SCC-DFTB: What is the proper degree of Self-Consistency?

The approximate SCC-DFTB method (Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, Th.; Suhai, S.; Seifert, G. Phys. Rev. B 1998, 58, 7260) is derived from DFT by a second-order expansion of the total energy expression. In this article, basic approximations and assumptions underlying the DFTB method are discussed in detail, and further extensions to include third-order terms are proposed. Further, the SCC-DFTB and semiempirical NDDO formalisms are compared to elucidate similarities and differences.