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lowed and if i j 4 n 2 they are photochemically allowed this is summarized in the following table i j thermally allowed photochemically allowed 4 n i s j a or i a j s i s j s or i a j a 4 n 2 i s j s or i a j a i s j a or i a j s cycloaddition reactions edit a general p q cycloaddition is a concerted addition reaction between two components one with p π electrons and one with q π electrons this reaction is symmetry allowed under the following conditions 17 for a supra supra or antara antara cycloaddition it is thermally allowed if p q 4 n 2 and photochemically allowed if p q 4 n for a supra antara cycloaddition it is thermally allowed if p q 4 n and photochemically allowed if p q 4 n 2 this is summarized 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 group transfer reactions edit a general double group transfer reaction which is synchronous can be represented as an interaction between a component with p π electrons and a component with q π electrons as shown generalized synchronous double group transfer reaction between a component with p π electrons and a component with q π electrons then the selection rules are the same as for the generalized cycloaddition reactions 17 that is for supra supra or antara antara double group transfers if p q 4 n 2 it is thermally allowed and if p q 4 n it is photochemically allowed for supra antara double group transfers if p q 4 n it is thermally allowed and if p q 4 n 2 it is photochemically allowed this is summarized 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 the case of q 0 corresponds to the thermal elimination of the transferred r groups there is evidence that the pyrolytic eliminations of dihydrogen and ethane from 1 4 cyclohexadiene and 3 3 6 6 tetramethyl 1 4 cyclohexadiene respectively represent examples of this type of pericyclic process the ene reaction is often classified as a type of group transfer process even though it does not involve the transfer of two σ bonded groups rather only one σ bond is transferred while a second σ bond is formed from a broken π bond as an all suprafacial process involving 6 electrons it is symmetry allowed under thermal conditions the woodward hoffmann symbol for the ene reaction is π 2 s π 2 s σ 2 s see below general formulation edit though the woodward hoffmann rules were first stated in terms of electrocyclic processes they were eventually generalized to all pericyclic reactions as the similarity and patterns in the above selection rules should indicate conrotatory motion is antarafacial while disrotatory motion is suprafacial in the generalized woodward hoffmann rules everything is characterized in terms of antarafacial and suprafacial bond topologies the terms conrotatory and disrotatory are sufficient for describing the relative sense of bond rotation in electrocyclic ring closing or opening reactions as illustrated on the right however they are unsuitable for describing the topologies of bond forming and breaking taking place in a general pericyclic reaction as described in detail below in the general formulation of the woodward hoffmann rules the bond rotation terms conrotatory and disrotatory are subsumed by the bond topology or faciality terms antarafacial and suprafacial respectively these descriptors can be used to characterize the topology of the bond forming and breaking that takes place in any pericyclic process woodward hoffmann notation edit a component is any part of a molecule or molecules that function as a unit in a pericyclic reaction a component consists of one or more atoms and any of the following types of associated orbitals an isolated p or sp x orbital unfilled or filled symbol ω a conjugated π system symbol π a σ bond symbol σ the electron count of a component is the number of electrons in the orbital s of the component the electron count of an unfilled ω orbital i e an empty p orbital is 0 while that of a filled ω orbital i e a lone pair is 2 the electron count of a conjugated π system with n double bonds is 2 n or 2 n 2 if a formal lone pair from a heteroatom or carbanion is conjugated thereto the electron count of a σ bond is 2 the bond topology of a component can be suprafacial and antarafacial the relationship is suprafacial symbol s when the interactions with the π system or p orbital occur on the same side of the nodal plane think syn for a σ bond it corresponds to interactions occurring on the two interior lobes or two exterior lobes of the bond the relationship is antarafacial symbol a when the interactions with the π system or p orbital occur on opposite sides of the nodal plane think anti for a σ bond it corresponds to interactions occurring on one interior lobe and one exterior lobe of the bond illustration of the assignment of orbital overlap as suprafacial or antarafacial for common pericyclic components using this notation all pericyclic reactions can be assigned a descriptor consisting of a series of symbols σ π ω n s a connected by signs and enclosed in brackets describing in order the type of orbital s number of electrons and bond topology involved for each component some illustrative examples follow the diels alder reaction a 4 2 cycloaddition is π 4 s π 2 s the 1 3 dipolar cycloaddition of ozone and an olefin in the first step of ozonolysis a 3 2 cycloaddition is π 4 s π 2 s the cheletropic addition of sulfur dioxide to 1 3 butadiene a 4 1 cheletropic addition is ω 0 a π 4 s ω 2 s π 4 s 27 the cope rearrangement a 3 3 sigmatropic shift is π 2 s σ 2 s π 2 s or π 2 a σ 2 s π 2 a the 1 3 alkyl migration with inversion at carbon discovered by berson a 1 3 sigmatropic shift is σ 2 a π 2 s the conrotatory electrocyclic ring closing of 1 3 butadiene a 4π electrocyclization is π 4 s the conrotatory electrocyclic ring opening of cyclobutene a reverse 4π electrocyclization is σ 2 a π 2 s or σ 2 s π 2 a the disrotatory electrocyclic ring closing of 1 3 cyclooctadien 5 ide anion a 6π electrocyclization is π 6 s a wagner meerwein shift of a carbocation a 1 2 sigmatropic shift is ω 0 s σ 2 s antarafacial and suprafacial are associated with conrotation or inversion and disrotation or retention respectively a single descriptor may correspond to two pericyclic processes that are chemically distinct that a reaction and its microscopic reverse are often described with two different descriptors and that a single process may have more than a one correct descriptor 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...
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