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How is DFT Calculated?

Published in Quantum Chemistry 5 mins read

Density Functional Theory (DFT) calculations involve solving the Kohn-Sham equations iteratively to determine the electronic structure of a system based on its electron density. Here's a breakdown of the process:

Core Principles of DFT

DFT is rooted in two fundamental theorems established by Hohenberg and Kohn:

  1. The first Hohenberg-Kohn theorem: The external potential (and hence the total energy) of a system is a unique functional of the ground state electron density. This means that all ground state properties are uniquely determined by the ground state electron density.
  2. The second Hohenberg-Kohn theorem: The ground state electron density minimizes the total energy functional of the system. This provides a variational principle for determining the ground state density.

The Kohn-Sham Equations

The practical implementation of DFT relies heavily on the Kohn-Sham equations. These equations map the problem of interacting electrons onto a problem of non-interacting electrons moving in an effective potential. The Kohn-Sham equations are:

[-ħ²/2m ∇² + V_ext(r) + V_H(r) + V_xc(r)] φ_i(r) = ε_i φ_i(r)

Where:

  • -ħ²/2m ∇² is the kinetic energy operator.
  • V_ext(r) is the external potential due to the nuclei.
  • V_H(r) is the Hartree potential, representing the classical electrostatic interaction between electrons: V_H(r) = e² ∫ n(r') / |r - r'| dr'
  • V_xc(r) is the exchange-correlation potential, accounting for the many-body effects (exchange and correlation) not captured by the Hartree term. This term is a functional derivative of the exchange-correlation energy functional: V_xc(r) = δE_xc[n(r)] / δn(r)
  • φ_i(r) are the Kohn-Sham orbitals.
  • ε_i are the Kohn-Sham orbital energies.
  • n(r) is the electron density, calculated as the sum of the squares of the Kohn-Sham orbitals: n(r) = Σ |φ_i(r)|² (summed over occupied orbitals).

Steps in a DFT Calculation

  1. Define the System: Specify the positions of the atoms, the unit cell (if a periodic system), and the charge of the system.

  2. Choose an Exchange-Correlation Functional: Select an appropriate approximation for the exchange-correlation functional (E_xc[n(r)]). Common choices include:

    • Local Density Approximation (LDA): Approximates the exchange-correlation energy density at a point based only on the electron density at that point. Suitable for many simple systems but often overestimates binding energies.
    • Generalized Gradient Approximation (GGA): Accounts for the gradient of the electron density in addition to the density itself. Often provides better accuracy than LDA. Examples include PBE and BLYP.
    • Meta-GGA: Includes the second derivative of the electron density or the kinetic energy density. Can provide further improvements in accuracy.
    • Hybrid Functionals: Incorporate a portion of exact exchange from Hartree-Fock theory. Examples include B3LYP and PBE0. Generally more accurate, but computationally more expensive.

  3. Set up the Basis Set: Choose a suitable basis set to represent the Kohn-Sham orbitals. Common choices include:

    • Plane waves: Well-suited for periodic systems (crystals, surfaces) and offer systematic convergence.
    • Localized atomic orbitals: More efficient for isolated molecules and offer a more intuitive chemical interpretation. Examples include Gaussian-type orbitals (GTOs) and Slater-type orbitals (STOs).

  4. Generate an Initial Guess for the Electron Density: An initial guess is needed to start the iterative process. This can be a superposition of atomic densities or a simpler approximation.

  5. Solve the Kohn-Sham Equations Iteratively (Self-Consistent Field - SCF):

    a. With the current electron density, calculate the Hartree potential V_H(r) and the exchange-correlation potential V_xc(r).

    b. Solve the Kohn-Sham equations to obtain a new set of Kohn-Sham orbitals φ_i(r) and orbital energies ε_i.

    c. Calculate the new electron density n(r) from the new Kohn-Sham orbitals.

    d. Compare the new electron density with the previous electron density. If the difference is below a certain threshold (convergence criterion), the calculation is considered converged. Otherwise, mix the old and new densities and repeat steps a-d.

  6. Calculate Properties: Once the self-consistent electron density is obtained, various properties can be calculated, such as the total energy, forces on atoms, electronic band structure, and vibrational frequencies.

Computational Considerations

  • Computational Cost: DFT calculations can be computationally demanding, especially for large systems or when using more accurate exchange-correlation functionals. The computational cost scales roughly as N3 or N4, where N is the number of basis functions.

  • Pseudopotentials: To reduce the computational cost, pseudopotentials are often used to represent the core electrons. These replace the strong potential of the nucleus and core electrons with a weaker, smoother potential that acts on the valence electrons.

  • Parallelization: DFT codes are often parallelized to take advantage of multi-core processors and clusters, allowing for the calculation of larger systems.

In summary, DFT is calculated by iteratively solving the Kohn-Sham equations until self-consistency is achieved. This process involves approximating the exchange-correlation functional, choosing a suitable basis set, and solving a set of one-electron equations in an effective potential.

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