Conjugated pi systems are fascinating molecular structures with alternating single and double bonds. These systems have unique properties due to their delocalized electrons, which form molecular orbitals spanning the entire molecule.

Understanding HOMO and LUMO in conjugated systems is crucial for predicting chemical reactivity and spectroscopic behavior. The energy gap between these orbitals determines absorption and emission wavelengths, influencing a molecule's color and reactivity.

Molecular Orbitals in Conjugated Pi Systems

HOMO and LUMO in conjugated systems

• HOMO represents highest energy orbital occupied by electrons in ground state
  • Donates electrons during chemical reactions (nucleophile)
• Electrons in HOMO excited to higher energy orbitals (LUMO) in excited state
  • Absorbs light energy (UV-Vis spectroscopy)
• LUMO represents lowest energy orbital unoccupied by electrons in ground state
  • Accepts electrons during chemical reactions (electrophile)
• Electrons from HOMO or lower orbitals occupy LUMO in excited state
  • Emits light energy (fluorescence, phosphorescence)
• Energy difference between HOMO and LUMO determines absorption and emission wavelengths
  • Smaller gap absorbs and emits longer wavelengths (red, infrared)
  • Larger gap absorbs and emits shorter wavelengths (blue, UV)
• Electron configuration influences HOMO-LUMO energy gap

Nodes and energy in polyenes

• Nodes are points where molecular orbital wavefunction equals zero
  • Electron density is zero at nodes
  • More nodes indicate higher energy orbitals
• Conjugated polyenes have alternating single and double bonds (1,3-butadiene, 1,3,5-hexatriene)
  • $\pi$ orbitals overlap to form delocalized molecular orbitals spanning entire molecule
• Number of nodes determines relative energies of molecular orbitals
  • $\psi_1$ has lowest energy with no nodes (bonding)
  • $\psi_2$ has one node and higher energy than $\psi_1$ (bonding)
  • $\psi_3$ has two nodes and higher energy than $\psi_2$ (antibonding)
  • $\psi_4$ has highest energy with three nodes (antibonding)
• Delocalization stabilizes conjugated systems by distributing electron density

Symmetry in pericyclic reactions

• Pericyclic reactions proceed through cyclic transition states (Diels-Alder, Cope rearrangement)
  • Concerted bond breaking and forming
• Molecular orbital symmetry key factor in determining reaction feasibility
  • Symmetric orbitals remain unchanged upon reflection (s orbitals, $\pi$ bonds)
  • Antisymmetric orbitals change sign upon reflection (p orbitals, $\sigma$ bonds)
• Woodward-Hoffmann rules predict allowed or disallowed pericyclic reactions
  • Based on conservation of orbital symmetry during reaction
  • Allowed if HOMO and LUMO symmetries match transition state symmetry
  • Disallowed if HOMO and LUMO symmetries mismatch transition state symmetry
• Thermal pericyclic reactions involve HOMO of one reactant and LUMO of another
  1. Allowed with $(4q+2)$ $\pi$ electrons (q = 0, 1, 2...)
  2. Example: Diels-Alder with diene (4 $\pi$) and dienophile (2 $\pi$)
• Photochemical pericyclic reactions involve HOMO and LUMO of same reactant
  1. Allowed with $4q$ $\pi$ electrons (q = 1, 2, 3...)
  2. Example: Disrotatory 4 $\pi$ electrocyclic ring opening of cyclobutene
• Aromaticity influences pericyclic reaction feasibility

Molecular Orbital Theory and Conjugated Systems

• Linear combination of atomic orbitals (LCAO) forms molecular orbitals
• Bonding and antibonding orbitals result from constructive and destructive interference
• Resonance structures represent electron delocalization in conjugated systems