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Si$_{y}$Ge$_{1-x-y}$(C,Sn,Pb)$_{x}$ alloys have attracted significantattention as a route to achieve a direct-gap group-IV semiconductor. Usingdensity functional theory (DFT) - employing local density approximation andhybrid Heyd-Scuzeria-Ernzerhof exchange-correlation functionals - we computethe lattice parameters, relaxed and inner elastic constants, and internalstrain (Kleinman) parameters for elemental (diamond) group-IV materials andzinc blende IV-IV compounds. Our DFT calculations support a little-knownexperimental re-evaluation of the $\alpha$-Sn elastic constants, and contradicta recent prediction of dynamic instability in selected IV-IV compounds.DFT-calculated structural and elastic properties are used in conjunction with arecently derived analytical parametrisation of a harmonic valence force field(VFF) [Phys. Rev. B 100, 094112 (2019)] to obtain a complete set of VFFpotentials for Si$_{y}$Ge$_{1-x-y}$(C,Sn,Pb)$_{x}$ and Si$_{x}$Ge$_{1-x}$alloys. The analytical parametrisation exactly reproduces the relaxed elasticconstants and Kleinman parameter without recourse to numerical fitting,allowing for accurate and computationally inexpensive lattice relaxation. Theaccuracy of the VFF potentials is demonstrated via comparison to the results ofDFT supercell relaxation for (i) ordered Si (Ge) alloy supercells containing asubstitutional C, Ge (Si), Sn or Pb impurity, where comparison is also made toa model analytical VFF, and (ii) disordered Si$_{x}$Ge$_{1-x}$ alloysupercells. The VFF potentials we present enable accurate and computationallyinexpensive relaxation of large-scale supercells representing bulk-likegroup-IV alloys or group-IV heterostructures, providing input to firstprinciples or empirical electronic structure calculations, and enablingstructural analysis and calculation of strain fields in heterostructures fordevice applications.

Fully analytic valence force fields for the relaxation of group-IV  semiconductor alloys: elastic properties of group-IV materials calculated  from first principles