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Using density-functional perturbation theory and the Gr\"uneisen formalism,we directly calculate the linear thermal expansion coefficients (TECs) of ahexagonal bulk system MoS$_2$ in the crystallographic $a$ and $c$ directions.The TEC calculation depends critically on the evaluation of atemperature-dependent quantity $I_i(T)$, which is the integral of the productof heat capacity and $\Gamma_i(\nu)$, of frequency $\nu$ and strain type $i$,where $\Gamma_i(\nu)$ is the phonon density of states weighted by theGr\"uneisen parameters. We show that to determine the linear TECs we may useminimally two uniaxial strains in the $z$ direction, and either the $x$ or $y$direction. However, a uniaxial strain in either the $x$ or $y$ directiondrastically reduces the symmetry of the crystal from a hexagonal one to abase-centered orthorhombic one. We propose to use an efficient and accuratesymmetry-preserving biaxial strain in the $xy$ plane to derive the same resultfor $\Gamma(\nu)$. We highlight that the Gr\"uneisen parameter associated witha biaxial strain may not be the same as the average of Gr\"uneisen parametersassociated with two separate uniaxial strains in the $x$ and $y$ directions dueto possible preservation of degeneracies of the phonon modes under a biaxialdeformation. Large anisotropy of TECs is observed where the linear TEC in the$c$ direction is about $1.8$ times larger than that in the $a$ or $b$ directionat high temperatures. Our theoretical TEC results are compared with experiment.The symmetry-preserving approach adopted here may be applied to a broad classof two lattice-parameter systems such as hexagonal, trigonal, and tetragonalsystems, which allows many complicated systems to be treated on afirst-principles level.

Direct calculation of the linear thermal expansion coefficients of MoS2 via symmetry-preserving deformations