Density Functional Theory Description Of The Theory

quantum mechanical approaches to the many-body electronic structure calculations of molecular and condensed matter systems. Within the framework of DFT, the practically unsolvable

1 Description of the Theory

In contrast to traditional methods like Hartree-Fock theory which are based on the complicated many-electron wavefunction DFT is written in terms of the electron density, giving this theory its name. DFT is an exact theory only for the free electron gas , while for the treatment of extended atomic systems various approximations have to be made. In many cases DFT gives quite satisfactory results in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum mechanical many-body problem.

DFT has been very popular for calculations in solid state physics since the 1970s. However, it was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined. DFT is now the leading method for electronic structure calculations in both fields.

However, there are still systems which are not described very well by DFT. One famous example is the false prediction of the band gap in semi-conductors. The method also fails to describe properly

2 Early Models

The first true density functional theory was developed by Thomas and Fermi in the 1920s. They calculated the energy of an atom by representing its kinetic energy as a functionalGenerally, functional refers to something with and able to fulfill its purpose or function. In medicine, the term functional is sometimes used to describe symptoms that have no organic basis, e. if they are a result of psychological or perceptual dysfunct of the electron density, combining this with the classical expressions for the nuclear-electron and electron-electron interactions (which can both also be represented in terms of the electron density).

Although this was an important first step, the Thomas-Fermi equation's accuracy was limited because it did not attempt to represent the exchange energy of an atom predicted by Hartree-Fock theory. An exchange energy functional was added by DiracPaul Adrien Maurice Dirac ( August 8 1902 October 20 1984) was a British theoretical physicist and a founder of the field of quantum physics. Biography Paul Dirac was born in the English city of Bristol. His father, Charles Dirac, was an immigrant from th in 1928.

However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications because it is difficult to represent kinetic energy with a density functional, and it neglects electron correlationIn probability theory and statistics, the correlation also called correlation coefficient between two random variables is found by dividing their covariance by the product of their standard deviations. It is defined only if these standard deviations are f entirely.

3 Derivation and Formalism

As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (Born-Oppenheimer approximation), generating a static external potential in which the electrons are moving. A stationary electronic state is then described by a wave function fulfilling the many-electron Schrödinger equationIn physics, the Schrodinger equation proposed by the Austrian physicist Erwin Schrodinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to New

where is the number of electrons and is the electron-electron interaction. The operators and are so-called universal operators as they are the same for any system, while is system dependent or non-universal. As one can see the actual difference between a single-particle problem and the much more complicated many-particle problem just arises from the interaction term . Now, there are many sophisticated methods for solving the many-body Schrödinger equationIn physics, the Schrodinger equation proposed by the Austrian physicist Erwin Schrodinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to New, e.g. there is diagrammatic perturbation theory in physics, while in quantum chemistry one often uses

configuration interaction (CI) methods, based on the systematic

expansion of the wave function in Slater determinantsA Slater determinant (named after the physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many- fermion system, which by construction satisfies the Pauli principle. The Slater determinant arises from the consideration. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger complex systems.

Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with , onto a single-body problem without . In DFT the key variable is the particle density

which is given by

Hohenberg and Kohn proved in 1964 [1] that the relation expressed above can be reversed, i.e. to a given ground state density it is in principle possible to calculate the corresponding ground state wave function . In other words,

is a unique functional of , i.e.

and consequently all other ground state observables are also functionals of

From this follows in particular, that also the ground state energy is a functional of

where the contribution of the external potential

can be written explicitly in terms of the density

The functionals and are called universal functionals while is obviously non-universal, as it depends on the system under study. Having specified a system, i.e. is known, one then has to minimise the functional

with respect to , assuming one has got reliable expressions for and . A successful minimisation of the energy functional will yield the ground state density and thus all other ground state observables.

The variational problem of minimising the energy functional can be solved by applying the Lagrangian method of undetermined multipliers, which was done by Kohn and Sham in 1965 [2]. Hereby, one uses the fact that the functional in the equation above can be written as a fictitious density functional of a non-interacting system

where denotes the non-interacting kinetic energy and is an external effective potential in which the particles are moving. Obviously, if is chosen to be

Thus, one can solve the so-called Kohn-Sham equations of this auxiliary non-interacting system

which yields the orbitals that reproduce the density

of the original many-body system

The effective single-particle potential can be written in more detail as

where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term is called exchange correlation potential. Here, includes all the many particle interactions. Since the Hartree term and depend on , which depends on the , which in turn depend on , the problem of solving the Kohn-Sham equation has to be done in a self-consistent way. Usually one starts with an initial guess for , then one calculates the corresponding and solves the Kohn-Sham equations for the . From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached.

4 Approximations

The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. The most widely used approximation is the local density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated. Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate. Using the latter (GGA) very good results for molecular geometries and ground state energies have been achieved. Many further incremental improvements have been made to DFT by developing better representations of the functionals.

5 Relativistic Generalization

The relativistic generalization of the DFT formalism leads to a current density functional theory.

6 Applications

In practice, Kohn-Sham theory can be applied in two distinct ways depending on what is being investigated. In the solid state, plane wave basis sets are used with periodic boundary conditions. Moreover, great emphasis is placed upon remaining consistent with the idealised model of a 'uniform electron gas', which exhibits similar behaviour to an infinite solid. In the gas and liquid phases, this emphasis is relaxed somewhat, as the uniform electron gas is a poor model for the behaviour of discrete atoms and molecules. Because of the relaxed constraints, a huge variety of exchange-correlation functionals have been developed for chemical applications. The most famous and popular of these is known as B3LYP [3-5]. The adjustable parameters of these functionals are generally fitted to a 'training set' of molecules. Unfortunately, although the results obtained with these functionals are usually relatively accurate for most applications, there is no systematic way of improving them (in contrast to some of the traditional wavefunction-based methods like configuration interaction or coupled cluster method). Hence, in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiment.

7 References

[1] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864
[2] W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133
[3] A. D. Becke, J. Chem. Phys. 38 (1998) 3089
[4] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37 (1988) 785
[5] P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem. 98 (1994) 11623

8 Literature

Klaus Capelle, A bird's-eye view of density-functional theory

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