Background on Kohn-Sham DFT

The Schrodinger equation is fundamental for describing the quantum mechanical electronic structure of matter, since it does not require any empirical input. However, the solution of the Schrodinger equation is exceedingly expensive, and this limits the size of systems that can be directly evaluated to tens of electrons. Numerous approaches have been proposed to reduce the computational cost of the solution of the Schrodinger equation. These approaches include the widely used DFT of Hohenberg and Kohn, for which Kohn won the Nobel prize in Chemistry in 1998.
In their seminal work, Hohenberg and Kohn proved the existence of a one-to-one correspondence between the ground state electron density and the ground state wavefunction of a many-particle system. By this correspondence, the electron density replaces the many-body electronic wavefunction as the fundamental unknown field, thereby greatly reducing the dimensionality and computational complexity of the problem. The most common present-day implementation of DFT is through the Kohn-Sham method, in which the intractable many-body problem of interacting electrons is reduced to a tractable problem of non-interacting electrons moving in an effective potential.
Over the past few decades, DFT has been extensively used for understanding and predicting a wide array of material behavior, including their electronic, mechanical, thermal, and optical properties. The tremendous popularity of DFT—free from any empirical parameters by virtue of its origins in the first principles of quantum mechanics—stems from its high accuracy to cost ratio when compared to other such ab-initio theories. However, the efficient and accurate solution of the DFT problem still remains a formidable task, which limits the length and time scales accessible to DFT simulations.