Density Functional Theory

What is Density Functional Theory?

Density Functional Theory (DFT) is a computational quantum mechanical modeling method widely used in chemistry, physics, and materials science. Unlike traditional methods, which focus on the wavefunction, DFT simplifies calculations by focusing on electron density as the primary quantity. This simplification makes DFT computationally efficient, allowing for accurate calculations of the electronic structure of many-body systems.

Fundamental Equations

At the heart of DFT are the Hohenberg-Kohn theorems, which establish that:

• The ground-state properties of a system are uniquely determined by its electron density.
• A unique energy functional exists, which provides the ground-state energy from the electron density.

The Kohn-Sham equations, another crucial part of DFT, decompose the electron density into a set of single-particle wavefunctions that are easier to compute:

∇²ψ_i(r) + V_eff(r)ψ_i(r) = ε_iψ_i(r)

Here, V_eff is an effective potential that includes terms for electron-electron interactions, exchange-correlation effects, and the external potential.

Exchange-Correlation Functionals

The exact exchange-correlation functional is unknown, so various approximations have been developed. These functionals balance accuracy and computational cost, allowing DFT to be applicable to many systems.

• Local Density Approximation (LDA): Assumes that the electron density is uniform.
• Generalized Gradient Approximation (GGA): Considers the density gradient to improve accuracy over LDA.
• Hybrid Functionals: Combine exact exchange from Hartree-Fock theory with DFT approximations (e.g., B3LYP, a widely used functional).

Basis Sets

Basis sets are used to represent the wavefunctions of electrons. Choosing an appropriate basis set affects the accuracy and computational cost of a DFT calculation.

• Minimal Basis Sets (e.g., STO-3G): Offer a basic approximation for quick calculations.
• Split-Valence Basis Sets (e.g., 6-31G): Provide higher accuracy by using more basis functions for valence electrons.
• Correlation-Consistent Basis Sets (e.g., cc-pVDZ): Designed for correlated methods, ideal for higher accuracy in molecular calculations.

Applications and Limitations

DFT is widely used to predict molecular properties, reaction energies, and surface interactions. While it’s powerful, DFT has limitations, especially with systems where van der Waals interactions or highly correlated electron behavior dominate.