Thomas-Fermi-Dirac functional

Thomas-Fermi total energy Eg.j.p [p] gives the so-called Thomas-Fermi-Dirac (TFD) energy functional. 

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

The minimization of this functional, which includes second order gradient corrections leads to the relativistic analogous of the Thomas-Fermi-Dirac-Weizsacker model and constitutes the state of the art in relativistic semiclassical approaches for many-electron systems. 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

Fia. 2-19.—Curves representing values of electron energies, as a function of atomic number. These curves were obtained by approximate solution of the wave equation by the Thomas-Fermi-Dirac method. 

The energy functional Etf[p] = Ttf[p] + ne />] + T[p] is known as Thomas-Fermi (TF) theory, inHiiHinv the Atp[p] exchange part (first derived by Block but commonly associated with the name of Dirac (constitutes the Thomas-Fermi-Dirac (TFD)... 

Response, wave function, 242 Thomas-Fermi (TF), Thomas-Fermi-Dirac Variational Principle, 407... 

Origins of Density Functional Theory the Thomas-Fermi-Dirac... 

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

Thomas-Fermi-Dirac-Weizsacker Density Functional Formalism Applied to the Study of Many-electron Atom Confinement by Open and Closed Boundaries... 

The first generation is the local density approximation (LDA). This estimation involves the Dirac functional for exchange, which is nothing else than the functional proposed by Dirac [15] in 1927 for the so-called Thomas-Fermi-Dirac model of the atoms. For the correlation energy, some parameterizations have been proposed, and the formula can be considered as the limit of what can be obtained at this level of approximation [16-18], The Xa approximation falls into this category, since a known proportion of the exchange energy approximates the correlation. 

The 1998 Nobel Prize for Chemistry, awarded to a physicist for inventing modem Density Functional Theory (DFT), signaled widespread recognition of DFT as the pre-eminent many-electron theory for predictive, materials-specific (chemically specific) calculation of extended and molecular systems. The original papers of modem DFT are those of Hohenberg and Kohn [1] and Kohn and Sham [2] (preceded by seminal work of Thomas, Fermi, Dirac, Slater, Caspar, Gombas, and others not of direct relevance). General references include [3-16]. 

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