This article addresses a fundamental problem faced by the community employing single-particle ab initio methods: the lack of an effective formalism for the rapid exploration and exchange of new methods. To rectify this, we introduce a new, basis-set independent, matrix-based formulation of generalized density functional theories which reduces the development, implementation, and dissemination of new techniques to the derivation and transcription of a few lines of algebra. This new framework enables us to concisely demystify the inner workings of fully functional, highly efficient modern ab initio codes and to give complete instructions for their construction for calculations employing arbitrary basis sets. Within this framework, we also discuss in full detail a variety of leading-edge techniques, minimization algorithms, and highly efficient computational kernels for use with scalar as well as shared and distributed-memory supercomputer architectures.