In KohnSham density functional theory (KSDFT), the ground state electron density is described using a single mathematical determinant, which leads to a set of equations that are similar to those used in HartreeFock theory and are known as KohnSham equations:
where is the external potential (such as the potential from the nuclei), is the Hartree potential (which describes the electrostatic interaction between the electrons), and is the exchangecorrelation potential (which describes the manybody effects in the system). The wavefunction (KS orbitals) and energy are the solutions to the KohnSham equation for the th electron.
The total energy is composed of the noninteracting kinetic energy , the energy due to the electrostatic interactions of the electrons with their charge density and with the nuclear cores , and the exchangecorrelation energy that accounts for the remaining electronic energy not included in the noninteracting kinetic and electrostatic terms like the nonclassical electronelectron repulsion and the difference between interacting and noninteracting kinetic energy .
Approximations
Here are some of the main approximations used in KohnSham DFT:

 In KSDFT, one avoids the need for a kinetic energy functional by evaluating the exact kinetic energy from the wavefunction (single Slater determinant) of a fictitious noninteracting system with the same density as that of the real one. Therefore, KSDFT calculates the kinetic energy in terms of orbitals. However, one still needs to approximate the (mentioned above). Since we are again working with orbitals in KSDFT (like in the case of HartreeFock theory), we need a set of basis functions (atomic orbitals) for each atom of our system whose linear combination corresponds to KS orbitals (molecular orbitals). This set of basis functions is called a basis set and makes the calculation accuracy dependent on the quality of basis sets used.
 Use of the exchangecorrelation functional, which approximates the exchange and correlation energy of a manyelectron system. The exchangecorrelation energy accounts for the remaining electronic energy not included in the noninteracting kinetic and electrostatic terms. Some of the common types are: local (LDA), semilocal (GGA), and hybrid (including some portion of exact exchange) functionals that differ in accuracy.
 Approximate exchangecorrelation functionals also lead to selfinteraction errors (SEI) which in turn lead to delocalization errors. In both HF theory and KSDFT, the potential field includes the Coulomb potential, which is the interaction of the electron with the entire electron density of the atom, molecule, or material. That is physically incorrect because an electron does not interact with itself. In HF theory, however, the exchange potential (nonlocal in nature) completely cancels the selfinteraction part of the Coulomb potential which is not the case for KSDFT, due to the approximate (not completely nonlocal) exchangecorrelation functionals. It has been known to be the primary cause for the underestimation of bandgaps in semiconductors.
 Ignoring relativistic effects, which can be important for certain types of systems. Standard DFT does not take into account relativistic effects like spinorbit coupling. However, many codes have the ability to account for it by using relativistic pseudopotentials.
 Use of BornOppenheimer approximation. The positions of the nuclear cores are considered parameters of the electronic Hamiltonian thus defining a potential energy surface. The electronic system is assumed to adiabatically follow changes in the nuclear coordinates relaxing to its ground state on much faster timescales than the timescales of nuclear motion (electronphonon coupling, e.g., is a correction to this adiabatic approximation).
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References:
 density functional theory – What correlation effects are included in KSDFT with LDA and GGA? – Matter Modeling Stack Exchange
 Perspective: KohnSham density functional theory descending a staircase: The Journal of Chemical Physics: Vol 145, No 13 (scitation.org)
 Status and Challenges of Density Functional Theory (umn.edu)
 SelfInteraction Error in Density Functional Theory: An Appraisal  The Journal of Physical Chemistry Letters (acs.org)
 computational chemistry – Striking examples where KohnSham orbitals clearly have no physical meaning – Chemistry Stack Exchange
 What Do the Kohn−Sham Orbitals and Eigenvalues Mean?  Journal of the American Chemical Society (acs.org)
 Slide 1 (southampton.ac.uk)
 Pantazis_DFTBasisSets (unipaderborn.de)
 basissets.pdf (gatech.edu)
 density functional theory – Spin–orbit interaction with DFT – Matter Modeling Stack Exchange
 Selfinteraction corrections in density functional theory: The Journal of Chemical Physics: Vol 140, No 18 (scitation.org)
Other useful resources:
Ph.D. researcher at FriedrichSchiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating lightmatter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.