The paper introduces DFTB3, an advanced version of the self-consistent-charge density-functional tight-binding (SCC-DFTB) method, which is an approximate quantum chemical method derived from density functional theory (DFT). The key improvements in DFTB3 include an enhanced Coulomb interaction between atomic partial charges and the inclusion of third-order terms in the Taylor series expansion of the DFT total energy. These modifications significantly enhance the method's performance, particularly for charged systems containing elements C, H, N, O, and P, in terms of hydrogen binding energies and proton affinities. The study also discusses the challenges and potential solutions for extending the method to more general applications. The computational details, including the calculation of proton affinities and the introduction of new parameters, are provided, along with benchmark results comparing DFTB3 with other methods and high-level ab initio calculations. The results show that DFTB3 improves the accuracy of geometries, binding energies, and proton affinities, especially for systems with complex charge distributions.The paper introduces DFTB3, an advanced version of the self-consistent-charge density-functional tight-binding (SCC-DFTB) method, which is an approximate quantum chemical method derived from density functional theory (DFT). The key improvements in DFTB3 include an enhanced Coulomb interaction between atomic partial charges and the inclusion of third-order terms in the Taylor series expansion of the DFT total energy. These modifications significantly enhance the method's performance, particularly for charged systems containing elements C, H, N, O, and P, in terms of hydrogen binding energies and proton affinities. The study also discusses the challenges and potential solutions for extending the method to more general applications. The computational details, including the calculation of proton affinities and the introduction of new parameters, are provided, along with benchmark results comparing DFTB3 with other methods and high-level ab initio calculations. The results show that DFTB3 improves the accuracy of geometries, binding energies, and proton affinities, especially for systems with complex charge distributions.