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axis and antisymmetric with respect to the σ mirror plane the same analysis can be carried out for the molecular orbitals of cyclobutene the result of both symmetry operations on each of the mos is shown to the left as the σ and σ orbitals lie entirely in the plane containing c 2 perpendicular to σ they are uniformly symmetric and antisymmetric respectively to both symmetry elements on the other hand π is symmetric with respect to reflection and antisymmetric with respect to rotation while π is antisymmetric with respect to reflection and symmetric with respect to rotation correlation lines are drawn to connect molecular orbitals in the starting material and the product that have the same symmetry with respect to the conserved symmetry element in the case of the conrotatory 4 electron electrocyclic ring closure of 1 3 butadiene the lowest molecular orbital ψ 1 is asymmetric a with respect to the c 2 axis so this molecular orbital is correlated with the π orbital of cyclobutene the lowest energy orbital that is also a with respect to the c 2 axis similarly ψ 2 which is symmetric s with respect to the c 2 axis is correlated with σ of cyclobutene the final two correlations are between the antisymmetric a molecular orbitals ψ 3 and σ and the symmetric s molecular orbitals ψ 4 and π 17 4 electron electrocyclization reaction correlation diagram with a conrotatory mechanism similarly there exists a correlation diagram for a disrotatory mechanism in this mechanism the symmetry element that persists throughout the entire mechanism is the σ mirror plane of reflection here the lowest energy mo ψ 1 of 1 3 butadiene is symmetric with respect to the reflection plane and as such correlates with the symmetric σ mo of cyclobutene similarly the higher energy pair of symmetric molecular orbitals ψ 3 and π correlate as for the asymmetric molecular orbitals the lower energy pair ψ 2 and π form a correlation pair as do ψ 4 and σ 17 4 electron electrocyclization reaction correlation diagram with a disrotatory mechanism evaluating the two mechanisms the conrotatory mechanism is predicted to have a lower barrier because it transforms the electrons from ground state orbitals of the reactants ψ 1 and ψ 2 into ground state orbitals of the product σ and π conversely the disrotatory mechanism forces the conversion of the ψ 1 orbital into the σ orbital and the ψ 2 orbital into the π orbital thus the two electrons in the ground state ψ 2 orbital are transferred to an excited antibonding orbital creating a doubly excited electronic state of the cyclobutene this would lead to a significantly higher transition state barrier to reaction 17 first excited state es 1 of butadiene however as reactions do not take place between disjointed molecular orbitals but electronic states the final analysis involves state correlation diagrams a state correlation diagram correlates the overall symmetry of electronic states in the starting material and product the ground state of 1 3 butadiene as shown above has 2 electrons in ψ 1 and 2 electrons in ψ 2 so it is represented as ψ 1 2 ψ 2 2 the overall symmetry of the state is the product of the symmetries of each filled orbital with multiplicity for doubly populated orbitals thus as ψ 1 is asymmetric with respect to the c 2 axis and ψ 2 is symmetric the total state is represented by a 2 s 2 to see why this particular product is mathematically overall s that s can be represented as 1 and a as 1 this derives from the fact that signs of the lobes of the p orbitals are multiplied by 1 if they are symmetric with respect to a symmetry transformation i e unaltered and multiplied by 1 if they are antisymmetric with respect to a symmetry transformation i e inverted thus a 2 s 2 1 2 1 2 1 s the first excited state es 1 is formed from promoting an electron from the homo to the lumo and thus is represented as ψ 1 2 ψ 2 ψ 3 as ψ 1 is a ψ 2 is s and ψ 3 is a the symmetry of this state is given by a 2 sa a now considering the electronic states of the product cyclobutene the ground state is given by σ 2 π 2 which has symmetry s 2 a 2 s the first excited state es 1 is again formed from a promotion of an electron from the homo to the lumo so in this case it is represented as σ 2 ππ the symmetry of this state is s 2 as a the ground state ψ 1 2 ψ 2 2 of 1 3 butadiene correlates with the ground state σ 2 π 2 of cyclobutene as demonstrated in the mo correlation diagram above ψ 1 correlates with π and ψ 2 correlates with σ thus the orbitals making up ψ 1 2 ψ 2 2 must transform into the orbitals making up σ 2 π 2 under a conrotatory mechanism however the state es 1 does not correlate with the state es 1 as the molecular orbitals do not transform into each other under the symmetry requirement seen in the molecular orbital correlation diagram instead as ψ 1 correlates with π ψ 2 correlates with σ and ψ 3 correlates with σ the state ψ 1 2 ψ 2 ψ 3 attempts to transform into π 2 σσ which is a different excited state so es 1 attempts to correlate with es 2 σπ 2 σ which is higher in energy than es 1 similarly es 1 σ 2 ππ attempts to correlate with es 2 ψ 1 ψ 2 2 ψ 4 these correlations can not actually take place due to the quantum mechanical rule known as the avoided crossing rule this says that energetic configurations of the same symmetry can not cross on an energy level correlation diagram in short this is caused by mixing of states of the same symmetry when brought close enough in energy so instead a high energetic barrier is formed between a forced transformation of es 1 into es 1 in the diagram below the symmetry preferred correlations are shown in dashed lines and the bold curved lines indicate the actual correlation with the high energetic barrier 17 25 4 electron electrocyclization state correlation diagram with a conrotatory mechanism the same analysis can be applied to the disrotatory mechanism to create the following state correlation diagram 17 25 4 electron electrocyclization state correlation diagram under disrotatory mechanism thus if the molecule is in the ground state it will proceed through the conrotatory mechanism i e under thermal control to avoid an electronic barrier however if the molecule is in the first excited state i e under photochemical control the electronic barrier is present in the conrotatory mechanism and the reaction will proceed through the disrotatory mechanism these are not completely distinct as both the conrotatory and disrotatory mechanisms lie on the same potential surface thus a more correct statement is that as a ground state molecule explores the potential energy surface it is more likely to achieve the activation barrier to undergo a conrotatory mechanism 25 cycloaddition reactions edit the woodward hoffmann rules can also explain bimolecular cycloaddition reactions through correlation diagrams 26 a π p π q cycloaddition brings together two components one with p π electrons and the other with q π electrons cycloaddition reactions are further characterized as suprafacial s or antarafacial a with respect to each of the π components see below general formulation for a detailed description of the generalization of wh notation to all pericyclic processes 2 2 cycloadditions edit for ordinary alkenes 2 2 cycloadditions are only observed under photochemical activation the rationale for the non observation of thermal 2 2 cycloadditions begins with the analysis of the four possible stereochemical consequences for the 2 2 cycloaddition π 2 s π 2 s π 2 a π 2 s π 2 s π 2 a π 2 a π 2 a the geometrically most plausible π 2 s π 2 s mode is forbidden under thermal conditions while the π 2 a π 2 s π 2 s π 2 a approaches are allowed from the point of view of symmetry but are rare due to an unfavorable strain and steric profile 17 the 2 s 2 s cycloaddition retains stereochemistry symmetry elements of the 2 2 cycloaddition considering the π 2 s π 2 s cycloaddition this mechanism leads to a retention of stereochemistry in the product as illustrated to the right two symmetry elements are present in the starting materials transition state and product σ 1 and σ 2 σ 1 is the mirror plane between the components perpendicular to the p orbitals σ 2 splits the molecules in half perpendicular to the σ bonds 26 these are both local symmetry elements in the case that the components are not identical to determine symmetry and asymmetry with respect to σ 1 and σ 2 the starting material molecular orbitals must be considered in tandem the figure to the right shows the molecular orbital correlation diagram for the π 2 s π 2 s cycloaddition the two π and π molecular orbitals of the starting materials are characterized by their symmetry with respect to first σ 1 and then σ 2 similarly the σ and σ molecular orbitals of the product are characterized by their symmetry in the correlation diagram molecular orbitals transformations over the course of the reaction must conserve the symmetry of the molecular orbitals thus π ss correlates with σ ss π as correlates with σ as π sa correlates with σ sa and finally π aa correlates with σ aa due to conservation of orbital symmetry the bonding orbital π as is forced to correlate with the antibonding orbital σ as thus a high barrier is predicted 17 25 26 this is made precise in the state correlation diagram below 17 25 the ground state in the starting materials is the electronic state where π ss and π as are both doubly populated i e the state ss 2 as 2 as such this state attempts to correlate with the electronic state in the product where both σ ss and σ as are doubly populated i e the state ss 2 as 2 however this state is neither the ground state ss 2 sa 2 of cyclobutane nor the first excited state es 1 ss 2 sa as where an electron is promoted from the homo to the lumo 4 2 cycloadditions edit the mirror plane is the only conserved symmetry element of the diels alder 4 2 cycloaddition a 4 2 cycloaddition is exemplified by the diels alder reaction the simplest case is the reaction of 1 3 butadiene with ethylene to form cyclohexene one symmetry element is conserved in this transformation the mirror plane through the center of the reactants as shown to the left the molecular orbitals of the reactants are the set ψ 1 ψ 2 ψ 3 ψ 4 of molecular orbitals of 1 3 butadiene shown above along with π and π of ethylene ψ 1 is symmetric ψ 2 is antisymmetric ψ 3 is symmetric and ψ 4 is antisymmetric with respect to the mirror plane similarly π is symmetric and π is antisymmetric with respect to the mirror plane the molecular orbitals of the product are the symmetric and antisymmetric combinations of the two newly formed σ and σ bonds and the π and π bonds as shown below correlating the pairs of orbitals in the starting materials and product of the same symmetry and increasing energy gives the correlation diagram to the right as this transforms the ground state bonding molecular orbitals of the starting materials into the ground state bonding orbitals of the product in a symmetry conservative manner this is predicted to not have the great energetic barrier present in the ground state 2 2 reaction above to make the analysis precise one can construct the state correlation diagram for the general 4 2 cycloaddition 25 as before the ground state is the electronic state depicted in the molecular orbital correlation diagram to the right this can be described as ψ 1 2 π 2 ψ 2 2 of total symmetry s 2 s 2 a 2 s this correlates with the ground state of the cyclohexene σ s σ a π 2 which is also s 2 s 2 a 2 s as such this ground state reaction is not predicted to have a high symmetry imposed barrier one can also construct the excited state correlations as is done above here there is a high energetic barrier to a photo induced diels alder reaction under a suprafacial suprafacial bond topology due to the avoided crossing shown below group transfer reactions edit transfer of a pair of hydrogen atoms from ethane to perdeuterioethylene the symmetry imposed barrier heights of group transfer reactions can also be analyzed using correlation diagrams a model reaction is the transfer of a pair of hydrogen atoms from ethane to perdeuterioethylene shown to the right the only conserved symmetry element in this reaction is the mirror plane through the center of the molecules as shown to the left conserved mirror plane in transfer reaction the molecular orbitals of the system are constructed as symmetric and antisymmetric combinations of σ and σ c h bonds in ethane and π and π bonds in the deutero substituted ethene thus the lowest energy mo is the symmetric sum of the two c h σ bond σ s followed by the antisymmetric sum σ a the two highest energy mos are formed from linear combinations of the σ ch antibonds highest is the antisymmetric σ a preceded by the symmetric σ a at a slightly lower energy in the middle of the energetic scale are the two remaining mos that are the π cc and π cc of ethene the full molecular orbital correlation diagram is constructed in by matching pairs of symmetric and asymmetric mos of increasing total energy as explained above as can be seen in the adjacent diagram as the bonding orbitals of the reactants exactly correlate with the bonding orbitals of the products this reaction is not predicted to have a high electronic symmetry imposed barrier 17 25 selection rules edit using correlation diagrams one can derive selection rules for the following generalized classes of pericyclic reactions each of these particular classes is further generalized in the generalized woodward hoffmann rules the more inclusive bond topology descriptors antarafacial and suprafacial subsume the terms conrotatory and disrotatory respectively antarafacial refers to bond making or breaking through the opposite face of a π system p orbital or σ bond while suprafacial refers to the process occurring through the same face a suprafacial transformation at a chiral center preserves stereochemistry whereas an antarafacial transformation reverses stereochemistry electrocyclic reactions edit the selection rule of electrocyclization reactions is given in the original statement of the woodward hoffmann rules if a generalized electrocyclic ring closure occurs in a polyene of 4 n π electrons then it is conrotatory under thermal conditions and disrotatory under photochemical conditions conversely in a polyene of 4 n 2 π electrons an electrocyclic ring closure is disrotatory under thermal conditions and conrotatory under photochemical conditions this result can either be derived via an fmo analysis based upon the sign of p orbital lobes of the homo of the polyene or with correlation diagrams taking first the first possibility in the ground state if a polyene has 4 n electrons the outer p orbitals of the homo that form the σ bond in the electrocyclized product are of opposite signs thus a constructive overlap is only produced under a conrotatory or antarafacial process conversely for a polyene with 4 n 2 electrons the outer p orbitals of the ground state homo ar...
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