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DESCRIPTION OF THE THEORY


Traditional methods in electronic structure theory, in particular Hartree-Fock Theory and Its Descendants , are based on the complicated many-electron Wavefunction . The main objective of density functional theory is to replace the many-body electronic wavefunction with the Electronic Density as the basic quantity. Whereas the many-body wavefunction is dependent on 3N variables, three spatial variables for each of the N electrons, the density is only a function of three variables and is a simpler quantity to deal with both conceptually and practically.

Although density functional theory has its conceptual roots in the Thomas - Fermi model, DFT was not put on a firm theoretical footing until the Hohenberg - Kohn (HK) theorems. The first of these demonstrates the existence of a one-to-one Mapping between the Ground State electron density and the ground state wavefunction of a many-particle system. Further, the second HK theorem proves that the ground state density minimizes the total electronic energy of the system. The original HK theorems held only for the ground state in the absence of magnetic field, although they have since been generalized. The first Hohenberg-Kohn theorem is only an existence theorem, stating that the mapping exists, but does not provide any such exact mapping. It is in these mappings that approximations are made. (The theorems can be extended to the time-dependent domain (TDDFT), which can be also used to determine excited states {Link without Title} .)

The most common implementation of density functional theory is through the Kohn-Sham Method . Within the framework of Kohn-Sham DFT, the intractable Many-body Problem of interacting electrons in a static external potential is reduced to a tractable problem of
non-interacting electrons moving in an effective Potential . The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g. the exchange and Correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas - Fermi model, and from fits to the correlation energy for a uniform electron gas.

DFT has been very popular for calculations in Solid State Physics since the 1970s. In many cases DFT with the local-density approximation gives quite satisfactory results, for solid-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum mechanical many-body problem. 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 to better model the exchange and correlation interactions. DFT is now a leading method for electronic structure calculations in both fields. Despite the improvements in DFT, there are still difficulties in using density functional theory to properly describe Intermolecular Interactions ,
especially Van Der Waals Forces (dispersion), or in calculations of the Band Gap in Semiconductors . Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g. interacting Noble Gas atoms) or where dispersion competes significantly with other effects (e.g. in Biomolecule s). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.


EARLY MODELS: THOMAS-FERMI MODEL


The predecessor to density functional theory was the ''Thomas-Fermi'' model, developed by Thomas and Fermi in 1927. They calculated the Energy of an atom by representing its Kinetic Energy as a Functional 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 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 .


DERIVATION AND FORMALISM


As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the Born-Oppenheimer Approximation ), generating a static external potential ''V'' in which the electrons are moving. A stationary electronic state is then described by a wave function \Psi( ec r_1,\dots, ec r_N) fulfilling the many-electron Schrödinger Equation

: H \Psi = \left[{T}+{V}+{U} ight]\Psi = \left[\sum_i^N - rac{\hbar^2}{2m}
abla_i^2 + \sum_i^N V( ec r_i) + \sum_{i

where ''H'' is the Electronic Molecular Hamiltonian , ''N'' is the number of electrons and ''U'' is the electron-electron interaction. The operators ''T'' and ''U'' are so-called universal operators as they are the same for any system, while ''V'' 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 ''U''. There are many sophisticated methods for solving the many-body Schrödinger Equation based on the expansion of the wave function in Slater Determinants . While the simplest one is the Hartree-Fock method, more sophisticated approaches are usually categorized as Post-Hartree-Fock methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more 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 ''U'', onto a single-body problem without ''U''. In DFT the key variable is the particle density n( ec r) which is given by

  • ( ec r, ec r_2,\dots, ec r_N) \Psi( ec r, ec r_2,\dots, ec r_N).


Hohenberg and Kohn proved in 1964 {Link without Title} that the relation expressed above can be reversed, i.e. to a given ground state density n_0( ec r) it is in principle possible to calculate the corresponding ground state wavefunction \Psi_0( ec r_1,\dots, ec r_N). In other words, \,\!\Psi_0 is a unique Functional of \,\!n_0, i.e.

:\,\!\Psi_0 = \Psi_0 {Link without Title}

and consequently all other ground state observables \,\!O are also functionals of \,\!n_0

  :<math>E 0 E = \left\langle \Psi_0[n_0 \left T+V+U ight \Psi_0[n_0] ight angle</math>,
  Where The Contribution Of The External Potential <math>\left\langle \Psi 0 "n_0]" class="copylinks" target="_blank">\leftV ight \Psi_0[n_0 ight angle</math> can be written explicitly in terms of the density
  :<math>E S "n]" class="copylinks" target="_blank">= \left\langle \Psi_s[n \left T_s+V_s ight \Psi_s[n] ight angle,</math>
  :<math>n( Ec R )\equiv N S( Ec R) \sum_i^N \left\phi_i( ec r) ight^2 </math>