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ptor one can verify using the pericyclic selection rule given below that all of these reactions are allowed processes original statement edit using this notation woodward and hoffmann state in their 1969 review the general formulation for all pericyclic reactions as follows a ground state pericyclic change is symmetry allowed when the total number of 4q 2 s and 4r a components is odd 17 here 4 q 2 s and 4 r a refer to suprafacial 4 q 2 electron and antarafacial 4 r electron components respectively moreover this criterion should be interpreted as both sufficient stated above as well as necessary not explicitly stated above see if and only if derivation of an alternative statement edit alternatively the general statement can be formulated in terms of the total number of electrons using simple rules of divisibility by a straightforward analysis of two cases first consider the case where the total number of electrons is 4 n 2 4 n 2 a 4 q 2 s b 4 p 2 a c 4 t s d 4 r a where a b c and d are coefficients indicating the number of each type of component this equation implies that one of but not both a or b is odd for if a and b are both even or both odd then the sum of the four terms is 0 mod 4 the generalized statement of the woodward hoffmann rules states that a d is odd if the reaction is allowed now if a is even then this implies that d is odd since b is odd in this case the number of antarafacial components b d is even likewise if a is odd then d is even since b even in this case the number of antarafacial components b d is again even thus regardless of the initial assumption of parity for a and b the number of antarafacial components is even when the electron count is 4 n 2 contrariwise b d is odd in the case where the total number of electrons is 4 n similar arguments omitted here lead to the conclusion that the number of antarafacial components b d must be odd in the allowed case and even in the forbidden case finally to complete the argument and show that this new criterion is truly equivalent to the original criterion one needs to argue the converse statements as well namely that the number of antarafacial components b d and the electron count 4 n 2 or 4 n implies the parity of a d that is given by the woodward hoffmann rules odd for allowed even for forbidden another round of somewhat tedious case analyses will easily show this to be the case the pericyclic selection rule states a pericyclic process involving 4n 2 or 4n electrons is thermally allowed if and only if the number of antarafacial components involved is even or odd respectively summary of the results of the equivalent dewar zimmerman aromatic transition state theory hückel möbius 4 n 2 e allowed aromatic forbidden anti aromatic 4 n e forbidden anti aromatic allowed aromatic in this formulation the electron count refers to the entire reacting system rather than to individual components as enumerated in woodward and hoffmann s original statement in practice an even or odd number of antarafacial components usually means zero or one antarafacial components respectively as transition states involving two or more antarafacial components are typically disfavored by strain as exceptions certain intramolecular reactions may be geometrically constrained in such a way that enforces an antarafacial trajectory for multiple components in addition in some cases e g the cope rearrangement the same not necessarily strained transition state geometry can be considered to contain two supra or two antara π components depending on how one draws the connections between orbital lobes this ambiguity is a consequence of the convention that overlap of either both interior or both exterior lobes of a σ component can be considered to be suprafacial this alternative formulation makes the equivalence of the woodward hoffmann rules to the dewar zimmerman analysis see below clear an even total number of phase inversions is equivalent to an even number of antarafacial components and corresponds to hückel topology requiring 4 n 2 electrons for aromaticity while an odd total number of phase inversions is equivalent to an odd number of antarafacial components and corresponds to möbius topology requiring 4 n electrons for aromaticity 28 to summarize aromatic transition state theory thermal pericyclic reactions proceed via 4 n 2 electron hückel or 4 n electron möbius transition states as a mnemonic the above formulation can be further restated as the following a ground state pericyclic process involving n electron pairs and a antarafacial components is symmetry allowed if and only if n a is odd alternative proof of equivalence edit the equivalence of the two formulations can also be seen by a simple parity argument without appeal to case analysis proposition the following formulations of the woodward hoffmann rules are equivalent a for a pericyclic reaction if the sum of the number of suprafacial 4q 2 components and antarafacial 4r components is odd then it is thermally allowed otherwise the reaction is thermally forbidden b for a pericyclic reaction if the total number of antarafacial components of a 4n 2 electron reaction is even or the total number of antarafacial components of a 4n electron reaction is odd then it is thermally allowed otherwise the reaction is thermally forbidden proof of equivalence index the components of a k component pericyclic reaction i 1 2 k displaystyle i 1 2 ldots k and assign component i with woodward hoffmann symbol σ π ω n s a the electron count and topology parity symbol n i p i i displaystyle n_ i p_ i i according to the following rules n i 0 n 0 m o d 4 1 n 2 m o d 4 a n d p i 0 i is supra 1 i is antara displaystyle n_ i begin cases 0 n equiv 0 mathrm mod 4 1 n equiv 2 mathrm mod 4 end cases quad mathrm and quad p_ i begin cases 0 i text is supra 1 i text is antara end cases we have a mathematically equivalent restatement of a a a collection of symbols n i p i i displaystyle n_ i p_ i i is thermally allowed if and only if the number of symbols with the property n i p i displaystyle n_ i neq p_ i is odd since the total electron count is 4 n 2 or 4 n precisely when i n i textstyle sum _ i n_ i the number of 4 q 2 electron components is odd or even respectively while i p i textstyle sum _ i p_ i gives the number of antarafacial components we can also restate b b a collection of symbols n i p i i displaystyle n_ i p_ i i is thermally allowed if and only if exactly one of i n i textstyle sum _ i n_ i or i p i textstyle sum _ i p_ i is odd it suffices to show that a and b are equivalent exactly one of i n i textstyle sum _ i n_ i or i p i textstyle sum _ i p_ i is odd if and only if i n i i p i i n i p i textstyle sum _ i n_ i sum _ i p_ i sum _ i n_ i p_ i is odd if n i p i displaystyle n_ i p_ i n i p i 0 m o d 2 displaystyle n_ i p_ i equiv 0 mathrm mod 2 holds hence omission of symbols with the property n i p i displaystyle n_ i p_ i from a collection will not change the parity of i n i p i textstyle sum _ i n_ i p_ i on the other hand when n i p i displaystyle n_ i neq p_ i we have n i p i 1 displaystyle n_ i p_ i 1 but n i p i 1 textstyle sum _ n_ i neq p_ i 1 simply enumerates the number of components with the property n i p i displaystyle n_ i neq p_ i therefore i n i p i n i p i n i p i n i p i 1 n i p i i n i p i m o d 2 displaystyle sum _ i n_ i p_ i equiv sum _ n_ i neq p_ i n_ i p_ i sum _ n_ i neq p_ i 1 n_ i p_ i i n_ i neq p_ i mathrm mod 2 thus i n i p i textstyle sum _ i n_ i p_ i and the number of symbols in a collection with the property n i p i displaystyle n_ i neq p_ i have the same parity since formulations a and b are equivalent so are a and b as claimed to give a concrete example a hypothetical reaction with the descriptor π 6 s π 4 a π 2 a would be assigned the collection 1 0 1 0 1 2 1 1 3 in the scheme above there are two components 1 0 1 and 0 1 2 with the property n i p i displaystyle n_ i neq p_ i so the reaction is not allowed by a likewise i n i 2 textstyle sum _ i n_ i 2 and i p i 2 textstyle sum _ i p_ i 2 are both even so b yields the same conclusion as it must the reaction is not allowed examples edit this formulation for a 2 component reaction is equivalent to the selection rules for a p q cycloaddition reactions shown in the following table p q thermally allowed photochemically allowed 4 n p s q a or p a q s p s q s or p a q a 4 n 2 p s q s or p a q a p s q a or p a q s if the total number of electrons is 4 n 2 then one is in the bottom row of the table the reaction is thermally allowed if it is suprafacial with respect to both components or antarafacial with respect to both components that is to say the number of antarafacial components is even it is 0 or 2 similarly if the total number of electrons is 4 n then one is in the top row of the table this is thermally allowed if it is suprafacial with respect to one component and antarafacial with respect to the other thus the total number of antarafacial components is always odd as it is always 1 the following are some common ground state i e thermal reaction classes analyzed in light of the generalized woodward hoffmann rules 2 2 cycloaddition edit a thermally allowed supra antara 2 2 dimerization of a strained trans olefin a 2 2 cycloaddition is a 4 electron process that brings together two components thus by the above general wh rules it is only allowed if the reaction is antarafacial with respect to exactly one component this is the same conclusion reached with correlation diagrams in the section above a rare but stereochemically unambiguous example of a π 2 s π 2 a cycloaddition is shown on the right the strain and steric properties of the trans double bond enables this generally kinetically unfavorable process cis trans 1 5 cyclooctadiene is also believed to undergo dimerization via this mode 17 ketenes are a large class of reactants favoring 2 2 cycloaddition with olefins the mo analysis of ketene cycloaddition is rendered complicated and ambiguous by the simultaneous but independent interaction of the orthogonal orbitals of the ketene but may involve a π 2 s π 2 a interaction as well 29 4 2 cycloaddition edit the synchronous 6π electron diels alder reaction is a π 4 s π 2 s cycloaddition i e suprafacial with respect to both components as exemplified by the reaction to the right the diels alder reaction is suprafacial with respect to both components thus as the total number of antarafacial components is 0 which is even the reaction is symmetry allowed 17 this prediction agrees with experiment as the diels alder reaction is a rather facile pericyclic reaction 4 n electrocyclic reaction edit a 4 n electron electrocyclic ring opening reaction can be considered to have 2 components the π system and the breaking σ bond with respect to the π system the reaction is suprafacial however with a conrotatory mechanism as shown in the figure above the reaction is antarafacial with respect to the σ bond conversely with a disrotatory mechanism it is suprafacial with respect to the breaking σ bond by the above rules for a 4 n electron pericyclic reaction of 2 components there must be one antarafacial component thus the reaction must proceed through a conrotatory mechanism 17 this agrees with the result derived in the correlation diagrams above 4 n 2 electrocyclic reaction edit a 4 n 2 electrocyclic ring opening reaction is also a 2 component pericyclic reaction which is suprafacial with respect to the π system thus in order for the reaction to be allowed the number of antarafacial components must be 0 i e it must be suprafacial with respect to the breaking σ bond as well thus a disrotatory mechanism is symmetry allowed 17 1 j sigmatropic rearrangement edit berson s classic 1967 example of a 1 3 sigmatropic alkyl shift proceeding with stereochemical inversion wh symbol σ 2 a π 2 s a 1 j sigmatropic rearrangement is also a two component pericyclic reaction one component is the π system the other component is the migrating group the simplest case is a 1 j hydride shift across a π system where j is odd in this case as the hydrogen has only a spherically symmetric s orbital the reaction must be suprafacial with respect to the hydrogen the total number of electrons involved is j 1 as there are j 1 2 π bond plus the σ bond involved in the reaction if j 4 n 1 then it must be antarafacial and if j 4 n 1 then it must be suprafacial 17 this agrees with experiment that 1 3 hydride shifts are generally not observed as the symmetry allowed antarafacial process is not feasible but 1 5 hydride shifts are quite facile for a 1 j alkyl shift where the reaction can be antarafacial i e invert stereochemistry with respect to the carbon center the same rules apply if j 4 n 1 then the reaction is symmetry allowed if it is either antarafacial with respect to the π system or inverts stereochemistry at the carbon if j 4 n 1 then the reaction is symmetry allowed if it is suprafacial with respect to the π system and retains stereochemistry at the carbon center 17 on the right is one of the first examples of a 1 3 sigmatropic shift to be discovered reported by berson in 1967 30 in order to allow for inversion of configuration as the σ bond breaks the c h d moiety twists around at the transition state with the hybridization of the carbon approximating sp 2 so that the remaining unhybridized p orbital maintains overlap with both carbons 1 and 3 equivalence of other theoretical models edit dewar zimmerman analysis edit main article möbius hückel concept hypothetical huckel versus mobius aromaticity the generalized woodward hoffmann rules first given in 1969 are equivalent to an earlier general approach the möbius hückel concept of zimmerman which was first stated in 1966 and is also known as aromatic transition state theory 3 31 32 as its central tenet aromatic transition state theory holds that allowed pericyclic reactions proceed via transition states with aromatic character while forbidden pericyclic reactions would encounter transition states that are antiaromatic in nature in the dewar zimmerman analysis one is concerned with the topology of the transition state of the pericyclic reaction if the transition state involves 4 n electrons the möbius topology is aromatic and the hückel topology is antiaromatic while if the transition state involves 4 n 2 electrons the hückel topology is aromatic and the möbius topology is antiaromatic the parity of the number of phase inversions described in detail below in the transition state determines its topology a möbius topology involves an odd number of phase inversions whereas a hückel topology involves an even number of phase inversions examples of dewar zimmerman analysis applied to common pericyclic reactions the red curves represent phase inversions in connection with woodward hoffmann terminology the number of antarafacial components and the number of phase inversions always have the same parity 28 consequently an odd number of antarafacial components gives möbius t...
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