Density functional theory

The density functional theory (DFT ) is a method for determining the quantum-mechanical ground state of a many-electron system which is based on the location-dependent electron density. The density functional theory is used to calculate fundamental properties of molecules and solids, such as, for example, used by bond lengths and energies.

The importance of this theory is that it is not necessary to her to solve the full Schrödinger equation for the many-electron system, whereby the cost of computing power decreases sharply and calculations of systems in general only be possible with well over ten electrons.

For the development of the density functional theory of the Nobel Prize for chemistry was awarded to Walter Kohn in 1998.

Basics

Basis of the density functional theory is the Hohenberg- Kohn theorem: The ground state of a system of electrons has a unique location-dependent electron density. In the density functional theory is now the electron density is determined in the ground state, from all other properties of the ground state can be determined in principle. These properties, such as the total energy, so are the functional density.

The calculations with the density functional theory are usually performed in the Born- Oppenheimer approximation, so only the electrons are treated quantum mechanically.

The Kohn -Sham functions

To determine the electron density, are one-electron wave functions, the so-called Kohn -Sham functions set (named after Kohn and Lu Jeu Sham ), the solutions of the Schrödinger equation in an effective potential function. This computational way is much less expensive than the solution of the Schrödinger equation with electrons at the same time, because it is independent solutions of the Schrödinger equation is ( no Slater determinant ). This one-electron Schrödinger equations are also called Kohn- Sham equations:

(Note: in this and the following equations are used for convenience atomic units in this unit system gives the energy in Hartree, which is about 27.2 eV.. )

The density is obtained from the sum of the electron densities of the Kohn -Sham functions:

The effective potential is dependent on the density:

Here, the first term, the external potential which substantially describes the attraction of the electrons by the atomic nuclei, and the second term describes the electrostatic interaction between the electrons ( Hartree- term). The third term, the so-called exchange - correlation potential ( "x " for English, " exchange", "c" for " correlation" ), is to ensure the correct treatment of the many-electron system.

Since the effective potential occurs on the one hand in the Kohn- Sham equations, on the other hand, the density and thus depends on the solutions of these equations, the solutions must be found iteratively. It is therefore with the potential new found (or a linear combination of the previous and the new potential ) solved the Kohn-Sham equation again, it determines a new potential and so on until a stable ( self-consistent ) solution is found.

Strictly speaking, the Kohn -Sham functions are pure operands and for himself alone no physical meaning. However, in practice they can be used as an approximation for actual electron states often, and their energies are used, for example, to calculate the band structure.

The exchange-correlation potential

With the Kohn-Sham formalism, the problem of many-electron system was really only moved to the exchange - correlation term, and not yet solved. Strictly speaking, depends on the electron density in all places, not just at the point, and can be calculated exactly only for very few trivial cases. But it turns out that it is often sufficient to find an approximate solution for this term:

  • Local density approximation (English local density approximation, LDA ): With this approximation, it is assumed that a function of the electron density at this location is. This method provides, in many cases a sufficiently accurate solution, especially if the density is all the same in any case approximately, for example for the conduction electrons in the metal. LDA calculations often lead to an overestimation of the bond strengths, the calculated bond lengths ( over binding engl. ) by about one to two percent too short.
  • Gradient approximation ( engl. generalized gradient approximation, GGA ) are taken into account not only the density but also their derivatives with respect to the site ( gradient). There are several different methods for this, which are generally named according to the authors of the method, for example, for the PW91 of Perdew and Wang 1991 presented method. The PBE functional, which is named after the physicists Perdew Burke Ernzerhof and should have the same shape as the PW91, but make do with considerably fewer parameters. The PBE is today still one of the most widely used functional, although there are already developments such as the revPBE or RPBE.
  • Hybrid Methods: Here are just a part of the exchange-correlation potential is calculated according to the density functional theory ( eg GGA ), a part is calculated as the exchange energy of the Kohn -Sham functions as in the Hartree -Fock method. These methods are more accurate than pure GGA calculations, especially for molecules, but the cost of the bills is much higher than in GGA. The most widely used method is known as hybrid B3LYP.

Most limitations and problems in the use of density functional theory are related to the exchange-correlation potential. Thus, for example, provide the various GGA potentials binding energies of simple molecules, which may differ from each other and from the experimental values ​​by more than 20 percent. Van der Waals bonds are of the " semi- local " features such as GGA not described at all, because they are based on long-range correlations of the charge distribution. Another problem is that the band gaps and HOMO- LUMO energy differences, which are calculated from the Kohn -Sham functions are generally too low for LDA and GGA.

Calculation method on the computer

Calculations of complex structures by means of density functional theory require high computing power, so an efficient implementation of the calculations is of great importance. The computational methods can be classified according to the basis functions for the Kohn- Sham equations:

Atomic wave functions ( engl: muffin - tin orbitals ) in a spherical environment around the nucleus (so-called muffin-tin region ) are well suited for the description of electrons near the nucleus. The advantage of the atomic wave functions is that under appropriate for their application problems usually very small basis sets satisfy ( one function per electron and angular momentum character ) for description. However, problems that nearly free electrons between atoms arise (eg, conduction electrons in metals, electrons at surfaces, etc. ) and the area of ​​overlap between the atoms to describe consistent.

Plane waves are good for describing the valence and conduction electrons in solids suitable, but the little spatially extended wave functions can be described close to the atomic nuclei bad. Plane waves have the advantage that efficient algorithms may be used for the Fourier transform and thereby solving the Kohn-Sham equations can be carried out very rapidly. They are also very flexible, as, for example, nearly free electrons can be described well to surfaces.

These methods are mainly used for calculations in solid state physics, therefore, combined by using plane waves for the region near the atomic nuclei but meets additional measures. This area can either be completely treated separately (English augmented plane waves ), there may be additional wave functions added ( engl. projector augmented waves ) or it may be a so-called pseudo-potential, which gives the correct wave functions only in the area of the outer electrons, but not near there of the nuclei can be used.

Gaussian functions and functions with tabulated function of the radius (starting from the atomic nucleus ) are often used for calculations on molecules.

With powerful computers systems can be treated up to about 1000 atoms using DFT calculations today. For larger systems, other approximation methods such as the tight-binding method or based on DFT calculations, approximate methods must be used.

Applications

DFT calculations give directly the total energy of the atomic configuration and therefore can serve one of several possible arrangements, the energetically most favorable out. It must normally to the geometry of each array can be optimized, that is, the atoms will be moved until the forces disappear or a minimum of the energy is found.

Calculated with DFT interaction potentials or forces on the atoms can serve as a basis for molecular dynamics calculations. Using molecular dynamics calculations accurate simulations of the behavior of liquids or the calculation of the atomic motions in chemical reactions are possible, for example. However, such calculations are limited due to the high cost of computing power to the simulation of very short periods of time ( picoseconds ).

From the forces on the atoms with a deflection from its rest position, the vibrational spectra can be determined and compared with the experiment. Also can be as far as the Kohn -Sham functions are sufficiently good approximation for the actual electron states, calculate photoelectron spectra.

Since the interaction of a molecule or solid is determined with electromagnetic fields mainly by the electrons, the frequency dependent dielectric constant and related quantities can be calculated by means of DFT.

Extensions

There are numerous extensions to the theory, such as spin density or current density functional theories, or about so-called dynamic density functional theories, although all are worth mentioning, but here in detail can not be discussed, since the area is very much in flux as before.

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