DFT

1. Density Functional Theory

2. Density Functional Theory (DFT)
• Schrodinger equation can be solved easily for one electron system.
• For multiple electron systems it is hard to solve the Schrodinger
equation.
• So we introduce the approximation of density functional theory.
• DFT calculations are like ab-initio and semi-empirical calculations,
based on the Schrodinger equation.
• However, unlike these two methods DFT does not calculate
conventional wave function, but it rather derives the electron
distribution (electron density function) directly.
• A functional is a mathematical entity related to a function.

3. • DFT is a computational quantum mechanical modelling method used
in physics, chemistry, & material science to investigate the electronic
structure ( ground state) of many body systems.
• Using this theory the properties of the many-electron system can be
determined by using FUNCTIONALS.
• In DFT instead of considering wave function we considered density
functional.
• DFT: work in terms of density
E=E[ρ(r)]

4. Figure: Electron density of an electronic defect condition at the surface
of an anatase crystal (TiO2) calculated with DFT.

5. DFT
• In DFT the functional is the electron density which is a function of
space and time.
• The electron density is used in DFT as a fundamental property, unlike
Hartree-Fock theory which deals with many-body wavefunction.
• Using the electron density significantly speeds up the calculation.
Whereas many-body electronic wave function is a function of 3N
variables. However the electron density is only a function of only
three variables i.e. x,y,z.

6. Historical Background of DFT
• Thomas, Fermi, and Dirac in 1927 imagined that the kinetic and
exchange energies of systems of many electrons could be locally
modeled by their uniform electron gas energy densities.
• Though wonderfully simple, this theory fails qualitatively because it is unable
to self-consistently reproduce atomic shell structure.
• Hartree-Fock (HF) presented a theory in late 1920s on the basis of
Pauli exclusion principle and, electron pairs in orthonormal orbitals
arranged in Slater determinants. Given the Hamiltonian operator, H,
for N electrons in an external (nuclear) potential vext:
• HF theory, while immensely more useful than Fermi-Dirac theory, is still not
accurate enough for energy predictions in chemistry. Bond energies are
significantly underestimated.

7. THE BIRTH OF DENSITY-FUNCTIONAL THEORY
• In their seminal 1964–1965 papers, Hohenberg, Kohn, and Sham
founded the rigorous theory that finally legitimized the intuitive leaps
of Thomas, Fermi, Dirac, and Slater.
• Thus, 1964 is widely accepted as the birth year of modern DFT.
• It was established in the 1964 paper of Hohenberg and Kohn that the
total electron density ρ completely and exactly determines all the
(ground-state) properties of an N-electron system.
• Thus, ρ can be used as the fundamental “variable” in electronic structure
theory. The much more complicated N-electron wave function is, in principle,
superfluous. The logic is subtle. It goes something like this:
𝑣𝑒𝑥𝑡→𝛹0→𝜌 or 𝑣𝑒𝑥𝑡→𝜌.

8. • In this case the total ground state energy of many-electron system is
functional of density.
• 1970s and early 80s: DFT becomes useful.
• 1998: Nobel prize awarded to walter kohn in Chemistry for
development of DFT.

9. HOHENBERG – KOHN THEOREMS
• Based on two fundamental theorems :
• Theorem 1: The external potential or the ground state energy E is a
unique function of electron density.
E=E[ρ(r)]
• Theorem 2 : The electron density that minimizes the energy of the
overall functional is the true ground state electron density
𝐸 𝑛 𝑟 ≥ 𝐸𝑜[𝑛𝑜 𝑟 ]