From Quantum Mechanics to Density Functional Theory

This chapter described the theoretical developments of first-principles quantum mechanics parallel to many-electron systems focused on DFT. It starts from the Schrödinger equation and describes its usefulness in single-electron systems, touches on the insurmountable computational difficulties that emerge when electron–electron interactions are included for multi-electrons which is due to curse of dimensionality. In order to address these difficulties, various approximation techniques are developed such as Hartree or Hartree–Fock methods which approximate the many body problem by mean field and post-Hartree–Fock methods including Configuration Interaction or Coupled Cluster approaches where electronic correlation can be included more accurately. Although wavefunction-based approaches are very powerful, their computational scaling is not favorable and it hinders their application on large and complex systems. This shortcoming motivates an ongoing tendency to density-based methods, where the electron density is instead the main property. The chapter, for this reason, develops the concept of electron density and expresses the Hohenberg–Kohn theorems that are based on which DFT rests. We further describe the Kohn–Sham formalism as a means to practical realization of DFT for efficient and accurate electronic structure calculations. Overall, the chapter provides a coherent conceptual bridge between traditional quantum mechanical methods and modern density functional approaches widely used in computational chemistry, condensed matter physics, and materials science.