Organic Chemistry · Orbitals and Organic Chemistry: Pericyclic Reactions
Molecular Orbitals of Conjugated Pi Systems
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A conjugated π system A chain of alternating double/single bonds whose p orbitals all overlap Full entry → is a chain of adjacent atoms in which every atom contributes one p orbital that overlaps with its neighbors. In 1,3-butadiene (SMILES C=CC=C), four sp²-hybridized carbons form a continuous line of overlapping p orbitals; in 1,3,5-hexatriene (C=CC=CC=C), six do. Because the electrons are no longer confined to individual double bonds, they are delocalized over the whole chain, which lowers the molecule's energy and changes its spectroscopy.
molecular orbital (MO) A wave function spanning several atoms, built from individual p orbitals Full entry → theory describes these delocalized electrons as wave functions built by combining the individual p orbitals. A chain of N p orbitals produces exactly N π MOs, each characterized by its number of nodes — planes where the wave function changes sign. More nodes mean higher energy, so the MOs form a predictable ladder: one with 0 nodes (all p orbitals in phase, most stable), one with 1 node A plane where the wave function changes sign Full entry →, and so on up to N − 1 nodes.
Two orbitals matter most. The HOMO Highest occupied molecular orbital (ground state) Full entry → (highest occupied molecular orbital) controls how the molecule reacts in its ground (thermal) state, and the LUMO Lowest unoccupied molecular orbital Full entry → (lowest unoccupied molecular orbital) controls its photochemical behavior. This chapter's subject — pericyclic reactions — follows from the symmetries of these two orbitals.
Why this matters
- UV–visible spectroscopy: the HOMO–LUMO gap shrinks as conjugation lengthens, so absorption shifts from the ultraviolet (butadiene, λmax ≈ 217 nm) toward the visible. That is why conjugated molecules are colored and why vision depends on retinal's conjugated chain.
- Pericyclic reactions: the Woodward–Hoffmann rules in this chapter are statements about the symmetry of the HOMO (thermal) or LUMO (photochemical). Without a feel for these orbitals, electrocyclic reactions, cycloadditions, and sigmatropic rearrangements seem like arbitrary rules.
- Photochemistry: which orbital an electron is promoted into determines which reaction path is allowed — the same molecule can react differently in light and in the dark.
The college version
Core Concepts
From p orbitals to molecular orbitals
Take 1,3-butadiene's four p orbitals, labeled C1 through C4 along the chain. Combining them gives four MOs, ψ1 through ψ4:
- ψ1 (0 nodes): every adjacent pair of p orbitals overlaps in phase — bonding everywhere, the most stable MO.
- ψ2 (1 node): the node passes through the central C2–C3 bond, so the C1–C2 and C3–C4 bonds are bonding while C2–C3 is antibonding.
- ψ3 (2 nodes): nodes pass through the two terminal bonds (C1–C2 and C3–C4), leaving the central bond bonding.
- ψ4 (3 nodes): every adjacent pair is out of phase — antibonding everywhere, highest in energy.
Each additional node raises the energy, so the order is always ψ1 < ψ2 < ψ3 < ψ4. The same recipe extends to any chain: hexatriene has six MOs with 0 through 5 nodes.
Filling the ladder: HOMO and LUMO
Electrons fill MOs from lowest energy upward, two per orbital (Aufbau principle Electrons fill MOs from lowest energy first, two per orbital Full entry →, Pauli exclusion). Butadiene has four π electrons, so ψ1 and ψ2 are filled: ψ2 is the HOMO and ψ3 is the LUMO. Hexatriene has six π electrons, filling ψ1–ψ3: its HOMO is ψ3 and its LUMO is ψ4.
Absorption of light promotes an electron from the HOMO to the LUMO. In the resulting excited state, the "frontier" electron now sits in the orbital that was formerly the LUMO — so photochemical reactions are controlled by an orbital whose symmetry is different from the ground-state HOMO. That single fact explains why thermal and photochemical pericyclic reactions give different products.
Symmetry: the property that governs reactions
Relative to the mirror plane that bisects the chain and lies perpendicular to the molecular plane, every π MO is either symmetric (S) — the two halves of the wave function have the same sign — or antisymmetric (A). For butadiene: ψ1 S, ψ2 A, ψ3 S, ψ4 A. For hexatriene: ψ1 S, ψ2 A, ψ3 S, ψ4 A, ψ5 S, ψ6 A. The labels alternate, S, A, S, A, ... with increasing energy.
The pattern is what matters: butadiene's HOMO (ψ2) is A, while hexatriene's HOMO (ψ3) is S. When the chain ends rotate to form a new σ bond (an electrocyclic reaction, Topic 2), bonding overlap needs conrotation for an A HOMO and disrotation for an S HOMO. Memorize the alternating S/A ladder and the chapter's rules become derivable.
Allyl cation, radical, and anion
The allyl system (three carbons, C=CC) has three MOs: ψ1 (bonding, 0 nodes), ψ2 (nonbonding, 1 node at the central carbon — its energy equals that of an isolated p orbital), and ψ3 (antibonding, 2 nodes). The electron count decides which are occupied:
- Allyl cation (2 π electrons): ψ1 only — stabilized relative to a simple alkene.
- Allyl radical (3 π electrons): ψ1 filled, ψ2 half-filled (the HOMO is the singly occupied ψ2).
- Allyl anion (4 π electrons): ψ1 and ψ2 filled — the HOMO is the nonbonding ψ2, which is why allylic anions are stable yet nucleophilic.
How It Works / Step-by-Step Process
- Count the p orbitals (N) in the conjugated chain.
- Generate N MOs with 0, 1, 2, ... up to N − 1 nodes, in order of increasing energy.
- Fill with the chain's π electrons, two per MO, lowest first.
- Identify the HOMO and LUMO, then assign S/A symmetry (alternating pattern).
- Use the HOMO symmetry for thermal questions and the LUMO symmetry for photochemical questions.
Common Confusions
| Do not confuse | With | Difference |
|---|---|---|
| A node | An antibonding orbital | ψ2 of butadiene has one node yet is still net bonding; antibonding character is about overall destabilization |
| The HOMO is the highest-energy MO | The HOMO is the highest occupied MO | The LUMO is higher in energy but empty; photochemistry promotes an electron into it |
| Longer chain = more stable orbitals | Longer chain = more MOs and a smaller gap | Adding atoms raises the HOMO and lowers the LUMO; the gap shrinks, so absorption shifts to longer λ |
| A π MO is localized on one double bond | π MOs span the entire chain | All π MOs of a conjugated system are delocalized; that is the point of conjugation |
| The model wavelength is exact | The model shows trends | Particle-in-a-box overestimates λ; compare 323 vs 217 nm (butadiene) |
Eli explains
The same idea, in plain words
Explain it like I’m 10
Picture a row of friends holding hands in a line. If everyone holds hands the same way, the line is calm and relaxed — that's the lowest-energy wave. If a few neighbors push their hands against each other instead of holding, that's a "node," and the line is tenser — higher energy. Electrons fill the calmest arrangements first. The most important electron is the one in the highest filled arrangement (the HOMO): its pattern decides how the whole molecule reacts.
Worked example
Example 1: HOMO–LUMO gap of 1,3-butadiene from the particle-in-a-box model
The free-electron model treats the π electrons as particles in a one-dimensional box spanning the conjugated chain. The allowed energies are
En = n2 h28 me L2
where n = 1, 2, 3, … is the quantum number, h = 6.626 × 10-34 J·s is Planck's constant, me = 9.109 × 10-31 kg is the electron mass, and L is the box length. For a polyene with N conjugated carbons the standard convention extends the box half a bond beyond each end: L = (N+1) × 1.40 Å. For butadiene, L = 5 × 1.40 Å = 7.00 Å = 7.00 × 10-10 m.
Butadiene's 4 π electrons fill n = 1 and n = 2, so HOMO n = 2 and LUMO n = 3. The transition energy is
ΔE = E3 - E2 = (32 - 22)h28 me L2 = 5h28 me L2
Substituting:
ΔE = 5(6.626 × 10-34 J·s)28(9.109 × 10-31 kg)(7.00 × 10-10 m)2 = 5(4.390 × 10-67)3.571 × 10-48 = 6.15 × 10-19 J
Unit check: J²·s² / (kg·m²) = J, since 1 J = 1 kg·m²·s⁻². Convert to wavelength with E = hc/λ, where c = 2.998 × 108 m/s:
λ= hcΔE = (6.626 × 10-34 J·s)(2.998 × 108 m/s)6.15 × 10-19 J = 3.23 × 10-7 m = 323 nm
The model predicts ~323 nm, but butadiene absorbs at λmax ≈ 217 nm: the box model overestimates the wavelength because it ignores bond-length alternation, the σ framework, and electron–electron repulsion. Use it for trends, not exact numbers.
Example 2: Why longer chains absorb longer wavelengths
Repeat for 1,3,5-hexatriene: 6 π electrons fill n = 1, 2, 3; HOMO n = 3, LUMO n = 4; box length L = 7 × 1.40 Å = 9.80 × 10-10 m.
ΔE = (42 - 32)h28 me L2 = 7h28 me L2 = 7(4.390 × 10-67 J2s2)8(9.109 × 10-31 kg)(9.80 × 10-10 m)2 = 4.39 × 10-19 J
λ= hcΔE = 1.986 × 10-25 J·m4.39 × 10-19 J = 4.52 × 10-7 m = 452 nm
Model: 323 → 452 nm from butadiene to hexatriene; observed: 217 → 258 nm. The trend is right even though absolute values are off: longer conjugation shrinks the gap and red-shifts absorption.
Example 3: Electron count in the allyl anion
The allyl anion has 4 π electrons over 3 MOs: ψ1 filled (2 e⁻), ψ2 filled (2 e⁻), ψ3 empty. Its HOMO is ψ2, the nonbonding orbital — the extra electron pair sits at approximately isolated-p-orbital energy, stable enough to form readily yet still able to donate electrons as a nucleophile. The cation (2 e⁻) has HOMO = ψ1; the radical (3 e⁻) has a singly occupied ψ2.
Key takeaways
- N conjugated p orbitals → N π MOs; energy rises with node count.
- Fill MOs from lowest up: butadiene HOMO = ψ2 (1 node), hexatriene HOMO = ψ3 (2 nodes).
- Symmetry alternates S, A, S, A with energy; HOMO symmetry dictates thermal reactivity, LUMO symmetry dictates photochemical reactivity.
- Conjugation narrows the HOMO–LUMO gap: butadiene λmax ≈ 217 nm, hexatriene λmax ≈ 258 nm; longer chains absorb longer wavelengths.
- Allyl systems: ψ2 is nonbonding; cation (2 e⁻), radical (3 e⁻), anion (4 e⁻).
Check yourself
5 review questions from the chapter. Try each one, then open the answer.
How many π MOs does 1,3,5-hexatriene have, and how many nodes are in its HOMO?
Show answer
Six π MOs (one per p orbital); the HOMO is ψ3 with 2 nodes.
Which orbital of 1,3-butadiene is the HOMO, and is it symmetric (S) or antisymmetric (A)?
Show answer
ψ2, with 1 node; it is antisymmetric (A) with respect to the central mirror plane.
Why does the HOMO–LUMO gap shrink as conjugation lengthens?
Show answer
The number of MOs increases and they pack closer together, so the gap between the highest filled and lowest empty level narrows.
How many π electrons does the allyl cation have, and which MO is its HOMO?
Show answer
Two π electrons; the HOMO is ψ1 (the only bonding MO).
Why do photochemical reactions follow the symmetry of the LUMO rather than the HOMO?
Show answer
Absorption of light promotes an electron out of the HOMO into the LUMO, so the frontier electron now occupies the former LUMO; its symmetry, not the ground-state HOMO's, controls the reaction.
Study tools & related lessonsKey vocabulary · Related
Key vocabulary
- conjugated π system
- A chain of alternating double/single bonds whose p orbitals all overlap
- molecular orbital (MO)
- A wave function spanning several atoms, built from individual p orbitals
- node
- A plane where the wave function changes sign
- HOMO
- Highest occupied molecular orbital (ground state)
- LUMO
- Lowest unoccupied molecular orbital
- Aufbau principle
- Electrons fill MOs from lowest energy first, two per orbital
- nonbonding orbital
- An MO with energy ≈ an isolated p orbital (e.g., allyl ψ2)
Sources & references
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