## Mapping and abinitio calculations

The parameters for empirical tight binding calculations are usually obtained by numerical fitting, This project, however, focuses on extracting tight binding parameters directly from ab-initio results thereby avoiding any numerical fitting. For this purpose, the procedure we developed is as follows:

First, wave functions and energy bands are obtained from Ab-initio calculations (e.g. density functional theory). The results serve as an input to the following parameter extraction. Then, the radial parts of the tight-binding basis functions are approximated. By choosing a small set of tight-binding basis function (e.g. 20 states in the case of a spin-less sp3d5s* model) a low rank approximation for the ab-initio results is defined. Consequently, the tight-binding Hamiltonian can be obtained via the rectangular basis transformation that describes the low rank approximation. To illustrate some results, the figures below show the assumed radial parts of the tight-binding basis functions (left figure) and the resulting band structure (right figure).