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Using {\it ab initio} density functional theory, we study the lattice phasetransition of quasi-one-dimensional (TaSe$_4$)$_2$I. In the undistorted state,the strongly anisotropic semimetal band structure presents two non-equivalentWeyl points. In previous efforts, two possible Ta-tetramerization patterns wereproposed to be associated with the low-temperature structure. Our phononcalculations indicate that the orthorhombic $F222$ CDW-I phase is the mostlikely ground state for this quasi-one-dimensional system. In addition, themonoclinic $C2$ CDW-II phase may also be stable according to the phonondispersion spectrum. Since these two phases have very similar energies in ourDFT calculations, both these Ta-tetramerization distortions likely compete orcoexist at low temperatures. The semimetal to insulator transition is inducedby a Fermi-surface-driven instability that supports the Peierls scenario, whichaffects the Weyl physics developed above $T_{\rm CDW}$. Furthermore, thespin-orbit coupling generates Rashba-like band splittings in the insulating CDWphases.

First-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe$_4$)$_2$I