How are elastic properties computed with DFT?

Elastic properties are computed by applying small strains to the crystal structure and measuring the resulting stress (or energy change).

Stress-Strain Method

1. Apply a set of small strains (typically 0.5-1%) to the unit cell
2. For each strained structure, relax the internal atomic positions while keeping the cell shape fixed
3. Compute the stress tensor from the DFT calculation
4. Fit the stress-strain relationship to extract the elastic stiffness tensor (Cij)

Derived Properties

From the full elastic tensor (Cij), several engineering quantities are derived:

• Bulk modulus (K): Voigt-Reuss-Hill average of the bulk modulus
• Shear modulus (G): Voigt-Reuss-Hill average of the shear modulus
• Young's modulus (E): Computed from K and G as E = 9KG/(3K+G)
• Poisson's ratio (v): Computed as v = (3K-2G)/(6K+2G)

Why Not All Materials Have Elastic Data

Elastic constant calculations require 6-21 separate strained calculations (depending on crystal symmetry), making them significantly more expensive than a standard relaxation. Not all materials in the databases have been processed through this workflow.

Accuracy

DFT elastic constants typically agree with experiment to within 10-20%. The agreement is best for simple metals and semiconductors and less reliable for strongly correlated or van der Waals bonded materials.