Elastic Tensor Skill

This skill calculates the full elastic tensor ($C_{ij}$) and derived mechanical properties of crystalline materials using Machine Learning Interatomic Potentials (MLIPs). It applies a set of normal and shear strains, computes the resulting stresses, and fits the elastic constants via least-squares regression using MatCalc's ElasticityCalc.

Goal

Calculate the elastic tensor ($C_{ij}$) of a material by applying systematic deformations (normal and shear strains), computing the stress response with an MLIP, and extracting the full Voigt elastic tensor along with:

• Bulk modulus $B$ (Voigt-Reuss-Hill average)
• Shear modulus $G$ (Voigt-Reuss-Hill average)
• Young's modulus $E$
• Poisson's ratio $\nu$

1. Prerequisites

• The appropriate MLIP wrapper must be available (MACEWrapper, MatGLWrapper, or FAIRCHEMWrapper).
• matcalc must be installed in the relevant conda environment.
• A structure file (CIF, POSCAR, or other ASE-readable format). The structure will be relaxed before deformation by default.

2. Choosing a Foundation Potential

Elastic tensor calculations require accurate stress predictions across multiple deformed structures.

[!IMPORTANT]

• Use OMAT or MatPES trained models: These models (e.g., MACE-OMAT-0-small, CHGNet-MatPES-PBE, TensorNet-MatPES-r2SCAN) are trained with stress labels and provide reliable stress predictions.
• Stress accuracy is critical: Unlike EOS (which only uses energies), elasticity calculations directly depend on stress tensors. Models trained without stress labels may give poor results.

Refer to the foundation-potentials skill for more details.

3. Calculation Workflow

To calculate the elastic tensor, use the calculate_elasticity.py script:

# Env: mace-agent
python .agents/skills/mat-elasticity/scripts/calculate_elasticity.py \
    --structure path/to/structure.cif \
    --model_type mace \
    --model_name MACE-OMAT-0-small \
    --norm_strains -0.01 -0.005 0.005 0.01 \
    --shear_strains -0.06 -0.03 0.03 0.06 \
    --relax_structure \
    --output_dir research/my_folder/elasticity

Key Parameters:

• --norm_strains: Normal strain magnitudes applied (default: ±0.5%, ±1.0%)
• --shear_strains: Shear strain magnitudes applied (default: ±3%, ±6%)
• --relax_structure: Relax the structure before applying strains (recommended)
• --relax_deformed / --no-relax_deformed (default on): re-minimise the ions inside each deformed cell, with the cell held fixed. See Relaxed-ion versus clamped-ion below — this flag selects which of two physically distinct quantities you get, and the difference is not small.
• --fmax: Force convergence tolerance for relaxation (default: 0.1 eV/Å)

[!TIP]

• For metals, the default strain magnitudes work well.
• For soft materials (polymers, molecular crystals), reduce strains to stay in the linear regime.
• For very hard materials (diamond, SiC), the default strains are fine since deformations remain small.

4. Output Files

• elasticity_results.json: Full results including:
  • elastic_tensor_GPa: 6×6 Voigt elastic tensor in GPa
  • bulk_modulus_vrh_GPa: Bulk modulus (VRH) in GPa
  • shear_modulus_vrh_GPa: Shear modulus (VRH) in GPa
  • youngs_modulus_GPa: Young's modulus in GPa
  • poissons_ratio: Poisson's ratio (dimensionless)
  • residuals_sum: Residual from the least-squares fit (lower is better)

5. Examples

See examples/Cu/ for a copper elastic tensor calculation using MACE-OMAT-0-small.

# Env: mace-agent
python .agents/skills/mat-elasticity/scripts/calculate_elasticity.py \
    --structure .agents/skills/mat-elasticity/examples/Cu/Cu.cif \
    --model_type mace \
    --model_name MACE-OMAT-0-small \
    --output_dir research/elasticity/Cu

6. Constraints

• Environment: Scripts require conda environments with MLIP packages installed:
  • mace-agent for MACE models
  • matgl-agent for MatGL/CHGNet models
  • fairchem-agent for FairChem/UMA models
• Structure Relaxation: two distinct stages, controlled by two different flags. --relax_structure (default on) relaxes the input cell before the strain scan, so the scan is centred on a stress-free reference — elastic constants are defined about zero stress, so this matters. --relax_deformed (default on) controls the per-deformation ion relaxation, which selects between two different physical quantities; see below.
• Linear Regime: Strains must be small enough to remain in the linear elastic regime. The default values are appropriate for most inorganic crystalline materials.
• Unit Conversion: MatCalc returns moduli in eV/ų (bulk, shear) and Pa (Young's). The script converts all to GPa.
• Symmetry: By default, symmetry reduction is disabled (--symmetry flag enables it). This means all 21 independent components are fitted independently.

Relaxed-ion versus clamped-ion

Applying a strain to a crystal leaves internal degrees of freedom that the strain does not itself fix — the fractional coordinates of atoms on general Wyckoff positions. What you do with them decides which elastic constant you compute:

	--relax_deformed (default)	--no-relax_deformed
ions in the deformed cell	re-minimised at fixed cell	carried rigidly by the affine strain
quantity	relaxed-ion, a.k.a. equilibrium	clamped-ion, a.k.a. frozen-ion
physical meaning	second derivative of the energy minimised over the internal coordinates — what a real crystal exhibits	second derivative at frozen internal coordinates
cost	one ionic relaxation per deformation	one energy/stress evaluation per deformation

Relaxed-ion is the default here because it is the macroscopic elastic constant: it is what experiment measures and what the Materials Project and atomate2 elastic workflows compute (ionic relaxation at fixed cell for every deformation). Note that matcalc's own ElasticityCalc defaults relax_deformed_structures=False, so inheriting that default silently gives the clamped-ion answer instead.

Clamped-ion is systematically stiffer, because freezing the ions suppresses the non-affine internal displacement that would otherwise relieve part of the strain. It is a reasonable fast screening choice, and it is exact only where symmetry leaves no internal degrees of freedom to relax (every atom on a special position, as in B1 or B2 binaries). Otherwise the gap is real: for Pnma CaMgSi it is 1.4% on the bulk modulus but 7.5% on the shear modulus, 7.3% on the Poisson ratio and 37% on the anisotropy index. Report which one you used.

Derived properties

Beyond the tensor and the VRH averages, the script reports the standard post-processing of an elastic tensor. Two of these are easy to get wrong by hand:

• Universal anisotropy index A^U = 5 G_V/G_R + B_V/B_R - 6 (Ranganathan & Ostoja-Starzewski, PRL 101, 055504 (2008)), zero only for an isotropic crystal. It needs the Voigt and Reuss bounds kept separate, so it cannot be recovered from the VRH averages; the Voigt and Reuss bulk and shear moduli are reported alongside it.
• Directional Young's moduli from E(n) = 1 / (S_ijkl n_i n_j n_k n_l): along [100], [010], [001], plus the global minimum and maximum over all directions with the directions they occur in. Two traps here. Expanding the Voigt compliance to S_ijkl requires a factor of 1/4 on shear-shear entries (S_1212 = S_66/4, not S_66) and 1/2 on normal-shear — the stiffness expands with no factors, so the two cannot share a helper. And the extrema of an anisotropic crystal need not lie on a crystal axis: for CaMgSi the stiffest direction sits ~40° off a in the a–c plane and is 13% stiffer than the stiffest axis, so scanning only the axes is wrong.
• Acoustic and Debye properties: density, longitudinal and transverse sound velocities, the Debye mean velocity and the Debye temperature via the Anderson relation Theta_D = (hbar/k_B)(6 pi^2 N/V)^(1/3) v_m. The mean is the harmonic-cube mean over one longitudinal and two transverse branches, not the arithmetic mean of the two branches (which runs ~20% high).
• Born stability from the eigenvalues of the tensor, and the Pugh ratio G/B.

---

Author: Bowen Deng Contact: GitHub @learningmatter-mit