3.3 Equation of state: fit a physically resolved minimum

These are unexecuted teaching inputs and starting models. Original diagrams are schematics, not calculated results. Validate version-specific syntax, licensed or authorized data, numerical convergence and the scientific model before using this workflow.

3.3.1 Model, units and provenance

PW cutoffs and energies use Ry, common force output uses Ry/bohr, and pressure uses kbar. Geometry cards state their coordinate units. Different executables have distinct grammars and time-unit conventions.

Shared inputs, conventions and evidence

Original schematic: Equation of state: fit a physically resolved minimum. No numerical results are claimed.

3.3.2 Unexecuted inputs and explicit deltas

Use the accompanying instructions to identify the parent calculation and placement of every delta; a snippet is not automatically a standalone input. Preserve all blank-line and file-provenance requirements.

3.3.2.1 Input block 1

# Independent Si-base SCFs, not a vc-relax trajectory:
volume_ratio = 0.94, 0.96, 0.98, 1.00, 1.02, 1.04, 1.06
A_each = A_reference * volume_ratio**(1/3)
V_primitive = A_each**3 / 4
# Store: V_Angstrom3, E_Ry_per_primitive_cell, pressure_kbar

3.3.3 Worked investigation

3.3.3.1 Intuition and prerequisites

An equation-of-state fit turns an energy–volume curve into equilibrium volume and curvature. Its apparent precision can greatly exceed the numerical quality of its input. Use the equilibrium neighborhood from 3.2, a fixed functional and pseudopotential, and cutoffs already checked for stress. For diamond Si, isotropic scaling preserves the ideal internal geometry.

3.3.3.2 Original scan definition

See input block 1 above.

3.3.3.3 Checks, pitfalls and exercise

Compare −dE/dV with computed pressure after aligning sign and units. A fit minimum outside the sampled interval is a request for more data, not a reliable extrapolation. Refit after removing the outermost points and compare parameter stability. Never feed energies per atom and volumes per cell into the same fit. Exercise: deliberately add a small synthetic noise vector to a copy of your computed data, clearly label it as synthetic, and study how curvature is less robust than the minimum position.