Density functional theory introductory material

DFT

1. Density Functional Theory (DFT)

3. Schrodinger Equation for a Two electron
system Based on HartreeFockMethod

4. Solving electron wave function for a muti-
electron molecule is a HerculianTask

5. Concept of Electron density in a molecule
instead of single electron wavefunctions

6. Electron Density Function is the main
focus in DFT

7. Electron density function
•Electron density around a point can be determined as the integral
of the square of the wavefunction for a one electron system.
•It is a measure of the probability of finding the electron in a small
volume dxdydz
•Wavefunction depends on 3coordinates for each electron –3n
coordinates for ‘n’ electrons
•DFT is based on the electron density 
xyzaround a point rather than
wavefunction.
•Electron density depends on 3coordinates for n electron system.
•No of integralswill be less for finding the total energy

9. Wilson’s Explanation of HK Theorem 1

11. Functional vs. function
•A functionis a rule that transforms a number into
another number:
•e.g.

13. Writing the Energy functional explicitly in terms of the
electron density function built from non interacting
orbitals(electrons) and applying the variationaltheorem
we find that the orbitals, which minimisethe energy,
satisfy the following set of equations;
Kohn-Sham DFT

15. Approximations to Exchange
correlation functional E
xc[ρ]
•Different approximations have been proposed
•local density approximation (LDA) functionals
•the generalized-gradient approximation (GGA)
functionals
•Hybrid functionals

16. Local Density approximation (LDA)
E
xcFunctionalsin DFT
The LDA is the most basic, non trivial, approximation
for the exchange correlation energy. We assume that
the XC energy for the system at a given point is equal
to that of a homogeneous electron gas with the same
density in that point
Egsof LDA functionals: VWN -3, VWN-5
Functionalsby Vosko,Wilkand Nusair

17. Generalized Gradient Approximations.
(GGA) E
xcFunctionalsin DFT)
The GGA is a class of functionalsthat not only take the
density at the given point into account but also the gradient
(change) of the density. This allows it to describe the
systems better (up to a point) and can help in the
description of certain systems.
Egsof GGA Functionals

18. Hybrid functionals
•Hybrid functionalsincorporate terms
obtained from
•LSDA, GGA, HF, And other approximations
•e.g. B3LYP
•The most popular hybrid functional,
developed by Beckein 1993, and modified by
Stevens in 1994 by introduction of the LYP
correlation-energy functional

19. B3LYP
•LSDA non-gradient-corrected exchange functional
•KS-orbital-based HF exchange energy functional
•Becke88 exchange functional
•Vosko, Wilk, Nusairfunction
•LYP correlation functional (Lee, Yang and Parr)
•(the last three terms are gradient corrected)
•a
0, a
xand a
care constants

20. Applications of DFT
•This method is good for the calculation of
•geometries of molecules and transition states,
•quantities relevant to thermodynamics and
kinetics,
•vibrational frequencies,
•dipole moments,
•charges and bond orders,
•UV and NMR spectra