380
The Steric Courses of Chemical Reactions.
I
W. G. Klempererl
Contribution f r o m the Department of Chemistry, Texas A & M University, College Station, Texas 77843. Received June 23, 1972 Abstract : Procedures for classifying, enumerating, and representing topologically the steric courses of chemical reactions which interconvert symmetric molecules are presented. Reactions are defined in terms of permutation operations, and the symmetry of reactant and product configurations is then used to define classes of symmetryequivalent reactions. The resulting classification schemes are shown to be readily interpretable in terms of traditional stereochemical concepts. Topological representations of reaction pathways are discussed, and procedures for calculating various properties are provided. Finally, some examples are treated which demonstrate the utility of this approach.
T
he steric course of a chemical reaction is defined by the stereochemical relationship between reactants products. Once this relationship is established, mechanistic information is obtained in that certain mechanisms are ruled out. Traditionally, mechanistic studies of tetrahedral substitution reactions have relied heavily on the stereochemical relationship between reactants and products. Since there exist only 3!/3 = 2 permutational isomers of a “tetrahedral” C I Smolecule RIMY, the substitution reaction R3MX
+Y
--f
R3MY
+X
may proceed either with “inversion of configuration” or “retention of configuration.” No other distinct possibility exists, assuming that both reactant and product have the same symmetry. For more complex systems, the situation is far less clear. Consider for example the case of octahedral substitution. Although there exist 5!/4 = 30 permutational isomers of Clo “octahedral” molecule R5MY, one might expect 30 distinct steric courses for the reaction R,MX
+ Y +RiMY + X
assuming both reactant and product have the same symmetry. It will be shown below, however, that this number is too large. This paper addresses the problem of defining, classifying, enumerating, and representing all distinct changes of stereochemistry which may accompany the reactions of symmetric molecules. In the first section, the concept of configuration is generalized and reactions are defined in terms of stereochemical change. Then two simple exchange reactions are examined closely in order to clarify the physical significance of these definitions. In the next section, the relationship between reaction mechanism and steric course is discussed. Finally, three examples are treated. I. Definitions When symmetric molecules undergo chemical reactions, the steric course is not uniquely defined by the stereochemical relationship between reactants and products unless labeling techniques are employed. This is because permutational isomers of molecules are not physically distinguishable unless chemically identical ligands are labeled. Labeling may be achieved by (1) National Science Foundation Predoctoral Fellow. Address correspondence to the author at Department of Chemistry, M.I.T., Cambridge, Mass. 02139.
nuclear spin states, isotopic substitution, or chemical substitution such that the effect of the labels on the chemical properties of a molecule is negligible. We say that the ligands are chemically identical but distinguishable by labels. Accordingly, the point group of a molecule is defined by neglecting the labeling of chemically identical ligands and considering only the symmetry of the unlabeled molecule. Molecules which are identical when labeling is ignored are called chemically identical molecules even though the labeled molecules may in fact be distinguishable by physical techniques. Consider for example Figure la, which may represent molecule (CH3)2permutational isomerization of the NPF,. Mathematically, the chemically identical F ligands are distinguishable by the indexed labels B2, B3, B4,and B5. Physically, the steric course of the reaction has been studied by nmr spectroscopy using the spin states of the I9F nuclei as labels2 Figure Ib may represent the cis-trans isomerization of Ru(PR&H2. Here again, the steric course might be studied by nmr spectroscopy using the I I P and ‘H nuclear spin states as labels. Figure I b may also represent the cis-trans isomerization of R U ( C O ) ~ ( S ~ C I ~ The ) ~ . steric course of this reaction might be studied using isotopic labeling, i.e., 13C0 and ‘ T O as labeled ligands. The dynamic stereochemistry of isomerization reactions has been investigated in detail.5-8 Reactions are defined by permutation operations, and sets of symmetry equivalent reactions, reactions formally nondifferentiable in a chiral environment, are defined such that the steric course of equivalent reactions is identical. Thus the number of classes of symmetry equivalent reactions is the number of theoretically measurable changes in stereochemistry which might accompany a given isomerization. Different definitions of symmetry equivalent reactions are necessary since the nature of the environment and the nature of the experimental measuring techniques used determine whether or not certain changes in stereochemistry may be differentiated. Figure lc-f may represent reactions of Mn(C0)SX (2) G. M. Whitesides and H. L. Mitchell, J . Amer. Chem. Soc., 91, 5384 (1969). (3) J. P. Jesson in “Transition Metal Hydrides,” E. L. Muetterties, Ed., Marcel Dekker, New York, N. Y., 1971, p 187. (4) R. K. Pomeroy and W. A. G. Graham, J. Amer. Chem. SOC.,94, 274 i1972). (5) W. G. Klemperer, J. Chem. Phys., 56, 5478 (1972). (6) W. G. Klemperer, J. Amer. Chem. Soc., 94, 6940 (1972). (7) W. G. Klemperer, Znorg. Chem., 11, 2668 (1972). (8) W. G. Klemperer, J . Amer. Chem. Soc., 94,8360 (1972).
Journal of the American Chemical Society 1 95:2 1 January 24, 1973
381
+ CO,g Pt(NH3)&1+ + C1-,l0 PC13 + ClZ,l1and SF, + SF4,I2 respectively. The types of reactions shown in Figure 1 are closely related. Two successive exchange reactions shown in (c) may result in permutational isomerization. In (d), forward reaction followed by reverse reaction may also result in permutational isomerization. Similarly, association followed by dissociation as in (e), (f), or (g) may result in a net exchange reaction. If the dissociation reaction shown in (f) is followed by association according to (g), polytopal isomerization reactions result. The purpose of this section is to express the relationship between these reactions in mathematical terms. Then the techniques already used to treat isomerization reactions may be applied to more complicated sequences of reactions. We begin by generalizing the definitions of concepts used in the treatment of isomerization reactions. A. Configurations and Their Symmetry. A configuration is defined here as a set of labeled ligands which have been assigned to labeled sites called skeletal positions. Assume that a set of n (unidentate) ligands contains n1 ligands of a given chemical identity, nz ligands of another chemical identity, and n3 ligands of a third chemical identity. Thus, n = HI n2 n3. These ligands are assigned labels from the set L = {AI, A29 . . .,An,, Bn1+1, Bn1+2, . . > Bnl+m, Cnr+n+l, Cn1+n2+2, . . ., C,} such that the letter indicates the chemical identity of each ligand and the integral subscript provides a unique index for each ligand. The skeletal positions of a configuration are assigned labels from szW,. . ., snW} such that ligands of the set SW = {slW, types A, B, and C occupy skeletal positions labeled by {siW,szW, . . . , snlWJ, {snl+iW, sn1+Zw, . . ., s , , + , ~ ~ ) and , {s ~W , s ,~, + , ~ +~ .~. ,. , ~ snW1, ~respectively. ~ Here, the superscript W identifies the geometry of the configuration, while the integral subscript provides a unique index for each skeletal position. Chemically speaking, a configuration is a set of labeled, rigid molecules having a definite orientation in space. Mathematically, a configuration is conveniently described by a 2 X n matrix
5
52
I
1
63
04 PERMUTATIONAL
POLYTOPAL ISOMERIZATION
8’
82
EXCHANOE
+ +
(f.)tw
SUBSTITUTION
A2 ASSOCIATION- DISSOCIATION
45
A? ASSOCIATION- OISSOCIATION
f’
123 . . . n = (jkp.. .q)
where ligand indices are listed in the top row, and below each ligand index is written the index of the skeletal position which that ligand occupies. The superscript W identifies the geometry of the configuration, and the subscript i is an integral label for each matrix. The reference configuration is defined by
(: : :3” Examples of this nomenclature are shown in Figure 2. Although a configuration is defined in terms of oriented molecules, we shall be concerned here with molecules which are free to rotate and translate in space. Therefore, different configurations having the same geometry may be equivalent in that they represent the same set of labeled molecules oriented differently in (9) T. L. Brown, Znorg. Chem., 7, 2673 (1968). and references therein. (10) F. Basolo and R. G. Pearson, “Mechanism of Inorganic Reactions,” Wiley, New York, N. Y . , 1967, Chapter 5. (11) R. Hoffmann, J. M. Howell, and E. L. Muetterties, J. Amer. Chem. SOC.,94, 3047 (1972), and references therein. (12) E. L. Muetterties and W. D. Phillips, ibid., 81, 1084 (1959).
lSOMERIZATlON
13 A5SOCIATION
A3
A7
:5
AI
A5
-
A3
47
A7 DISSOCIATION
Figure 1. Some representative types of chemical reactions interconverting symmetric molecules.
space. Such is the case in Figure 2 for the pairs of configurations shown in (b), (f), (j), (l), and (n). Equivalence between two configurations and (f),W may be defined by a permutation operation which represents translation and/or rotation of molecules in the configuration. By “translation” we mean permutation of chemically identical molecules. Since the indices of the skeletal positions serve as “ligand coordinates,” we let the permutation operations act on these indices. Thus the permutation represents a “coordinate transformation.” For example, the equivalence of configurations (t)eQ and (:)IQ shown in Figure 2a is expressed by the relation
(s)iW
(:)eQ
= 9PQ
(;y
where y ,QQ = (1)(24)(35)QQ,acting on the indices of the skeletal positions, represents a twofold rotation operaKlemperer / Steric Courses of Chemical Reactions
382
10,
,+
’.
Q :c
P
:e
6
*
same letter is used to identify this group and the geometry of the configurations which the group acts on. Since wWw represents rotations and/or translations of molecules, we now seek a more precise definition of wWw in terms of permutation group representations of the rotational point groups of the molecules in a configuration. If a configuration contains only one molecule, then the proper configurational symmetry group is a permutation group representation of that molecule’s rotational point group which acts on the indices of the skeletal positions. The rotational point group of the molecules in configurations shown in Figure 2f is Ca. It contains two operations: the identity operation u1 and a twofold rotation operation up. When u1 and up act on the indices of the skeletal positions shown in Figure 2e, the operations uluu = (1)(2)(3)(4>(5>(6)(7X8>uu and uzuu = (18)(27)(36)(4)(5)UU are generated. These two operations form the group Uuu, and we say U’u = CZuU.
If a configuration having geometry W contains m chemically nonidentical molecules, and R r is the rotational point group of the ith molecule, then Www is generated by letting all possible combinations of permutations in RIWW,RZWW,. . ., and RmWWoperate. Letting /AI denote the number of elements in an arbitrary set A, we have / W W W ~ = ~ R l W W ~ ~ ~ R Z W W ~ . . . RmWW1. Formally, the proper configurational symmetry group is defined by
V
,n;, I
2..
(,/
X
WWW
=
5 RiWW i= 1
%-’.
.,----I,
where the summation implies direct sums.13 For the configurations shown in Figure 21, R 1 = C, and Rz = D,. C3yycontains the operations (1)(2)(3)yy, (123)yy, and (132)yy, while DmYy contains the operations (4)(5)yy and (45)yy. Thus yYy = C I Y y D m Y ycontains [ c,YYI 1 D , Y Y / = six operations.
+
-
YIYY
= (1>(2>(3>(4)(5)yy
yzYy = (1)(2)(3)(45)yy ~
Figure 2. Six pairs of configurations having different geometries are shown in the two columns on the right. To the left of each pair is shown the indexing of skeletal positions which allows definition of the appropriate (i) matrices.
3
ydYy = (123)(45)” y 5 y y = (1 32>(4)(5>yy ~ 6 “
tion. When qtQQacts on the bottom row of (:),Q, the bottom row of (“,eQ is generated
12345 (12345)
(,,,,,)
= (1)(24)(35)QQ 12345
Q
Mathematically, the permutation operation (1)(24)(35) means “leave 1 fixed, replace 2 with 4, replace 4 with 2, replace 3 with 5, and replace 5 with 3.” Chemically speaking, the operation (1)(24)(35)QQmeans “the ligand in position slQis left fixed, the ligand in position s2Qis moved to position s4Q, the ligand in position s4Q is moved to position sZQ, the ligand in position s3Qis moved to position sgQ,and the ligand in position ssQis moved to position s3Q.” All the operations wiWW which represent rotation and/or translation of molecules in a configuration called the having geometry W form a group VW proper conjgurational symmetry group. Note that the
= ~ (1 ‘ 23)(4)(5)yy
= (132)(45)yy
Assume a configuration having geometry W contains m molecules and all m molecules are chemically identical, Therefore all these molecules have the same rotational point group R, and the group Www will contain 1 RWWImoperations representing all possible combinations of proper rotations of molecules in the configuration. In addition, however, there will be m ! operations in wWw which represent all possible permutations of the m chemically identical molecules. Since any combination of rotations can be combined with any one of the m ! permutations of entire molecules, the group WwW will contain a total of m!lRWWlm operations. Using standard notation, W”” = Sm[RWW],the composition of “S, around RWW,” l 3 where S, is the symmetric permutation group degree m (13) For a rigorous definition of this operation see: R. W. Robinson, J. Combinatorial Theory, 4, 184 (1968), or F. Harary, “Graph Theory,” Addison-Wesley, Reading, Mass., 1969, pp 163-164.
Journal of the American Chemical Society / 95:2 / January 24, 1973
383
which contains all m! possible permutations of the m chemically identical molecules. The configurations shown in Figure 2n contain two chemically identical molecules having rotational point groups C2. Here, m = 2 and R = C,. Thus Zzz = Sz[Czzzlcontains
The first four operations represent rotations of the molecules in the configuration, while the last four represent rotations combined with permutation of the two chemically identical molecules, i.e., rotations combined with translations. In the most general case, a configuration having geometry W will contain m molecules having different chemical identities and mi molecules of chemical identity i having rotational point group Ri. Thus Zi,lmmi equals the number of molecules in the configuration. Applying the definitions given in the preceding paragraphs, the proper configurational symmetry group is defined by m
and the number of operations in this group is m
1 WWIVI
=
a
1=1
R,”Tyni
(2)
Two configurations (:)iw and (:),” having the same ligand set and the same geometry W are defined to be equivalent if
(3) for some whww E Www. These configurations are nonequivalent if eq 3 does not hold. Another useful permutation group may be generated which represents the operations in the full point groups of molecules in a configuration. The full conjgurational symmetry group TTww is defined by (4) where 8,is the full point group of molecules in the configuration having chemical identity i. The number of operations in T T W ” is
one or more molecules in the configuration. If a i W W represents improper rotation (and possibly translation) of all molecules in the configuration, we say a i w w represents inversion of conjguration. Accordingly, two nonequivalent configurations (“,iw and (i),w are defined to be enantiomeric if eq 6 holds for some @kww E TPww which represents inversion of configuration.
If eq 6 does not hold, then nonequivalent configurations (:)(” and (i),w are diastereomeric. Two examples should make the physical significance of these definitions clear. For the configurations shown in Figure 2d, the operation tiTT = (18x27)(36)(45)TT represents inversion of configuration. Since = ZiTT(:)lT and (:)eT # tlTT(:)lT for any tjTT E FT, the configurations and are enantiomeric. For the configurations shown in Figure 2j, the operation %*XX = (1)(2)(34)(5)xx represents inversion of configuration, here, reflection operations acting on both molecules. l 4 However, the operation ( 1)(2)(34)(5)xx also represents rotation of Configuration. Thus the con= 3ixx(:)~x are equivalent configurations (i)lx and figurations. Whenever all molecules in a configuration are planar, operations which represent rotation of configuration also represent inversion of configuration. B. Reactions and Their Symmetry. A reaction is defined here as a transformation which converts one configuration into another configuration such that stereochemical change occurs. Mathematically, this transformation is defined by an operation hi“fv which transforms (”,,”, the reactant conjguration, into (f)kW, the product conjguration, by letting the permutation operation hi act on the indices of the skeletal positions listed ,I’ replacing the superin the bottom row of (:)and script V by W.I5 The reactant and product configurations may have the same geometry, i.e., the case V = W is allowed. The only restriction is that reactant and product configurations must have the same ligand set. The reactant and product configurations of the reaction shown in Figure l a both have the same geometry. Using the indexing of skeletal positions defined in Figure 2a, the reactant configuration is defined by =
12345 (12345)
Q
and the product configuration is defined by
/I\ (S)~
Q
=
/12345\Q (13452)
The reaction is defined by hiQQ = (1)(2345)QQ since (i)zQis generated when hi acts on the indices of the indices of the skeletal positions listed in the bottom row of (9eQ 12345 Q 12345 Q (1)(2345)QQ(12345) = (1 3452) Chemically speaking, (1)(2345)QQmeans “the ligand in
Clearly, Www is a subgroup of TfwW If wiww‘ E Www, we say wiwW represents rotation of configuration. If miww E ITww, then miww may represent improper rotation (and possibly translation) of
(14) It is assumed that ligands may be represented by points and therefore must have reflection symmetry, Le., be achiral. (15) In common usage, the word “reaction” may refer collectively to a set of “reactions,” i.e., permutations, as defined here. The intended meaning of the word should be clear from context.
Klemperer 1 Steric Courses of Chemical Reactions
384
position slQis left fixed, the ligand in position szQ is moved to position s3Q,the ligand in position s3Qis moved to position s4Q, the ligand in position s4Qis moved to position s5Q,and the ligand in position s5Q is moved to position szQ.” In Figure Id, the reactant configuration is defined by
and the product configuration by
(6)lx
=
12345 (12435)
if skeletal positions are labeled as in Figure 2g and i, respectively. The reaction is defined by hixV = ( 1)(2)(34)(5)xv since
We describe the reaction hlXV = (1)(2)(34)(5)xv by saying “the ligand in position bIv is moved to position six, the ligand in position szvis moved to position szx, the ligand in position saVis moved to position sqX, the ligand in position sqvis moved to position sax,and the ligand in position siv is moved to position sgx.” The reader may verify that the reactions shown in Figure If and g are defined by (12)(3)(456)(7)(8)uz and ( 12564)(3)(7)(8)TZ,respectively, if skeletal positions are labeled as shown in Figure 2. Given the set of n ligands labeled by L = {A1, Az,. Am, Bm+tl, B n l + ~ , . . > Bni+m, Cnl+n+l, Cnl+n+~, , , ., C,) as above, we define a permutation group H , the group of allowedpermutations, which acts on the set of numbers { 1, 2,. . . , n ] . Operations in H permute 1, nl the numbers in the sets { 1, 2,. . . , n l ] , { n l 2,. . ., nl n z ] , and { m n2 1, n2 2 , . . ., n ) among themselves in all possible ways. H therefore contains nl!nz!n3!operations. Formally, H is defined by ’ 3
+ +
+
HE=
(:)ew
IHM‘”l= nl!nz!n3! different configurations are generated, and this set of configurations contains all possible configurations having geometry W. There is therefore a one-to-one correspondence between configurations having geometry W and operations in the group PW. Two configurations hcWW(f)eWand hlWW(i)eware equivalent if
12345 = (12345)
(
Hww is a group with a product operation defined by htWW.hlWW = (hi.h5)Ww. HwWcontains wWw and TPWWas subgroups. If each operation in HwWacts on
+
+ +
for some whwW E WWw. A complete set of equivalent configurations therefore contains 1 wWWl configurations. can be partitioned into disjoint sets of The group HWW operations wWwhrWWsuch that each set corresponds to a complete set of equivalent configurations. The set of permutations Ww”hiww is called a right coset of wWW in HWw. If we select one element from each right coset of WWw in HwW,we form a set of coset representatives Cw. Each ci” e C” thus uniquely specifies a complete set of equivalent configurations, and the set C” represents a complete set of nonequivalent configurations having geometry W. Since a complete set of equivalent configurations contains 1 WU‘”~ operations, if we let Iw be the total number of nonequivalent configurations having geometry W, then Iw.l WWW/ = /H””j = n l ! n z ! n 3 ! . Accordingly, we define I , to be the conjiguration count and
+
s,, + s n , + s,,
where S,< is the symmetric group degree nr and the sums imply direct sums. The set of all possible reactions hiwv which transform configurations having geometry V into configurations having geometry W is defined by the permutation operations in H, the group of allowed permutations. This set is called Hwv,the superscript indicating the geometry of product and reactant configurations. Unless V = W, products of the type hiBrv.hjWvare undefined, since
using eq 2 to calculate I WwWI. Equation 7 may be rigorously derived noting that ZW = I Cwl and following well-known procedures outlined in ref 6 . Since reactions are defined in terms of oriented molecules but reactant and product molecules are assumed here to be free to rotate and translate in space, different reactions may in fact represent the same change in stereochemistry. For example, let
and hovW(~)pW=
(3 P
(9)
If and hlwV by definition may only act on configurations having geometry V. Thus Hwv is not a group. The inverse of h2vv, called the reverse reaction of hiWV,is defined by
(hi”’)-
1
and
E (h,-’)VTT’
The product of hiwy and h5yT5is defined by h,\YV, hj\-lY (h,. hj)TVll’ and indicates reaction hjv” followed by reaction hiWv. Journal of the American Chemical Society
95:2
then hivW and hoVWrepresent the same change in stereochemistry since the reactant configuratons as well as product configurations in eq 8 and 9 are equivalent.
January 24, I973
385 Mathematically, we substitute eq 10 and 11 into eq 8
This implies
and comparing with eq 9 heVW
=
(v 1 . h . w,)’~
In general, hiVWand hjvw are defined to be nondijferentiahle in a chiral environment if eq 12 holds for some vnVv e V‘ and wlww E VVm-. hi = oI;.h.jwl (12) The reactions hivJv and hjvW are differentiable in a chiral environment if eq 12 does not hold. In less precise chemical terms, reactions which are nondifferentiable in a chiral environment represent the same stereochemical change, i.e., have the same steric course. Just as the concept of equivalent configurations implies the concept of reactions differentiable in a chiral environment, the concept of enantiomeric configurations implies a corresponding relationship between reand (“Sjw as well as and (:)ov actions. Let (6)$” be enantiomeric configurations. Thus
where
f i J P w ~ ~ ’ represents
where
figvv
inversion of configuration, and
represents inversion of configuration.
If
and
then reactions formally nondifferentiable from hTVw and hsvW occur with equal probability in an achiral environment. Substituting eq 13 and 14 into eq 15 we obtain
h, = fi+.h,.@, (17) hQv“ are also differentiable in a chiral environment, they are defined to be enantiomeric reactions or simply enantiomeric. We then say that hDvw (or hQvW)is a chiral reaction. If hpvw is nondifferentiable from its “mirror images” in a chiral environment, then hpvW is an achiral reaction. If two reactions are differentiable in a chiral environment and are not enantiomeric, they are defined to be diastereomeric reactions or simply diastereomeric. One further type of symmetry equivalence between reactions is of interest. Two reactions hivw and h j v W are nondijferentiable in a totally symmetric environment if eq 18 holds for some gkvx’ E Pvv and some @,ww E TTww. If eq 18 does not hold, h T W and hlVw are h,
hsVW
=
(Tg- 1. h, . W,)vw
Accordingly, a “mirror image” hPv” of a reaction havw is defined by eq 17, where fi2Vv and @jww both represent inversion of configuration. If hpT’w and
fi,*hj.fiJl
(18)
direrentiable in a totally symmetric environment. Note both repthat eq 17 and 18 differ in that givv and resent inversion of configuration, while Okvv and m2Ww are arbitrary operations in Pvvand lrww, respectively. As was discussed above, the set Hvw contains all possible reactions which interconvert configurations having geometries V and W. Since we are concerned here with the steric course of reactions, we are interested primarily in different sets of reactions nondifferentiable in a chiral environment as defined by eq 12. The number of these disjoint sets contained in Hv” is called the number of reactions differentiable in a chiral environment. A complete set of reactions differentiable in a chiral environment is generated by selecting one reaction from each set of reactions nondifferentiable in a chiral environment. A complete set of reactions differentiable in a totally symmetric environment and a complete set of diastereomeric reactions may be similarly generated. It is often a great help when dealing with complex problems in dynamic stereochemistry to determine the number of diastereomeric reactions or reactions differentiable in a chiral or totally symmetric environment before actually identifying the reactions. Accordingly, formulas are provided here which enumerate these reactions. These formulas will not be discussed in detail at this point. Their application and significance will be explained later in the context of‘ specific examples. First, we define the generalized cyclic type (jl,j,, . . . , j,,; kl, kz,. . . , kn2;11, I,,. . . , Ins) of a permutation hi E H. Recall that elements in H permute numbers in the 1 , nl 2,. . ., nl n 2 ) , and sets { 1, 2,. . ,, nl], { n l ( n l nz 1, nl nz 2,. . ., nl n2 n3j among themselves. The array (j1, j,,. . ., jnl; kl, kz,. . ., kn*; 11, h,. . . , Ins) indicates that the permutation hi contains j , cycles of length p which permute the numbers { l , 2,. . . , n l } among themselves, k, cycles of length q which 1, nl 2,. . . , nl nz} permute the numbers (nl among themselves, and I, cycles of length r which pern2 1, nl n2 2,. . . , nl mute the numbers { nl n2 n 8 }among themselves. T o count the number of reactions hiJrw in the set Hvw differentiable in a chiral environment, eq 19 is used if V # W, i.e., the reactant and product configurations have different geometries.
+ +
+
Comparing this result with eq 16
=
+
+ +
+ + +
+
+ +
+ + +
+
+
+
(16) Equations 19-21 may be derived following procedures outlined in ref 5, 7, and 8. Equation 22 is derived in the Appendix.
Klemperer / Steric Courses of ChemicaI Reactions
386 Table I. Definitions Relating the Structural Stereochemistry of Configurations and the Dynamic Stereochemistry of Reactions Configurationsa
Reactionsb
,b
I. Equivalent
tc
I. “differentiable
in a Chiral
Environment p= 1
q=l
r=l
hpVW
Here, D v w is the number of reactions differentiable in a 11. Nonequivalent chiral environment, the summation extends over all generalized cyclic types of permutations in Yvv and wWw, and hVjlj2...jnl,k1k2... . k n 2 , z ~ z 2..zn8 . and hWj~~~...~nl,Xlkz...knz, ZIZZ. .. I,, are the numbers of operations in Yvv and Vw, A. Enantiomeric respectively, having cyclic type ( j l , j z , . . . , j n l ; k ~ , kz, . . . , k n z ; 11, 1 2 , . . . , LJ. If reactant and product configurations have the same B. Diasteromeric geometry W, the number of reactions differentiable in a chiral environment is D r W w D’ww = D w w - 1
(20)
where D w w is calculated using eq 19. T o count the number of reactions hivW e Hvwdifferentiable in a totally symmetric environment, eq 21 is D -vw - -
=
okVV.
h V W , whWW
11. Differentiable in a Chiral Environment h,Vw
#
L’kVV. h V W . whWW
A. Enantiomeric hPVW
=
l1 vw. &WW
omW.
B. Diastereomeric hpVW
# gmVY. h p V W . a n W W
0 (f);“ and (:)jW are two configurations having the same geometry W. * whWW and okVv are elements in the proper configurational symmetry groups WWw and V V V , respectively. a n W W and fimVV are operation in the full configurational symmetry groups LVWW and pvV, respectively, which represent inversion of configuration. c hPVW and hqvw are reactions in the set Hvw.
C. Summary. Table I summarizes the more important definitions provided in this section. The definitions relating the structural stereochemistry of configurations retain the physical significance of the words equivalent, nonequivalent, enantiomeric, and diastereomeric as conventionally used to describe the p=l q=l r=1 stereoisomeric relationships between molecules or used if reactant and product configurations have differEquivalent configurations are groups in molecules. ent geometries. In eq 21, Dpw is the number of reacphysically indistinguishable if molecules in the contions differentiable in a totally symmetric environment, figuration are free to rotate and translate in space, while and all other symbols were defined above for eq 19. nonequivalent configurations are in theory distinguishWhen reactant and product configurations have the able. Within a set of nonequivalent configurations, same geometry W, D i i ; ~is given by eq 21 if TT”‘” # however, certain pairs of configurations are physically www and D‘ w- w- -- D { if Ifww ~ ~= www indistinguishable in an achiral environment. These If reactant and product configurations have different configurations are called enantiomeric. Diastereogeometries, eq 22 is used to count the number of diameric configurations are nonequivalent configurations stereomeric reactions in the set Hvw. Here, D y l w t is which are in theory physically distinguishable in an achiral environment. Dylwl = The definitions relating the dynamic stereochemistry of reactions are based on the same physical criteria, as shall be made clear in the next section. Reactions nondifferentiable in a chiral environment are precisely those reactions which are physically indistinguishable if reactant and product molecules are free to rotate and translate in space. Reactions differentiable in a chiral environment are in theory distinguishable by physical methods and may be described as representing differthe number of diasteromeric reactions; 1 Yrvv) and I W’u‘wI are the number of operations in VVv and TfJvW, ent steric courses. Within a set of reactions differentiable in a chiral environment, certain pairs of reacrespectively, which represent inversion of configuration ; tions are physically indistinguishable in an achiral enthe summation extends over all generalized cyclic types r which represent rotation and/ vironment in that they must occur with equal probabilof operations in P and T ity. These reactions are called enantiomeric. Diaor inversion o f configuration; hv’,~,~z.,,~~l,~lkz.,.k~z,zl~z.,. z,3 stereomeric reactions are reactions differentiable in a andh”‘j,j,.,.j,,,hlii~. , . k n n , Z1zz. ,. are the numbersof operations chiral environment which are in theory physically disin and TT, respectively, having cyclic type (j1, j 2 , .. . , tinguishable in an achiral environment. j z , ; k., k 2 , .. . knz; 11, 12,. . . , ln3) which represent inversion of configuration; and all other symbols were de11. Discussion fined above for eq 19. When reactant and product conIn this section we shall examine the steric courses of figurations have the same geometry, D ’ w ~ w ,= Dwtwf some very simple exchange reactions in order to dem- 1 is the number of diastereomeric reactions. From the symmetry of eq 19, 21, and 22, it is clear (1 7) K. Mislow, “Introduction to Stereochemistry,” W. A. Benjamin, and DVtwt = D w t v ~ . that DwTv = Dvw, D e p = D,, New York, N. Y . , 1965. Journal of the American Chemical Society
1 95:2 1 January 24, 1973
387
onstrate the physical significance of the definitions given above. We consider first the exchange reactions interconverting configurations having the geometry shown in Figure 3a. Here, the ligands are labeled by L = (A1, A2, A3, B4, BS}and the skeletal positions are indexed as shown in Figure 3b. The proper configurational symRsWW,18and the full metry group is WW” = C3IvIv configurational symmetry group is ~ f w w= c ~ ~ R31WW. l8 These groups are representations of the r defined by the following permutation groups W and T operations.
+
w1 = ~2
=
~3
=
a1
=
+
4
w
W
~
(1)(2)(3)(4)(5)
= (123)(4)(5 )
a3
= (132)(4)(5)
a 4
= (12)(3)(4)(5)
a 5
= (13)(2)(4)(5)
0 6
=
43
*3
(1)(23)(4)(5)
Operations wlWw, wzwW, and w3ww represent rotation z ~ and~ mGWw ~ repre~ , of configuration, while a4ww, sent inversion of configuration. The group of allowed permutations H = S 3 SZcontains 3! .2! = 12 operations. Three of the 12 operations in Hwwrepresent rota43 A3 tion of configuration and the remaining nine operations represent exchange reactions and permutational isomerization reactions. The configuration count Iw equals 3! .2!/3 = 4 (see eq 7). A complete set of four nonequivalent configurations is shown in Figure IC. Note that a complete set of nonequivalent configurations is not uniquely defined, but represents an arbitrary choice of one configuration from each of the Iw 43 A3 sets of equivalent configurations. Figure 3. Configurations and reactions defining the stereochemisThe number of reactions differentiable in a chiral try of exchange reactions discussed in the text. environment, DIww, equals Dww - 1 (see eq 20). T o calculate Dww using eq 19, we need the generalized j2, cyclic types of operations in W w W : for wlWW, (jl, pathways but merely provide a convenient description j 3 ; kl, k2) = (3, 0: 0; 2, 0) and for w2ww and wsWw> of the stereochemical change involved. We see that ( j 1 , j 2 , j 8 ;kl, k2) = (0, 0, 1; 2 , O ) . Therefore h W 3 c c , 2 c = hlww implies “retention of configuration” while htWw 1 and hW001,20 = 2. Substituting in eq 19 implies “inversion of configuration.” For completeness’ sake, we note that both hlWWand Dww = {l/(3.3)]{l.l.3!.13.2!.1*. hzww are achiral since 2 * 2 * 1 ! ~ 3 ’ . 2 ! ~ 1=* 4) a 4 W W . h WW.@4WW = h W W 1 1 and therefore D t w w = 3. Since we are only interested a i M 7 W . hzWW,a 4 U ’ W= h z W W in exchange reactions, not permutational isomerization = 1, the number of perreactions, we calculate DtCnCa Thus hlww and hzww are diastereomeric reactions and mutational isomerization reactions differentiable in a form a complete set of diastereomeric exchange rechiral environment’ which interconvert the permutaactions. We verify this by calculating the number of tional isomers of the CSvmolecule in the configuration. diastereomeric reactions in HwW,D ’ w ~ w t= D w ~ w-~ Hwwtherefore contains only D l W w - DICsCa= 2 ex1, and calculating D’CatC3s = DCatCat 1, the number change reactions differentiable in a chiral environment. of diastereomeric permutational isomerization reactions Figure 3d shows a complete set of exchange reactions must equal two. in HwW. Then D’wtwt - DtCa,Cst differentiable in a chiral environment. Note that a From eq 22, complete set of reactions differentiable in a chiral enDwtwf = (1/(3.3 3.3))((1.1 O.0)*3!.l3.2!.l2+ vironment is not uniquely defined, but represents an arbitrary choice of one reaction from each set of reac( 2 . 2 0.0).1!*31.2!.1’ tions nondifferentiable in a chiral environment. The (0.0 3.3).1!-1’.1!.21*2!12) = 4 arrows drawn in Figure 3d connecting ligands in the Dc3~Cs~ = 2 reactant configurations do not represent mechanistic
+
+
-
+ +
(18) Rsi is the infinite point group consisting of all proper rotations, all reflections and inversion at a point. R S is the subgroup of Rat which contains only proper rotations.
+
+
+
and therefore
Dwtwr - D‘CstCa’= 3 Klemperer
-
1 = 2
1 Steric Courses of Chemical Reactions
388
interest here is the set of operations generated when the reaction hBUwis followed by the reaction h3WU. If the molecules in the configuration having geometry U are assumed to be free to rotate and translate in space, this set of operations is defined by h3wUUuUh3uW = (h3Uh3)WW = U-. The permutation group U,defined by the representation Uuu, contains these operations.
A13
FI
184
/
I.
u1
.
= (1)(2)(3)(4)(5)
u-2 = (123)(4)(5)
u3 = (132)(4)(5) u4 = (1)(2)(3)(45) ~5
= (123)(45)
246
= (132)(45)
~7
= (12)(3)(45)
4
I
I
U
ua = ( 13)(2)(45)
V
ug = (1)(23)(45)
Figure 4. Association-dissociation reactions implying exchange reactions shown in Figure 3.
u10
= (12)(3)(4)(5)
uii = (13)(2)(4)(5)
The number of reactions differentiable in a totally symmetric environment equals D + ~since lrww# Www. From eq 21 D,,
=
+
{1/(6~6)~~l~1~3!.13~2!~12
2.2.1!.31.2!.12
+ 3.3.1!.11.1!.21.2!.12} = 1/36{12
+ 24 + 3 6 ) = 2
The number of permutational isomerization reactions differentiable in a totally symmetric environment,’ DcBVc3.,is one. Hence, DGW - D,,,,,, = 1, and all exchange reactions are nondifferentiable in a totally symmetric environment. The reactions hlww and hzww are since z ~ ~ = hZWW. ~ ~ In general, . h reactions ~ ~ nondifferentiable in a totally symmetric environment are those which cannot be distinguished by physical techniques incapable of discerning enantiomeric molecules. Thus the steric course of two reactions nondifferentiable in a totally symmetric environment may be different, but this difference will not be measurable by certain physical techniques. It is well known in carbon chemistry that “tetrahedral” exchange reactions of the type being discussed may proceed with “inversion” (hzWW) or “racemization” (a combination of hlWWand hzww) depending on whether the mechanism is associative or dissociative. To show the formal relationship between association-dissociation reactions and exchange reactions, we will discuss these two possibilities in detail. Consider first the sets of reactions Huw and flu which interconvert configurations having geometries W and U shown in Figure 4a. Skeletal positions are Sz. indexed in Figures 3b and 4c. Uuu = D3uc [R3U”] and therefore 1 Uuu/ = 1 D Q U U2!. [ . IR3uuI *= 6 . 2 = 12. Also, Zu = /HUU//UuU[=3!.2!/12= 1. This implies that Dwc = DEWmust equal one and all reactions hzUw 6 H“” are nondifferentiable in a chiral environment. Discussion may therefore proceed in terms of the achiral dissociation reaction h3un’ = ( 1)(2)(3)(4)(5)Uw and association reaction ( h ~ ~ ~ = )-l (h3-1)WU = (1)(2)(3)(4)(5)Tvushown in Figure 4a. Of
+
Journal of the American Chemical Society / 95:2
Ul2
= (1)(23)(4)(5)
We now examine the set of operations Uww. This set contains operations in W- as well as reactions in IPW.Three operations are identical with the three operations in Www: uJww = wlWw, uzww = wzww: and uIWw= w3ww. There are in addition three sets of operations, each set containing three reactions nondifferentiable in a chiralenvironment, namely: u4ww, ujwwI and usww; uTWw,usWw, and ueWW; uloww, ullwW, and u1zWw. Each of these sets may be represented by a reaction in HWW, Le., uqww = hlWw, u T w w = hzww, and uloww 5 h4wW. We thus partition U W W into four sets, each containing three operations. This result is summarized ~ in the relationship 9\$‘w((h3-1)v‘U;
h3Uw) =
3eWW
+ 3h4ww+ 3hlww +
3hzww (23)
Here ‘kWw((h3-l)WU; h3uw) represents the unique partition of (h3- l)wu UUUh3uWinto sets containing operations in Www or reactions in Z P w which are nondifferentiable in a chiral environment. Let us consider the situation where the configurations having geometry U represent metastable reaction intermediates and the ligands occupying skeletal positions squ and sju are present in equal concentrations. Since the coefficients of the terms in 9WW((h8-1)wu; h3Uw) are identical, each of the following possibilities will occur with equal probability every time dissociation is followed by association: (i) operations in Www (no net reaction), (ii) reactions nondifferentiable from hdWW in a chiral environment (permutational isomerization reactions), (iii) reactions nondifferentiable from hzWWin a chiral environment (exchange reactions with “retention of configuration”), (iv) reactions nondifferentiable from hzWWin a chiral environment (exchange reactions with “inversion of configuration”). If the ratios of the coefficients of the terms in \Tr-((h3-1)WU; hauw) d o not represent the actual probablilities, i.e., do not have physical signifiance, we say that “memory effects” are present. Crudely, this means that a molecule remem-
1 January 24, 1973
389
bers its past history and because of this it is more likely to follow one reaction path than another. The most usual type of memory effect for dissociation-association reactions is incomplete dissociation. Here, the molecules in the configurations having geometry U are not free to rotate and translate in space and therefore the proper configurational symmetry group Ucc is not physically meaningful. We now turn to the association-dissociation reactions of the type shown in Figure 4b. The skeletal positions of the configurations shown in Figure 4b are indexed as shown in Figures 3b and 4d. We examine only the operations implied by the achiral association reaction h3VW = (1)(2)(3)(4)(5)”w followed by the reverse reaction (h3-1)”v = ( 1)(2)(3)(4)(5)wv = h3”v, i.e., the set of operations h3J1rvV/vvh3vw= Vww. The permutation group V , defined by the representation Yvv = Dovv, contains these operations. u1
3
,c3
c4
4
X
= (1)(2)(3)(4)(5)
uz = (123)(4)(5)
(132)(4)(5)
03
=
04
= (1)(23)(45)
u5
= (1 2)(3)(45)
u6
= (13)(2)(45)
Following the procedure used above, we find \EW”((h3-l)WI’;
h3vW) =
3eWW
+ 3hzWW
(24)
This means that if association is followed by dissociation as shown in Figure 4b, the only stereochemical change possible is that implied by h2WW,i.e., “inversion of configuration.’’ Permutational isomerization and exchange with “retention of configuration” are ruled out. A slightly more complicated example should clarify the physical signifiance of enantiomeric reastions. For configurations having the geometry shown in Figure 5a, the configurational symmetry group Xxx = Clxx R3xx contains but one operation, xlxx = (1)(2)(3)(4)(5),xx and Xsx= Csxx R3iXXcontains two operations, alXx = xlxx and Q , =~ (1)(2)(34)(5)xx, ~ if skeletal positions are indexed as in Figure 5b. The ligand set is labeled by L = { Ai, B2,C3,C4,C 5 ]and thereSI fore the group of allowed permutations H = S, S3 contains nl!np!n3! = 1!.1!.3! = 6 operations. Since X X X contains only one operation, Ix = I H X X I / /Pyx\ = 6. D’xx = 5 : DICIc, = 1, and therefore Hxx contains only four exchange reactions differentiable in a chiral environment. A complete set of exchange reactions differentiable in a chiral environment is shown in Figure 5c. If we choose to view “tetrahedral” exchange reactions in terms of “inversion or retention of configuration.” hlXX and hzXX both represent “retention” while h3XXand h,xX both represent “inversion.” Reactions hlxx and hZxx as well as h3Xx and hdXXnevertheless represent different changes in stereochemistry (steric courses) since ligands in skeletal positions sgx and sJx are enantiotopic and are therefore not equivalent in a chiral environment. Chiral catalysts or enzymes may provide a chiral environment which allows preferential substitution of one of the enantiotopic ligands. l9
+
+
+ +
(19) CY. K. Mislow and M. Raban, Top. Stereochem., 4, 127 (1969)-
Figure 5. Configurations and reactions defining the stereochemistry of exchange reactions discussed in the text.
represents inversion of configura-
The operation tion and
hzXX =
a 2 X X . h l X X . &XX
=
(a,. hi. Q ~ ) X X
h3xX = 3 , x X . h4Xx.9,XX = (a,. A,.
3,)Xx
Thus hiXXand hzxx as well as hBxx and h4XXare enantiomeric reactions. As was pointed out above, enantiomeric reactions have equal probabilities of occurring in an achiral environment and are physically indistinguishable. We see that in the case at hand, this fact is related to the enantiotopic relationship between ligands occupying positions sgX and sqX. As in the previous example, we now examine the reactions implied by dissociation followed by association. Consider the configurations shown in Figure 6a. If skeletal positions are labeled as in Figures 5b and 6b, FY, and Pyyare representations the groups Xxx, of the groups X , 8, Y , and P defined by the following operations.
xxx,
XI
Yl
=
=
fi
91
= (1)(2)(3)(4)(5) =
(1)(2)(34)(5)
=
(1)(2)(3)(4)(5)
Y Z = yz = (1)(2)(3)(45) Klemperer
/
Steric Courses of Chemical Reactions
390 1
c3
followed by association. These operations are ( h . 5 . y i . h ~=) ~(1)(2)(3)(4)(5)xx ~ = Xixx (hs*y2*h.5)XX = (1)(2)(3)(45)xx = hzXX ( h ~ * Y i . h j=) ~(1)(2)(34)(5)xx ~ = h5XX = h4XX ( h 6 . ~ 2 * h 5=) ~(1)(2)(354)xx ~
( h j . y i *hj)Xx = (1)(2)(3)(4)(5)XX = xiXx (hj*yz.h5)XX= (1)(2)(35)(4)xx = h1xX ( h ~ . y i * h=~ (1)(2)(34)(5)Xx )~~ = h 5 xX ( h 5 * ~ 2 . h s= ) ~ (1)(2)(345)xx ~ = h 3xX
and therefore
+
+
hlYX) = 2eWW+ 2hEWn‘+hi Ww hzwW h3WW h4WW
SIX”((hl-l)”Y; I
c3
+
111. Reaction Mechanism and Steric Course A chemical reaction mechanism is usually defined in terms of a potential energy surface in multidimensional space which indicates the potential energy of every possible geometric arrangement of the atoms comprising the system in question. “Low regions” on this surface c3 correspond to stable or metastable configurations, and the mechanistic pathways are defined by “paths” on this surface which interconnect the low regions. It is \ c4 possible that these “paths” are “forked” and there exist “junctions” at which more than two paths meet. Instead of actually considering the relevant parts of the potential energy surface and defining reaction mechac3 nisms in terms of “paths” and “junctions,” it is more convenient from a stereochemical point of view to represent reaction mechanisms topologically by con\ structing a graph where points represent junctions, and c4 lines represent the “paths” which interconnect them. Figure 6. Dissociation-association reactions implying exchange These lines may be further subdivided by points which reactions shown in Figure 5. represent other configurations of interest. Such topological representations20s2 1 may be described in terms of the formalisms developed above. xIxX,ylYy,and y z y y represent rotation of configuration, Consider the topological representation of a simple while R ~ Tiyy, ~ and ~ yzyy , represent inversion of consystem where each point represents equivalent configuration. Using eq 19 figurations having geometry V, W, or X. These points are labeled by coset representatives from the sets Cv, D x y = D y x = (1/(1~2))(1~1~1!~1’1!~11~3!~13‘ I = 3 Cy: and Cx, and the total number of points thus equals I, Iw Ix. Assume that lines connect points is the number of reactions differentiable in a chiral representing configurations having geometry V and W environment. Complete sets of reactions differentiable as well as points representing configurations having W in a chiral environment are shown in Figure 6c. Using and X, but no (single) lines connect points representing eq 22, we find Dxtyt = DytX! = 2. The reader may configurations having geometries V and X. Such a verify that heyX and hBYxas well as hsXy and h j X Y are graph represents the case where reactant configurations, enantiomeric reactions, while AByx and h4XYare achiral. intermediate configurations, and product configurations The fact that hSYXand hjYX are enantiomeric again rehave geometry V. W, and X, respectively. If only one flects the enantiotopic relationship between ligands mechanism interconvei ts reactant and intermediate occupying positions saX and s4X, while the fact that configurations, and only one mechanism interconverts hsxy and h j X Yare enantiomeric reflects the enantiotopic intermediate and product configurations, then two rerelationship between the two faces of the three-coactions, hPJ1-”and htXJy,define the lines of the topologiordinate molecule in the configuration having geometry cal representation: points cZv and c>v are connected Y.19 by a line if and only if Assuming that the reactions take place in an achiral htJiV . om\ V . c , V = wAJ5W , cj\J7 (25) environment, we examine the reactions implied by the I
I
+
dissociation reaction h S y x followed by the reverse reaction h g X Y . Since hBYXand hjYX as well as hcXY and hjx’ must occur with equal probability in an achiral environment, the set of operations (h6Yhfi)xX,(h6Y h 5 ) x X , (h5Y h j ) x x , and ( h , yh6)xx is implied by dissociation
+
holds for some u , V V E YVv,some wkWWE WwW,and h,”V E H J y V where h,”v = h,Wv if h,WV is achiral and (20) E. L. Muetterties, J . Amer. Chem. SOC.,91, 1636 (1969). (21) For a recent review, see: M. Gielen, Ind. Chim. Belge, 36, 815 (1971).
Journal of the American Chemical Society f 95:2 / January 24, 1973
391
hlWV = hPwv or a “mirror image” of hpwV if hPwv is chiral. Points cIw and ckx are connected by a line if and only if h lXTv . wm\\-w . c,“ = x,=cI;x (26)
holds for some wmWw e Www, some x a X Xe Xxx, and hLXWe Hxv where hlXw = hqxw if hqXwis achiral and hlxn’ = hqXTVor a “mirror image” of hqX” if hqXwis chiral. As was shown in ref 6 for the case of permutational isomerization reactions, eq 25 and 26 define topological representations such that two reactions are represented topologically in an identical fashion if the following conditions exist. (a) The reactions are nondifferentiable in a chiral environment. Reactions nondifferentiable in a chiral environment must occur with equal probability if molecules rotate and translate freely in their environment. (b) One reaction and the reverse reaction of the other reaction are nondifferentiable in a chiral environment. If a given reaction occurs in an equilibrium situation, the principle of microscopic reversibility demands that its reverse reaction also occur. (c) Any “mirror image” of one reaction is nondifferentiable from the other reaction or its reverse reaction in a chiral environment. When reactions occur in an achiral environment, a reaction and its “mirror images” must occur with equal probability. The connectivity 6, of a point ctV is defined as the total number of lines which meet at that point. The number of these (single) lines which connect different points representing configurations having geometry W to the point cLvis denoted 6,, and thus for the case under discussion, 6, = awl,. For the connectivity 6w, however, 6, = 6vw 6 x w , since is connected by single lines to points representing configurations having geometry V as well as points representing configurations having geometry X. Also, 6, = 6wx. Connectivities may be calculated using formulas derived elsewherees If hp’vv is achiral
+
and if hPJyv is chiral
Here, / V i is the number of operations in the group V and I V n A,-] Wh,l is the number of operations which the groups V and h,-lWh, have in common. Note that in general, avw # tiwv. If h,”” represents the transformation of configurations having geometry V into configurations having geometry W, then by microscopic reversibility, (h,” v)- = (h,- 1)va7 represents the transformation of configurations having geometry W into configurations having geometry V. If (hp-l)v\y is achiral 6“W
=
i WI W f l h,Vh,-ll
and comparing with eq 27, 6,, does not necessarily equal aWv. In eq 23 and 24 we provided means of expressing the steric course of exchange reactions which proceeded via intermediate configurations having connectivities
greater than two. If we wish to express the net stereofolchemical changes implied by the sequence h,”V lowed by hqXWdiscussed above, we define Dx v
qm-(hqX“;
h,\V’)
=
a=1
(29)
a,h,XV
Here, Dxv is the number of reactions in Hxydifferentiable in a chiral environment, and ai is the relative probability of reactions nondifferentiable from h,xV occurring if these reactions occur rjia intermediate configurations having geometry W. Note that if X = V as in eq 23 and 24, S’XX(hqX’v; hPTvX)includes an extra term aoexx which indicates the relative probability of no net reaction occurring each time the sequence h,”>followed by hqXJTT occurs. If “memory effects” are ruled out, X€’Xv(h,X”-; hpTyv) may be calculated as follows. (i) Assuming hgXJVand hpTyv are both achiral, W‘v\ThpTJ-v= (hqWh,)Xvis the set of 1 W / reactions hqXTy partitioned into subsets of reactions nondifferentiable in a chiral environment. Then a i is the number of reactions in the subset containing reactions nondifferentiable from hixv in a chiral environment. (ii) If hqXJyis achiral but h,Tyv is chiral, the coefficients ai are deri\ed by partitioning the set of 21 W ( reactions in ( h , Wh,)XT’ and (hqWh,)xv where is a “mirror image” of hpXvv. (iii) If hPTyyis achiral, but hOXvis chiral and AsXv is a “mirror image” of hpXv, then the coefficients ai are derived from the set of 21 Wl reactions in (h,Wh,)Xv and (hsWlip)XT-.(iv) If both hpTyvand hGX\‘-are chiral, is a “mirror image” of hpTyT-,and hSXvis a “mirror image” of hQXJv,then the coefficients ai are derived from the set of 4jWI reactions in (h,Wh,)’vv, (h,Wh,)Xv, (h, WhJXv? and (h,WhJXV. To consider more complicated sequences of I eactions, e.g., hlUTfollowed by h,yu. . . , followed by hkX’”
may be derived in a similar fashion. IV. Examples A. Trigonal-Bipyramidal Substitution. This example concerns reactions of the type shown in Figure 7a, where reactant configurations have geometry V and product configurations have geometry X. Skeletal positions are indexed in Figure 7b and c. The proper configurational symmetry groups are defined by I/”” = Czv” + R3Yv and X X X = C2XX + R 3 X X , while ?vv = Cz>J’ + R 3 $ v and gXX = CZvXX R3tXX. The groups V , 7,X,and X contain the following operations
+
(1)(2)(3)(4)(5)(6)
U I = ti1 = XI = 3 1 = UZ
=
ti2
= XZ = 22 =
(14)(23)(5)(6)
zT1, = 33 = (14)(2)(3)(5)(6) 64
= 34 =
(1)(23)(4)(5)(6)
+ +
The group of allowed permutations H = S? SI SI contains 4!. l ! . I! = 24 operations. Configuration counts Zv and Zx both equal 24/2 = 12. The number of reactions differentiable in a chiral environment, D x v , is calculated using eq 19 Dxp =
{ 1/(2.2)]{ 1.1.4!. 1 4 .I!. 1’. l! * 1’ f 1 * 1 ~ 2 ! . 2 2 ~ 1 ! . 1 ! . 1= ’) 8 Klemperer
1 Steric Courses of Chemical Reactions
392 I,
I
.
4
V
X
3
/
4
Figure 8. Dissociation and association reactions implying substitution reactions shown in Figure 7.
chiral environment. Eight differentiable reactions selected from this set of 12 reactions are shown in Figure 7d. The arrows drawn in the reactant configuration connecting the ligand labels crudely represent the permutation of ligands which each reaction implies. Since the reactant and product configurations have different geometries, these arrows are somewhat ill-defined from a mathematical and physical point of view, but they are very useful for quickly identifying the stereochemical relationships between various reactions. For example, they indicate the enantiomeric relationship between hsxv and heXVas well as hTXv and hsXV,which is verified by the fact that 13
?I\
$ ( -%
h~~l1231141151161xy c6+
'4
I
:,.
h,XT
= 3 , X X . h 6X T . g 3Y V
h7XV
= .j&XX.
and h X V .g,XV
. 1
Figure 7. Configurations and reactions defining the stereochemistry of "trigonal bipyramidal" substitution reactons discussed in the text.
Eight appropriate reactions are most easily derived following a procedure justified elsewhere:8 the set of 12 operations which convert into the 12 nonequivalent configurations having geometry X must contain a complete set of reactions differentiable in a Journal of the American ChemicaI Society 1 95:2
Using eq 22 Dxlvt =
+
~(l~1+0~0)~4!~14~1!~11~1!~11 (2.2+2.2) (1.1 0.0).2!.22.1!.1'.1!1'+ ( 0 . 0 + 2.2).1!.12.1!.2'.1!.1'.1!.1') = 6
+
and therefore the reactions hlXV,haxv, hBXv, hdXV,AsXV,
January 24, 1973
393
and h7xv form a complete set of diasteromeric reactions. We now examine the steric course of substitution reactions for two cases where intermediate configurations have a connectivity greater than two. First, consider the sequence of reactions shown in Figure 8a and b. If the skeletal positions of the reactant, intermediate, and product configurations are indexed as shown in Figures 7b, 8c, and 7c, respectively, the dissociation reaction is described by hlwv = ( 1)(2)(3)(4)(5)(6)wv, while hlXW = ( 1)(2)(3)(4)(5)(6)xw describes the subsequent association reaction. The groups Www = DdWw R3WW R3ww and IfWw= DdhWW R3iWW R3iWw are representations of the groups W and J v defined by the operations
+
+
mi
wi =
+
'3
6
i
5
3
V
X
+
= (1)(2)(3)(4)(5)(6)
~2
= fi& = (1243)(5)(6)
w3
=
w4
=
a 3
'3
=
(1342)(5)(6)
I).
C,
= (14)(23)(5)(6) = (1)(23)(4)(5)(6)
wj = w6
=
m6
= (14)(2)(3)(5)(6)
w7
=
2717
= ( 12)(34)( 5)(6)
WE
=
= (13)(24)(5)(6)
Since lW"wi = 8, Iw = 2418 = 3. Reactions hlwv and hlXWare achiral since hlWV = @ J ~hlWv.?jIPv ~ ~ ~ and . hlXW = % , ~ x x . h l X W ~ ~ , g WFrom W . eq 27
'3
A3
+
andconsequently, G V = G W v = 1, G W = 8vw Gxw = 8, and G x = aWx = 1. These results may be checked using a relation derived elsewhere,* IvGwv = IwSvw and IxSwx = ZwSxw. The isomer counts and connectivities just calculated are of great help in constructing a topological representation. We shall however determine the steric course of this reaction sequence using the techniques presented in the previous section. The substitution reactions implied by hlWVfollowed by hlXWare contained in the set of reactions hlXW.WWW.hllT'v= Wxv. Consulting Figure 7d, we note that wlxv = hlXV, w2XV = h 4 X V . U 2 X V , w I X V = h 4 X V w g X V = h l X V . u2VV, w6XY = h XV w6XV = h XV . u2VT' w7XV = h X V 2 , 3 ,and wgXP = hI.uzVV,and therefore *xV(hgxw; h,"')
+ 2hzXv + 2haxv + 2hdxv
= 2hlXV
Note that the implicit assumption has been made that the reaction sequence is "irreversible," Le., dissociation always leads to substitution. Now consider an alternative reaction sequence shown in Figure 8e and f. Here, the skeletal positions of reactant, intermediate, and product configurations are indexed as shown in Figures 7b, 8d, and 7c, respectively.
Figure 9. Configurations and reactions defining the stereochemistry of "octahedral" substitution reactions discussed in the text.
Klemperer
Steric Courses of Chemical Reactions
394
16
:e
43
42
:5
77
4
I
; X
I
+ /.’*
3
IC1
I
V 85
I
.e .I
2
W Figure 10. Dissociation and association reactions implying substitution reactions shown in Figure 9.
The dissociation reaction is described by hlYV = (1)(2)(3)(4)(5)(6)yv and the subsequent association reaction by hlXY = (1)(2)(3)(4)(5)(6)xy. The proper configurational symmetry group of the intermediate configuration is Y y y= TYy RsYY RQYY. Following the procedure used in treating the reaction sequence examined in the preceding paragraphs, the reader may verfy that
+
Wv(hlXY; hlyv)
=
+
+ 2hlXV+
2hlXV
4h7XV
+ 4h8XV
B. Octahedral Substitution. The type of “octahedral” substitution reactions considered here is shown in Figure 9a. Skeletal positions are labeled as in Figure 9b and c. The group of allowed permutations, H = S5 SI SI, contains 5!. I!. l! = 120 operations. The groups VV = CdVV R3VV, X x x = C 4 X X + R B X X VVV = C4 0V V + R 3 , V V and X X X = C 10XX R 3 2 X Xare representations of the groups V , X, P, and 8 defined by
+ +
+
+
DI = fii =
xi
~z = ijz =
xz = Z2 = (1245)(3)(6)(7)
21 = (1)(2)(3)(4)(5)(6)(7)
(1542)(3)(6)(7)
~3
=
=
~3
=
U?
= 8, =
~4
= 24 = (14)(25)(3)(6)(7)
ij3
%3
=
= ( 15)(24)(3)(6)(7)
g6 =
$5
fip
26 = (12)(3)(45)(6)(7)
fi7
=
37
= (14)(2)(3)(5)(6)(7)
e8
=
28
=
=
I
IO
(:)(25)(3)(4)(6)(7)
I W Figure 11. Configurations describing isomerization reactions of IrHdPRds.
lowed in the previous example. Such a set is shown in Figure 9d. The number of diastereomeric reactions, D x ~ v is~ ,
+
+
+
+
+ +
+
As the arrows drawn in Figure 9d suggest, hsXV and hXVas well as heXV and hsXV are enantiomeric. The seven reactions hlXV,hzXV,hXV, A d X V , h j X V ,heXV, and hsXV thus form a complete set of diastereomeric reactions. We now determine the stereochemical implications of substitution which proceeds by dissociation as shown in Figure 10a followed by association as shown in Figure lob. The skeletal positions are indexed as shown in Figures 9b, lOc, and 9c, and therefore hpWv = (1)(2)(3)(4)(5)(6)(7)wv describes the dissociation reaction, while h,X” = ( 1)(2)(3)(4)(5)(6)(7)xw describes the association reaction. The group Www = D3ww Rzww RsWwis a representation of the group W defined by the operations w1 = (1)(2)(3>(4)(5>(6>(7) wz = (134)(2)(5)(6)(7) =
(1?3)(2)(5)(6)(7)
Wa =
(1)(25)(34)(6)(7)
=
(14)(3)(25)(6)(7)
~3
+
~5
1.1.2!. 2Z.1!.1’.1!.1’.1!.1’) = 9
A complete set of reactions differentiable in a chiral environment may be derived using the procedure fol95:2
+
Dxivt = (1/(4.4 4 . 4 ) ) f ( l . l 0.0). 5!.15.1!*1’.1!.1’ (2.2 0 * 0 ) . 1 ! . 4 ’ . 1 ! . 1’- 1!- l’-l! .1‘ (1.1 2,2)2! .22*1!.1’. l ! 11.1!. I’ (0.0 2 ~ 2 ) 3 ! ~ 1 3 1 ! ~ 2 1 * 1 ! ~ 1 ’ . 1 ! . 1=’ ) 7
{ 1/(4.4)] { 1+5! .16.1!.1’.1! .1’ 2.2.1!.4’.1!.1’.1!.1’.1!.1’ +
Journal of the American Chemical Society
I
+
Thus configuration counts I , and I, both equal 12014 = 30. Dxv, the number of reactions differentiable in a chiral environment, is calculated using eq 19
Dxv
5
+
w6 = (13)(4)(25)(6)(7) Both hPTvVand hgX” are achiral, and the steric course of this reaction sequence is therefore derived from the set
January 24, 1973
395
Acknowledgment. I am grateful to Dr. Bertram Frenz for his critical comments and to Mrs. Irene Casih , X Y . u 2 V V w3XV = x 2 X X . h j X ” . u 3 Y v w4Xv = x 3 X X . h,Xv. miro for carefully and efficiently preparing the manufor publication. u 3 V V , w j X V = hlXV.U4YV, and w6Xv = x ~ X X . ~ ~ X V . V ~ V script V Consequently Appendix lpX’(h,X”, h W V ) = 2hlXV + 4 p v In this section eq 22, which enumerates diastereomeric reactions, is derived. A similar result has been deC. Polytopal Isomerization of I I - H ~ ( P R ~ )Certain ~. phosphino iridium hydrides of the type I T H ~ ( P Rexist ~ ) ~ rived by Ruch, Hasselbarth, and RichtereZ4 The derivation provided here is analogous to the proofs of in two polytopal forms, facial (see Figure I l a and theorems given in the Appendices of ref 5 and 7 which meridial (see Figure 11b). 7 2 Upon heating in solution, provide formulas which enumerate differentiable perthe facial isomer rearranges to the meridial isomer, mutational isomerization reactions. Familiarity with and Muetterties’ has suggested that this process may these proofs should aid in understanding the present proceed via five-coordinate intermediates, IrH3(PR&. derivation. Although eq 22 is a general result, the We shall examine the possibility of checking the validity following derivation does not allow for the case where of this mechanism using labeling techniques. all operations in PVvand TVvWW which represent inHydride ligands are labeled AI, A2, and ABand phosversion of configuration also represent rotation of phine ligands are labeled B4, B;, and Be. The skeletal configuration. In this case, two reactions are diastereopositions are indexed in Figure I l c and d. Assuming if and only if they are differentiable in a chiral meric that the isomers have C3$ and C2, symmetry, Vvv = environment, and the limitation is therefore of no conCSvVand XXX= CzXx. The groups V and X contain sequence. the operations We shall assume that the group of allowed permuta01 = (1)(2)(3)(4)(5)(6) tions is defined by 02 = (123)(456) 1 HE S, ~3 = (132)(465) i=l of six reactions, h,Xm’WU‘”h,TvV = WXV. Consulting Figure 9d, we note that wIXV = hlXV, wlXV = x3XX.
x1
=
(1)(2)(3)(4)(5)(6)
x2
=
(1)(23)(45)(6)
The final step of Muetterties’ proposed mechanism is shown in Figure lle, and the skeletal positions of the postulated intermediate configuration are indexed in Figure 1. Thus WWMT= DQWW RBWW, and W is defined by
+
w1
=
(1)(2)(3)(4)(5)(6)
w o = (123)(4)(5)(6)
(132)(4)(5)(6)
w3
=
w4
= (1)(23)(45)(6)
wj = (13)(2)(45)(6) w6
=
(12)(3)(45)(6)
Neglecting any further intermediate configurations, to ascertain the poswe examine the reactions in HWMr sible changes in stereochemistry which might accompany transformations of the reactant configurations into intermediate configurations having geometry W. Using eq 19 and 22 Dwv = { 1 / ( 3 ~ 6 ) ) { 1 ~ 1 ~ 3 ! ~ 1 3 * = 3 !2~ 1 3 ~ Dwrvf = { 1/(3.6
+ 3.6)] X ((1.1
+ O*0)3!.l3.3!.l3]= 1
Thus regardless of the mechanistic pathway connecting the reactant and the postulated intermediate configurations, only two steric courses exist, and in an achiral environment they will be indistinguishable. Ligands will become completely “scrambled,” and should the postulated intermediate exist, the net isomerization reaction will completely lack stereospecificity. (22) J. Chatt, R. W. Coffey, and B. L. Shaw, J . Chem. Soc., 7391 (1965). (23) E. L. Muetterties, Accounts Chem. Res., 3,266 (1970).
I 5 3, but the case for an arbitrary I > 3 may be treated in the same fashion. Before proceding with the derivation, we must express the relationship between diastereomeric reactions in a mathematically concise manner. If reactions in the set HV” are to be enumerated, V and W are the permutation groups whose representations Vvv and WM-ware proper configurational symmetry groups. We define the sets V’ and W’ to be subsets of the groups P and T f , respectively, such that ut‘ E V ‘ and wj’ E W‘ if and only if u t t v v and wjlWw represent inversion of configuration. Next, we mj) E ll’( 7, define a permutation group II’( 7,TT), ll’(fii, TF), whose elements permute elements hkVW in Hv” according to II’(fii, m,.>hh- b*’h,.m,-] where fit E 7 and mu E T f must either both be elements in V and W , respectively, or both be elements in V’ and W’, respectively. Thus
In’(7,Ti-)[
/v~(wI
+ iV’i\W’!
(‘41) The reader may verify that II(7, lf) is a well defined group with a product operation defined by =
ll’(G, mj).n’(a,,
a,)= II’@i.fi,, mj.a,)
The group II’(7, Tr) may be used to rigorously define diastereomeric reactions; hivn- and hlV’5- are diastereomeric reactions if and only if rI’(fit,
m’j)ht # h;
for all II’(fii,aj)E II’( 8:lf). The number of diasteromeric reactions, Dvtwt, is the number of equivalency classes generated in H if hk, h , e H a r e equivalent when rI’(b(, fij)hk = h ,
for some II’(fit,
@jj) E
II’(7, lr). The number D1rtwf
(24) E. Ruch, W. Hasselbarth, and B. Richter, Theor. Chim. Acta, 19, 288 (1970); E. Ruch and W. Hasselbarth, private communication.
Klemperer 1 Steric Courses of Chemical Reactions
396
may be calculated using Burnside’s Lemma25
where x(fit,aJ) is the number of h, in H which II’(fi$,m j ) leaves fixed, Le., the number of h, in H for which
h, = II’(fi$,aj)hk E “ j . h i , * ~ ~ ~ ~ - ’ or equivalently U , = h,.w,.h,-’ Accordingly, ~ ( f i , , m,) = 0 unless fit and a, have identical generalized cyclic type, and if 8, and a, both have , jnl;kl, kz, , knz; generalized cyclic type (il, j z , Ii, 1 2 , , In,), then p= 1
r=l
q=l
(25) F. Harary, “Graph Theory,” Addison-Wesley, Reading, Mass,, 1969, p 181.
Since ~ ( f i , , 19,) is dependent only on the generalized cyclic type of fit and a,, and x(fii,a!)# 0 if and only if f i t and aj have identical generalized cyclic type, the summation in eq A2 may be changed to one over the generalized cyclic types of the operations in V , W , V’, and W‘. Let hVjljz...ln,,ii,kZ... knz,ll~2...lna, hV ’; I ~ Z . . . h l klk?. I . .k,,, 1112.. . in3, hWjljz.. ,klkz.. .k,?, 1112. . . in,’ and hW’jljz l,ii1k?...Rn2r~1~2...lns be the number of operations in V , V’, W , and W’, respectively, having generalized cyclic type ( 4 ,j,, . . . ,j n 1 ;k:, kz, . . . , kn2; 11,12, . . . , ln8). Then by substituting eq AI and A3 into eq A2 and using the new summation extending over all generalized cyclic types of permutations in V , W , V’, and W’, we obtain eq 22. During the course of this derivation we have assumed that all elements in Hvw represent reactions. For the case V = W, some elements in Hwwd o not represent reactions but represent rotation of configuration, and therefore the number of diastereomeric reactions is - 1. Dwlw1
Effects of Additional Ring Fusions and Binding to Metal Atoms upon the Cyclooctatriene-Bicyclooctadiene Equilibrium’ F. A. Cotton*2 and G. Deganelloa Contribution f r o m the Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts Received M a y 22, 1972
02139.
Abstract: The hydrocarbons bicyclo[6.3.0]undeca-2,4,6-triene(1) and bicyclo[6.4.0]dodeca-2,4,6-triene (3) have been prepared by reaction of dilithiocyclooctatetraene with 1,3-dibromopropane and 1,4-dibromobutane, respectively. 1 and 3 could not be isolated pure, but instead 1 was obtained mixed with its tautomer tricyclo-
[6.3.O.O2~7]undeca-3,5-diene (2) and 3 was obtained mixed with its tautomer tricyc1o[6.4.0.0*J]dodeca-3,5-diene (4). The kinetics and equilibrium of the 1 + 2 conversion have been studied. The rate constant at 20” (extraposec-l. Assuming a frequency factor of 10lathe Arrhenius lated from measurements at 34 and 58”) is 1.6 x The equilibrium activation energy is 24 =k 1 kcal/mol. At 58’ the equilibrium constant, K = [2]/[1], is -33. molar ratio 3 :4 at 114’ is approximately unity. From pmr spectra and other considerations the probable molecular structures and ring conformations are deduced. The systems 1-2 and 3-4 react with Fe2(CO)S and Mo(C0)3(NCCH& to form a variety of crystalline derivatives, all of which have been characterized as to gross structure (ix.,connexity of bonds but not conformational details) by ir and pmr spectra. These derivatives, which will be suitable for X-ray crystallographic studies of structural details, are the following: (2)Fe(C0)3, (2)2Mo(C0)~, (l)Mo(C0)8,(4)Fe(C0)3, (4)2Mo(C0)2,(3)Fez(C0)6,(3)Mo(C0)3. Yields of the various derivatives run parallel to the intrinsic relative stabilities of the tautomers in each of the pairs, 1-2 and 3-4.
I
t was nearly 20 years ago that Cope and coworkers4 first studied an equilibrium of the type
a
b
(1) Supported in part by the National Science Foundation. (2) Address correspondence to this author at the Department of
Chemistry, The Texas A&M University, College Station, Texas 77843. (3) On leave from Ceiitro di Chimica e Tecnologia dei Composti Metallorganici degli Elementi di Transizione del C.N.R., Bologna, Italy. (4) A. C. Cope, A. C. Haven, F. L. Ramp, and E. R. TrumbuI1, J . Amer. Chem. SOC.,74, 4876 (1952).
for the case where X = X’ = CH2, i.e., for cyclooctatriene and its tautomer bicyclooctadiene. They reported that the equilibrium ratio a/b is -6 at 80-100”. More recently, Huisgen and coworkers5 reinvestigated this system (for which they found a ratio of -8 at 60”) and 11 others with a variety of X and X’ groups. Variations in X and X’ were found to influence the ratio greatly, changing it over a range of -lo4. Huisgen and coworkers considered several factors which might be expected to influence the position of equilibrium, but concluded that no entirely satisfactory explanation for the observed facts could be found within the framework of their considerations. (5) R. Huisgen, G. Boche, A. Dahmen, and W . Hechtl, Tetrahedron Lett., 5215 (1968).
Journal of the American Chemical Society 1 95:2 1 January 24, 1973
