  This is the second and the final part of the review on density functional
theory (DFT), referred to as DFT-II. In the first review, DFT-I, we have
discussed wavefunction-based methods, their complexity, and the basic of
density functional theory. In DFT-II, we focus on fundamentals of DFT and their
implications for the betterment of the theory. We start our presentation with
the exact DFT result followed by the concept of exchange-correlation (xc) or
Fermi-Coulomb hole and its relation with xc energy functional. We also provide
the exact conditions for the xc-hole, xc-energy and xc-potential along with
their physical interpretation. Next, we describe the extension of DFT for
non-integer numbers of electrons, the piecewise linearity of total energy and
discontinuity of chemical potential at integer particle numbers, and derivative
discontinuity of the xc potential, which has consequences on fundamental gap of
solids. After that, we present how one obtain more accurate xc energy
functionals by going beyond LDA. We discuss the gradient expansion
approximation (GEA), generalized gradient approximation (GGA), and hybrid
functional approaches to designing better xc energy functionals that give
accurate total energies but fail to predict properties like the ionization
potential and the band gap. Thus, we describe different methods of modeling
these potentials and the results of their application for the calculation of
the band gaps of different solids to highlight accuracy of different xc
potential. Finally, we conclude with a glimpse on orbital-free density
functional theory and the machine learning approach .
