The Kohn-Sham formalism is a method used in density functional theory (DFT) to study the electronic structure of many-body systems, particularly atoms and molecules. It simplifies complex interactions by introducing a set of non-interacting particles that have the same electron density as the real system. This approach allows for the calculation of properties like energy and electron distribution more efficiently. In this framework, the total energy of a system is expressed as a functional of the electron density, which is a key concept in DFT. The Kohn-Sham equations are derived from this functional, enabling researchers to solve for the electron density and gain insights into the behavior of materials and chemical reactions.