A definition of the Grüneisen parameters for anisotropic materials is derived based on the response of phonon frequencies to uniaxial stress perturbations. This Grüneisen model relates the thermal expansion in a given direction (α_ii) to one element of the elastic compliance tensor, which corresponds to the Young's modulus in that direction (Y_ii). The model is tested through ab initio prediction of thermal expansion in zinc, graphite, and calcite using density functional perturbation theory,

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... dicating that it could lead to increased accuracy for structurally complex systems. The direct dependence of α_ii on Y_ii suggests that materials which are flexible along their principal axes but rigid in other directions will generally display both positive and negative thermal expansion.