  Single-layer transition metal dichalcogenides (TMDCs) can adopt two distinct
structures corresponding to different coordination of the metal atoms. TMDCs
adopting the T-type structure exhibit a rich and diverse set of phenomena,
including charge density waves (CDW) in a $\sqrt{13}\times\sqrt{13}$ supercell
pattern in TaS$_2$ and TaSe$_2$, and a possible excitonic insulating phase in
TiSe$_2$. These properties make the T-TMDCs desirable components of layered
heterostructure devices. In order to predict the emergent properties of
combinations of different layered materials, one needs simple and accurate
models for the constituent layers which can take into account potential effects
of lattice mismatch, relaxation, strain, and structural distortion. Previous
studies have developed ab initio tight-binding Hamiltonians for H-type TMDCs
[arXiv:1709.07510]. Here we extend this work to include T-type TMDCs. We
demonstrate the capabilities of our model using three example systems: a
1-dimensional sinusoidal ripple, the 2$\times$2 CDW in TiSe$_2$, and the
$\sqrt{13}\times\sqrt{13}$ CDW in TaS$_2$. Using the technique of band
unfolding we compare the electronic structure of the distorted crystals to the
pristine band structure and find excellent agreement with direct DFT
calculations, provided the magnitude of the distortions remains in the linear
regime.
