![• 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)]](https://image.slidesharecdn.com/densityfunctionaltheory-220725043224-6a6c312b/85/Density-Functional-Theory-pptx-3-320.jpg)
![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 ]](https://image.slidesharecdn.com/densityfunctionaltheory-220725043224-6a6c312b/85/Density-Functional-Theory-pptx-9-320.jpg)
Density Functional Theory.pptx
- 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 ]