Density Functional Theory by Tayyab Shabir

Density Functional Theory

1. Density Functional Theory (DFT) 1 1998 : Nobel prize awarded to Walter Kohn Walter Kohn a Austrian-American theoretical physicist and theoretical chemist To solve many body problems by Schrödinger's equation. H  = E

2. Working of DFT 2

3. The Hamiltonian for N-Particle system BORN OPPENHEIMER APPROXIMATION   Reduced dimension from 3 N e to 3 by considering nuclei is static . HOHENBERG – KOHN THEOREMS Theorem: 1 “The external potential v ext or the ground state energy E is a unique functional of electron density”. Theorem: 2 “The electron density that minimizes the energy of the overall functional is the true ground state electron density”. Limitations of HK Theorems They do provide method of finding in practice however these theorems were not very helpful in real calculation. Two other scientists Kohn and Sham gave an equation which turned DFT into a practical tool.   17

4. KOHN- SHAM SCHEME 4 The K-S equations turned the DFT one of the practical tool for the calculation of ground state density. The Kohn-Sham total energy equation is, kinetic energy of electrons External potential due to nucleus Coulomb energy due to electron-electron interaction Exchange and correlation energy term No theoretical tool is determined and hence approximations are needed to calculate this energy

5. 5 Approximations Local density approximation (LDA) Generalized Gradient Approximation (GGA) It gives more appropriate result for that material whose density will be same during reactions. GGA is improved form of LDA. In GGA E xc in any microscopic volume not only depend on the local density in that volume but on the density of adjacent volumes as well. By using gradient of density together with constant density calculations of LDA can be improved .

6. Amsterdam Density Functional 6   Developed in 1970 Vrije University of Amsterdam and university of Calgary , Canada. Structure, Reactivity and spectra of molecules. Transition metal complexes  and molecules with heavy atoms. 20

7. Computational Detail Structure is build from Space Group Fm-3m (no.225) Lattice Parameters a= 5.984Å Miller Indices of (001) is used to cut slab from bulk. Geometry Optimization LDA GGA-mPBE approximation is used . 7

8. Results and Discussion 8