Thomas-Fermi-Dirac density

This form is obtained from the small r behavior of the Thomas-Fermi-Dirac density and an exponential decreasing behavior. This is a very simple form that approximates reasonably well the values of p2 r) obtained from the numerical resolution mentioned above. 

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... 

In summary, the original Thomas-Fermi-Dirac DFT was unable to give binding in molecules. This was corrected by Kohn-Sham, [11] who chose to use an orbital rather than density evaluation of the kinetic energy. By the virial theorem, = —E, so this was a necessity to obtain realistic results for energies. Next, it was shown that the exact exchange requires an orbital-dependent form, too. [47,48] The future seems to demand an orbital-dependent form for the correlation. 

Gradient corrections to the energy density for improving the Thomas-Fermi-(Dirac)... 

Two types of corrections to the Thomas-Fermi-Dirac non-relativistic energy density appear. The first is the correction to the kinetic energy given by the mass-variation term ... 

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... 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

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

Electron-Electron Scaling with Constant Density Thomas-Fermi-Dirac Limit... 

The Thomas-Fermi-Dirac (TFD) ground-state (GS) energy corresponding to different A and a definite density p0(r)> is now introduced as the minimum obtained from... 

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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