DFT in Solid-State Physics

Density Functional Theory (DFT) is a widely used and highly effective method that has revolutionized condensed matter physics and materials science by enabling the study of material properties at the atomistic and electronic levels. By describing the interaction of electrons within the framework of quantum mechanics, no other approach except DFT describes a wide range of physical properties in materials at a minimum computational cost. The core principle is that the ground-state properties of a stationary many-body system may be expressed only via the ground-state density ( \(\rho \) (r)). In DFT, \(\rho \) (r) is the basic quantity where it is a function of only three spatial coordinates. Therefore, it is computationally viable for the many-particle wave function, which contains 3N coordinates for the N-particle system. The basic formulation of DFT is based upon the two theorems of Hohenberg and Kohn (HK) [1] and Kohn-Sham equations (KS) [2]. KS equation is the one-particle Schrödinger equation of a non-interacting KS system generating the same \(\rho \) (r) as in the case of interacting inhomogeneous systems [2]. These are a set of self-consistent equations based on the formalism as provided by HK [1]. These equations mainly consist of external potential, Hartree potential, and exchange-correlation (XC) potential. To date, the XC potential has no exact form, which is responsible for explaining the quantum mechanical features of fermions. The quest for exact XC functionals has been a significant challenge in DFT for a long time.