Vibrational Free Energy
Vibrational free energy is the contribution of lattice vibrations (phonons) to the free energy of a solid. In alloys it can differ between competing structures and shift phase boundaries and transition temperatures, but computing full phonon spectra for many configurations is expensive, so approximate models such as transferable bond-length-dependent force constants are often used.
Stiffness versus bond length for Cu-Pd fcc and B2 (2017)
Geng, Boes and Kitchin estimated vibrational free energies for Cu-Pd from nearest-neighbor force constants fitted as functions of bond length on a few ordered fcc and bcc structures, with a quasi-harmonic 1% volume expansion. Stretching stiffness fell with bond length for Cu-Cu, Cu-Pd and Pd-Pd bonds, while bending terms were small and nearly constant and were neglected. The ordered bcc structures had lower vibrational free energy than ordered fcc structures between 25 and 50 at% Pd, which stabilized the B2 phase and widened its region of the computed phase diagram toward the experimental one, while lowering the order-disorder temperature by about 20 K (Geng et al. 2017). Summary: geng2017.
Vibrations, phonons and thermochemistry in ASE (2017)
ASE computes molecular vibrations from finite-difference Hessians (6N+1 force evaluations), correcting the egg-box effect of real-space grids by imposing momentum conservation, and computes phonons by the same displacement approach in a supercell (Hjorth Larsen et al. 2017). Its thermochemistry module converts these into free energies in three limits: HarmonicThermo for adsorbates treated as independent harmonic modes, IdealGasThermo with translational, rotational and vibrational terms for gases, and CrystalThermo integrating a phonon density of states for solids. The paper cautions that users must judge whether these approximations hold, giving the adsorption free energy as an ideal-gas Gibbs energy minus a harmonic Helmholtz energy as an example. Summary: hjorthlarsen2017.
ML Gibbs corrections and translational entropy of adsorbates (2025)
Using machine-learned Hessians, the harmonic Gibbs correction at 300 K for OC20 adsorbates was predicted with an MAE of 0.042 eV after a constant bias correction (0.031 eV after fine-tuning), better than reusing a per-adsorbate correction computed on one surface (0.058 eV) and slightly better than a table of mean per-adsorbate frequencies (0.045 eV). Complete potential energy sampling with the same potentials showed the harmonic approximation misses on average 0.16 eV (up to 0.27 eV) of translational entropy contribution at 300 K, with 94% of 100 sampled systems above 0.1 eV (Wander et al. 2025). Summary: wander2025.