilstein lists five isomeric forms. The p bromo-isocrotonic acid has apparently never been prepared. But since the a and p forms of both chloro-crotonic and chloro-isocrotonic acids are known (vide Beilstein), the number six may be considered to be sufficiently well denonstrated.
I050
A . C. LCSS A S D J. K . SESIOR
(p. I O L , ~ ) is a formula for calculating the number of isomers in any set Sv(p) where the group of the skeleton is known. Conversely, there arises the possibility of reversing formula (A) and determining the group G from a complete set of pIT~(p)s. Combining these two processes, it might be possible to determine G from certain experimentally known values of S y ( p ) , and then to use this value of G to calculate the other values of S v ( p ) . The values thus calculated could be coinpared with experimental results and would thus serve as a check on the correctness of the determination of G . In exactly the same way H G ’ ( ~and ) Zc’(p) are associated respectively with S’yip) and S ’ d p ) : also H G ” ( ~and ) Z,”(p) are associated respectively with S”&) and S”Y(P). Hence formula (A) may be used in like fashion to calculate K’v(p) and S ” V ( P ) ;G’ and G” might also be determined from a sufficient number of values of IV’y(p) and ?;”~(p)respectively. In the succeeding sections of this paper, these various operations will be discussed in greater detail. Group and Con$yumfion.-But the most important consequence of the association of a permutation group with a molecule is the light which this association throws on the so-called configuration of the molecule in questionthat is on the space relat’ions of its component atoms. Every geometrical configuration consisting of d points which are subject to stipulated relatims of invariance determines a permutation group of degree d on these points. But it is not true that an assembly of d points and a permutation group of degree d on these points determines a space configuration for the assembly in question. In the first place, many permutation groups are compat,ible with a number of different space configurations. But what is more important’, geometrical configurations are only one of a variet’y of ways representing or illustrating permutation groups. I n fact, such a group has much the same relation to a space configuration as the mathematical law of inverse squares has to the physical laws of gravitation (Sewtonian form), the Coulomb law of electrostatic attractions and repulsions, the law of the attractions and repulsions of magnetic poles, etc. Suppose that a system be composed of two bodies, and suppose that an attractive force inversely proportional to the square of their distance from one another be known to exist between them. I n default of further information, no decision as to the gravitational, electrostatic or magnetic nature of the force is justified. S o r can any item in the behavior of the system, the mathematical analogue of which is inherent in the abstract law of inverse squares be used to determine the physical nature of the force in question. The mathematical analogue of S y ( p ) is ZC.(p)-a property not exclusively of the space configuration of the molecule, but of a permutation group on the univalent substituents. And all suitable space configurations as a class give only one out of various possible modes of illustrating such a group. For this reason, no inference as to the space configuration of the skeleton or of the molecule can be drawn simply from a value of S ~ i p or ) even from a complete set of S v ( p ) s . It is trur that a space configuration (rigid or non-rigid) is likely to be available for uic 3 s an illustration of the permutation group in
ISOUERISM A S D CONFIGCRATIOS
IOjI
question; it is true that a geometrical diagram is usually the most convenient method of plotting the data; but the fact that the diagram is geometrical does not imply that the relations plotted are space relations. To assume that the relations are space relations cannot be in conflict with the known values of S r ( p ) (which are properties of the group), but neither is the special assumption justified by these stated facts. And the continued adherence to a particular mode of representation, instead of clarifying the situation, is apt to result in confusion between the properties of the phenomena observed and the properties of the particular method of representation adopted. Various instances where the group of the molecule throws more light on its properties than does its so-called configuration will be pointed out in the present paper.
IV. Determination and Applications of G and Nv(p) Cnlcuiation of the Swnber of Cnimlent Substifution Isomers.--?;v(p) has been defined as the number of univalent substitution isomers determined by the skeleton 1-and the partition (p), taking account of structural isomerism and stereo-isomerism, both diameric and enantiomorphous. I t has already been shown (see preceding section) that, since every skeleton 1- is associated with a group G, and since each value of S d p ) is identical with the corresponding value of Zr(p), formula (A) (p. 104j) may be used to calculate the value of S ~ i p wherever ) G is known. The formula as given is in its general form; but for certain special cases, it may be considerably simplified. If every operation of a permutation group (except the identity) involves every letter of the group, then the “class” of the group is equal to its degree.*‘ If a group of this kind is transitive, it must be regular; if it is intransitive and has n sets of transitixlty, it must exhibit a I : I , , . . . . I : I isomorphism between n regular groups on different sets of letters. For this reason, permutation groups of which the class is equal to the degree will hereafter be referred to as “niultiregular” groups. In multiregular groups, every type (t) of operation consists of identical integers and e is equal to d. Consequent’ly,for such groups, every value of I< must either contain negative integers or else it must be of the form o , o . . . .o,o. Under these conditions K(p,t) (for all values of (p) and (t)) is equal t o the number of values of R which contain no negative integer. For this limited kind of groups, formula (A) reduces to the simplified form (B).
where n, is the number of operations of order c in G , and c is any common factor of g and d l , d ? .. . .di. Kauffmann’s formulae for the numbers of subtitution derivatives of quinoline, quinosaline and naphthalene are special instances of this restricted formula. (See S o t e D j . 21 For this technical use of t h e word clilss. see 1Iiller. Blichfeldt. and Dickson:“Finite Groups,” p. 47, (1916).
A. C. LUSN AND J. K. S E S I O R
I052
Another special form of (AI applies only where (p) is form (C) is zc
(I,I,. , . . , I , I ) =
(I,I.
d! g
. . . I , I ) . This
(C)
The significance of formula (C) will be brought out later on.
T h e Determination of the Group G.-The problem of determining G from certain values of ?;~(p)is usually of more int’erest than the converse problem just considered. This question may be resolved into a sequence of three: (I)
(2)
(3)
Does a complete set of S r i p ) s uniquely determine a permutation group G? If so, is the complete set of SV(p)s necessary for this unique determination?
If the complete set of iY~(p)sis not necessary for this purpose, how many and Tvhich values of ? ; ~ ( p )are sufficient?
The importance of the first question is self-evident. The reasons for the importance of the other two are as follows. For most skeletons, the value of iXv(p) becomes fairly great as (p) becomes more and more complex That is to say, in most of t’he chemically interesting cases, d!jg is large. Where G is (abcdef)I* for example, Z ( I , I , I , I , I , I ) is 6 0 ; and to check this value experimentally would require the preparation and investigation of 60 different compounds. It will probably be readily admitted that such a program is usually out of the question. Consequently, it becomes important to know just how many and which values of Xv(p) are sufficient to determine G uniquely. I n order to be of practical interest, the sufficient values must lie among the less complex values of (p) where the values of S v ( p ) are small enough to permit of experimental verification.
T h e Significance o j Literal Conjormality.-The structure of formula (A) serves to indicate where part of the answer to the first question in the preceding paragraph is to be sought. From this formula, it is clear that if two or more permutation groups consist of equal numbers of operations of like type, they must give rise to identical sets of Z,(p)s. And one part of the problem is therefore that of finding such sets of groups, if any exist. Permutation groups which satisfy the stated restriction must be isomorphous respectively with abstract groups which are either identical or conformal. But furthermore, they must exhibit an additional close relation as permutation groups. Apparently no name has ever been applied by mathematicians to sets of groups which exhibit this relationship. It is suggested that the members of such a set be called “literally conformal.” The mathematical questions connected with literal conformality need not be discussed in detail in this paper. It is hoped to treat them more adequately elsewhere. The proof that sets of literally conformal groups exist is
ISOMERISM A S D CONFIGURATION
I053
easy, but a search of the tables shows that there are not many of degree less than 9. Probably the simplest set of this kind is the following pair.
I = (abcd)de)(j) consisting of the operations
1
I1 = { (ab)(4(ej ) POS consisting of the operations
I
1
ab.cd.e.j
a b.cd .e.f ab.ej.c.d cd.ej.a.b
ac.bd.e.j ad.bc.e.j
Both of these groups of degree 6 consist of the identity and three operations of type ( 2 , ~ ) . It is therefore clear that both must have the same set of Zc(p)s. Consequently, if a molecule were associated with either one of them, it would be impossible to decide from the set of Nv(p)s alone which one was the group of the molecule in question. The existence of literally conformal groups certainly shows that a complete set of ZG(p)s is not always a means of uniquely defining a permutation group, and so the method loses some of its mathematical interest; but for chemistry it is nevertheless of great importance. The reason for this is not far to seek. The statement that with every molecular skeleton there is associated a permutation group is not likely t o be a t fault, but its converse is to say the least dubious. Indeed the very nature of chemical synthetic methods is such that it is hard to see how they could be used to prepare molecules to correspond to certain classes of permutation groups. The dicyclic groups, for example, seem to be chemically quite out of reach. It may perhaps turn out that the sets of literally conformal groups lie for the most part in that class of groups which is chemically inaccessible, and if this is so, then the method of determining G by its set of Z ~ ( p ) swill be of much greater use chemically than it is mathematically. The existence of such a possibility calls attention to the urgent need for an abstract definition of the kind of groups which is chemically accessible. This subject is, however, beyond the scope of the present paper. I t has been shown that literal conformality is a sufficient condition for ambiguity in the determination of G from its set of ZG(p)s. Whether it is a necessary condition or not is still unknown, since no cases of ambiguity which are not also literally conformal have so far been found. However, no member of any set of literally conformal groups yet discovered appears to be associated with a known molecular skeleton, so that, for the present, the whole question of literal conformality seems rather academic from the chemical point of view. For every molecular skeleton so far investigated, a complete set of Kv(p)s uniquely determines G.
The Szcficient l’alues of Sr(p).-The second and third questions raised on p. 1 0 j 2 are more easily.disposed of. Mere examination of Table I1 reveals immediately that in many cases it is not necessary t o have the complete s order to determine G. But for the number of values of Nv(p) set of S ~ ( p )in sufficient in each particular case, no rule has yet been found. In the table
1054
A. C. L U S S A S D J. K . SENIOR
for degree j, the last value of S v ( p ) (counting from the top) necessary for the unique determination of G is underlined. The sufficient number varies from one to six. T h e Group G of Benzene.-The method of determining G from part or all of a set of S ~ ( p )and s t’hevalue of the results obtained by this procedure may be illust’ratedby a few examples. Suppose that the skeleton in question is the six carbon atom ring of benzene. Then d is 6 and (I)
S
(j,I)
=
(2)
5
(3)
s
(4,2) (4,I,I)
= 3 = 3
I
The degree of G is determined by d- in the present case 6. Since there are only 56 groups of degree 6, the choice of G is t’hus restricted. Equation ( I ) shows that G is a transitive group, and since only 16 of the groups of degree 6 are transit,ive, the choice of G is still further cut down. Of these 16 transitive groups, only 2 , (abcdef)cyc and (abcdef)12, agree with equation ( z ) , and of these two only (abcdef),:! is in agreement with equation (3). The group G of benzene is thus uniquely defined by these three equations, and further values of ;?;v(p) may therefore be calculated from this value of G. The predicted values are 5 (3 3) = 3
S
(3,2,1)
?: (3,1,1,1)
x
(2,2,2)
S
(2,2,1,1)
K
(z,I,I,I,I~
=
6
= IO = I1 = 16 = 30
r\’ ( I > I , I , I , I , I )= 60
The experimental data are in agreement with the first four of these equations and compatible with the last three.?? The configuration of the benzene molecule has been a matter of discussion for the last fifty years and the question is still far from settled. But with only a small amount of isomerism data which has been almost unquestioned throughout that period, the group G of benzene can be quickly fixed beyond a shadow of doubt. And from this group may be derived much of the information which it was hoped t o obtain from the configuration so long sought. Moreover the group may be used as a basis for certain conclusions which the configuration would tend to obscure. These statements will be substantiated in the course of the present paper. T h e Group G oj” Ethane.-To give another example of the results which may be obtained by using the group G associated with a molecular skeleton it is instructive to consider the skeleton consisting of two carbon atoms united by a single bond. Here d is 6 and ethane is an example of the case where ?* I n Richter’s “Lexikon der Kohlenstoffverbindungen” (3rd Edition) there are listed ten isomeric forms of nitro-amino-toluene and eleven isomeric forms of dinitro-xylene. SO case has been found where the full number of sixteen forms required by the partition (z,z,I,I) have all been prepared.
ISO?,IERIS>I A S D C O S F I G U R h T I O S
I0
. ( P ~=) ZG:(P~) it is improbable that the adjacency relations between HGjip.) and HGI(pb) ~ ~ )HG2(pb). And so if the genetic will be the same as those between H G * ( and relations are worked out for Srip,) and Svfpb), they may suffice to eliminate either G1 or G2. If the number of groups in agreement with all known values of Kv(p) is large, such genetic relations may not be sufficient t o identify G uniquely, but the number of possible groups is often considerably reduced. As was mentioned in section IT, certain sets of literally conformal groups have been found, the members of which cannot be distinguished by complete sets of ZG(p)s. It is interesting to note that in all cases so far investigated, each group of this kind has been found t o be identifiable by the adjacency relations of the HG(p)s. That is, so far as known, any permutation group may be uniquely identified, either by a sufficient number of values of Z,(p) alone, or else by these values considered in connection with the adjacency relations between the HG(P)s. It seems probable, therefore, that it is possible with sufficient type data to define uniquely the group associated with any molecular skeleton.** Taking the genetic and enantiomorphous relations together, it is frequently possible (as in the case cited in full) to establish the correspondence , between each isomer in &(p) and each set of transitivity in H G ( ~ )except that only the pair of sets corresponding t o a pair of enantiomorphs may be determined. It is not always possible to work out so complete a system of correspondence, but in every case yet investigated a large number may easily be found. Cis and T r a m Derwatioes of Ethylene.--It is instructive to examine a case where reasoning of this sort is inadequate, in order to illustrate the fate of problems in which conclusions drawn from type properties are insufficient for a unique solution. If the skeleton V consists of two carbon atoms united by a double bond, then 28 It bv no means follows from this statement that G uniquely determines V. For example, in"the dioxan and dithian molecules, if V is so chosen that d is 8, then G D ~ ~ is W identical with GDithian. And other instances where widely differing values of V determine the same G might be cited. The relations of the different skeletons n-hich give rise to the same G form a n intricate problem in connexity.
1063
ISOMERISM A S D CONFIGURATIOS
S
(3,1)
s(2,2) s (Z,I,I)
= I = 3 = 3
S’ (3,1)
=
N’ ( 2 , 2 )
= 3 = 3
” (2,1,1)
h” (3,1) N” ( 2 , 2 )
I
K!’
(2,I,I)
= I = 2 = 2
S (I,I,I,I) = 6 K’ ( I , I , I , I ) = 6 N” ( I , I , I , I ) = 3 These data determine both G and G’ as (abcd)d and G” as (abcd)*. Since there are no two enantiomorphous skeletons which correspond to the above definition of T:, enantiomorphism is lacking among the substitution derivatives of ethylene. The problem to be solved is the correlation of each one of the three isomers in Sy(z,z) with one of the three sets of transitivity in HG(z,z).The group G is generated by the operations ub.cd and uc.bd; the operation ac extends this group to (abcd)s which is G“. Let ab = h bc = D ac = B bd = E ad = C cd = F Then H ~ ( 2 , 2is) generated by the operations BE.CD and hF.CD; it is therefore the group [fAFj~BE)(CD)]pos.This group contains three sets of transitivity-which agrees with XV(Z,Z)= 3. H ~ ” ( 2 , z )is generated by BE.CD, h F . C D , and hD.CF;it is therefore the group [ (.1FCD)8pos(BE)]dim. This group contains only two sets of transitivity-in agreement wit,h S ” ~ - - ( z , z=) 2 . These results indicate that one of the three isomers in Sv(z,2) is related by structural isomerism xith the other two which are stereo-isomers. And since these two are not enantiomorphs, they must be diamers. (BE) is the set of transitivity in H ~ ( 2 , 2 )which corresponds to the structurally related isomer in 3 v ( 2 , 2 ) ; (hF) and (CD) correspond to the diameric pair. The next step is to decide which member of this pair corresponds to (-IF) and which to (CD). I t has been shown that the rclations of G and G” are insufficient to establish the correspondence. Since G = G‘, no help is t o be obtained from enantiomorphism. Hence it is necessary to have recourse to the genetic relations. A complete diagram of the genetic and isomeric relations between substitution derivatives of ethylene for all values of (p) is given below.
(PI
= (4)
K(p) =
I
”’(p)
=
I
h-(p) =
I
S”(p) =
I
Sip) = 3
S”fp) =
2
Sfp)= 3
K”(p) =
2
S f p ) = 6 S”(p) = 3
h
1064
A . C . LUNN AND
J. K . SENIOR
The presence of a horizontal double arrow indicates a diameric relation between two isomers; the absence of such an arrow indicates that the relation is one of structural isomerism. The two diamers under discussion are I and 11, and the table shows at a glance that there is in it no genetic relation which serves to distinguish one of these isomers from the other. Since the relations between G,G’, and G” also fail to differentiate these two substances, only one conclusion is possible. Such a pair of isomers cannot be distinguished by any type property now known. And the same is true of the diameric pair 111, IV. This conclusion is entirely in agreement with experimental practice. Khenever diameric pairs of disubstitution derivatives of ethylene have been investigated, it has been necessary to fall back on the specific properties of the molecules in question in order to decide which one is the cis and which one t,he trans isomer. The classic example of this sort of a pair of diamers is the case of maleic and fumaric acids. Here the decision is based on the relative tendency of the two dibasic acids to yield cyclic anhydrides. Of course this anhydride formation is a specific property of the two carboxyl groups. X o decision on similar grounds is possible in the case of stilbene and iso-stilbene where no univalent substituents with ring-forming properties are present. As to the value and permanence of conclusions based on such specific properties, it is sufficient to review the history of the maleic-fumaric acid question. This pair of acids was first thoroughly discussed in 1889 by Wislicenus,29 who decided that maleic acid was the cis and fumaric acid the trans form. From that time on the discussion has continued. I n 1926, Kuhn published a paper gravely questioning the correctness of Wislicenus’s reasoning. His work called forth at least two controversial answers, to which he replied with new arguments supporting his previous p u b l i ~ a t i o n . ~ A~reasonable prediction is that argument on this subject will go on until the discovery of a type property which differentiates the two substances in question. It will then cease once and for all, just as argument about the ortho, meta and para positions ceased when Korner published his results in 1874. VII. Structurally Isomeric Univalent Substitution Isomers The Calculation of N”I.(p).-The number of structurally isomeric sets of univalent substitution isomers determined by a given skeleton V and a given value of (p) has been defined as ?r”’v(p). And just as a complete set of S v ( p ) s or X’v(p)s usually determines and is always compatible with a permutation group G or G’, so in like manner a complete set of X ” ~ ( p )usually s determines and is always compatible with a permutation group G”. Hence formulae ( A ) , (B) and (C) can be used to calculate K”v(p) if G” is known. These formulae can also be reversed (as in the preceding sections) and used to determine G” from a sufficient number of values of N”v(p). N”v(p) again has its analogue in the properties of the group H G # , ( ~and ) , is similarly independent of the space configuration of the molecule. 29 30
n’islicenus: Abhand. math. physik. Iilasse sachs. hkad. Wiss., 14, I (1889). I h h n and Ebel: Ber., 58, 919, 2088: Meisenheimer: 1491;Boeseken: 1470 (1925).
ISOMERISM AND CONFIGURATION
1065
T h e Relations of G and G”.-The relations of G and G’ to G” (when V is constant) offer some fascinating and puzzling- problems, most of which lie beyond the scope of the present paper. It is hoped to return to them in a later article. But one particular instance of such relations will be considered in some detail because it shows how the number of isomers in &(pj which go to make up each one of the subsets in S”v(pj may be determined. Where V is the three carbon atom ring of cyclopropane, S ~ ( 4 , j2is 4 and N ” v ( 4 , z j is 2 . One of the subsets in S”v(4,z) contains three of the isomers in S V ( 4 , z ) ; the other contains but one. For this skeleton ”’( 5 > 1) K“(4,2) K”(4,1>1)
=
5“(3,3) N”(3,2,1)
= 3
2
-
These data3I are in agreement with the idea that G ” c is (abcdefjra, ~ ~ ~ a group of which ( ~ b c d e fis) ~a subgroup.3* As stated on p. 1059, ( ~ b c d e f ) ~ The (which is generated by abc.def and ad.bf.ce) is the group GCselopropnne. operation ad extends (abcdefj6 ( = G ) to ( ~ b c d e f )( =~ G ~ ” ) . On p. 1059 it was also shown that for the value of V here considered, H ~ ( 4 , 2is) of the type [(CHL.DIJ.EGKjall(AFBMOS)sl,:,. Where (p) is ( 4 , 2 ) , the extending operation ad becomes AG.BJ.DM.EN, and H ~ ” ( 4 , z )is of the type [(CHL) all(C1JEGKAFBMOK)48],:8.In other words, the effect of extending H ~ ( 4 , z to ) H ~ ” ( 4 , zis ) to make the three sets of transitivity (DIJ), (EGK) and (AFBMON) coalesce into the single set (DIJEGKAFBMON). These three sets therefore represent the three isomers which are distinct as long as stereo-isomerism is taken into account, but which become identical when this form of isomerism is no longer considered. The fourth set (CHL) represents the isomer which retains its individuality even when stereoisomerism is disregarded and which is thus proved to be structurally isomeric with the other three. Since ( D I J ) and (EGK) have already been shown, p. 1059, t o represent a pair of enantiomorphs, the relations developed are sufficient to permit the correlation of each one of the sets of transitivity in H ~ ( 4 , jz with one of the isomers in S ~ ( 4 , 2 )except , that there is no means of deciding which one of the enantiomorphous pair is represented by (DI Jj. Reasoning of the sort here employed is independent of the nature of 1and (pj, and consequently whenever G and G” are known, such reasoning may be used t o determine the number of isomers in &(p) which go to make 31For S“ (4,1,1j = z see Marburg: Ann., 294, 131 (1897); also Kohn a n 3 Mendelewitsch: Monatscheft , 4 2 , 241 (1921). For S “ (3,3) = 2 see Conrad and Gnthzeit: Ber., 17, 1186 (1884); also Perkin and Ing: J. Chem. SOC. 125 1816 (1924). For X ” (3,;,1) 3 see Goss, Ingold and Thorpe: J. Chem. Soc., 123, 3353 (1923); also hlarburg: Ann., 294, 1 1 2 (1897);also Staudinger et al.: Helv. Chim. Acta, 7,401 f1924). 3* The other values of G ” which are in agreement with these data are excluded on grounds which will not here be discussed.
~
~
~
1066
A. C. LUNN A S D J. K. SESIOR
up each one of the subsets in S”y(p). Hence these numbers are again properties of the groups G and G”, and therefore independent of the space configuration. VIII. Conclusions The properties emphasized in t’his paper arc the following: (I) The number of univalent substitution isomers in the set Sy.(p) determined by a given skeleton V and a given value of ( p ) . The number of enantiomorphous pairs in Sv(p). (2) (3) The number of structurally isomeric subsets in Sy!pj. (4) The number of members of Sv(p) in each of the structurally isomeric subsets of Sv(p). ( 5 ) The genetic relations between adjacent S ~ ( p ) s . All these five properties have mathematical analogues inherent in the groups G, G’ and G” and therefore need no consideration of space configuration. Since these complete the known list of type properties which have to do with univalent substitution isomerism, it may be stated that the known type properties related to such isomers cannot justifiably be used to determine the space configuration of the molecule in question. In view of the fact that the type properties mentioned are all expressed by pure numbers, this conclusion should occasion no particular surprise. The great majority of space configurations in the chemical literature are, however, based principally on just these five properties. I n fact, were the conclusions founded on such data eliminated, knowledge of molecular space configurations would be reduced to a small fraction of what is now supposed to be known. It would be necessary t o rely entirely on specific properties for information in this field. There are, it is true, certain theories based on the specific properties of molecules which bear on the matter of space configuration. Sucharethe Baeyer strain theory, the theory of steric hindrance, the dissociation theory of Bjerrum, the theories of dipole moments, band spectra, etc. These theories have been relied on t o confirm (where they do not contradict) the evidence based on type properties. Whether it is possible to develop from such ideas alone a consistent system of molecular space configurations is a matter of individual opinion, since the question is certainly open to doubt. If the term “configuration” (meaning space configuration) were to vanish from a large part of chemical theory, and the term “permutation group” were to take its place, such a change might well be regarded as a step forward. In the first place, it is possible to deduce from the permutation group most of what has customarily been deduced from the configuration. Secondly, whatever cannot be deduced from the permutation group cannot be deduced from the type properties of the molecule in question, or at least not from those type properties which have t o do with univalent substitution isomerism. And since experience has shown that type properties hare furnished a foundation for reasoning very much surer than that furnished by specific properties, it would be possible by use of the concept “permutation group” to distin-
ISOMERISJI A S D COSFIGURATIOS
1067
guish certain parts of chemical theory which are relatively certain from others which are relatively questionable. Thirdly, by fixing attention on t’he permutation groups associated with a given molecule, it is possible in some instances to elicit from the facts more information than is obtainable by exclusive consideration of the space configuration usually attributed to the molecule in question. AIoreorer, the use of permutation groups in place of space configurations ~ o u l dput an end to much of the confusion which has recently arisen in this field. For many years the determination of molecular space configurabions lay in the hands of organic and inorganic chemists. The results which they obtained formed a system almost completely self-consistent and of very wide scope. But recently physicists, using the X-ray spectrograms of crystals, and physical chemists, using dipole moments, band spectra, etc., have entered the field; and the results obtained by the new methods are often a t variance with those previously obtained under the guidance of stereo-chemical theories. A way of escape from this dilemma is clear. If two independent investigators determine the space configuration of the same molecule under like conditions and arrive at contradict,ory results! a t least one conclusion must be in error. Both may be. But if the same two investigators set out to determine the permutation group of the molecule in question and arrive at different results, it is entirely possible that both conclusions may be correct. -1molecule is associated not with one but with many permutation groups. In the present article, the three which are most commonly considered by organic chemists (that is G , G’ and G”)have been discussed. S o doubt there are many ot’hers. Khich of these groups emerges upon investigation depends on various considerations among which are the definition adopted for the term “pure substance” and the defining agreement as to the class of isomers to be enumerated. In fact, every permutation group so determined is to be regarded as an aspect of the molecule under discussion. Occasionally two fornially different aspects of one molecule may turn out to be the same (as in the case where G = G’), but such instances are the exception and not the rule. If t v o so-called configurations of one molecule arrived at by different methods turn out to be the same, the identity requires explanation. If they turn out to be different, this result is only what would usually be expected. The definitions of pure substance, and the defining agreements as to the classes of isomers to be enumerated used (sometimes without explicit statement) by physicists and physical chemists are not synonymous with those customarily used (also for the most part without explicit statement) by or33 Hence it, is not strange that the new results do not agree with those previously regarded as well established. If the claim is made that these new views add to the knowledge of the substances involved, it is difficult to see how this claim can be denied or why anyone should wish to deny it. But there is no ground for the conclusion that the recent results obtained by physical methods invalidate any part of the classic theory of stereo-chemistry. Keissenherg: Ber., 59,
1j26 (1926‘1; Henri:
Chern. Reviews., 4, 189 !192;).
1068
A . C. L E N S A S D J. K. SEKIOR
M. Notes A. In certain cases (and particularly among unsaturated aliphatic cc.npounds), where there are only two diameric stereoisomers, it is customary to refer to the relation between these compounds as one of “geometrical isomerism,” and to call the two substances respectively the “cis” and “trans’! forms. I n other cases, where there are three stereo-isomers of which two are enantiomorphs, the third is commonly called the “meso” form. Among ring compounds, the distinction between these two varieties of diamerism breaks down. The optically inactive, unresolvable hexahydro-phthalic acid may be called either the cis or the ineso form. And when more complex sets of stereo-isomers (such as the truxillic acids) are considered, the unqualified terms meso, cis and trans cease to have any unique significance. On account of the limited applicability of this terminology, it will not be used in the present article, which aims at a treatment as general as circumstances will permit. Any two stereo-isomers must be either diameric or enantiomorphous. Whether or not they happen to fall within the limited classes to which the above mentioned names apply is a matter which will here be disregarded.
B. There is one small but important class of exceptions to the rule that the number of enantiomorphous pairs in &(p) is a type property. Among certain restricted varieties of diphenyl derivatives, this number appears to depend on the specific nature of the univalent’ substituents. The most widely accepted theory on this subject is that of Mills.34 If his reasoning is followed, it is to be predicted that octabromo-diphenic acid should exist in enantiomorphous forms but that diphenic acid should not. If the skeleton chosen in both of these cases be the united carbon rings of the diphenyl molecule, then d is I O and (p) is (8,~).As the distribution of the 8 and z univalent substituents is also the same in each case, it is obvious that the difference in the number of enantiomorphous pairs must be due to specific differences in the univalent substituents t’hemselvea. Since the revolutionary nature of Mills’ theory has not been sufficiently recognized, it is hoped to treat these remarkable compounds in greater detail later on. For the present, it is enough to point out that in this small class of cases, the symbol KV(p) has no definite meaning. I n these instances, a ‘ and (p) is not sufficient to permit the calculation of the nuniknowledge of 1 ber of isomers. The additional item necessary is probably the number of hydrogen atoms among t,he univalent substituents. Consequently such compounds are explicitly ruled out from all that follows in regard to the value of Nv(p). The class of compounds in which the number of univalent substitution isomers ceases to be a type property has been extended beyond the limits 34 Mills: Chemistry and Industry, 4 5 , 884, 905 (1926). Cf. also lleisenheimer and Horing: Ber., 6 0 , 1425 (1927); 1Iascarelh: Atti. .Iccad. Lincei, ( 6 ) ,6, 60 (1927).
1069
ISOMERISM AND CONFIGURATION
of the diphenyl derivative^.^^ Probably the number of instances of this sort of “specific enantiomorphism” is destined to increase considerably. Kevertheless, for the present a t least, the whole class forms only a small exception to the general rule that the value of S v ( p ) is independent of the specific nature of the unix ,lent substituents. And it is further to be noted that even in these cases, N’v(p) and h”’v(p) are true type properties. C. From this point on, it is necessary to assume on the part of the reader an understanding of the elementary portion of the theory of groups, as well as a n acquaintance with the terminology used by workers in this field. All that is necessary in this line may be obtained from any standard work on group the0ry.3~ To use the formulae herein developed, it is necessary to have access to the lists of permutation groups of given degree. Easton in his bibliography gives a resume of the literature in this field.37 All the permutation groups of degree less than 12 have been determined, and all the transitive permutation groups of degree less than 16 likewise. The lists of groups are given in the notation devised by Cayley and explained by him in the first paper ~ i t e d . ~ The 8 articles noted contain all that is needful for a working knowledge of the subject. But, for the purposes of the present paper, the number of permutation groups of degree n ( n < 12) cannot be taken just as Easton gives it. .A group x where given as of degree n must also be considered as a group of degree n x is any positive integer. Hence the following table:
+
Degree
Kumber of groups (as given by Easton)
Number of groups (here considered)
Degree
7 8
I
I
I
2
I
2
3
2
4
9
4 5
7 8 37
I1
IO
19 56
I1
6
Kumber of groups (as given by Easton)
40 2 00
258 I039 I500
?;umber of groups (here considered)
96 296 554 I593 3093
A simple extension of the Cayley notation serves to identify the additional groups thus introduced. The symbol ( a b ~ d ) ~ (f) ( c ) indicates the group (ab&)*, usually thought of as of degree 4, but here considered as of degree 6 ; the symbol [(abcd)all(ef)]dim(g) indicates the group [ (abcd)all(ef)]dim, usually thought of as of degree 6, but here considered as of degree 7 ; etc., etc. I n this sense, the identity group on five letters (for example) is (a)(b)(c)(d)(e). Such symbols are used in Table 11. 35 36
Mills: J. Chem. SOC.,1928, 1291;Kuhn and -4lbrecht: Ann., 464, 91;465, 282 (1928),; See e.g. Miller, Blichfeldt and Dickson: “Theory and Apphcations of Finite Groups
(1916). 3’ 38
Easton: “The Constructive Development of Group Theory,” pp. 77, 78 (1902).
Cayley: Quart. Math., 25, 7 1 , 137;Cole: 26, 372; 27, 39; Miller: 27, 99; 28, 193;29, 32, 342; Am. J. Math. 21, 287; Proc. London Math. Soc., 28, 533; Bull. Am. Math. Soc., 1, 67; Kuhn: 6, 260.
224;
A . C. L U N S A S D J .
1070
Ei. S E S I O R
D. The formulae developed by Iiauffmann3Qfor the number of univalent substitution derivatives of quinoline, quinosaline and naphthalene are all special cases of formula (B). The groups are as follows: Sfolecule
Group
Quinoline ( a )ib) i c j ( d ) !e) !S)(8) Quinosaline (Ub.Cd.Cf) Saphthalene (abcd.ejglij All of these groups are niultiregulur. The group of naphthalene will be considered in more detail as an example. It consists of the four following operations: I
ah.cd.ef.gh ac.hd.eg .jh
ad.hc.eh.fg The value of d is 8 and that of g is 4. I n the equation
c has successively the values of all the comnion factors of g and dl, da. . , . di. And since, in the case under consideration, g is 4,c can have only the values I , 2 and 4. But nl = I n2
= 3
n4 = o and so the term where c is 4 always vanishes. There is always a term \vhere c is I and nl is I no matter what the values of dl,dp.. . .df, but the term xhere c is z and n is 3 appears only where the set dt,d2., . . d i is composed entirely of even integers. Hence the formula for the number of naphthalene derivatives becomes
a9 Kauffman: Ber.. 33, 2131 (1900,; ‘,Die Valenzlehre,” pp. 127-130, (1911); “Seues Handworterbuch der Chernie, 7; 564-j6j (19oj). See also Soelting; $Ion. sci., (4) 8, 178, (1894); and Key: Ber., 33: 1910 ~~1900:.
ISOMERISM A S D C O S F I G T R A T I O S
IOjI
it being understood that the second term is applied only where dl,d2.. . .di are all even integers. This expression is essentially identical with Kauffmann‘s formula. It is difficult to determine from Kauffmann‘s published work just how general a setting he had in mind for the formulae he used. Although he does not make explicit use of the group concept, he speaks of the “symmetry lines” of his diagrams and recognizes their importance in determining the constants of his formulae. But he does not say just how he arrives at these constants. There is, for example, no statement of how he got the constant j which appears in the second term of his formula for the number of naphthalene derivatives. The group analysis explains it by the fact that there are 3 operations of the second order in the group (abcd.efgh!4. KauffmannlO does make the definite statement that his formulae applv only to molecules where no hydrogen atom lies on any sg-nimetry line of the diagram-which in his cases is the necessary and sufficient condition that the group of the skeleton be multiregular. He refers4‘ to the case of benzene (where the group is not multiregular) as a more difficult, problem, and s t a t e that in conjunction with Hell he arrived at a formula for the numbers of substitution derivatives of this substance also. But according to a privatc communication, this result has not yet been published. In developing his formulae Kauffniann makes use of a function Fin) which is such that F(n) = n! where n has any real, positive, integral value. (I) F(n) = = where n has any other value. (2) To this function he attaches great importance, and states (Berichte, p. 2134) that it will probably suffice for the solution of the problem of the number of isomers among paraffin derivatives. But in the opinion of the present authors, this function Fin) or any similar function is superfluoua. In cases like the formula for the number of naphthalene derivatives, its use may be obviated merely by defining c as a common factor of g and dl,d2.. . . di. Xmong paraffin derivatives, similar simple definitions serve to accomplish the same purpose. The key to the problems which Kauffmann attacked is the theory of permutation groups, and the considerable success which attended his efforts is due to the fact that he used in effect certain principles of group theory, although apparently unconsciously. The F function which he introduced tends to obscure the essentially Diophantine nature of the problem; hence it seems better to avoid it. If, however, it is desired to use such a function, the conditions (I) and ( 2 ) cited above serve as a sufficient definition. To define the function by the use of expressions containing continued products is an unnecessary complication. It may be possible to express the required F(n) in this way, but the development given by Kauffmann (Handworterbuch, Hell and Haussermann: loc. cit.) is unintelligible because of the divergence of the infinite products used. 4o
4
“Die Valenzlehre,” p. 129 (1911). Ber., 33, 2134 (1900).
A. C . LUNN AND J. K . SENIOR
1072
X. Tables Table I -Values of K. The omitted values of K in the upper right hand half of each section of this table are all equal to zero. In each column of values for (t), the vacuous set of integers which represent the identity is indicated by a dash. Table I1 --Values of Z,(p). In the section for degree five the underlined number in each column is the last integer (counting from the top) necessary for the unique determination of G.
TABLE I - VALUESOF K 5
I
4
I
I
3,2
I
0
g
4 13 2 3 1 I - 2I 1 1I
1I
I
I
I
d =3 I
I
t
1
0
I 2
2
I
4
6 12 24
4
3
2
1
5
2 2
d=4
2
2
1 I
1
1 1 I
d=5
I
ISOMERISM AND CONFIGURATION
w
I H
w J
m
e
I073
a ---d--4 w 2
;?loop
8
NNN-
m----
I074
A.
C. L U S N A S D J . K. SESIOR
TABLE
I-
VALUES OF
K
I
I
I
1 0 1
0 I O O 2 2 2 4 6 0 1 2 1 1 2 3 4 5 1
I O I
I
~
0 0 0 2 12 0 2 0 6
0 0 0'2 I O 0 2 I 0 0 0 2 0 2 2 2 0 2 2 2 2 4 I2 2 4 2 4 6 I2 24 2 0 0 0 0 0 0 0 2 2 0 0 4 4 0 0 0 2 4 3 2 0 2 1 2 0 0 4 0 2 5 6 6 6 9 6 6 0 8 l 2 6 6
I I 1 1 1 1 1 0 ~ 0 l l I 0 4 0 0 0 0 I 2 4 2 6 6 0 I 4 8 IZ I2 ZOU
~ ~ ~ 2 5 W 6 0 ~ ~ 4 0 S 6 0 Q Q
6 0 I2 0 0 0 0 0 0 0 24 6 6 U 6 0 I t 0 I? 0 0 2412
U 2 8 f L Y 2 4 4 0 4 8 5 6 4 8 0 'It7248
d=8
ISOMERISJI AND COSFIGURATIOX
I
19
1
I8
1
7. 2
1 0 1
7
1 2 1 2 I 0 0 011 1 1 1 0 1 I 3 3 6 , l 1 0 0 0 0 I I O O i l I 0 2 0.0
6. 3 6. 2
6 5, 4
' 5. 3 5.2.2
4,
3
4.2,z
4 2 A
3,3, 3
3.9, 2 3. 3 3.2.2.2
1
3 6 0 x 1 0 O l l
I
0 011 0 2
I 2 1 2 ' 1 3 0 3 2 0 0 0 4 l I I 2 0 2 2 012 0 2 0 0 2 2 I S A 6 1 6 6 4 6 6 6 0 6 4 I s oiol03Y 6 m o ~ l z o 1 oO I 0 0 0'3 0 010 0 0 12 0 i O 0 I I I 013 I 0 2 2 0 0 O ! O 2 I 3 3 6l3 9 6 6 6 0 0 0 t? 6 I 0 3 0 ; I 0 013 0 6 0 0 0 3
2
02 6 6 6 Ma6ol 0 0 0 0 0 0 I2 0 0
d=9
20 0 6
0 1 62 O i 6 6 12
0 0 0 Oi0 0 0 6
I075
1076
A . C. LUSN AND J. K. S E N O R
TABLE I - VALUES OF K I I 1 1 0 1
1 2 1 2 1 0 0 0 1
1 1 1 0 1
I
1 3 3 6 1 3 6
I
O 0 0 0 0 011 0 0 10011
I I
I l 0 2 0 0 0 0 l 0 t I 2 2 2 2 2 011 2 2 2
I I 1 0 2 1 0 2 1 0 0 0 2 1 3 3 6 2 3 6 4 3 0 0 0 6 I O 3 0 0 0 0 4 0 6 0 0 0 I 2 3 2 4 4 0 4 4 6 4 0 4 I 4 7 I2 8 16 24 8 16 18 24 24 8 I 6 15 30 20 60 120 Ib 60 80 ID 3W 112
1 1 0 0 0 0
9 3 6 0 0 0 0 0 0 0 0 0 4 4 0 4 0 0 lb I6 24 24 24 0 36 63 120is0 360 ?a
1 1 0 0 3 0 0 ) 3 3 0 0 0 / 0 0 0 0 0 0 0
0 8 8 8 7 7 l b G 6 I Z I 3 2 1 4 3 2 I
I
1
1
d =10
6 6 5 5 5 2 1 5 4 3 1 2 1 1 , t / I
5
555
3 2 2 l
Z
i
I l
1
1
1
1
ISOMERISM AND CONFIGURATION
TABLE I - VALUES OF K 4.
4, 2
I2
4. 4
4, 3, 3 4. 3, 2 4. 3 4 . e, 2 , 2 A, 2 2
4, 2 4 3. 3. 3 3
3
2 . 2
3
3
2
3
3
3 3
2 2.2 2. 2 3. 2
70 I10 70 I15 180 90 110 120 0 120 ID0 IbO 210 4) 120 llJ
3
2 1 0 4 2 0 2 1 i ) 5 1 5 ~ ~ ~ 6 ) 0 24$2I0~4 ~ 2 0~8~1 0 b ~6 3 0 1 , ~
2. 2, 2. 2. 2
&6,,
2 , 2. 2. 2 , z. 2 2. 2
2
z 72 72 96 12012 Ib8144 72 0
114192 I t 4 0 2 4 0 i i 1 0 0
170420IMb008(0 7109b0&720
720tr*&br:u1$&t:a
%&,b (&
&Qo:07&7k;m
$28
6%
2 :,P,&,I:o&t!2e33
A & LLiG P % ZE &E-:8:
0
JDOZBSl4i
&:b
:LC:,::%%
E 2: 2% 2: $0
4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 4 4 3 3 3 2 2 2 1 3 3 3 3 2 2 2 1 2 2 z z I
1 I
2 1 3 2 1 2 L 1 1 3 2 2 1 1 2 1 1 2 2 2 1 1 I I I / 2 1 I I 1 2 1 I 2 1 I 1 2 2 1 1 I I I I I I 1 1 ; 2 1 I I I I I I l l I I l l I I 1 I 1 I I 1 1 I 1 / I I 1 1 I I I 1
1 1
d-10 (cont’d)
TABLE
I
I
l
l
I
l
l
1
1
1
1
4 3 2 2 2 1 1 2 1 1 1 6 4 4 2 3 3 2 2 2 1 1
d =3
d =a
1078
A . C. L C S X B S D J . K . SENIOR
n-VALU E S
TABLE
OF
ZG(P)
d=5 The underllned figure ln each column is the last integer (counting from the top) necesssry for the unique determination of G.
6
5, I 4. 2 4, I,!
3%3 3,2,1 3, I, I, I 2.2.2 2 . 2 I, I 2, I, I, I I
I
I
I
l
l
J
6 5 4 3 4 2 I5 I I 9 9 ? 5 3 0 2 1 I6 I5 1410 2 0 1 4 I? 1 0 8 8 60 38 32 30 22 X , I20 72 60 €0 4 . 40 90 Y 48 48 39 30 I80 IO? 92 90 60 60 260 192 180 180 110 RO
l
l
4 3 3 3 3 8 6 5 7 6 14 9 9 I1 9 IO U 6 6 8 24 I8 I6 h) I6 42 U, 30 36 30 33 27 24 30 27 S 48 46 52 I 8 102 90 90 96 90
1
l
I
I
2 6 8 6 16
30 27 46
90
I
l
7 2 5 3 8 6 6 4 !4 12 30 24 24 18 46 36
/
4 3 ? 5 13 9 8 6
I
l
l
I
I
3 2 1 I 5 4 3 4 8 6 5 5 6 6 4 4 19 I4 13 12 IO IO 34 26 22 20 20 20 24 18 I8 I8 16 I8 42 34 37 32 30 30 90 72 R SA 60 60 M)
ISOhIERIShl A X D C O S F I G U R A T I O S
I I I I, I, I, I
90 90 90 90 90 90 90 SO 72 60 60 bo 60 45 40 40 00 36
TABLE II - VALu E s OF 2 G ( ~ )
6
l
5, I
I I 2 1 1 2 1 2 1 I 2 I I I 4 3 3 2 2 2 3 2 2 3 2 2 1 2 2 1 1 1
4,2 4.1.I 3.3
3.2. I 3, 1, 1, I 2.2,2 2.2. I, I 2. I. I. I, I I. I. I, I. I. I
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
3 2 2 1
7 4 4 2 4 3 3 2 8 5 5 3 1 3 a 7 9 6 6 6 l 4 9 9 8 21 6 I5 15 30 30 u) 30
2 3 4 5 6 9 15
3 3 5 5 1 1 18
4 2 2 4 2 3 1 2 3 1 1 1 4 2 2 3 2 2 2 2 2 1 I 1 6 3 3 5 3 3 2 3 3 1 I I 8 8 4 4 7 4 4 2 4 4 1 I I 7 4 5 6 5 3 3 4 3 2 1 1 1 l 0 6 6 8 6 4 4 5 4 2 I I 14 IO IO 1 1 9 6 6 7 5 3 I I 30 30 x ) 2020 I5 15 I2 12 IO 6 6 2 I
d =6 (CONTID)
