Q1(m). Zero-point energy in joule and in cm⁻¹

Answer in brief

The energy of the lowest vibrational level (v = 0), kept even at 0 K: E₀ = ½hν J, ε₀ = ½ν̄ cm⁻¹.

Key points

• Zero-point energy: the energy of the ground vibrational level above the potential minimum; it cannot be removed, even at 0 K.
• In joule: E₀ = ½hν = (h/4π)√(k/μ), with ν = (1/2π)√(k/μ), k = force constant, μ = reduced mass.
• In cm⁻¹ (divide by hc): ε₀ = ½ν̄ = (1/4πc)√(k/μ); anharmonic: G(0) = ½ω̄e − ¼ω̄exe.
• E = 0 would fix the atoms at rest at re (Δx = 0 and Δp = 0), which the uncertainty principle forbids.
• HCl: G(0) = 1482 cm⁻¹ = 17.7 kJ mol⁻¹; hence D₀ = De − E₀.