Normal Modes of a Linear Triatomic (e.g. CO₂)
For a symmetric linear triatomic A–B–A, mass m at ends, mass M in center, the normal coordinates decouple the 3N−5 = 4 vibrational degrees of freedom into independent harmonic oscillators.
ν₁ Symmetric Stretch
ω₁ = √(2k/m)
Both outer atoms move out/in simultaneously. IR inactive (no dipole change in CO₂).
ν₃ Antisymmetric Stretch
ω₃ = √(k(2m+M)/mM)
Outer atoms move opposite. IR active. Always ω₃ > ω₁.
ν₂ Bending
ω₂ = √(k_b/μ)
Doubly degenerate. Lowest frequency. Key for greenhouse gas absorption.