chemx-props — molecular properties and vibrational analysis.

8a — One-electron properties

• [dipole::dipole_moment] — electronic + nuclear dipole (a.u. and Debye).
• [population::population_analysis] — Mulliken and Löwdin atomic charges, Mayer bond orders.

8b — Harmonic frequencies

• [hessian::numerical_hessian] — central-difference Hessian from any gradient.
• [frequencies::harmonic_frequencies] — mass-weighting, Eckart projection, diagonalization → cm⁻¹.

8c — RRHO thermochemistry

• [thermo::rrho_thermochemistry] — ZPE, H, S, G at 298.15 K (ideal gas).