Kohn-Sham Equations

The foundational insight of Kohn-Sham theory is mapping the real, interacting many-electron system onto a fictitious system of non-interacting electrons that yields the exact same ground-state electron density. This avoids the direct calculation of the complex many-body wavefunction, which scales factorially with system size. The non-interacting electrons occupy Kohn-Sham orbitals, and their kinetic energy is calculated exactly, with all remaining quantum complexity folded into the exchange-correlation functional.

This is the sole source of approximation in the Kohn-Sham framework. The exchange-correlation (XC) functional encapsulates all the non-classical electron-electron interactions: exchange (due to Pauli exclusion) and correlation (due to Coulombic avoidance). The exact form is unknown, leading to a 'Jacob's Ladder' of approximations:

• LDA: Based on the uniform electron gas.
• GGA: Includes the gradient of the density (e.g., PBE).
• Meta-GGA: Includes kinetic energy density (e.g., SCAN).
• Hybrids: Mix in exact Hartree-Fock exchange (e.g., B3LYP).

The Kohn-Sham equations are solved iteratively because the effective potential depends on the electron density, which itself is calculated from the orbitals. The Self-Consistent Field (SCF) procedure begins with an initial guess for the density, constructs the Hamiltonian, solves for new orbitals, computes a new density, and repeats until the input and output densities converge below a defined threshold. This cycle is the primary computational bottleneck in DFT calculations.

Each non-interacting electron feels a single, local effective potential (v_eff) rather than instantaneous interactions with other individual electrons. This potential is composed of three terms:

• External Potential (v_ext): The Coulomb attraction to the nuclei.
• Hartree Potential (v_H): The classical electrostatic repulsion from the total electron density cloud.
• Exchange-Correlation Potential (v_xc): The functional derivative of the XC energy, capturing all quantum corrections to the Hartree term.

The rigorous theoretical bedrock upon which the Kohn-Sham equations stand. The first Hohenberg-Kohn theorem proves that the ground-state electron density uniquely determines the external potential and thus all properties of the system. The second theorem establishes a variational principle for the energy as a functional of the density, guaranteeing that minimizing the energy functional yields the true ground-state density. These theorems legitimize the use of density as the central variable.

A key practical advantage is the formal scaling of O(N³) with system size, where N is the number of basis functions, driven by matrix diagonalization. This is dramatically cheaper than wavefunction methods like Coupled Cluster. In practice, linear-scaling or O(N) methods exploit the 'nearsightedness' of electronic matter to achieve near-linear scaling for large systems, enabling calculations on thousands of atoms, bridging the gap to classical molecular dynamics length scales.

The only unknown term in the Kohn-Sham formalism. This functional approximates the complex quantum mechanical exchange and correlation energy of electrons. The accuracy of a DFT calculation is entirely dependent on the quality of this approximation.

• LDA: Local Density Approximation, depends only on local electron density.
• GGA: Generalized Gradient Approximation, includes the density gradient.
• Hybrids: Mix in a fraction of exact Hartree-Fock exchange (e.g., B3LYP).

The iterative algorithm used to solve the Kohn-Sham equations. Since the effective potential depends on the electron density, which itself depends on the orbitals, the solution must be found iteratively.

• Guess: Start with an initial trial density.
• Construct: Build the Kohn-Sham Hamiltonian.
• Diagonalize: Solve for new orbitals and energies.
• Mix: Combine old and new densities to ensure stability.
• Converge: Repeat until the input and output densities are consistent.

A set of mathematical functions used to represent the Kohn-Sham orbitals. The choice of basis set represents a trade-off between computational cost and accuracy.

• Plane Waves: Natural for periodic solids, used with pseudopotentials.
• Gaussian-Type Orbitals (GTOs): Dominant in quantum chemistry for molecules.
• Numerical Atomic Orbitals (NAOs): Highly efficient, used in codes like FHI-aims.
• Augmented Methods: APW+lo/LAPW, the 'gold standard' for solids.

Replaces the strong Coulomb potential of the nucleus and the chemically inert core electrons with a weaker, smooth effective potential. This dramatically reduces the number of basis functions needed.

• Frozen Core: Assumes core electrons do not participate in bonding.
• Valence Only: Only the chemically active valence electrons are treated explicitly.
• PAW: The Projector Augmented Wave method reconstructs the true all-electron wavefunction, combining the efficiency of pseudopotentials with the accuracy of all-electron methods.

A computational technique to accelerate the calculation of the Coulomb and exchange-correlation contributions in the Kohn-Sham scheme. The electron density is expanded in an auxiliary basis set.

• RI-J: Resolution of Identity for the Coulomb term, reduces the four-center two-electron integrals to three-center ones.
• RI-JK: Extends the approximation to the exchange term, crucial for hybrid functionals.
• Speedup: Can reduce computational cost by an order of magnitude with negligible loss in accuracy.

The time-dependent extension of the Kohn-Sham framework for calculating excited-state properties. Applies a time-dependent external electric field to probe the linear response of the electron density.

• Linear Response: Calculates excitation energies and oscillator strengths for UV/Vis spectra.
• Casida's Equation: The standard matrix formulation for solving the TD-DFT eigenvalue problem.
• Real-Time Propagation: An alternative approach that explicitly propagates the density in time, useful for strong fields and non-linear optics.