Introduction to Density Functional Theory
(DFT)
Overview
• Motivation for DFT
• Hohenberg-Kohn Theorems
• Kohn-Sham Equations
• Exchange-Correlation Functional
• Jacob’s Ladder
• Types of Functionals
• Applications and Limitations
Why Density Functional Theory?
• Solves many-electron problems with reduced computational cost
• Replaces the many-body wavefunction with electron density
• Widely used in chemistry, materials science, and condensed matter physics
Pre-DFT era
• Thomas-Fermi model (1927) –
crude density-based approach – failed for chemistry problems
Jellium can’t bond!
A system of electrons is spread uniformly
over space. The positive charges
(typically from atomic nuclei) are smeared
out into a uniform positive background.
The system is electrically neutral overall.
• Slater’s Xa model (1950)
added exchange interaction from HF
DFT Era
Hohenberg-Kohn Theorem – I (1964-65)
• The ground-state energy of a many-electron system is a unique
functional of the electron density.
Hohenberg-Kohn Theorem –II
• The ground-state density minimizes the energy functional.
Kohn-Sham Formalism (1965)
• Introduces non-interacting as reference system, which has the same ground-state density as
interacting system
• Solves for single-particle orbitals (Kohn-Sham Orbitals)
Kohn-Sham Equations
Kohn-Sham Self-Consistent Loop
Kohn-Sham
Self-Consistent
Loop
Strengths of KS approach:
• Instead of approximating the whole energy functional, we only
approximate the exchange-correlation part.
• The kinetic energy is computed exactly (for the non-interacting
system).
• This gives much better accuracy than Thomas-Fermi or Xα models.
Exchange-Correlation Functional
• Accounts for all many-body effects not included in VH and Vext
• Includes
– exchange,
– correlation,
– kinetic energy correction
– self-interaction corrections
How to get Exchange Functional
1. Uniform Electron Gas (UEG) — Local Density Approximation (LDA)
2. Generalized Gradient Approximation (GGA)
3. Meta – GGA (go beyond gradient)
KE density Not only how the density changes in space, but
also how sharp or diffuse the orbitals are
How to get Correlation Functional
1. Uniform Electron Gas (UEG) and LDA Correlation
Exact correlation energy of UEG was known from Quantum Monte Carlo (QMC)
(Ceperley-Alder, 1980). Fit this to get analytic correlation functional.
Examples: Perdew-Zunger (PZ81) and Vosko-Wilk-Nusair (VWN)
2. GGA Correlation Functionals
3. meta-GGA Correlation Functionals
Some popular names!!
Jacob's
Ladder
of DFT
towards
Chemical
Accuracy.
•Hartree–Fock exchange is non-local and captures exchange interactions well but lacks
electron correlation.
•DFT functionals are good at capturing correlation but often approximate exchange poorly.
The Game Changer: Hybrid functional
A Hybrid functional in DFT mixes Hartree–Fock (HF) exact exchange with a DFT exchange-correlation functional. These are
more accurate than standard (pure) DFT methods like LDA or GGA, especially for reaction thermochemistry, kinetics, and
spectroscopy.
B3LYP the magic pill
Hybrids of all kinds!
Applications of DFT
• Molecular structures
• Reaction energies
• Band structures
• Vibrational spectra
• NMR shifts and more
Limitations of DFT
• Self-interaction errors
• van der Waals interactions
• Strongly correlated systems
• Band gap underestimation
Summary
• DFT replaces complex many-electron wavefunction with
density
• Relies on approximations for E_xc[n]
• Jacob’s ladder guides improvement of functionals
• Widely used with some limitations
