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ptual density functional theory dft 9 33 the key to the analysis is the dual descriptor function proposed by christophe morell andré grand and alejandro toro labbé 34 f 2 r 2 ρ r n 2 displaystyle f 2 r frac partial 2 rho r partial n 2 the second derivative of the electron density ρ r displaystyle rho r with respect to the number of electrons n displaystyle n this response function is important as the reaction of two components a and b involving a transfer of electrons will depend on the responsiveness of the electron density to electron donation or acceptance i e the derivative of the fukui function f r ρ r n displaystyle f r frac partial rho r partial n in fact from a simplistic viewpoint the dual descriptor function gives a readout on the electrophilicity or nucleophilicity of the various regions of the molecule for f 2 0 displaystyle f 2 0 the region is electrophilic and for f 2 0 displaystyle f 2 0 the region is nucleophilic using the frontier molecular orbital assumption and a finite difference approximation of the fukui function one may write the dual descriptor as f 2 r f r n ϕ lumo r 2 ϕ homo r 2 displaystyle f 2 r frac partial f r partial n cong phi _ text lumo r 2 phi _ text homo r 2 this makes intuitive sense as if a region is better at accepting electrons than donating then the lumo must dominate and dual descriptor function will be positive conversely if a region is better at donating electrons then the homo term will dominate and the descriptor will be negative notice that although the concept of phase and orbitals are replaced simply by the notion of electron density this function still takes both positive and negative values dual descriptor coloring red 0 blue 0 of electron density in the diels alder supra supra transition state the woodward hoffmann rules are reinterpreted using this formulation by matching favorable interactions between regions of electron density for which the dual descriptor has opposite signs this is equivalent to maximizing predicted favorable interactions and minimizing repulsive interactions for the case of a 4 2 cycloaddition a simplified schematic of the reactants with the dual descriptor function colored red positive blue negative is shown in the optimal supra supra configuration to the left this method correctly predicts the wh rules for the major classes of pericyclic reactions exceptions edit in chapter 12 of the conservation of orbital symmetry entitled violations woodward and hoffmann famously stated there are none nor can violations be expected of so fundamental a principle of maximum bonding this pronouncement notwithstanding it is important to recognize that the woodward hoffmann rules are used to predict relative barrier heights and thus likely reaction mechanisms and that they only take into account barriers due to conservation of orbital symmetry thus it is not guaranteed that a wh symmetry allowed reaction actually takes place in a facile manner conversely it is possible upon enough energetic input to achieve an anti woodward hoffmann product this is especially prevalent in sterically constrained systems where the wh product has an added steric barrier to overcome for example in the electrocyclic ring opening of the dimethylbicyclo 0 2 3 heptene derivative 1 a conrotatory mechanism is not possible due to resulting angle strain and the reaction proceeds slowly through a disrotatory mechanism at 400 o c to give a cycloheptadiene product 2 violations may also be observed in cases with very strong thermodynamic driving forces the decomposition of dioxetane 1 2 dione to two molecules of carbon dioxide famous for its role in the luminescence of glowsticks has been scrutinized computationally in the absence of fluorescers the reaction is now believed to proceed in a concerted though asynchronous fashion via a retro 2 2 cycloaddition that formally violates the woodward hoffmann rules 35 anti wh product via disrotatory mechanism induced by ring strain computationally predicted products of 4e electrocyclic ring opening under thermal photo and mechanical control similarly a recent paper describes how mechanical stress can be used to reshape chemical reaction pathways to lead to products that apparently violate woodward hoffman rules 36 in this paper they use ultrasound irradiation to induce a mechanical stress on link functionalized polymers attached syn or anti on the cyclobutene ring computational studies predict that the mechanical force resulting from friction of the polymers induces bond lengthening along the reaction coordinate of the conrotatory mechanism in the anti bisubstituted cyclobutene and along the reaction coordinate of the disrotatory mechanism in the syn bisubstituted cyclobutene thus in the syn bisubstituted cyclobutene the anti wh product is predicted to be formed this computational prediction was backed up by experiment on the system below link functionalized polymers were conjugated to cis benzocyclobutene in both syn and anti conformations as predicted both products gave the same z z product as determined by quenching by a stereospecific diels alder reaction with the substituted maleimide in particular the syn substituted product gave the anti wh product presumably as the mechanical stretching along the coordinate of the disrotatory pathway lowered the barrier of the reaction under the disrotatory pathway enough to bias that mechanism see also edit woodward s rules for calculating uv absorptions torquoselectivity notes edit references edit havinga e schlatmann j l m a january 1961 remarks on the specificities of the photochemical and thermal transformations in the vitamin d field tetrahedron 16 1 4 146 152 doi 10 1016 0040 4020 61 80065 3 1 2 3 woodward r b hoffmann roald 1965 stereochemistry of electrocyclic reactions journal of the american chemical society 87 2 395 bibcode 1965jachs 87 395w doi 10 1021 ja01080a054 1 2 zimmerman h e 1966 on molecular orbital correlation diagrams the occurrence of möbius systems in cyclization reactions and factors controlling ground and excited state reactions i journal of the american chemical society 88 7 1564 1565 bibcode 1966jachs 88 1564z doi 10 1021 ja00959a052 zimmerman howard e 1971 08 01 moebius hueckel concept in organic chemistry application of organic molecules and reactions accounts of chemical research 4 8 272 280 doi 10 1021 ar50044a002 issn 0001 4842 dewar m j s 1966 01 01 a molecular orbital theory of organic chemistry viii romaticity and electrocyclic reactions tetrahedron 22 75 92 doi 10 1016 s0040 4020 01 82171 2 issn 0040 4020 dewar michael j s 1971 11 01 aromaticity and pericyclic reactions angewandte chemie international edition in english 10 11 761 776 doi 10 1002 anie 197107611 issn 1521 3773 fukui kenichi 1965 01 01 stereoselectivity associated with noncycloaddition to unsaturated bonds tetrahedron letters 6 28 2427 2432 doi 10 1016 s0040 4039 00 90203 x issn 0040 4039 fukui kenichi 1982 role of frontier orbitals in chemical reactions science 218 4574 747 754 bibcode 1982sci 218 747f doi 10 1126 science 218 4574 747 jstor 1689733 pmid 17771019 1 2 geerlings paul ayers paul w toro labbé alejandro chattaraj pratim k de proft frank 2012 the woodward hoffmann rules reinterpreted by conceptual density functional theory accounts of chemical research 45 5 683 95 doi 10 1021 ar200192t hdl 10533 131820 pmid 22283422 the nobel prize in chemistry 1981 nobelprize org corey e j 2004 impossible dreams the journal of organic chemistry 69 9 2917 9 doi 10 1021 jo049925d pmid 15104426 johnson carolyn y march 1 2005 whose idea was it boston globe 1 2 hoffmann roald 2004 a claim on the development of the frontier orbital explanation of electrocyclic reactions angewandte chemie international edition 43 48 6586 90 doi 10 1002 anie 200461440 pmid 15558636 corey e j hortmann alfred g 1963 total synthesis of dihydrocostunolide journal of the american chemical society 85 24 4033 bibcode 1963jachs 85 4033c doi 10 1021 ja00907a030 corey e j hortmann alfred g 1965 the total synthesis of dihydrocostunolide journal of the american chemical society 87 24 5736 42 bibcode 1965jachs 87 5736c doi 10 1021 ja00952a037 pmid 5845424 the original statement given by woodward and hoffmann is somewhat more elaborate a ground state pericyclic change is symmetry allowed when the total number of 4q 2 s and 4r a components is odd however the statement given here is mathematically equivalent for a proof see https ia601607 us archive org 11 items woodhoff2025 woodhoff2025 pdf 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 woodward r b hoffmann roald 1969 the conservation of orbital symmetry angew chem int ed 8 11 781 853 doi 10 1002 anie 196907811 criegee rudolf noll klaus 1959 umsetzungen in der reihe des 1 2 3 4 tetramethyl cyclobutans justus liebigs annalen der chemie 627 1 14 doi 10 1002 jlac 19596270102 although the figure below shows both ends rotating clockwise it is important to recognize that orbital symmetry only distinguishes between rotation in the same direction or opposing directions conrotation vs disrotation the formation of the z z isomer from ring opening of 1 both ends rotating counterclockwise is therefore also symmetry allowed that the z z isomer was not observed is likely due to unfavorable steric interactions between the inwardly rotating methyl groups in other cases the preference for clockwise or counterclockwise bond rotation may be controlled by stereoelectronic factors see torquoselectivity srinivasan r 1968 07 01 mechanism of the photochemical valence tautomerization of 1 3 butadienes journal of the american chemical society 90 16 4498 4499 bibcode 1968jachs 90 4498s doi 10 1021 ja01018a080 issn 0002 7863 winter rudolph ernst k 1965 01 01 the preparation and isomerization of cis and trans 3 4 dimethylcyclobutene tetrahedron letters 6 17 1207 1212 doi 10 1016 s0040 4039 01 83997 6 issn 0040 4039 longuet higgins h c abrahamson e w 1965 the electronic mechanism of electrocyclic reactions journal of the american chemical society 87 9 2045 doi 10 1021 ja01087a033 woodward r b hoffmann roald 1971 the conservation of orbital symmetry 3rd printing 1st ed weinheim brd verlag chemie gmbh brd and academic press usa pp 1 178 isbn 978 1483256153 p r bunker and p jensen 2005 fundamentals of molecular symmetry crc press isbn 0 7503 0941 5 see section 10 4 1 2 3 4 5 6 7 8 9 10 11 carroll felix 1998 perspectives on structure and mechanism in organic chemistry brooks cole pp 710 794 isbn 0534249485 1 2 3 hoffmann roald woodward r b 1965 selection rules for concerted cycloaddition reactions j am chem soc 87 9 2046 bibcode 1965jachs 87 2046h doi 10 1021 ja01087a034 because so 2 has orthogonal lone pair and antibonding orbitals the allyl like antibonding orbital of so 2 is analyzed here as a simple unoccupied p orbital the interaction of these orbitals with those of 1 3 butadiene must be analyzed separately in general for a composite process like this cheletropic reaction both of the separate orbital interactions have to allowed for the reaction to be allowed the fmo approach probably gives the most intuitive picture here 1 2 it is possible to produce a dewar zimmerman analysis with at most one phase inversion and that the outcome depends on the parity of the number of antarafacial components the process involves shading in the first orbital of any component arbitrarily and completing the shading of the rest of the component so that no phase inversion takes place within it one draws in the dewar zimmerman connections between components based on the bond topology supra or antara of each component using these connections it is possible to continue the shading onto neighboring components such that no inversion is present between components or within a component this is continued until all interacting orbitals are shaded in and only a final dewar zimmerman connection needs to be made to complete the cycle no phase inversion has been introduced up to this point crucial observation the dewar zimmerman connections extending from the two ends of an antarafacial suprafacial component to the neighboring components will connect lobes of opposite the same shading hence an odd number of antarafacial components will force a single phase inversion in order to complete the cycle while an even number of antarafacial components will allow a cycle to be completed with no phase inversion since the number of inversions modulo 2 is invariant with respect to orbital phasing these two possibilities an odd or an even number of antarafacial components fix whether a möbius one inversion or hückel no inversion system applies respectively yamabe shinichi kuwata kayoko minato tsutomu 1999 frontier orbital analyses of ketene 2 2 cycloadditions theoretical chemistry accounts theory computation and modeling 102 1 6 139 146 doi 10 1007 s002140050484 s2cid 206899145 berson jerome a nelson george l 1967 10 01 inversion of configuration in the migrating group of a thermal 1 3 sigmatropic rearrangement journal of the american chemical society 89 21 5503 5504 bibcode 1967jachs 89 5503b doi 10 1021 ja00997a065 issn 0002 7863 zimmerman h e 1966 on molecular orbital correlation diagrams möbius systems and factors controlling ground and excited state reactions ii j am chem soc 88 7 1566 156 bibcode 1966jachs 88 1566z doi 10 1021 ja00959a053 zimmerman h e 2006 five decades of mechanistic and exploratory organic photochemistry pure appl chem 78 12 2193 2203 doi 10 1351 pac200678122193 s2cid 37436155 ayers paul w morell christophe de proft frank geerlings paul 5 october 2007 understanding the woodward hoffmann rules by using changes in electron density chemistry a european journal 13 29 8240 8247 doi 10 1002 chem 200700365 pmid 17639522 morell christophe grand andré toro labbé alejandro 1 january 2005 new dual descriptor for chemical reactivity the journal of physical chemistry a 109 1 205 212 bibcode 2005jpca 109 205m doi 10 1021 jp046577a hdl 10533 176692 pmid 16839107 farahani pooria baader wilhelm j 2017 02 16 unimolecular decomposition mechanism of 1 2 dioxetanedione concerted or biradical that is the question the journal of physical chemistry a 121 6 1189 1194 bibcode 2017jpca 121 1189f doi 10 1021 acs jpca 6b10365 issn 1089 5639 pmid 28094939 hickenboth charles r moore jeffrey s white scott r sottos nancy r baudry jerome wilson scott r 2007 biasing reaction pathways with mechanical force nature 446 7134 423 7 bibcode 2007natur 446 423h doi 10 1038 nature05681 pmid 17377579 s2cid 4427747 journal articles understanding the woodward hoffmann rules by using changes in electron density 1 v t e topics in organic reactions addition reaction elimination reaction polymerization reagents rearrangement reaction redox reaction regioselectivity stereoselectivity stereospecificity substitution reaction a value alpha effect annulene anomeric effect antiaromaticity aromatic ring current aromaticity baird s rule baker nathan effect baldwin s rules bema 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