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Density functional theory (DFT) is a Quantum Mechanical method used in Physics and Chemistry to investigate the Electronic Structure of many-body systems, in particular molecules and the Condensed Phase s. DFT is among the most popular and versatile methods available in condensed matter physics, Computational Physics , and Computational Chemistry .

PREAMBLE


Thomas-Fermi model

The predecessor to density functional theory was the Thomas-Fermi model, developed by Thomas and Fermi in 1927. They used a statistical model to approximate the distribution of electrons in an atom. The mathematical basis used was to postulate that electrons are distributed uniformly in phase space with two electrons in every h3 of volumeParr and Yang page 47. For each element of coordinate space volume d3r we can fill out a sphere of momentum space up to the fermi momentum pfMarch, N. H. page 24

:(4/3)\pi p_f^3(r)

equating the number of electrons in coordinate space to that in phase space gives:

:n(r)= rac{8\pi}{3h^3}p_f^3(r)

solving for pf and substituting in the classical kinetic energy formula then leads directly to a Kinetic Energy represented as a Functional of the electron density:

:T_{TF} {Link without Title} =C_F\int n^{5/3}(r) d^3r.

As such they were able to calculate the Energy of an atom using this kinetic energy functional combined 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 is limited because the resulting kinetic energy functional is only approximate, and also the method does not attempt to represent the Exchange Energy of an atom as a conclusion of the Pauli Principle . An exchange energy functional was added by Dirac in 1928.

However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and due to the complete neglect of Electron Correlation .

Teller (1962) showed that Thomas-Fermi theory cannot describe molecular bonding. This can however be overcome by improving the kinetic energy functional.

The kinetic energy functional can be improved by adding the Weizsäcker (1935) correction:

:V_s = V + U + \left(T - T_s ight).

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

:\left[- rac{\hbar^2}{2m}
abla^2+V_s( ec r) ight] \phi_i( ec r) = \epsilon_i \phi_i( ec r),

which yields the Orbital s \,\!\phi_i that reproduce the density n( ec r) of the original many-body system


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


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. In physics 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:

:E_{XC} {Link without Title} =\int\epsilon_{XC}(n)n (r) { m d}^3r.

The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron Spin :

:E_{XC} {Link without Title} =\int\epsilon_{XC}(n_\uparrow,n_\downarrow)n (r){ m d}^3r.

Highly accurate formulae for the exchange-correlation energy density
\epsilon_{XC}(n_\uparrow,n_\downarrow) have been constructed
from Quantum Monte Carlo simulations of a free-electron gas.1

Generalized gradient approximations (GGA) are still local but also take into account the Gradient of the density at the same coordinate:

:E_{XC} {Link without Title} =\int\epsilon_{XC}(n_\uparrow,n_\downarrow, ec{
abla}n_\uparrow, ec{
abla}n_\downarrow)
n (r) { m d}^3r.

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.


GENERALIZATIONS TO INCLUDE MAGNETIC FIELDS

The DFT formalism above breaks down in the presence of a vector potential, i.e. a Magnetic Field . In such a case, the one-to-one mapping between electron density and external potential breaks down. Generalizations to include the effects of magnetic fields have led to two different theories: current density functional theory and magnetic field functional theory. In both these theories, the functional used for the exchange and correlation must be generalized to include more than just the electron density. In current density functional theory, developed by Vignale and Rasolt, the functionals become dependent on both the electron density and the current density. In magnetic field density functional theory, developed by Salsbury, Grayce and Harris, the functionals depend on the electron density and the magnetic field, and the functional form can depend on the form of the magnetic field. In both of these theories it has been difficult to develop functionals beyond their equivalent to LDA, which are also readily implementable computationally.


APPLICATIONS

In practice, Kohn-Sham theory can be applied in several distinct ways depending on what is being investigated. In solid state calculations, the local density approximations are still commonly used along with Plane Wave basis sets, as an electron gas approach is more appropriate for an infinite solid. In molecular calculations, however, more sophisticated functionals are needed, and a huge variety of exchange-correlation functionals have been developed for chemical applications. Some of these are inconsistent with the uniform electron gas approximation, however, they must reduce to LDA in the electron gas limit. Among physicists, probably the most widely used functional is the revised Perdew-Burke-Ernzerhof exchange model (a direct generalized-gradient parametrization of the free electron gas with no free parameters); however, this is not sufficiently calorimetrically accurate for gas-phase molecular calculations. In the chemistry community, one popular functional is known as BLYP (from the name Becke for the exchange part and Lee, Yang and Parr for the correlation part). Even more widely used is B3LYP {Link without Title} which is a Hybrid Functional in which the exchange energy, in this case from Becke's exchange functional, is combined with the exact energy from Hartree-Fock theory. Along with the component exchange and correlation funсtionals, three parameters define the hybrid functional, specifying how much of the exact exchange is mixed in. The adjustable parameters in hybrid functionals are generally fitted to a 'training set' of molecules. Unfortunately, although the results obtained with these functionals are usually sufficiently 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 theory). 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.


SOFTWARE SUPPORTING DFT







BOOKS ON DFT


  • R. Dreizler, E. Gross, Density Functional Theory. (Plenum Press, New York, 1995).

  • W. Koch, M. C. Holthausen, A Chemist's Guide to Density Functional Theory. (Wiley-VCH, Weinheim, ed. 2, 2002).

  • R. G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules. (Oxford University Press, New York, 1989).



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