Thomas-Fermi-Dirac theory

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... 

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. 

TF) theory, including the Ko[p exchange part (first derived by Block but commonly associated with the name of Dirac" (constitutes the Thomas-Fermi-Dirac (TFD) model. 

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). 

Cowan RD, Ashkin J (1957) Extension of the Thomas-Fermi-Dirac statistical theory of the atom to finite temperatures. Phys Rev 105 144-157... 

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... 

The first statistical models of these interactions are the well-known Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) theories based on the idea of approximating the behavior of electrons by that of the uniform negatively charged gas. Some authors (Sheldon, 1955 Teller, 1962 Balazs, 1967 Firsov, 1953,1957 Townsend and Handler, 1962 Townsend and Keller, 1963 Goodisman, 1971) proved that these theories provide an adequate description of purely repulsive diatomic interactions. Abraham-son (1963, 1964) and Konowalow et al. (Konowalow, 1969 Konowalow and Zakheim, 1972) extended this region to intermediate internuclear distances, but Gombas (1949) and March (1957) showed that the Abraham-son approach is incorrect, and so the question of how adequately the TFD theory provides diatomic interactions for closed-shell atoms is still open. Here we need to note that until recently, there has existed only work by Sheldon (1955), as far as we know, in which the TFD interaction potential is actually calculated by solving the TFD equation for a series of internuclear distances (see also, Kaplan, 1982). 

Thomas-Fermi model [18], or with embellishments by Dirac, the Thomas-Fermi-Dirac model [18], 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). 

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]. 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

One can consider the cumulative case of the Thomas-Fermi-Dirac-von Weizsaecker (TFDW) theory in which the total energy as the density electronic functional will have the general expression... 

TFDW Thomas-Fermi-Dirac-von Weizsaecker theory... 

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