A concrete RealQM-based statistical-mechanics calculation.

Step 1. Compute the H₂ ground-state energy E(R) at several internuclear separations R using RealQM (via sweep_h2_adaptive.html or repeated runs of mol_fast_h2.html).
Step 2. Fit a parabola near the minimum: E(R) ≈ E₀ + ½k(R−R₀)². Extract k = E″(R₀) and the harmonic frequency ω = √(k/μ), with reduced mass μ = m_p/2 = 918.08 m_e (atomic units).
Step 3. Compute the canonical vibrational partition function and thermodynamics:

Z_vib(T) = e^(−βℏω/2) / (1 − e^(−βℏω))
U_vib(T) = (ℏω/2) coth(βℏω/2)
C_vib(T) = k_B (βℏω/2)² / sinh²(βℏω/2)

where β = 1/(k_B T). All formulas exact for a harmonic oscillator.

Input: E(R) data points

Defaults are KW exact reference values. Replace with RealQM-computed E(R) to see how the fit and thermodynamics shift.

R (a.u.)	E (Ha) — KW	E (Ha) — RealQM (optional)
1.2	-1.16493
1.3	-1.17234
1.4	-1.17448
1.5	-1.17285
1.6	-1.16858

Use KW (default) Use RealQM column

Fit results

R₀ (eq. bond length)	— a.u.
E(R₀) minimum	— Ha
k = E″(R₀)	— Ha/a.u.²
ω = √(k/μ)	— Ha
ω (cm⁻¹)	— cm⁻¹
Experimental ω	4401 cm⁻¹
ZPE = ℏω/2	— Ha

Thermodynamics vs T

T (K)	U_vib (Ha)	U_vib (kJ/mol)	C_vib (k_B)