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Molecular Orbitals — Ethylene, Acetylene & Benzene

Molecular Orbital Theory — π Systems

Ethylene · Acetylene · Benzene  |  HF/STO-3G & B3LYP/3-21G  |  Undergraduate General Chemistry

Ethylene — C₂H₄

Overview

Ethylene contains a C=C double bond consisting of one σ bond and one π bond. It has 8 valence electrons occupying 4 molecular orbitals (MOs 3–8 are the valence MOs; MOs 1–2 are C 1s core).

The highest occupied MO is MO 8 (π, A″ symmetry) — the π bond formed by overlap of the two carbon pz orbitals perpendicular to the molecular plane.

The lowest unoccupied MO is MO 9 (π*, A″) — the antibonding π* orbital with a nodal plane between the carbons.

Bond Order
C=C Bond Order Calculation
Bond Order = ½ × (bonding e⁻ − antibonding e⁻)
σ (MO 3) ½ × (2 − 0) = 1
π (MO 8) ½ × (2 − 0) = 1
Total C=C bond order 2
→ Double bond: 1σ + 1π
Explore MOs Interactively
MO Energy Diagram (valence MOs, HF/STO-3G)

Ethylene — Molecular Orbital Energies

au 0.0 +0.5 −0.5 3 σ (A′) −0.977 au ↑↓ 4 σ (A′) −0.743 au ↑↓ 5 σ (A′) −0.603 au ↑↓ 6 σ (A′) −0.528 au ↑↓ 7 σ (A′) −0.463 au ↑↓ 8 π HOMO (A″) −0.324 au ↑↓ HOMO HOMO-LUMO gap 9 π* LUMO (A″) +0.319 au LUMO Energy

Acetylene — C₂H₂

Overview

Acetylene has a C≡C triple bond: one σ bond and two degenerate π bonds. With 7 occupied MOs, the two HOMOs are MO 6 and MO 7 (both π, degenerate at −0.349 au) — one in the xz plane and one in the yz plane.

These two perpendicular π bonds together form a cylindrical electron cloud around the C≡C axis, characteristic of the linear geometry.

The LUMOs are MO 8 and MO 9 (π*, degenerate at +0.397 au), the antibonding counterparts.

Bond Order
C≡C Bond Order Calculation
Bond Order = ½ × (bonding e⁻ − antibonding e⁻)
σ (MO 5) ½ × (2 − 0) = 1
π (MO 6) ½ × (2 − 0) = 1
π (MO 7) ½ × (2 − 0) = 1
Total C≡C bond order 3
→ Triple bond: 1σ + 2π
Explore MOs Interactively
MO Energy Diagram (valence MOs, HF/STO-3G)

Acetylene — Molecular Orbital Energies

0.0 Energy 3 σ (Σg) −0.954 au ↑↓ 4 σ (Σu) −0.705 au ↑↓ 5 σ (Σg) −0.607 au ↑↓ 6 π HOMO (Πu) −0.349 au ↑↓ 7 π HOMO (Πu) ↑↓ HOMO (degen.) 8,9 π* LUMO (Πg) +0.397 au LUMO (degen.)

Benzene — C₆H₆

Overview

Benzene has 21 occupied MOs (6 C 1s core + 15 valence). The six carbon pz orbitals combine to form 6 π MOs: three bonding and three antibonding.

MO 12 is a σ C–H bonding orbital delocalized over the ring.
MO 14 is a σ C–C bonding orbital. These represent the σ framework holding the ring together.

MO 17 (a2u) is the fully bonding π MO with no nodal planes through the ring — all pz lobes in phase.
MO 23 (e2u) is the π* LUMO, degenerate with MO 22 (+0.005 au), with three nodal planes through the ring.
MO 27 (e2u) is a higher π* antibonding virtual orbital with three nodal planes.

All 6 π electrons are delocalized over the entire ring, giving a π bond order of 1.5 per C–C bond, consistent with equal bond lengths.

Bond Order
C–C Bond Order (per bond)
σ framework 1 per C–C bond 1.0
π contribution 6 π e⁻ ÷ (2 × 6 bonds) 0.5
Total C–C bond order 1.5
→ Aromatic: all 6 C–C bonds equivalent
Explore MOs Interactively
Selected MO Energy Diagram (B3LYP/3-21G)

Benzene — Selected MO Energies (MOs 12, 14, 17, 20/21, 22/23, 27)

Energy 0.0 12 σ C–H −0.526 au ↑↓ 14 σ C–C −0.454 au ↑↓ 17 π (a2u) −0.368 au ↑↓ 20,21 π HOMO (e1g) −0.252 au ↑↓ ↑↓ HOMO HOMO-LUMO gap 22,23 π* LUMO (e2u) +0.005 au LUMO 27 π* (e2u) +0.166 au 6 π electrons · aromatic (4n+2, n=1)

Comparison — Ethylene, Acetylene & Benzene

Summary Table
Property Ethylene
C₂H₄		 Acetylene
C₂H₂		 Benzene
C₆H₆
Level of theory HF/STO-3G HF/STO-3G B3LYP/3-21G
Total occupied MOs 8 7 21
Number of π bonds 1 2 (degenerate) 3 (delocalized)
π electrons 2 4 6
HOMO symmetry A″ (π) Πᵤ (π, ×2) e₁g (π, ×2)
HOMO energy −0.324 au −0.349 au −0.252 au (MO 20/21)
LUMO symmetry A″ (π*) Πg (π*, ×2) e₂u (π*, ×2) — MO 22/23
LUMO energy +0.319 au +0.397 au +0.005 au
HOMO–LUMO gap 0.643 au (17.5 eV) 0.746 au (20.3 eV) 0.258 au (7.0 eV) ★
C–C bond order 2.0 (double) 3.0 (triple) 1.5 (aromatic)
Geometry Planar, sp² Linear, sp Planar, sp²
Aromaticity No No Yes (4n+2, n=1)
Interactive model Open ↗ Open ↗ Open ↗
★ Benzene's small HOMO–LUMO gap (relative to the other two) reflects its aromatic stabilization and explains its characteristic UV absorption and reactivity patterns. au = atomic units (Hartrees); 1 au ≈ 27.21 eV.
Level of Theory — HF/STO-3G vs B3LYP/3-21G
Aspect HF/STO-3G
(ethylene, acetylene)		 B3LYP/3-21G
(benzene)
Method type Ab initio (Hartree-Fock) DFT (Density Functional Theory)
Electron correlation ✗ Ignored ✓ Included (approximately)
Basis set flexibility Minimal — 1 function per orbital Split-valence — 2 functions per valence orbital
Bond length accuracy Usually too short Close to experiment
Orbital energies Systematically too negative More physically meaningful
Computational cost Very fast Moderate
MO shapes & ordering Both give correct qualitative MO topology — suitable for understanding bonding, nodes, and symmetry
⚠ Because ethylene/acetylene use HF/STO-3G and benzene uses B3LYP/3-21G, their orbital energies cannot be directly compared. For a fair numerical comparison all molecules should be calculated at the same level of theory.