Semi-empirical methods, for example Density Functional Tight Binding (DFTB), are a potential strong alternative quantum approach. The DFTB Hamiltonian is derived directly from an expansion of the Kohn-Sham DFT total energy, yielding a good balance between approximate quantum mechanics and empiricism. This can result in calculations requiring only a small fraction of the computational cost compared to DFT or other high-level quantum approaches. Here, the DFTB total energy is written as: