3.4 Build a controlled equation of state
These educational templates have not been executed and contain no fabricated numerical results. Inputs and filenames follow ABACUS 3.9.0 documentation; recheck every interface when changing version.
3.4.1 Question and declared model
Can a small set of fixed-volume energies independently locate silicon's equilibrium volume and quantify its resistance to compression? Build a volume series using the accepted first PW silicon model. The target is a zero-temperature electronic equation of state for that Hamiltonian. Thermal expansion, zero-point motion and experimental bulk moduli are distinct corrections or comparisons.
3.4.2 Physical interpretation
Scale every lattice vector by s=(V/V_ref)^(1/3) while retaining fractional coordinates. A volume change of 6 percent is not a lattice-length change of 6 percent. For diamond silicon, symmetry fixes the ideal fractional sites within the chosen family. For a less symmetric material, relax internal coordinates at each fixed volume and label the curve as relaxed-ion rather than clamped-ion.
The curvature requires more stringent consistency than locating a broad minimum. A smooth fit can hide an unconverged point, and too wide a range can include a different structural or magnetic branch. A local quadratic model is a useful diagnostic near the minimum; a physically motivated equation of state can be considered afterwards with its fitted parameter uncertainty and fit-window sensitivity.
Original course illustration; conceptual geometry and curves, not calculated results.
3.4.3 Worked input and file changes
# Same converged INPUT at every volume; replace matching keys.
calculation scf
cal_force 1
cal_stress 1
scf_thr 1e-9
suffix eos_v100
# At each point write a distinct STRU and suffix.
# V/Vref: 0.94 0.96 0.98 1.00 1.02 1.04 1.06
# Scale LATTICE_VECTORS by (V/Vref)**(1/3).
# Keep Direct fractional atom positions unchanged.
# Keep identical PP checksum, accepted cutoff and KPT mesh.
This delta uses the full parent INPUT/STRU/KPT, not invented pseudopotentials. Use first PW silicon and its converged controls. Create seven separate directories and never overwrite the only copy of the reference structure. Compute V from the determinant of the physical lattice matrix; record Angstrom cubed and the atom count. If the parent uses Bohr for its lattice scale, convert before combining volume with energies in eV. A two-atom primitive energy and volume may both be divided by two to obtain per-atom quantities; dividing only energy changes the fitted modulus incorrectly. Keep the mesh topology constant and test its adequacy at the smallest and largest volumes.
3.4.4 Run and inspect the evidence
For every directory, inspect OUT.
3.4.5 Observable-specific convergence
Repeat the fit with five central points and all seven points. A proposed target is less than 0.1 percent change in V0 and less than 2 percent in B0 between acceptable fit windows and a stricter numerical setup; choose a tighter budget if the intended comparison is smaller. Check residuals against the SCF and basis noise floor, not against how attractive the curve looks. A point that fails SCF is excluded with a reason and rerun, not silently filled by interpolation. Verify that the fit minimum agrees with the independently relaxed cell within the combined uncertainty. Disagreement prompts checks of volume units, atom normalization, cutoff stress and coordinate constraints.
3.4.6 Acceptance worksheet and provenance
Construct the extraction sheet before fitting. Each row needs volume, atom count, final energy, stress-derived pressure, SCF status and source-log location. For this two-atom primitive cell use either total-cell energy with total-cell volume or per-atom energy with per-atom volume. Keep one convention throughout. Subtract a constant energy only for numerical conditioning or plotting, saving the offset; it changes neither pressure nor curvature.
For a local quadratic fit use the declared volume center Vc to reduce coefficient correlation. Check that the fitted curvature is positive, that V0 lies inside the sample range and that residuals do not follow a systematic trend. Estimate slope-based pressure at each volume and compare with the sign-corrected stress quantity. Large disagreement can identify a bad volume conversion before it appears as an implausible modulus. Repeat the calculation of B0 using both normalization conventions as an algebraic self-check: a physical modulus must be invariant when energy and volume are divided by the same atom count.
| V/Vref / physical V | Converged E (eV) | Pressure (kbar) | Fit residual / status |
|---|---|---|---|
| 0.94, 0.96, 0.98 | — | — | — |
| 1.00 | — | — | — |
| 1.02, 1.04, 1.06 | — | — | — |
The grouped blank rows are a compact plan; expand to one row per actual calculation. Archive raw data and fitted-window definitions, not just a curve image.
3.4.7 Pitfalls and limits
A seven-point curve is not evidence of convergence by itself. Changing the cutoff between volumes destroys cancellation and can create artificial curvature. Mixing energies from different pseudopotential valence spaces is invalid. A cubic-cell lattice parameter is not the volume of the primitive cell. Do not estimate pressure using differences of rounded plotted energies. A fitted negative modulus or minimum outside the interval is a diagnostic failure, not a new material property.
3.4.8 Exercises with answer guidance
- You divide energies by two but retain primitive volumes. What happens to the fitted B0? Guidance: its curvature is halved while volume is unchanged, so the modulus is spuriously halved; normalize energy and volume together.
- A higher cutoff moves every energy by almost the same constant. Does B0 necessarily change? Guidance: a constant shift has zero derivatives, but establish that the shift is constant within the curvature error budget rather than assuming it.
3.4.9 Versioned references
The university-authored guide provides teaching context; older pages can predate 3.9. Use the linked official version for exact syntax. This is original instructional synthesis, and reference outputs are not presented as results of this course.