30.1 Molecular Orbitals of Conjugated Pi Systems

Molecular Orbitals of Conjugated Pi Systems

Molecular orbital (MO) theory explains how conjugated pi systems behave during pericyclic reactions. By understanding how p orbitals combine across a conjugated framework, you can predict which reactions are thermally allowed, which require light, and why. This topic lays the groundwork for every pericyclic reaction you'll encounter in this unit.

Molecular Orbitals in Conjugated Pi Systems

Building Molecular Orbitals: LCAO

Molecular orbitals in conjugated systems are built using linear combination of atomic orbitals (LCAO). Each carbon in the conjugated system contributes one p orbital, and these p orbitals combine to produce the same number of molecular orbitals.

• Constructive overlap (p orbitals in phase) creates bonding MOs, which are lower in energy than the original atomic orbitals.
• Destructive overlap (p orbitals out of phase) creates antibonding MOs, which are higher in energy.

For a system with n contributing p orbitals, you get n molecular orbitals. Half (or roughly half) are bonding, and the rest are antibonding. Electrons fill these MOs from lowest energy to highest, just like atomic orbitals.

Nodes and Energy in Polyenes

A node is a point where the orbital wavefunction passes through zero, meaning electron density is zero there. The number of nodes directly determines an orbital's energy: more nodes = higher energy.

Take 1,3-butadiene as an example. It has four p orbitals combining to give four molecular orbitals:

• $\psi_1$: 0 nodes, lowest energy (bonding)
• $\psi_2$: 1 node, next lowest (bonding)
• $\psi_3$: 2 nodes, higher energy (antibonding)
• $\psi_4$: 3 nodes, highest energy (antibonding)

For 1,3,5-hexatriene (six p orbitals), you get six MOs: $\psi_1$ through $\psi_6$, with 0 through 5 nodes respectively. The pattern always holds: the lowest MO has zero nodes, and each successive MO adds one more.

Delocalization across the conjugated system stabilizes these molecules. Spreading electron density over multiple atoms lowers the overall energy compared to isolated double bonds.

HOMO and LUMO

Two specific molecular orbitals control reactivity:

• HOMO (Highest Occupied Molecular Orbital): the highest-energy MO that contains electrons in the ground state. It acts as the electron donor in reactions.
• LUMO (Lowest Unoccupied Molecular Orbital): the lowest-energy MO with no electrons in the ground state. It acts as the electron acceptor.

For 1,3-butadiene (4 pi electrons filling $\psi_1$ and $\psi_2$), the HOMO is $\psi_2$ and the LUMO is $\psi_3$. For 1,3,5-hexatriene (6 pi electrons filling $\psi_1$, $\psi_2$, and $\psi_3$), the HOMO is $\psi_3$ and the LUMO is $\psi_4$.

The HOMO-LUMO gap determines spectroscopic and photochemical behavior:

• A smaller gap means the molecule absorbs longer-wavelength (lower-energy) light.
• A larger gap means it absorbs shorter-wavelength (higher-energy) light.

As conjugation length increases, the HOMO rises in energy and the LUMO drops, shrinking the gap. That's why longer polyenes absorb visible light and appear colored, while shorter ones absorb only in the UV.

When a molecule absorbs light, an electron is promoted from the HOMO to the LUMO. This excited state has a different HOMO than the ground state, which is critical for photochemical pericyclic reactions.

Symmetry in Pericyclic Reactions

Pericyclic reactions are concerted, meaning all bond-breaking and bond-forming happens simultaneously through a cyclic transition state. Whether a given pericyclic reaction is allowed depends on the symmetry of the frontier molecular orbitals involved.

Orbital symmetry properties:

Each MO is classified as symmetric (S) or antisymmetric (A) with respect to a relevant symmetry element (a mirror plane or a $C_2$ rotation axis, depending on the reaction type).

• A symmetric orbital looks the same after the symmetry operation.
• An antisymmetric orbital changes sign (lobes swap phase) after the operation.

The symmetry of the HOMO and LUMO at the terminal carbons is what matters most, because that's where new bonds form.

The Woodward-Hoffmann rules use conservation of orbital symmetry to predict whether a pericyclic reaction is allowed or forbidden:

Thermal vs. photochemical selection rules:

1. Thermal reactions use the ground-state HOMO. They are allowed when the system has $(4q+2)$ pi electrons (where $q = 0, 1, 2...$) for suprafacial processes. Example: the Diels-Alder reaction, with 4 pi electrons from the diene + 2 from the dienophile = 6 total $(q = 1)$.

2. Photochemical reactions use the excited-state HOMO (which is the ground-state LUMO). They are allowed when the system has $4q$ pi electrons for suprafacial processes. Example: the photochemical disrotatory electrocyclic ring closure of butadiene (4 pi electrons, $q = 1$).

These complementary rules mean that a reaction forbidden under heat is often allowed under light, and vice versa. The underlying reason is that promoting an electron to the LUMO changes the symmetry of the highest occupied orbital, flipping which pathway has constructive overlap in the transition state.