**4. Recent developments**

This section focuses on the evolution of new functionals in DFT during the last decades.

## **4.1 Random phase approximation (RPA-type Functionals)**

The exchange correlation energy (Exc) can be calculated using DFT fluctuation dissipation in the form of coupling constant and frequency [24–26]. The direct random-phase approximation (RPA) [27, 28] or time dependent TD-Hartree, are the results of ignoring the exchange kernel of TDFT. A fifth-rung approximation is generated as a result of this methodology, and this can be expensive to examine, although the relative burden is always reducing [29, 30]. It only examines bubble diagrams in the many-body expansion of the energy, so direct RPA over-correlates systems by ignoring extra contributions at higher levels that diminish correlation. It also has issues with self-interaction since, even when just one electron is involved, it yields low correlation energies, and the dissociation energies of molecules are erroneous [23].

#### **4.2 Meta-GGA's**

The meta-GGA [31] is a novel component that extends beyond density and gradient, and is commonly used to indicate the KS orbitals' kinetic energy density. The objective of a successful meta-GGA is to achieve hybrid accuracy without incurring the computational expense of the exact exchange contribution. The incorporation of atom-centered basis functions, the cost of accurate exchange is reasonable, however, it can be costly while using periodic boundary conditions in addition of basis sets. Perdew and colleagues, and plenty of others, have worked on meta-GGAs for decades, with multiple failed attempts [32]. SCAN (strongly constrained and suitably normed semi-local density functional) [33], the most current effort has undergone a number of conventional tests and looks to have a good chance of becoming part of the pantheon of widely employed functionals. The G3 data-set [34] is a common collection of chemical compounds that LDA overbinds around 3 eV, while PBE reduces it to approximately 1 eV, and SCAN around 1/4 eV. On the S22 data-set [35] of weakly bonded systems, SCAN has 2–3 times less errors than PBE does, while SCAN decreases miscalculations of lattice constant and other parameters on the LC20 data [36] set around 0.05 Å, and to around 0.01 Å in PBE. The PBE [37], on contrary to SCAN, only improves underestimation of chemical barrier height by 30 percent, while hybrids on the other hand are frequently 2–3 times superior with conventional varieties. Therefore, one can conclude that SCAN achieves accuracies comparable to hybrid functionals for several characteristics at a fraction of the computing cost [38].

#### **4.3 Range separated hybrids (RSH)**

Andreas Savin was the first to create the range separation hypothesis, which is quite precise [39, 40], through which coulomb repulsion may be easily expressed by combining a short-ranged input with a long-ranged involvement that do not have coulomb singularity at zero separation, and decays quicker than the inverse of the separating distance. In KS equation generalizations, one contribution is treated as an interaction, while the other is compensated by a redefined XC contribution. The HSE06 functional [41] is a hybrid with a range separation that manages long-ranged *Fundamentals of Density Functional Theory: Recent Developments, Challenges and Future… DOI: http://dx.doi.org/10.5772/intechopen.99019*

exchanges with an approximation, short-ranged exchanges with accuracy in an extended insulator [42], and this combination frequently yields exact gaps for moderate-gap semiconductors and insulators [37].
