A theory that uses the fact that the ground-state energy is determined by the electron density, solving the Kohn–Sham equations to obtain the electronic states and energy of a material. The core of first-principles calculation.
Japanese name: 密度汎関数理論
Density functional theory appears in 7 practice questions across 1 topic, including Computational materials science.
The properties of materials are determined by the states of their electrons, but the wave function of many mutually interacting electrons has too many variables to be solved as it is. Hohenberg and Kohn showed that the ground-state energy is determined by the electron density alone, a function of only three dimensions. Kohn and Sham replaced this with a problem in which fictitious non-interacting electrons move in an effective potential, arranged so that their electron density is the same as that of the real system. The complicated interactions between electrons are gathered into a term called the exchange-correlation energy.
DFT takes the kinds and arrangement of atoms as input and gives the total energy, the forces on the atoms, the electronic band structure and so on. Moving the atoms until the forces are zero gives the stable structure and lattice parameter; comparing energies of different structures gives phase stability; supercell calculations give the formation energies of defects such as vacancies; and calculations with applied strain give elastic constants. Because it uses no experimental data, it is also called first-principles calculation, and it is used to predict materials not yet made and to generate data for the CALPHAD method and for interatomic potentials.
The exact form of the exchange-correlation energy is unknown, so approximations such as the local density approximation (LDA) and generalized gradient approximation (GGA) are used. These have limits: band gaps come out smaller than experiment, van der Waals forces are described poorly, and the properties of oxides with localized d or f electrons are sometimes wrong; hybrid functionals and correction methods are used to address them. Ordinary calculations are also of the ground state at 0 K and handle a few hundred atoms. For finite-temperature properties, phonon calculations or molecular dynamics are combined.
Answer: A. Based on the electron density determining the ground-state energy, it solves the Kohn–Sham equations and treats electron interactions with an approximate exchange-correlation functional
The wave function of many electrons has too many variables to be solved as it is. DFT takes the electron density as its main variable, based on the Hohenberg–Kohn theorem that the ground-state energy is determined by the electron density, a function of only three dimensions. In practice, the Kohn–Sham equations, in which fictitious non-interacting electrons move in an effective potential, are solved repeatedly until the electron density no longer changes (a self-consistent calculation). The complicated interactions between electrons are gathered into the exchange-correlation functional, for which approximations such as LDA and GGA are used, so the results are not exact and their accuracy differs with the material and property. Solving equations of motion with an empirical interatomic potential is classical molecular dynamics.
Answer: B. They tend to underestimate the band gaps of semiconductors and insulators compared with experiment; for silicon GGA gives about 0.6 eV
It is well known that with LDA or GGA, a band gap read from the difference of eigenvalues of the Kohn–Sham equations comes out smaller than experiment. For silicon, GGA gives about 0.6 eV, roughly half the experimental 1.12 eV, so the statement that they overestimate gets the trend backwards. When accurate values are needed, hybrid functionals or the GW approximation are used. On the other hand, quantities tied to the total energy of the ground state, such as lattice parameters, elastic moduli and formation energies, are obtained with good accuracy for many materials. Ordinary DFT is a calculation of the ground state at 0 K, so obtaining finite-temperature properties requires combining it with calculations of lattice vibrations (phonons) or molecular dynamics. Standard functionals also describe van der Waals forces poorly, so corrections are added for layered materials and adsorption.
First-principles calculation is a general term for methods that calculate from the basic laws of quantum mechanics without empirical parameters, and in materials science calculations by DFT are by far the most widely used. The two are therefore often used almost interchangeably, but more rigorous methods of quantum chemistry are also included in first-principles calculation.
Because the LDA and GGA exchange-correlation energies retain an error in which an electron interacts with itself, and they do not correctly describe how the energy changes with the number of electrons. Another reason is that the eigenvalues of the Kohn–Sham equations do not, in principle, represent excitation energies. The GW approximation and hybrid functionals improve this.
Ordinary calculations handle tens to hundreds of atoms, and the cost grows roughly as the cube of the number of atoms. Because periodic boundary conditions are used, a crystal can be represented by its unit cell or a supercell of a few hundred atoms, but problems needing large regions, such as dislocations and grain boundaries, are handled by combining DFT with molecular dynamics using interatomic potentials and similar methods.
DFT takes the electron density as its main variable, based on the Hohenberg–Kohn theorem that the ground-state energy is determined by the electron density, a function of only three dimensions.
Which statement about the ideas behind density functional t…Ordinary DFT is a calculation of the ground state at 0 K, so obtaining finite-temperature properties requires combining it with calculations of lattice vibrations (phonons) or molecular dynamics.
Which is a correct tendency of the results of standard DFT …DFT, which solves for the electronic states by quantum mechanics, usually handles tens to hundreds of atoms, and can follow time evolution only up to some tens of picoseconds.
Which pairing of a materials computation method with the ra…Machine-learning potentials trained on DFT results are also widely used.
Which statement about the interatomic potentials used in mo…Activation energies calculated by DFT are often used.
Which statement about the kinetic Monte Carlo (kMC) method …Molecular dynamics Interatomic potential Vacancy Phonon Phase-field method Finite element method Monte Carlo method CALPHAD method Gibbs energy Materials informatics