A Generalized Selection Rule for Pericyclic Reactions Fu-Cheng He Chengdu University of Science and Technology, Chengdu. Sichuan 610065, The People's Republic of China Gary V. Pfelffer Ohio University. Athens, OH 45701 Woodward and Hoffmann introduced a n important selection rule for pericyclic reactions from detailed ksiderations of electrocvlic reactions, cvcloadditions, and related processes (I). ~ h e r e ; sno question but that the insights of these authors have transformed our understanding of a major segment of organic chemistry. Several articles have appeared in THIS JOURNAL which have urged the assimilation of the Woodward-Hoffmann rules into the undergraduate organic curriculum (2,3). Important, alternative viewpoints of reaction mechanisms have been advanced by Dewar (4) and Zimmerman (5).These authors focused attention on the oresence or absence of aromatic stability in transition states as the indicator or the allowedness or iorbiddenncss of a .ornoosed mechanism. This . type of selection rule considers transition states and whether thev are examoles of Huckel svstems ( f i ) in which maximum stability is ach'ie\,ed when the c u t e contains 4n T 2 electrons or whether thev arr Midius syscems (7)in which 4n electrons stability. hen has previously discussed the give the pedagogial usefulness of the Zimmerman-Dewar selection rule (8). Recent calculations of Tang and co-workers have suggested the validity of a new selection rule for pericyclic reactions (9). This rule, which will he referred t o as the Odd-Even Rule, is similar to, hut simpler in application than, previous selection rules. Statement of the Odd-Even Rule By means of calculations hased on molecular orbital theory Tang and co-workers studied the relative energies of various conformations arisine durine the course of certain cvcloadditions, sigmatropic reactions, and other pericyclic reactions. T o summarize the results of these calculations the followine Odd-Even Rule is proposed:
-
A mound-state ~ericvclicreaction is thermallv allowed when onehaif of the totafnumi~rof electronsthat takioart in the reaction is odd (wen, and the tom1 nurnhrr ofantarafarinl components is even (odd),with the fdlowmy ransideratiml 1) If [he total number oieleetrnna mkmg pnn m the reaction isdd. the reaction can be analyzed by adding one to the total number ~~~~~
~
~
~
~~
~
~
~
of electrons involved and then applying the rule.
2) An sntarafacial component in a pericyclic reaction is defined as a species in which bonds are broken or formed which lie on opposite faces ofa given plane of that species (10).
In applying the Odd-Even Rule three points should be noted: first. this rule can he applied regardless of whether the numher of ilectrons taking pait in the'reaction is even or odd (in practice the Woodwanl-Iloffmann rulecan heapplied only .. to even-number electron sysrrms.); second, ifa ground stare pericyrlic reaction is thermally iorl~idden,then it is photochemically allowed; third, zeroshould he regarded as & even numher. Relatlonshlp of the Odd-Even Rule and the WoodwardHoflmann Rule The relationship between the Odd-Even Rule and Zimmerman's rule is straightforward, but the connection between
948
Journal of Chemical Education
the Odd-Even Rule and the Woodward-Hoffmann rule is worth detailed discussion. 'l'he Woodward-Hoffmann rule can in fact be derived from the Odd-Even Rule. In the derivation of the Woodnard-Hoffmann rule only those comDonents in a reaction which have an even numher of electrons need be considered. Therefore, the totalnumber of comoonents in a ~ericvclicreaction is d v e n by the sum of the foliowing four numbers:
+
Here n(49 2), represents the number of suprafacial components (a suprafacial component in a pericyclic reaction is defined as a species in which bonds are broken or formed which lie on the same face of a given plane of that species (10)) with 49 2 (q = 0,1,2,. . .) electrons involved in the reaction; n(4r), represents the numher of antarafacial components with 4r (r = 1,2,. . .) electrons involved in the reaction; n(4q 2). and n(4r), have similar meanings. Since each of these four numbers may be even or odd there are sixteen possible combinations of their collective odd or even character. The term n(4r)., however, need not be considered in detail in applying the Odd-Even Rule because: (1) since 4r/2 = 2r is alwavs an even number the term n(4r), always adds a n even number of electrons to one-half of the sum of electronstakina Dart in the reaction and thus cannot affrrt the oddness or i e n n e s s of components one-half of the sum: (2) the number of su~rafacial does not enter the o d d - ~ v e nRule. Table 1shows the remaining eight possible combinations for n(4q 2)., n(4r)., and n(49 2).. In addition Table 1 shows values for the quantity H which represents one-half of the total number of electrons involved in the reaction (recall that if the sum of electrons is odd then one "ghost" electron is added to the sum before dividing by two) and for the quantity A which represents the total number of antarafacial components in the same reaction. The odd or even character of A can he easily determined n(4q 2).. In order to from the simple sum, A = n(4r). determine the odd or even character of H, one-half of the total numher of electrons is calculated hv summine the vroducts formed from the numher of comp&ents of each iype and one-half of the number of electrons in each component:
+
+
+
+
+
+
+
+
+
+
where n(4q 2), = Zjn(4j 2),, n(49 2). = Z j n(4j 21. a n d j = 0 , 1 , 2 ,... ; k = 1 , 2 , . . ..Sincetermswith2jor2kas multipliers will always he equal to even numbers, the odd or even character of H is really determined by the oddness or evenness of 4 4 9 2), and n(4q 2).. If n(4q 2). and n(4q 2). are either both odd or both even, then H will be even; otherwise H will be odd.
+
+
+
+
Table 1. The Odd-Even Rule and the Woodward-Hoflmann Rule n(4q
+ 2.)
odd even odd even odd even odd even
H=
even
even
odd
even
OW
odd odd
even
odd
wen even odd
odd
odd
odd
even
even odd odd
even odd even
A= thermally
12.
+
n(4q+ 21.
H
thermally
odd
allowed
even
I,
+ 2. + 2.1
even
12+ 25
P. 2 J H=2,A=Z
H=Z,A=l
thermally forbidden
themally e l i d
Figure 1. The dimeriatian of ethylene.
12
+ 2 + 21 Cydoaddltlons
[2.
+ 2. + 2.1
[2.
+
1% +%I
H=Z,A=O
rbidden
even
Table 2. Mode
n(4q+ 2). n(4&
A
n(4r).
+ Z + 2.1
12.
+ 2. + 2.1
3
3
3
3
0 allowed
1 forbiien
2 allowed
3 forbidden
+ +
From Tahle 1i t is evident that the quantity, n(49 2), n(4r)., is odd when H and A differ in odd or even character. From the Odd-Even Rule we conclude: a ground-state pericyclic reaction is thermally allowed when the sum, n(4q 2). n(4r)., is odd. This is also the usual statement of the Woodward-Hoffmann rule. However. statine the selection rule in this form can he perplexing to theuser when some reaction components are present which contain odd numbers of electrons. In contrast, by focusing on the total numhrr of elertrons and explicitly considering odd electron systems the Odd-Even Rule is hothcompletelygeneral and very simple to employ.
+
+
Applications The Odd-Even Rule is not only more general than the Woodward-Hoffmann rule. but also i t is verv convenient for practical use. The followingexamples, whic6 have been used ).rive ~reviuuslvto illustrate the Woodward.Hoffmann rule (.1 ,.some indkation of the range of application of the Odd-Even Rule.
IU
H=2,A=0
thennaliy forbidden Figure 2. The electrocyclie reaction of butadiene.
~~~
Figure 1shows three distinct bonding schemes for the dimerization of ethylene to yield cyclohutane. The labels [2. 2.1, etc. indicate hoth the numher of electrons in each component and whether bonds to that component are formed in a suprafacial (s) or antarafacial (a) manner. In Fimre 1,H = 2 as there are four electrons involved in the reactiori Since H i s an even numher only the [2, 2J scheme, which has an odd number of antarafacial components, is 21. with zero antarafacial thermally allowed. Both [2. components and [2. 2$ with two antarafacial components are thermallv forbidden accordine to the Odd-Even Rule. For 12 + ;2 21 cycloadditions tiere are four possible reaction modes. Table 2 illustrates how readilv the Odd-Even Rule leads to correct predictions of the allowedness or forbiddenness of these modes.
+
IL + SJ
X=3,A=1
thermally forbidden
Figure 3. 11.51 Sigmalropic hydragen shin.
As another example consider the dirrotatory, electrocyclic reaction of dimethylallyl cation,
+
+
+
Elecfrocyclic Reactions Electrocyclic reactions may be regarded as single-component "cycloadditions." The conversion of hutadiene into cyclohutene can occur by the two modes shown in Figure 2. Disrotatory motion of the terminal orbitals leads to sigma hond formation in the suprafacial sense and thus the component is also labeled as suprafacial. Conrotatory motion of the terminal orbitals leads to hond formation in the antarafacial sense and the component is now labeled as antarafacial. Applying the Odd-Even Rule, with H = 2 and A = 0 [4,] or 1 [4.], clearly indicates that only the conrotatory process is thermally allowed.
Is this reaction thermally allowed? The proposed mode may he labeled [3,+] indicating three atoms with only two electrons and the hond between the terminal orbitals being formed in the suprafacial sense. Since H = 1and A = 0, the Odd-Even Rule predicts correctly that the proposed path is thermally allowed. For an electrocvclic reaction involvine an allvl radical there would he three el&rons involved and an additional, "ghost," electron is added to give H = 2. Now. the conrotatow Drocess. which leads to antarifacial hond formation becomes ;hermall; allowed ( A = 1) bv the Odd-Even Rule. Becauseof the useof the "ghok" election the reaction process for the ally1 anion will he the same as for the radical. Sigmatropic Reactions Sigmatropic reactions are topologically equivalent to cycloadditions of two components. A typical example is the [1,5] hydrogen shift reaction shown schematically in Figure 3. Since the s orbital of the hydrogen atom is spherically symmetric, the migrating hydrogen always presents the same Volume 61 Number 11 November 1984
949
IZ+Is+Z+l.I,H=3.A-O thermally allowed
Flgure 5. The concerted transfer of two hydrogen atoms between ethane and ethylene molecules. Figure 4. Carbon atom In the migrating group.
"face" when breakine and formine bonds. The hvdrwen . .. atom is, therefore, considered to be a ~"prafacialcomponent. In the six electron svstem shown in Figure 3.H = 3 while A is either zero as in the-[1, 5J mode or = 1 in the [I, 5J mode. It is the former nath, with H odd and A even, that the OddEven Rule predicts; again correctly, to b e thermally allowed. Woodward and Hoffmann have discussed sigmatropic shifts involving carbon cencers in the migrating group on the basis of the symmetry of the transition state \ I / ) . In the present paper emphasis is placed instead on the suprafacial or antaiafacial nature of t h e mieratine erouo and the residual r system. For the ?r system, it is obvious upon inspection whether the mieratine.... erouo . has remained on the same face (suprafacial) or has moved to the opposite face (antarafacialj. The nature of the mirratineerouo. is determined bv whether retention or inversion of configuration is observed in the group following the shift. If retention is observed then the same hybrid orbital in the migrating group was used in both the old and new sigma bonds which were broken and formed during the shift. In this case the migrating group is considered a suprafacial component. On the other hand, if inversion of configuration is observed then the broken sigma hond and the newly sicma bond must have utilized hybrid orbitals on oppositk sides of the migrating group. In this case the migrating group is considered as an antarafacial component. Figure 4 illustrates this type of sigmatropic shift for two migrations. Both shifts are suprafacial with respect t o the r system; the left drawing illustrates retention and the suprafacial character of the migrating group while the right drawing illustrates inversion and the antarafacial character of the migrating group. I t is the former process (H = 3, A = 0) which the Odd-Even Rule predicts will he the allowed route for a 1,5shift.
+
+
- -- .
-
--
Group Transfer Reactions The concerted transfer of two groups between two molecules is topologically equivalent to a cycloaddition of four componen&. lilustrations of such reactions are shown in Figures 5 and 6. In these examples the transferring groups are hyhrogen atoms which, as in the previous examples, are always treated as suprafacial components. Cheletropic Reactions Cheletropic reactions have been defmed by Woodward and Hoffmann (12) as processes in which two sigma bonds that terminate at a single atom are made or broken in concert. 11lustrations of this type of reaction are shown in Figure 7. The molecule VUW is shown being expelled from the parent compound by both "linear" and "non-linear" routes while leaving an m-electron ?r-system in its original conformation. The sigma bonds C1-U and C,-U are broken and the pair of electrons from these bonds is considered to be localized in the atomic orbital pictured in Figure 7 as being centered on atom u. The total number of electrons to be considered is m 2
.,
950
Figure 6. The concerted transfer of two hydrogen atoms between ethane and butadlene molecules.
+
Journal of Chemical Education
(bl
-
F l a w 7. la1 b . . A "linear" dlssociatlon In which the molecule VUW remains in t YZplane. lo) A "nan-linear" dissocation in which the mlscuie VUW finishes in the XY plans. In bolh react ons the C, . . . C, palysne remains in lhe XZ plane.
from summine the ?r-electrons and the lone nair on VUW. T o apply the 0 d 2 - ~ v e nRule the number of intarafacial comoonents must be counted. I t is perhaps easiest to consider the ieverse reactions in Figure l a and 7b and note that these can be thought of as cycloaddition reactions. In the linear case (Fig. l a ) the molecule VUW is considered a suprafacial component because the sigma bonds C1-U and C,-U are formed on the same side of the plane of symmetry (the XY plane) which bisects the lone pair orbital. In the non-linear case (Fig. 7b) the C1-U bond is formed on a different face of the plane of symmetry (the YZ plane) bisecting the lone pair orbital when comnared to the C.,...-U hond. In this case then the W W molecule is acting as an antarafacial component. The antarafacial or suorafacial character of the Ca . . . C , polyene is determined by the manner in which the terminal ?r orbitals interact with the aooroachine (or departin4 VUW molecule. If the orbitals rot&; in a disiotator; manner then the polyene will be asuprafacial component while a conrota~~~~~
~~
.~ ~
Table 3.
m
H = (m
+ 2)/2
4s
2q+ 1 (odd)
4q+2
2q+ 2 (even)
Cheletrmlc Readlons Allowed (jrwnd State Reactions Llnear Reaction Nonlinear Reactlon
+ -1.
Dismtatw [2. A =0 Conrotatw [2. A=l
+ ma],
Conmtatow 12. A =2 Disrotatwy 12. A = l
+61.
+ m].
tory motion would lead to the polyene heing classified as an antarafacial comDonent. The allowed rotation modes for the polyene dependupon the number of a electrons and the manner of departure of the W W molecule. Table 3 lists the allowed proc&ses for ground state reactions. The reaction of butadiene with singlet methylene provides a simple example of a cheletropic reaction:
In this example m + 2 = 6; therefore H = 3 and either a linear, disrotatory or a non-linear, conrotatory process will he thermally allowed. Summary This paper describs a convenient procedure, the Odd-Even Rule, for predicting the allowedness or forhiddenness of ground-state, pericyclic reactions. This selection rule has been
applied to a number of specific reactions. In contrast to the Woodward-Hoffmann approach the application to each reaction is always the same. Only two, easily deduced, parameters are employed: one-half of the total number of electrons taking part in the cycle and the number of antarafacial components. If the odd or even character of these two parameters differ the reaction is predicted to be thermally allowed. Importantly, also, systems with an odd number of electrons in the cycle are explicitly considered by adding a "ghost" electron to the ring before proceeding exactly as above. Acknowledgment The authors are indebted to Sun Jia-zhong for discussing some problems related to this subject and to Stephen Young for his editorial assistance. We would also like to thank William Huntsman for helpful discussions and his careful reading of the paper.
(1) W o o d d , R . B . . a n d Hoffmaoo, R.,"ThaColusrsstionofOlbiWSymmshy.(.Acad e m i e P ~ glac., , N e w Y 0 4 1970. Vollmer, J., and Servia, K., J. CHeM.WUC., 45,214 (1966);47,491(1970). Caserio,M., J. CHEM.EDUC..48,782 (19711. Dsvar, M. J. S., Tetrahedron Suppl.,8,15 (1%). Zimmerman.H. E.. J A m r Chem. Soc., 88,1561,1566 (1966). Hkkei,E., 2.Phy~.,10,204(1931); 16,628 (19321; 83,832 (1933). Hcilbrooner, E. T&ahsdran Left.. 1923 (1964). Shen. K. W.. J. C W . Eouc., 60.238 (1973). Tang. Ao-qing, Jiang, Yuan-sheng,ct aL;'MalecularhbiWGnphThmry)I.Sdea~ Press.Baiging. 1980. (10) Woadward and Hoffmano. Ref. (l).P.BS. (11) Hoffmann, R.. Woodward, R. B..S&ncr, 167.625 (19701. (12) Woodward and Hoffmann. Ref. ( I ) , p. 152. (2) (3) (4) (5) (6) (1) (8) (9)
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