täthetsfunktional

A täthetsfunktional, often translated as density functional, is a central concept in density-functional theory (DFT). DFT is a quantum mechanical method used in physics and chemistry to investigate the electronic structure of materials. The core idea of DFT is that the ground-state properties of a many-electron system, such as its total energy, can be determined solely by its ground-state electron density, which is a function of three spatial coordinates.

The täthetsfunktional, often denoted as E[n], represents the total energy of the system as a functional of the electron density n(r). This means that the energy is a unique value for each specific electron density distribution. The Hohenberg-Kohn theorems, the foundational theorems of DFT, prove that the ground-state energy and all other ground-state properties are uniquely determined by the ground-state electron density. The first Hohenberg-Kohn theorem states that the external potential, which defines the system, is uniquely determined by the ground-state electron density up to a constant. The second Hohenberg-Kohn theorem states that the täthetsfunktional provides the exact ground-state energy for the true ground-state density.

In practice, the exact form of the täthetsfunktional is unknown. Therefore, approximations are used to represent the exchange-correlation energy, which is a crucial part of the total energy functional. The total energy functional is typically decomposed into the kinetic energy of the non-interacting electrons, the classical electrostatic interaction energy (Hartree energy), the interaction energy with the external potential, and the exchange-correlation energy. The accuracy of DFT calculations heavily relies on the quality of the approximation used for the exchange-correlation täthetsfunktional. Various approximations exist, such as the local density approximation (LDA), generalized gradient approximations (GGAs), and hybrid functionals.