Thermoelastic scientific validation

Validation scope

The thermoelastic validation matrix separates four questions that must not be collapsed into one numerical comparison:

  1. Can the CRYSTAL interface recover the original PRESSURE value, final structure, energy, volume, density and Wallace stiffness tensor?

  2. Does the cold finite-strain implementation reproduce an independently assembled linear least-squares problem?

  3. Does the quasi-static reconstruction preserve historical Quantas results when the same tensor frame and inputs are used?

  4. Do the adiabatic tensors satisfy the independent thermodynamic identity, tensor symmetry, positive-semidefinite rank-one correction and exact zero-temperature limit?

Two real reference systems are used:

MgO

Cubic periclase with eight CRYSTAL SOEC calculations spanning applied pressures from -16 to 16 GPa and a QHA grid spanning 0–20 GPa and 0–2000 K.

dolomite

Low-trigonal CaMg(CO3)2 with eleven CRYSTAL SOEC calculations spanning -6 to 20 GPa and a QHA grid spanning 0–10 GPa and 0–1700 K. This system exercises non-zero C14 and C15 terms, tensor-frame normalization and anisotropic thermal expansion.

The large electronic-structure and QHA files are not distributed in the test suite. Instead, a compact frozen fixture stores the exact source SHA-256 identifiers, volume-dependent observations, fitted targets and selected P–T states. This keeps routine tests small while preserving traceability to the full phase-validation archive.

Validated invariants

The real-data tests require all of the following:

  • direct CRYSTAL parsing agrees with the generated YAML to the text precision of the output files;

  • every source contains an explicit PRESSURE keyword and its value agrees with the pressure attached to the corrected elastic tensor;

  • an independent NumPy least-squares construction recovers every fitted C0 and Cprime parameter;

  • the frozen normalized isothermal reference tensors are reproduced to numerical roundoff;

  • the adiabatic correction agrees with a separate evaluation of (T V / C_V) outer(C^T alpha, C^T alpha);

  • the correction is symmetric, positive semidefinite and of numerical rank no greater than one;

  • C^S = C^T exactly at 0 K;

  • cubic MgO receives equal corrections to C11 and C12 and no correction to C44;

  • all states in the declared validation domains remain mechanically stable.

Reference results

The frozen reference calculations give the following principal checkpoints:

System

P–T states

Minimum Wallace eigenvalue

Maximum \(|C^S-C^T|\)

Elastic extrapolations

MgO

4221

102.277 GPa

30.2562 GPa

126

Dolomite, normalized frame

3591

26.7303 GPa

5.14159 GPa

0

The MgO extrapolations occur only where high-temperature QHA expansion places volumes above the largest static elastic sample. They are retained and marked; they are not silently clipped.

Input-frame normalization

Thermoelastic YAML schema 1.0 is the pre-release normalized schema. Every elastic point must store its source lattice, right-polar-decomposition co-rotation, removed rotation angle, principal logarithmic strain, and ordered-atom correspondence diagnostic. Earlier unnormalized schemas are no longer accepted. Users must regenerate them from the original CRYSTAL outputs with quantas thermoelasticity inpgen.

The dolomite validation demonstrates why this is required: removing rigid rotations up to 1.7392 degrees changes some reconstructed stiffness entries by as much as 1.1254 GPa. No automatic numerical migration is attempted because the missing frame provenance cannot be reconstructed reliably from an old YAML alone.

Interpretation

This matrix validates parsing, finite-strain fitting, tensor reconstruction, uncertainty-preserving backward compatibility, frame migration, adiabatic conversion and mechanical stability for two independent symmetry classes. It does not establish universal accuracy of the quasi-static approximation. A QSA calculation omits explicit vibrational derivatives with respect to strain, and its physical accuracy must still be assessed against full strain-dependent QHA, experiment or another finite-temperature method for the material of interest.

Literature context

The MgO case is appropriate because its three independent single-crystal elastic constants and their pressure dependence have been extensively studied experimentally and by first-principles methods. The dolomite case provides a low-symmetry carbonate whose ambient single-crystal tensor has also been measured by Brillouin spectroscopy.

  • Karki, B. B., Stixrude, L., Clark, S. J., Warren, M. C., Ackland, G. J., and Crain, J. (1997). Structure and elasticity of MgO at high pressure. American Mineralogist, 82, 51–60. https://doi.org/10.2138/am-1997-1-207

  • Sinogeikin, S. V. and Bass, J. D. (1999). Single-crystal elasticity of MgO at high pressure. Physical Review B, 59, R14141–R14144. https://doi.org/10.1103/PhysRevB.59.R14141

  • Jiang, F., Speziale, S., and Duffy, T. S. (2006). Elasticity of magnesite and dolomite from a genetic algorithm for inverting Brillouin spectroscopy measurements. Physics of the Earth and Planetary Interiors, 155, 1–20. https://doi.org/10.1016/j.pepi.2005.08.004