eos-fitting

EOS Fitting

Description

This skill covers equation-of-state (EOS) fitting for crystalline materials from DFT or ML-potential calculations: generating energy-volume datasets via fixed-volume DFT relaxations, fitting Birch-Murnaghan, Vinet, and Murnaghan EOS forms, extracting equilibrium volume V₀, ground-state energy E₀, bulk modulus B₀, and pressure derivative B'₀, quantifying fitting uncertainty via bootstrap resampling, detecting outliers and phase-transition contamination, and comparing DFT versus MLP EOS for potential validation. Invoke this skill when extracting mechanical properties of crystalline materials, validating an ML potential against DFT ground-state properties, or generating reference EOS data for a materials database.

Domain Context

The equation of state relates the total energy (or pressure) of a crystal to its volume at fixed composition and temperature. For a DFT EOS at 0 K, the energy-volume curve E(V) is the Born-Oppenheimer ground-state total energy as a function of isotropic compression and dilation of the unit cell, with atomic positions and cell shape (but not volume) optimized at each volume. The EOS encodes the two most important bulk mechanical properties: the equilibrium volume V₀ and the isothermal bulk modulus B₀ = −V (∂P/∂V)_T = V (∂²E/∂V²)_T.

Several physical and numerical constraints govern EOS accuracy:

• Pulay stress. Plane-wave DFT has an incomplete basis set at finite ENCUT. When the cell volume changes, the number and character of plane waves changes discontinuously, introducing a systematic error in the Hellmann-Feynman stress tensor. For a fixed ENCUT, the computed stress at a strained volume is systematically too small (tensile) relative to the true DFT stress. This shifts the apparent equilibrium volume. Pulay stress is reduced by increasing ENCUT to ≥ 1.3 × ENMAX (VASP) or verifying that residual stress from a static single-point is < 1 kbar after relaxation at the converged ENCUT.
• Isotropic vs. anisotropic strain. The simplest EOS applies to isotropic (hydrostatic) volume change: all lattice vectors scaled uniformly by a factor η = (V/V₀)^{1/3}. For anisotropic crystals (tetragonal, orthorhombic, monoclinic, triclinic), isotropic scaling distorts the equilibrium shape. A more accurate protocol scales the volume while simultaneously optimizing the cell shape at each volume (ISIF=4 in VASP). The resulting EOS characterizes the ground-state energy as a function of volume at the optimal shape for each volume, not at a fixed metric tensor.
• ISIF settings for fixed-volume relaxations. ISIF controls which degrees of freedom are relaxed in VASP: ISIF=2 (ions only, fixed cell), ISIF=4 (ions + cell shape, fixed volume), ISIF=3 (ions + cell shape + volume, full optimization). For EOS generation, ISIF=4 is the correct setting: it relaxes the internal structure and cell shape but keeps the volume fixed. Using ISIF=2 introduces shape-stress artifacts for non-cubic crystals. Using ISIF=3 would allow the volume to relax to equilibrium rather than to the target value.
• Volume range and density. A typical EOS fit uses 7–15 volume points spanning ±5–15% of the equilibrium volume. Too narrow a range (±3%) underdetermines the curvature and gives an unreliable B'₀. Too wide a range (> ±20%) risks entering a different structural phase or a high-pressure regime where the 3rd-order Birch-Murnaghan EOS is invalid. For soft materials (polymorphs, molecular crystals, or materials with shallow E-V curves), even ±10% may cross a transition.
• Magnetic state consistency. For magnetic materials (Fe, Ni, Co, Mn oxides, rare-earth intermetallics), the magnetic ground state may change with volume. A ferromagnetic ground state at equilibrium may become antiferromagnetic or paramagnetic at large compressions. If this transition occurs within the fitting range, the E-V curve has a kink and no single EOS form fits the entire range. The spin state must be monitored at each volume and the fit range restricted to a single-phase, single-spin-state region.