Introduction to Density Functional Theory in Computational Physics

By Nishi Prabhat Hazarika January 15, 2025 8 min read

DFT Theory Computational Physics First Principles

Density Functional Theory (DFT) is one of the most important theoretical frameworks in modern computational physics and materials science. It enables quantitative predictions of electronic, structural, and magnetic properties of materials directly from quantum mechanics, without fitting parameters to experiments.

What is Density Functional Theory?

DFT reformulates the quantum many-electron problem in terms of the electron density instead of the many-body wavefunction. This idea dramatically reduces computational complexity, replacing an exponentially difficult problem with a tractable one defined in three-dimensional space.

“The ground-state properties of a many-electron system are uniquely determined by its electron density.”

Hohenberg–Kohn Theorems

The theoretical foundation of DFT rests on two key theorems:

Uniqueness: The external potential, and therefore all ground-state properties, are unique functionals of the electron density.
Variational Principle: The true ground-state density minimizes the total energy functional.

E[ρ] = T[ρ] + V_ext[ρ] + V_ee[ρ]

The Kohn–Sham Formalism

The practical success of DFT comes from the Kohn–Sham approach, which maps the interacting electron system onto an equivalent non-interacting system with the same ground-state density.

[-½∇² + v_eff(r)]ψ_i(r) = ε_iψ_i(r)

The effective potential includes nuclear attraction, Hartree repulsion, and the exchange–correlation term, which contains all many-body effects.

Exchange–Correlation Functionals

Approximations to the exchange–correlation functional determine the accuracy of DFT:

Local Density Approximation (LDA)

E_xc^LDA[ρ] = ∫ ρ(r) ε_xc(ρ(r)) dr

Generalized Gradient Approximation (GGA)

E_xc^GGA[ρ] = ∫ ρ(r) ε_xc(ρ, ∇ρ) dr

Hybrid Functionals

Hybrid functionals combine exact Hartree–Fock exchange with DFT correlation, improving band gaps and electronic structure accuracy.

Applications of DFT

Electronic band structures and density of states
Structural optimization and phase stability
Magnetic ordering and spin textures
Phonons and thermal properties
Surfaces, interfaces, and defects

Computational Workflow

Choice of basis set and pseudopotentials
k-point sampling and energy cutoffs
Self-consistent field iterations
Convergence and validation

Widely Used DFT Codes

VASP
Quantum ESPRESSO
ABINIT
GAUSSIAN
FHI-aims

Limitations

Band-gap underestimation
Weak van der Waals interactions
Strongly correlated materials
Excited-state limitations

Future Directions

Machine-learning-enhanced DFT
Time-dependent DFT
Meta-GGA and advanced hybrids
GW and Bethe–Salpeter approaches

Conclusion

Density Functional Theory remains the cornerstone of first-principles materials modeling. Its balance between accuracy and efficiency makes it indispensable for understanding and designing quantum materials.