DFT

DFT most commonly refers to density functional theory, a quantum mechanical modeling method used to investigate the electronic structure of many-body systems in physics, chemistry, and materials science. The foundational idea is that the ground-state properties of a many-electron system are determined by its electron density, rather than its many-body wavefunction, as stated in the Hohenberg-Kohn theorems. The practical approach is the Kohn-Sham formulation, which represents the interacting system by a fictitious non-interacting reference system of orbitals that reproduce the true density. The total energy is expressed as a functional of density and minimized self-consistently, with an approximate exchange-correlation functional.

In routine use, the exact exchange-correlation functional is unknown, so a hierarchy of approximations is employed. Local Density Approximation (LDA) treats exchange and correlation as uniform in space, while Generalized Gradient Approximation (GGA, e.g., PBE) includes density gradients. Meta-GGA functionals further incorporate kinetic-energy density, and hybrid functionals mix a portion of exact exchange with conventional functionals. Calculations typically use pseudopotentials or all-electron approaches and rely on basis sets or plane-wave representations. DFT scales approximately between O(N^3) and O(N log N) with system size, making it efficient for relatively large systems. It is widely used for geometry optimization, reaction energetics, electronic properties, and materials screening. Limitations include self-interaction errors, imperfect treatment of van der Waals forces without corrections, and tendency to underestimate band gaps and excitation energies in standard formulations.

The acronym DFT may also refer to the discrete Fourier transform, a mathematical transform used in signal processing to decompose a finite sequence of samples into its frequency components. The transform is computed by summing the samples multiplied by complex exponentials. In practice it is often evaluated with the fast Fourier transform (FFT) algorithm, which reduces computational complexity and enables efficient spectral analysis, filtering, and image processing.