Orbital Transformations in Diels-Alder Reaction

The Diels–Alder reaction is a [4+2]-cycloaddition of a conjugated diene (such as butadiene) and a dienophile (such as ethylene):
This is classified as an electrocyclic reaction, which involves the four -electrons of the diene and the two -electrons of the dieneophile. It is a concerted reaction, with no transition state, in which electrons are transferred usually from the highest occupied molecular orbital (HOMO) of the diene into the lowest unoccupied molecular orbital (LUMO) of the dienophile.
The "orbitals" graphic shows the configurations of atomic orbitals in the -electron states of the diene and the dienophile. The blue lobes of the -orbitals correspond to positive values of the wavefunction; the yellow lobes correspond to negative values (or vice versa; only the relative signs are significant). The "energy levels" graphic shows the ground electronic states of the two molecules. In the diene, the two lowest -orbitals, and , are doubly occupied, each with a pair of electrons with opposite spin. The orbital is the HOMO. In the dienophile, the lowest -orbital is doubly occupied, while the empty is the LUMO.
With "reaction" selected, the slider "reaction parameter" traces out the course of the reaction. The four adjacent -orbitals on the two reactant molecules transform into two new -bonds, which are energetically more stable than -bonds. Thus a cyclic product molecule (cyclohexene) is formed.

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The significance of frontier orbitals, HOMOs and LUMOs, in the reactivity of aromatic hydrocarbons was first recognized by Kenichi Fukui in 1952. Frontier orbitals are analogous to valence electrons for atoms.
The Diels–Alder reaction conforms to the conservation of orbital symmetry, a set of principles proposed by Woodward and Hoffmann in 1965. In a concerted reaction, the molecular orbitals of the starting materials are transformed into the molecular orbitals of the products in a smooth, continuous way. Stereospecificity of the substituents on all carbon centers is preserved, which is important in the synthesis of complex molecules, particularly natural products.
References
[1] Wikipedia. "Diels–Alder Reaction." (Sep 7, 2018) en.wikipedia.org/wiki/Diels%E2%80%93Alder_reaction.
[2] Wikipedia. "Frontier Molecular Orbital Theory." (Sep 7, 2018) en.wikipedia.org/wiki/Frontier_molecular_orbital _theory.
[3] Wikipedia. "Woodward–Hoffmann Rules." (Sep 7, 2018) en.wikipedia.org/wiki/Woodward%E2%80%93Hoffmann_rules.
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